NASA Astrophysics Data System (ADS)
Xu, Huan; Li, Yongsheng; Zhai, Xiaoping
2016-04-01
In this paper, we first prove the local well-posedness of the 2-D incompressible Navier-Stokes equations with variable viscosity in critical Besov spaces with negative regularity indices, without smallness assumption on the variation of the density. The key is to prove for p ∈ (1 , 4) and a ∈ B˙p, 1 2/p (R2) that the solution mapping Ha : F ↦ ∇Π to the 2-D elliptic equation div ((1 + a) ∇Π) = div F is bounded on B˙p, 1 2/p - 1 (R2).
NASA Astrophysics Data System (ADS)
Abidi, Hammadi; Zhang, Ping
2015-10-01
Given solenoidal vector u0 ∈H ˙ - 2 δ ∩H1 (R2), ρ0 - 1 ∈L2 (R2), and ρ0 ∈L∞ ∩W ˙ 1, r (R2) with a positive lower bound for δ ∈ (0, 1/2) and 2 < r < 2/1 - 2 δ, we prove that 2-D incompressible inhomogeneous Navier-Stokes system (1.1) has a unique global solution provided that the viscous coefficient μ (ρ0) is close enough to 1 in the L∞ norm compared to the size of δ and the norms of the initial data. With smoother initial data, we can prove the propagation of regularities for such solutions. Furthermore, for 1 < p < 4, if (ρ0 - 1, u0) belongs to the critical Besov spaces B˙p, 1 2/p (R2) × ( B˙p, 1 - 1 +2/p ∩L2 (R2)) and the B˙p, 1 2/p (R2) norm of ρ0 - 1 is sufficiently small compared to the exponential of ‖u0‖L2 2 +‖u0 ‖ B˙p, 1 - 1 +2/p, we prove the global well-posedness of (1.1) in the scaling invariant spaces. Finally for initial data in the almost critical Besov spaces, we prove the global well-posedness of (1.1) under the assumption that the L∞ norm of ρ0 - 1 is sufficiently small.
Incompressible Navier-Stokes computations of rotating flows
NASA Astrophysics Data System (ADS)
Kiris, Cetin; Chang, Leon; Kwak, Dochan; Rogers, Stuart
1993-01-01
Flow through pump components, such as an inducer and an impeller, is efficiently simulated by solving the incompressible Navier-Stokes equations. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. Current computations use one-equation Baldwin-Barth turbulence model which is derived from a simplified form of the standard k-epsilon model equations. The resulting computer code is applied to the flow analysis inside a generic rocket engine pump inducer, a fuel pump impeller, and SSME high-pressure fuel turbopump impeller. Numerical results of inducer flow are compared with experimental measurements. Flow analyses at 80-, 100-, and 120-percent of design conditions are presented.
Incompressible Navier-Stokes calculations in pump flows
NASA Astrophysics Data System (ADS)
Kiris, Cetin; Chang, Leon; Kwak, Dochan
1993-07-01
Flow through pump components, such as the SSME-HPFTP Impeller and an advanced rocket pump impeller, is efficiently simulated by solving the incompressible Navier-Stokes equations. The solution method is based on the pseudo compressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. The equations are solved in steadily rotating reference frames and the centrifugal force and the Coriolis force are added to the equation of motion. Current computations use one-equation Baldwin-Barth turbulence model which is derived from a simplified form of the standard k-epsilon model equations. The resulting computer code is applied to the flow analysis inside an 11-inch SSME High Pressure Fuel Turbopump impeller, and an advanced rocket pump impeller. Numerical results of SSME-HPFTP impeller flow are compared with experimental measurements. In the advanced pump impeller, the effects of exit and shroud cavities are investigated. Flow analyses at design conditions will be presented.
Incompressible Navier-Stokes Calculations in Pump Flows
NASA Technical Reports Server (NTRS)
Kiris, Cetin; Chang, Leon; Kwak, Dochan
1993-01-01
Flow through pump components, such as the SSME-HPFTP Impeller and an advanced rocket pump impeller, is efficiently simulated by solving the incompressible Navier-Stokes equations. The solution method is based on the pseudo compressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. The equations are solved in steadily rotating reference frames and the centrifugal force and the Coriolis force are added to the equation of motion. Current computations use one-equation Baldwin-Barth turbulence model which is derived from a simplified form of the standard k-epsilon model equations. The resulting computer code is applied to the flow analysis inside an 11-inch SSME High Pressure Fuel Turbopump impeller, and an advanced rocket pump impeller. Numerical results of SSME-HPFTP impeller flow are compared with experimental measurements. In the advanced pump impeller, the effects of exit and shroud cavities are investigated. Flow analyses at design conditions will be presented.
Exponential integrators for the incompressible Navier-Stokes equations.
Newman, Christopher K.
2004-07-01
We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L{sup 2} inner product. We examine this L{sup 2} projection and an H{sup 1} projection induced by the H{sup 1} semi-inner product. The H
The Minkowski dimension of interior singular points in the incompressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Koh, Youngwoo; Yang, Minsuk
2016-09-01
We study the possible interior singular points of suitable weak solutions to the three dimensional incompressible Navier-Stokes equations. We present an improved parabolic upper Minkowski dimension of the possible singular set, which is bounded by 95/63. The result also continue to hold for the three dimensional incompressible magnetohydrodynamic equations without any difficulty.
NASA Astrophysics Data System (ADS)
Samtaney, Ravi; Mohamed, Mamdouh; Hirani, Anil
2015-11-01
We present examples of numerical solutions of incompressible flow on 2D curved domains. The Navier-Stokes equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. A conservative discretization of Navier-Stokes equations on simplicial meshes is developed based on discrete exterior calculus (DEC). The discretization is then carried out by substituting the corresponding discrete operators based on the DEC framework. By construction, the method is conservative in that both the discrete divergence and circulation are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step. Numerical examples include Taylor vortices on a sphere, Stuart vortices on a sphere, and flow past a cylinder on domains with varying curvature. Supported by the KAUST Office of Competitive Research Funds under Award No. URF/1/1401-01.
NASA Technical Reports Server (NTRS)
Kiris, Cetin
1995-01-01
Development of an incompressible Navier-Stokes solution procedure was performed for the analysis of a liquid rocket engine pump components and for the mechanical heart assist devices. The solution procedure for the propulsion systems is applicable to incompressible Navier-Stokes flows in a steadily rotating frame of reference for any general complex configurations. The computer codes were tested on different complex configurations such as liquid rocket engine inducer and impellers. As a spin-off technology from the turbopump component simulations, the flow analysis for an axial heart pump was conducted. The baseline Left Ventricular Assist Device (LVAD) design was improved by adding an inducer geometry by adapting from the liquid rocket engine pump. The time-accurate mode of the incompressible Navier-Stokes code was validated with flapping foil experiment by using different domain decomposition methods. In the flapping foil experiment, two upstream NACA 0025 foils perform high-frequency synchronized motion and generate unsteady flow conditions for a downstream larger stationary foil. Fairly good agreement was obtained between unsteady experimental data and numerical results from two different moving boundary procedures. Incompressible Navier-Stokes code (INS3D) has been extended for heat transfer applications. The temperature equation was written for both forced and natural convection phenomena. Flow in a square duct case was used for the validation of the code in both natural and forced convection.
NASA Technical Reports Server (NTRS)
Biringen, S.; Cook, C.
1988-01-01
Pressure boundary conditions satisfying the normal momentum equation at solid boundaries with second-order accuracy are developed. Implementation of these conditions in an explicit numerical procedure for the two-dimensional incompressible Navier-Stokes equations enables convergent and accurate solutions for the driven cavity problem provided that the integral constraint of the Neumann boundary condtions is satisfied.
NASA Technical Reports Server (NTRS)
Thompson, C. P.; Leaf, G. K.; Vanrosendale, J.
1991-01-01
An algorithm is described for the solution of the laminar, incompressible Navier-Stokes equations. The basic algorithm is a multigrid based on a robust, box-based smoothing step. Its most important feature is the incorporation of automatic, dynamic mesh refinement. This algorithm supports generalized simple domains. The program is based on a standard staggered-grid formulation of the Navier-Stokes equations for robustness and efficiency. Special grid transfer operators were introduced at grid interfaces in the multigrid algorithm to ensure discrete mass conservation. Results are presented for three models: the driven-cavity, a backward-facing step, and a sudden expansion/contraction.
Computation of 2D Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Chakrabartty, Sunil Kumar
Two schemes for computing two-dimensional Navier-Stokes equations are described and applied to laminar flow over a flat plate and viscous flow over a NACA0012 airfoil. The variation of local skin-friction coefficient with local Reynolds number is compared with the Blasius solution and that of Swanson and Turkel (1985). The effect of free-stream Mach number on the temperature profile is shown, and a comparison is made of velocity profile at M(infinity) = 0.50 and Re = 500, with no artificial viscosity used for stability. Pressure distributions, local skin friction distributions, and velocity profiles on the airfoil and wake are presented.
Accuracy of Projection Methods for the Incompressible Navier-Stokes Equations
Brown, D L
2001-06-12
Numerous papers have appeared in the literature over the past thirty years discussing projection-type methods for solving the incompressible Navier-Stokes equations. A recurring difficulty encountered is the choice of boundary conditions for the intermediate or predicted velocity in order to obtain at least second order convergence. A further issue is the formula for the pressure correction at each timestep. A simple overview is presented here based on recently published results by Brown, Cortez and Minion [2].
An Incompressible Navier-Stokes with Particles Algorithm andParallel Implementation
Martin, Daniel F.; Colella, Phillip; Keen, Noel D.
2006-11-28
We present a variation of an adaptive projection method forcomputing solutions to the incompressible Navier-Stokes equations withsuspended particles. To compute the divergence-free component of themomentum forcing due to the particle drag, we employ an approach whichexploits the locality and smoothness of the Laplacian of the projectionoperator applied to the discretized particle drag force. We presentconvergence and performance results to demonstrate the effectiveness ofthis approach.
INS3D: An incompressible Navier-Stokes code in generalized three-dimensional coordinates
NASA Technical Reports Server (NTRS)
Rogers, S. E.; Kwak, D.; Chang, J. L. C.
1987-01-01
The operation of the INS3D code, which computes steady-state solutions to the incompressible Navier-Stokes equations, is described. The flow solver utilizes a pseudocompressibility approach combined with an approximate factorization scheme. This manual describes key operating features to orient new users. This includes the organization of the code, description of the input parameters, description of each subroutine, and sample problems. Details for more extended operations, including possible code modifications, are given in the appendix.
Incomplete Augmented Lagrangian Preconditioner for Steady Incompressible Navier-Stokes Equations
Tan, Ning-Bo; Huang, Ting-Zhu; Hu, Ze-Jun
2013-01-01
An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids. PMID:24235888
Incomplete augmented Lagrangian preconditioner for steady incompressible Navier-Stokes equations.
Tan, Ning-Bo; Huang, Ting-Zhu; Hu, Ze-Jun
2013-01-01
An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids. PMID:24235888
A fully vectorized numerical solution of the incompressible Navier-Stokes equations. Ph.D. Thesis
NASA Technical Reports Server (NTRS)
Patel, N.
1983-01-01
A vectorizable algorithm is presented for the implicit finite difference solution of the incompressible Navier-Stokes equations in general curvilinear coordinates. The unsteady Reynolds averaged Navier-Stokes equations solved are in two dimension and non-conservative primitive variable form. A two-layer algebraic eddy viscosity turbulence model is used to incorporate the effects of turbulence. Two momentum equations and a Poisson pressure equation, which is obtained by taking the divergence of the momentum equations and satisfying the continuity equation, are solved simultaneously at each time step. An elliptic grid generation approach is used to generate a boundary conforming coordinate system about an airfoil. The governing equations are expressed in terms of the curvilinear coordinates and are solved on a uniform rectangular computational domain. A checkerboard SOR, which can effectively utilize the computer architectural concept of vector processing, is used for iterative solution of the governing equations.
Evaluation of a Multigrid Scheme for the Incompressible Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Swanson, R. C.
2004-01-01
A fast multigrid solver for the steady, incompressible Navier-Stokes equations is presented. The multigrid solver is based upon a factorizable discrete scheme for the velocity-pressure form of the Navier-Stokes equations. This scheme correctly distinguishes between the advection-diffusion and elliptic parts of the operator, allowing efficient smoothers to be constructed. To evaluate the multigrid algorithm, solutions are computed for flow over a flat plate, parabola, and a Karman-Trefftz airfoil. Both nonlifting and lifting airfoil flows are considered, with a Reynolds number range of 200 to 800. Convergence and accuracy of the algorithm are discussed. Using Gauss-Seidel line relaxation in alternating directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of Runge-Kutta and implicit upwind based multigrid methods.
A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Ho, Lee-Wing; Maday, Yvon; Patera, Anthony T.; Ronquist, Einar M.
1989-01-01
A high-order Lagrangian-decoupling method is presented for the unsteady convection-diffusion and incompressible Navier-Stokes equations. The method is based upon: (1) Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem; (2) implicit high-order backward-differentiation finite-difference schemes for integration along characteristics; (3) finite element or spectral element spatial discretizations; and (4) mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high order accuracy, and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.
On the supnorm form of Leray's problem for the incompressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Schütz, Lineia; Zingano, Janaína P.; Zingano, Paulo R.
2015-07-01
We show that t3/4 ↑u(ṡ,t)↑ L∞ (R3) → 0 as t → ∞ for all Leray-Hopf's global weak solutions u(ṡ, t) of the incompressible Navier-Stokes equations in ℝ3. It is also shown that t ↑u(ṡ,t) - eΔt u0↑ L∞ (R3) → 0 as t → ∞, where eΔt is the heat semigroup, as well as other fundamental new results. In spite of the complexity of the questions, our approach is elementary and is based on standard tools like conventional Fourier and energy methods.
Direct Coupling Method for Time-Accurate Solution of Incompressible Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Soh, Woo Y.
1992-01-01
A noniterative finite difference numerical method is presented for the solution of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The solution of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct.
Fischer, P.F.
1996-12-31
Efficient solution of the Navier-Stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems. For unsteady incompressible flows, the pressure operator is the leading contributor to stiffness, as the characteristic propagation speed is infinite. In the context of operator splitting formulations, it is the pressure solve which is the most computationally challenging, despite its elliptic origins. We seek to improve existing spectral element iterative methods for the pressure solve in order to overcome the slow convergence frequently observed in the presence of highly refined grids or high-aspect ratio elements.
Spectral solution of the incompressible Navier-Stokes equations on the Connection Machine 2
NASA Technical Reports Server (NTRS)
Tomboulian, Sherryl; Streett, Craig; Macaraeg, Michele
1989-01-01
The issue of solving the time-dependent incompressible Navier-Stokes equations on the Connection Machine 2 is addressed, for the problem of transition to turbulence of the steady flow in a channel. The spectral algorithm used serially requires O(N(4)) operations when solving the equations on an N x N x N grid; using the massive parallelism of the CM, it becomes an O(N(2)) problem. Preliminary timings of the code, written in LISP, are included and compared with a corresponding code optimized for the Cray-2 for a 128 x 128 x 101 grid.
Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Hafez, Mohammed; Dacles, Jennifer
1989-01-01
The numerical analysis of the incompressible Navier-Stokes equations are becoming important tools in the understanding of some fluid flow problems which are encountered in research as well as in industry. With the advent of the supercomputers, more realistic problems can be studied with a wider choice of numerical algorithms. An alternative formulation is presented for viscous incompressible flows. The incompressible Navier-Stokes equations are cast in a velocity/vorticity formulation. This formulation consists of solving the Poisson equations for the velocity components and the vorticity transport equation. Two numerical algorithms for the steady two-dimensional laminar flows are presented. The first method is based on the actual partial differential equations. This uses a finite-difference approximation of the governing equations on a staggered grid. The second method uses a finite element discretization with the vorticity transport equation approximated using a Galerkin approximation and the Poisson equations are obtained using a least squares method. The equations are solved efficiently using Newton's method and a banded direct matrix solver (LINPACK). The method is extended to steady three-dimensional laminar flows and applied to a cubic driven cavity using finite difference schemes and a staggered grid arrangement on a Cartesian mesh. The equations are solved iteratively using a plane zebra relaxation scheme. Currently, a two-dimensional, unsteady algorithm is being developed using a generalized coordinate system. The equations are discretized using a finite-volume approach. This work will then be extended to three-dimensional flows.
A comparison of two incompressible Navier-Stokes algorithms for unsteady internal flow
NASA Technical Reports Server (NTRS)
Wiltberger, N. Lyn; Rogers, Stuart E.; Kwak, Dochan
1993-01-01
A comparative study of two different incompressible Navier-Stokes algorithms for solving an unsteady, incompressible, internal flow problem is performed. The first algorithm uses an artificial compressibility method coupled with upwind differencing and a line relaxation scheme. The second algorithm uses a fractional step method with a staggered grid, finite volume approach. Unsteady, viscous, incompressible, internal flow through a channel with a constriction is computed using the first algorithm. A grid resolution study and parameter studies on the artificial compressibility coefficient and the maximum allowable residual of the continuity equation are performed. The periodicity of the solution is examined and several periodic data sets are generated using the first algorithm. These computational results are compared with previously published results computed using the second algorithm and experimental data.
NASA Astrophysics Data System (ADS)
Sahin, Mehmet
2005-11-01
A new semi-staggered finite volume method is presented for the solution of the incompressible Navier-Stokes equations on all-quadrilateral (2D)/hexahedral (3D) meshes. The velocity components are defined at element node points while the pressure term is defined at element centroids. The continuity equation is satisfied exactly within each elements. The checkerboard pressure oscillations are prevented using a special filtering matrix as a preconditioner for the saddle-point problem resulting from second-order discretization of the incompressible Navier-Stokes equations. The preconditioned saddle-point problem is solved using block preconditioners with GMRES solver. In order to achieve higher performance FORTRAN source code is based on highly efficient PETSc and HYPRE libraries. As test cases the 2D/3D lid-driven cavity flow problem and the 3D flow past array of circular cylinders are solved in order to verify the accuracy of the proposed method.
NASA Astrophysics Data System (ADS)
Serson, D.; Meneghini, J. R.; Sherwin, S. J.
2016-07-01
This paper presents methods of including coordinate transformations into the solution of the incompressible Navier-Stokes equations using the velocity-correction scheme, which is commonly used in the numerical solution of unsteady incompressible flows. This is important when the transformation leads to symmetries that allow the use of more efficient numerical techniques, like employing a Fourier expansion to discretize a homogeneous direction. Two different approaches are presented: in the first approach all the influence of the mapping is treated explicitly, while in the second the mapping terms related to convection are treated explicitly, with the pressure and viscous terms treated implicitly. Through numerical results, we demonstrate how these methods maintain the accuracy of the underlying high-order method, and further apply the discretisation strategy to problems where mixed Fourier-spectral/hp element discretisations can be applied, thereby extending the usefulness of this discretisation technique.
NASA Astrophysics Data System (ADS)
Jee, SolKeun; Moser, Robert D.
2012-08-01
This study provides a simple moving-grid scheme which is based on a modified conservative form of the incompressible Navier-Stokes equations for flow around a moving rigid body. The modified integral form is conservative and seeks the solution of the absolute velocity. This approach is different from previous conservative differential forms [1-3] whose reference frame is not inertial. Keeping the reference frame being inertial results in simpler mathematical derivation to the governing equation which includes one dyadic product of velocity vectors in the convective term, whereas the previous [2,3] needs to obtain the time derivative with respect to non-inertial frames causing an additional dyadic product in the convective term. The scheme is implemented in a second-order accurate Navier-Stokes solver and maintains the order of the accuracy. After this verification, the scheme is validated for a pitching airfoil with very high frequencies. The simulation results match very well with the experimental results [4,5], including vorticity fields and a net thrust force. This airfoil simulation also provides detailed vortical structures near the trailing edge and time-evolving aerodynamic forces that are used to investigate the mechanism of the thrust force generation and the effects of the trailing edge shape. The developed moving-grid scheme demonstrates its validity for a rapid oscillating motion.
A least-squares finite element method for 3D incompressible Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Jiang, Bo-Nan; Lin, T. L.; Hou, Lin-Jun; Povinelli, Louis A.
1993-01-01
The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.
A Split-Step Scheme for the Incompressible Navier-Stokes
Henshaw, W; Petersson, N A
2001-06-12
We describe a split-step finite-difference scheme for solving the incompressible Navier-Stokes equations on composite overlapping grids. The split-step approach decouples the solution of the velocity variables from the solution of the pressure. The scheme is based on the velocity-pressure formulation and uses a method of lines approach so that a variety of implicit or explicit time stepping schemes can be used once the equations have been discretized in space. We have implemented both second-order and fourth-order accurate spatial approximations that can be used with implicit or explicit time stepping methods. We describe how to choose appropriate boundary conditions to make the scheme accurate and stable. A divergence damping term is added to the pressure equation to keep the numerical dilatation small. Several numerical examples are presented.
NASA Astrophysics Data System (ADS)
Mohamed, Mamdouh S.; Hirani, Anil N.; Samtaney, Ravi
2016-05-01
A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.
An implicit velocity decoupling procedure for the incompressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Kim, Kyoungyoun; Baek, Seung-Jin; Sung, Hyung Jin
2002-01-01
An efficient numerical method to solve the unsteady incompressible Navier-Stokes equations is developed. A fully implicit time advancement is employed to avoid the Courant-Friedrichs-Lewy restriction, where the Crank-Nicolson discretization is used for both the diffusion and convection terms. Based on a block LU decomposition, velocity-pressure decoupling is achieved in conjunction with the approximate factorization. The main emphasis is placed on the additional decoupling of the intermediate velocity components with only nth time step velocity. The temporal second-order accuracy is preserved with the approximate factorization without any modification of boundary conditions. Since the decoupled momentum equations are solved without iteration, the computational time is reduced significantly. The present decoupling method is validated by solving several test cases, in particular, the turbulent minimal channel flow unit. Copyright
NASA Astrophysics Data System (ADS)
Chen, Ying; Shen, Jie
2016-03-01
In this paper we develop a fully adaptive energy stable scheme for Cahn-Hilliard Navier-Stokes system, which is a phase-field model for two-phase incompressible flows, consisting a Cahn-Hilliard-type diffusion equation and a Navier-Stokes equation. This scheme, which is decoupled and unconditionally energy stable based on stabilization, involves adaptive mesh, adaptive time and a nonlinear multigrid finite difference method. Numerical experiments are carried out to validate the scheme for problems with matched density and non-matched density, and also demonstrate that CPU time can be significantly reduced with our adaptive approach.
NASA Technical Reports Server (NTRS)
Goodrich, John W.
1991-01-01
An algorithm is presented for unsteady two-dimensional incompressible Navier-Stokes calculations. This algorithm is based on the fourth order partial differential equation for incompressible fluid flow which uses the streamfunction as the only dependent variable. The algorithm is second order accurate in both time and space. It uses a multigrid solver at each time step. It is extremely efficient with respect to the use of both CPU time and physical memory. It is extremely robust with respect to Reynolds number.
NASA Technical Reports Server (NTRS)
Goodrich, John W.
1991-01-01
An algorithm is presented for unsteady two-dimensional incompressible Navier-Stokes calculations. This algorithm is based on the fourth order partial differential equation for incompressible fluid flow which uses the streamfunction as the only dependent variable. The algorithm is second order accurate in both time and space. It uses a multigrid solver at each time step. It is extremely efficient with respect to the use of both CPU time and physical memory. It is extremely robust with respect to Reynolds number.
Dynamics and Control of a Reduced Order System of the 2-d Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Smaoui, Nejib; Zribi, Mohamed
2014-11-01
The dynamics and control problem of a reduced order system of the 2-d Navier-Stokes (N-S) equations is analyzed. First, a seventh order system of nonlinear ordinary differential equations (ODE) which approximates the dynamical behavior of the 2-d N-S equations is obtained by using the Fourier Galerkin method. We show that the dynamics of this ODE system transforms from periodic solutions to chaotic attractors through a sequence of bifurcations including a period doubling scenarios. Then three Lyapunov based controllers are designed to either control the system of ODEs to a desired fixed point or to synchronize two ODE systems obtained from the truncation of the 2-d N-S equations under different conditions. Numerical simulations are presented to show the effectiveness of the proposed controllers. This research was supported and funded by the Research Sector, Kuwait University under Grant No. SM02/14.
Three-step H-P adaptve strategy for the incompressible Navier-Stokes equations
Oden, J.T.; Wu, W.; Ainsworth, M.
1995-12-31
Recently, a reliable a posteriori error estimate was developed, mainly based on the element residual method, for a class of steady state incompressible Navier-Stokes equations. In this paper, using this error estimate, a three-step h-p adaptive strategy is developed to solve incompressible flow problems. The goal of developing an h-p adaptive strategy is to obtain accurate approximate solutions while minimizing computational costs. The basic idea of the three-step h-p adaptive strategy is to solve for the system on the three consecutive meshes, i.e. an initial mesh, an intermediate h-p adaptive mesh, and a final h-p adaptive mesh. Each new adaptive mesh is obtained by estimating the error on the previous mesh and executing a single h- or p- refinement procedure on the previous mesh according to the results of the adaptive strategy. Numerical results indicate that the proposed three-step adaptive strategy produces accurate solutions while keeping the total computational costs under control.
Dynamics and Control of the 2-d Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Smaoui, Nejib; Zribi, Mohamed
2013-11-01
The control problem of the dynamics of the two-dimensional (2-d) Navier-Stokes (N-S) equations with spatially periodic and temporally steady forcing is studied. First, we devise a dynamical system of several nonlinear differential equations by a truncation of the 2-d N-S equations. Then, we study the dynamics of the obtained Galerkin system by analyzing the system's attractors for different values of the Reynolds number, Re. By applying the symmetry of the equation on one of the system's attractors, a symmetric limit trajectory that is part of the dynamics is obtained. Next, a control strategy to drive the dynamics from one attractor to another attractor for a given Re is designed. Finally, numerical simulations are undertaken to validate the theoretical developments. This work was supported and funded by Kuwait University Research Grant No. SM02/13.
Towards A Fast High-Order Method for Unsteady Incompressible Navier-Stokes Equations using FR/CPR
NASA Astrophysics Data System (ADS)
Cox, Christopher; Liang, Chunlei; Plesniak, Michael
2014-11-01
A high-order compact spectral difference method for solving the 2D incompressible Navier-Stokes equations on unstructured grids is currently being developed. This method employs the gGA correction of Huynh, and falls under the class of methods now refered to as Flux Reconstruction/Correction Procedure via Reconstruction. This method and the artificial compressibility method are integrated along with a dual time-integration scheme to model unsteady incompressible viscous flows. A lower-upper symmetric Gauss-Seidel scheme and a backward Euler scheme are used to efficiently march the solution in pseudo time and physical time, respectively. We demonstrate order of accuracy with steady Taylor-Couette flow at Re = 10. We further validate the solver with steady flow past a NACA0012 airfoil at zero angle of attack at Re = 1850 and unsteady flow past a circle at Re = 100. The implicit time-integration scheme for the pseudo time derivative term is proved efficient and effective for the classical artificial compressibility treatment to achieve the divergence-free condition of the continuity equation. We greatly acknowledge financial support from The George Washington University under the Presidential Merit Fellowship.
NASA Technical Reports Server (NTRS)
Thompson, D. S.
1980-01-01
The full Navier-Stokes equations for incompressible turbulent flow must be solved to accurately represent all flow phenomena which occur in a high Reynolds number incompressible flow. A two layer algebraic eddy viscosity turbulence model is used to represent the Reynolds stress in the primitive variable formulation. The development of the boundary-fitted coordinate systems makes the numerical solution of these equations feasible for arbitrarily shaped bodies. The nondimensional time averaged Navier-Stokes equations, including the turbulence mode, are represented by finite difference approximations in the transformed plane. The resulting coupled system of nonlinear algebraic equations is solved using a point successive over relaxation iteration. The test case considered was a NACA 64A010 airfoil section at an angle of attack of two degrees and a Reynolds number of 2,000,000.
Roberts, Nathan V.; Demkowiz, Leszek; Moser, Robert
2015-11-15
The discontinuous Petrov-Galerkin methodology with optimal test functions (DPG) of Demkowicz and Gopalakrishnan [18, 20] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. Whereas Bubnov-Galerkin methods use identical trial and test spaces, Petrov-Galerkin methods allow these function spaces to differ. In DPG, test functions are computed on the fly and are chosen to realize the supremum in the inf-sup condition; the method is equivalent to a minimum residual method. For well-posed problems with sufficiently regular solutions, DPG can be shown to converge at optimal rates—the inf-sup constants governing the convergence are mesh-independent, and of the same order as those governing the continuous problem [48]. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. We employ DPG to solve the steady incompressible Navier-Stokes equations in two dimensions, building on previous work on the Stokes equations, and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh. We apply our approach to a manufactured solution due to Kovasznay as well as the lid-driven cavity flow, backward-facing step, and flow past a cylinder problems.
NASA Technical Reports Server (NTRS)
Rogers, S. E.; Kwak, D.; Chang, J. L. C.
1986-01-01
Numerically solving the incompressible Navier-Stokes equations is known to be time consuming and expensive. Testing of the INS3D computers code, which solves these equations with the use of the pseudocompressibility method, shows this method to be an efficient way to obtain the steady state solution. The effects of the waves introduced by the pseudocompressibility method are analyzed and criteria are set and tested for the choice of the pseudocompressibility parameter which governs the artificial sound speed. The code is tested using laminar flow over a two dimensional backward-facing step, and laminar flow over a two dimensional circular cylinder. The results of the computations over the backward-facing step are in excellent agreement with experimental results. The transient solution of the flow over the cylinder impulsively started from rest is in good agreement with experimental results. However, the computed frequency of periodic shedding of vortices behind the cylinder is not in agreement with the experimental value. For a three dimensional test case, computations were conducted for a cylinder end wall junction. The saddle point separation and horseshoe vortex system appear in the computed field. The solution also shows secondary vortex filaments which wrap around the cylinder and spiral up in the wake.
A direct algorithm for solution of incompressible three-dimensional unsteady Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Osswald, G. A.; Ghia, K. N.; Ghia, U.
1987-01-01
A direct, implicit, numerical solution algorithm, with second-order accuracy in space and time, is constructed for the three-dimensional unsteady incompressible Navier-Stokes equations formulated in terms of velocity and vorticity, using generalized orthogonal coordinates to achieve the accurate solution of complex viscous flow configurations. A numerically stable, efficient, direct inversion procedure is developed for the computationally intensive divergence-curl elliptic velocity problem. This overdetermined partial differential operator is first formulated as a uniquely determined, nonsingular matrix-vector problem; this aspect of the procedure is a unique feature of the present analysis. The three-dimensional vorticity-transport equation is solved by a modified factorization technique which completely eliminates the need for any block-matrix inversions and only scalar tridiagonal matrices need to be inverted. The method is applied to the test problem of the three-dimensional flow within a shear-driven cubical box. Coherent streamwise vortex structures are observed within the steady-state flow at Re = 100.
An iteration free backward semi-Lagrangian scheme for solving incompressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Piao, Xiangfan; Bu, Sunyoung; Bak, Soyoon; Kim, Philsu
2015-02-01
A backward semi-Lagrangian method based on the error correction method is designed to solve incompressible Navier-Stokes equations. The time derivative of the Stokes equation is discretized with the second order backward differentiation formula. For the induced steady Stokes equation, a projection method is used to split it into velocity and pressure. Fourth-order finite differences for partial derivatives are used to the boundary value problems for the velocity and the pressure. Also, finite linear systems for Poisson equations and Helmholtz equations are solved with a matrix-diagonalization technique. For characteristic curves satisfying highly nonlinear self-consistent initial value problems, the departure points are solved with an error correction strategy having a temporal convergence of order two. The constructed algorithm turns out to be completely iteration free. In particular, the suggested algorithm possesses a good behavior of the total energy conservation compared to existing methods. To assess the effectiveness of the method, two-dimensional lid-driven cavity problems with large different Reynolds numbers are solved. The doubly periodic shear layer flows are also used to assess the efficiency of the algorithm.
NASA Astrophysics Data System (ADS)
Roberts, Nathan V.; Demkowicz, Leszek; Moser, Robert
2015-11-01
The discontinuous Petrov-Galerkin methodology with optimal test functions (DPG) of Demkowicz and Gopalakrishnan [18,20] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. Whereas Bubnov-Galerkin methods use identical trial and test spaces, Petrov-Galerkin methods allow these function spaces to differ. In DPG, test functions are computed on the fly and are chosen to realize the supremum in the inf-sup condition; the method is equivalent to a minimum residual method. For well-posed problems with sufficiently regular solutions, DPG can be shown to converge at optimal rates-the inf-sup constants governing the convergence are mesh-independent, and of the same order as those governing the continuous problem [48]. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. We employ DPG to solve the steady incompressible Navier-Stokes equations in two dimensions, building on previous work on the Stokes equations, and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh. We apply our approach to a manufactured solution due to Kovasznay as well as the lid-driven cavity flow, backward-facing step, and flow past a cylinder problems.
NASA Technical Reports Server (NTRS)
Rosenfeld, Moshe; Kwak, Dochan; Vinokur, Marcel
1988-01-01
A solution method based on a fractional step approach is developed for obtaining time-dependent solutions of the three-dimensional, incompressible Navier-Stokes equations in generalized coordinate systems. The governing equations are discretized conservatively by finite volumes using a staggered mesh system. The primitive variable formulation uses the volume fluxes across the faces of each computational cell as dependent variables. This procedure, combined with accurate and consistent approximations of geometric parameters, is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain favorable convergence properties of the Poisson solver. The discretized equations are second-order-accurate in time and space and no smoothing terms are added. An approximate-factorization scheme is implemented in solving the momentum equations. A novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two and three-dimensional solutions are compared with other numerical and experimental results to validate the present method.
Approximate factorization for incompressible flow. Ph.D. Thesis; [Navier-Stokes equation
NASA Technical Reports Server (NTRS)
Bernard, R. S.
1981-01-01
For computational solution of the incompressible Navier-Stokes equations, the approximate factorization (AF) algorithm is used to solve the vectorized momentum equation in delta form based on the pressure calculated in the previous time step. The newly calculated velocities are substituted into the pressure equation (obtained from a linear combination of the continuity and momentum equation), which is then solved by means of line SOR. Computational results are presented for the NACA 66 sub 3 018 airfoil at Reynolds numbers of 1000 and 40,000 and attack angles of 0 and 6 degrees. Comparison with wind tunnel data for Re = 40,000 indicates good qualitative agreement between measured and calculated pressure distributions. Quantitative agreement is only fair, however, with the calculations somewhat displaced from the measurements. Furthermore, the computed velocity profiles are unrealistically thick around the airfoil, due to the excessive amount of artificial viscosity needed for stability. Based on the performance of the algorithm with regard to stability, it is concluded that AF/SOR is suitable for calculations at Reynolds numbers less than 10,000. Speedwise, the method is faster than point SOR by at least a factor of two.
A fast Poisson solver for unsteady incompressible Navier-Stokes equations on the half-staggered grid
NASA Technical Reports Server (NTRS)
Golub, G. H.; Huang, L. C.; Simon, H.; Tang, W. -P.
1995-01-01
In this paper, a fast Poisson solver for unsteady, incompressible Navier-Stokes equations with finite difference methods on the non-uniform, half-staggered grid is presented. To achieve this, new algorithms for diagonalizing a semi-definite pair are developed. Our fast solver can also be extended to the three dimensional case. The motivation and related issues in using this second kind of staggered grid are also discussed. Numerical testing has indicated the effectiveness of this algorithm.
NASA Astrophysics Data System (ADS)
Ha, Sanghyun; You, Donghyun
2015-11-01
Utility of the computational power of Graphics Processing Units (GPUs) is elaborated for solutions of both incompressible and compressible Navier-Stokes equations. A semi-implicit ADI finite-volume method for integration of the incompressible and compressible Navier-Stokes equations, which are discretized on a structured arbitrary grid, is parallelized for GPU computations using CUDA (Compute Unified Device Architecture). In the semi-implicit ADI finite-volume method, the nonlinear convection terms and the linear diffusion terms are integrated in time using a combination of an explicit scheme and an ADI scheme. Inversion of multiple tri-diagonal matrices is found to be the major challenge in GPU computations of the present method. Some of the algorithms for solving tri-diagonal matrices on GPUs are evaluated and optimized for GPU-acceleration of the present semi-implicit ADI computations of incompressible and compressible Navier-Stokes equations. Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning Grant NRF-2014R1A2A1A11049599.
NASA Astrophysics Data System (ADS)
Ahn, Hyung Taek
2005-12-01
A new incompressible Navier-Stokes method is developed for unstructured general hybrid meshes which contain all four types of elements in a single computational domain, namely tetrahedra, pyramids, prisms, and hexahedra. Various types of general hybrid meshes are utilized and appropriate numerical flux computation schemes are presented. The artificial compressibility method with a dual time-stepping scheme is used for the time-accurate solution of the incompressible Navier-Stokes equations. The Spalart-Allmaras turbulence model is also presented in the dual time-stepping form and is solved in a strongly coupled manner with the incompressible Navier-Stokes equations. The developed scheme is applied to the study of the inflow turbulence effect on the hydrodynamic forces exerted on a circular cylinder. In order to accommodate possible structural and mesh motion, the method is extended to the arbitrary Lagrangian-Eulerian (ALE) frame of reference. The geometric conservation law is satisfied with the proposed ALE scheme in moving mesh simulations. The developed ALE scheme is applied to the vortex induced vibration of a cylinder. A strong coupling of fluid and structure interaction based on the predictor-corrector method is presented. The superior stability property of the strong coupling is demonstrated by a comparison with the weak coupling. Finally, the developed methods are parallelized for distributed memory machines using partitioned general hybrid meshes and an efficient parallel communication scheme to minimize CPU time.
ARC2D - EFFICIENT SOLUTION METHODS FOR THE NAVIER-STOKES EQUATIONS (DEC RISC ULTRIX VERSION)
NASA Technical Reports Server (NTRS)
Biyabani, S. R.
1994-01-01
ARC2D is a computational fluid dynamics program developed at the NASA Ames Research Center specifically for airfoil computations. The program uses implicit finite-difference techniques to solve two-dimensional Euler equations and thin layer Navier-Stokes equations. It is based on the Beam and Warming implicit approximate factorization algorithm in generalized coordinates. The methods are either time accurate or accelerated non-time accurate steady state schemes. The evolution of the solution through time is physically realistic; good solution accuracy is dependent on mesh spacing and boundary conditions. The mathematical development of ARC2D begins with the strong conservation law form of the two-dimensional Navier-Stokes equations in Cartesian coordinates, which admits shock capturing. The Navier-Stokes equations can be transformed from Cartesian coordinates to generalized curvilinear coordinates in a manner that permits one computational code to serve a wide variety of physical geometries and grid systems. ARC2D includes an algebraic mixing length model to approximate the effect of turbulence. In cases of high Reynolds number viscous flows, thin layer approximation can be applied. ARC2D allows for a variety of solutions to stability boundaries, such as those encountered in flows with shocks. The user has considerable flexibility in assigning geometry and developing grid patterns, as well as in assigning boundary conditions. However, the ARC2D model is most appropriate for attached and mildly separated boundary layers; no attempt is made to model wake regions and widely separated flows. The techniques have been successfully used for a variety of inviscid and viscous flowfield calculations. The Cray version of ARC2D is written in FORTRAN 77 for use on Cray series computers and requires approximately 5Mb memory. The program is fully vectorized. The tape includes variations for the COS and UNICOS operating systems. Also included is a sample routine for CONVEX
ARC2D - EFFICIENT SOLUTION METHODS FOR THE NAVIER-STOKES EQUATIONS (CRAY VERSION)
NASA Technical Reports Server (NTRS)
Pulliam, T. H.
1994-01-01
ARC2D is a computational fluid dynamics program developed at the NASA Ames Research Center specifically for airfoil computations. The program uses implicit finite-difference techniques to solve two-dimensional Euler equations and thin layer Navier-Stokes equations. It is based on the Beam and Warming implicit approximate factorization algorithm in generalized coordinates. The methods are either time accurate or accelerated non-time accurate steady state schemes. The evolution of the solution through time is physically realistic; good solution accuracy is dependent on mesh spacing and boundary conditions. The mathematical development of ARC2D begins with the strong conservation law form of the two-dimensional Navier-Stokes equations in Cartesian coordinates, which admits shock capturing. The Navier-Stokes equations can be transformed from Cartesian coordinates to generalized curvilinear coordinates in a manner that permits one computational code to serve a wide variety of physical geometries and grid systems. ARC2D includes an algebraic mixing length model to approximate the effect of turbulence. In cases of high Reynolds number viscous flows, thin layer approximation can be applied. ARC2D allows for a variety of solutions to stability boundaries, such as those encountered in flows with shocks. The user has considerable flexibility in assigning geometry and developing grid patterns, as well as in assigning boundary conditions. However, the ARC2D model is most appropriate for attached and mildly separated boundary layers; no attempt is made to model wake regions and widely separated flows. The techniques have been successfully used for a variety of inviscid and viscous flowfield calculations. The Cray version of ARC2D is written in FORTRAN 77 for use on Cray series computers and requires approximately 5Mb memory. The program is fully vectorized. The tape includes variations for the COS and UNICOS operating systems. Also included is a sample routine for CONVEX
NASA Technical Reports Server (NTRS)
Jiang, Bo-Nan; Sonnad, Vijay
1991-01-01
A p-version of the least squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady state incompressible viscous flow problems. The resulting system of symmetric and positive definite linear equations can be solved satisfactorily with the conjugate gradient method. In conjunction with the use of rapid operator application which avoids the formation of either element of global matrices, it is possible to achieve a highly compact and efficient solution scheme for the incompressible Navier-Stokes equations. Numerical results are presented for two-dimensional flow over a backward facing step. The effectiveness of simple outflow boundary conditions is also demonstrated.
Numerical Solution of Incompressible Navier-Stokes Equations Using a Fractional-Step Approach
NASA Technical Reports Server (NTRS)
Kiris, Cetin; Kwak, Dochan
1999-01-01
A fractional step method for the solution of steady and unsteady incompressible Navier-Stokes equations is outlined. The method is based on a finite volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (3rd and 5th) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds Numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when 5th-order upwind differencing and a modified production term in the Baldwin-Barth one-equation turbulence model are used with adequate grid resolution.
Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations
Dascaliuc, Radu; Thomann, Enrique; Waymire, Edward C.; Michalowski, Nicholas
2015-07-15
The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group (0, ∞), naturally arise as a result of this investigation.
Reynolds-Averaged Navier-Stokes Simulation of a 2D Circulation Control Wind Tunnel Experiment
NASA Technical Reports Server (NTRS)
Allan, Brian G.; Jones, Greg; Lin, John C.
2011-01-01
Numerical simulations are performed using a Reynolds-averaged Navier-Stokes (RANS) flow solver for a circulation control airfoil. 2D and 3D simulation results are compared to a circulation control wind tunnel test conducted at the NASA Langley Basic Aerodynamics Research Tunnel (BART). The RANS simulations are compared to a low blowing case with a jet momentum coefficient, C(sub u), of 0:047 and a higher blowing case of 0.115. Three dimensional simulations of the model and tunnel walls show wall effects on the lift and airfoil surface pressures. These wall effects include a 4% decrease of the midspan sectional lift for the C(sub u) 0.115 blowing condition. Simulations comparing the performance of the Spalart Allmaras (SA) and Shear Stress Transport (SST) turbulence models are also made, showing the SST model compares best to the experimental data. A Rotational/Curvature Correction (RCC) to the turbulence model is also evaluated demonstrating an improvement in the CFD predictions.
An efficient non-linear multigrid procedure for the incompressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Sivaloganathan, S.; Shaw, G. J.
An efficient Full Approximation multigrid scheme for finite volume discretizations of the Navier-Stokes equations is presented. The algorithm is applied to the driven cavity test problem. Numerical results are presented and a comparison made with PACE, a Rolls-Royce industrial code, which uses the SIMPLE pressure correction method as an iterative solver.
Three-dimensional full Navier-Stokes solvers for incompressible flows past arbitrary geometries
NASA Astrophysics Data System (ADS)
Deng, G. B.; Piquet, J.; Queutey, P.; Visonneau, M.
1991-05-01
The computation of the three-dimensional viscous flow past several geometries is investigated. An iterative technique resting on the fully elliptic mode is applied to the Reynolds-Averaged Navier-Stokes Equations written in a nonorthogonal curvilinear body-fitted coordinate system. Results of the computation are compared with available experiments.
NASA Technical Reports Server (NTRS)
Rosenfeld, Moshe
1990-01-01
The development, validation and application of a fractional step solution method of the time-dependent incompressible Navier-Stokes equations in generalized coordinate systems are discussed. A solution method that combines a finite-volume discretization with a novel choice of the dependent variables and a fractional step splitting to obtain accurate solutions in arbitrary geometries was previously developed for fixed-grids. In the present research effort, this solution method is extended to include more general situations, including cases with moving grids. The numerical techniques are enhanced to gain efficiency and generality.
NASA Technical Reports Server (NTRS)
Zeng, S.; Wesseling, P.
1993-01-01
The performance of a linear multigrid method using four smoothing methods, called SCGS (Symmetrical Coupled GauBeta-Seidel), CLGS (Collective Line GauBeta-Seidel), SILU (Scalar ILU), and CILU (Collective ILU), is investigated for the incompressible Navier-Stokes equations in general coordinates, in association with Galerkin coarse grid approximation. Robustness and efficiency are measured and compared by application to test problems. The numerical results show that CILU is the most robust, SILU the least, with CLGS and SCGS in between. CLGS is the best in efficiency, SCGS and CILU follow, and SILU is the worst.
Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains
NASA Astrophysics Data System (ADS)
Brzeźniak, Z.; Caraballo, T.; Langa, J. A.; Li, Y.; Łukaszewicz, G.; Real, J.
We show that the stochastic flow generated by the 2-dimensional Stochastic Navier-Stokes equations with rough noise on a Poincaré-like domain has a unique random attractor. One of the technical problems associated with the rough noise is overcomed by the use of the corresponding Cameron-Martin (or reproducing kernel Hilbert) space. Our results complement the result by Brzeźniak and Li (2006) [10] who showed that the corresponding flow is asymptotically compact and also generalize Caraballo et al. (2006) [12] who proved existence of a unique attractor for the time-dependent deterministic Navier-Stokes equations.
NASA Astrophysics Data System (ADS)
Chen, De-Xiang; Xu, Zi-Li; Liu, Shi; Feng, Yong-Xin
2014-03-01
Modern least squares finite element method (LSFEM) has advantage over mixed finite element method for non-self-adjoint problem like Navier-Stokes equations, but has problem to be norm equivalent and mass conservative when using C0 type basis. In this paper, LSFEM with non-uniform B-splines (NURBS) is proposed for Navier-Stokes equations. High order continuity NURBS is used to construct the finite-dimensional spaces for both velocity and pressure. Variational form is derived from the governing equations with primitive variables and the DOFs due to additional variables are not necessary. There is a novel k-refinement which has spectral convergence of least squares functional. The method also has same advantages as isogeometric analysis like automatic mesh generation and exact geometry representation. Several benchmark problems are solved using the proposed method. The results agree well with the benchmark solutions available in literature. The results also show good performance in mass conservation.
NASA Astrophysics Data System (ADS)
Codina, Ramon; Blasco, Jordi; Buscaglia, Gustavo C.; Huerta, Antonio
2001-10-01
We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier-Stokes equations which allows the use of equal order velocity-pressure interpolations. The method consists in introducing the projection of the pressure gradient and adding the difference between the pressure Laplacian and the divergence of this new field to the incompressibility equation, both multiplied by suitable algorithmic parameters. The main purpose of this paper is to discuss how to deal with the new variable in the implementation of the algorithm. Obviously, it could be treated as one extra unknown, either explicitly or as a condensed variable. However, we take for granted that the only way for the algorithm to be efficient is to uncouple it from the velocity-pressure calculation in one way or another. Here we discuss some iterative schemes to perform this uncoupling of the pressure gradient projection (PGP) from the calculation of the velocity and the pressure, both for the stationary and the transient Navier-Stokes equations. In the first case, the strategies analyzed refer to the interaction of the linearization loop and the iterative segregation of the PGP, whereas in the second the main dilemma concerns the explicit or implicit treatment of the PGP. Copyright
NASA Technical Reports Server (NTRS)
Yoon, Seokkwan; Chang, Leon; Kwak, Dochan
1989-01-01
A numerical method is developed for solving the incompressible Navier-Stokes equations using the concept of pseudocompressibility. A lower-upper symmetric-Gauss-Seidel implicit scheme is developed for three-dimensional incompressible viscous flow computations. The present algorithm offers additional advantages when solving the flow equations with source terms. Complete vectorizability of the algorithm on oblique planes of sweep in three-dimensions is accomplished in a new flow solver, INS3D-LU code. Spatial differencing is a second-order accurate semi-discrete finite-volume method augmented by a third-order accurate numerical dissipation model which is based on spectral-radii. Comparison of numerical solutions for a curved duct with experimental data shows good agreement. The method is applied to calculate the inducer flow of the Space Shuttle Main Engine turbopump.
Shadid, John Nicolas; Elman, Howard; Shuttleworth, Robert R.; Howle, Victoria E.; Tuminaro, Raymond Stephen
2007-04-01
In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [25]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code.
NASA Astrophysics Data System (ADS)
Ren, Dandan; Ou, Yaobin
2016-08-01
In this paper, we prove the incompressible limit of all-time strong solutions to the three-dimensional full compressible Navier-Stokes equations. Here the velocity field and temperature satisfy the Dirichlet boundary condition and convective boundary condition, respectively. The uniform estimates in both the Mach number {ɛin(0,overline{ɛ}]} and time {tin[0,∞)} are established by deriving a differential inequality with decay property, where {overline{ɛ} in(0,1]} is a constant. Based on these uniform estimates, the global solution of full compressible Navier-Stokes equations with "well-prepared" initial conditions converges to the one of isentropic incompressible Navier-Stokes equations as the Mach number goes to zero.
Theoretical study of the incompressible Navier-Stokes equations by the least-squares method
NASA Technical Reports Server (NTRS)
Jiang, Bo-Nan; Loh, Ching Y.; Povinelli, Louis A.
1994-01-01
Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis.
NASA Technical Reports Server (NTRS)
Davis, J. E.
1980-01-01
A second-order time-accurate and spatially factored algorithm was used in a finite difference scheme for the numerical solution of the time-dependent, incompressible, two dimensional Navier-Stokes equations in conservation-law form using vorticity and stream function variables. The systems of equations are solved at each time step by an iterative technique. Numerical results were obtained for a circular cylinder at a Reynolds number of 15, and an NACA 0012 airfoil at zero angle of attack at Reynolds numbers of 10 to the third and 10 to the fourth powers. The results are in agreement with another numerical technique, and the computing time required to obtain the steady state solution at the Reynolds number of 10 to the 4th power was 49.7 sec on CDC 7600 computer using a 65 x 84 computational grind.
NASA Technical Reports Server (NTRS)
Rogers, S. E.; Kwak, D.; Chang, J. L. C.
1986-01-01
The method of pseudocompressibility has been shown to be an efficient method for obtaining a steady-state solution to the incompressible Navier-Stokes equations. Recent improvements to this method include the use of a diagonal scheme for the inversion of the equations at each iteration. The necessary transformations have been derived for the pseudocompressibility equations in generalized coordinates. The diagonal algorithm reduces the computing time necessary to obtain a steady-state solution by a factor of nearly three. Implicit viscous terms are maintained in the equations, and it has become possible to use fourth-order implicit dissipation. The steady-state solution is unchanged by the approximations resulting from the diagonalization of the equations. Computed results for flow over a two-dimensional backward-facing step and a three-dimensional cylinder mounted normal to a flat plate are presented for both the old and new algorithms. The accuracy and computing efficiency of these algorithms are compared.
NASA Technical Reports Server (NTRS)
Rosenfeld, Moshe; Kwak, Dochan; Vinokur, Marcel
1991-01-01
The time-dependent, three-dimensional incompressible Navier-Stokes equations are presently solved in generalized coordinate systems by means of a fractional-step method whose primitive variable formulation uses as dependent variables, in place of the Cartesian components of the velocity: (1) pressure (defined at the center of the computational cell), and (2) volume fluxes across the faces of the cells. The momentum equations are solved by means of an approximate factorization method. A novel 'ZEBRA' scheme incorporating four-color ordering efficiently solves the Poisson equation. Illustrative two- and three-dimensional laminar flow test cases are computed and evaluated relative to extant numerical and experimental results, and good agreement is obtained.
Boergers, C.; Peskin, C.S.
1987-06-01
In the Lagrangian fractional step method introduced in this paper, the fluid velocity and pressure are defined on a collection of N fluid markers. At each time step, these markers are used to generate a Voronoi diagram, and this diagram is used to construct finite-difference operators corresponding to the divergence, gradient, and Laplacian. The splitting of the Navier--Stokes equations leads to discrete Helmholtz and Poisson problems, which we solve using a two-grid method. The nonlinear convection terms are modeled simply by the displacement of the fluid markers. We have implemented this method on a periodic domain in the plane. We describe an efficient algorithm for the numerical construction of periodic Voronoi diagrams, and we report on numerical results which indicate the the fractional step method is convergent of first order. The overall work per time step is proportional to N log N. copyright 1987 Academic Press, Inc.
NASA Technical Reports Server (NTRS)
Mizukami, M.; Saunders, J. D.
1995-01-01
The supersonic diffuser of a Mach 2.68 bifurcated, rectangular, mixed-compression inlet was analyzed using a two-dimensional (2D) Navier-Stokes flow solver. Parametric studies were performed on turbulence models, computational grids and bleed models. The computer flowfield was substantially different from the original inviscid design, due to interactions of shocks, boundary layers, and bleed. Good agreement with experimental data was obtained in many aspects. Many of the discrepancies were thought to originate primarily from 3D effects. Therefore, a balance should be struck between expending resources on a high fidelity 2D simulation, and the inherent limitations of 2D analysis. The solutions were fairly insensitive to turbulence models, grids and bleed models. Overall, the k-e turbulence model, and the bleed models based on unchoked bleed hole discharge coefficients or uniform velocity are recommended. The 2D Navier-Stokes methods appear to be a useful tool for the design and analysis of supersonic inlets, by providing a higher fidelity simulation of the inlet flowfield than inviscid methods, in a reasonable turnaround time.
Controlling the Dynamics of the Five-Mode Truncation System of the 2-d Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Smaoui, Nejib; Zribi, Mohamed
2015-11-01
The dynamics and the control problem of the two dimensional (2-d) Navier-Stokes (N-S) equations with spatially periodic and temporally steady forcing is addressed. At first, the Fourier Galerkin method is applied to the 2-d N-S equations to obtain a fifth order system of nonlinear ordinary differential equations (ODE) that approximates the behavior of these equations. Simulation studies indicate that the obtained ODE system captures the behavior of the 2-d N-S equations. Then, a control law is proposed to drive the states of the ODE system to a desired fixed point. Next, a second control law is developed to synchronize two reduced order ODE models of the 2-d N-S equations having the same Reynolds number and starting from different initial conditions. Finally, simulation results are undertaken to validate the theoretical developments. This research was supported and funded by the Research Sector, Kuwait University under Grant No. SM 05/15.
NASA Astrophysics Data System (ADS)
Zhai, Cuili; Zhang, Ting
2015-09-01
In this article, we consider the global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity. More precisely, assuming a 0 ∈ B˙ q , 1 /3 q ( R 3 ) and u 0 = ( u0 h , u0 3 ) ∈ B˙ p , 1 - 1 + /3 p ( R 3 ) for p, q ∈ (1, 6) with sup ( /1 p , /1 q ) ≤ /1 3 + inf ( /1 p , /1 q ) , we prove that if C a↑0↑ B˙q1/3 q α (↑u0 3↑ B˙ p , 1 - 1 + /3 p/μ + 1 ) ≤ 1 , /C μ (↑u0 h↑ B˙ p , 1 - 1 + /3 p + ↑u03↑ B˙ p , 1 - 1 + /3 p 1 - α ↑u0h↑ B˙ p , 1 - 1 + /3 p α) ≤ 1 , then the system has a unique global solution a ∈ C ˜ ( [ 0 , ∞ ) ; B˙ q , 1 /3 q ( R 3 ) ) , u ∈ C ˜ ( [ 0 , ∞ ) ; B˙ p , 1 - 1 + /3 p ( R 3 ) ) ∩ L 1 ( R + ; B˙ p , 1 1 + /3 p ( R 3 ) ) . It improves the recent result of M. Paicu and P. Zhang [J. Funct. Anal. 262, 3556-3584 (2012)], where the exponent form of the initial smallness condition is replaced by a polynomial form.
NASA Technical Reports Server (NTRS)
Rosenfeld, Moshe; Kwak, Dochan; Vinokur, Marcel
1992-01-01
A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.
Williams, P.T.
1993-09-01
As the field of computational fluid dynamics (CFD) continues to mature, algorithms are required to exploit the most recent advances in approximation theory, numerical mathematics, computing architectures, and hardware. Meeting this requirement is particularly challenging in incompressible fluid mechanics, where primitive-variable CFD formulations that are robust, while also accurate and efficient in three dimensions, remain an elusive goal. This dissertation asserts that one key to accomplishing this goal is recognition of the dual role assumed by the pressure, i.e., a mechanism for instantaneously enforcing conservation of mass and a force in the mechanical balance law for conservation of momentum. Proving this assertion has motivated the development of a new, primitive-variable, incompressible, CFD algorithm called the Continuity Constraint Method (CCM). The theoretical basis for the CCM consists of a finite-element spatial semi-discretization of a Galerkin weak statement, equal-order interpolation for all state-variables, a 0-implicit time-integration scheme, and a quasi-Newton iterative procedure extended by a Taylor Weak Statement (TWS) formulation for dispersion error control. Original contributions to algorithmic theory include: (a) formulation of the unsteady evolution of the divergence error, (b) investigation of the role of non-smoothness in the discretized continuity-constraint function, (c) development of a uniformly H{sup 1} Galerkin weak statement for the Reynolds-averaged Navier-Stokes pressure Poisson equation, (d) derivation of physically and numerically well-posed boundary conditions, and (e) investigation of sparse data structures and iterative methods for solving the matrix algebra statements generated by the algorithm.
Kempka, S.N.; Strickland, J.H.; Glass, M.W.; Peery, J.S.; Ingber, M.S.
1995-04-01
formulation to satisfy velocity boundary conditions for the vorticity form of the incompressible, viscous fluid momentum equations is presented. The tangential and normal components of the velocity boundary condition are satisfied simultaneously by creating vorticity adjacent to boundaries. The newly created vorticity is determined using a kinematical formulation which is a generalization of Helmholtz` decomposition of a vector field. Though it has not been generally recognized, these formulations resolve the over-specification issue associated with creating voracity to satisfy velocity boundary conditions. The generalized decomposition has not been widely used, apparently due to a lack of a useful physical interpretation. An analysis is presented which shows that the generalized decomposition has a relatively simple physical interpretation which facilitates its numerical implementation. The implementation of the generalized decomposition is discussed in detail. As an example the flow in a two-dimensional lid-driven cavity is simulated. The solution technique is based on a Lagrangian transport algorithm in the hydrocode ALEGRA. ALEGRA`s Lagrangian transport algorithm has been modified to solve the vorticity transport equation and the generalized decomposition, thus providing a new, accurate method to simulate incompressible flows. This numerical implementation and the new boundary condition formulation allow vorticity-based formulations to be used in a wider range of engineering problems.
Yang, S.L.; Chang, Y.L.; Arici, O. . Mechanics Dept.)
1994-11-01
The purpose of this paper is to present a numerical study of flow fields for the NREL S805 and S809 airfoils using a spatially second-order symmetric total variational diminishing scheme. The steady two-dimensional flow is modeled as turbulent, viscous, and incompressible and is formulated in the pseudo-compressible form. The turbulent flow is closed by the Baldwind-Lomax algebraic turbulence model. Numerical solutions are obtained by the implicit approximate-factorization method. Numerical solutions are obtained by the implicit approximate-factorization method. The accuracy of the numerical results is compared with the Delft two-dimensional wind tunnel test data. For comparison, the Eppler code results are also included. Numerical solutions of pressure and life coefficients show good agreement with the experimental data, but not the drag coefficients. To properly simulate the post-stall flow field, a better turbulence model should be used.
Control Theory based Shape Design for the Incompressible Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Cowles, G.; Martinelli, L.
2003-12-01
A design method for shape optimization in incompressible turbulent viscous flow has been developed and validated for inverse design. The gradient information is determined using a control theory based algorithm. With such an approach, the cost of computing the gradient is negligible. An additional adjoint system must be solved which requires the cost of a single steady state flow solution. Thus, this method has an enormous advantage over traditional finite-difference based algorithms. The method of artificial compressibility is utilized to solve both the flow and adjoint systems. An algebraic turbulence model is used to compute the eddy viscosity. The method is validated using several inverse wing design test cases. In each case, the program must modify the shape of the initial wing such that its pressure distribution matches that of the target wing. Results are shown for the inversion of both finite thickness wings as well as zero thickness wings which can be considered a model of yacht sails.
A spectrally accurate method for overlapping grid solution of incompressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Merrill, Brandon E.; Peet, Yulia T.; Fischer, Paul F.; Lottes, James W.
2016-02-01
An overlapping mesh methodology that is spectrally accurate in space and up to third-order accurate in time is developed for solution of unsteady incompressible flow equations in three-dimensional domains. The ability to decompose a global domain into separate, but overlapping, subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries. The methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally-accurate spatial interpolation and high-order accurate temporal extrapolation, and requires few, if any, iterations, yet maintains the global accuracy and stability of the underlying flow solver. The overlapping mesh methodology is thoroughly validated using two-dimensional and three-dimensional benchmark problems in laminar and turbulent flows. The spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver. The influence of long integration times, as well as inflow-outflow global boundary conditions on the performance of the overlapping grid solver is assessed. In a turbulent benchmark of fully-developed turbulent pipe flow, the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data. Scaling tests are presented that show near linear strong scaling, even for moderately large processor counts.
NASA Technical Reports Server (NTRS)
Kiris, Cetin C.; Kwak, Dochan; Rogers, Stuart E.
2002-01-01
This paper reviews recent progress made in incompressible Navier-Stokes simulation procedures and their application to problems of engineering interest. Discussions are focused on the methods designed for complex geometry applications in three dimensions, and thus are limited to primitive variable formulation. A summary of efforts in flow solver development is given followed by numerical studies of a few example problems of current interest. Both steady and unsteady solution algorithms and their salient features are discussed. Solvers discussed here are based on a structured-grid approach using either a finite -difference or a finite-volume frame work. As a grand-challenge application of these solvers, an unsteady turbopump flow simulation procedure has been developed which utilizes high performance computing platforms. In the paper, the progress toward the complete simulation capability of the turbo-pump for a liquid rocket engine is reported. The Space Shuttle Main Engine (SSME) turbo-pump is used as a test case for evaluation of two parallel computing algorithms that have been implemented in the INS3D code. The relative motion of the grid systems for the rotorstator interaction was obtained using overact grid techniques. Unsteady computations for the SSME turbo-pump, which contains 114 zones with 34.5 million grid points, are carried out on SCSI Origin 3000 systems at NASA Ames Research Center. The same procedure has been extended to the development of NASA-DeBakey Ventricular Assist Device (VAD) that is based on an axial blood pump. Computational, and clinical analysis of this device are presented.
NASA Technical Reports Server (NTRS)
Rogers, Stuart E.
1990-01-01
The current work is initiated in an effort to obtain an efficient, accurate, and robust algorithm for the numerical solution of the incompressible Navier-Stokes equations in two- and three-dimensional generalized curvilinear coordinates for both steady-state and time-dependent flow problems. This is accomplished with the use of the method of artificial compressibility and a high-order flux-difference splitting technique for the differencing of the convective terms. Time accuracy is obtained in the numerical solutions by subiterating the equations in psuedo-time for each physical time step. The system of equations is solved with a line-relaxation scheme which allows the use of very large pseudo-time steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. Numerous laminar test flow problems are computed and presented with a comparison against analytically known solutions or experimental results. These include the flow in a driven cavity, the flow over a backward-facing step, the steady and unsteady flow over a circular cylinder, flow over an oscillating plate, flow through a one-dimensional inviscid channel with oscillating back pressure, the steady-state flow through a square duct with a 90 degree bend, and the flow through an artificial heart configuration with moving boundaries. An adequate comparison with the analytical or experimental results is obtained in all cases. Numerical comparisons of the upwind differencing with central differencing plus artificial dissipation indicates that the upwind differencing provides a much more robust algorithm, which requires significantly less computing time. The time-dependent problems require on the order of 10 to 20 subiterations, indicating that the elliptical nature of the problem does require a substantial amount of computing effort.
NASA Astrophysics Data System (ADS)
Ge, Liang; Sotiropoulos, Fotis
2007-08-01
A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g. the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics 207 (2005) 457-492.]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow
Adaptive parallel multigrid for Euler and incompressible Navier-Stokes equations
Trottenberg, U.; Oosterlee, K.; Ritzdorf, H.
1996-12-31
The combination of (1) very efficient solution methods (Multigrid), (2) adaptivity, and (3) parallelism (distributed memory) clearly is absolutely necessary for future oriented numerics but still regarded as extremely difficult or even unsolved. We show that very nice results can be obtained for real life problems. Our approach is straightforward (based on {open_quotes}MLAT{close_quotes}). But, of course, reasonable refinement and load-balancing strategies have to be used. Our examples are 2D, but 3D is on the way.
NASA Astrophysics Data System (ADS)
Tavelli, Maurizio; Dumbser, Michael
2016-08-01
In this paper we propose a novel arbitrary high order accurate semi-implicit space-time discontinuous Galerkin method for the solution of the three-dimensional incompressible Navier-Stokes equations on staggered unstructured curved tetrahedral meshes. As is typical for space-time DG schemes, the discrete solution is represented in terms of space-time basis functions. This allows to achieve very high order of accuracy also in time, which is not easy to obtain for the incompressible Navier-Stokes equations. Similarly to staggered finite difference schemes, in our approach the discrete pressure is defined on the primary tetrahedral grid, while the discrete velocity is defined on a face-based staggered dual grid. While staggered meshes are state of the art in classical finite difference schemes for the incompressible Navier-Stokes equations, their use in high order DG schemes is still quite rare. A very simple and efficient Picard iteration is used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. From numerical experiments we find that the linear system seems to be reasonably well conditioned, since all simulations shown in this paper could be run without the use of any preconditioner, even up to very high polynomial degrees. For a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point, the resulting system is, in addition, symmetric and positive definite. This allows us to use even faster iterative solvers, like the conjugate gradient method. The flexibility and accuracy of high order space-time DG methods on curved
NASA Technical Reports Server (NTRS)
Cain, Michael D.
1999-01-01
The goal of this thesis is to develop an efficient and robust locally preconditioned semi-coarsening multigrid algorithm for the two-dimensional Navier-Stokes equations. This thesis examines the performance of the multigrid algorithm with local preconditioning for an upwind-discretization of the Navier-Stokes equations. A block Jacobi iterative scheme is used because of its high frequency error mode damping ability. At low Mach numbers, the performance of a flux preconditioner is investigated. The flux preconditioner utilizes a new limiting technique based on local information that was developed by Siu. Full-coarsening and-semi-coarsening are examined as well as the multigrid V-cycle and full multigrid. The numerical tests were performed on a NACA 0012 airfoil at a range of Mach numbers. The tests show that semi-coarsening with flux preconditioning is the most efficient and robust combination of coarsening strategy, and iterative scheme - especially at low Mach numbers.
Gorshkov, Aleksei V
2012-09-30
The problem of stabilizing a solution of the 2D Navier-Stokes system defined in the exterior of a bounded domain with smooth boundary is investigated. For a given initial velocity field a control on the boundary of the domain must be constructed such that the solution stabilizes to a prescribed vortex solution or trivial solution at the rate of 1/t{sup k}. On the way, related questions are investigated, concerning the behaviour of the spectrum of an operator under a relatively compact perturbation and the existence of attracting invariant manifolds. Bibliography: 21 titles.
Richard C. Martineau; Ray A. Berry; Aurélia Esteve; Kurt D. Hamman; Dana A. Knoll; Ryosuke Park; William Taitano
2009-01-01
This report illustrates a comparative study to analyze the physical differences between numerical simulations obtained with both the conservation and incompressible forms of the Navier-Stokes equations for natural convection flows in simple geometries. The purpose of this study is to quantify how the incompressible flow assumption (which is based upon constant density advection, divergence-free flow, and the Boussinesq gravitational body force approximation) differs from the conservation form (which only assumes that the fluid is a continuum) when solving flows driven by gravity acting upon density variations resulting from local temperature gradients. Driving this study is the common use of the incompressible flow assumption in fluid flow simulations for nuclear power applications in natural convection flows subjected to a high heat flux (large temperature differences). A series of simulations were conducted on two-dimensional, differentially-heated rectangular geometries and modeled with both hydrodynamic formulations. From these simulations, the selected characterization parameters of maximum Nusselt number, average Nusselt number, and normalized pressure reduction were calculated. Comparisons of these parameters were made with available benchmark solutions for air with the ideal gas assumption at both low and high heat fluxes. Additionally, we generated body force, velocity, and divergence of velocity distributions to provide a basis for further analysis. The simulations and analysis were then extended to include helium at the Very High Temperature gas-cooled Reactor (VHTR) normal operating conditions. Our results show that the consequences of incorporating the incompressible flow assumption in high heat flux situations may lead to unrepresentative results. The results question the use of the incompressible flow assumption for simulating fluid flow in an operating nuclear reactor, where large temperature variations are present. The results show that the use of
Richard C. Martineau; Ray A. Berry; Aur´elia Esteve; Kurt D. Hamman; Dana A. Knoll; Ryosuke Park; William Taitano
2010-06-01
This manuscript illustrates a comparative study to analyze the physical differences between numerical simulations obtained with both the conservation and incompressible forms of the Navier-Stokes equations for natural convection flows in simple geometries. The purpose of this study is to quantify how the incompressible flow assumption (which is based upon constant density advection, divergence-free flow, and the Boussinesq gravitational body force approximation) differs from the conservation form (which only assumes that the fluid is a continuum) when solving flows driven by gravity acting upon density variations resulting from local temperature gradients. Driving this study is the common use of the incompressible flow assumption in fluid flow simulations for nuclear power applications in natural convection flows subjected to a high heat flux (large temperature differences). A series of simulations were conducted on two-dimensional, differentially-heated rectangular geometries and modeled with both hydrodynamic formulations. From these simulations, the selected characterization parameters of maximum Nusselt number, average Nusselt number, and normalized pressure reduction were calculated. Comparisons of these parameters were made with available benchmark solutions for air with the ideal gas assumption at both low and high heat fluxes. Additionally, we generated specific force quantities and velocity and temperature distributions to provide a basis for further analysis. The simulations and analysis were then extended to include helium at the Very High Temperature gas-cooled Reactor (VHTR) normal operating conditions. Our results show that the consequences of incorporating the incompressible flow assumption in high heat flux situations may lead to unrepresentative results. The results question the use of the incompressible flow assumption for simulating fluid flow in an operating nuclear reactor, where large temperature variations are present.
About the Regularized Navier Stokes Equations
NASA Astrophysics Data System (ADS)
Cannone, Marco; Karch, Grzegorz
2005-03-01
The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier Stokes system. The Marcinkiewicz space L3,∞ is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical “regularized” Navier Stokes systems. The first one was introduced by J. Leray and consists in “mollifying” the nonlinearity. The second one was proposed by J.-L. Lions, who added the artificial hyper-viscosity (-Δ)ℓ/ 2, ℓ > 2 to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t → ∞ toward solutions of the original Navier Stokes system.
Martin, D.F.; Colella, P.; Graves, D.T.
2007-09-25
We present a method for computing incompressible viscousflows in three dimensions using block-structured local refinement in bothspace and time. This method uses a projection formulation based on acell-centered approximate projection, combined with the systematic use ofmultilevel elliptic solvers to compute increments in the solutiongenerated at boundaries between refinement levels due to refinement intime. We use an L_0-stable second-order semi-implicit scheme to evaluatethe viscous terms. Results are presentedto demonstrate the accuracy andeffectiveness of this approach.
NASA Astrophysics Data System (ADS)
Tsuzuki, Yutaka
2015-09-01
This paper is concerned with a system of heat equations with hysteresis and Navier-Stokes equations. In Tsuzuki (J Math Anal Appl 423:877-897, 2015) an existence result is obtained for the problem in a 2-dimensional domain with the Navier-Stokes equation in a weak sense. However the result does not include uniqueness for the problem due to the low regularity for solutions. This paper establishes existence and uniqueness in 2- and 3-dimensional domains with the Navier-Stokes equation in a stronger sense. Moreover this work decides required height of regularity for the initial data by introducing the fractional power of the Stokes operator.
NASA Technical Reports Server (NTRS)
Sheng, Chunhua; Hyams, Daniel G.; Sreenivas, Kidambi; Gaither, J. Adam; Marcum, David L.; Whitfield, David L.
2000-01-01
A multiblock unstructured grid approach is presented for solving three-dimensional incompressible inviscid and viscous turbulent flows about complete configurations. The artificial compressibility form of the governing equations is solved by a node-based, finite volume implicit scheme which uses a backward Euler time discretization. Point Gauss-Seidel relaxations are used to solve the linear system of equations at each time step. This work employs a multiblock strategy to the solution procedure, which greatly improves the efficiency of the algorithm by significantly reducing the memory requirements by a factor of 5 over the single-grid algorithm while maintaining a similar convergence behavior. The numerical accuracy of solutions is assessed by comparing with the experimental data for a submarine with stem appendages and a high-lift configuration.
Ge, Liang; Sotiropoulos, Fotis
2008-01-01
A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus. PMID:19194533
Ge, Liang; Sotiropoulos, Fotis
2007-08-01
A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus. PMID:19194533
NASA Technical Reports Server (NTRS)
Brown, James L.
2014-01-01
Examined is sensitivity of separation extent, wall pressure and heating to variation of primary input flow parameters, such as Mach and Reynolds numbers and shock strength, for 2D and Axisymmetric Hypersonic Shock Wave Turbulent Boundary Layer interactions obtained by Navier-Stokes methods using the SST turbulence model. Baseline parametric sensitivity response is provided in part by comparison with vetted experiments, and in part through updated correlations based on free interaction theory concepts. A recent database compilation of hypersonic 2D shock-wave/turbulent boundary layer experiments extensively used in a prior related uncertainty analysis provides the foundation for this updated correlation approach, as well as for more conventional validation. The primary CFD method for this work is DPLR, one of NASA's real-gas aerothermodynamic production RANS codes. Comparisons are also made with CFL3D, one of NASA's mature perfect-gas RANS codes. Deficiencies in predicted separation response of RANS/SST solutions to parametric variations of test conditions are summarized, along with recommendations as to future turbulence approach.
Gao, Hang; Bijnens, Nathalie; Coisne, Damien; Lugiez, Mathieu; Rutten, Marcel; D'hooge, Jan
2015-01-01
Despite the availability of multiple ultrasound approaches to left ventricular (LV) flow characterization in two dimensions, this technique remains in its childhood and further developments seem warranted. This article describes a new methodology for tracking the 2-D LV flow field based on ultrasound data. Hereto, a standard speckle tracking algorithm was modified by using a dynamic kernel embedding Navier-Stokes-based regularization in an iterative manner. The performance of the proposed approach was first quantified in synthetic ultrasound data based on a computational fluid dynamics model of LV flow. Next, an experimental flow phantom setup mimicking the normal human heart was used for experimental validation by employing simultaneous optical particle image velocimetry as a standard reference technique. Finally, the applicability of the approach was tested in a clinical setting. On the basis of the simulated data, pointwise evaluation of the estimated velocity vectors correlated well (mean r = 0.84) with the computational fluid dynamics measurement. During the filling period of the left ventricle, the properties of the main vortex obtained from the proposed method were also measured, and their correlations with the reference measurement were also calculated (radius, r = 0.96; circulation, r = 0.85; weighted center, r = 0.81). In vitro results at 60 bpm during one cardiac cycle confirmed that the algorithm properly measures typical characteristics of the vortex (radius, r = 0.60; circulation, r = 0.81; weighted center, r = 0.92). Preliminary qualitative results on clinical data revealed physiologic flow fields. PMID:25438850
NASA Technical Reports Server (NTRS)
Kwak, D.
1994-01-01
INS3D computes steady-state solutions to the incompressible Navier-Stokes equations. The INS3D approach utilizes pseudo-compressibility combined with an approximate factorization scheme. This computational fluid dynamics (CFD) code has been verified on problems such as flow through a channel, flow over a backwardfacing step and flow over a circular cylinder. Three dimensional cases include flow over an ogive cylinder, flow through a rectangular duct, wind tunnel inlet flow, cylinder-wall juncture flow and flow through multiple posts mounted between two plates. INS3D uses a pseudo-compressibility approach in which a time derivative of pressure is added to the continuity equation, which together with the momentum equations form a set of four equations with pressure and velocity as the dependent variables. The equations' coordinates are transformed for general three dimensional applications. The equations are advanced in time by the implicit, non-iterative, approximately-factored, finite-difference scheme of Beam and Warming. The numerical stability of the scheme depends on the use of higher-order smoothing terms to damp out higher-frequency oscillations caused by second-order central differencing. The artificial compressibility introduces pressure (sound) waves of finite speed (whereas the speed of sound would be infinite in an incompressible fluid). As the solution converges, these pressure waves die out, causing the derivation of pressure with respect to time to approach zero. Thus, continuity is satisfied for the incompressible fluid in the steady state. Computational efficiency is achieved using a diagonal algorithm. A block tri-diagonal option is also available. When a steady-state solution is reached, the modified continuity equation will satisfy the divergence-free velocity field condition. INS3D is capable of handling several different types of boundaries encountered in numerical simulations, including solid-surface, inflow and outflow, and far
NASA Astrophysics Data System (ADS)
Bruno, Oscar P.; Cubillos, Max
2016-02-01
This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples (h , Δt)of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form (0 ,Mh) × (0 ,Mt). In other words, for each fixed value of Δt below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second-order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second-order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.
Development of advanced Navier-Stokes solver
NASA Technical Reports Server (NTRS)
Yoon, Seokkwan
1994-01-01
The objective of research was to develop and validate new computational algorithms for solving the steady and unsteady Euler and Navier-Stokes equations. The end-products are new three-dimensional Euler and Navier-Stokes codes that are faster, more reliable, more accurate, and easier to use. The three-dimensional Euler and full/thin-layer Reynolds-averaged Navier-Stokes equations for compressible/incompressible flows are solved on structured hexahedral grids. The Baldwin-Lomax algebraic turbulence model is used for closure. The space discretization is based on a cell-centered finite-volume method augmented by a variety of numerical dissipation models with optional total variation diminishing limiters. The governing equations are integrated in time by an implicit method based on lower-upper factorization and symmetric Gauss-Seidel relaxation. The algorithm is vectorized on diagonal planes of sweep using two-dimensional indices in three dimensions. Convergence rates and the robustness of the codes are enhanced by the use of an implicit full approximation storage multigrid method.
Verification of the proteus two-dimensional Navier-Stokes code for flat plate and pipe flows
NASA Technical Reports Server (NTRS)
Conley, Julianne M.; Zeman, Patrick L.
1991-01-01
The Proteus Navier-Stokes Code is evaluated for 2-D/axisymmetric, viscous, incompressible, internal, and external flows. The particular cases to be discussed are laminar and turbulent flows over a flat plate, laminar and turbulent developing pipe flows, and turbulent pipe flow with swirl. Results are compared with exact solutions, empirical correlations, and experimental data. A detailed description of the code set-up, including boundary conditions, initial conditions, grid size, and grid packing is given for each case.
NASA Astrophysics Data System (ADS)
Guo, Z.; Lin, P.; Lowengrub, J. S.
2014-11-01
In this paper, we investigate numerically a diffuse interface model for the Navier-Stokes equation with fluid-fluid interface when the fluids have different densities [48]. Under minor reformulation of the system, we show that there is a continuous energy law underlying the system, assuming that all variables have reasonable regularities. It is shown in the literature that an energy law preserving method will perform better for multiphase problems. Thus for the reformulated system, we design a C0 finite element method and a special temporal scheme where the energy law is preserved at the discrete level. Such a discrete energy law (almost the same as the continuous energy law) for this variable density two-phase flow model has never been established before with C0 finite element. A Newton method is introduced to linearise the highly non-linear system of our discretization scheme. Some numerical experiments are carried out using the adaptive mesh to investigate the scenario of coalescing and rising drops with differing density ratio. The snapshots for the evolution of the interface together with the adaptive mesh at different times are presented to show that the evolution, including the break-up/pinch-off of the drop, can be handled smoothly by our numerical scheme. The discrete energy functional for the system is examined to show that the energy law at the discrete level is preserved by our scheme.
Exact solutions of the generalized Navier- Stokes equations for benchmarking
NASA Astrophysics Data System (ADS)
Bourchtein, Andrei
2002-08-01
The generalized Navier- Stokes equations for incompressible viscous flows through isotropic granular porous medium are studied. Some analytical classic solutions of the Navier- Stokes equations are generalized to the case of the considered equations. Obtained solutions of generalized equations reduce to classic ones as porosity effect disappears. Average velocity of generalized solutions is calculated and evaluated in two limiting regimes of flow. In the shallow conduit, the generalized flow rate approximates the free (without porous medium) flow rate and in the case of removed boundaries this approaches Darcy's law. The use of the derived exact solutions for benchmarking purposes is described. Copyright
On multigrid methods for the Navier-Stokes Computer
NASA Technical Reports Server (NTRS)
Nosenchuck, D. M.; Krist, S. E.; Zang, T. A.
1988-01-01
The overall architecture of the multipurpose parallel-processing Navier-Stokes Computer (NSC) being developed by Princeton and NASA Langley (Nosenchuck et al., 1986) is described and illustrated with extensive diagrams, and the NSC implementation of an elementary multigrid algorithm for simulating isotropic turbulence (based on solution of the incompressible time-dependent Navier-Stokes equations with constant viscosity) is characterized in detail. The present NSC design concept calls for 64 nodes, each with the performance of a class VI supercomputer, linked together by a fiber-optic hypercube network and joined to a front-end computer by a global bus. In this configuration, the NSC would have a storage capacity of over 32 Gword and a peak speed of over 40 Gflops. The multigrid Navier-Stokes code discussed would give sustained operation rates of about 25 Gflops.
Lattice-gas automata for the Navier-Stokes equation
NASA Astrophysics Data System (ADS)
Frisch, U.; Hasslacher, B.; Pomeau, Y.
1986-04-01
It is shown that a class of deterministic lattice gases with discrete Boolean elements simulates the Navier-Stokes equations, and can be used to design simple, massively parallel computing machines. A hexagonal lattice gas (HLG) model consisting of a triangular lattice with hexagonal symmetry is developed, and is shown to lead to the two-dimensional Navier-Stokes equations. The three-dimensional formulation is obtained by a splitting method in which the nonlinear term in the three-dimensional Navier-Stokes equation is recasts as the sum of two terms, each containing spurious elements and each realizable on a different lattice. Freed slip and rigid boundary conditions are easily implemented. It is noted that lattice-gas models must be run at moderate Mach numbers to remain incompressible, and to avoid spurious high-order nonlinear terms. The model gives a concrete hydrodynamical example of how cellular automata can be used to simulate classical nonlinear fields.
Global small solutions of 2-D incompressible MHD system
NASA Astrophysics Data System (ADS)
Lin, Fanghua; Xu, Li; Zhang, Ping
2015-11-01
In this paper, we consider the global wellposedness of 2-D incompressible magneto-hydrodynamical system with smooth initial data which is close to some non-trivial steady state. It is a coupled system between the Navier-Stokes equations and a free transport equation with a universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation, we first reformulate our system (1.1) in the Lagrangian coordinates (2.19). Then we employ anisotropic Littlewood-Paley analysis to establish the key a prioriL1 (R+ ; Lip (R2)) estimate for the Lagrangian velocity field Yt. With this estimate, we can prove the global wellposedness of (2.19) with smooth and small initial data by using the energy method. We emphasize that the algebraic structure of (2.19) is crucial for the proofs to work. The global wellposedness of the original system (1.1) then follows by a suitable change of variables.
A dual potential formulation of the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Gegg, S. G.; Pletcher, R. H.; Steger, J. L.
1989-01-01
A dual potential formulation for numerically solving the Navier-Stokes equations is developed and presented. The velocity field is decomposed using a scalar and vector potential. Vorticity and dilatation are used as the dependent variables in the momentum equations. Test cases in two dimensions verify the capability to solve flows using approximations from potential flow to full Navier-Stokes simulations. A three-dimensional incompressible flow formulation is also described. An interesting feature of this approach to solving the Navier-Stokes equations is the decomposition of the velocity field into a rotational part (vector potential) and an irrotational part (scalar potential). The Helmholtz decomposition theorem allows this splitting of the velocity field. This approach has had only limited use since it increases the number of dependent variables in the solution. However, it has often been used for incompressible flows where the solution scheme is known to be fast and accurate. This research extends the usage of this method to fully compressible Navier-Stokes simulations by using the dilatation variable along with vorticity. A time-accurate, iterative algorithm is used for the uncoupled solution of the governing equations. Several levels of flow approximation are available within the framework of this method. Potential flow, Euler and full Navier-Stokes solutions are possible using the dual potential formulation. Solution efficiency can be enhanced in a straightforward way. For some flows, the vorticity and/or dilatation may be negligible in certain regions (e.g., far from a viscous boundary in an external flow). It is possible to drop the calculation of these variables then and optimize the solution speed. Also, efficient Poisson solvers are available for the potentials. The relative merits of non-primitive variables versus primitive variables for solution of the Navier-Stokes equations are also discussed.
A locally stabilized immersed boundary method for the compressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Brehm, C.; Hader, C.; Fasel, H. F.
2015-08-01
A higher-order immersed boundary method for solving the compressible Navier-Stokes equations is presented. The distinguishing feature of this new immersed boundary method is that the coefficients of the irregular finite-difference stencils in the vicinity of the immersed boundary are optimized to obtain improved numerical stability. This basic idea was introduced in a previous publication by the authors for the advection step in the projection method used to solve the incompressible Navier-Stokes equations. This paper extends the original approach to the compressible Navier-Stokes equations considering flux vector splitting schemes and viscous wall boundary conditions at the immersed geometry. In addition to the stencil optimization procedure for the convective terms, this paper discusses other key aspects of the method, such as imposing flux boundary conditions at the immersed boundary and the discretization of the viscous flux in the vicinity of the boundary. Extensive linear stability investigations of the immersed scheme confirm that a linearly stable method is obtained. The method of manufactured solutions is used to validate the expected higher-order accuracy and to study the error convergence properties of this new method. Steady and unsteady, 2D and 3D canonical test cases are used for validation of the immersed boundary approach. Finally, the method is employed to simulate the laminar to turbulent transition process of a hypersonic Mach 6 boundary layer flow over a porous wall and subsonic boundary layer flow over a three-dimensional spherical roughness element.
From Petrov-Einstein to Navier-Stokes
NASA Astrophysics Data System (ADS)
Lysov, Vyacheslav
The fluid/gravity correspondence relates solutions of the incompressible Navier-Stokes equation to metrics which solve the Einstein equations. We propose propose two possible approaches to establish this correspondence: perturbative expansion for shear modes and large mean curvature expansion for algebraically special metrics. We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in p+1 dimensions, there is an associated "dual" solution of the vacuum Einstein equations in p+2 dimensions. The dual geometry has an intrinsically flat time-like boundary segment whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a "near-horizon" limit in which hypersurface becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. It is shown that imposing a Petrov type I condition on the hypersurface geometry reduces the degrees of freedom in the extrinsic curvature to those of a fluid. Moreover, expanding around a limit in which the mean curvature of the embedding diverges, the leading-order Einstein constraint equations on hypersurface are shown to reduce to the non-linear incompressible Navier-Stokes equation for a fluid moving in hypersurface. We extend the fluid/gravity correspondence to include the magnetohydrodynamics/gravity correspondence, which translates solutions of the equations of magnetohydrodynamics (describing charged fluids) into geometries that satisfy the Einstein-Maxwell equations. We present an explicit example of this new correspondence in the context of flat Minkowski space. We show that a perturbative deformation of the Rindler wedge satisfies the Einstein-Maxwell equations provided that the parameters appearing in the expansion, which we interpret as fluid fields, satisfy the
NASA Astrophysics Data System (ADS)
Cox, Christopher; Liang, Chunlei; Plesniak, Michael W.
2016-06-01
We report development of a high-order compact flux reconstruction method for solving unsteady incompressible flow on unstructured grids with implicit dual time stepping. The method falls under the class of methods now referred to as flux reconstruction/correction procedure via reconstruction. The governing equations employ Chorin's classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems. An implicit non-linear lower-upper symmetric Gauss-Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation within the context of the high-order flux reconstruction method. This compact high-order method is very suitable for parallel computing and can easily be extended to moving and deforming grids.
NASA Astrophysics Data System (ADS)
Xie, Bin; , Satoshi, Ii; Ikebata, Akio; Xiao, Feng
2014-11-01
A robust and accurate finite volume method (FVM) is proposed for incompressible viscous fluid dynamics on triangular and tetrahedral unstructured grids. Differently from conventional FVM where the volume integrated average (VIA) value is the only computational variable, the present formulation treats both VIA and the point value (PV) as the computational variables which are updated separately at each time step. The VIA is computed from a finite volume scheme of flux form, and is thus numerically conservative. The PV is updated from the differential form of the governing equation that does not have to be conservative but can be solved in a very efficient way. Including PV as the additional variable enables us to make higher-order reconstructions over compact mesh stencil to improve the accuracy, and moreover, the resulting numerical model is more robust for unstructured grids. We present the numerical formulations in both two and three dimensions on triangular and tetrahedral mesh elements. Numerical results of several benchmark tests are also presented to verify the proposed numerical method as an accurate and robust solver for incompressible flows on unstructured grids.
Optimal control of thermally coupled Navier Stokes equations
NASA Technical Reports Server (NTRS)
Ito, Kazufumi; Scroggs, Jeffrey S.; Tran, Hien T.
1994-01-01
The optimal boundary temperature control of the stationary thermally coupled incompressible Navier-Stokes equation is considered. Well-posedness and existence of the optimal control and a necessary optimality condition are obtained. Optimization algorithms based on the augmented Lagrangian method with second order update are discussed. A test example motivated by control of transport process in the high pressure vapor transport (HVPT) reactor is presented to demonstrate the applicability of our theoretical results and proposed algorithm.
Finite element methods and Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Cuvelier, C.; Segal, A.; van Steenhoven, A. A.
This book is devoted to two and three-dimensional FEM analysis of the Navier-Stokes (NS) equations describing one flow of a viscous incompressible fluid. Three different approaches to the NS equations are described: a direct method, a penalty method, and a method that constructs discrete solenoidal vector fields. Subjects of current research which are important from the industrial/technological viewpoint are considered, including capillary-free boundaries, nonisothermal flows, turbulence, and non-Newtonian fluids.
Navier Stokes Theorem in Hydrology
NASA Astrophysics Data System (ADS)
Narayanan, M.
2005-12-01
In a paper presented at the 2004 AGU International Conference, the author outlined and stressed the importance of studying and teaching certain important mathematical techniques while developing a course in Hydrology and Fluid Mechanics. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. In an article entitled "Corrections to Fluid Dynamics" R. F. Streater, (Open Systems and Information Dynamics, 10, 3-30, 2003.) proposes a kinetic model of a fluid in which five macroscopic fields, the mass, energy, and three components of momentum, are conserved. The dynamics is constructed using the methods of statistical dynamics, and results in a non-linear discrete-time Markov chain for random fields on a lattice. In the continuum limit he obtains a non-linear coupled parabolic system of field equations, showing a correction to the Navier-Stokes equations. In 2001, David Hoff published an article in Journees Equations aux derivees partielles. (Art. No. 7, 9 p.). His paper is entitled : Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions. In his paper, David Hoff proves the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions with piecewise smooth initial data. These solutions remain piecewise smooth for all time, retaining simple jump discontinuities in the density and in the divergence of the velocity across a smooth curve, which is convected with the flow. The strengths of these discontinuities are shown to decay exponentially in time
NASA Astrophysics Data System (ADS)
Cox, Christopher; Liang, Chunlei; Plesniak, Michael
2015-11-01
This paper reports development of a high-order compact method for solving unsteady incompressible flow on unstructured grids with implicit time stepping. The method falls under the class of methods now referred to as flux reconstruction/correction procedure via reconstruction. The governing equations employ the classical artificial compressibility treatment, where dual time stepping is needed to solve unsteady flow problems. An implicit non-linear lower-upper symmetric Gauss-Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time-stepping scheme. Three-dimensional results computed on many processing elements will be presented. The high-order method is very suitable for parallel computing and can easily be extended to moving and deforming grids. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation within the context of the high-order flux reconstruction method. Financial support provided under the GW Presidential Merit Fellowship.
An investigation of DTNS2D for use as an incompressible turbulence modelling test-bed
NASA Technical Reports Server (NTRS)
Steffen, Christopher J., Jr.
1992-01-01
This paper documents an investigation of a two dimensional, incompressible Navier-Stokes solver for use as a test-bed for turbulence modelling. DTNS2D is the code under consideration for use at the Center for Modelling of Turbulence and Transition (CMOTT). This code was created by Gorski at the David Taylor Research Center and incorporates the pseudo compressibility method. Two laminar benchmark flows are used to measure the performance and implementation of the method. The classical solution of the Blasius boundary layer is used for validating the flat plate flow, while experimental data is incorporated in the validation of backward facing step flow. Velocity profiles, convergence histories, and reattachment lengths are used to quantify these calculations. The organization and adaptability of the code are also examined in light of the role as a numerical test-bed.
Use of Navier-Stokes analysis in section design
NASA Astrophysics Data System (ADS)
Nguyen, Phuc N.
1990-12-01
The Navier-Stokes analysis method and a design technique based on conformal mapping are combined to develop new 2-D thick sections. The Eppler-Somers design technique allows for fast design of arbitrary section shape. The well-validated David Taylor Navier-Stokes code is used to optimize the thickness of the section. From previous experimental results, the turbulence characteristics in the near-wake region correlate with the pressure spectra on the trailing edge of a 2-D lifting surface. Therefore, the turbulent kinetic energy, and the Reynolds shear stress are used as design parameters to develop new 2-D sections. Minimizing these parameters is assumed to provide desirable boundary layer and wake characteristics. The characteristics of one new section are compared with those of a baseline section to demonstrate the new foil design method.
Transonic airfoil and wing design using Navier-Stokes codes
NASA Technical Reports Server (NTRS)
Yu, N. J.; Campbell, R. L.
1992-01-01
An iterative design method has been implemented into 2D and 3D Navier-Stokes codes for the design of airfoils or wings with given target pressure distributions. The method begins with the analysis of an initial geometry, and obtains the analysis pressure distributions of that geometry. The differences between analysis pressures and target pressures are used to drive geometry changes through the use of a streamline curvature method. This paper describes the procedure that makes the iterative design method work for Navier-Stokes codes. Examples of 2D airfoil design, and 3D wing design are included. It is demonstrated that the method is highly effective for airfoil or wing design at flow conditions where no substantial separation occurs. Problems encountered in the airfoil design with shock induced flow separations are discussed.
Navier-Stokes computations of separated vortical flows past prolate spheroid at incidence
NASA Technical Reports Server (NTRS)
Wong, Tin-Chee; Kandil, Osama A.; Liu, C. H.
1989-01-01
The problem of steady incompressible viscous flow past prolate spheroids at incidence is formulated using the unsteady incompressible and compressible thin-layer Navier-Stokes equations. The two sets of Navier-Stokes equations are solved using a pseudotime stepping of the implicit flux-difference splitting scheme on a curvilinear grid, which is generated by a transfinite grid generator. The Baldwin and Lomax (1978) algebraic eddy-viscosity model is used to model the turbulent flow. The computational applications cover a 6:1 prolate spheroid at different angles of attack and Reynolds numbers. The results are compared with experimental data.
Turbulent solution of the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Deissler, R. G.
1980-01-01
The unaveraged Navier-Stokes equations are solved numerically in order to study the nonlinear physics of incompressible turbulent flow. Initial three dimensional cosine velocity fluctuations and periodic boundary conditions are used. No mean gradients are present. The three components of the mean square velocity fluctuations are equal for the initial conditions chosen. The resulting solution shows characteristics of turbulence, such as the nonlinear excitation of small scale fluctuations. For the higher Reynolds numbers the initially nonrandom flow develops into an apparently random turbulence.
Attractors of three-dimensional fast-rotating Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Trahe, Markus
The three-dimensional (3-D) rotating Navier-Stokes equations describe the dynamics of rotating, incompressible, viscous fluids. In this work, they are considered with smooth, time-independent forces and the original statements implied by the classical "Taylor-Proudman Theorem" of geophysics are rigorously proved. It is shown that fully developed turbulence of 3-D fast-rotating fluids is essentially characterized by turbulence of two-dimensional (2-D) fluids in terms of numbers of degrees of freedom. In this context, the 3-D nonlinear "resonant limit equations", which arise in a non-linear averaging process as the rotation frequency O → infinity, are studied and optimal (2-D-type) upper bounds for fractal box and Hausdorff dimensions of the global attractor as well as upper bounds for box dimensions of exponential attractors are determined. Then, the convergence of exponential attractors for the full 3-D rotating Navier-Stokes equations to exponential attractors for the resonant limit equations as O → infinity in the sense of full Hausdorff-metric distances is established. This provides upper and lower semi-continuity of exponential attractors with respect to the rotation frequency and implies that the number of degrees of freedom (attractor dimension) of 3-D fast-rotating fluids is close to that of 2-D fluids. Finally, the algebraic-geometric structure of the Poincare curves, which control the resonances and small divisor estimates for partial differential equations, is further investigated; the 3-D nonlinear limit resonant operators are characterized by three-wave interactions governed by these curves. A new canonical transformation between those curves is constructed; with far-reaching consequences on the density of the latter.
On the Dynamic Programming Approach for the 3D Navier-Stokes Equations
Manca, Luigi
2008-06-15
The dynamic programming approach for the control of a 3D flow governed by the stochastic Navier-Stokes equations for incompressible fluid in a bounded domain is studied. By a compactness argument, existence of solutions for the associated Hamilton-Jacobi-Bellman equation is proved. Finally, existence of an optimal control through the feedback formula and of an optimal state is discussed.
Time-accurate Navier-Stokes calculations with multigrid acceleration
NASA Technical Reports Server (NTRS)
Melson, N. D.; Sanetrik, Mark D.; Atkins, Harold L.
1993-01-01
An efficient method for calculating unsteady flows is presented, with emphasis on a modified version of the thin-layer Navier-Stokes equations. Fourier stability analysis is used to illustrate the effect of treating the source term implicitly instead of explicity, as well as to illustrate other algorithmic choices. A 2D circular cylinder (with a Reynolds number of 1200 and a Mach number of 0.3) is calculated. The present scheme requires only about 10 percent of the computer time required by global minimum time stepping.
Airfoil design method using the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Malone, J. B.; Narramore, J. C.; Sankar, L. N.
1991-01-01
An airfoil design procedure is described that was incorporated into an existing 2-D Navier-Stokes airfoil analysis method. The resulting design method, an iterative procedure based on a residual-correction algorithm, permits the automated design of airfoil sections with prescribed surface pressure distributions. The inverse design method and the technique used to specify target pressure distributions are described. It presents several example problems to demonstrate application of the design procedure. It shows that this inverse design method develops useful airfoil configurations with a reasonable expenditure of computer resources.
Preconditioned conjugate gradient methods for the Navier-Stokes equations
Ajmani, K.; Ng, Wing Fai ); Liou, Meng Sing )
1994-01-01
A preconditioned Krylov subspace method (GMRES) is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux-split formulations. Several preconditioning techniques are investigated to enhance the efficiency and convergence rate of the implicit solver based on the GMRES algorithm. The superiority of the new solver is established by comparisons with a (LGSR). Computational test results for low-speed (incompressible flow over a backward-facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade), and hypersonic flow (shock-on-shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the Mach 0.1 case, overall speedup factors of up to 17 (in terms of time-steps) and 15 (in terms of CPU times on a CRAY-YMP/8) are found in favor of the preconditioned GMRES solver, when compared with the LGSR solver. The corresponding speedup factors for the transonic flow cases are 17 and 23, respectively. The hypersonic flow case shows slightly lower speedup factors of 9 and 13, respectively. The study of preconditioners conducted in this research reveals that a new LUSGS-type preconditioner is much more efficient than a conventional incomplete LU-type preconditioner. 34 refs., 15 figs.
Preconditioned conjugate gradient methods for the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing
1994-01-01
A preconditioned Krylov subspace method (GMRES) is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux-split formulation. Several preconditioning techniques are investigated to enhance the efficiency and convergence rate of the implicit solver based on the GMRES algorithm. The superiority of the new solver is established by comparisons with a conventional implicit solver, namely line Gauss-Seidel relaxation (LGSR). Computational test results for low-speed (incompressible flow over a backward-facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade), and hypersonic flow (shock-on-shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the Mach 0.1 case, overall speedup factors of up to 17 (in terms of time-steps) and 15 (in terms of CPU time on a CRAY-YMP/8) are found in favor of the preconditioned GMRES solver, when compared with the LGSR solver. The corresponding speedup factors for the transonic flow case are 17 and 23, respectively. The hypersonic flow case shows slightly lower speedup factors of 9 and 13, respectively. The study of preconditioners conducted in this research reveals that a new LUSGS-type preconditioner is much more efficient than a conventional incomplete LU-type preconditioner.
Algorithms for the Euler and Navier-Stokes equations for supercomputers
NASA Technical Reports Server (NTRS)
Turkel, E.
1985-01-01
The steady state Euler and Navier-Stokes equations are considered for both compressible and incompressible flow. Methods are found for accelerating the convergence to a steady state. This acceleration is based on preconditioning the system so that it is no longer time consistent. In order that the acceleration technique be scheme-independent, this preconditioning is done at the differential equation level. Applications are presented for very slow flows and also for the incompressible equations.
Automatic differentiation and Navier-Stokes.
Bischof, C.; Hovland, P.; Mohammadi, B.
1997-12-17
We describe the use of automatic differentiation (AD) to enhance a compressible Navier-Stokes model. With the solver, AD is used to accelerate convergence by more than an order of magnitude. Outside the solver, AD is used to compute the derivatives needed for optimization. We emphasize the potential for performance gains if the programmer does not treat AD as a black box, but instead utilizes high-level knowledge about the nature of the application.
Towards Optimal Multigrid Efficiency for the Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Swanson, R. C.
2001-01-01
A fast multigrid solver for the steady incompressible Navier-Stokes equations is presented. Unlike time-marching schemes, this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Numerical solutions are shown for flow over a flat plate and a Karman-Trefftz airfoil. Using collective Gauss-Seidel line relaxation in both the vertical and horizontal directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of a Runge-Kutta based multigrid method.
Numerical solutions of the complete Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Hassan, H. A.
1988-01-01
The physical phenomena within supersonic flows that sustain chemical reactions are investigated. An earlier study to develop accurate physical models for supersonic reacting flowfields focused on 2-D laminar shear layers. The objective is to examine the mixing and subsequent combustion within turbulent reacting shear layers. To conduct this study, a computer program has been written to solve the axisymmetric Reynolds averaged Navier-Stokes equations. The numerical method uses a cell-centered finite volume approach and a Runge Kutta time stepping scheme. The Reynolds averaged equations are closed using the eddy viscosity concept. Several zero-equation models have been tested by making calculations for an H2-air nonreacting coaxial jet flow. Comparisons made with experimental data show that Cohen's eddy viscosity model provides best agreement. The finite rate chemistry model used in the study of 2-D laminar shear layers is incorporated into the computer program and data is compared from a recent experiment performed at NASA Langley.
The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Leslie, T. M.; Shvydkoy, R.
2016-09-01
We consider the incompressible inhomogeneous Navier-Stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero. We show that the energy balance relation for this system holds for weak solutions if the velocity, density, and pressure belong to a range of Besov spaces of smoothness 1/3. A density-dependent version of the classical Kármán-Howarth-Monin relation is derived.
Coupling Boltzmann and Navier-Stokes equations by friction
Bourgat, J.F.; Le Tallec, P. |; Tidriri, M.D.
1996-09-01
The aim of this paper is to introduce and validate a coupled Navier-Stokes Boltzman approach for the calculation of hypersonic rarefied flows around maneuvering vehicles. The proposed strategy uses locally a kinetic model in the boundary layer coupled through wall friction forces to a global Navier-Stokes solver. Different numerical experiments illustrate the potentialities of the method. 29 refs., 24 figs.
Multigrid solution of the Navier-Stokes equations on highly stretched grids with defect correction
NASA Technical Reports Server (NTRS)
Sockol, Peter M.
1993-01-01
Relaxation-based multigrid solvers for the steady incompressible Navier-Stokes equations are examined to determine their computational speed and robustness. Four relaxation methods with a common discretization have been used as smoothers in a single tailored multigrid procedure. The equations are discretized on a staggered grid with first order upwind used for convection in the relaxation process on all grids and defect correction to second order central on the fine grid introduced once per multigrid cycle. A fixed W(1,1) cycle with full weighting of residuals is used in the FAS multigrid process. The resulting solvers have been applied to three 2D flow problems, over a range of Reynolds numbers, on both uniform and highly stretched grids. In all cases the L(sub 2) norm of the velocity changes is reduced to 10(exp -6) in a few 10's of fine grid sweeps. The results from this study are used to draw conclusions on the strengths and weaknesses of the individual relaxation schemes as well as those of the overall multigrid procedure when used as a solver on highly stretched grids.
Unified semi-analytical wall boundary conditions applied to 2-D incompressible SPH
NASA Astrophysics Data System (ADS)
Leroy, A.; Violeau, D.; Ferrand, M.; Kassiotis, C.
2014-03-01
This work aims at improving the 2-D incompressible SPH model (ISPH) by adapting it to the unified semi-analytical wall boundary conditions proposed by Ferrand et al. [10]. The ISPH algorithm considered is as proposed by Lind et al. [25], based on the projection method with a divergence-free velocity field and using a stabilising procedure based on particle shifting. However, we consider an extension of this model to Reynolds-Averaged Navier-Stokes equations based on the k-ɛ turbulent closure model, as done in [10]. The discrete SPH operators are modified by the new description of the wall boundary conditions. In particular, a boundary term appears in the Laplacian operator, which makes it possible to accurately impose a von Neumann pressure wall boundary condition that corresponds to impermeability. The shifting and free-surface detection algorithms have also been adapted to the new boundary conditions. Moreover, a new way to compute the wall renormalisation factor in the frame of the unified semi-analytical boundary conditions is proposed in order to decrease the computational time. We present several verifications to the present approach, including a lid-driven cavity, a water column collapsing on a wedge and a periodic schematic fish-pass. Our results are compared to Finite Volumes methods, using Volume of Fluids in the case of free-surface flows. We briefly investigate the convergence of the method and prove its ability to model complex free-surface and turbulent flows. The results are generally improved when compared to a weakly compressible SPH model with the same boundary conditions, especially in terms of pressure prediction.
The Proteus Navier-Stokes code
NASA Technical Reports Server (NTRS)
Towne, Charles E.; Bui, Trong T.; Cavicchi, Richard H.; Conley, Julianne M.; Molls, Frank B.; Schwab, John R.
1992-01-01
An effort is currently underway at NASA Lewis to develop two- and three-dimensional Navier-Stokes codes, called Proteus, for aerospace propulsion applications. The emphasis in the development of Proteus is not algorithm development or research on numerical methods, but rather the development of the code itself. The objective is to develop codes that are user-oriented, easily-modified, and well-documented. Well-proven, state-of-the-art solution algorithms are being used. Code readability, documentation (both internal and external), and validation are being emphasized. This paper is a status report on the Proteus development effort. The analysis and solution procedure are described briefly, and the various features in the code are summarized. The results from some of the validation cases that have been run are presented for both the two- and three-dimensional codes.
Simulations of transition and turbulence on the Navier-Stokes computer
NASA Technical Reports Server (NTRS)
Krist, S. E.; Zang, T. A.
1987-01-01
The Navier-Stokes Computer (NSC) consists of multiple local memory parallel processors interconnected in a hypercube network. Efficient implementation of algorithms on the NSC thus requires the effective utilization of both the coarse and fine grain paralelism inherent in the architectural design. The basic approach to implementing an algorithm on the NSC is presented herein. The particular finite-difference algorithm considered was developed for performing transition and turbulence simulations by direct solution of the time-dependent incompressible Navier-Stokes equations. The suitability of this algorithm for performing simulations of the isotropic turbulence problem is verified from computations performed on a Cray 2. Projected timing results for the algorithm on the NSC itself are presented for both the isotropic turbulence and laminar turbulent transition problems.
Solution of the Navier-Stokes equations for a driven cavity
NASA Astrophysics Data System (ADS)
Semeraro, B. D.; Sameh, Ahmed
1991-03-01
The flow field in a lid driven cavity is determined by integration of the incompressible Navier-Stokes equations. The numerical integration is accomplished via an operator splitting method known as the theta-scheme. This splitting separates the problem into the solution of a quasi-stokes problem and a nonlinear convection problem. Some details of solution methods used for the two subproblems and results obtained for the driven cavity are described. The schemes developed for the quasi-Stokes problem are more advanced at this stage than those for the nonlinear problem. However, the approaches used for both parts are outlined. As a model problem, a two dimensional square cavity with sides of unit length and a lid moving with unit velocity from left to right is considered. The Navier-Stokes equations are discretized in space on a uniform staggered or MAC mesh. The time discretization is accomplished via the theta-scheme.
Solution of the Navier-Stokes equations for a driven cavity
NASA Technical Reports Server (NTRS)
Semeraro, B. D.; Sameh, Ahmed
1991-01-01
The flow field in a lid driven cavity is determined by integration of the incompressible Navier-Stokes equations. The numerical integration is accomplished via an operator splitting method known as the theta-scheme. This splitting separates the problem into the solution of a quasi-stokes problem and a nonlinear convection problem. Some details of solution methods used for the two subproblems and results obtained for the driven cavity are described. The schemes developed for the quasi-Stokes problem are more advanced at this stage than those for the nonlinear problem. However, the approaches used for both parts are outlined. As a model problem, a two dimensional square cavity with sides of unit length and a lid moving with unit velocity from left to right is considered. The Navier-Stokes equations are discretized in space on a uniform staggered or MAC mesh. The time discretization is accomplished via the theta-scheme.
Finite element modified method of characteristics for the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Allievi, Alejandro; Bermejo, Rodolfo
2000-02-01
An algorithm based on the finite element modified method of characteristics (FEMMC) is presented to solve convection-diffusion, Burgers and unsteady incompressible Navier-Stokes equations for laminar flow. Solutions for these progressively more involved problems are presented so as to give numerical evidence for the robustness, good error characteristics and accuracy of our method. To solve the Navier-Stokes equations, an approach that can be conceived as a fractional step method is used. The innovative first stage of our method is a backward search and interpolation at the foot of the characteristics, which we identify as the convective step. In this particular work, this step is followed by a conjugate gradient solution of the remaining Stokes problem. Numerical results are presented for:aConvection-diffusion equation. Gaussian hill in a uniform rotating field.bBurgers equations with viscosity.
Stabilization and scalable block preconditioning for the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Cyr, Eric C.; Shadid, John N.; Tuminaro, Raymond S.
2012-01-01
This study compares several block-oriented preconditioners for the stabilized finite element discretization of the incompressible Navier-Stokes equations. This includes standard additive Schwarz domain decomposition methods, aggressive coarsening multigrid, and three preconditioners based on an approximate block LU factorization, specifically SIMPLEC, LSC, and PCD. Robustness is considered with a particular focus on the impact that different stabilization methods have on preconditioner performance. Additionally, parallel scaling studies are undertaken. The numerical results indicate that aggressive coarsening multigrid, LSC and PCD all have good algorithmic scalability. Coupling this with the fact that block methods can be applied to systems arising from stable mixed discretizations implies that these techniques are a promising direction for developing scalable methods for Navier-Stokes.
Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Nielsen, Eric J.; Anderson, W. Kyle
1998-01-01
A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences. Several simplifying approximations to the complete linearization of the residual are also presented, and the resulting accuracy of the derivatives is examined. Demonstration optimizations for both compressible and incompressible flows are given.
Chaos Synchronization in Navier-Stokes Turbulence
NASA Astrophysics Data System (ADS)
Lalescu, Cristian; Meneveau, Charles; Eyink, Gregory
2013-03-01
Chaos synchronization (CS) has been studied for some time now (Pecora & Carroll 1990), for systems with only a few degrees of freedom as well as for systems described by partial differential equations (Boccaletti et al 2002). CS in general is said to be present in coupled dynamical systems when a specific property of each system has the same time evolution for all, even though the evolution itself is chaotic. The Navier-Stokes (NS) equations describe the velocity for a wide range of fluids, and their solutions are usually called turbulent if fluctuation amplitudes decrease as a power of their wavenumber. There have been some studies of CS for continuous systems (Kocarev et al 1997), but CS for NS turbulence seems not to have been investigated so far. We focus on the synchronization of the small scales of a turbulent flow for which the time history of large scales is prescribed. Our DNS results show that high-wavenumbers in turbulence are fully slaved to modes with wavenumbers up to a critical fraction of the Kolmogorov dissipation wavenumber. The motivation for our work is to study deeply sub-Kolmogorov scales in fully developed turbulence (Schumacher 2007), which we found to be recoverable even at very high Reynolds number from simulations with moderate resolutions. This work is supported by the National Science Foundation's CDI-II program, project CMMI-0941530
Chaos Synchronization in Navier-Stokes Turbulence
NASA Astrophysics Data System (ADS)
Lalescu, Cristian C.; Meneveau, Charles; Eyink, Gregory L.
2012-11-01
Chaos synchronization (CS) has been studied for some time now (Pecora & Carroll 1990), for systems with only a few degrees of freedom as well as for systems described by partial differential equations (Boccaletti et al. 2002). CS in general is said to be present in a pair of coupled dynamical systems when a specific property of each system has the same time evolution for both, even though the evolution itself is chaotic. There have been some studies of CS for systems with an infinite number of degrees of freedom (Kocarev et al. 1997), but CS for Navier-Stokes (NS) turbulence seems not to have been investigated so far. We focus on the synchronization of the small scales of a turbulent flow for which the time history of large scales is prescribed. We present DNS results which show that high-wavenumbers in turbulence are fully slaved to modes with wavenumbers up to a critical fraction of the Kolmogorov dissipation wavenumber. We compare our results with related ideas of ``approximate inertial manifolds.'' The motivation for our work is to study deeply sub-Kolmogorov scales in fully developed turbulence (Schumacher 2007), which we show are recoverable even at very high Reynolds number from simulations that only resolve down to about the Kolmogorov scale. This work is supported by the National Science Foundation's CDI-II program, project CMMI-0941530.
Navier-Stokes Computations on Commodity Computers
NASA Technical Reports Server (NTRS)
Vatsa, Veer N.; Faulkner, Thomas R.
1998-01-01
In this paper we discuss and demonstrate the feasibility of solving high-fidelity, nonlinear computational fluid dynamics (CFD) problems of practical interest on commodity machines, namely Pentium Pro PC's. Such calculations have now become possible due to the progress in computational power and memory of the off-the-shelf commodity computers, along with the growth in bandwidth and communication speeds of networks. A widely used CFD code known as TLNS3D, which was developed originally on large shared memory computers was selected for this effort. This code has recently been ported to massively parallel processor (MPP) type machines, where natural partitioning along grid blocks is adopted in which one or more blocks are distributed to each of the available processors. In this paper, a similar approach is adapted to port this code to a cluster of Pentium Pro computers. The message passing among the processors is accomplished through the use of standard message passing interface (MPI) libraries. Scaling studies indicate fairly high level of parallelism on such clusters of commodity machines, thus making solutions to Navier-Stokes equations for practical problems more affordable.
Scaling Navier-Stokes equation in nanotubes
NASA Astrophysics Data System (ADS)
Gǎrǎjeu, Mihail; Gouin, Henri; Saccomandi, Giuseppe
2013-08-01
On one hand, classical Monte Carlo and molecular dynamics simulations have been very useful in the study of liquids in nanotubes, enabling a wide variety of properties to be calculated in intuitive agreement with experiments. On the other hand, recent studies indicate that the theory of continuum breaks down only at the nanometer level; consequently flows through nanotubes still can be investigated with Navier-Stokes equations if we take suitable boundary conditions into account. The aim of this paper is to study the statics and dynamics of liquids in nanotubes by using methods of nonlinear continuum mechanics. We assume that the nanotube is filled with only a liquid phase; by using a second gradient theory the static profile of the liquid density in the tube is analytically obtained and compared with the profile issued from molecular dynamics simulation. Inside the tube there are two domains: a thin layer near the solid wall where the liquid density is non-uniform and a central core where the liquid density is uniform. In the dynamic case a closed form analytic solution seems to be no more possible, but by a scaling argument it is shown that, in the tube, two distinct domains connected at their frontiers still exist. The thin inhomogeneous layer near the solid wall can be interpreted in relation with the Navier length when the liquid slips on the boundary as it is expected by experiments and molecular dynamics calculations.
Navier-Stokes computations for circulation control airfoils
NASA Technical Reports Server (NTRS)
Pulliam, Thomas H.; Jespersen, Dennis C.; Barth, Timothy J.
1987-01-01
Navier-Stokes computations of subsonic to transonic flow past airfoils with augmented lift due to rearward jet blowing over a curved trailing edge are presented. The approach uses a spiral grid topology. Solutions are obtained using a Navier-Stokes code which employs an implicit finite difference method, an algebraic turbulence model, and developments which improve stability, convergence, and accuracy. Results are compared against experiments for no jet blowing and moderate jet pressures and demonstrate the capability to compute these complicated flows.
Compressible Navier-Stokes Equations with Revised Maxwell's Law
NASA Astrophysics Data System (ADS)
Hu, Yuxi; Racke, Reinhard
2016-05-01
We investigate the compressible Navier-Stokes equations where the constitutive law for the stress tensor given by Maxwell's law is revised to a system of relaxation equations for two parts of the tensor. The global well-posedness is proved as well as the compatibility with the classical compressible Navier-Stokes system in the sense that, for vanishing relaxation parameters, the solutions to the Maxwell system are shown to converge to solutions of the classical system.
On relaxation times in the Navier-Stokes-Voigt model
NASA Astrophysics Data System (ADS)
Layton, William J.; Rebholz, Leo G.
2013-03-01
We study analytically and numerically the relaxation time of flow evolution governed by the Navier-Stokes-Voigt (NSV) model. We first show that for the Taylor-Green vortex decay problem, NSV admits an exact solution which evolves slower than true fluid flow. Secondly, we show numerically for a channel flow test problem using standard discretisation methods that although NSV provides more regular solutions compared to usual Navier-Stokes solutions, NSV approximations take significantly longer to reach the steady state.
Navier-Stokes computations for circulation controlled airfoils
NASA Technical Reports Server (NTRS)
Pulliam, T. H.; Jesperen, D. C.; Barth, T. J.
1986-01-01
Navier-Stokes computations of subsonic to transonic flow past airfoils with augmented lift due to rearward jet blowing over a curved trailing edge are presented. The approach uses a spiral grid topology. Solutions are obtained using a Navier-Stokes code which employs an implicit finite difference method, an algebraic turbulence model, and developments which improve stability, convergence, and accuracy. Results are compared against experiments for no jet blowing and moderate jet pressures and demonstrate the capability to compute these complicated flows.
From Petrov-Einstein-Dilaton-Axion to Navier-Stokes equation in anisotropic model
NASA Astrophysics Data System (ADS)
Pan, Wen-Jian; Tian, Yu; Wu, Xiao-Ning
2016-01-01
In this paper we generalize the previous works to the case that the near-horizon dynamics of the Einstein-Dilaton-Axion theory can be governed by the incompressible Navier-Stokes equation via imposing the Petrov-like boundary condition on hypersurfaces in the non-relativistic and near-horizon limit. The dynamical shear viscosity η of such dual horizon fluid in our scenario, which isotropically saturates the Kovtun-Son-Starinet (KSS) bound, is independent of both the dilaton field and axion field in that limit.
A solution to the Navier-Stokes equations based upon the Newton Kantorovich method
NASA Technical Reports Server (NTRS)
Davis, J. E.; Gabrielsen, R. E.; Mehta, U. B.
1977-01-01
An implicit finite difference scheme based on the Newton-Kantorovich technique was developed for the numerical solution of the nonsteady, incompressible, two-dimensional Navier-Stokes equations in conservation-law form. The algorithm was second-order-time accurate, noniterative with regard to the nonlinear terms in the vorticity transport equation except at the earliest few time steps, and spatially factored. Numerical results were obtained with the technique for a circular cylinder at Reynolds number 15. Results indicate that the technique is in excellent agreement with other numerical techniques for all geometries and Reynolds numbers investigated, and indicates a potential for significant reduction in computation time over current iterative techniques.
On a Modified Form of Navier-Stokes Equations for Three-Dimensional Flows
Venetis, J.
2015-01-01
A rephrased form of Navier-Stokes equations is performed for incompressible, three-dimensional, unsteady flows according to Eulerian formalism for the fluid motion. In particular, we propose a geometrical method for the elimination of the nonlinear terms of these fundamental equations, which are expressed in true vector form, and finally arrive at an equivalent system of three semilinear first order PDEs, which hold for a three-dimensional rectangular Cartesian coordinate system. Next, we present the related variational formulation of these modified equations as well as a general type of weak solutions which mainly concern Sobolev spaces. PMID:25918743
Navier-Stokes simulations of WECS airfoil flowfields
Homicz, G.F.
1994-06-01
Sandia National Laboratories has initiated an effort to apply Computational Fluid Dynamics (CFD) to the study of WECS aerodynamics. Preliminary calculations are presented for the flow past a SAND 0018/50 airfoil. The flow solver used is F3D, an implicitly, finite-difference code which solves the Thin-Layer Navier-airfoil. The flow solver used is F3D, an implicit, finite-difference code which solves the Thin-Layer Navier-Stokes equations. 2D steady-state calculations are presented at various angles of attack, {alpha}. Sectional lift and drag coefficient, as well as surface pressure distributions, are compared with wind tunnel data, and exhibit reasonable agreement at low to moderate angles of attack. At high {alpha}, where the airfoil is stalled, a converged solution to the steady-state equations could not be obtained. The flowfield continued to change with successive iterations, which is consistent with the fact that the actual flow is inherently transient, and requires the solution of the full unsteady form of the equations.
Navier-Stokes computations of aft end flow fields
NASA Astrophysics Data System (ADS)
Weinberg, B. C.; McDonald, H.; Shamroth, S. J.
1982-05-01
A Navier-Stokes code to solve the aft end flow field of missile type configurations is presented. The consistently split linearized block implicit method of McDonald and Briley is employed in modified form to handle L-shaped domains with sharp reentrant corners. Appropriate boundary conditions are applied for the supersonic flow in particular at the outer boundary so that waves generated within the flow field are allowed to pass out of the computational domain without reflecting back into it. An adaptive grid option has been incorporated into the code and has been exercised by following the shear layer in a model backstep problem. Results are presented for the supersonic turbulent flow over a nozzle boattail configuration with and without jet exhaust and the results are compared with experiment. Calculations of the 2-D turbulent supersonic flow over a right angle back step with shear layer reattachment on a 20 deg ramp are also shown, and compared with experiments. The computation shows the qualitative physical behavior of the flows and there is generally good agreement with the experimental velocity profiles through most of the free shear layer and the ramp reattachment zone.
Implementation and Validation of the Chien k-epsilon Turbulence Model in the Wind Navier-Stokes Code
NASA Technical Reports Server (NTRS)
Yoder, Dennis A.; Georgiadis, Nicholas J.
1999-01-01
The two equation k-epsilon turbulence model of Chien has been implemented in the WIND Navier-Stokes flow solver. Details of the numerical solution algorithm, initialization procedure, and stability enhancements are described. Results obtained with this version of the model are compared with those from the Chien k-epsilon model in the NPARC Navier-Stokes code and from the WIND SST model for three validation cases: the incompressible flow over a smooth flat plate, the incompressible flow over a backward facing step, and the shock-induced flow separation inside a transonic diffuser. The k-epsilon model results indicate that the WIND model functions very similarly to that in NPARC, though the WIND code appears to he slightly more accurate in the treatment of the near-wall region. Comparisons of the k-epsilon model results with those from the SST model were less definitive, as each model exhibited strengths and weaknesses for each particular case.
Verification of the Proteus two-dimensional Navier-Stokes code for flat plate and pipe flows
NASA Technical Reports Server (NTRS)
Conley, Julianne M.; Zeman, Patrick L.
1991-01-01
The Proteus Navier-Stokes Code is evaluated for two-dimensional/axisymmetric, viscous, incompressible, internal and external flows. The particular cases to be discussed are laminar and turbulent flows over a flat plate, laminar and turbulent dveloping pipe flows and turbulent pipe flow with swirl. Results are compared with exact solutions, empirical correlations and experimental data. A detailed description of the code set-up, including boundary conditions, intitial conditions, grid size and grid packing is given for each case.
Numerical solutions of the complete Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Hassan, H. A.
1986-01-01
Using ideas from the kinetic theory, the Navier-Stokes equations are modified in such a way that they can be cast as a set of first order hyperbolic equations. This is achieved by incorporating time dependent terms into the definition of the stress tensor and the heat flux vectors. The boundary conditions are then determined from the theory of characteristics. Because the resulting equations reduce to the traditional Navier-Stokes equations when the steady state is reached, the present approach provides a straightforward scheme for the determination of inflow and outflow boundary conditions. The method is validated by comparing its predictions with known exact solutions of the steady Navier-Stokes equations.
Navier-Stokes and viscous-inviscid interaction
NASA Technical Reports Server (NTRS)
Steger, Joseph L.; Vandalsem, William R.
1989-01-01
Some considerations toward developing numerical procedures for simulating viscous compressible flows are discussed. Both Navier-Stokes and boundary layer field methods are considered. Because efficient viscous-inviscid interaction methods have been difficult to extend to complex 3-D flow simulations, Navier-Stokes procedures are more frequently being utilized even though they require considerably more work per grid point. It would seem a mistake, however, not to make use of the more efficient approximate methods in those regions in which they are clearly valid. Ideally, a general purpose compressible flow solver that can optionally take advantage of approximate solution methods would suffice, both to improve accuracy and efficiency. Some potentially useful steps toward this goal are described: a generalized 3-D boundary layer formulation and the fortified Navier-Stokes procedure.
What do the Navier-Stokes equations mean?
NASA Astrophysics Data System (ADS)
Schneiderbauer, Simon; Krieger, Michael
2014-01-01
The Navier-Stokes equations are nonlinear partial differential equations describing the motion of fluids. Due to their complicated mathematical form they are not part of secondary school education. A detailed discussion of fundamental physics—the conservation of mass and Newton’s second law—may, however, increase the understanding of the behaviour of fluids. Based on these principles the Navier-Stokes equations can be derived. This article attempts to make these equations available to a wider readership, especially teachers and undergraduate students. Therefore, in this article a derivation restricted to simple differential calculus is presented. Finally, we try to give answers to the questions ‘what is a fluid?’ and ‘what do the Navier-Stokes equations mean?’.
Some recent applications of Navier-Stokes codes to rotorcraft
NASA Technical Reports Server (NTRS)
Mccroskey, W. J.
1992-01-01
Many operational limitations of helicopters and other rotary-wing aircraft are due to nonlinear aerodynamic phenomena incuding unsteady, three-dimensional transonic and separated flow near the surfaces and highly vortical flow in the wakes of rotating blades. Modern computational fluid dynamics (CFD) technology offers new tools to study and simulate these complex flows. However, existing Euler and Navier-Stokes codes have to be modified significantly for rotorcraft applications, and the enormous computational requirements presently limit their use in routine design applications. Nevertheless, the Euler/Navier-Stokes technology is progressing in anticipation of future supercomputers that will enable meaningful calculations to be made for complete rotorcraft configurations.
Pseudo-time algorithms for the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Swanson, R. C.; Turkel, E.
1986-01-01
A pseudo-time method is introduced to integrate the compressible Navier-Stokes equations to a steady state. This method is a generalization of a method used by Crocco and also by Allen and Cheng. We show that for a simple heat equation that this is just a renormalization of the time. For a convection-diffusion equation the renormalization is dependent only on the viscous terms. We implement the method for the Navier-Stokes equations using a Runge-Kutta type algorithm. This permits the time step to be chosen based on the inviscid model only. We also discuss the use of residual smoothing when viscous terms are present.
Algorithm implementation on the Navier-Stokes computer
NASA Technical Reports Server (NTRS)
Krist, Steven E.; Zang, Thomas A.
1987-01-01
The Navier-Stokes Computer is a multi-purpose parallel-processing supercomputer which is currently under development at Princeton University. It consists of multiple local memory parallel processors, called Nodes, which are interconnected in a hypercube network. Details of the procedures involved in implementing an algorithm on the Navier-Stokes computer are presented. The particular finite difference algorithm considered in this analysis was developed for simulation of laminar-turbulent transition in wall bounded shear flows. Projected timing results for implementing this algorithm indicate that operation rates in excess of 42 GFLOPS are feasible on a 128 Node machine.
Factorization of the Compressible Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Roberts, Thomas W.
2005-01-01
The Navier-Stokes equations for a Newtonian ideal gas are examined to determine the factorizable form of the equations relevant to the construction of a factorizable relaxation scheme. The principal linearization of the equations is found by examining the relative magnitude of the terms for short-wavelength errors. The principal part of the operator is then found. Comparison of the factors of the Navier-Stokes and Euler equations differ qualitatively because of the coupling of entropy and pressure through thermal diffusion. Special cases of the factorization are considered.
NASA Astrophysics Data System (ADS)
Li, Zhaorui; Livescu, Daniel
2014-11-01
By using the second-law of thermodynamics and the Onsager reciprocal method for irreversible processes, we have developed a set of physically consistent multicomponent compressible generalized Cahn-Hilliard Navier-Stokes (CGCHNS) equations from basic thermodynamics. The new equations can describe not only flows with pure miscible and pure immiscible materials but also complex flows in which mass diffusion and surface tension or Korteweg stresses effects may coexist. Furthermore, for the first time, the incompressible generalized Cahn-Hilliard Navier-Stokes (IGCHNS) equations are rigorously derived from the incompressible limit of the CGCHNS equations (as the infinite sound speed limit) and applied to the immiscible Rayleigh-Taylor instability problem. Extensive good agreements between numerical results and the linear stability theory (LST) predictions for the Rayleigh-Taylor instability are achieved for a wide range of wavenumber, surface tension, and viscosity values. The late-time results indicate that the IGCHNS equations can naturally capture complex interface topological changes including merging and breaking-up and are free of singularity problems.
The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow
NASA Astrophysics Data System (ADS)
Desvillettes, Laurent; Golse, François; Ricci, Valeria
2008-06-01
We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω⊂ℝ3 for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman's force consists of a source term 6 π ν j where j is the current density of the particles, and of a friction term 6 π ν ρ u where ρ is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Ω minus the disjoint union of N balls of radius ɛ=1/ N in the large N limit with no-slip boundary condition. The number density ρ and current density j are obtained from the limiting phase space empirical measure 1/Nsum_{1le kle N}δ_{xk,vk} , where x k is the center of the k-th ball and v k its instantaneous velocity. This can be seen as a generalization of Allaire's result in [Arch. Ration. Mech. Anal. 113:209-259, [1991
Wavelet regularization of the 2D incompressible Euler equations
NASA Astrophysics Data System (ADS)
Nguyen van Yen, Romain; Farge, Marie; Schneider, Kai
2009-11-01
We examine the viscosity dependence of the solutions of two-dimensional Navier-Stokes equations in periodic and wall-bounded domains, for Reynolds numbers varying from 10^3 to 10^7. We compare the Navier-Stokes solutions to those of the regularized two-dimensional Euler equations. The regularization is performed by applying at each time step the wavelet-based CVS filter (Farge et al., Phys. Fluids, 11, 1999), which splits turbulent fluctuations into coherent and incoherent contributions. We find that for Reynolds 10^5 the dissipation of coherent enstrophy tends to become independent of Reynolds, while the dissipation of total enstrophy decays to zero logarithmically with Reynolds. In the wall-bounded case, we observe an additional production of enstrophy at the wall. As a result, coherent enstrophy diverges when Reynolds tends to infinity, but its time derivative seems to remain bounded independently of Reynolds. This indicates that a balance may have been established between coherent enstrophy dissipation and coherent enstrophy production at the wall. The Reynolds number for which the dissipation of coherent enstrophy becomes independent on the Reynolds number is proposed to define the onset of the fully-turbulent regime.
The space-time solution element method: A new numerical approach for the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Scott, James R.; Chang, Sin-Chung
1995-01-01
This paper is one of a series of papers describing the development of a new numerical method for the Navier-Stokes equations. Unlike conventional numerical methods, the current method concentrates on the discrete simulation of both the integral and differential forms of the Navier-Stokes equations. Conservation of mass, momentum, and energy in space-time is explicitly provided for through a rigorous enforcement of both the integral and differential forms of the governing conservation laws. Using local polynomial expansions to represent the discrete primitive variables on each cell, fluxes at cell interfaces are evaluated and balanced using exact functional expressions. No interpolation or flux limiters are required. Because of the generality of the current method, it applies equally to the steady and unsteady Navier-Stokes equations. In this paper, we generalize and extend the authors' 2-D, steady state implicit scheme. A general closure methodology is presented so that all terms up through a given order in the local expansions may be retained. The scheme is also extended to nonorthogonal Cartesian grids. Numerous flow fields are computed and results are compared with known solutions. The high accuracy of the scheme is demonstrated through its ability to accurately resolve developing boundary layers on coarse grids. Finally, we discuss applications of the current method to the unsteady Navier-Stokes equations.
On solving the compressible Navier-Stokes equations for unsteady flows at very low Mach numbers
NASA Technical Reports Server (NTRS)
Pletcher, R. H.; Chen, K.-H.
1993-01-01
The properties of a preconditioned, coupled, strongly implicit finite difference scheme for solving the compressible Navier-Stokes equations in primitive variables are investigated for two unsteady flows at low speeds, namely the impulsively started driven cavity and the startup of pipe flow. For the shear-driven cavity flow, the computational effort was observed to be nearly independent of Mach number, especially at the low end of the range considered. This Mach number independence was also observed for steady pipe flow calculations; however, rather different conclusions were drawn for the unsteady calculations. In the pressure-driven pipe startup problem, the compressibility of the fluid began to significantly influence the physics of the flow development at quite low Mach numbers. The present scheme was observed to produce the expected characteristics of completely incompressible flow when the Mach number was set at very low values. Good agreement with incompressible results available in the literature was observed.
Analysis of regularized Navier-Stokes equations, 2
NASA Technical Reports Server (NTRS)
Ou, Yuh-Roung; Sritharan, S. S.
1989-01-01
A practically important regularization of the Navier-Stokes equations was analyzed. As a continuation of the previous work, the structure of the attractors characterizing the solutins was studied. Local as well as global invariant manifolds were found. Regularity properties of these manifolds are analyzed.
Symmetric approximations of the Navier-Stokes equations
Kobel'kov, G M
2002-08-31
A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as {epsilon}{yields}0 of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established.
Symmetric approximations of the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Kobel'kov, G. M.
2002-08-01
A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as \\varepsilon\\to0 of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established.
NASA Astrophysics Data System (ADS)
Miao, Sha; Hendrickson, Kelli; Liu, Yuming; Subramani, Hariprasad
2015-11-01
This work presents a novel and efficient Cartesian-grid based simulation capability for the study of an incompressible, turbulent gas layer over a liquid flow with disparate Reynolds numbers in two phases. This capability couples a turbulent gas-flow solver and a liquid-layer based on a second-order accurate Boundary Data Immersion Method (BDIM) at the deformable interface. The turbulent gas flow solver solves the incompressible Navier-Stokes equations via direct numerical simulation or through turbulence closure (unsteady Reynolds-Averaged Navier-Stokes Models) for Reynolds numbers O(106). In this application, a laminar liquid layer solution is obtained from depth-integrated Navier-Stokes equations utilizing shallow water wave assumptions. The immersed boundary method (BDIM) enforces the coupling at the deformable interface, the boundary conditions to turbulence closure equations and defines the domain geometry on the Cartesian grid. Validations are made for the turbulent gas channel flow over high-viscosity liquid. This simulation capability can be applied to problems in the oil and industrial sector such as channel and pipe flows with heavy oils as well as wind wave generation in shallow waters. Sponsored by the Chevron Energy Technology Company.
Numerical Investigation of the “Poor Man’s Navier-Stokes Equations” with Darcy and Forchheimer Terms
NASA Astrophysics Data System (ADS)
Tang, Tingting; Li, Zhiyong; McDonough, J. M.; Hislop, P. D.
In this paper, a discrete dynamical system (DDS) is derived from the generalized Navier-Stokes equations for incompressible flow in porous media via a Galerkin procedure. The main difference from the previously studied poor man’s Navier-Stokes equations is the addition of forcing terms accounting for linear and nonlinear drag forces of the medium — Darcy and Forchheimer terms. A detailed numerical investigation focusing on the bifurcation parameters due to these additional terms is provided in the form of regime maps, time series, power spectra, phase portraits and basins of attraction, which indicate system behaviors in agreement with expected physical fluid flow through porous media. As concluded from the previous studies, this DDS can be employed in subgrid-scale models of synthetic-velocity form for large-eddy simulation of turbulent flow through porous media.
Inverse airfoil design procedure using a multigrid Navier-Stokes method
NASA Technical Reports Server (NTRS)
Malone, J. B.; Swanson, R. C.
1991-01-01
The Modified Garabedian McFadden (MGM) design procedure was incorporated into an existing 2-D multigrid Navier-Stokes airfoil analysis method. The resulting design method is an iterative procedure based on a residual correction algorithm and permits the automated design of airfoil sections with prescribed surface pressure distributions. The new design method, Multigrid Modified Garabedian McFadden (MG-MGM), is demonstrated for several different transonic pressure distributions obtained from both symmetric and cambered airfoil shapes. The airfoil profiles generated with the MG-MGM code are compared to the original configurations to assess the capabilities of the inverse design method.
Space-Time Error Representation and Estimation in Navier-Stokes Calculations
NASA Technical Reports Server (NTRS)
Barth, Timothy J.
2006-01-01
The mathematical framework for a-posteriori error estimation of functionals elucidated by Eriksson et al. [7] and Becker and Rannacher [3] is revisited in a space-time context. Using these theories, a hierarchy of exact and approximate error representation formulas are presented for use in error estimation and mesh adaptivity. Numerical space-time results for simple model problems as well as compressible Navier-Stokes flow at Re = 300 over a 2D circular cylinder are then presented to demonstrate elements of the error representation theory for time-dependent problems.
A multidimensional flux function with applications to the Euler and Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Rumsey, Christopher L.; Van Leer, Bram; Roe, Philip L.
1993-01-01
In the present grid-independent approximate Riemann solver for 2D and 3D flows that are governed by the Euler or Navier-Stokes equations, fluxes on grid faces are obtained by wave decomposition; the assumption of information-propagation in the velocity-difference directions leads to a more accurate resolution of shear and shock waves, when these are are oblique to the grid. The model, which yields significantly greater accuracy in both supersonic and subsonic first-order spatially accurate computations, describes the difference in states at each grid interface by the action of five waves.
Navier-Stokes simulation of the flow around an airfoil in Darrieus motion
Tchon, K.F.; Paraschivoiu, I. . Dept. of Mechanical Engineering)
1994-12-01
In order to study the dynamic stall phenomenon on a Darrieus wind turbine, the incompressible flow field around a moving airfoil is simulated using a noninertial stream function-vorticity formulation of the two-dimensional unsteady navier-Stokes equations. Spatial discretization is achieved by the streamline upwind Petrov-Galerkin finite element method on a hybrid mesh composed of a structured region of quadrilateral elements in the vicinity of solid boundaries, an unstructured region of triangular elements elsewhere, and a layer of infinite elements surrounding the domain and projecting the external boundary to infinity. Temporal discretization is achieved by an implicit second order finite difference scheme. At each time step, a nonlinear algebraic system is solved by a Newton method. To accelerate computations, the generalized minimum residual method with an incomplete triangular factorization preconditioning is used to solve the linearized Newton systems. The solver is applied to simulate the flow around a NACA 0015 airfoil in Darrieus motion and the results are compared to experimental observations. To the authors' knowledge, it is the first time that the simulation of such a motion has been performed using the Navier-Stokes equations.
Implementation and analysis of a Navier-Stokes algorithm on parallel computers
NASA Technical Reports Server (NTRS)
Fatoohi, Raad A.; Grosch, Chester E.
1988-01-01
The results of the implementation of a Navier-Stokes algorithm on three parallel/vector computers are presented. The object of this research is to determine how well, or poorly, a single numerical algorithm would map onto three different architectures. The algorithm is a compact difference scheme for the solution of the incompressible, two-dimensional, time-dependent Navier-Stokes equations. The computers were chosen so as to encompass a variety of architectures. They are the following: the MPP, an SIMD machine with 16K bit serial processors; Flex/32, an MIMD machine with 20 processors; and Cray/2. The implementation of the algorithm is discussed in relation to these architectures and measures of the performance on each machine are given. The basic comparison is among SIMD instruction parallelism on the MPP, MIMD process parallelism on the Flex/32, and vectorization of a serial code on the Cray/2. Simple performance models are used to describe the performance. These models highlight the bottlenecks and limiting factors for this algorithm on these architectures. Finally, conclusions are presented.
Partial Averaged Navier-Stokes approach for cavitating flow
NASA Astrophysics Data System (ADS)
Zhang, L.; Zhang, Y. N.
2015-01-01
Partial Averaged Navier Stokes (PANS) is a numerical approach developed for studying practical engineering problems (e.g. cavitating flow inside hydroturbines) with a resonance cost and accuracy. One of the advantages of PANS is that it is suitable for any filter width, leading a bridging method from traditional Reynolds Averaged Navier-Stokes (RANS) to direct numerical simulations by choosing appropriate parameters. Comparing with RANS, the PANS model will inherit many physical nature from parent RANS but further resolve more scales of motion in great details, leading to PANS superior to RANS. As an important step for PANS approach, one need to identify appropriate physical filter-width control parameters e.g. ratios of unresolved-to-total kinetic energy and dissipation. In present paper, recent studies of cavitating flow based on PANS approach are introduced with a focus on the influences of filter-width control parameters on the simulation results.
Turbomachinery blade optimization using the Navier-Stokes equations
Chand, K.K.; Lee, K.D.
1997-12-01
A method is presented to perform aerodynamic design optimization of turbomachinery blades. The method couples a Navier-Stokes flow solver with a grid generator and numerical optimization algorithm to seek improved designs for transonic turbine blades. A fast and efficient multigrid, finite-volume flow solver provides accurate performance evaluations of potential designs. Design variables consist of smooth perturbations to the blade surface. A unique elliptic-hyperbolic grid generation method is used to regenerate a Navier-Stokes grid after perturbations have been added to the geometry. Designs are sought which improve a design objective while remaining within specified constraints. The method is demonstrated with two transonic turbine blades with different types and numbers of design variables.
A Continuation and Bifurcation Technique for Navier-Stokes Flows
NASA Astrophysics Data System (ADS)
Sanchez, J.; Marques, F.; Lopez, J. M.
2002-07-01
An efficient numerical bifurcation and continuation method for the Navier-Stokes equations in cylindrical geometries is presented and applied to a nontrivial fluid dynamics problem, the flow in a cylindrical container driven by differential rotation. The large systems that result from discretizing the Navier-Stokes equations, especially in regimes where inertia is important, necessitate the use of iterative solvers which in turn need preconditioners. We use incomplete lower-upper decomposition (ILU) as an effective preconditioner for such systems and show the significant gain in efficiency when an incomplete LU of the full Jacobian is used instead of using only the Stokes operator. The computational cost, in terms of CPU time, grows with the size of the system (i.e., spatial resolution) according to a power law with exponent around 1.7, which is very modest compared to direct methods, indicating the appropriateness of the schemes for large nonlinear partial differential equation problems.
Compressible Navier Stokes Model with Inflow-Outflow Boundary Conditions
NASA Astrophysics Data System (ADS)
Novo, Sébastien
2005-11-01
In the paper [7], author gives a definition of weak solution to the nonsteady Navier Stokes system of equations which describes compressible and isentropic flows in some bounded region Ω with influx of fluid through a part of the boundary ∂Ω. Here, we present a way for proving existence of such solutions in the same situation as in [7] under the sole hypothesis γ > 3/2 for the adiabatic constant.
Cornetti, G.M.
1995-12-31
The 3D Navier-Stokes equations are solved via the Characteristic-Galerkin method extended to free boundary problems. A temporal discretization procedure is proposed for the case where a preferential direction to move mesh point exists, as in thin domains. Using a single layer of finite elements, the numerical results cover the so-called shallow water 2D approximation, showing the same wave propagation speed.
A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Scott, James R.; Chang, Sin-Chung
1993-01-01
A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel
NASA Astrophysics Data System (ADS)
Pan, Xinghong
2016-06-01
Consider an axisymmetric suitable weak solution of 3D incompressible Navier-Stokes equations with nontrivial swirl, v =vrer +vθeθ +vzez. Let z denote the axis of symmetry and r be the distance to the z-axis. If the solution satisfies a slightly supercritical assumption (that is, | v | ≤ C(ln | ln r |)/α r for α ∈ [ 0 , 0.028 ] when r is small), then we prove that v is regular. This extends the results in [6,16,18] where regularities under critical assumptions, such as | v | ≤Cr, were proven. As a useful tool in the proof of our main result, an upper-bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper.
Preconditioning for the Navier-Stokes equations with finite-rate chemistry
NASA Technical Reports Server (NTRS)
Godfrey, Andrew G.; Walters, Robert W.; Van Leer, Bram
1993-01-01
The preconditioning procedure for generalized finite-rate chemistry and the proper preconditioning for the one-dimensional Navier-Stokes equations are presented. Eigenvalue stiffness is resolved and convergence-rate acceleration is demonstrated over the entire Mach-number range from the incompressible to the hypersonic. Specific benefits are realized at low and transonic flow speeds. The extended preconditioning matrix accounts for thermal and chemical non-equilibrium and its implementation is explained for both explicit and implicit time marching. The effect of higher-order spatial accuracy and various flux splittings is investigated. Numerical analysis reveals the possible theoretical improvements from using proconditioning at all Mach numbers. Numerical results confirm the expectations from the numerical analysis. Representative test cases include flows with previously troublesome embedded high-condition-number regions.
Probabilistic Aspects of Equation of Motion of Forced Burgers and Navier-Stokes Turbulence
NASA Astrophysics Data System (ADS)
Nakazawa, H.
1980-11-01
Physical requirements and limitations on the force terms of the equations of motion for forced Burgers turbulence and for a class of forced, incompressible Navier-Stokes turbulence are discussed from probabilistic point of view. A basic problem, to determine the appropriate normalization of equations of motion, is answered. The normalization and the physical requirements are shown to stipulate that the force terms must bear Gaussian and white character for their time dependence as an exclusive consequence of the central limit theorem of Rosenblatt. A range of physical phenomena is thus pointed out to substantialize Kraichnan-Wyld-Edwards type of equations of motion for turbulence. A problem is found in the definition, as stochastic partial differential equations, of such equations with Gaussian-white-noise forces in the inviscid limit, and a possible way to circumvent the difficulty is shown to be inherent in the central limit theorem itself.
The dual variable method for finite element discretizations of Navier/Stokes equations
NASA Astrophysics Data System (ADS)
Hall, C. A.; Peterson, J. S.; Porsching, T. A.; Sledge, F. R.
1985-05-01
The dual-variable method of Amit et al. (1981) and Hall et al. (1980) is applied to the numerical solution of the transient Navier-Stokes equations for two-dimensional incompressible flows. The basic procedures of the method are reviewed, including determining the rank of the discrete divergence matrix, obtaining a particular solution of the discrete continuity equation, and defining the null space of the discrete divergence operator. Finite-element algorithms based on quadrilateral piecewise-bilateral-velocity/constant-pressure elements are developed and demonstrated for Poiseuille flow, a lid-driven cavity, and flow past a semicircular obstacle. The results are presented in tables and graphs and compared with those of a primitive-variable method, and the dual-variable approach is found to yield significant savings in dynamic memory and computation time.
Accuracy of least-squares methods for the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Bochev, Pavel B.; Gunzburger, Max D.
1993-01-01
Recently there has been substantial interest in least-squares finite element methods for velocity-vorticity-pressure formulations of the incompressible Navier-Stokes equations. The main cause for this interest is the fact that algorithms for the resulting discrete equations can be devised which require the solution of only symmetric, positive definite systems of algebraic equations. On the other hand, it is well-documented that methods using the vorticity as a primary variable often yield very poor approximations. Thus, here we study the accuracy of these methods through a series of computational experiments, and also comment on theoretical error estimates. It is found, despite the failure of standard methods for deriving error estimates, that computational evidence suggests that these methods are, at the least, nearly optimally accurate. Thus, in addition to the desirable matrix properties yielded by least-squares methods, one also obtains accurate approximations.
NASA Astrophysics Data System (ADS)
Kandula, M.; Pearce, D. G.
1991-06-01
A steady incompressible three-dimensional viscous flow analysis has been conducted for the Space Shuttle external tank/orbiter propellant feed line disconnect flapper valves with upstream elbows. The Navier-Stokes code, INS3D, is modified to handle interior obstacles and a simple turbulence model. The flow solver is tested for stability and convergence in the presence of interior flappers. An under-relaxation scheme has been incorporated to improve the solution stability. Important flow characteristics such as secondary flows, recirculation, vortex and wake regions, and separated flows are observed. Computed values for forces, moments, and pressure drop are in satisfactory agreement with water flow test data covering a maximum tube Reynolds number of 3.5 million. The predicted hydrodynamical stability of the flappers correlates well with the measurements.
A Modular Approach to Model Oscillating Control Surfaces Using Navier Stokes Equations
NASA Technical Reports Server (NTRS)
Guruswamy, Guru P.; Lee, Henry
2014-01-01
The use of active controls for rotorcraft is becoming more important for modern aerospace configurations. Efforts to reduce the vibrations of helicopter blades with use of active-controls are in progress. Modeling oscillating control surfaces using the linear aerodynamics theory is well established. However, higher-fidelity methods are needed to account for nonlinear effects, such as those that occur in transonic flow. The aeroelastic responses of a wing with an oscillating control surface, computed using the transonic small perturbation (TSP) theory, have been shown to cause important transonic flow effects such as a reversal of control surface effectiveness that occurs as the shock wave crosses the hinge line. In order to account for flow complexities such as blade-vortex interactions of rotor blades higher-fidelity methods based on the Navier-Stokes equations are used. Reference 6 presents a procedure that uses the Navier-Stokes equations with moving-sheared grids and demonstrates up to 8 degrees of control-surface amplitude, using a single grid. Later, this procedure was extended to accommodate larger amplitudes, based on sliding grid zones. The sheared grid method implemented in EulerlNavier-Stokes-based aeroelastic code ENS AERO was successfully applied to active control design by industry. Recently there are several papers that present results for oscillating control surface using Reynolds Averaged Navier-Stokes (RANS) equations. References 9 and 10 report 2-D cases by filling gaps with overset grids. Reference 9 compares integrated forces with the experiment at low oscillating frequencies whereas Ref. 10 reports parametric studies but with no validation. Reference II reports results for a 3D case by modeling the gap region with a deformed grid and compares force results with the experiment only at the mid-span of flap. In Ref. II grid is deformed to match the control surface deflections at the section where the measurements are made. However, there is no
Numerical study on comparison of Navier-Stokes and Burgers equations
NASA Astrophysics Data System (ADS)
Ohkitani, Koji; Dowker, Mark
2012-05-01
We compare freely decaying evolution of the Navier-Stokes equations with that of the 3D Burgers equations with the same kinematic viscosity and the same incompressible initial data by using direct numerical simulations. The Burgers equations are well-known to be regular by a maximum principle [A. A. Kiselev and O. A. Ladyzenskaya, "On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid," Izv. Akad. Nauk SSSR Ser. Mat. 21, 655 (1957); A. A. Kiselev and O. A. Ladyzenskaya, Am. Math. Soc. Transl. 24, 79 (1957)] unlike the Navier-Stokes equations. It is found in the Burgers equations that the potential part of velocity becomes large in comparison with the solenoidal part which decays more quickly. The probability distribution of the nonlocal term -{u}\\cdot nabla p, which spoils the maximum principle, in the local energy budget is studied in detail. It is basically symmetric, i.e., it can be either positive or negative with fluctuations. Its joint probability density functions with 1/2|{u}|^2 and with 1/2|{ω }|^2 are also found to be symmetric, fluctuating at the same times as the probability density function of -{u}\\cdot nabla p. A power-law relationship is found in the mathematical bound for the enstrophy growth dfrac{dQ}{dt} + 2 ν P ∝ left(Q^a P^bright)^α , where Q and P denote the enstrophy and the palinstrophy, respectively, and the exponents a and b are determined by calculus inequalities. We propose to quantify nonlinearity depletion by the exponent α on this basis.
NASA Astrophysics Data System (ADS)
Dutta, Vimala
1993-07-01
An implicit finite volume nodal point scheme has been developed for solving the two-dimensional compressible Navier-Stokes equations. The numerical scheme is evolved by efficiently combining the basic ideas of the implicit finite-difference scheme of Beam and Warming (1978) with those of nodal point schemes due to Hall (1985) and Ni (1982). The 2-D Navier-Stokes solver is implemented for steady, laminar/turbulent flows past airfoils by using C-type grids. Turbulence closure is achieved by employing the algebraic eddy-viscosity model of Baldwin and Lomax (1978). Results are presented for the NACA-0012 and RAE-2822 airfoil sections. Comparison of the aerodynamic coefficients with experimental results for the different test cases presented here establishes the validity and efficiency of the method.
NASA Astrophysics Data System (ADS)
Kouhi, Mohammad; Oñate, Eugenio
2015-07-01
A new implicit stabilized formulation for the numerical solution of the compressible Navier-Stokes equations is presented. The method is based on the finite calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors Kouhi, Oñate (Int J Numer Methods Fluids 74:872-897, 2014). In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier-Stokes equations using an implicit scheme is presented.
Navier-Stokes turbine heat transfer predictions using two-equation turbulence
NASA Technical Reports Server (NTRS)
Ameri, Ali A.; Arnone, Andrea
1992-01-01
Navier-Stokes calculations were carried out in order to predict the heat transfer rates on turbine blades. The calculations were performed using TRAF2D which is a two-dimensional, explicit, finite volume mass-averaged Navier-Stokes solver. Turbulence was modeled using q-omega and k-epsilon two-equation models and the Baldwin-Lomax algebraic model. The model equations along with the flow equations were solved explicitly on a non-periodic C grid. Implicit residual smoothing (IRS) or a combination of multigrid technique and IRS was applied to enhance convergence rates. Calculations were performed to predict the Stanton number distributions on the first stage vane and blade row as well as the second stage vane row of the Rocketdyne Space Shuttle Main Engine (SSME) high pressure fuel turbine. The comparison with the experimental results, although generally favorable, serves to highlight the weaknesses of the turbulence models and the possible areas of improving these models for use in turbomachinery heat transfer calculations.
Flux Based Surface Boundary Conditions for Navier-Stokes Simulations
NASA Astrophysics Data System (ADS)
Fertig, M.; Auweter-Kurtz, M.
2005-02-01
During re-entry high thermal combined with mechanical loads arise at the TPS surface of a re-entry vehicle. Due to low gas density, high Knudsen Numbers arise, which indicate rarefaction effects such as thermo-chemical non-equilibrium as well as temperature and velocity slip. With increasing altitude, local Knudsen Numbers predict the failure of continuum equations starting in the bow shock and at the surface. While local failure of the equations in the shock can be neglected for the determination of surface loads, local failure at the surface is not negligible. The validity of continuum models can be extended by emploing surface boundary equations accounting for temperature and velocity slip. A new flux based model has been developed originating on the Boltzmann Equation. Making use of the Enskog Method perturbed partition functions for a multi-component gas are determined from the Boltzmann Equation. By introduction of the moments of Boltzmann's Equation, Maxwell's Transport Equation can be obtained. Particles approaching the surface are distinguished from particles leaving the surface depending on their molecular velocities. Hence, mass, momentum and energy fluxes to the surface can be determined employing the collisional invariants. Reactive as well as scattering models can be easily introduced in order to compute the fluxes from the surface. Finally, flux differences are balanced with the continuum fluxes from the Navier-Stokes equations. Hence, the model is able to predict temperature and velocity slip at the surface of a re-entry vehicle under rarefied conditions. Moreover, it is valid in the continuum regime as well. The boundary equations are solved fully implicit and fully coupled with the non-equilibrium Navier-Stokes Code URANUS. Results are compared to DSMC simulations for the re-entry of the US Space Shuttle orbiter at high altitudes. Key words: Navier-Stokes; re-entry; slip; non-equilibrium.
Application of Aeroelastic Solvers Based on Navier Stokes Equations
NASA Technical Reports Server (NTRS)
Keith, Theo G., Jr.; Srivastava, Rakesh
2001-01-01
The propulsion element of the NASA Advanced Subsonic Technology (AST) initiative is directed towards increasing the overall efficiency of current aircraft engines. This effort requires an increase in the efficiency of various components, such as fans, compressors, turbines etc. Improvement in engine efficiency can be accomplished through the use of lighter materials, larger diameter fans and/or higher-pressure ratio compressors. However, each of these has the potential to result in aeroelastic problems such as flutter or forced response. To address the aeroelastic problems, the Structural Dynamics Branch of NASA Glenn has been involved in the development of numerical capabilities for analyzing the aeroelastic stability characteristics and forced response of wide chord fans, multi-stage compressors and turbines. In order to design an engine to safely perform a set of desired tasks, accurate information of the stresses on the blade during the entire cycle of blade motion is required. This requirement in turn demands that accurate knowledge of steady and unsteady blade loading is available. To obtain the steady and unsteady aerodynamic forces for the complex flows around the engine components, for the flow regimes encountered by the rotor, an advanced compressible Navier-Stokes solver is required. A finite volume based Navier-Stokes solver has been developed at Mississippi State University (MSU) for solving the flow field around multistage rotors. The focus of the current research effort, under NASA Cooperative Agreement NCC3- 596 was on developing an aeroelastic analysis code (entitled TURBO-AE) based on the Navier-Stokes solver developed by MSU. The TURBO-AE code has been developed for flutter analysis of turbomachine components and delivered to NASA and its industry partners. The code has been verified. validated and is being applied by NASA Glenn and by aircraft engine manufacturers to analyze the aeroelastic stability characteristics of modem fans, compressors
Smooth solutions of the Navier-Stokes equations
Pokhozhaev, S I
2014-02-28
We consider smooth solutions of the Cauchy problem for the Navier-Stokes equations on the scale of smooth functions which are periodic with respect to x∈R{sup 3}. We obtain existence theorems for global (with respect to t>0) and local solutions of the Cauchy problem. The statements of these depend on the smoothness and the norm of the initial vector function. Upper bounds for the behaviour of solutions in both classes, which depend on t, are also obtained. Bibliography: 10 titles.
Navier-Stokes solutions for rotating 3-D duct flows
NASA Astrophysics Data System (ADS)
Srivastava, B. N.
1988-07-01
This paper deals with the computation of three-dimensional viscous turbulent flow in a rotating rectangular duct of low aspect ratio using thin-layer Navier-Stokes equations. Scalar form of an approximate factorization implicit scheme along with a modified q-omega turbulence model has been utilized for mean flow predictions. The predicted mean flow behavior has been favorably compared with the experimental data for mean axial velocity, channel pressure and cross-flow velocities at a flow Mach number of 0.05 and a rotational speed of 300 rpm.
A visual programming environment for the Navier-Stokes computer
NASA Technical Reports Server (NTRS)
Tomboulian, Sherryl; Crockett, Thomas W.; Middleton, David
1988-01-01
The Navier-Stokes computer is a high-performance, reconfigurable, pipelined machine designed to solve large computational fluid dynamics problems. Due to the complexity of the architecture, development of effective, high-level language compilers for the system appears to be a very difficult task. Consequently, a visual programming methodology has been developed which allows users to program the system at an architectural level by constructing diagrams of the pipeline configuration. These schematic program representations can then be checked for validity and automatically translated into machine code. The visual environment is illustrated by using a prototype graphical editor to program an example problem.
Towards an ideal preconditioner for linearized Navier-Stokes problems
Murphy, M.F.
1996-12-31
Discretizing certain linearizations of the steady-state Navier-Stokes equations gives rise to nonsymmetric linear systems with indefinite symmetric part. We show that for such systems there exists a block diagonal preconditioner which gives convergence in three GMRES steps, independent of the mesh size and viscosity parameter (Reynolds number). While this {open_quotes}ideal{close_quotes} preconditioner is too expensive to be used in practice, it provides a useful insight into the problem. We then consider various approximations to the ideal preconditioner, and describe the eigenvalues of the preconditioned systems. Finally, we compare these preconditioners numerically, and present our conclusions.
Navier-Stokes analysis of radial turbine rotor performance
NASA Technical Reports Server (NTRS)
Larosiliere, L. M.
1993-01-01
An analysis of flow through a radial turbine rotor using the three-dimensional, thin-layer Navier-Stokes code RVC3D is described. The rotor is a solid version of an air-cooled metallic radial turbine having thick trailing edges, shroud clearance, and scalloped-backface clearance. Results are presented at the nominal operating condition using both a zero-clearance model and a model simulating the effects of the shroud and scalloped-backface clearance flows. A comparison with the available test data is made and details of the internal flow physics are discussed, allowing a better understanding of the complex flow distribution within the rotor.
On Bi-Grid Local Mode Analysis of Solution Techniques for 3-D Euler and Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Ibraheem, S. O.; Demuren, A. O.
1994-01-01
A procedure is presented for utilizing a bi-grid stability analysis as a practical tool for predicting multigrid performance in a range of numerical methods for solving Euler and Navier-Stokes equations. Model problems based on the convection, diffusion and Burger's equation are used to illustrate the superiority of the bi-grid analysis as a predictive tool for multigrid performance in comparison to the smoothing factor derived from conventional von Neumann analysis. For the Euler equations, bi-grid analysis is presented for three upwind difference based factorizations, namely Spatial, Eigenvalue and Combination splits, and two central difference based factorizations, namely LU and ADI methods. In the former, both the Steger-Warming and van Leer flux-vector splitting methods are considered. For the Navier-Stokes equations, only the Beam-Warming (ADI) central difference scheme is considered. In each case, estimates of multigrid convergence rates from the bi-grid analysis are compared to smoothing factors obtained from single-grid stability analysis. Effects of grid aspect ratio and flow skewness are examined. Both predictions are compared with practical multigrid convergence rates for 2-D Euler and Navier-Stokes solutions based on the Beam-Warming central scheme.
Modeling Vortex Generators in a Navier-Stokes Code
NASA Technical Reports Server (NTRS)
Dudek, Julianne C.
2011-01-01
A source-term model that simulates the effects of vortex generators was implemented into the Wind-US Navier-Stokes code. The source term added to the Navier-Stokes equations simulates the lift force that would result from a vane-type vortex generator in the flowfield. The implementation is user-friendly, requiring the user to specify only three quantities for each desired vortex generator: the range of grid points over which the force is to be applied and the planform area and angle of incidence of the physical vane. The model behavior was evaluated for subsonic flow in a rectangular duct with a single vane vortex generator, subsonic flow in an S-duct with 22 corotating vortex generators, and supersonic flow in a rectangular duct with a counter-rotating vortex-generator pair. The model was also used to successfully simulate microramps in supersonic flow by treating each microramp as a pair of vanes with opposite angles of incidence. The validation results indicate that the source-term vortex-generator model provides a useful tool for screening vortex-generator configurations and gives comparable results to solutions computed using gridded vanes.
Reliability enhancement of Navier-Stokes codes through convergence acceleration
NASA Technical Reports Server (NTRS)
Merkle, Charles L.; Dulikravich, George S.
1995-01-01
Methods for enhancing the reliability of Navier-Stokes computer codes through improving convergence characteristics are presented. The improving of these characteristics decreases the likelihood of code unreliability and user interventions in a design environment. The problem referred to as a 'stiffness' in the governing equations for propulsion-related flowfields is investigated, particularly in regard to common sources of equation stiffness that lead to convergence degradation of CFD algorithms. Von Neumann stability theory is employed as a tool to study the convergence difficulties involved. Based on the stability results, improved algorithms are devised to ensure efficient convergence in different situations. A number of test cases are considered to confirm a correlation between stability theory and numerical convergence. The examples of turbulent and reacting flow are presented, and a generalized form of the preconditioning matrix is derived to handle these problems, i.e., the problems involving additional differential equations for describing the transport of turbulent kinetic energy, dissipation rate and chemical species. Algorithms for unsteady computations are considered. The extension of the preconditioning techniques and algorithms derived for Navier-Stokes computations to three-dimensional flow problems is discussed. New methods to accelerate the convergence of iterative schemes for the numerical integration of systems of partial differential equtions are developed, with a special emphasis on the acceleration of convergence on highly clustered grids.
Algorithmic Enhancements to the VULCAN Navier-Stokes Solver
NASA Technical Reports Server (NTRS)
Litton, D. K.; Edwards, J. R.; White, J. A.
2003-01-01
VULCAN (Viscous Upwind aLgorithm for Complex flow ANalysis) is a cell centered, finite volume code used to solve high speed flows related to hypersonic vehicles. Two algorithms are presented for expanding the range of applications of the current Navier-Stokes solver implemented in VULCAN. The first addition is a highly implicit approach that uses subiterations to enhance block to block connectivity between adjacent subdomains. The addition of this scheme allows more efficient solution of viscous flows on highly-stretched meshes. The second algorithm addresses the shortcomings associated with density-based schemes by the addition of a time-derivative preconditioning strategy. High speed, compressible flows are typically solved with density based schemes, which show a high level of degradation in accuracy and convergence at low Mach numbers (M less than or equal to 0.1). With the addition of preconditioning and associated modifications to the numerical discretization scheme, the eigenvalues will scale with the local velocity, and the above problems will be eliminated. With these additions, VULCAN now has improved convergence behavior for multi-block, highly-stretched meshes and also can solve the Navier-Stokes equations for very low Mach numbers.
Navier-Stokes analysis of muzzle-blast-type waves
NASA Astrophysics Data System (ADS)
Baysal, O.
1986-05-01
A Navier-Stokes solution is presented as a mathematical model to muzzle-blast-type waves. The study has two novel features. First, it is a combined internal/external analysis relating barrel flow parameters to muzzle environment parameters. Second, the dissipative and dispersive effects of viscosity on the propagation phenomenon are captured. The investigation also serves as a numerical analysis of axisymmetric, high-pressure waves in an unsteady, viscous flow. Conservation-form Navier-Stokes equations are integrated by a two-step, explicit finite-difference scheme. The shocks are captured and treated by the inclusion of artificial dissipative terms. Turbulence is accounted for by an algebraic eddy-viscosity model. The internal flow is solved by a predictor-corrector method of characteristics with the shock fitted in; its results compare very well with the experimental data available. The numerical results obtained simulate the muzzle blast waves and show the effects of viscosity. Comparison with the classical spherical blast wave theory shows the deviation in propagation patterns of the axisymmetric and spherical waves.
Time-accurate Navier-Stokes calculations with multigrid acceleration
NASA Technical Reports Server (NTRS)
Melson, N. Duane; Atkins, Harold L.; Sanetrik, Mark D.
1993-01-01
A numerical scheme to solve the unsteady Navier-Stokes equations is described. The scheme is implemented by modifying the multigrid-multiblock version of the steady Navier-Stokes equations solver, TLNS3D. The scheme is fully implicit in time and uses TLNS3D to iteratively invert the equations at each physical time step. The design objective of the scheme is unconditional stability (at least for first- and second-order discretizations of the physical time derivatives). With unconditional stability, the choice of the time step is based on the physical phenomena to be resolved rather than limited by numerical stability which is especially important for high Reynolds number viscous flows, where the spatial variation of grid cell size can be as much as six orders of magnitude. An analysis of the iterative procedure and the implementation of this procedure in TLNS3D are discussed. Numerical results are presented to show both the capabilities of the scheme and its speed up relative to the use of global minimum time stepping. Reductions in computational times of an order of magnitude are demonstrated.
NASA Astrophysics Data System (ADS)
Li, Zhilin; Xiao, Li; Cai, Qin; Zhao, Hongkai; Luo, Ray
2015-09-01
In this paper, a new Navier-Stokes solver based on a finite difference approximation is proposed to solve incompressible flows on irregular domains with open, traction, and free boundary conditions, which can be applied to simulations of fluid structure interaction, implicit solvent model for biomolecular applications and other free boundary or interface problems. For some problems of this type, the projection method and the augmented immersed interface method (IIM) do not work well or does not work at all. The proposed new Navier-Stokes solver is based on the local pressure boundary method, and a semi-implicit augmented IIM. A fast Poisson solver can be used in our algorithm which gives us the potential for developing fast overall solvers in the future. The time discretization is based on a second order multi-step method. Numerical tests with exact solutions are presented to validate the accuracy of the method. Application to fluid structure interaction between an incompressible fluid and a compressible gas bubble is also presented.
NASA Astrophysics Data System (ADS)
Johnson, Philip; Johnsen, Eric
2015-11-01
The Recovery discontinuous Galerkin (DG) method is a highly accurate approach to computing diffusion problems, which achieves up to 3p+2 convergence rates on Cartesian cells, where p is the order of the polynomial basis. Based on the construction of a unique and differentiable solution across cell interfaces, Recovery DG has mostly been investigated on periodic domains. However, whether such accuracy can be sustained for Dirichlet and Neumann boundary conditions has not been thoroughly explored. We present boundary treatments for Recovery DG on 2D Cartesian geometry that exhibit up to 3p+2 convergence rates and are stable. We demonstrate the efficiency of Recovery DG in context with other commonly used approaches using scalar shear diffusion problems and apply it to the compressible Navier-Stokes equations. The extension of the method to perturbed quadrilateral cells, rather than Cartesian, will also be discussed.
Navier-Stokes analysis of turbine flowfield and external heat transfer
NASA Technical Reports Server (NTRS)
Luo, J.; Lakshminarayana, B.
1993-01-01
An explicit 2D Navier-Stokes code has been modified and used to analyze the aerodynamics and heat transfer of a transonic turbine cascade. This code is based on a four-stage Runge-Kutta scheme. An algebraic Reynolds stress model (ARSM) and two versions of low Reynolds number (LRN) two-equation turbulence models, Chien's (1982) LRN k-epsilon model and Coakley's (1983) LRN q-omega model, have been employed in the computations. The surface pressure distributions and wake profiles are predicted well by all the models. The k-epsilon model and the k-epsilon/ARSM model yield better predictions of heat transfer than the q-omega model. The k-epsilon/ARSM solutions show some significant, though not dramatic, differences in the predicted skin friction coefficients, heat transfer rates, and performance parameters, as compared to the k-epsilon model. The predicted semiwake width is consistent with the measurement and correlation.
Multigrid Solution of the Navier-Stokes Equations at Low Speeds with Large Temperature Variations
NASA Technical Reports Server (NTRS)
Sockol, Peter M.
2002-01-01
Multigrid methods for the Navier-Stokes equations at low speeds and large temperature variations are investigated. The compressible equations with time-derivative preconditioning and preconditioned flux-difference splitting of the inviscid terms are used. Three implicit smoothers have been incorporated into a common multigrid procedure. Both full coarsening and semi-coarsening with directional fine-grid defect correction have been studied. The resulting methods have been tested on four 2D laminar problems over a range of Reynolds numbers on both uniform and highly stretched grids. Two of the three methods show efficient and robust performance over the entire range of conditions. In addition none of the methods have any difficulty with the large temperature variations.
NASA Technical Reports Server (NTRS)
Saleem, M.; Pulliam, T.; Cheer, A. Y.
1993-01-01
Implicit difference operator spectra are presently computed by applying eigensystem analysis techniques to finite-difference formulations of 2D Euler and Navier-Stokes equations, and attention is given to these iterative methods' convergence and stability characteristics by taking into account the effects of grid geometry, time-step, numerical viscosity, and boundary conditions. On the basis of the eigenvalue distributions for various flow configurations, the feasibility of applying such convergence-acceleration techniques as eigenvalue annihilation and relaxation is discussed. Spectrum-shifting is applied to NASA-Ames' ARC2D flow code, achieving a 20-33 percent efficiency.
NASA Technical Reports Server (NTRS)
Anderson, B. H.; Reddy, D. R.; Kapoor, K.
1993-01-01
A three-dimensional implicit Full Navier-Stokes (FNS) analysis and a 3D Reduced Navier-Stokes (RNS) initial value space marching solution technique has been applied to a class of separate flow problems within a diffusing S-duct configuration characterized as vortex-liftoff. Both Full Navier-Stokes and Reduced Navier-Stokes solution techniques were able to capture the overall flow physics of vortex lift-off, however more consideration must be given to the development of turbulence models for the prediction of the locations of separation and reattachment. This accounts for some of the discrepancies in the prediction of the relevant inlet distortion descriptors, particularly circumferential distortion. The 3D RNS solution technique adequately described the topological structure of flow separation associated with vortex lift-off.
Calculation of Two-Phase Navier-Stokes Flows Using Phase-Field Modeling
NASA Astrophysics Data System (ADS)
Jacqmin, David
1999-10-01
Phase-field models provide a way to model fluid interfaces as having finite thickness. This can allow the computation of interface movement and deformation on fixed grids. This paper applies phase-field modeling to the computation of two-phase incompressible Navier-Stokes flows. The Navier-Stokes equations are modified by the addition of the continuum forcing -C∇→φ, where C is the composition variable and φ is C's chemical potential. The equation for interface advection is replaced by a continuum advective-diffusion equation, with diffusion driven by C's chemical potential gradients. The paper discusses how solutions to these equations approach those of the original sharp-interface Navier-Stokes equations as the interface thickness ɛ and the diffusivity both go to zero. The basic flow-physics of phase-field interfaces is discussed. Straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion. On the other hand, too large a diffusion will overly damp the flow. These two constraints result in an upper bound for the diffusivity of O(ɛ) and a lower bound of O(ɛ2). Within these two bounds, the phase-field Navier-Stokes equations appear to generate an O(ɛ) error relative to the exact sharp-interface equations. An O(h2/ɛ2) numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing. An O(h4/ɛ4) compact scheme is introduced that takes advantage of the asymptotic, comparatively smooth, behavior of the chemical potential. For O(ɛ) accurate phase-field models the optimum path to convergence for this scheme appears to be ɛ∝h4/5. The asymptotic rate of convergence corresponding to this is O(h4/5) but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear. Extensive analysis and computations show that
Investigation of Navier-Stokes Code Verification and Design Optimization
NASA Technical Reports Server (NTRS)
Vaidyanathan, Rajkumar
2004-01-01
With rapid progress made in employing computational techniques for various complex Navier-Stokes fluid flow problems, design optimization problems traditionally based on empirical formulations and experiments are now being addressed with the aid of computational fluid dynamics (CFD). To be able to carry out an effective CFD-based optimization study, it is essential that the uncertainty and appropriate confidence limits of the CFD solutions be quantified over the chosen design space. The present dissertation investigates the issues related to code verification, surrogate model-based optimization and sensitivity evaluation. For Navier-Stokes (NS) CFD code verification a least square extrapolation (LSE) method is assessed. This method projects numerically computed NS solutions from multiple, coarser base grids onto a freer grid and improves solution accuracy by minimizing the residual of the discretized NS equations over the projected grid. In this dissertation, the finite volume (FV) formulation is focused on. The interplay between the xi concepts and the outcome of LSE, and the effects of solution gradients and singularities, nonlinear physics, and coupling of flow variables on the effectiveness of LSE are investigated. A CFD-based design optimization of a single element liquid rocket injector is conducted with surrogate models developed using response surface methodology (RSM) based on CFD solutions. The computational model consists of the NS equations, finite rate chemistry, and the k-6 turbulence closure. With the aid of these surrogate models, sensitivity and trade-off analyses are carried out for the injector design whose geometry (hydrogen flow angle, hydrogen and oxygen flow areas and oxygen post tip thickness) is optimized to attain desirable goals in performance (combustion length) and life/survivability (the maximum temperatures on the oxidizer post tip and injector face and a combustion chamber wall temperature). A preliminary multi-objective optimization
Investigation of Navier-Stokes code verification and design optimization
NASA Astrophysics Data System (ADS)
Vaidyanathan, Rajkumar
With rapid progress made in employing computational techniques for various complex Navier-Stokes fluid flow problems, design optimization problems traditionally based on empirical formulations and experiments are now being addressed with the aid of computational fluid dynamics (CFD). To be able to carry out an effective CFD-based optimization study, it is essential that the uncertainty and appropriate confidence limits of the CFD solutions be quantified over the chosen design space. The present dissertation investigates the issues related to code verification, surrogate model-based optimization and sensitivity evaluation. For Navier-Stokes (NS) CFD code verification a least square extrapolation (LSE) method is assessed. This method projects numerically computed NS solutions from multiple, coarser base grids onto a finer grid and improves solution accuracy by minimizing the residual of the discretized NS equations over the projected grid. In this dissertation, the finite volume (FV) formulation is focused on. The interplay between the concepts and the outcome of LSE, and the effects of solution gradients and singularities, nonlinear physics, and coupling of flow variables on the effectiveness of LSE are investigated. A CFD-based design optimization of a single element liquid rocket injector is conducted with surrogate models developed using response surface methodology (RSM) based on CFD solutions. The computational model consists of the NS equations, finite rate chemistry, and the k-epsilonturbulence closure. With the aid of these surrogate models, sensitivity and trade-off analyses are carried out for the injector design whose geometry (hydrogen flow angle, hydrogen and oxygen flow areas and oxygen post tip thickness) is optimized to attain desirable goals in performance (combustion length) and life/survivability (the maximum temperatures on the oxidizer post tip and injector face and a combustion chamber wall temperature). A preliminary multi
NASA Technical Reports Server (NTRS)
Anderson, B. H.; Reddy, D. R.; Kapoor, K.
1993-01-01
A three-dimensional implicit Full Navier-Stokes (FNS) analysis and a 3D Reduced Navier Stokes (RNS) initial value space marching solution technique has been applied to a class of separated flow problems within a diffusing S-duct configuration characterized by vortex-liftoff. Both the FNS and the RNS solution technique were able to capture the overall flow physics of vortex lift-off, and gave remarkably similar results which agreed reasonably well with the experimental measured averaged performance parameters of engine face total pressure recovery and distortion. However, the Full Navier-Stokes and Reduced Navier-Stokes also consistently predicted separation further downstream in the M2129 inlet S-duct than was indicated by experimental data, thus compensating errors were present in the two Navier-Stokes analyses. The difficulties encountered in the Navier-Stokes separations analyses of the M2129 inlet S-duct center primarily on turbulence model issues, and these focused on two distinct but different phenomena, namely, (1) characterization of low skin friction adverse pressure gradient flows, and (2) description of the near wall behavior of flows characterized by vortex lift-off.
NASA Technical Reports Server (NTRS)
Bui, Trong T.
1992-01-01
The implementation and validation of the Chien low Reynolds number k-epsilon turbulence model in the two dimensional axisymmetric version Proteus, a compressible Navier-Stokes computer code, are presented. The set of k-epsilon equations are solved by marching in time using a coupled alternating direction implicit (ADI) solution procedure with generalized first or second order time differencing. To validate Proteus and the k-epsilon turbulence model, laminar and turbulent computations were done for several benchmark test cases: incompressible fully developed 2-D channel flow; fully developed axisymmetric pipe flow; boundary layer flow over a flat plate; and turbulent Sajben subsonic transonic diffuser flows. Proteus results from these test cases showed good agreement with analytical results and experimental data. Detailed comparisons of both mean flow and turbulent quantities showed that the Chien k-epsilon turbulence model given good results over a wider range of turbulent flow than the Baldwin-Lomax turbulence model in the Proteus code with no significant CPU time penalty for more complicated flow cases.
SSME thrust chamber simulation using Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Przekwas, A. J.; Singhal, A. K.; Tam, L. T.
1984-01-01
The capability of the PHOENICS fluid dynamics code in predicting two-dimensional, compressible, and reacting flow in the combustion chamber and nozzle of the space shuttle main engine (SSME) was evaluated. A non-orthogonal body fitted coordinate system was used to represent the nozzle geometry. The Navier-Stokes equations were solved for the entire nozzle with a turbulence model. The wall boundary conditions were calculated based on the wall functions which account for pressure gradients. Results of the demonstration test case reveal all expected features of the transonic nozzle flows. Of particular interest are the locations of normal and barrel shocks, and regions of highest temperature gradients. Calculated performance (global) parameters such as thrust chamber flow rate, thrust, and specific impulse are also in good agreement with available data.
Application of Navier-Stokes analysis to stall flutter
NASA Technical Reports Server (NTRS)
Wu, J. C.; Srivastava, R.; Sankar, L. N.
1988-01-01
A solution procedure was developed to investigate the two-dimensional, one- or two-dimensional flutter characteristics of arbitrary airfoils. This procedure requires a simultaneous integration in time of the solid and fluid equations of motion. The fluid equations of motion are the unsteady compressible Navier-Stokes equations, solved in a body-fitted moving coordinate system using an approximate factorization scheme. The solid equations of motion are integrated in time using an Euler implicit scheme. Flutter is said to occur if small disturbances imposed on the airfoil attitude lead to divergent oscillatory motions at subsequent times. The flutter characteristics of airfoils in subsonic speed at high angles of attack and airfoils in high subsonic and transonic speeds at low angles of attack are investigated. The stall flutter characteristics are also predicted using the same procedure.
Convergence analysis of WLS based solution of Navier Stokes equation
NASA Astrophysics Data System (ADS)
Kosec, G.
2016-06-01
A numerical solution of a Navier-Stokes problem based on the Weighted Least Squares (WLS) approximation of velocity and pressure fields is presented in this paper. The approximation function is constructed over the local support, i.e., a sub cluster of computational nodes. Besides local approximation of the fields also the pressure-velocity algorithm is constructed locally. The presented solution procedure is demonstrated on two classical fluid-flow benchmark tests, i.e., lid-driven cavity and backward-facing step problem. The method is validated through comparison against already published data on regular nodal distributions and convergence analyses. In addition the method is also tested on irregular nodal distributions. Results are presented in terms of cross-section velocity profiles and convergence plots.
The Navier-Stokes Equations in Nonendpoint Borderline Lorentz Spaces
NASA Astrophysics Data System (ADS)
Phuc, Nguyen Cong
2015-12-01
It is shown both locally and globally that {L_t^{∞}(L_x^{3,q})} solutions to the three-dimensional Navier-Stokes equations are regular provided {q≠∞}. Here {L_x^{3,q}}, {0 < q ≤∞}, is an increasing scale of Lorentz spaces containing {L^3_x}. Thus the result provides an improvement of a result by Escauriaza et al. (Uspekhi Mat Nauk 58:3-44, 2003; translation in Russ Math Surv 58, 211-250, 2003), which treated the case q = 3. A new local energy bound and a new {ɛ}-regularity criterion are combined with the backward uniqueness theory of parabolic equations to obtain the result. A weak-strong uniqueness of Leray-Hopf weak solutions in {L_t^{∞}(L_x^{3,q})}, {q≠∞}, is also obtained as a consequence.
Perturbation of eigenvalues of preconditioned Navier-Stokes operators
Elman, H.C.
1996-12-31
We study the sensitivity of algebraic eigenvalue problems associated with matrices arising from linearization and discretization of the steady-state Navier-Stokes equations. In particular, for several choices of preconditioners applied to the system of discrete equations, we derive upper bounds on perturbations of eigenvalues as functions of the viscosity and discretization mesh size. The bounds suggest that the sensitivity of the eigenvalues is at worst linear in the inverse of the viscosity and quadratic in the inverse of the mesh size, and that scaling can be used to decrease the sensitivity in some cases. Experimental results supplement these results and confirm the relatively mild dependence on viscosity. They also indicate a dependence on the mesh size of magnitude smaller than the analysis suggests.
Aerodynamics of thrust vectoring by Navier-Stokes solutions
NASA Technical Reports Server (NTRS)
Tseng, Jing-Biau; Lan, C. Edward
1991-01-01
Induced aerodynamics from thrust vectoring are investigated by a computational fluid dynamic method. A thin-layer Reynolds-averaged Navier-Stokes code with multiblock capability is used. Jet properties are specified on the nozzle exit plane to simulate the jet momentum. Results for a rectangular jet in a cross flow are compared with data to verify the code. Further verification of the calculation is made by comparing the numerical results with transonic data for a wing-body combination. Additional calculations were performed to elucidate the following thrust vectoring effects: the thrust vectoring effect on shock and expansion waves, induced effects on nearby surfaces, and the thrust vectoring effect on the leading edge vortex.
Navier-Stokes analysis of turbine blade heat transfer
NASA Technical Reports Server (NTRS)
Boyle, R. J.
1990-01-01
Comparisons with experimental heat transfer and surface pressures were made for seven turbine vane and blade geometries using a quasi-three-dimensional thin-layer Navier-Stokes analysis. Comparisons are made for cases with both separated and unseparated flow over a range of Reynolds numbers and freestream turbulence intensities. The analysis used a modified Baldwin-Lomax turbulent eddy viscosity mode. Modifications were made to account for the effects of: (1) freestream turbulence on both transition and leading edge heat transfer; (2) strong favorable pressure gradients on relaminarization; and (3) variable turbulent Prandtl number heat transfer. In addition, the effect of heat transfer on the near wall model of Deissler is compared with the Van Driest model.
Navier-Stokes Dynamics by a Discrete Boltzmann Model
NASA Technical Reports Server (NTRS)
Rubinstein, Robet
2010-01-01
This work investigates the possibility of particle-based algorithms for the Navier-Stokes equations and higher order continuum approximations of the Boltzmann equation; such algorithms would generalize the well-known Pullin scheme for the Euler equations. One such method is proposed in the context of a discrete velocity model of the Boltzmann equation. Preliminary results on shock structure are consistent with the expectation that the shock should be much broader than the near discontinuity predicted by the Pullin scheme, yet narrower than the prediction of the Boltzmann equation. We discuss the extension of this essentially deterministic method to a stochastic particle method that, like DSMC, samples the distribution function rather than resolving it completely.
Navier-Stokes analysis of turbomachinery blade external heat transfer
NASA Technical Reports Server (NTRS)
Gorla, Rama S. R.
1991-01-01
The two-dimensional, compressible, thin-layer Navier-Stokes and energy equations were solved numerically to obtain heat transfer rates on turbomachinery blades. The Baldwin-Lomax algebraic model and the q-omega low Reynolds number two-equation model were used for modeling of turbulence. For the numerical solution of the governing equations a four-stage Runge-Kutta solver was employed. The turbulence model equations were solved using an implicit scheme. Numerical solutions are presented for two-dimensional flow within two vane cascades. The heat transfer results and the pressure distributions were compared with published experimental data. The agreement between the numerical calculations and the experimental values were found to be generally favorable. The position of transition from laminar to turbulent flow was also predicted accurately.
Iterative methods for compressible Navier-Stokes and Euler equations
Tang, W.P.; Forsyth, P.A.
1996-12-31
This workshop will focus on methods for solution of compressible Navier-Stokes and Euler equations. In particular, attention will be focused on the interaction between the methods used to solve the non-linear algebraic equations (e.g. full Newton or first order Jacobian) and the resulting large sparse systems. Various types of block and incomplete LU factorization will be discussed, as well as stability issues, and the use of Newton-Krylov methods. These techniques will be demonstrated on a variety of model transonic and supersonic airfoil problems. Applications to industrial CFD problems will also be presented. Experience with the use of C++ for solution of large scale problems will also be discussed. The format for this workshop will be four fifteen minute talks, followed by a roundtable discussion.
A Navier-Stokes solver for turbomachinery applications
Arnone, A.; Swanson, R.C. )
1993-04-01
A computer code for solving the Reynolds-averaged full Navier-Stokes equations has been developed and applied using H- and C-type grids. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. The integration in time is based on an explicit four-stage Runge-Kutta scheme. Local time stepping, variable coefficient implicit residual smoothing, and a full multigrid method have been implemented to accelerate steady-state calculations. A grid independence analysis is presented for a transonic rotor blade. Comparisons with experimental data show that the code is an accurate viscous solver and can give very good blade-to-blade predictions for engineering applications.
Time Integration Schemes for the Unsteady Navier-stokes Equations
NASA Technical Reports Server (NTRS)
Bijl, Hester; Carpenter, Mark H.; Vatsa, Veer N.
2001-01-01
The efficiency and accuracy of several time integration schemes are investigated for the unsteady Navier-Stokes equations. This study focuses on the efficiency of higher-order Runge-Kutta schemes in comparison with the popular Backward Differencing Formulations. For this comparison an unsteady two-dimensional laminar flow problem is chosen, i.e., flow around a circular cylinder at Re = 1200. It is concluded that for realistic error tolerances (smaller than 10(exp -1)) fourth-and fifth-order Runge-Kutta schemes are the most efficient. For reasons of robustness and computer storage, the fourth-order Runge-Kutta method is recommended. The efficiency of the fourth-order Runge-Kutta scheme exceeds that of second-order Backward Difference Formula by a factor of 2.5 at engineering error tolerance levels (10(exp -1) to 10(exp -2)). Efficiency gains are more dramatic at smaller tolerances.
Navier-Stokes analysis of turbomachinery blade external heat transfer
NASA Astrophysics Data System (ADS)
Ameri, A. A.; Sockol, P. M.; Gorla, R. S. R.
1992-04-01
The two-dimensional, compressible, thin-layer Navier-Stokes and energy equations were solved numerically to obtain heat transfer rates on turbomachinery blades. The Baldwin-Lomax algebraic model and the q - omega low Reynolds number, two-equation model were used for modeling of turbulence. For the numerical solution of the governing equations a four-stage Runge-Kutta solver was employed. The turbulence model equations were solved using an implicit scheme. Numerical solutions are presented for two-dimensional flow within two vane cascades. The heat transfer results and the pressure distributions were compared with published experimental data. The agreement between the numerical calculations and the experimental values were found to be generally favorable. The position of transition from laminar to turbulent flow was also predicted accurately.
NASA Astrophysics Data System (ADS)
Chae, Dongho; Constantin, Peter; Wu, Jiahong
2014-09-01
We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time for all time. In addition, we introduce a variant of the 2D Boussinesq equations which is perhaps a more faithful companion of the 3D axisymmetric Euler equations than the usual 2D Boussinesq equations.
On the Global Regularity of a Helical-Decimated Version of the 3D Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Biferale, Luca; Titi, Edriss S.
2013-06-01
We study the global regularity, for all time and all initial data in H 1/2, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution of Navier-Stokes (NS) equations into the subspace where helicity (the L 2-scalar product of velocity and vorticity) is sign-definite. The presence of a second (beside energy) sign-definite inviscid conserved quadratic quantity, which is equivalent to the H 1/2-Sobolev norm, allows us to demonstrate global existence and uniqueness, of space-periodic solutions, together with continuity with respect to the initial conditions, for this decimated 3D model. This is achieved thanks to the establishment of two new estimates, for this 3D model, which show that the H 1/2 and the time average of the square of the H 3/2 norms of the velocity field remain finite. Such two additional bounds are known, in the spirit of the work of H. Fujita and T. Kato (Arch. Ration. Mech. Anal. 16:269-315, 1964; Rend. Semin. Mat. Univ. Padova 32:243-260, 1962), to be sufficient for showing well-posedness for the 3D NS equations. Furthermore, they are directly linked to the helicity evolution for the dNS model, and therefore with a clear physical meaning and consequences.
NASA Technical Reports Server (NTRS)
Bui, Trong T.
1993-01-01
New turbulence modeling options recently implemented for the 3-D version of Proteus, a Reynolds-averaged compressible Navier-Stokes code, are described. The implemented turbulence models include: the Baldwin-Lomax algebraic model, the Baldwin-Barth one-equation model, the Chien k-epsilon model, and the Launder-Sharma k-epsilon model. Features of this turbulence modeling package include: well documented and easy to use turbulence modeling options, uniform integration of turbulence models from different classes, automatic initialization of turbulence variables for calculations using one- or two-equation turbulence models, multiple solid boundaries treatment, and fully vectorized L-U solver for one- and two-equation models. Validation test cases include the incompressible and compressible flat plate turbulent boundary layers, turbulent developing S-duct flow, and glancing shock wave/turbulent boundary layer interaction. Good agreement is obtained between the computational results and experimental data. Sensitivity of the compressible turbulent solutions with the method of y(sup +) computation, the turbulent length scale correction, and some compressibility corrections are examined in detail. The test cases show that the highly optimized one-and two-equation turbulence models can be used in routine 3-D Navier-Stokes computations with no significant increase in CPU time as compared with the Baldwin-Lomax algebraic model.
Calculations of High-Temperature Jet Flow Using Hybrid Reynolds-Average Navier-Stokes Formulations
NASA Technical Reports Server (NTRS)
Abdol-Hamid, Khaled S.; Elmiligui, Alaa; Giriamaji, Sharath S.
2008-01-01
Two multiscale-type turbulence models are implemented in the PAB3D solver. The models are based on modifying the Reynolds-averaged Navier Stokes equations. The first scheme is a hybrid Reynolds-averaged- Navier Stokes/large-eddy-simulation model using the two-equation k(epsilon) model with a Reynolds-averaged-Navier Stokes/large-eddy-simulation transition function dependent on grid spacing and the computed turbulence length scale. The second scheme is a modified version of the partially averaged Navier Stokes model in which the unresolved kinetic energy parameter f(sub k) is allowed to vary as a function of grid spacing and the turbulence length scale. This parameter is estimated based on a novel two-stage procedure to efficiently estimate the level of scale resolution possible for a given flow on a given grid for partially averaged Navier Stokes. It has been found that the prescribed scale resolution can play a major role in obtaining accurate flow solutions. The parameter f(sub k) varies between zero and one and is equal to one in the viscous sublayer and when the Reynolds-averaged Navier Stokes turbulent viscosity becomes smaller than the large-eddy-simulation viscosity. The formulation, usage methodology, and validation examples are presented to demonstrate the enhancement of PAB3D's time-accurate turbulence modeling capabilities. The accurate simulations of flow and turbulent quantities will provide a valuable tool for accurate jet noise predictions. Solutions from these models are compared with Reynolds-averaged Navier Stokes results and experimental data for high-temperature jet flows. The current results show promise for the capability of hybrid Reynolds-averaged Navier Stokes and large eddy simulation and partially averaged Navier Stokes in simulating such flow phenomena.
Navier-Stokes simulations of unsteady transonic flow phenomena
NASA Technical Reports Server (NTRS)
Atwood, C. A.
1992-01-01
Numerical simulations of two classes of unsteady flows are obtained via the Navier-Stokes equations: a blast-wave/target interaction problem class and a transonic cavity flow problem class. The method developed for the viscous blast-wave/target interaction problem assumes a laminar, perfect gas implemented in a structured finite-volume framework. The approximately factored implicit scheme uses Newton subiterations to obtain the spatially and temporally second-order accurate time history of the blast-waves with stationary targets. The inviscid flux is evaluated using either of two upwind techniques, while the full viscous terms are computed by central differencing. Comparisons of unsteady numerical, analytical, and experimental results are made in two- and three-dimensions for Couette flows, a starting shock-tunnel, and a shock-tube blockage study. The results show accurate wave speed resolution and nonoscillatory discontinuity capturing of the predominantly inviscid flows. Viscous effects were increasingly significant at large post-interaction times. While the blast-wave/target interaction problem benefits from high-resolution methods applied to the Euler terms, the transonic cavity flow problem requires the use of an efficient scheme implemented in a geometrically flexible overset mesh environment. Hence, the Reynolds averaged Navier-Stokes equations implemented in a diagonal form are applied to the cavity flow class of problems. Comparisons between numerical and experimental results are made in two-dimensions for free shear layers and both rectangular and quieted cavities, and in three-dimensions for Stratospheric Observatory For Infrared Astronomy (SOFIA) geometries. The acoustic behavior of the rectangular and three-dimensional cavity flows compare well with experiment in terms of frequency, magnitude, and quieting trends. However, there is a more rapid decrease in computed acoustic energy with frequency than observed experimentally owing to numerical
Stable boundary conditions and difference schemes for Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Dutt, P.
1985-01-01
The Navier-Stokes equations can be viewed as an incompletely elliptic perturbation of the Euler equations. By using the entropy function for the Euler equations as a measure of energy for the Navier-Stokes equations, it was possible to obtain nonlinear energy estimates for the mixed initial boundary value problem. These estimates are used to derive boundary conditions which guarantee L2 boundedness even when the Reynolds number tends to infinity. Finally, a new difference scheme for modelling the Navier-Stokes equations in multidimensions for which it is possible to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation was proposed.
Mathematical analysis of the Navier-Stokes equations with non standard boundary conditions
NASA Technical Reports Server (NTRS)
Tidriri, M. D.
1995-01-01
One of the major applications of the domain decomposition time marching algorithm is the coupling of the Navier-Stokes systems with Boltzmann equations in order to compute transitional flows. Another important application is the coupling of a global Navier-Stokes problem with a local one in order to use different modelizations and/or discretizations. Both of these applications involve a global Navier-Stokes system with nonstandard boundary conditions. The purpose of this work is to prove, using the classical Leray-Schauder theory, that these boundary conditions are admissible and lead to a well posed problem.
Blow-up of Critical Besov Norms at a Potential Navier-Stokes Singularity
NASA Astrophysics Data System (ADS)
Gallagher, Isabelle; Koch, Gabriel S.; Planchon, Fabrice
2016-04-01
We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space {dot B^{-1+ 3/p}_{p,q}({R}3)}, with {3 < p, q < ∞}, gives rise to a strong solution with a singularity at a finite time {T > 0}, then the norm of the solution in that Besov space becomes unbounded at time T. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza et al. (Uspekhi Mat Nauk 58(2(350)):3-44, 2003) concerning suitable weak solutions blowing up in {L3({R}3)}. Our proof uses profile decompositions and is based on our previous work (Gallagher et al., Math. Ann. 355(4):1527-1559, 2013), which provided an alternative proof of the {L3({R}3)} result. For very large values of p, an iterative method, which may be of independent interest, enables us to use some techniques from the {L3({R}3)} setting.
Convergence of Time Averages of Weak Solutions of the Three-Dimensional Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Foias, Ciprian; Rosa, Ricardo M. S.; Temam, Roger M.
2015-08-01
Using the concept of stationary statistical solution, which generalizes the notion of invariant measure, it is proved that, in a suitable sense, time averages of almost every Leray-Hopf weak solution of the three-dimensional incompressible Navier-Stokes equations converge as the averaging time goes to infinity. This system of equations is not known to be globally well-posed, and the above result answers a long-standing problem, extending to this system a classical result from ergodic theory. It is also shown that, from a measure-theoretic point of view, the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit. Finally, any Borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that Borel set and for at least one Leray-Hopf weak solution starting with that initial condition, the corresponding orbit is recurrent to that Borel subset and its mean sojourn time within that Borel subset is strictly positive.
Manufactured solutions for steady-flow Reynolds-averaged Navier-Stokes solvers
NASA Astrophysics Data System (ADS)
Eça, L.; Hoekstra, M.; Vaz, G.
2012-06-01
This paper presents manufactured solutions (MS's) for code verification of incompressible flow solvers based on the Reynolds-averaged Navier-Stokes (RANS) equations. The proposed solutions mimic statistically steady, two-dimensional or three-dimensional near-wall turbulent flows in a simple domain (rectangle or rectangular box) at a given Reynolds number. The proposed analytical functions cover the mean flow quantities and the dependent variables of several eddy-viscosity turbulence models. Namely, the undamped eddy-viscosity of the Spalart and Allmaras and Menter one-equations models, from the one (SKL) and two-equation (KSKL) models proposed by Menter, the turbulence kinetic energy and the turbulence frequency included in two-equation k - ω models. A basic flow field resembling a turbulent flat plate flow is constructed with the turbulence quantities defined from 'automatic wall functions' that are supposed to reproduce more or less the normal behaviour of these variables. Alternative flow fields are constructed superposing a perturbation flow field that creates a 'recirculation zone'. However, the near-wall solution of the basic flow is kept to avoid zero friction at the wall. Three-dimensional MS's are obtained from the blending of the basic two-dimensional MS's in the transverse direction. All flow fields satisfy mass conservation, i.e. mean velocity fields are divergence-free. The source functions required for the balancing of momentum and turbulence quantities transport equations and all the dependent variables and their derivatives are available in Fortran 90 modules.
On Nonlocal Cahn-Hilliard-Navier-Stokes Systems in Two Dimensions
NASA Astrophysics Data System (ADS)
Frigeri, Sergio; Gal, Ciprian G.; Grasselli, Maurizio
2016-08-01
We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the weak-strong uniqueness in the case of viscosity depending on the order parameter, provided that either the mobility is constant and the potential is regular or the mobility is degenerate and the potential is singular. In the case of constant viscosity, on account of the uniqueness results, we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor. The latter is established even in the case of variable viscosity, constant mobility and regular potential.
NASA Astrophysics Data System (ADS)
Dowker, Mark; Ohkitani, Koji
2012-11-01
We study space-time integrals, which appear in the Caffarelli-Kohn-Nirenberg (CKN) theory for the Navier-Stokes equations analytically and numerically. The key quantity is written in standard notations δ (r)=1/(ν r)int _{Q_r}left|nabla {u}right|^2 d{{x}} dt, which can be regarded as a local Reynolds number over a parabolic cylinder Qr. First, by re-examining the CKN integral, we identify a cross-over scale r_* ∝ Lleft( overline{Vert nabla {u} Vert ^2_{L^2}} /Vert nabla {u Vert ^2_{L^infty }} right)^{1/3}, at which the CKN Reynolds number δ(r) changes its scaling behavior. This reproduces a result on the minimum scale rmin in turbulence: r_min^2 Vert nabla {u}Vert _infty ∝ ν , consistent with a result of Henshaw et al. ["On the smallest scale for the incompressible Navier-Stokes equations," Theor. Comput. Fluid Dyn. 1, 65 (1989), 10.1007/BF00272138]. For the energy spectrum E(k) ∝ k-q (1 < q < 3), we show that r* ∝ νa with a=4/3(3-q)-1. Parametric representations are then obtained as Vert nabla {u}Vert _infty ∝ ν ^{-(1+3a)/2} and rmin ∝ ν3(a+1)/4. By the assumptions of the regularity and finite energy dissipation rate in the inviscid limit, we derive lim _{p rArr infty }ζ _p/p=1 - ζ _2 for any phenomenological models on intermittency, where ζp is the exponent of pth order (longitudinal) velocity structure function. It follows that ζp ⩽ (1 - ζ2)(p - 3) + 1 for any p ⩾ 3 without invoking fractal energy cascade. Second, we determine the scaling behavior of δ(r) in direct numerical simulations of the Navier-Stokes equations. In isotropic turbulence around Rλ ≈ 100 starting from random initial conditions, we have found that δ(r) ∝ r4throughout the inertial range. This can be explained by the smallness of a ≈ 0.26,with a result that r* is in the energy-containing range. If the β-model is perfectly correct, the intermittency parameter a must be related to the dissipation correlation exponent μ as μ =4a/1+a ≈ 0.8, which is larger
Hypersonic Navier Stokes Comparisons to Orbiter Flight Data
NASA Technical Reports Server (NTRS)
Campbell, Charles H.; Nompelis, Ioannis; Candler, Graham; Barnhart, Michael; Yoon, Seokkwan
2009-01-01
Hypersonic chemical nonequilibrium simulations of low earth orbit entry flow fields are becoming increasingly commonplace as software and computational capabilities become more capable. However, development of robust and accurate software to model these environments will always encounter a significant barrier in developing a suite of high quality calibration cases. The US3D hypersonic nonequilibrium Navier Stokes analysis capability has been favorably compared to a number of wind tunnel test cases. Extension of the calibration basis for this software to Orbiter flight conditions will provide an incremental increase in confidence. As part of the Orbiter Boundary Layer Transition Flight Experiment and the Hypersonic Thermodynamic Infrared Measurements project, NASA is performing entry flight testing on the Orbiter to provide valuable aerothermodynamic heating data. An increase in interest related to orbiter entry environments is resulting from this activity. With the advent of this new data, comparisons of the US3D software to the new flight testing data is warranted. This paper will provide information regarding the framework of analyses that will be applied with the US3D analysis tool. In addition, comparisons will be made to entry flight testing data provided by the Orbiter BLT Flight Experiment and HYTHIRM projects. If data from digital scans of the Orbiter windward surface become available, simulations will also be performed to characterize the difference in surface heating between the CAD reference OML and the digitized surface provided by the surface scans.
A Data Parallel Multizone Navier-Stokes Code
NASA Technical Reports Server (NTRS)
Jespersen, Dennis C.; Levit, Creon; Kwak, Dochan (Technical Monitor)
1995-01-01
We have developed a data parallel multizone compressible Navier-Stokes code on the Connection Machine CM-5. The code is set up for implicit time-stepping on single or multiple structured grids. For multiple grids and geometrically complex problems, we follow the "chimera" approach, where flow data on one zone is interpolated onto another in the region of overlap. We will describe our design philosophy and give some timing results for the current code. The design choices can be summarized as: 1. finite differences on structured grids; 2. implicit time-stepping with either distributed solves or data motion and local solves; 3. sequential stepping through multiple zones with interzone data transfer via a distributed data structure. We have implemented these ideas on the CM-5 using CMF (Connection Machine Fortran), a data parallel language which combines elements of Fortran 90 and certain extensions, and which bears a strong similarity to High Performance Fortran (HPF). One interesting feature is the issue of turbulence modeling, where the architecture of a parallel machine makes the use of an algebraic turbulence model awkward, whereas models based on transport equations are more natural. We will present some performance figures for the code on the CM-5, and consider the issues involved in transitioning the code to HPF for portability to other parallel platforms.
Exponential Mixing of the 3D Stochastic Navier-Stokes Equations Driven by Mildly Degenerate Noises
Albeverio, Sergio; Debussche, Arnaud; Xu Lihu
2012-10-15
We prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes being forced) via a Kolmogorov equation approach.
Design efficiency evaluation for transonic airfoil optimization - A case for Navier-Stokes design
NASA Technical Reports Server (NTRS)
Hager, J. O.; Eyi, S.; Lee, K. D.
1993-01-01
A constrained-optimization design method which improves the aerodynamic performance of transonic airfoils is evaluated from a design-quality and design-efficiency viewpoint. Design efficiency is a measure of the performance improvement and the design time (CPU time). Total-airfoil design and upper-surface design are performed using the Euler and Navier-Stokes equations with several grids, and are evaluated using the Navier-Stokes equations to determine the anticipated physical design response. Even though the cost of the Euler design is lower than Navier-Stokes design, the Navier-Stokes evaluation indicates that the Euler design does not necessarily improve the aerodynamic performance. Therefore, the design optimization should be based on an accurate flow simulation to achieve an actual performance improvement, and the design time is a secondary concern.
NASA Technical Reports Server (NTRS)
Liu, C. H.; Wong, T. C.; Kandil, O. A.
1988-01-01
The two-dimensional flow over a blunt leading-edge plate is simulated on the basis of an Euler/Navier-Stokes zonal scheme. The scheme uses an implicit upwind finite-volume scheme, which is based on the van Leer flux-vector splitting. It is shown that the Euler/Navier-Stokes zonal scheme with downstream boundary-layer compatibility conditions is accurate and efficient.
Multi-zonal Navier-Stokes code with the LU-SGS scheme
NASA Technical Reports Server (NTRS)
Klopfer, G. H.; Yoon, S.
1993-01-01
The LU-SGS (lower upper symmetric Gauss Seidel) algorithm has been implemented into the Compressible Navier-Stokes, Finite Volume (CNSFV) code and validated with a multizonal Navier-Stokes simulation of a transonic turbulent flow around an Onera M6 transport wing. The convergence rate and robustness of the code have been improved and the computational cost has been reduced by at least a factor of 2 over the diagonal Beam-Warming scheme.
NASA Astrophysics Data System (ADS)
Kim, Bong-Sik
Three dimensional (3D) Navier-Stokes-alpha equations are considered for uniformly rotating geophysical fluid flows (large Coriolis parameter f = 2O). The Navier-Stokes-alpha equations are a nonlinear dispersive regularization of usual Navier-Stokes equations obtained by Lagrangian averaging. The focus is on the existence and global regularity of solutions of the 3D rotating Navier-Stokes-alpha equations and the uniform convergence of these solutions to those of the original 3D rotating Navier-Stokes equations for large Coriolis parameters f as alpha → 0. Methods are based on fast singular oscillating limits and results are obtained for periodic boundary conditions for all domain aspect ratios, including the case of three wave resonances which yields nonlinear "2½-dimensional" limit resonant equations for f → 0. The existence and global regularity of solutions of limit resonant equations is established, uniformly in alpha. Bootstrapping from global regularity of the limit equations, the existence of a regular solution of the full 3D rotating Navier-Stokes-alpha equations for large f for an infinite time is established. Then, the uniform convergence of a regular solution of the 3D rotating Navier-Stokes-alpha equations (alpha ≠ 0) to the one of the original 3D rotating NavierStokes equations (alpha = 0) for f large but fixed as alpha → 0 follows; this implies "shadowing" of trajectories of the limit dynamical systems by those of the perturbed alpha-dynamical systems. All the estimates are uniform in alpha, in contrast with previous estimates in the literature which blow up as alpha → 0. Finally, the existence of global attractors as well as exponential attractors is established for large f and the estimates are uniform in alpha.
Single-grid spectral collocation for the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Bernardi, Christine; Canuto, Claudio; Maday, Yvon; Metivet, Brigitte
1988-01-01
The aim of the paper is to study a collocation spectral method to approximate the Navier-Stokes equations: only one grid is used, which is built from the nodes of a Gauss-Lobatto quadrature formula, either of Legendre or of Chebyshev type. The convergence is proven for the Stokes problem provided with inhomogeneous Dirichlet conditions, then thoroughly analyzed for the Navier-Stokes equations. The practical implementation algorithm is presented, together with numerical results.
Navier-Stokes and potential theory solutions for ahelicopter fuselage and comparison with experiment
NASA Technical Reports Server (NTRS)
Chaffin, Mark S.; Berry, John D.
1994-01-01
A thin-layer Navier-Stokes code and a panel method code are used to predict the flow over a generic helicopter fuselage. The computational results are compared with pressure data at four experimental conditions. Both methods produce results that agree with the experimental pressure data. However, separation patterns and other viscous flow features from the Navier-Stokes code solution are shown that cannot be easily modeled with the panel method.
Navier-Stokes solution for steady two-dimensional transonic cascade flows
Kwon, O.K.
1987-01-01
A robust, time-marching Navier-Stokes solution procedure based on the explicit hopscotch method is presented for solution of steady, two-dimensional, transonic turbine cascade flows. The method is applied to the strong conservation form of the unsteady Navier-Stokes equations written in arbitrary curvilinear coordinates. Cascade flow solutions are obtained on an orthogonal, body-conforming ''O'' grid with the standard k-epsilon turbulence model. Computed results are presented and compared with experimental data.
Adapting a Navier-Stokes code to the ICL-DAP
NASA Technical Reports Server (NTRS)
Grosch, C. E.
1985-01-01
The results of an experiment are reported, i.c., to adapt a Navier-Stokes code, originally developed on a serial computer, to concurrent processing on the CL Distributed Array Processor (DAP). The algorithm used in solving the Navier-Stokes equations is briefly described. The architecture of the DAP and DAP FORTRAN are also described. The modifications of the algorithm so as to fit the DAP are given and discussed. Finally, performance results are given and conclusions are drawn.
NASA Technical Reports Server (NTRS)
Thompson, J. F.; Mastin, C. W.; Thames, F. C.; Shanks, S. P.
1975-01-01
A procedure for numerical solution of the time-dependent, two-dimensional incompressible Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple airfoils, and submerged hydrofoils, as naturally as it can deal with the flow about simple bodies. The solution is based on a method of automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multiconnected region containing any number of arbitrarily shaped bodies. The curvilinear coordinates are generated as the solution of two elliptical partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other being specified along the boundaries. The solution compares excellently with the Blasius boundary layer solution for the flow past a semiinfinite flat plate.
NASA Astrophysics Data System (ADS)
Cervone, A.; Manservisi, S.; Scardovelli, R.
2010-09-01
A multilevel VOF approach has been coupled to an accurate finite element Navier-Stokes solver in axisymmetric geometry for the simulation of incompressible liquid jets with high density ratios. The representation of the color function over a fine grid has been introduced to reduce the discontinuity of the interface at the cell boundary. In the refined grid the automatic breakup and coalescence occur at a spatial scale much smaller than the coarse grid spacing. To reduce memory requirements, we have implemented on the fine grid a compact storage scheme which memorizes the color function data only in the mixed cells. The capillary force is computed by using the Laplace-Beltrami operator and a volumetric approach for the two principal curvatures. Several simulations of axisymmetric jets have been performed to show the accuracy and robustness of the proposed scheme.
Quasi-3D Navier-Stokes model for a rotating airfoil
Shen, W.Z.; Soerensen, J.N.
1999-04-10
The design of blade shapes for wind turbines is typically based on employing the blade-element momentum-theory (BEM) with lift and drag forces determined from 2D measurements. The results obtained are reasonable in the vicinity of the design point, but in stalled conditions the BEM is known to underpredict the forces acting on the blades. Here, a quasi-3D model of the unsteady Navier-Stokes equations in a rotating frame of reference has been developed. The equations governing the flow past a rotating blade are approximated using an order of magnitude analysis on the spanwise derivatives. The model takes into account rotational effects and spanwise outflow at computing expenses in the order of what is typical for similar 2D calculations. Results are presented for both laminar and turbulent flows past blades in pure rotation. In the turbulent case the influence of small-scale turbulence is modelled by the one-equation Baldwin-Barth turbulence model. The computations demonstrate that the main influence of rotation is to increase the maximum lift.
A Navier-Stokes-Based Approach for Mean Flow Perturbation Analysis
NASA Astrophysics Data System (ADS)
Bhaumik, Swagata; Gaitonde, Datta; Waindim, Mbu; The Ohio State University Team
2014-11-01
The manner in which a basic state, obtained from a time-averaged unsteady flowfield, processes perturbations can provide significant insight into the cause and evolution of instabilities. A widely used approach is based on Parabolized Stability Equations (PSE), which limits streamwise mean flow variation and is often applied to 2-D base flows. To avoid some of these issues, we advance a Navier-Stokes-based method, which can address non-trivial three-dimensional fields. The method stems from that employed by Touber and Sandham (Theor. Comput. Fluid. Dyn., 23, 79, 2009) to analyze global modes in nominally 2-D shock-wave turbulent-boundary layer interactions (STBLI). We first develop its theoretical underpinnings by examining conditions under which it degenerates to traditional methods. We then illustrate the application by considering perturbations to an entropy layer at Mach 6, a turbulent supersonic jet at Mach 1.3 and STBLI at Mach 2.3. For the entropy layer and jet cases, known linear stability and PSE results are successfully reproduced, while global modes are obtained for STBLI. The results not only validate the proposed technique, but also demonstrate its suitability in analyzing instabilities for any general 3D basic state, including impulse response. Sponsored by AFOSR.
Global smooth flows for compressible Navier-Stokes-Maxwell equations
NASA Astrophysics Data System (ADS)
Xu, Jiang; Cao, Hongmei
2016-08-01
Umeda et al. (Jpn J Appl Math 1:435-457, 1984) considered a rather general class of symmetric hyperbolic-parabolic systems: A0zt+sum_{j=1}nAjz_{xj}+Lz=sum_{j,k=1}nB^{jk}z_{xjxk} and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices {L} and {B^{jk}(j,k=1,ldots,n)} are both assumed to be real symmetric. So far there are no general results in case that {L} and {B^{jk}} are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier-Stokes-Maxwell (N-S-M) equations arising in plasmas physics, which is a concrete example of hyperbolic-parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N-S-M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of {L1({mathbb{R}}^3)}-{L^2({mathbb{R}}^3)} type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N-S-M equations, with aid of {L^p({mathbb{R}}^n)}-{Lq({mathbb{R}}^n)}-{Lr({mathbb{R}}^n)} estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.
Laser engine simulation using pressure based Navier-Stokes solver
NASA Astrophysics Data System (ADS)
Youssef, Hazim Saad
1994-03-01
Analysis of the flow field in a laser engine represents a difficult computational problem involving combinations of complex physical and gas-dynamical processes. Following a brief discussion of these processes a calculation procedure using primitive variables formulation on a nonstaggered grid system is introduced. Based on this procedure, a pressure based Navier-Stokes solver (PBNS) is developed using a generalized curvilinear coordinate system. The solver is first tested in application to a subsonic compressible flow over an insulated flat plate and to a flow in an axisymmetric converging-diverging nozzle. Next, the PBNS code is used to analyze the flowfield and performance of a laser thruster. The physical/numerical model includes the geometric ray tracing for the laser beam, beam power absorption, plasma radiation losses, and plasma thermophysical and optical properties. Equilibrium hydrogen is used as a flowing gas and its properties are calculated using the Hydrogen Properties Calculation (HPC) based on the methods of statistical thermodynamics. Two thrustor configurations, two laser types (CO2 and iodide), various laser power levels, and various injection conditions are tested. The results of these tests include the temperature, pressure, velocity, and Mach number contours, as well as tables of the laser beam power absorbed, radiation losses to the thrustor walls, thrust level, and specific impulse. The maximum specific impulse obtained in these tests is 1537 sec for a CO2 laser thruster and 827 sec for an iodide laser thruster. Up to 100% power absorption can be achieved; however, radiation losses from the hot plasma are quite high disallowing a full conversion of the absorbed power into the thermal energy of the propellant. The PBNS code can be used to study the effects of various design parameters on the performance of a laser thruster and provide guidelines for the preliminary design of a laser engine.
NASA Astrophysics Data System (ADS)
Stück, Arthur
2015-11-01
Inconsistent discrete expressions in the boundary treatment of Navier-Stokes solvers and in the definition of force objective functionals can lead to discrete-adjoint boundary treatments that are not a valid representation of the boundary conditions to the corresponding adjoint partial differential equations. The underlying problem is studied for an elementary 1D advection-diffusion problem first using a node-centred finite-volume discretisation. The defect of the boundary operators in the inconsistently defined discrete-adjoint problem leads to oscillations and becomes evident with the additional insight of the continuous-adjoint approach. A homogenisation of the discretisations for the primal boundary treatment and the force objective functional yields second-order functional accuracy and eliminates the defect in the discrete-adjoint boundary treatment. Subsequently, the issue is studied for aerodynamic Reynolds-averaged Navier-Stokes problems in conjunction with a standard finite-volume discretisation on median-dual grids and a strong implementation of noslip walls, found in many unstructured general-purpose flow solvers. Going out from a base-line discretisation of force objective functionals which is independent of the boundary treatment in the flow solver, two improved flux-consistent schemes are presented; based on either body wall-defined or farfield-defined control-volumes they resolve the dual inconsistency. The behaviour of the schemes is investigated on a sequence of grids in 2D and 3D.
NASA Astrophysics Data System (ADS)
Court, Sébastien; Fournié, Michel
2015-05-01
The paper extends a stabilized fictitious domain finite element method initially developed for the Stokes problem to the incompressible Navier-Stokes equations coupled with a moving solid. This method presents the advantage to predict an optimal approximation of the normal stress tensor at the interface. The dynamics of the solid is governed by the Newton's laws and the interface between the fluid and the structure is materialized by a level-set which cuts the elements of the mesh. An algorithm is proposed in order to treat the time evolution of the geometry and numerical results are presented on a classical benchmark of the motion of a disk falling in a channel.
Convergence Acceleration of Runge-Kutta Schemes for Solving the Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Swanson, Roy C., Jr.; Turkel, Eli; Rossow, C.-C.
2007-01-01
The convergence of a Runge-Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge-Kutta scheme, CFL numbers as high as 1000 can be used. The implicit preconditioner addresses the stiffness in the discrete equations associated with stretched meshes. This RK/implicit scheme is used as a smoother for multigrid. Fourier analysis is applied to determine damping properties. Numerical dissipation operators based on the Roe scheme, a matrix dissipation, and the CUSP scheme are considered in evaluating the RK/implicit scheme. In addition, the effect of the number of RK stages is examined. Both the numerical and computational efficiency of the scheme with the different dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier-Stokes equations. Turbulent flows over an airfoil and wing at subsonic and transonic conditions are computed. The effects of the cell aspect ratio on convergence are investigated for Reynolds numbers between 5:7 x 10(exp 6) and 100 x 10(exp 6). It is demonstrated that the implicit preconditioner can reduce the computational time of a well-tuned standard RK scheme by a factor between four and ten.
Navier-Stokes computation of compressible turbulent flows with a second order closure, part 1
NASA Technical Reports Server (NTRS)
Haminh, Hieu; Kollmann, Wolfgang; Vandromme, Dany
1990-01-01
A second order closure turbulence model for compressible flows is developed and implemented in a 2D Reynolds-averaged Navier-Stokes solver. From the beginning where a kappa-epsilon turbulence model was implemented in the bidiagonal implicit method of MACCORMACK (referred to as the MAC3 code) to the final stage of implementing a full second order closure in the efficient line Gauss-Seidel algorithm, numerous work was done, individually and collectively. Besides the collaboration itself, the final product of this work is a second order closure derived from the Launder, Reece, and Rodi model to account for near wall effects, which has been called FRAME model, which stands for FRench-AMerican-Effort. During the reporting period, two different problems were worked out. The first was to provide Ames researchers with a reliable compressible boundary layer code including a wide collection of turbulence models for quick testing of new terms, both in two equations and in second order closure (LRR and FRAME). The second topic was to complete the implementation of the FRAME model in the MAC5 code. The work related to these two different contributions is reported. dilatation in presence of stron shocks. This work, which has been conducted during a work at the Center for Turbulence Research with Zeman aimed also to cros-check earlier assumptions by Rubesin and Vandromme.
A variational level set method for the topology optimization of steady-state Navier Stokes flow
NASA Astrophysics Data System (ADS)
Zhou, Shiwei; Li, Qing
2008-12-01
The smoothness of topological interfaces often largely affects the fluid optimization and sometimes makes the density-based approaches, though well established in structural designs, inadequate. This paper presents a level-set method for topology optimization of steady-state Navier-Stokes flow subject to a specific fluid volume constraint. The solid-fluid interface is implicitly characterized by a zero-level contour of a higher-order scalar level set function and can be naturally transformed to other configurations as its host moves. A variational form of the cost function is constructed based upon the adjoint variable and Lagrangian multiplier techniques. To satisfy the volume constraint effectively, the Lagrangian multiplier derived from the first-order approximation of the cost function is amended by the bisection algorithm. The procedure allows evolving initial design to an optimal shape and/or topology by solving the Hamilton-Jacobi equation. Two classes of benchmarking examples are presented in this paper: (1) periodic microstructural material design for the maximum permeability; and (2) topology optimization of flow channels for minimizing energy dissipation. A number of 2D and 3D examples well demonstrated the feasibility and advantage of the level-set method in solving fluid-solid shape and topology optimization problems.
Navier-Stokes analysis of airfoils with leading edge ice accretions
NASA Technical Reports Server (NTRS)
Potapczuk, Mark G.
1993-01-01
A numerical analysis of the flowfield characteristics and the performance degradation of an airfoil with leading edge ice accretions was performed. The important fluid dynamic processes were identified and calculated. Among these were the leading edge separation bubble at low angles of attack, complete separation on the low pressure surface resulting in premature shell, drag rise due to the ice shape, and the effects of angle of attack on the separated flow field. Comparisons to experimental results were conducted to confirm these calculations. A computer code which solves the Navier-Stokes equations in two dimensions, ARC2D, was used to perform the calculations. A Modified Mixing Length turbulence model was developed to produce grids for several ice shape and airfoil combinations. Results indicate that the ability to predict overall performance characteristics, such as lift and drag, at low angles of attack is excellent. Transition location is important for accurately determining separation bubble shape. Details of the flowfield in and downstream of the separated regions requires some modifications. Calculations for the stalled airfoil indicate periodic shedding of vorticity that was generated aft of the ice accretion. Time averaged pressure values produce results which compare favorably with experimental information. A turbulence model which accounts for the history effects in the flow may be justified.
Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Flyer, Natasha; Barnett, Gregory A.; Wicker, Louis J.
2016-07-01
Polynomials are used together with polyharmonic spline (PHS) radial basis functions (RBFs) to create local RBF-finite-difference (RBF-FD) weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences (FD). The presented method replicates the convergence properties of FD but for arbitrary node layouts. It is tested on the 2D compressible Navier-Stokes equations at low Mach number, relevant to atmospheric flows. Test cases are taken from the numerical weather prediction community and solved on bounded domains. Thus, attention is given on how to handle boundaries with the RBF-FD method, as well as a novel implementation for hyperviscosity. Comparisons are done on Cartesian, hexagonal, and quasi-uniform node layouts. Consideration and guidelines are given on PHS order, polynomial degree and stencil size. The main advantages of the present method are: 1) capturing the basic physics of the problem surprisingly well, even at very coarse resolutions, 2) high-order accuracy without the need of tuning a shape parameter, and 3) the inclusion of polynomials eliminates stagnation (saturation) errors. A MATLAB code is given to calculate the differentiation weights for this novel approach.
Marsden, O; Bogey, C; Bailly, C
2014-03-01
The feasibility of using numerical simulation of fluid dynamics equations for the detailed description of long-range infrasound propagation in the atmosphere is investigated. The two dimensional (2D) Navier Stokes equations are solved via high fidelity spatial finite differences and Runge-Kutta time integration, coupled with a shock-capturing filter procedure allowing large amplitudes to be studied. The accuracy of acoustic prediction over long distances with this approach is first assessed in the linear regime thanks to two test cases featuring an acoustic source placed above a reflective ground in a homogeneous and weakly inhomogeneous medium, solved for a range of grid resolutions. An atmospheric model which can account for realistic features affecting acoustic propagation is then described. A 2D study of the effect of source amplitude on signals recorded at ground level at varying distances from the source is carried out. Modifications both in terms of waveforms and arrival times are described. PMID:24606252
NASA Technical Reports Server (NTRS)
Swanson, R. Charles; Radespiel, Rolf; Mccormick, V. Edward
1989-01-01
The two-dimensional (2-D) and three-dimensional Navier-Stokes equations are solved for flow over a NAE CAST-10 airfoil model. Recently developed finite-volume codes that apply a multistage time stepping scheme in conjunction with steady state acceleration techniques are used to solve the equations. Two-dimensional results are shown for flow conditions uncorrected and corrected for wind tunnel wall interference effects. Predicted surface pressures from 3-D simulations are compared with those from 2-D calculations. The focus of the 3-D computations is the influence of the sidewall boundary layers. Topological features of the 3-D flow fields are indicated. Lift and drag results are compared with experimental measurements.
Hybridizable discontinuous Galerkin projection methods for Navier-Stokes and Boussinesq equations
NASA Astrophysics Data System (ADS)
Ueckermann, M. P.; Lermusiaux, P. F. J.
2016-02-01
Schemes for the incompressible Navier-Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit-Explicit Runge-Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed-element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.
Fast solvers for finite difference approximations for the Stokes and Navier-Stokes equations
Shin, D.
1992-01-01
The authors consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The pressure equation method presented here for the first time, apparently, and the method, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes and incompressible Navier-Stokes at low Reynolds number. The inf-sup conditions resulting from three finite difference approximations of the Stokes equations are proven. These conditions are used to prove that the Schur complement Q[sub h] of the linear system generated by each of these approximations is bounded uniformly away from zero. For the pressure equation method, this guarantees that the conjugate gradient method applied to Q[sub h] converges in a finite number of iterations which is independent of mesh size. The fact that Q[sub h] is bounded below is used to prove convergence estimates for the solutions generated by these finite difference approximations. One of the estimates is for a staggered grid and the estimate of the scheme shows that both the pressure and the velocity parts of the solution are second-order accurate. Iterative methods are compared by the use of the regularized central differencing introduced by Strikwerda. Several finite difference approximations of the Stokes equations by the SOR method are compared and the excellence of the approximations by the regularized central differencing over the other finite difference approximation is mentioned. This difference gives rise to a linear equation with a matrix which is slightly non-symmetric. The convergence of the typical steepest descent method and conjugate gradient method, which is almost as same as the typical conjugate gradient method, applied to slightly non-symmetric positive definite matrices are proven.
A gas-kinetic BGK scheme for the compressible Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Xu, Kun
2000-01-01
This paper presents an improved gas-kinetic scheme based on the Bhatnagar-Gross-Krook (BGK) model for the compressible Navier-Stokes equations. The current method extends the previous gas-kinetic Navier-Stokes solver developed by Xu and Prendergast by implementing a general nonequilibrium state to represent the gas distribution function at the beginning of each time step. As a result, the requirement in the previous scheme, such as the particle collision time being less than the time step for the validity of the BGK Navier-Stokes solution, is removed. Therefore, the applicable regime of the current method is much enlarged and the Navier-Stokes solution can be obtained accurately regardless of the ratio between the collision time and the time step. The gas-kinetic Navier-Stokes solver developed by Chou and Baganoff is the limiting case of the current method, and it is valid only under such a limiting condition. Also, in this paper, the appropriate implementation of boundary condition for the kinetic scheme, different kinetic limiting cases, and the Prandtl number fix are presented. The connection among artificial dissipative central schemes, Godunov-type schemes, and the gas-kinetic BGK method is discussed. Many numerical tests are included to validate the current method.
NASA Technical Reports Server (NTRS)
Deese, J. E.; Agarwal, R. K.
1989-01-01
Computational fluid dynamics has an increasingly important role in the design and analysis of aircraft as computer hardware becomes faster and algorithms become more efficient. Progress is being made in two directions: more complex and realistic configurations are being treated and algorithms based on higher approximations to the complete Navier-Stokes equations are being developed. The literature indicates that linear panel methods can model detailed, realistic aircraft geometries in flow regimes where this approximation is valid. As algorithms including higher approximations to the Navier-Stokes equations are developed, computer resource requirements increase rapidly. Generation of suitable grids become more difficult and the number of grid points required to resolve flow features of interest increases. Recently, the development of large vector computers has enabled researchers to attempt more complex geometries with Euler and Navier-Stokes algorithms. The results of calculations for transonic flow about a typical transport and fighter wing-body configuration using thin layer Navier-Stokes equations are described along with flow about helicopter rotor blades using both Euler/Navier-Stokes equations.
NASA Technical Reports Server (NTRS)
Gabrielsen, R. E.; Karel, S.
1975-01-01
An algorithm for solving the nonlinear stationary Navier-Stokes problem is developed. Explicit error estimates are given. This mathematical technique is potentially adaptable to the separation problem.
Navier-Stokes Analysis of the Flowfield Characteristics of an Ice Contaminated Aircraft Wing
NASA Technical Reports Server (NTRS)
Chung, J.; Choo, Y.; Reehorst, A.; Potapczuk, M.; Slater, J.
1999-01-01
An analytical study was performed as part of the NASA Lewis support of a National Transportation Safety Board (NTSB) aircraft accident investigation. The study was focused on the performance degradation associated with ice contamination on the wing of a commercial turbo-prop-powered aircraft. Based upon the results of an earlier numerical study conducted by the authors, a prominent ridged-ice formation on the subject aircraft wing was selected for detailed flow analysis using 2-dimensional (2-D), as well as, 3-dimensional (3-D) Navier-Stokes computations. This configuration was selected because it caused the largest lift decrease and drag increase among all the ice shapes investigated in the earlier study. A grid sensitivity test was performed to find out the influence of grid spacing on the lift, drag, and associated angle-of-attack for the maximum lift (C(sub lmax)). This study showed that grid resolution is important and a sensitivity analysis is an essential element of the process in order to assure that the final solution is independent of the grid. The 2-D results suggested that a severe stability and control difficulty could have occurred at a slightly higher angle-of-attack (AOA) than the one recorded by the Flight Data Recorder (FDR). This stability and control problem was thought to have resulted from a decreased differential lift on the wings with respect to the normal loading for the configuration. The analysis also indicated that this stability and control problem could have occurred whether or not natural ice shedding took place. Numerical results using an assumed 3-D ice shape showed an increase of the angle at which this phenomena occurred of about 4 degrees. As it occurred with the 2-D case, the trailing edge separation was observed but started only when the AOA was very close to the angle at which the maximum lift occurred.
Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in R
NASA Astrophysics Data System (ADS)
Zhang, Guojing; Li, Hai-Liang; Zhu, Changjiang
In this paper, the compressible non-isentropic Navier-Stokes-Poisson (NSP) system is considered in R and the influences of internal electric field on the qualitative behaviors of solutions are analyzed. We observe that the electric field leads to the rotating phenomena in charge transport and reduces the speed of fluid motion, but it does not influence the transport of charge density and the heat diffusion. Indeed, we show that both density and temperature of the NSP system converge to their equilibrium state at the same rate (1 as the non-isentropic compressible Navier-Stokes system, but the momentum decays at the rate (1, which is slower than the rate (1 for the pure compressible Navier-Stokes system. These convergence rates are also shown to be optimal for the non-isentropic compressible NSP system.
Existence and Regularity of the Pressure for the Stochastic Navier-Stokes Equations
Langa, Jose A. Real, Jose Simon, Jacques
2003-10-15
We prove, on one hand, that for a convenient body force with value sin the distribution space (H{sup -1}(D)){sup d}, where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier-Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V' of the divergence free subspace V of (H{sup 1}{sub 0}(D)){sup d},in general it is not possible to solve the stochastic Navier-Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier-Stokes equations could be meaningful for them.
Partially-Averaged Navier Stokes Model for Turbulence: Implementation and Validation
NASA Technical Reports Server (NTRS)
Girimaji, Sharath S.; Abdol-Hamid, Khaled S.
2005-01-01
Partially-averaged Navier Stokes (PANS) is a suite of turbulence closure models of various modeled-to-resolved scale ratios ranging from Reynolds-averaged Navier Stokes (RANS) to Navier-Stokes (direct numerical simulations). The objective of PANS, like hybrid models, is to resolve large scale structures at reasonable computational expense. The modeled-to-resolved scale ratio or the level of physical resolution in PANS is quantified by two parameters: the unresolved-to-total ratios of kinetic energy (f(sub k)) and dissipation (f(sub epsilon)). The unresolved-scale stress is modeled with the Boussinesq approximation and modeled transport equations are solved for the unresolved kinetic energy and dissipation. In this paper, we first present a brief discussion of the PANS philosophy followed by a description of the implementation procedure and finally perform preliminary evaluation in benchmark problems.
High speed transport cruise drag. [scaling laws using Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Roberts, Leonard
1992-01-01
This report provides scaling laws for the cruise aerodynamics of high speed transport wings based on the results of Navier-Stokes computations. Expressions for the various drag components are found, together with the corresponding values (L/D)(sub m) for various values of the geometric parameter s/l which allow for simple optimization of the wing configurations with respect to the span. It is found that linear theory expressions can be used for this purpose provided the coefficients of these experiments for C(sub D) and (L/D)(sub m) are available using Navier-Stokes results.
Numerical solution of 3D Navier-Stokes equations with upwind implicit schemes
NASA Technical Reports Server (NTRS)
Marx, Yves P.
1990-01-01
An upwind MUSCL type implicit scheme for the three-dimensional Navier-Stokes equations is presented. Comparison between different approximate Riemann solvers (Roe and Osher) are performed and the influence of the reconstructions schemes on the accuracy of the solution as well as on the convergence of the method is studied. A new limiter is introduced in order to remove the problems usually associated with non-linear upwind schemes. The implementation of a diagonal upwind implicit operator for the three-dimensional Navier-Stokes equations is also discussed. Finally the turbulence modeling is assessed. Good prediction of separated flows are demonstrated if a non-equilibrium turbulence model is used.
Cao, Yong; Chu, Yuchuan; He, Xiaoming; Wei, Mingzhen
2013-01-01
This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method.
An Exact Mapping from Navier-Stokes Equation to Schr"odinger Equation via Riccati Equation
NASA Astrophysics Data System (ADS)
Christianto, Vic; Smarandache, Florentin
2010-03-01
In the present article we argue that it is possible to write down Schr"odinger representation of Navier-Stokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R. M. Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of Navier-Stokes equation, for instance via Kravchenko's and Gibbon's route. Further observation is of course recommended in order to refute or verify this proposition.
An adaptive-mesh finite-difference solution method for the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Luchini, Paolo
1987-02-01
An adjustable variable-spacing grid is presented which permits the addition or deletion of single points during iterative solutions of the Navier-Stokes equations by finite difference methods. The grid is designed for application to two-dimensional steady-flow problems which can be described by partial differential equations whose second derivatives are constrained to the Laplacian operator. An explicit Navier-Stokes equations solution technique defined for use with the grid incorporates a hybrid form of the convective terms. Three methods are developed for automatic modifications of the mesh during calculations.
Weighted bounds for the velocity and the vorticity for the Navier Stokes equations
NASA Astrophysics Data System (ADS)
Kukavica, Igor; Torres, J. J.
2006-02-01
We study decay in space and time for derivatives of solutions of the Navier-Stokes equations in {\\Bbb R}^{n} . The main results concern the ranges of exponents b, c and d for validity of the bounds \\[ \\begin{equation*}\\Vert |x|^{b}\\partial_\\alpha u\\Vert_{L^p} \\le \\frac{C} {(1+t)^{c}}\\end{equation*} \\] and \\[ \\begin{equation*}\\Vert |x|^{b}\\partial_\\alpha \\omega\\Vert_{L^p} \\le \\frac{C}{(1+t)^{d}},\\end{equation*} \\] where u is a solution of the Navier-Stokes equation and ω is its vorticity.
Navier-Stokes Neutral and Plasma Fluid Modelling in 3D
Riemann, J; Borchardt, M; Schneider, R; Mutzke, A; Rognlien, T; Umansky, M
2004-05-17
The 3D finite volume transport code BoRiS is applied to a system of coupled plasma and neutral fluid equations in a slab. Demonstrating easy implementation of new equations, a new parallel BoRiS version is tested on three different models for the neutral fluid - diffusive, parallel Navier-Stokes and full Navier-Stokes - and the results are compared to each other. Typical effects like density enhancement by ionization of recycled neutrals in front of a target plate can be seen and differences are linked to the neutral models in use.
A stable penalty method for the compressible Navier-Stokes equations. 1: Open boundary conditions
NASA Technical Reports Server (NTRS)
Hesthaven, J. S.; Gottlieb, D.
1994-01-01
The purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible Navier-Stokes equations in three spatial dimensions. The treatment uses the conservation form of the Navier-Stokes equations and utilizes linearization and localization at the boundaries based on these variables. The proposed boundary conditions are applied through a penalty procedure, thus ensuring correct behavior of the scheme as the Reynolds number tends to infinity. The versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder.
NASA Astrophysics Data System (ADS)
Spyropoulos, John T.
This thesis extends earlier research in numerical analysis and computational fluid dynamics (CFD) to obtain a novel finite element method for the transient, 3-D, incompressible Navier-Stokes equations, along with efficient, parallelizable algorithms to carry out an implementation of the method in such a fashion as to be useful in mainstream industrial settings. This new finite element procedure employs alternating-direction operator splittings to model problems of increasing complexity in a step-by-step and natural manner. The scheme employs a characteristic-Galerkin method for the numerical treatment of the nonlinear advection operator. Non-overlapping domain decomposition schemes are employed for the solution of linear Stokes-type subproblems and for the matching of the inviscid and viscous solutions in different subdomains. These problems are solved by Bramble-Pasciak-Schatz wirebasket domain decomposition methods in a stabilized mixed finite element method formulation. The scheme is coupled to an existing grid generator code that provides globally unstructured, but locally structured grids, within each subdomain. Numerical results obtained include incompressible viscous flows over a backward facing steps at various Reynolds numbers and show very good to excellent agreement with experiments as well as other published numerical results.
An Explicit Upwind Algorithm for Solving the Parabolized Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Korte, John J.
1991-01-01
An explicit, upwind algorithm was developed for the direct (noniterative) integration of the 3-D Parabolized Navier-Stokes (PNS) equations in a generalized coordinate system. The new algorithm uses upwind approximations of the numerical fluxes for the pressure and convection terms obtained by combining flux difference splittings (FDS) formed from the solution of an approximate Riemann (RP). The approximate RP is solved using an extension of the method developed by Roe for steady supersonic flow of an ideal gas. Roe's method is extended for use with the 3-D PNS equations expressed in generalized coordinates and to include Vigneron's technique of splitting the streamwise pressure gradient. The difficulty associated with applying Roe's scheme in the subsonic region is overcome. The second-order upwind differencing of the flux derivatives are obtained by adding FDS to either an original forward or backward differencing of the flux derivative. This approach is used to modify an explicit MacCormack differencing scheme into an upwind differencing scheme. The second order upwind flux approximations, applied with flux limiters, provide a method for numerically capturing shocks without the need for additional artificial damping terms which require adjustment by the user. In addition, a cubic equation is derived for determining Vegneron's pressure splitting coefficient using the updated streamwise flux vector. Decoding the streamwise flux vector with the updated value of Vigneron's pressure splitting improves the stability of the scheme. The new algorithm is applied to 2-D and 3-D supersonic and hypersonic laminar flow test cases. Results are presented for the experimental studies of Holden and of Tracy. In addition, a flow field solution is presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of 1 degree. The computed results compare well to both experimental data and numerical results from other algorithms. Computational times required
Applications of the contravariant form of the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Katsanis, T.
1983-01-01
The contravariant Navier-Stokes equations in weak conservation form are well suited to certain fluid flow analysis problems. Three dimensional contravariant momentum equations may be used to obtain Navier-Stokes equations in weak conservation form on a nonplanar two dimensional surface with varying streamsheet thickness. Thus a three dimensional flow can be simulated with two dimensional equations to obtain a quasi-three dimensional solution for viscous flow. When the Navier-Stokes equations on the two dimensional nonplanar surface are transformed to a generalized body fitted mesh coordinate system, the resulting equations are similar to the equations for a body fitted mesh coordinate system on the Euclidean plane. Contravariant momentum components are also useful for analyzing compressible, three dimensional viscous flow through an internal duct by parabolic marching. This type of flow is efficiently analyzed by parabolic marching methods, where the streamwise momentum equation is uncoupled from the two crossflow momentum equations. This can be done, even for ducts with a large amount of turning, if the Navier-Stokes equations are written with contravariant components.
NASA Technical Reports Server (NTRS)
Biedron, Robert T.; Vatsa, Veer N.; Atkins, Harold L.
2005-01-01
We apply an unsteady Reynolds-averaged Navier-Stokes (URANS) solver for unstructured grids to unsteady flows on moving and stationary grids. Example problems considered are relevant to active flow control and stability and control. Computational results are presented using the Spalart-Allmaras turbulence model and are compared to experimental data. The effect of grid and time-step refinement are examined.
Solving Navier-Stokes' equation using Castillo-Grone's mimetic difference operators on GPUs
NASA Astrophysics Data System (ADS)
Abouali, Mohammad; Castillo, Jose
2012-11-01
This paper discusses the performance and the accuracy of Castillo-Grone's (CG) mimetic difference operator in solving the Navier-Stokes' equation in order to simulate oceanic and atmospheric flows. The implementation is further adapted to harness the power of the many computing cores available on the Graphics Processing Units (GPUs) and the speedup is discussed.
Properties of the Residual Stress of the Temporally Filtered Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Pruett, C. D.; Gatski, T. B.; Grosch, C. E.; Thacker, W. D.
2002-01-01
The development of a unifying framework among direct numerical simulations, large-eddy simulations, and statistically averaged formulations of the Navier-Stokes equations, is of current interest. Toward that goal, the properties of the residual (subgrid-scale) stress of the temporally filtered Navier-Stokes equations are carefully examined. Causal time-domain filters, parameterized by a temporal filter width 0 less than Delta less than infinity, are considered. For several reasons, the differential forms of such filters are preferred to their corresponding integral forms; among these, storage requirements for differential forms are typically much less than for integral forms and, for some filters, are independent of Delta. The behavior of the residual stress in the limits of both vanishing and in infinite filter widths is examined. It is shown analytically that, in the limit Delta to 0, the residual stress vanishes, in which case the Navier-Stokes equations are recovered from the temporally filtered equations. Alternately, in the limit Delta to infinity, the residual stress is equivalent to the long-time averaged stress, and the Reynolds-averaged Navier-Stokes equations are recovered from the temporally filtered equations. The predicted behavior at the asymptotic limits of filter width is further validated by numerical simulations of the temporally filtered forced, viscous Burger's equation. Finally, finite filter widths are also considered, and a priori analyses of temporal similarity and temporal approximate deconvolution models of the residual stress are conducted.
Remark on boundary data for inverse boundary value problems for the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Imanuvilov, O. Yu; Yamamoto, M.
2015-10-01
In this note, we prove that for the Navier-Stokes equations, a pair of Dirichlet and Neumann data and pressure uniquely correspond to a pair of Dirichlet data and surface stress on the boundary. Hence the two inverse boundary value problems in Imanuvilov and Yamamoto (2015 Inverse Probl. 31 035004) and Lai et al (Arch. Rational Mech. Anal.) are the same.
Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces
Padmanabhan, T.
2011-02-15
It has been known for several decades that Einstein's field equations, when projected onto a null surface, exhibit a structure very similar to the nonrelativistic Navier-Stokes equation. I show that this result arises quite naturally when gravitational dynamics is viewed as an emergent phenomenon. Extremizing the spacetime entropy density associated with the null surfaces leads to a set of equations which, when viewed in the local inertial frame, becomes identical to the Navier-Stokes equation. This is in contrast to the usual description of the Damour-Navier-Stokes equation in a general coordinate system, in which there appears a Lie derivative rather than a convective derivative. I discuss this difference, its importance, and why it is more appropriate to view the equation in a local inertial frame. The viscous force on fluid, arising from the gradient of the viscous stress-tensor, involves the second derivatives of the metric and does not vanish in the local inertial frame, while the viscous stress-tensor itself vanishes so that inertial observers detect no dissipation. We thus provide an entropy extremization principle that leads to the Damour-Navier-Stokes equation, which makes the hydrodynamical analogy with gravity completely natural and obvious. Several implications of these results are discussed.
Hong Luo; Luqing Luo; Robert Nourgaliev; Vincent A. Mousseau
2010-01-01
A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. The RDG method, originally developed for the compressible Euler equations, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations using a so-called inter-cell reconstruction, where a smooth solution is locally reconstructed using a least-squares method from the underlying discontinuous DG solution. Similar to the recovery-based DG (rDG) methods, this reconstructed DG method eliminates the introduction of ad hoc penalty or coupling terms commonly found in traditional DG methods. Unlike rDG methods, this RDG method does not need to judiciously choose a proper form of a recovered polynomial, thus is simple, flexible, and robust, and can be used on arbitrary grids. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is able to deliver the same accuracy as the well-known Bassi-Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier-Stokes equations, clearly demonstrating its superior performance over the existing DG methods for solving the compressible Navier-Stokes equations.
An implicit flux-split algorithm for the compressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Thomas, James L.; Rumsey, Christopher L.; Walters, Robert W.; van Leer, Bram
An implicit upwind scheme for the compressible Navier-Stokes equations is described and applied to the internal flow in a dual-throat nozzle. The method is second-order accurate spatially and naturally dissipative. A spatially-split approximate factorization method is used to obtain efficient steady-state solutions on the NASA Langley VPS-32 (CYBER 205) supercomputer.
Turbomachinery Heat Transfer and Loss Modeling for 3D Navier-Stokes Codes
NASA Technical Reports Server (NTRS)
DeWitt, Kenneth; Ameri, Ali
2005-01-01
This report's contents focus on making use of NASA Glenn on-site computational facilities,to develop, validate, and apply models for use in advanced 3D Navier-Stokes Computational Fluid Dynamics (CFD) codes to enhance the capability to compute heat transfer and losses in turbomachiney.
Comparing nearly singular vorticity moments in Euler and Navier-Stokes numerical solutions
NASA Astrophysics Data System (ADS)
Kerr, Robert M.
2013-11-01
The inviscid growth of a range of vorticity moments in Navier-Stokes and Euler calculations are compared for simulations using a new anti-parallel initial condition. One goal is to understand the origins of a new hierarchy of rescaled vorticity moments in several Navier-Stokes calculations where the rescaled moments obey Dm >=Dm + 1 , the reverse of the usual Ωm + 1 >=Ωm Hölder ordering. Two temporal phases have been identified for the Euler calculations. In the first phase the 1 < m < ∞ vorticity moments are ordered as Dm >=Dm + 1 , as in the Navier-Stokes case and grow in a manner that skirts possible singular growth with Dm2 --> sup | ω | ~Am (Tc - t)-1 with the Am are nearly independent of m . In the second phase, the new Dm ordering breaks down and the Ωm converge towards super-exponential growth for all m , shown by plotting log (dlogΩm / dt) . The transition is identified using new inequalities for the upper bounds for the -dDm- 2 / dt . The Navier-Stokes solutions while showing less growth in the Dm , surprisingly obey Dm >=Dm + 1 for all times.
A Navier-Stokes solver using the LU-SSOR TVD algorithm
NASA Technical Reports Server (NTRS)
Yoon, Seokkwan
1987-01-01
A new Navier-Stokes solver is developed by combining the efficiency of the LU-SSOR scheme and the accuracy of the flux-limited dissipation scheme. Application to laminar and turbulent flows and hypersonic flows proves the reliability of the new algorithm.
NASA Astrophysics Data System (ADS)
Asgharzadeh, Hafez; Borazjani, Iman
2014-11-01
Time step-size restrictions and low convergence rates are major bottle necks for implicit solution of the Navier-Stokes in simulations involving complex geometries with moving boundaries. Newton-Krylov method (NKM) is a combination of a Newton-type method for super-linearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations, which can theoretically address both bottle necks. The efficiency of this method vastly depends on the Jacobian forming scheme e.g. automatic differentiation is very expensive and Jacobian-free methods slow down as the mesh is refined. A novel, computationally efficient analytical Jacobian for NKM was developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered curvilinear grids with immersed boundaries. The NKM was validated and verified against Taylor-Green vortex and pulsatile flow in a 90 degree bend and efficiently handles complex geometries such as an intracranial aneurysm with multiple overset grids, pulsatile inlet flow and immersed boundaries. The NKM method is shown to be more efficient than the semi-implicit Runge-Kutta methods and Jabobian-free Newton-Krylov methods. We believe NKM can be applied to many CFD techniques to decrease the computational cost. This work was supported partly by the NIH Grant R03EB014860, and the computational resources were partly provided by Center for Computational Research (CCR) at University at Buffalo.
NASA Technical Reports Server (NTRS)
Liu, N. S.; Shamroth, S. J.; Mcdonald, H.
1983-01-01
The multidimensional ensemble averaged compressible time dependent Navier Stokes equations in conjunction with mixing length turbulence model and shock capturing technique were used to study the terminal shock type of flows in various flight regimes occurring in a diffuser/inlet model. The numerical scheme for solving the governing equations is based on a linearized block implicit approach and the following high Reynolds number calculations were carried out: (1) 2 D, steady, subsonic; (2) 2 D, steady, transonic with normal shock; (3) 2 D, steady, supersonic with terminal shock; (4) 2 D, transient process of shock development and (5) 3 D, steady, transonic with normal shock. The numerical results obtained for the 2 D and 3 D transonic shocked flows were compared with corresponding experimental data; the calculated wall static pressure distributions agree well with the measured data.
Post-stall Navier-Stokes computations of the NREL airfoil using a {kappa}-{omega} turbulence model
Yang, S.L.; Chang, Y.L.; Arici, O.
1995-09-01
This paper presents a two-dimensional numerical simulation of the turbulent flow fields for the NREL (National Renewable Energy Laboratory) S809 airfoil in the post-stall region. The flow is modeled as steady, viscous, turbulent, and incompressible. The pseudo-compressible formulation is used for the time-averaged Navier-Stokes equations so that a time marching scheme developed for the compressible flow can be applied directly. The turbulent flow is simulated using Wilcox`s modified {kappa}-{omega} model to account for the low-Reynolds number effects near solid wall and the model`s sensitivity to the free stream conditions. The governing equations are solved by an implicit approximate-factorization scheme. To correctly model the convection terms in the mean-flow and turbulence model equations, the symmetric TVD (Total Variational Diminishing) scheme is incorporated. The methodology developed is then applied to analyze the NREL S809 airfoil at various angles of attack ({alpha}) from 1 to 45 degrees. The accuracy of the numerical results is compared with the available Delft wind tunnel test data. For comparison, two Eppler code results at low angles of attack are also included. Depending on the value of {alpha}, preliminary results show excellent to fairly good agreement with the experimental data. Directions for future work are also included.
Yang, S.L.; Chang, Y.L.; Arici, O.
1995-11-01
This paper presents a two-dimensional numerical simulation of the turbulent flow fields for the NREL (National Renewable Energy Laboratory) S809 airfoil. The flow is modeled as steady, viscous, turbulent, and incompressible. The pseudo-compressible formulation is used for the time-averaged Navier-Stokes equations so that a time marching scheme developed for the compressible flow can be applied directly. The turbulent flow is simulated using Wilcox`s modified {kappa}-{omega} model to account for the low Reynolds number effects near a solid wall and the model`s sensitivity to the freestream conditions. The governing equations are solved by an implicit approximate-factorization scheme. To correctly model the convection terms in the mean-flow and turbulence model equations, the symmetric TVD (Total Variational Diminishing) scheme is incorporated. The methodology developed is then applied to analyze the NREL S809 airfoil at various angles of attack ({alpha}) from 1 to 45 degrees. The accuracy of the numerical results is compared with the available Delft wind tunnel test data. For comparison, two Eppler code results at low angles of attack are also included. Depending on the value of {alpha}, preliminary results show excellent to fairly good agreement with the experimental data. Directions for future work are also discussed.
NASA Astrophysics Data System (ADS)
Avrin, Joel
In the hyper-viscous Navier-Stokes equations of incompressible flow, the operator A=- Δ is replaced by Aα, a, b≡ aAα+ bA for real numbers α, a, b with α⩾1 and b⩾0. We treat here the case a>0 and equip A (and hence Aα, a, b) with periodic boundary conditions over a rectangular solid Ω⊂R n. For initial data in L p(Ω) with α⩾ n/(2 p)+1/2 we establish local existence and uniqueness of strong solutions, generalizing a result of Giga/Miyakawa for α=1 and b=0. Specializing to the case p=2, which holds a particular physical relevance in terms of the total energy of the system, it is somewhat interesting to note that the condition α⩾ n/4+1/2 is sufficient also to establish global existence of these unique regular solutions and uniform higher-order bounds. For the borderline case α= n/4+1/2 we generalize standard existing (for n=3) "folklore" results and use energy techniques and Gronwall's inequality to obtain first a time-dependent Hα-bound, and then convert to a time-independent global exponential Hα-bound. This is to be expected, given that uniform bounds already exist for n=2, α=1 ([6, pp. 78-79]), and the folklore bounds already suggest that the α⩾ n/4+1/2 cases for n⩾3 should behave as well as the n=2 case. What is slightly less expected is that the n⩾3 cases are easier to prove and give better bounds, e.g. the uniform bound for n⩾3 depends on the square of the data in the exponential rather than the fourth power for n=2. More significantly, for α> n/4+1/2 we use our own entirely semigroup techniques to obtain uniform global bounds which bootstrap directly from the uniform L2-estimate and are algebraic in terms of the uniform L2-bounds on the initial and forcing data. The integer powers on the square of the data increase without bound as α↓ n/4+1/2, thus "anticipating" the exponential bound in the borderline case α= n/4+1/2. We prove our results for the case a=1 and b=0; the general case with a>0 and b⩾0 can be recovered by
Bypass Transitional Flow Calculations Using a Navier-Stokes Solver and Two-Equation Models
NASA Technical Reports Server (NTRS)
Liuo, William W.; Shih, Tsan-Hsing; Povinelli, L. A. (Technical Monitor)
2000-01-01
Bypass transitional flows over a flat plate were simulated using a Navier-Stokes solver and two equation models. A new model for the bypass transition, which occurs in cases with high free stream turbulence intensity (TI), is described. The new transition model is developed by including an intermittency correction function to an existing two-equation turbulence model. The advantages of using Navier-Stokes equations, as opposed to boundary-layer equations, in bypass transition simulations are also illustrated. The results for two test flows over a flat plate with different levels of free stream turbulence intensity are reported. Comparisons with the experimental measurements show that the new model can capture very well both the onset and the length of bypass transition.
Falling paper: Navier-Stokes solutions, model of fluid forces, and center of mass elevation.
Pesavento, Umberto; Wang, Z Jane
2004-10-01
We investigate the problem of falling paper by solving the two dimensional Navier-Stokes equations subject to the motion of a free-falling body at Reynolds numbers around 10(3). The aerodynamic lift on a tumbling plate is found to be dominated by the product of linear and angular velocities rather than velocity squared, as appropriate for an airfoil. This coupling between translation and rotation provides a mechanism for a brief elevation of center of mass near the cusplike turning points. The Navier-Stokes solutions further provide the missing quantity in the classical theory of lift, the instantaneous circulation, and suggest a revised model for the fluid forces. PMID:15524800
The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Choi, Young-Pil; Kwon, Bongsuk
2016-07-01
We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in the hydrodynamic limit, from the particle-fluid equations that are frequently used to study the medical sprays, aerosols and sedimentation problems. For the proposed system, we first construct the local-in-time classical solutions in an appropriate L2 Sobolev space. We also establish the a priori large-time behavior estimate by constructing a Lyapunov functional measuring the fluctuation of momentum and mass from the averaged quantities, and using this together with the bootstrapping argument, we obtain the global classical solution. The large-time behavior estimate asserts that the velocity functions of the pressureless Euler and the compressible Navier-Stokes equations are aligned exponentially fast as time tends to infinity.
Programming the Navier-Stokes computer: An abstract machine model and a visual editor
NASA Technical Reports Server (NTRS)
Middleton, David; Crockett, Tom; Tomboulian, Sherry
1988-01-01
The Navier-Stokes computer is a parallel computer designed to solve Computational Fluid Dynamics problems. Each processor contains several floating point units which can be configured under program control to implement a vector pipeline with several inputs and outputs. Since the development of an effective compiler for this computer appears to be very difficult, machine level programming seems necessary and support tools for this process have been studied. These support tools are organized into a graphical program editor. A programming process is described by which appropriate computations may be efficiently implemented on the Navier-Stokes computer. The graphical editor would support this programming process, verifying various programmer choices for correctness and deducing values such as pipeline delays and network configurations. Step by step details are provided and demonstrated with two example programs.
A comparison of numerical flux formulas for the Euler and Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Van Leer, Bram; Thomas, James L.; Roe, Philip L.; Newsome, Richard W.
1987-01-01
Numerical flux formulas for the convection terms in the Euler or Navier-Stokes equations are analyzed with regard to their accuracy in representing steady nonlinear and linear waves (shocks and entropy/shear waves, respectively). Numerical results are obtained for a one-dimensional conical Navier-Stokes flow including both a shock and a boundary layer. Analysis and experiments indicate that for an accurate representation of both layers the flux formula must include information about all different waves by which neighboring cells interact, as in Roe's flux-difference splitting. In comparison, Van Leer's flux-vector splitting, which ignores the linear waves, badly diffuses the boundary layer. The results of MacCormack's scheme, if properly tuned, are significantly better. The use of a sufficiently detailed flux formula appears to reduce the number of cells required to resolve a boundary layer by a factor 1/2 to 1/4 and thus pays off.
Transonic Navier-Stokes solutions of three-dimensional afterbody flows
NASA Technical Reports Server (NTRS)
Compton, William B., III; Thomas, James L.; Abeyounis, William K.; Mason, Mary L.
1989-01-01
The performance of a three-dimensional Navier-Stokes solution technique in predicting the transonic flow past a nonaxisymmetric nozzle was investigated. The investigation was conducted at free-stream Mach numbers ranging from 0.60 to 0.94 and an angle of attack of 0 degrees. The numerical solution procedure employs the three-dimensional, unsteady, Reynolds-averaged Navier-Stokes equations written in strong conservation form, a thin layer assumption, and the Baldwin-Lomax turbulence model. The equations are solved by using the finite-volume principle in conjunction with an approximately factored upwind-biased numerical algorithm. In the numerical procedure, the jet exhaust is represented by a solid sting. Wind-tunnel data with the jet exhaust simulated by high pressure air were also obtained to compare with the numerical calculations.
High-order ENO methods for the unsteady compressible Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Atkins, H. L.
1991-01-01
The adaptive stencil concepts of ENO (Essentially Non-Oscillatory) methods are applied to the laminar Navier-Stokes equations to yield a high-order, time-accurate algorithm with a shock-capturing capability. The method targets problems in the areas of nonlinear acoustics, compressible transition, and turbulence which, due to the presence of shocks or complex geometries, are not easily solved by spectral methods. The present approach has been implemented and tested for the full three-dimensional Navier-Stokes equations in a transformed curvilinear coordinate system. Validation results are presented for a variety of problems which verify the method's accuracy properties and shock capturing capabilities, as well as demonstrate its use as a direct simulation tool.
NASA Technical Reports Server (NTRS)
Chen, S. C.; Liu, N. S.; Kim, H. D.
1992-01-01
An algorithm utilizing a first order upwind split flux technique and the diagonally dominant treatment is proposed to be the temporal operator for solving the Navier-Stokes equations. Given the limit of a five point stencil, the right hand side flux derivatives are formulated by several commonly used central and upwind schemes. Their performances are studied through a test case of free vortex convection in a uniform stream. From these results, a superior treatment for evaluating the flux term is proposed and compared with the rest. The application of the proposed algorithm to the full Navier-Stokes equations is demonstrated through a calculation of flow over a backward facing step. Results are compared against the calculation done by using the fourth order central differencing scheme with artificial damping.
Large-time behavior for the Vlasov/compressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Choi, Young-Pil
2016-07-01
We establish the large-time behavior for the coupled kinetic-fluid equations. More precisely, we consider the Vlasov equation coupled to the compressible isentropic Navier-Stokes equations through a drag forcing term. For this system, the large-time behavior shows the exponential alignment between particles and fluid velocities as time evolves. This improves the previous result by Bae et al. [Discrete Contin. Dyn. Syst. 34, 4419-4458 (2014)] in which they considered the Vlasov/Navier-Stokes equations with nonlocal velocity alignment forces for particles. Employing a new Lyapunov functional measuring the fluctuations of momentum and mass from the averaged quantities, we refine assumptions for the large-time behavior of the solutions to that system.
Thickening oscillation of a delta wing using Navier-Stokes and Navier-displacement equations
NASA Technical Reports Server (NTRS)
Chuang, Hsin-Kung A.; Kandil, Osama A.
1989-01-01
The problem of unsteady, supersonic, locally conical, vortical flow around a delta wing undergoing thickening oscillation is solved using the unsteady, thin-layer Navier-Stokes equations and the unsteady linearized Navier-displacement equations. The unsteady, thin-layer Navier-Stokes equations are solved using the implicit approximate-factorization finite-volume scheme to compute the conservative components of the flow vector field. With the conservative components known at any time step, the linearized, Navier-displacement equations are solved using the alternating, direction-implicit scheme to obtain the grid points displacements due to known displacement boundary conditions. A grid-displacements limiter, in the form of a low mesh Reynolds number, is used to limit grid-folding in regions of highly reversed flow.
NASA Technical Reports Server (NTRS)
Truong, K. V.; Tobak, M.
1990-01-01
The indicial response approach is recast in a form appropriate to the study of vortex induced oscillations phenomena. An appropriate form is demonstrated for the indicial response of the velocity field which may be derived directly from the Navier-Stokes equations. On the basis of the Navier-Stokes equations, it is demonstrated how a form of the velocity response to an arbitrary motion may be determined. To establish its connection with the previous work, the new approach is applied first to the simple situation wherein the indicial response has a time invariant equilibrium state. Results for the aerodynamic response to an arbitrary motion are shown to confirm to the form obtained previously.
A cell-vertex multigrid method for the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Radespiel, R.
1989-01-01
A cell-vertex scheme for the Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing, a multigrid method, and carefully controlled artificial dissipative terms, very good convergence rates are obtained for a wide range of two- and three-dimensional flows over airfoils and wings. The accuracy of the code is examined by grid refinement studies and comparison with experimental data. For an accurate prediction of turbulent flows with strong separations, a modified version of the nonequilibrium turbulence model of Johnson and King is introduced, which is well suited for an implementation into three-dimensional Navier-Stokes codes. It is shown that the solutions for three-dimensional flows with strong separations can be dramatically improved, when a nonequilibrium model of turbulence is used.
Renumbering Methods to Unleash Multi-Threaded Approaches for a General Navier-Stokes Implementation
NASA Astrophysics Data System (ADS)
Vezolle, Pascal; Fournier, Yvan; Tallet, Nicolas; Heymans, Jerrold; D'Amora, Bruce
2010-09-01
Our investigation leverages the general industrial Navier-Stokes open-source Computational Fluid Dynamics (CFD) application, Code_Saturne, developed by Électricité de France (EDF). We deal with how to take advantage of the emerging processor features such as many-cores, Simultaneous Multi-Threading (SMT) and Thread Level Speculation (TLS), through a mixed MPI/multithreads approach. We focus here on the per-node performance improvements and present the constraints for a multithreads implementation to solve the general 3D Navier-Stokes equations using a finite volume discretization into polyhedral cells. We describe a simple and efficient mesh numbering scheme allowing us to introduce OpenMP and Thread Level Speculation implementations with minimal impact to overall code structure.
Prediction of Business Jet Airloads Using The Overflow Navier-Stokes Code
NASA Technical Reports Server (NTRS)
Bounajem, Elias; Buning, Pieter G.
2001-01-01
The objective of this work is to evaluate the application of Navier-Stokes computational fluid dynamics technology, for the purpose of predicting off-design condition airloads on a business jet configuration in the transonic regime. The NASA Navier-Stokes flow solver OVERFLOW with Chimera overset grid capability, availability of several numerical schemes and convergence acceleration techniques was selected for this work. A set of scripts which have been compiled to reduce the time required for the grid generation process are described. Several turbulence models are evaluated in the presence of separated flow regions on the wing. Computed results are compared to available wind tunnel data for two Mach numbers and a range of angles-of-attack. Comparisons of wing surface pressure from numerical simulation and wind tunnel measurements show good agreement up to fairly high angles-of-attack.
Numerical study of singularity formation in a class of Euler and Navier-Stokes flows
NASA Astrophysics Data System (ADS)
Ohkitani, Koji; Gibbon, John D.
2000-12-01
We study numerically a class of stretched solutions of the three-dimensional Euler and Navier-Stokes equations identified by Gibbon, Fokas, and Doering (1999). Pseudo-spectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover, this singularity apparently persists in the Navier-Stokes case. Independent evidence for the existence of a singularity is given by a Taylor series expansion in time. The mechanism underlying the formation of this singularity is the two-dimensionalization of the vorticity vector under strong compression; that is, the intensification of the azimuthal components associated with the diminishing of the axial component. It is suggested that the hollowing of the vortex accompanying this phenomenon may have some relevance to studies in vortex breakdown.
Flowfield of a lifting hovering rotor: A Navier-Stokes simulation
NASA Technical Reports Server (NTRS)
Srinivasan, G. R.; Baeder, J. D.; Obayashi, S.; Mccroskey, W. J.
1990-01-01
The viscous, three-dimensional flowfield of a lifting helicopter rotor in hover is calculated by using an upwind, implicit, finite-difference numerical method for solving the thin layer Navier-Stokes equations. The induced effects of the wake, including the interaction of tip vortices with successive blades, are calculated as a part of the overall flowfield solution without using any ad hoc wake models. Comparison of the numerical results for the subsonic and transonic conditions show good agreement with the experimental data and with the previously published Navier-Stokes calculations using a simple wake model. Some comparisons with Euler calculations are also presented, along with some discussions of the grid refinement studies.
Navier-Stokes calculations on multi-element airfoils using a chimera-based solver
NASA Technical Reports Server (NTRS)
Jasper, Donald W.; Agrawal, Shreekant; Robinson, Brian A.
1993-01-01
A study of Navier-Stokes calculations of flows about multielement airfoils using a chimera grid approach is presented. The chimera approach utilizes structured, overlapped grids which allow great flexibility of grid arrangement and simplifies grid generation. Calculations are made for two-, three-, and four-element airfoils, and modeling of the effect of gap distance between elements is demonstrated for a two element case. Solutions are obtained using the thin-layer form of the Reynolds averaged Navier-Stokes equations with turbulence closure provided by the Baldwin-Lomax algebraic model or the Baldwin-Barth one equation model. The Baldwin-Barth turbulence model is shown to provide better agreement with experimental data and to dramatically improve convergence rates for some cases. Recently developed, improved farfield boundary conditions are incorporated into the solver for greater efficiency. Computed results show good comparison with experimental data which include aerodynamic forces, surface pressures, and boundary layer velocity profiles.
Numerical solution of transonic wing flows using an Euler/Navier-Stokes zonal approach
NASA Technical Reports Server (NTRS)
Holst, T. L.; Gundy, K. L.; Thomas, S. D.; Chaderjian, N. M.; Flores, J.
1985-01-01
Transonic flow fields about wing geometries are computed using an Euler/Navier-Stokes approach in which the flow field is divided into several zones. The grid zones immediately adjacent to the wing surface are suitably clustered and solved with the Navier-Stokes equations. Grid zones removed from the wing are less finely clustered and are solved with the Euler equations. Wind tunnel wall effects are easily and accurately modeled with the new grid-zoning algorithm because the wind tunnel grid is constructed as an exact subset of the corresponding free-air grid. Solutions are obtained that are in good agreement with experiment, including cases with significant wind tunnel wall effects and shock-induced separation on the upper wing surface.
NASA Technical Reports Server (NTRS)
Holst, T. L.; Thomas, S. D.; Kaynak, U.; Gundy, K. L.; Flores, J.; Chaderjian, N. M.
1985-01-01
Transonic flow fields about wing geometries are computed using an Euler/Navier-Stokes approach in which the flow field is divided into several zones. The flow field immediately adjacent to the wing surface is resolved with fine grid zones and solved using a Navier-Stokes algorithm. Flow field regions removed from the wing are resolved with less finely clustered grid zones and are solved with an Euler algorithm. Computational issues associated with this zonal approach, including data base management aspects, are discussed. Solutions are obtained that are in good agreement with experiment, including cases with significant wind tunnel wall effects. Additional cases with significant shock induced separation on the upper wing surface are also presented.
On lower bounds for possible blow-up solutions to the periodic Navier-Stokes equation
Cortissoz, Jean C. Montero, Julio A. Pinilla, Carlos E.
2014-03-15
We show a new lower bound on the H{sup .3/2} (T{sup 3}) norm of a possible blow-up solution to the Navier-Stokes equation, and also comment on the extension of this result to the whole space. This estimate can be seen as a natural limiting result for Leray's blow-up estimates in L{sup p}(R{sup 3}), 3 < p < ∞. We also show a lower bound on the blow-up rate of a possible blow-up solution of the Navier-Stokes equation in H{sup .5/2} (T{sup 3}), and give the corresponding extension to the case of the whole space.
Application of multi-grid methods for solving the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Demuren, A. O.
1989-01-01
The application of a class of multi-grid methods to the solution of the Navier-Stokes equations for two-dimensional laminar flow problems is discussed. The methods consist of combining the full approximation scheme-full multi-grid technique (FAS-FMG) with point-, line-, or plane-relaxation routines for solving the Navier-Stokes equations in primitive variables. The performance of the multi-grid methods is compared to that of several single-grid methods. The results show that much faster convergence can be procured through the use of the multi-grid approach than through the various suggestions for improving single-grid methods. The importance of the choice of relaxation scheme for the multi-grid method is illustrated.
Application of multi-grid methods for solving the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Demuren, A. O.
1989-01-01
This paper presents the application of a class of multi-grid methods to the solution of the Navier-Stokes equations for two-dimensional laminar flow problems. The methods consists of combining the full approximation scheme-full multi-grid technique (FAS-FMG) with point-, line- or plane-relaxation routines for solving the Navier-Stokes equations in primitive variables. The performance of the multi-grid methods is compared to those of several single-grid methods. The results show that much faster convergence can be procured through the use of the multi-grid approach than through the various suggestions for improving single-grid methods. The importance of the choice of relaxation scheme for the multi-grid method is illustrated.
Three-Dimensional Navier-Stokes Calculations Using the Modified Space-Time CESE Method
NASA Technical Reports Server (NTRS)
Chang, Chau-lyan
2007-01-01
The space-time conservation element solution element (CESE) method is modified to address the robustness issues of high-aspect-ratio, viscous, near-wall meshes. In this new approach, the dependent variable gradients are evaluated using element edges and the corresponding neighboring solution elements while keeping the original flux integration procedure intact. As such, the excellent flux conservation property is retained and the new edge-based gradients evaluation significantly improves the robustness for high-aspect ratio meshes frequently encountered in three-dimensional, Navier-Stokes calculations. The order of accuracy of the proposed method is demonstrated for oblique acoustic wave propagation, shock-wave interaction, and hypersonic flows over a blunt body. The confirmed second-order convergence along with the enhanced robustness in handling hypersonic blunt body flow calculations makes the proposed approach a very competitive CFD framework for 3D Navier-Stokes simulations.
Development of an Aeroelastic Code Based on an Euler/Navier-Stokes Aerodynamic Solver
NASA Technical Reports Server (NTRS)
Bakhle, Milind A.; Srivastava, Rakesh; Keith, Theo G., Jr.; Stefko, George L.; Janus, Mark J.
1996-01-01
This paper describes the development of an aeroelastic code (TURBO-AE) based on an Euler/Navier-Stokes unsteady aerodynamic analysis. A brief review of the relevant research in the area of propulsion aeroelasticity is presented. The paper briefly describes the original Euler/Navier-Stokes code (TURBO) and then details the development of the aeroelastic extensions. The aeroelastic formulation is described. The modeling of the dynamics of the blade using a modal approach is detailed, along with the grid deformation approach used to model the elastic deformation of the blade. The work-per-cycle approach used to evaluate aeroelastic stability is described. Representative results used to verify the code are presented. The paper concludes with an evaluation of the development thus far, and some plans for further development and validation of the TURBO-AE code.
Three-dimensional Navier-Stokes heat transfer predictions for turbine blade rows
NASA Technical Reports Server (NTRS)
Boyle, Robert J.; Giel, Paul W.
1992-01-01
Results are shown for a three-dimensional Navier-Stokes analysis of both the flow and the surface heat transfer for turbine applications. Heat transfer comparisons are made with the experimental shock-tunnel data of Dunn and Kim, and with the data of Blair for the rotor of the large scale rotating turbine. The analysis was done using the steady-state, three-dimensional, thin-layer Navier-Stokes code developed by Chima, which uses a multistage Runge-Kutta scheme with implicit residual smoothing. An algebraic mixing length turbulence model is used to calculate turbulent eddy viscosity. The variation in heat transfer due to variations in grid parameters is examined. The effects of rotation, tip clearance, and inlet boundary layer thickness variation on the predicted blade and endwall heat transfer are examined.
Application of the implicit MacCormack scheme to the parabolized Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Lawrence, J. L.; Tannehill, J. C.; Chaussee, D. S.
1984-01-01
MacCormack's implicit finite-difference scheme was used to solve the two-dimensional parabolized Navier-Stokes (PNS) equations. This method for solving the PNS equations does not require the inversion of block tridiagonal systems of algebraic equations and permits the original explicit MacCormack scheme to be employed in those regions where implicit treatment is not needed. The advantages and disadvantages of the present adaptation are discussed in relation to those of the conventional Beam-Warming scheme for a flat plate boundary layer test case. Comparisons are made for accuracy, stability, computer time, computer storage, and ease of implementation. The present method was also applied to a second test case of hypersonic laminar flow over a 15% compression corner. The computed results compare favorably with experiment and a numerical solution of the complete Navier-Stokes equations.
Parabolized Navier-Stokes solutions of separation and trailing-edge flows
NASA Technical Reports Server (NTRS)
Brown, J. L.
1983-01-01
A robust, iterative solution procedure is presented for the parabolized Navier-Stokes or higher order boundary layer equations as applied to subsonic viscous-inviscid interaction flows. The robustness of the present procedure is due, in part, to an improved algorithmic formulation. The present formulation is based on a reinterpretation of stability requirements for this class of algorithms and requires only second order accurate backward or central differences for all streamwise derivatives. Upstream influence is provided for through the algorithmic formulation and iterative sweeps in x. The primary contribution to robustness, however, is the boundary condition treatment, which imposes global constraints to control the convergence path. Discussed are successful calculations of subsonic, strong viscous-inviscid interactions, including separation. These results are consistent with Navier-Stokes solutions and triple deck theory.
NASA Technical Reports Server (NTRS)
Chen, Y. K.; Henline, W. D.
1993-01-01
The general boundary conditions including mass and energy balances of chemically equilibrated or nonequilibrated gas adjacent to ablating surfaces have been derived. A computer procedure based on these conditions was developed and interfaced with the Navier-Stokes solver for predictions of the flow field, surface temperature, and surface ablation rates over re-entry space vehicles with ablating Thermal Protection Systems (TPS). The Navier-Stokes solver with general surface thermochemistry boundary conditions can predict more realistic solutions and provide useful information for the design of TPS. A test case with a proposed hypersonic test vehicle configuration and associated free stream conditions was developed. Solutions with various surface boundary conditions were obtained, and the effect of nonequilibrium gas as well as surface chemistry on surface heating and ablation rate were examined. The solutions of the GASP code with complete ablating surface conditions were compared with those of the ASC code. The direction of future work is also discussed.
The development of a solution-adaptive 3D Navier-Stokes solver for turbomachinery
NASA Astrophysics Data System (ADS)
Dawes, W. N.
1991-06-01
This paper describes the early stages in the development of a solution-adaptive fully three-dimensional Navier-Stokes solver. The compressible, Navier-Stokes equations, closed with k-epsiton turbulence modeling, are discretized on an unstructured mesh formed from tetrahedral computational control volumes. At the mesh generation stage and at stages during the solution process itself, mesh refinement is carried out by flagging cells which satisfy particular critera. These criteria include geometric features such as proximity to wetted surfaces and features associated with the particular flowfield, such as fractional variation of a flow variable over cell faces. Solutions are presented for the highly three-dimensional flows associated with a truncated cylinder in a cross flow, a three-dimensional swept transonic bump, and the corner stall and secondary flow in a transonic compressor cascade.
Navier-Stokes cascade analysis with a stiff Kappa-Epsilon turbulence solver
NASA Technical Reports Server (NTRS)
Liu, Jong-Shang; Sockol, Peter M.; Prahl, Joseph M.
1987-01-01
The two dimensional, compressible, thin layer Navier-Stokes equations with the Baldwin-Lomax turbulence model and the kinetic energy-energy dissipation (k-epsilon) model are solved numerically to simulate the flow through a cascade. The governing equations are solved for the entire flow domain, without the boundary layer assumptions. The stiffness of the k-epsilon equations is discussed. A semi-implicit, Runge-Kutta, time-marching scheme is developed to solve the k-epsilon equations. The impact of the k-epsilon solver on the explicit Runge-Kutta Navier-Stokes solver is discussed. Numerical solutions are presented for two dimensional turbulent flow over a flat plate and a double circular arc cascade and compared with experimental data.
Navier-Stokes simulation of transonic wing flow fields using a zonal grid approach
NASA Technical Reports Server (NTRS)
Chaderjian, Neal M.
1988-01-01
The transonic Navier-Stokes code was used to simulate flow fields about isolated wings for workshop wind-tunnel and free-air cases using the thin-layer Reynolds-averaged Navier-Stokes equations. An implicit finite-difference scheme based on a diagonal version of the Beam-Warming algorithm was used to integrate the governing equations. A zonal grid approach was used to allow efficient grid refinement near the wing surface. The flow field was sensitive to the turbulent transition model, and flow unsteadiness was observed for a wind-tunnel case but not for the corresponding free-air case. The specification of experimental pressure at the wind-tunnel exit plane is the primary reason for the difference of these two numerical solutions.
Patched-grid calculations with the Euler and Navier-Stokes equations: Theory and application
NASA Technical Reports Server (NTRS)
Rai, M. M.
1986-01-01
The Rai (1984,85) patch-boundary scheme for the Euler equations is described. The integration methods used to update the interior grid points are are discussed. Stability of patch-boundary schemes and the use of these schemes in Navier-Stokes calculations are mentioned. Results for inviscid, supersonic flow over a cylinder, blast wave diffraction by ramp, and the motion of a vortex in a freestream are presented. These test cases demonstrate the quality of solutions possible with the scheme.
On Energy Cascades in the Forced 3D Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Dascaliuc, R.; Grujić, Z.
2016-02-01
We show—in the framework of physical scales and (K_1,K_2) -averages—that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale is sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws—in terms of Grashof number—for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.
On Energy Cascades in the Forced 3D Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Dascaliuc, R.; Grujić, Z.
2016-06-01
We show—in the framework of physical scales and (K_1,K_2)-averages—that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale is sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws—in terms of Grashof number—for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.
Navier-Stokes simulation of wind-tunnel flow using LU-ADI factorization algorithm
NASA Technical Reports Server (NTRS)
Obayashi, Shigeru; Fujii, Kozo; Gavali, Sharad
1988-01-01
The three dimensional Navier-Stokes solution code using the LU-ADI factorization algorithm was employed to simulate the workshop test cases of transonic flow past a wing model in a wind tunnel and in free air. The effect of the tunnel walls is well demonstrated by the present simulations. An Amdahl 1200 supercomputer having 128 Mbytes main memory was used for these computations.
Some observations on a new numerical method for solving Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Kumar, A.
1981-01-01
An explicit-implicit technique for solving Navier-Stokes equations is described which, is much less complex than other implicit methods. It is used to solve a complex, two-dimensional, steady-state, supersonic-flow problem. The computational efficiency of the method and the quality of the solution obtained from it at high Courant-Friedrich-Lewy (CFL) numbers are discussed. Modifications are discussed and certain observations are made about the method which may be helpful in using it successfully.
Complete Galilean-Invariant Lattice BGK Models for the Navier-Stokes Equation
NASA Technical Reports Server (NTRS)
Qian, Yue-Hong; Zhou, Ye
1998-01-01
Galilean invariance has been an important issue in lattice-based hydrodynamics models. Previous models concentrated on the nonlinear advection term. In this paper, we take into account the nonlinear response effect in a systematic way. Using the Chapman-Enskog expansion up to second order, complete Galilean invariant lattice BGK models in one dimension (theta = 3) and two dimensions (theta = 1) for the Navier-Stokes equation have been obtained.
Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method
NASA Technical Reports Server (NTRS)
Chen, Hudong; Chen, Shiyi; Matthaeus, William H.
1992-01-01
A lattice Boltzmann model is presented which gives the complete Navier-Stokes equation and may provide an efficient parallel numerical method for solving various fluid problems. The model uses the single-time relaxation approximation and a particular Maxwell-type distribution. The model eliminates exactly (1) the non-Galilean invariance caused by a density-dependent coefficient in the convection term and (2) a velocity-dependent equation of state.
Time-Filtered Navier-Stokes Approach and Emulation of Turbulence-Chemistry Interaction
NASA Technical Reports Server (NTRS)
Liu, Nan-Suey; Wey, Thomas; Shih, Tsan-Hsing
2013-01-01
This paper describes the time-filtered Navier-Stokes approach capable of capturing unsteady flow structures important for turbulent mixing and an accompanying subgrid model directly accounting for the major processes in turbulence-chemistry interaction. They have been applied to the computation of two-phase turbulent combustion occurring in a single-element lean-direct-injection combustor. Some of the preliminary results from this computational effort are presented in this paper.
Generalized INF-SUP condition for Chebyshev approximation of the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Bernardi, Christine; Canuto, Claudio; Maday, Yvon
1986-01-01
An abstract mixed problem and its approximation are studied; both are well-posed if and only if several inf-sup conditions are satisfied. These results are applied to a spectral Galerkin method for the Stokes problem in a square, when it is formulated in Chebyshev weighted Sobolev spaces. Finally, a collocation method for the Navier-Stokes equations at Chebyshev nodes is analyzed.
NASA Technical Reports Server (NTRS)
Carpenter, Mark H.; Fisher, Travis C.; Nielsen, Eric J.; Frankel, Steven H.
2013-01-01
Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the new schemes are applicable to a broad class of linear and nonlinear conservation laws, emphasis herein is placed on the entropy stability of the compressible Navier-Stokes equations.
Large time behavior of the isentropic compressible Navier-Stokes-Maxwell system
NASA Astrophysics Data System (ADS)
Chen, Yan; Li, Fucai; Zhang, Zhipeng
2016-08-01
In this paper, we study the large time behavior of the isentropic compressible Navier-Stokes-Maxwell system introduced by Jiang and Li (Nonlinearity 25(6):1735-1752, 2012) in the whole space {{{R}}^3} when the initial data are a small perturbation of some given constant state. We obtain the desired result through taking the refined analysis on the time decay property and Green's function of the linearized system. Moreover, we also obtain the optimal time rate of the solution.
NASA Technical Reports Server (NTRS)
Mehta, Unmeel; Lomax, Harvard
1981-01-01
During the past five years, numerous pioneering archival publications have appeared that have presented computer solutions of the mass-weighted, time-averaged Navier-Stokes equations for transonic problems pertinent to the aircraft industry. These solutions have been pathfinders of developments that could evolve into a major new technological capability, namely the computational Navier-Stokes technology, for the aircraft industry. So far these simulations have demonstrated that computational techniques, and computer capabilities have advanced to the point where it is possible to solve forms of the Navier-Stokes equations for transonic research problems. At present there are two major shortcomings of the technology: limited computer speed and memory, and difficulties in turbulence modelling and in computation of complex three-dimensional geometries. These limitations and difficulties are the pacing items of the continuing developments, although the one item that will most likely turn out to be the most crucial to the progress of this technology is turbulence modelling. The objective of this presentation is to discuss the state of the art of this technology and suggest possible future areas of research. We now discuss some of the flow conditions for which the Navier-Stokes equations appear to be required. On an airfoil there are four different types of interaction of a shock wave with a boundary layer: (1) shock-boundary-layer interaction with no separation, (2) shock-induced turbulent separation with immediate reattachment (we refer to this as a shock-induced separation bubble), (3) shock-induced turbulent separation without reattachment, and (4) shock-induced separation bubble with trailing edge separation.
Comparison of generalized Reynolds and Navier Stokes equations for flow of a power law fluid
NASA Technical Reports Server (NTRS)
Mullen, R. L.; Prekwas, A.; Braun, M. J.; Hendricks, R. C.
1987-01-01
This paper compares a finite element solution of a modified Reynolds equation with a finite difference solution of the Navier-Stokes equation for a power law fluid. Both the finite element and finite difference formulation are reviewed. Solutions to spiral flow in parallel and conical geometries are compared. Comparison with experimental results are also given. The effects of the assumptions used in the Reynolds equation are discussed.
NASA Technical Reports Server (NTRS)
Mahajan, Aparajit J.; Dowell, Earl H.; Bliss, Donald B.
1991-01-01
A Lanczos procedure is presently applied to a Navier-Stokes (N-S) solver for eigenvalues and eigenvectors associated with the small-perturbation analysis of the N-S equations' finite-difference representation for airfoil flows; the matrix used is very large, sparse, real, and nonsymmetric. The Lanczos procedure is shown to furnish complete spectral information for the eigenvalues, as required for transient-stability analysis of N-S solvers.
A new mixed basis Navier-Stokes formulation for incompressible flows over complex geometries
NASA Astrophysics Data System (ADS)
Murali, Avinaash; Rajagopalan, R. G.
2016-02-01
Numerical modeling of complex geometries necessitates the use of curvilinear body fitted coordinates. This article proposes a novel mixed basis formulation of the governing conservation equations for general curvilinear non-orthogonal grids with the physical covariant velocity as the primary solution variable. This results in an algorithm which has many advantages of orthogonal equations. The conservation equations written in this form retains the diagonal dominance of the pressure equation. The newly formed conservation equations are solved on a structured grid using the SIMPLER algorithm and are shown to converge well for non-orthogonal grids. Standard K-ɛ model is used for the turbulence closure.
Partially-averaged Navier-Stokes method for turbulent thermal plume
NASA Astrophysics Data System (ADS)
Kumar, Rajesh; Dewan, Anupam
2015-12-01
In this paper, the partially-averaged Navier-Stokes (PANS) simulation is performed for a turbulent thermal plume. The aim of the paper is to assess the PANS method for modeling buoyancy-driven flows at a reasonable computational cost. PANS is a turbulence closure model which is developed to be used as a bridging model ranging from the direct numerical simulation to the Reynolds-averaged Navier-Stokes simulation by varying the level of resolution. The PANS computations are performed for various values of the filter-width to evaluate the sensitivity of the filter-widths to the computed flow statistics. The present simulations have been carried out employing a source code buoyantPimpleFOAM based on the OpenFOAM platform. In order to capture the effect of buoyancy on turbulence, the generalized gradient diffusion hypothesis is employed to model the production of turbulence due to buoyancy. A detailed comparison of the time-averaged and turbulent statistics obtained from the PANS simulations with the experimental data and LES results reported in the literature has been presented. The present results have also been compared with the results of the unsteady Reynolds-averaged Navier-Stokes solutions. The PANS model is shown to enhance the computing capability significantly in predicting buoyancy-driven flows compared with those of URANS model. Finally, various important unsteady flow structures of turbulent thermal plume have been visualized from the instantaneous flow statistics obtained using the PANS simulations.
NASA Technical Reports Server (NTRS)
Carpenter, Mark H.; Parsani, Matteo; Fisher, Travis C.; Nielsen, Eric J.
2015-01-01
Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for Burgers' and the compressible Navier-Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [1, 2], extends the applicable set of points from tensor product, Legendre-Gauss-Lobatto (LGL) to a combination of tensor product Legendre-Gauss (LG) and LGL points. The new semi-discrete operators discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality for both Burgers' and the compressible Navier-Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly to implement. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinearly stability proof for the compressible Navier-Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators).
NASA Astrophysics Data System (ADS)
Yang, Xiaoquan; Cheng, Jian; Liu, Tiegang; Luo, Hong
2015-11-01
The direct discontinuous Galerkin (DDG) method based on a traditional discontinuous Galerkin (DG) formulation is extended and implemented for solving the compressible Navier-Stokes equations on arbitrary grids. Compared to the widely used second Bassi-Rebay (BR2) scheme for the discretization of diffusive fluxes, the DDG method has two attractive features: first, it is simple to implement as it is directly based on the weak form, and therefore there is no need for any local or global lifting operator; second, it can deliver comparable results, if not better than BR2 scheme, in a more efficient way with much less CPU time. Two approaches to perform the DDG flux for the Navier- Stokes equations are presented in this work, one is based on conservative variables, the other is based on primitive variables. In the implementation of the DDG method for arbitrary grid, the definition of mesh size plays a critical role as the formation of viscous flux explicitly depends on the geometry. A variety of test cases are presented to demonstrate the accuracy and efficiency of the DDG method for discretizing the viscous fluxes in the compressible Navier-Stokes equations on arbitrary grids.
NASA Technical Reports Server (NTRS)
Ahmad, Jasim; Aiken, Edwin, W. (Technical Monitor)
1998-01-01
Helicopter flowfields are highly unsteady, nonlinear and three-dimensional. In forward flight and in hover, the rotor blades interact with the tip vortex and wake sheet developed by either itself or the other blades. This interaction, known as blade-vortex interactions (BVI), results in unsteady loading of the blades and can cause a distinctive acoustic signature. Accurate and cost-effective computational fluid dynamic solutions that capture blade-vortex interactions can help rotor designers and engineers to predict rotor performance and to develop designs for low acoustic signature. Such a predictive method must preserve a blade's shed vortex for several blade revolutions before being dissipated. A number of researchers have explored the requirements for this task. This paper will outline some new capabilities that have been added to the NASA Ames' OVERFLOW code to improve its overall accuracy for both vortex capturing and unsteady flows. To highlight these improvements, a number of case studies will be presented. These case studies consist of free convection of a 2-dimensional vortex, dynamically pitching 2-D airfoil including light-stall, and a full 3-D unsteady viscous solution of a helicopter rotor in forward flight In this study both central and upwind difference schemes are modified to be more accurate. Central difference scheme is chosen for this simulation because the flowfield is not dominated by strong shocks. The feature of shock-vortex interaction in such a flow is less important than the dominant blade-vortex interaction. The scheme is second-order accurate in time and solves the thin-layer Navier-Stokes equations in fully-implicit manner at each time-step. The spatial accuracy is either second and fourth-order central difference or third-order upwind difference using Roe-flux and MUSCLE scheme. This paper will highlight and demonstrate the methods for several sample cases and for a helicopter rotor. Preliminary computations on a rotor were performed
A comparative study of Navier-Stokes codes for high-speed flows
NASA Technical Reports Server (NTRS)
Rudy, David H.; Thomas, James L.; Kumar, Ajay; Gnoffo, Peter A.; Chakravarthy, Sukumar R.
1989-01-01
A comparative study was made with four different codes for solving the compressible Navier-Stokes equations using three different test problems. The first of these cases was hypersonic flow through the P8 inlet, which represents inlet configurations typical of a hypersonic airbreathing vehicle. The free-stream Mach number in this case was 7.4. This 2-D inlet was designed to provide an internal compression ratio of 8. Initial calculations were made using two state-of-the-art finite-volume upwind codes, CFL3D and USA-PG2, as well as NASCRIN, a code which uses the unsplit finite-difference technique of MacCormack. All of these codes used the same algebraic eddy-viscosity turbulence model. In the experiment, the cowl lip was slightly blunted; however, for the computations, a sharp cowl leading edge was used to simplify the construction of the grid. The second test problem was the supersonic (Mach 3.0) flow in a three-dimensional corner formed by the intersection of two wedges with equal wedge angles of 9.48 degrees. The flow in such a corner is representative of the flow in the corners of a scramjet inlet. Calculations were made for both laminar and turbulent flow and compared with experimental data. The three-dimensional versions of the three codes used for the inlet study (CFL3D, USA-PG3, and SCRAMIN, respectively) were used for this case. For the laminar corner flow, a fourth code, LAURA, which also uses recently-developed upwind technology, was also utilized. The final test case is the two-dimensional hypersonic flow over a compression ramp. The flow is laminar with a free-stream Mach number of 14.1. In the experiment, the ramp angle was varied to change the strength of the ramp shock and the extent of the viscous-inviscid interaction. Calculations were made for the 24-degree ramp configuration which produces a large separated-flow region that extends upstream of the corner.
NASA Astrophysics Data System (ADS)
Fakhar, K.; Hayat, T.; Yi, Cheng; Amin, N.
2010-03-01
This note looks at the two similarity solutions of the Navier-Stokes equations in polar coordinates. In the second solution an initial value problem is reduced into generalized stationary KDV and hence integrable.
NASA Technical Reports Server (NTRS)
Moitra, A.
1982-01-01
An implicit finite-difference algorithm is developed for the numerical solution of the incompressible three dimensional Navier-Stokes equations in the non-conservative primitive-variable formulation. The flow field about an airfoil spanning a wind-tunnel is computed. The coordinate system is generated by an extension of the two dimensional body-fitted coordinate generation techniques of Thompson, as well as that of Sorenson, into three dimensions. Two dimensional grids are stacked along a spanwise coordinate defined by a simple analytical function. A Poisson pressure equation for advancing the pressure in time is arrived at by performing a divergence operation on the momentum equations. The pressure at each time-step is calculated on the assumption that continuity be unconditionally satisfied. An eddy viscosity coefficient, computed according to the algebraic turbulence formulation of Baldwin and Lomax, simulates the effects of turbulence.
AN IMMERSED BOUNDARY METHOD FOR COMPLEX INCOMPRESSIBLE FLOWS
An immersed boundary method for time-dependant, three- dimensional, incompressible flows is presented in this paper. The incompressible Navier-Stokes equations are discretized using a low-diffusion flux splitting method for the inviscid fluxes and a second order central differenc...
A lattice-Boltzmann scheme of the Navier-Stokes equations on a 3D cuboid lattice
NASA Astrophysics Data System (ADS)
Min, Haoda; Peng, Cheng; Wang, Lian-Ping
2015-11-01
The standard lattice-Boltzmann method (LBM) for fluid flow simulation is based on a square (in 2D) or cubic (in 3D) lattice grids. Recently, two new lattice Boltzmann schemes have been developed on a 2D rectangular grid using the MRT (multiple-relaxation-time) collision model, by adding a free parameter in the definition of moments or by extending the equilibrium moments. Here we developed a lattice Boltzmann model on 3D cuboid lattice, namely, a lattice grid with different grid lengths in different spatial directions. We designed our MRT-LBM model by matching the moment equations from the Chapman-Enskog expansion with the Navier-Stokes equations. The model guarantees correct hydrodynamics. A second-order term is added to the equilibrium moments in order to restore the isotropy of viscosity on a cuboid lattice. The form and the coefficients of the extended equilibrium moments are determined through an inverse design process. An additional benefit of the model is that the viscosity can be adjusted independent of the stress-moment relaxation parameter, thus improving the numerical stability of the model. The resulting cuboid MRT-LBM model is then validated through benchmark simulations using laminar channel flow, turbulent channel flow, and the 3D Taylor-Green vortex flow.
NASA Astrophysics Data System (ADS)
Debbi, Latifa
2016-03-01
In this work, we introduce and study the well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on bounded domains and on the torus (briefly dD-FSNSE). For the subcritical regime, we establish thresholds for which a maximal local mild solution exists and satisfies required space and time regularities. We prove that under conditions of Beale-Kato-Majda type, these solutions are global and unique. These conditions are automatically satisfied for the 2D-FSNSE on the torus if the initial data has H 1-regularity and the diffusion term satisfies growth and Lipschitz conditions corresponding to H 1-spaces. The case of 2D-FSNSE on the torus is studied separately. In particular, we established thresholds for the global existence, uniqueness, space and time regularities of the weak (strong in probability) solutions in the subcritical regime. For the general regime, we prove the existence of a martingale solution and we establish the uniqueness under a condition of Serrin's type on the fractional Sobolev spaces.
Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 2: User's guide
NASA Technical Reports Server (NTRS)
Towne, Charles E.; Schwab, John R.; Bui, Trong T.
1993-01-01
A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the User's Guide, and describes the program's features, the input and output, the procedure for setting up initial conditions, the computer resource requirements, the diagnostic messages that may be generated, the job control language used to run the program, and several test cases.
Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 1: Analysis description
NASA Technical Reports Server (NTRS)
Towne, Charles E.; Schwab, John R.; Bui, Trong T.
1993-01-01
A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the Analysis Description, and presents the equations and solution procedure. The governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models are described in detail.
PROTEUS two-dimensional Navier-Stokes computer code, version 1.0. Volume 3: Programmer's reference
NASA Technical Reports Server (NTRS)
Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady
1990-01-01
A new computer code was developed to solve the 2-D or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating-direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 3 is the Programmer's Reference, and describes the program structure, the FORTRAN variables stored in common blocks, and the details of each subprogram.
Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 3: Programmer's reference
NASA Technical Reports Server (NTRS)
Towne, Charles E.; Schwab, John R.; Bui, Trong T.
1993-01-01
A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. The Programmer's Reference contains detailed information useful when modifying the program. The program structure, the Fortran variables stored in common blocks, and the details of each subprogram are described.
Optimal Decay Rate of the Compressible Navier-Stokes-Poisson System in {mathbb {R}^3}
NASA Astrophysics Data System (ADS)
Li, Hai-Liang; Matsumura, Akitaka; Zhang, Guojing
2010-05-01
The compressible Navier-Stokes-Poisson (NSP) system is considered in {mathbb {R}^3} in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L 2-rate {(1+t)^{-frac {3}{4}}} or L ∞-rate (1 + t)-3/2 respectively as the compressible Navier-Stokes system, but the momentum of the NSP system decays at the L 2-rate {(1+t)^{-frac {1}{4}}} or L ∞-rate (1 + t)-1 respectively, which is slower than the L 2-rate {(1+t)^{-frac {3}{4}}} or L ∞-rate (1 + t)-3/2 for compressible Navier-Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737-758, 2007; Liu and Wang, in Comm Math Phys 196:145-173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67-104, 1980] and the L ∞-rate (1 + t)- p with {p in (1, 3/2)} for irrotational Euler-Poisson system [Guo, in Comm Math Phys 195:249-265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.
Chaotic Behaviuor of the Navier-Stokes Solutions, Gyroscopes and Storm Surging
NASA Astrophysics Data System (ADS)
Tchiguirinskaia, Ioulia; Schertzer, Daniel
2015-04-01
Storm surges are phenomena inflicting wide damages all over the planet. Unfortunately they are badly represented in classical forecast model schemes because their multiscale nature is at odd with the scale truncation of these models. For similar reasons, classical data analysis often compelled to considered them as 'outliers' of the normal atmospheric activity, whereas as in fact they result from the same physical mechanisms that create less extreme behavior. A better representation of storm surges requires a multicale understanding of how a cascade of seemingly harmless instabilities can generate major ones. This correspond to the conjectured, outstanding intermittency.of the chaotic behaviour of the Navier-Stokes solutions. However, our limited, mathematical understanding of the Navier-Stokes equations prevent us to directly use them to investigate this question. We therefore use the most relevant cascade model to theoretically tackle this question of intermittency, i.e. the Scaling Gyroscopes Cascade (SGC). Indeed, this model is obtained with the help of a non trivial tree-decomposition of the Lie structure of the Navier-Stokes equations. the SGC model is deduced from these equations by preserving only a certain type of direct interactions, while the resulting indirect interactions are built dynamically along the tree-structure of the cascade. Because its fundamental element corresponds to a 'top' -i.e., an object with which almost anyone began to discover the puzzling nonlinear properties of rotation!- the SGC model remains rather simple, yet not simplistic! In particular, the SGC model enables us to investigate in details the occurrence of the critical singularity of a first order multifractal phase transition, which theoretically define storm surges. Overall, these theoretical findings could significantly reduce numerous uncertainties of environmental risk assessments.
Prediction of Unsteady Transitional Layers in Turbomachinery Using Navier Stokes Equations
NASA Technical Reports Server (NTRS)
Lakshminarayana, B.; Chernobrovkin, A.; Kang, D. J.
1998-01-01
The objective of the research reported in this presentation is to develop computational techniques for the prediction of unsteady transitional flows associated with the rotor stator interaction in turbomachinery. Three low-Reynolds number turbulence models are incorporated in two unsteady Navier-Stokes codes (one is pressure based and the other is time marching with Runge-Kutta time stepping) and evaluated for accuracy in predicting the onset and the end of unsteady transitional patches due to wake passing. The best model is then used for modification and improvement for the leading edge effect. An existing steady Navier-Stokes code was modified to include pseudo-time stepping, which provided acceleration from 5 to 25 times that of the original code. A systematic validation procedure was implemented to assess the effects of the grid, artificial dissipation, physical, and the pseudo-time step for an accurate prediction of transitional flows resulting from the rotor-stator interaction. The ability of the Navier-Stokes code to predict the unsteady transitional flow on a turbomachinery blade is demonstrated. The unsteady pressure and velocity fields are in good agreement with the experimental data and the prediction from the Euler/boundary layer approach. The numerical solver was able to capture all zones (wake induced transitional strip, wake induced turbulent strip, calmed region, etc.) associated with wake induced transition in a compressor cascade. Another significant step is the assessment of k-epsilon turbulence models, including the leading edge modifications. Best results were obtained from the FLB model. The LB model predicted earlier inception of the transition and shorter transition length. Modification of the k-epsilon model was found to be essential for an accurate prediction of the unsteady transitional flow in a compressor cascade. The CH model failed to predict the unsteady transitional flow. Predicted boundary layer was turbulent from the leading edge
Decay rates to viscous contact waves for the compressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Ma, Shixiang; Wang, Jing
2016-02-01
In this paper, we study the large-time asymptotic behavior of contact wave for the Cauchy problem of one-dimensional compressible Navier-Stokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, and proves that it is nonlinearly stable provided that the strength of the contact discontinuity and the perturbation of the initial data are suitably small.
Evaluation of a research circulation control airfoil using Navier-Stokes methods
NASA Technical Reports Server (NTRS)
Shrewsbury, George D.
1987-01-01
The compressible Reynolds time averaged Navier-Stokes equations were used to obtain solutions for flows about a two dimensional circulation control airfoil. The governing equations were written in conservation form for a body-fitted coordinate system and solved using an Alternating Direction Implicit (ADI) procedure. A modified algebraic eddy viscosity model was used to define the turbulent characteristics of the flow, including the wall jet flow over the Coanda surface at the trailing edge. Numerical results are compared to experimental data obtained for a research circulation control airfoil geometry. Excellent agreement with the experimental results was obtained.
Three-dimensional multigrid Navier-Stokes computations for turbomachinery applications
NASA Technical Reports Server (NTRS)
Subramanian, S. V.
1989-01-01
The fully three-dimensional, time-dependent compressible Navier-Stokes equations in cylindrical coordinates are presently used, in conjunction with the multistage Runge-Kutta numerical integration scheme for solution of the governing flow equations, to simulate complex flowfields within turbomechanical components whose pertinent effects encompass those of viscosity, compressibility, blade rotation, and tip clearance. Computed results are presented for selected cascades, emphasizing the code's capabilities in the accurate prediction of such features as airfoil loadings, exit flow angles, shocks, and secondary flows. Computations for several test cases have been performed on a Cray-YMP, using nearly 90,000 grid points.
A Navier-Stokes phase-field crystal model for colloidal suspensions.
Praetorius, Simon; Voigt, Axel
2015-04-21
We develop a fully continuous model for colloidal suspensions with hydrodynamic interactions. The Navier-Stokes Phase-Field Crystal model combines ideas of dynamic density functional theory with particulate flow approaches and is derived in detail and related to other dynamic density functional theory approaches with hydrodynamic interactions. The derived system is numerically solved using adaptive finite elements and is used to analyze colloidal crystallization in flowing environments demonstrating a strong coupling in both directions between the crystal shape and the flow field. We further validate the model against other computational approaches for particulate flow systems for various colloidal sedimentation problems. PMID:25903907
NASA Technical Reports Server (NTRS)
Vatsa, Veer N.; Turkel, Eli
2006-01-01
We apply an unsteady Reynolds-averaged Navier-Stokes (URANS) solver for the simulation of a synthetic jet created by a single diaphragm piezoelectric actuator in quiescent air. This configuration was designated as Case 1 for the CFDVAL2004 workshop held at Williamsburg, Virginia, in March 2004. Time-averaged and instantaneous data for this case were obtained at NASA Langley Research Center, using multiple measurement techniques. Computational results for this case using one-equation Spalart-Allmaras and two-equation Menter's turbulence models are presented along with the experimental data. The effect of grid refinement, preconditioning and time-step variation are also examined in this paper.
Upwind differencing and LU factorization for chemical non-equilibrium Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Shuen, Jian-Shun
1992-01-01
By means of either the Roe or the Van Leer flux-splittings for inviscid terms, in conjunction with central differencing for viscous terms in the explicit operator and the Steger-Warming splitting and lower-upper approximate factorization for the implicit operator, the present, robust upwind method for solving the chemical nonequilibrium Navier-Stokes equations yields formulas for finite-volume discretization in general coordinates. Numerical tests in the illustrative cases of a hypersonic blunt body, a ramped duct, divergent nozzle flows, and shock wave/boundary layer interactions, establish the method's efficiency.
Nonlinear Aeroelastic Analysis Using a Time-Accurate Navier-Stokes Equations Solver
NASA Technical Reports Server (NTRS)
Kuruvila, Geojoe; Bartels, Robert E.; Hong, Moeljo S.; Bhatia, G.
2007-01-01
A method to simulate limit cycle oscillation (LCO) due to control surface freeplay using a modified CFL3D, a time-accurate Navier-Stokes computational fluid dynamics (CFD) analysis code with structural modeling capability, is presented. This approach can be used to analyze aeroelastic response of aircraft with structural behavior characterized by nonlinearity in the force verses displacement curve. A limited validation of the method, using very low Mach number experimental data for a three-degrees-of-freedom (pitch/plunge/flap deflection) airfoil model with flap freeplay, is also presented.
NASA Technical Reports Server (NTRS)
Kandil, Osama A.
1993-01-01
Research on Navier-Stokes, dynamics, and aeroelastic computations for vortical flows, buffet, and flutter applications was performed. Progress during the period from 1 Oct. 1992 to 30 Sep. 1993 is included. Papers on the following topics are included: vertical tail buffet in vortex breakdown flows; simulation of tail buffet using delta wing-vertical tail configuration; shock-vortex interaction over a 65-degree delta wing in transonic flow; supersonic vortex breakdown over a delta wing in transonic flow; and prediction and control of slender wing rock.
Navier-Stokes simulation of a hypersonic generic wing/fuselage
NASA Technical Reports Server (NTRS)
Wai, John C.; Dao, Samuel C.; Chaussee, Denny S.
1987-01-01
An unsteady thin-layer Navier-Stokes code is used to calculate a generic wing/fuselage configuration at a Mach number of 25 and freestream conditions corresponding to an altitude of 220,000 feet. Calculations were performed with the assumptions of a perfect gas and with chemical equilibrium, and the boundary layer was assumed to be turbulent and to have a surface temperature prescribed at 1255 K. Results for the two different gas assumptions were compared in terms of distributions of pressure, density, temperature, Mach number, ratio of specific heat, and heat transfer. Numerical problems arising in the calculations were identified.
Three-dimensional multigrid Navier-Stokes computations for turbomachinery applications
NASA Astrophysics Data System (ADS)
Subramanian, S. V.
1989-07-01
The fully three-dimensional, time-dependent compressible Navier-Stokes equations in cylindrical coordinates are presently used, in conjunction with the multistage Runge-Kutta numerical integration scheme for solution of the governing flow equations, to simulate complex flowfields within turbomechanical components whose pertinent effects encompass those of viscosity, compressibility, blade rotation, and tip clearance. Computed results are presented for selected cascades, emphasizing the code's capabilities in the accurate prediction of such features as airfoil loadings, exit flow angles, shocks, and secondary flows. Computations for several test cases have been performed on a Cray-YMP, using nearly 90,000 grid points.
Large Scale Flutter Data for Design of Rotating Blades Using Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Guruswamy, Guru P.
2012-01-01
A procedure to compute flutter boundaries of rotating blades is presented; a) Navier-Stokes equations. b) Frequency domain method compatible with industry practice. Procedure is initially validated: a) Unsteady loads with flapping wing experiment. b) Flutter boundary with fixed wing experiment. Large scale flutter computation is demonstrated for rotating blade: a) Single job submission script. b) Flutter boundary in 24 hour wall clock time with 100 cores. c) Linearly scalable with number of cores. Tested with 1000 cores that produced data in 25 hrs for 10 flutter boundaries. Further wall-clock speed-up is possible by performing parallel computations within each case.
Explicit and implicit solution of the Navier-Stokes equations on a massively parallel computer
NASA Technical Reports Server (NTRS)
Levit, Creon; Jespersen, Dennis
1988-01-01
The design, implementation, and performance of a two-dimensional time-accurate Navier-Stokes solver for the CM2 supercomputer are described. The program uses a single processor for each grid point. Two different time-stepping methods have so far been implemented: an explicit third-order Runge-Kutta method and an implicit approximation-factorization method. The CM2 results are checked against those of a mature well-vectorized Cray 2 program, both for correctness and performance. The code is found to be correct, and the performance in some cases is up to several times that of the Cray 2.
Application of a Scalable, Parallel, Unstructured-Grid-Based Navier-Stokes Solver
NASA Technical Reports Server (NTRS)
Parikh, Paresh
2001-01-01
A parallel version of an unstructured-grid based Navier-Stokes solver, USM3Dns, previously developed for efficient operation on a variety of parallel computers, has been enhanced to incorporate upgrades made to the serial version. The resultant parallel code has been extensively tested on a variety of problems of aerospace interest and on two sets of parallel computers to understand and document its characteristics. An innovative grid renumbering construct and use of non-blocking communication are shown to produce superlinear computing performance. Preliminary results from parallelization of a recently introduced "porous surface" boundary condition are also presented.
Iterative solvers for Navier-Stokes equations: Experiments with turbulence model
Page, M.; Garon, A.
1994-12-31
In the framework of developing software for the prediction of flows in hydraulic turbine components, Reynolds averaged Navier-Stokes equations coupled with {kappa}-{omega} two-equation turbulence model are discretized by finite element method. Since the resulting matrices are large, sparse and nonsymmetric, strategies based on CG-type iterative methods must be devised. A segregated solution strategy decouples the momentum equation, the {kappa} transport equation and the {omega} transport equation. These sets of equations must be solved while satisfying constraint equations. Experiments with orthogonal projection method are presented for the imposition of essential boundary conditions in a weak sense.
NASA Technical Reports Server (NTRS)
Beam, R. M.; Warming, R. F.
1979-01-01
An attempt is made to establish a connection between linear multistep methods for applications to ordinary differential equations and their extension (by approximate factorization) to alternating direction implicit methods for partial differential equations. An earlier implicit factored scheme for the compressible Navier-Stokes equations is generalized by innovations that (1) increase the class of temporal difference schemes to include all linear multistep methods, (2) optimize the class of unconditionally stable factored schemes by a new choice of unknown variable, and (3) improve the computational efficiency by the introduction of quasi-one-leg methods.
NASA Technical Reports Server (NTRS)
Srinivasan, G. R.; Baeder, J. D.
1991-01-01
This paper outlines some recent advances in the application of the Euler and Navier-Stokes computational fluid dynamics methods to analyze nonlinear problems of helicopter aerodynamics and acoustics. A complete flowfield simulation of helicopters is currently not feasible with these methods. However, the use of the state-of-the-art numerical algorithms in conjunction with powerful supercomputers, like the Cray-2, have enabled notable progress to be made in modeling several individual components of this complex flow in hover and forward flight.
Nearly Interactive Parabolized Navier-Stokes Solver for High Speed Forebody and Inlet Flows
NASA Technical Reports Server (NTRS)
Benson, Thomas J.; Liou, May-Fun; Jones, William H.; Trefny, Charles J.
2009-01-01
A system of computer programs is being developed for the preliminary design of high speed inlets and forebodies. The system comprises four functions: geometry definition, flow grid generation, flow solver, and graphics post-processor. The system runs on a dedicated personal computer using the Windows operating system and is controlled by graphical user interfaces written in MATLAB (The Mathworks, Inc.). The flow solver uses the Parabolized Navier-Stokes equations to compute millions of mesh points in several minutes. Sample two-dimensional and three-dimensional calculations are demonstrated in the paper.
Parallelization of Unsteady Adaptive Mesh Refinement for Unstructured Navier-Stokes Solvers
NASA Technical Reports Server (NTRS)
Schwing, Alan M.; Nompelis, Ioannis; Candler, Graham V.
2014-01-01
This paper explores the implementation of the MPI parallelization in a Navier-Stokes solver using adaptive mesh re nement. Viscous and inviscid test problems are considered for the purpose of benchmarking, as are implicit and explicit time advancement methods. The main test problem for comparison includes e ects from boundary layers and other viscous features and requires a large number of grid points for accurate computation. Ex- perimental validation against double cone experiments in hypersonic ow are shown. The adaptive mesh re nement shows promise for a staple test problem in the hypersonic com- munity. Extension to more advanced techniques for more complicated ows is described.
Hypersonic nonequilibrium Navier-Stokes solutions over an ablating graphite nosetip
Chen, Y.K.; Henline, W.D.
1994-09-01
The general boundary conditions, including mass and energy balances, of chemically equilibrated or nonequilibrated gas adjacent to ablating surfaces have been derived. A computer procedure based on these conditions was developed and interfaced with the Navier-Stokes solver GASP (General Aerodynamics Simulation Program). A test case with a proposed hypersonic test-vehicle configuration and associated freestream conditions was developed. The solutions of the GASP code with various surface boundary conditions were obtained and compared with those of the ASCC (ABRES Shape Change) code, and the effect of nonequilibrium gas as well as surface chemistry on surface heating and ablation rate were examined. 22 refs.
Hypersonic nonequilibrium Navier-Stokes solutions over an ablating graphite nosetip
NASA Astrophysics Data System (ADS)
Chen, Y.-K.; Henline, W. D.
1994-09-01
The general boundary conditions, including mass and energy balances, of chemically equilibrated or nonequilibrated gas adjacent to ablating surfaces have been derived. A computer procedure based on these conditions was developed and interfaced with the Navier-Stokes solver GASP (General Aerodynamics Simulation Program). A test case with a proposed hypersonic test-vehicle configuration and associated freestream conditions was developed. The solutions of the GASP code with various surface boundary conditions were obtained and compared with those of the ASCC (ABRES Shape Change) code, and the effect of nonequilibrium gas as well as surface chemistry on surface heating and ablation rate were examined.
NASA Technical Reports Server (NTRS)
Chan, J. S.; Freeman, J. A.
1984-01-01
The viscous, axisymmetric flow in the thrust chamber of the space shuttle main engine (SSME) was computed on the CRAY 205 computer using the general interpolants method (GIM) code. Results show that the Navier-Stokes codes can be used for these flows to study trends and viscous effects as well as determine flow patterns; but further research and development is needed before they can be used as production tools for nozzle performance calculations. The GIM formulation, numerical scheme, and computer code are described. The actual SSME nozzle computation showing grid points, flow contours, and flow parameter plots is discussed. The computer system and run times/costs are detailed.
Navier-Stokes Simulation of Homogeneous Turbulence on the CYBER 205
NASA Technical Reports Server (NTRS)
Wu, C. T.; Ferziger, J. H.; Chapman, D. R.; Rogallo, R. S.
1984-01-01
A computer code which solves the Navier-Stokes equations for three dimensional, time-dependent, homogenous turbulence has been written for the CYBER 205. The code has options for both 64-bit and 32-bit arithmetic. With 32-bit computation, mesh sizes up to 64 (3) are contained within core of a 2 million 64-bit word memory. Computer speed timing runs were made for various vector lengths up to 6144. With this code, speeds a little over 100 Mflops have been achieved on a 2-pipe CYBER 205. Several problems encountered in the coding are discussed.
Propulsion-related flowfields using the preconditioned Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Venkateswaran, S.; Weiss, J. M.; Merkle, C. L.; Choi, Y.-H.
1992-01-01
A previous time-derivative preconditioning procedure for solving the Navier-Stokes is extended to the chemical species equations. The scheme is implemented using both the implicit ADI and the explicit Runge-Kutta algorithms. A new definition for time-step is proposed to enable grid-independent convergence. Several examples of both reacting and non-reacting propulsion-related flowfields are considered. In all cases, convergence that is superior to conventional methods is demonstrated. Accuracy is verified using the example of a backward facing step. These results demonstrate that preconditioning can enhance the capability of density-based methods over a wide range of Mach and Reynolds numbers.
A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations
Kupiainen, M; Sjogreen, B
2008-03-21
We here generalize the embedded boundary method that was developed for boundary discretizations of the wave equation in second order formulation in [6] and for the Euler equations of compressible fluid flow in [11], to the compressible Navier-Stokes equations. We describe the method and we implement it on a parallel computer. The implementation is tested for accuracy and correctness. The ability of the embedded boundary technique to resolve boundary layers is investigated by computing skin-friction profiles along the surfaces of the embedded objects. The accuracy is assessed by comparing the computed skin-friction profiles with those obtained by a body fitted discretization.
Quasiconservation laws for compressible three-dimensional Navier-Stokes flow.
Gibbon, J D; Holm, D D
2012-10-01
We formulate the quasi-Lagrangian fluid transport dynamics of mass density ρ and the projection q=ω·∇ρ of the vorticity ω onto the density gradient, as determined by the three-dimensional compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of q cannot cross a level set of ρ. That is, in this formulation, level sets of ρ (isopycnals) are impermeable to the transport of the projection q. PMID:23214709
3-D Navier-Stokes Analysis of Blade Root Aerodynamics for a Tiltrotor Aircraft In Cruise
NASA Technical Reports Server (NTRS)
Romander, Ethan
2006-01-01
The blade root area of a tiltrotor aircraft's rotor is constrained by a great many factors, not the least of which is aerodynamic performance in cruise. For this study, Navier-Stokes CFD techniques are used to study the aerodynamic performance in cruise of a rotor design as a function of airfoil thickness along the blade and spinner shape. Reducing airfoil thickness along the entire blade will be shown to have the greatest effect followed by smaller but still significant improvements achieved by reducing the thickness of root airfoils only. Furthermore, altering the shape of the spinner will be illustrated as a tool to tune the aerodynamic performance very near the blade root.
Stochastic Galerkin methods for the steady-state Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Sousedík, Bedřich; Elman, Howard C.
2016-07-01
We study the steady-state Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems.
A multistage time-stepping scheme for the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Swanson, R. C.; Turkel, E.
1985-01-01
A class of explicit multistage time-stepping schemes is used to construct an algorithm for solving the compressible Navier-Stokes equations. Flexibility in treating arbitrary geometries is obtained with a finite-volume formulation. Numerical efficiency is achieved by employing techniques for accelerating convergence to steady state. Computer processing is enhanced through vectorization of the algorithm. The scheme is evaluated by solving laminar and turbulent flows over a flat plate and an NACA 0012 airfoil. Numerical results are compared with theoretical solutions or other numerical solutions and/or experimental data.
Three-dimensional computational study of asymmetric flows using Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Cheung, Y. K. (Editor); Lee, J. H. W. (Editor); Leung, A. Y. T. (Editor); Wong, Tin-Chee; Kandil, Osama A.; Liu, C. H.
1992-01-01
The unsteady, compressible, thin-layer Navier-Stokes equations are used to obtain three-dimensional, asymmetric, vortex-flow solutions around cones and cone-cylinder configurations. The equations are solved using an implicit, upwind, flux-difference splitting, finite-volume scheme. The computational applications cover asymmetric flows around a 5 semi-apex angle cone of unit length at various Reynolds number. Next, a cylindrical afterbody of various length is added to the conical forebody to study the effect of the length of cylindrical afterbody on the flow asymmetry. All the asymmetric flow solutions are obtained by using a short-duration side-slip disturbance.
Three-dimensional multigrid Navier-Stokes computations for turbomachinery applications
Subramanian, S. V.
1989-01-01
The fully three-dimensional, time-dependent compressible Navier-Stokes equations in cylindrical coordinates are presently used, in conjunction with the multistage Runge-Kutta numerical integration scheme for solution of the governing flow equations, to simulate complex flowfields within turbomechanical components whose pertinent effects encompass those of viscosity, compressibility, blade rotation, and tip clearance. Computed results are presented for selected cascades, emphasizing the code's capabilities in the accurate prediction of such features as airfoil loadings, exit flow angles, shocks, and secondary flows. Computations for several test cases have been performed on a Cray-YMP, using nearly 90,000 grid points. 18 refs.
A three-dimensional structured/unstructured hybrid Navier-Stokes method for turbine blade rows
NASA Astrophysics Data System (ADS)
Tsung, F.-L.; Loellbach, J.; Kwon, O.; Hah, C.
1994-12-01
A three-dimensional viscous structured/unstructured hybrid scheme has been developed for numerical computation of high Reynolds number turbomachinery flows. The procedure allows an efficient structured solver to be employed in the densely clustered, high aspect-ratio grid around the viscous regions near solid surfaces, while employing an unstructured solver elsewhere in the flow domain to add flexibility in mesh generation. Test results for an inviscid flow over an external transonic wing and a Navier-Stokes flow for an internal annular cascade are presented.
Compressible Navier-Stokes equations: A study of leading edge effects
NASA Technical Reports Server (NTRS)
Hariharan, S. I.; Karbhari, P. R.
1987-01-01
A computational method is developed that allows numerical calculations of the time dependent compressible Navier-Stokes equations.The current results concern a study of flow past a semi-infinite flat plate.Flow develops from given inflow conditions upstream and passes over the flat plate to leave the computational domain without reflecting at the downstream boundary. Leading edge effects are included in this paper. In addition, specification of a heated region which gets convected with the flow is considered. The time history of this convection is obtained, and it exhibits a wave phenomena.
A Derivation of the Magnetohydrodynamic System from Navier-Stokes-Maxwell Systems
NASA Astrophysics Data System (ADS)
Arsénio, Diogo; Ibrahim, Slim; Masmoudi, Nader
2015-06-01
We provide a full and rigorous derivation of the standard viscous magnetohydrodynamic system (MHD) as the asymptotic limit of Navier-Stokes-Maxwell systems when the speed of light is infinitely large. We work in the physical setting provided by the natural energy bounds and therefore mainly consider Leray solutions of fluid dynamical systems. Our methods are based on a direct analysis of frequencies and we are able to establish the weak stability of a crucial nonlinear term (the Lorentz force), neither assuming any strong compactness of the components nor applying standard compensated compactness methods (which actually fail in this case).
Accurate Navier-Stokes results for the hypersonic flow over a spherical nosetip
Blottner, F.G.
1989-01-01
The unsteady thin-layer Navier-Stokes equations for a perfect gas are solved with a linearized block Alternating Direction Implicit finite-difference solution procedure. Solution errors due to numerical dissipation added to the governing equations are evaluated. Errors in the numerical predictions on three different grids are determined where Richardson extrapolation is used to estimate the exact solution. Accurate computational results are tabulated for the hypersonic laminar flow over a spherical body which can be used as a benchmark test case. Predictions obtained from the code are in good agreement with inviscid numerical results and experimental data. 9 refs., 11 figs., 3 tabs.
Time-dependent Navier-Stokes computations for flow-induced vibrations of vanes
NASA Astrophysics Data System (ADS)
Liu, B. L.; O'Farrel, J. M.; Holt, J. B.; Dougherty, N. S.
Flows over two curved vane configurations were computed using a time-accurate compressible Navier-Stokes flow model. One configuration showed the presence of strong flow-induced vibrations at Strouhal numbers near 0.19 and 0.38 for bending and torsional excitation. In the other configuration, a simple modification reduced both types of response. Laminar flows were analyzed for the effects of flow-induced vibrations, and flow fields were solved for a rigid vane and a vane undergoing forced vibrations at prescribed amplitude and frequency simulating vibration response to a coupled vortex-shedding/elastic motion feedback cycle.
NASA Technical Reports Server (NTRS)
Jentink, Thomas Neil; Usab, William J., Jr.
1990-01-01
An explicit, Multigrid algorithm was written to solve the Euler and Navier-Stokes equations with special consideration given to the coarse mesh boundary conditions. These are formulated in a manner consistent with the interior solution, utilizing forcing terms to prevent coarse-mesh truncation error from affecting the fine-mesh solution. A 4-Stage Hybrid Runge-Kutta Scheme is used to advance the solution in time, and Multigrid convergence is further enhanced by using local time-stepping and implicit residual smoothing. Details of the algorithm are presented along with a description of Jameson's standard Multigrid method and a new approach to formulating the Multigrid equations.
Reynolds-Averaged Navier-Stokes Analysis of Zero Efflux Flow Control Over a Hump Model
NASA Technical Reports Server (NTRS)
Rumsey, Christopher L.
2006-01-01
The unsteady flow over a hump model with zero efflux oscillatory flow control is modeled computationally using the unsteady Reynolds-averaged Navier-Stokes equations. Three different turbulence models produce similar results, and do a reasonably good job predicting the general character of the unsteady surface pressure coefficients during the forced cycle. However, the turbulent shear stresses are underpredicted in magnitude inside the separation bubble, and the computed results predict too large a (mean) separation bubble compared with experiment. These missed predictions are consistent with earlier steady-state results using no-flow-control and steady suction, from a 2004 CFD validation workshop for synthetic jets.
Reynolds-Averaged Navier-Stokes Analysis of Zero Efflux Flow Control over a Hump Model
NASA Technical Reports Server (NTRS)
Rumsey, Christopher L.
2006-01-01
The unsteady flow over a hump model with zero efflux oscillatory flow control is modeled computationally using the unsteady Reynolds-averaged Navier-Stokes equations. Three different turbulence models produce similar results, and do a reasonably good job predicting the general character of the unsteady surface pressure coefficients during the forced cycle. However, the turbulent shear stresses are underpredicted in magnitude inside the separation bubble, and the computed results predict too large a (mean) separation bubble compared with experiment. These missed predictions are consistent with earlier steady-state results using no-flow-control and steady suction, from a 2004 CFD validation workshop for synthetic jets.
An investigation of cell centered and cell vertex multigrid schemes for the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Radespiel, R.; Swanson, R. C.
1989-01-01
Two efficient and robust finite-volume multigrid schemes for solving the Navier-Stokes equations are investigated. These schemes employ either a cell centered or a cell vertex discretization technique. An explicit Runge-Kutta algorithm is used to advance the solution in time. Acceleration techniques are applied to obtain faster steady-state convergence. Accuracy and convergence of the schemes are examined. Computational results for transonic airfoil flows are essentially the same, even for a coarse mesh. Both schemes exhibit good convergence rates for a broad range of artificial dissipation coefficients.
An implicit three-dimensional Navier-Stokes solver for compressible flow
NASA Technical Reports Server (NTRS)
Yoon, Seokkwan; Kwak, Dochan
1991-01-01
A three-dimensional numerical method based on the lower-upper symmetric-Gauss-Seidel implicit scheme in conjunction with the flux-limited dissipation model is developed for solving the compressible Navier-Stokes equations. A new computer code which is based on this method requires only 9 microsec per grid-point per iteration on a single processor of a Cray YMP computer and executes at the sustained rate of 170 MFLOPS. A reduction of 4 orders of magnitude in the residual for a high Reynolds number flow using 230 K grid points is obtained in 24 minutes. The computational results compare well with available experimental data.
Navier-Stokes computations past a prolate spheroid at incidence. II - High incidence case
NASA Astrophysics Data System (ADS)
Piquet, J.; Queutey, P.
1993-01-01
A fully elliptic numerical method for the solution of Reynolds-averaged Navier-Stokes equations (RANSEs) is developed and applied to the lifting flow past a prolate spheroid. The method uses a system of numerically generated curvilinear coordinates; it retains the Cartesian velocity components, a nonstaggered grid, and a segregated approach in which a pressure solver couples velocity and pressure fields. It is shown that the RANSEs could describe a strongly concentrated vortical flow past a 6:1 prolate spheroid at high incidence (30 deg) and a Reynolds number of 7.2 x 10 exp 6.
NASA Astrophysics Data System (ADS)
Dauenhauer, Eric C.; Majdalani, Joseph
2003-06-01
This article describes a self-similarity solution of the Navier-Stokes equations for a laminar, incompressible, and time-dependent flow that develops within a channel possessing permeable, moving walls. The case considered here pertains to a channel that exhibits either injection or suction across two opposing porous walls while undergoing uniform expansion or contraction. Instances of direct application include the modeling of pulsating diaphragms, sweat cooling or heating, isotope separation, filtration, paper manufacturing, irrigation, and the grain regression during solid propellant combustion. To start, the stream function and the vorticity equation are used in concert to yield a partial differential equation that lends itself to a similarity transformation. Following this similarity transformation, the original problem is reduced to solving a fourth-order differential equation in one similarity variable η that combines both space and time dimensions. Since two of the four auxiliary conditions are of the boundary value type, a numerical solution becomes dependent upon two initial guesses. In order to achieve convergence, the governing equation is first transformed into a function of three variables: The two guesses and η. At the outset, a suitable numerical algorithm is applied by solving the resulting set of twelve first-order ordinary differential equations with two unspecified start-up conditions. In seeking the two unknown initial guesses, the rapidly converging inverse Jacobian method is applied in an iterative fashion. Numerical results are later used to ascertain a deeper understanding of the flow character. The numerical scheme enables us to extend the solution range to physical settings not considered in previous studies. Moreover, the numerical approach broadens the scope to cover both suction and injection cases occurring with simultaneous wall motion.
Higher-Order Compact Schemes for Numerical Simulation of Incompressible Flows
NASA Technical Reports Server (NTRS)
Wilson, Robert V.; Demuren, Ayodeji O.; Carpenter, Mark
1998-01-01
A higher order accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for 2D or 3D fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth and sixth order compact finite-difference schemes for spatial discretization. The particular difficulty of satisfying the divergence-free velocity field required in incompressible fluid flow is resolved by solving a Poisson equation for pressure. It is demonstrated that for consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Special care is also required to achieve the formal temporal accuracy of the Runge-Kutta schemes. The accuracy of the present procedure is demonstrated by application to several pertinent benchmark problems.
NASA Technical Reports Server (NTRS)
Mcmillin, S. Naomi; Thomas, James L.; Murman, Earll M.
1990-01-01
An Euler flow solver and a thin layer Navier-Stokes flow solver were used to numerically simulate the supersonic leeside flow fields over delta wings which were observed experimentally. Three delta wings with 75, 67.5, and 60 deg leading edge sweeps were computed over an angle-of-attack range of 4 to 20 deg at a Mach number 2.8. The Euler code and Navier-Stokes code predict equally well the primary flow structure where the flow is expected to be separated or attached at the leading edge based on the Stanbrook-Squire boundary. The Navier-Stokes code is capable of predicting both the primary and the secondary flow features for the parameter range investigated. For those flow conditions where the Euler code did not predict the correct type of primary flow structure, the Navier-Stokes code illustrated that the flow structure is sensitive to boundary layer model. In general, the laminar Navier-Stokes solutions agreed better with the experimental data, especially for the lower sweep delta wings. The computational results and a detailed re-examination of the experimental data resulted in a refinement of the flow classifications. This refinement in the flow classification results in the separation bubble with the shock flow type as the intermediate flow pattern between separated and attached flows.
NASA Astrophysics Data System (ADS)
Suzuki, Kensuke
A new analysis tool, an unsteady Hybrid Navier-Stokes/Vortex Model, for a horizontal axis wind turbine (HAWT) in yawed flow is presented, and its convergence and low cost computational performance are demonstrated. In earlier work, a steady Hybrid Navier-Stokes/Vortex Model was developed with a view to improving simulation results obtained by participants of the NASA Ames blind comparison workshop, following the NREL Unsteady Aerodynamics Experiment. The hybrid method was shown to better predict rotor torque and power over the range of wind speeds, from fully attached to separated flows. A decade has passed since the workshop was held and three dimensional unsteady Navier-Stokes analyses have become available using super computers. In the first chapter, recent results of unsteady Euler and Navier-Stokes computations are reviewed as standard references of what is currently possible and are contrasted with results of the Hybrid Navier-Stokes/Vortex Model in steady flow. In Chapter 2, the computational method for the unsteady Hybrid model is detailed. The grid generation procedure, using ICEM CFD, is presented in Chapter 3. Steady and unsteady analysis results for the NREL Phase IV rotor and for a modified "swept NREL rotor" are presented in Chapter 4-Chapter 7.
A unified multigrid solver for the Navier-Stokes equations on mixed element meshes
NASA Technical Reports Server (NTRS)
Mavriplis, D. J.; Venkatakrishnan, V.
1995-01-01
A unified multigrid solution technique is presented for solving the Euler and Reynolds-averaged Navier-Stokes equations on unstructured meshes using mixed elements consisting of triangles and quadrilaterals in two dimensions, and of hexahedra, pyramids, prisms, and tetrahedra in three dimensions. While the use of mixed elements is by no means a novel idea, the contribution of the paper lies in the formulation of a complete solution technique which can handle structured grids, block structured grids, and unstructured grids of tetrahedra or mixed elements without any modification. This is achieved by discretizing the full Navier-Stokes equations on tetrahedral elements, and the thin layer version of these equations on other types of elements, while using a single edge-based data-structure to construct the discretization over all element types. An agglomeration multigrid algorithm, which naturally handles meshes of any types of elements, is employed to accelerate convergence. An automatic algorithm which reduces the complexity of a given triangular or tetrahedral mesh by merging candidate triangular or tetrahedral elements into quadrilateral or prismatic elements is also described. The gains in computational efficiency afforded by the use of non-simplicial meshes over fully tetrahedral meshes are demonstrated through several examples.
Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains
Persson, P.-O.; Bonet, J.; Peraire, J.
2009-01-13
We describe a method for computing time-dependent solutions to the compressible Navier-Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time varying domain, By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration, The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge-Kutta method, For general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high order approximations on complex geometries.
Adaptively Refined Euler and Navier-Stokes Solutions with a Cartesian-Cell Based Scheme
NASA Technical Reports Server (NTRS)
Coirier, William J.; Powell, Kenneth G.
1995-01-01
A Cartesian-cell based scheme with adaptive mesh refinement for solving the Euler and Navier-Stokes equations in two dimensions has been developed and tested. Grids about geometrically complicated bodies were generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells were created using polygon-clipping algorithms. The grid was stored in a binary-tree data structure which provided a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations were solved on the resulting grids using an upwind, finite-volume formulation. The inviscid fluxes were found in an upwinded manner using a linear reconstruction of the cell primitives, providing the input states to an approximate Riemann solver. The viscous fluxes were formed using a Green-Gauss type of reconstruction upon a co-volume surrounding the cell interface. Data at the vertices of this co-volume were found in a linearly K-exact manner, which ensured linear K-exactness of the gradients. Adaptively-refined solutions for the inviscid flow about a four-element airfoil (test case 3) were compared to theory. Laminar, adaptively-refined solutions were compared to accepted computational, experimental and theoretical results.
Tetrahedral finite-volume solutions to the Navier-Stokes equations on complex configurations
NASA Astrophysics Data System (ADS)
Frink, N. T.; Pirzadeh, S. Z.
1999-09-01
A review of the algorithmic features and capabilities of the unstructured-grid flow solver USM3Dns is presented. This code, along with the tetrahedral grid generator, VGRIDns, is being extensively used throughout the USA for solving the Euler and Navier-Stokes equations on complex aerodynamic problems. Spatial discretization is accomplished by a tetrahedral cell-centered finite-volume formulation using Roe's upwind flux difference splitting. The fluxes are limited by either a Superbee or MinMod limiter. Solution reconstruction within the tetrahedral cells is accomplished with a simple, but novel, multidimensional analytical formula. Time is advanced by an implicit backward-Euler time-stepping scheme. Flow turbulence effects are modeled by the Spalart-Allmaras one-equation model, which is coupled with a wall function to reduce the number of cells in the near-wall region of the boundary layer. The issues of accuracy and robustness of USM3Dns Navier-Stokes capabilities are addressed for a flat-plate boundary layer, and a full F-16 aircraft with external stores at transonic speed.
A Galerkin-free model reduction approach for the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Shinde, Vilas; Longatte, Elisabeth; Baj, Franck; Hoarau, Yannick; Braza, Marianna
2016-03-01
Galerkin projection of the Navier-Stokes equations on Proper Orthogonal Decomposition (POD) basis is predominantly used for model reduction in fluid dynamics. The robustness for changing operating conditions, numerical stability in long-term transient behavior and the pressure-term consideration are generally the main concerns of the Galerkin Reduced-Order Models (ROM). In this article, we present a novel procedure to construct an off-reference solution state by using an interpolated POD reduced basis. A linear interpolation of the POD reduced basis is performed by using two reference solution states. The POD basis functions are optimal in capturing the averaged flow energy. The energy dominant POD modes and corresponding base flow are interpolated according to the change in operating parameter. The solution state is readily built without performing the Galerkin projection of the Navier-Stokes equations on the reduced POD space modes as well as the following time-integration of the resulted Ordinary Differential Equations (ODE) to obtain the POD time coefficients. The proposed interpolation based approach is thus immune from the numerical issues associated with a standard POD-Galerkin ROM. In addition, a posteriori error estimate and a stability analysis of the obtained ROM solution are formulated. A detailed case study of the flow past a cylinder at low Reynolds numbers is considered for the demonstration of proposed method. The ROM results show good agreement with the high fidelity numerical flow simulation.
Ducted-Fan Engine Acoustic Predictions using a Navier-Stokes Code
NASA Technical Reports Server (NTRS)
Rumsey, C. L.; Biedron, R. T.; Farassat, F.; Spence, P. L.
1998-01-01
A Navier-Stokes computer code is used to predict one of the ducted-fan engine acoustic modes that results from rotor-wake/stator-blade interaction. A patched sliding-zone interface is employed to pass information between the moving rotor row and the stationary stator row. The code produces averaged aerodynamic results downstream of the rotor that agree well with a widely used average-passage code. The acoustic mode of interest is generated successfully by the code and is propagated well upstream of the rotor; temporal and spatial numerical resolution are fine enough such that attenuation of the signal is small. Two acoustic codes are used to find the far-field noise. Near-field propagation is computed by using Eversman's wave envelope code, which is based on a finite-element model. Propagation to the far field is accomplished by using the Kirchhoff formula for moving surfaces with the results of the wave envelope code as input data. Comparison of measured and computed far-field noise levels show fair agreement in the range of directivity angles where the peak radiation lobes from the inlet are observed. Although only a single acoustic mode is targeted in this study, the main conclusion is a proof-of-concept: Navier-Stokes codes can be used both to generate and propagate rotor/stator acoustic modes forward through an engine, where the results can be coupled to other far-field noise prediction codes.
Navier-Stokes solutions with surface catalysis for Martian atmospheric entry
NASA Technical Reports Server (NTRS)
Chen, Y.-K.; Henline, W. D.; Stewart, D. A.; Candler, G. V.
1992-01-01
In this study numerical solutions have been obtained for two-dimensional axisymmetric hypersonic nonequilibrium CO2 flow over a high angle blunt cone with appropriate surface boundary conditions to account for energy and mass conservation at the body surface. The flowfield is described by the Navier-Stokes equations and multicomponent conservation laws which account for both translational and internal vibrational nonequilibrium effects. Complete forebody solutions have been obtained for the peak heating point of the Mars entry trajectory specified in the proposed NASA MESUR (Mars Environmental Survey) project. In these solutions, radiative equilibrium wall temperature and surface heating distributions are determined over the MESUR aeroshell forebody for entry velocity equal to 7 km/sec with varying degrees of surface catalysis. The effects of gas kinetics, surface catalysis, transport properties, and vibrational relaxation times on the surface heating are examined. The results identify some important issues in the prediction of surface heating for flows in thermochemical nonequilibrium and show that the Navier-Stokes code used herein is effective for thermal protection system design and materials selection.
Fast non-symmetric iterations and efficient preconditioning for Navier-Stokes equations
Silvester, D.; Elman, H.
1994-12-31
Discretisation of the steady-state Navier-Stokes equations: (u.grad)u-{nu}{del}{sup 2}u + grad p = 0; div u = 0 [1]. in some flow domain {Omega} {contained_in} IR{sup d}, (d = 2 or 3), gives a system of non-linear algebraic equations for discretised variables u (the velocity), and p (the pressure). The authors assume that appropriate boundary conditions are imposed. The non-linear equation system can be linearised using a fixed-point (Picard) iteration to give a matrix system which must be solved at every iteration. Part of this matrix is block diagonal, and consists of d convection-diffusion operators, one for each component of velocity. Two difficulties arise when solving this matrix equation. Firstly, the block diagonal part is not symmetric, although under certain conditions the symmetric part is positive definite. Secondly, the overall system is indefinite. This makes the design of fast and efficient iterative solvers for discretised Navier-Stokes operators an extremely challenging task.
Tetrahedral Finite-Volume Solutions to the Navier-Stokes Equations on Complex Configurations
NASA Technical Reports Server (NTRS)
Frink, Neal T.; Pirzadeh, Shahyar Z.
1998-01-01
A review of the algorithmic features and capabilities of the unstructured-grid flow solver USM3Dns is presented. This code, along with the tetrahedral grid generator, VGRIDns, is being extensively used throughout the U.S. for solving the Euler and Navier-Stokes equations on complex aerodynamic problems. Spatial discretization is accomplished by a tetrahedral cell-centered finite-volume formulation using Roe's upwind flux difference splitting. The fluxes are limited by either a Superbee or MinMod limiter. Solution reconstruction within the tetrahedral cells is accomplished with a simple, but novel, multidimensional analytical formula. Time is advanced by an implicit backward-Euler time-stepping scheme. Flow turbulence effects are modeled by the Spalart-Allmaras one-equation model, which is coupled with a wall function to reduce the number of cells in the near-wall region of the boundary layer. The issues of accuracy and robustness of USM3Dns Navier-Stokes capabilities are addressed for a flat-plate boundary layer, and a full F-16 aircraft with external stores at transonic speed.
Solution of Reynolds-averaged Navier-Stokes equations by discontinuous Galerkin method
NASA Astrophysics Data System (ADS)
Kang, Sungwoo; Yoo, Jung Yul
2007-11-01
Discontinuous Galerkin method is a finite element method that allows discontinuities at inter-element boundaries. The discontinuities in the method are treated by approximate Riemann solvers. One important feature of the method is that it obtains high-order accuracy for unstructured mesh with no difficulty. Due to this feature, it can be useful for various practical applications to turbulence and aeroacoustics, but there are few problems to be solved before the method is applicable to practical flow problems. Due to discontinuous approximations in discontinuous Galerkin method, the treatments of viscous terms are complicated and expensive. Moreover, careful treatments of source terms in turbulence model equations are necessary for Reynolds-averaged Navier-Stokes equations to prevent blow-up of high-order-accurate simulations. In this study, we compare high-order accurate discontinuous Galerkin method with different viscous treatments and stabilization of source terms for compressible Reynold-averaged Navier-Stokes equations. Spalart-Allmaras or k-φ model is used for turbulence model. To compare the implemented formulations, steady turbulent flow over a flat plate and unsteady turbulent flow over cavity are solved.
Computation of transonic separated wing flows using an Euler/Navier-Stokes zonal approach
NASA Technical Reports Server (NTRS)
Kaynak, Uenver; Holst, Terry L.; Cantwell, Brian J.
1986-01-01
A computer program called Transonic Navier Stokes (TNS) has been developed which solves the Euler/Navier-Stokes equations around wings using a zonal grid approach. In the present zonal scheme, the physical domain of interest is divided into several subdomains called zones and the governing equations are solved interactively. The advantages of the Zonal Grid approach are as follows: (1) the grid for any subdomain can be generated easily; (2) grids can be, in a sense, adapted to the solution; (3) different equation sets can be used in different zones; and, (4) this approach allows for a convenient data base organization scheme. Using this code, separated flows on a NACA 0012 section wing and on the NASA Ames WING C have been computed. First, the effects of turbulence and artificial dissipation models incorporated into the code are assessed by comparing the TNS results with other CFD codes and experiments. Then a series of flow cases is described where data are available. The computed results, including cases with shock-induced separation, are in good agreement with experimental data. Finally, some futuristic cases are presented to demonstrate the abilities of the code for massively separated cases which do not have experimental data.
Navier-Stokes analysis of three-dimensional flow and heat transfer inside turbine blade rows
NASA Technical Reports Server (NTRS)
Hah, C.
1993-01-01
A numerical method for solving the three-dimensional, Navier-Stokes equations for unsteady, viscous flow and heat transfer through multiple turbomachinery blade rows is presented. The method solves the fully three-dimensional Navier-Stokes equations with an implicit scheme which is based on a control volume approach. A two-equation turbulence model with a low Reynolds number modification is employed. A third-order accurate upwinding scheme is used to approximate convection terms while a second order accurate central difference scheme is used for the discretization of viscous terms. A second-order accurate scheme is employed for the temporal discretization. The numerical method is applied to study the unsteady flow and heat transfer field of the High Pressure Fuel side Turbo-Pump (HPFTP) of the Space Shuttle Main Engine (SSME). The stage calculation is performed by coupling the stator and the rotor flow fields at each time step through an over-laid grid. Numerical results for the complete geometry with the vane trailing edge cutback are presented and compared with the available experimental data.
NASA Astrophysics Data System (ADS)
Almazán, Lidia; Carrillo, José A.; Salueña, Clara; Garzó, Vicente; Pöschel, Thorsten
2013-04-01
A numerical study that aims to analyze the thermal mechanisms of unsteady, supersonic granular flow by means of hydrodynamic simulations of the Navier-Stokes granular equation is reported in this paper. For this purpose, a paradigmatic problem in granular dynamics such as the Faraday instability is selected. Two different approaches for the Navier-Stokes transport coefficients for granular materials are considered, namely the traditional Jenkins-Richman theory for moderately dense quasi-elastic grains and the improved Garzó-Dufty-Lutsko theory for arbitrary inelasticity, which we also present here. Both the solutions are compared with event-driven simulations of the same system under the same conditions, by analyzing the density, temperature and velocity field. Important differences are found between the two approaches, leading to interesting implications. In particular, the heat transfer mechanism coupled to the density gradient, which is a distinctive feature of inelastic granular gases, is responsible for a major discrepancy in the temperature field and hence in the diffusion mechanisms.
Flowfield Comparisons from Three Navier-Stokes Solvers for an Axisymmetric Separate Flow Jet
NASA Technical Reports Server (NTRS)
Koch, L. Danielle; Bridges, James; Khavaran, Abbas
2002-01-01
To meet new noise reduction goals, many concepts to enhance mixing in the exhaust jets of turbofan engines are being studied. Accurate steady state flowfield predictions from state-of-the-art computational fluid dynamics (CFD) solvers are needed as input to the latest noise prediction codes. The main intent of this paper was to ascertain that similar Navier-Stokes solvers run at different sites would yield comparable results for an axisymmetric two-stream nozzle case. Predictions from the WIND and the NPARC codes are compared to previously reported experimental data and results from the CRAFT Navier-Stokes solver. Similar k-epsilon turbulence models were employed in each solver, and identical computational grids were used. Agreement between experimental data and predictions from each code was generally good for mean values. All three codes underpredict the maximum value of turbulent kinetic energy. The predicted locations of the maximum turbulent kinetic energy were farther downstream than seen in the data. A grid study was conducted using the WIND code, and comments about convergence criteria and grid requirements for CFD solutions to be used as input for noise prediction computations are given. Additionally, noise predictions from the MGBK code, using the CFD results from the CRAFT code, NPARC, and WIND as input are compared to data.
Navier-Stokes flow in the weighted Hardy space with applications to time decay problem
NASA Astrophysics Data System (ADS)
Okabe, Takahiro; Tsutsui, Yohei
2016-08-01
The asymptotic expansions of the Navier-Stokes flow in Rn and the rates of decay are studied with aid of weighted Hardy spaces. Fujigaki and Miyakawa [12], Miyakawa [28] proved the nth order asymptotic expansion of the Navier-Stokes flow if initial data decays like (1 + | x |)-n-1 and if nth moment of initial data is finite. In the present paper, it is clarified that the moment condition for initial data is essential in order to obtain higher order asymptotic expansion of the flow and to consider the rapid time decay problem. The second author [39] established the weighted estimates of the strong solutions in the weighted Hardy spaces with small initial data which belongs to Ln and a weighed Hardy space. Firstly, the refinement of the previous work [39] is achieved with alternative proof. Then the existence time of the solution in the weighted Hardy spaces is characterized without any Hardy norm. As a result, in two dimensional case the smallness condition on initial data is completely removed. As an application, the rapid time decay of the flow is investigated with aid of asymptotic expansions and of the symmetry conditions introduced by Brandolese [3].
A point implicit unstructured grid solver for the Euler and Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Thareja, Rajiv R.; Stewart, James R.; Hassan, Obey; Morgan, Ken; Peraire, Jaime
1988-01-01
An upwind finite element technique that uses cell centered quantities and implicit and/or explicit time marching has been developed for computing hypersonic laminar viscous flows using adaptive unstructured triangular grids. A structured grid of quadrilaterals is laid out near the body surface. For inviscid flows the method is stable at Courant numbers of over 100,000. A first order basic scheme and a higher order flux corrected transport (FCT) scheme have been implemented. This technique has been applied to the problem of predicting type III and IV shock wave interactions on a cylinder, with a view of simulating the pressure and heating rate augmentation caused by an impinging shock on the leading edge of a cowl lip of an engine inlet. The predictions of wall pressure and heating rates compare very well with experimental data. The flow features are very distinctly captured with a sequence of adaptively generated grids. The adaptive mesh generator and the upwind Navier-Stokes solver are combined in a set of programs called LARCNESS, an acronym for Langley Adaptive Remeshing Code and Navier-Stokes Solver.
Analysis of spurious oscillation modes for the shallow water and Navier-Stokes equations
Walters, R.A.; Carey, G.F.
1983-01-01
The origin and nature of spurious oscillation modes that appear in mixed finite element methods are examined. In particular, the shallow water equations are considered and a modal analysis for the one-dimensional problem is developed. From the resulting dispersion relations we find that the spurious modes in elevation are associated with zero frequency and large wave number (wavelengths of the order of the nodal spacing) and consequently are zero-velocity modes. The spurious modal behavior is the result of the finite spatial discretization. By means of an artificial compressibility and limiting argument we are able to resolve the similar problem for the Navier-Stokes equations. The relationship of this simpler analysis to alternative consistency arguments is explained. This modal approach provides an explanation of the phenomenon in question and permits us to deduce the cause of the very complex behavior of spurious modes observed in numerical experiments with the shallow water equations and Navier-Stokes equations. Furthermore, this analysis is not limited to finite element formulations, but is also applicable to finite difference formulations. ?? 1983.
Source Term Model for Steady Micro Jets in a Navier-Stokes Computer Code
NASA Technical Reports Server (NTRS)
Waithe, Kenrick A.
2005-01-01
A source term model for steady micro jets was implemented into a non-proprietary Navier-Stokes computer code, OVERFLOW. The source term models the mass flow and momentum created by a steady blowing micro jet. The model is obtained by adding the momentum and mass flow created by the jet to the Navier-Stokes equations. The model was tested by comparing with data from numerical simulations of a single, steady micro jet on a flat plate in two and three dimensions. The source term model predicted the velocity distribution well compared to the two-dimensional plate using a steady mass flow boundary condition, which was used to simulate a steady micro jet. The model was also compared to two three-dimensional flat plate cases using a steady mass flow boundary condition to simulate a steady micro jet. The three-dimensional comparison included a case with a grid generated to capture the circular shape of the jet and a case without a grid generated for the micro jet. The case without the jet grid mimics the application of the source term. The source term model compared well with both of the three-dimensional cases. Comparisons of velocity distribution were made before and after the jet and Mach and vorticity contours were examined. The source term model allows a researcher to quickly investigate different locations of individual or several steady micro jets. The researcher is able to conduct a preliminary investigation with minimal grid generation and computational time.
Further Development of a New, Flux-Conserving Newton Scheme for the Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Scott, James R.
1996-01-01
This paper is one of a series of papers describing the development of a new numerical approach for solving the steady Navier-Stokes equations. The key features in the current development are (1) the discrete representation of the dependent variables by way of high order polynomial expansions, (2) the retention of all derivatives in the expansions as unknowns to be explicitly solved for, (3) the automatic balancing of fluxes at cell interfaces, and (4) the discrete simulation of both the integral and differential forms of the governing equations. The main purpose of this paper is, first, to provide a systematic and rigorous derivation of the conditions that are used to simulate the differential form of the Navier-Stokes equations, and second, to extend our previously-presented internal flow scheme to external flows and nonuniform grids. Numerical results are presented for high Reynolds number flow (Re = 100,000) around a finite flat plate, and detailed comparisons are made with the Blasius flat plate solution and Goldstein wake solution. It is shown that the error in the streamwise velocity decreases like r(sup alpha)(Delta)y(exp 2), where alpha approx. 0.25 and r = delta(y)/delta(x) is the grid aspect ratio.
A new nonlinear turbulence model based on Partially-Averaged Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Liu, J. T.; Wu, Y. L.; Cai, C.; Liu, S. H.; Wang, L. Q.
2013-12-01
Partially-averaged Navier-Stokes (PANS) Model was recognized as a Reynolds-averaged Navier-Stokes (RANS) to direct numerical simulation (DNS) bridging method. PANS model was purported for any filter width-from RANS to DNS. PANS method also shared some similarities with the currently popular URANS (unsteady RANS) method. In this paper, a new PANS model was proposed, which was based on RNG k-ε turbulence model. The Standard and RNG k-ε turbulence model were both isotropic models, as well as PANS models. The sheer stress in those PANS models was solved by linear equation. The linear hypothesis was not accurate in the simulation of complex flow, such as stall phenomenon. The sheer stress here was solved by nonlinear method proposed by Ehrhard. Then, the nonlinear PANS model was set up. The pressure coefficient of the suction side of the NACA0015 hydrofoil was predicted. The result of pressure coefficient agrees well with experimental result, which proves that the nonlinear PANS model can capture the high pressure gradient flow. A low specific centrifugal pump was used to verify the capacity of the nonlinear PANS model. The comparison between the simulation results of the centrifugal pump and Particle Image Velocimetry (PIV) results proves that the nonlinear PANS model can be used in the prediction of complex flow field.
The Proteus Navier-Stokes code. [two and three dimensional computational fluid dynamics
NASA Technical Reports Server (NTRS)
Towne, Charles E.; Schwab, John R.
1992-01-01
An effort is currently underway at NASA Lewis to develop two and three dimensional Navier-Stokes codes, called Proteus, for aerospace propulsion applications. Proteus solves the Reynolds-averaged, unsteady, compressible Navier-Stokes equations in strong conservation law form. Turbulence is modeled using a Baldwin-Lomax based algebraic eddy viscosity model. In addition, options are available to solve thin layer or Euler equations, and to eliminate the energy equation by assuming constant stagnation enthalpy. An extensive series of validation cases have been run, primarily using the two dimensional planar/axisymmetric version of the code. Several flows were computed that have exact solution such as: fully developed channel and pipe flow; Couette flow with and without pressure gradients; unsteady Couette flow formation; flow near a suddenly accelerated flat plate; flow between concentric rotating cylinders; and flow near a rotating disk. The two dimensional version of the Proteus code has been released, and the three dimensional code is scheduled for release in late 1991.
Calculation of a simulated 3-D high speed inlet using the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Knight, D. D.
1983-01-01
A hybrid numerical algorithm, developed to solve the full three-dimensional Navier-Stokes equations, is applied to the computation of the flowfield in a simulated three-dimensional high speed aircraft inlet at a Mach number of 2.5 and Reynolds number of 1.4 x 10 to the 7th based on inlet length. The numerical algorithm incorporates a coordinate transformation in order to handle general flow geometries, and utilizes the algebraic turbulent eddy viscosity model of Baldwin and Lomax. The hybrid algorithm has been vectorized on the CDC CYBER 203 computer using the SL/1 vector programming language developed at NASA Langley. The computed results are compared with experimental measurements of the ramp and cowl static pressures, and boundary layer pitot profiles. The results are also compared with a previous two-dimensional Navier-Stokes computation of the same configuration. The agreement with the experimental data is generally good; however, additional improvements in turbulence modeling are needed.
Turbulent solution of the Navier-Stokes equations for uniform shear flow
NASA Technical Reports Server (NTRS)
Deissler, R. G.
1981-01-01
To study the nonlinear physics of uniform turbulent shear flow, the unaveraged Navier-Stokes equations are solved numerically. This extends our previous work in which mean gradients were absent. For initial conditions, modified three-dimensional-cosine velocity fluctuations are used. The boundary conditions are modified periodic conditions on a stationary three-dimensional numerical grid. A uniform mean shear is superimposed on the initial and boundary conditions. The three components of the mean-square velocity fluctuations are initially equal for the conditions chosen. As in the case of no shear the initially nonrandom flow develops into an apparently random turbulence at higher Reynolds number. Thus, randomness or turbulence can apparently arise as a consequence of the structure of the Navier-Stokes equations. Except for an initial period of adjustment, all fluctuating components grow with time. The initial equality of the three intensity components is destroyed by the shear, the transverse components becoming smaller than the longitudinal one, in agreement with experiment. Also, the shear creates a small-scale structure in the turbulence. The nonlinear solutions are compared with linearized ones.
NASA Astrophysics Data System (ADS)
Luo, Dahai; Yan, Chao; Wang, Xiaoyong
2015-02-01
Separation commonly exists in the flows around flight vehicles and also in the internal combustor flows. Simulation of high-speed turbulent-separated flows using a reliable computational design tool is crucial for the development of supersonic and hypersonic vehicles. In this paper, we present the computational results of supersonic base and ramped-cavity flows at high Reynolds numbers using the partially averaged Navier-Stokes (PANS) method. The current PANS models are based on the Menter SST turbulence model and also the Wilcox k-ω model. Results from PANS simulations are compared in detail with the available experimental data. The effect of the resolution control parameter fk (the ratio of unresolved-to-total kinetic energy) relevant to the PANS method is investigated. More turbulent flow structures are resolved as expected with decreasing fk, but it does not mean better results can be obtained. Spatially varying and dynamically updated fk in PANS simulations has been performed. Results from variable fk PANS simulations show good agreement with the experiment and great improvement when compared to Reynolds averaged Navier-Stokes (RANS) computation and constant fk PANS simulations.
Looking for O(N) Navier-Stokes solutions on non-structured meshes
NASA Technical Reports Server (NTRS)
Morano, Eric; Dervieux, Alain
1993-01-01
Multigrid methods are good candidates for the resolution of the system arising in numerical fluid dynamics. However, the question is to know if those algorithms which are efficient for the Poisson equation on structured meshes will still apply well to the Euler and Navier-Stokes equations on unstructured meshes. The study of elliptic problems leads us to define the conditions where a full multigrid strategy has O(N) complexity. The aim of this paper is to build a comparison between the elliptic theory and practical CFD problems. First, as an introduction, we will recall some basic definitions and theorems applied to a model problem. The goal of this section is to point out the different properties that we need to produce an FMG algorithm with O(N) complexity. Then, we will show how we can apply this theory to the fluid dynamics equations such as Euler and Navier-Stokes equations. At last, we present some results which are 2nd-order accurate and some explanations about the behavior of the FMG process.
Looking for O(N) Navier-Stokes solutions on non-structured meshes
NASA Technical Reports Server (NTRS)
Morano, Eric; Dervieux, Alain
1993-01-01
Multigrid methods are good candidates for the resolution of the system arising in Numerical Fluid Dynamics. However, the question is to know if those algorithms which are efficient for the Poissan equation on structured meshes will still apply well to the Euler and Navier-Stokes equations on unstructured meshes. The study of elliptic problems leads us to define the conditions where a Full Multigrid strategy has O(N) complexity. The aim of this paper is to build a comparison between the elliptic theory and practical CFD problems. First, as an introduction, we will recall some basic definitions and theorems applied to a model problem. The goal of this section is to point out the different properties that we need to produce an FMG algorithm with O(N) complexity. Then, we will show how we can apply this theory to the fluid dynamics equations such as Euler and Navier-Stokes equations. At last, we present some results which are 2nd-order accurate and some explanations about the behavior of the FMG process.
Using Navier-Stokes to Characterize Re-Entry of Microscale Vehicles
NASA Astrophysics Data System (ADS)
Thiruvenkadam, Sudharsan; Ben, Harris
2013-11-01
Atmospheric reentry vehicles experience different flow regimes during flight due to the change in atmospheric density. This change in density creates non-equilibrium regions on the order of one mean free path, called as Knudsen layer. In the design of atmospheric reentry vehicles, the flux variations near solid surface are of critical importance. The traditional CFD simulations which use Navier Stokes equations fail to predict the flow in Knudsen layer. These areas where the rarefaction effects begin to dominate can be quantified by the Knudsen breakdown parameter. The Direct Simulation Monte Carlo (DSMC) method, although accurate for all flow regimes, it is computationally expensive as the number of simulating molecules increases. We developed a method that models the Knudsen Layer by using Navier Stokes equations with Maxwell-Smoluchowski slip boundary conditions and DSMC for low (Kn < 0.1) and high (Kn > 0.1) Knudsen numbers respectively. This study investigates the surface properties of a flat plate with Nitrogen gas flow from continuum to rarefied regimes. Computational fluid dynamics and DSMC results are obtained for different test conditions. The results demonstrate that the Knudsen layer can be predicted with DSMC and continuum approach for all flow regimes.
Numerical simulation of jet aerodynamics using the three-dimensional Navier-Stokes code PAB3D
NASA Technical Reports Server (NTRS)
Pao, S. Paul; Abdol-Hamid, Khaled S.
1996-01-01
This report presents a unified method for subsonic and supersonic jet analysis using the three-dimensional Navier-Stokes code PAB3D. The Navier-Stokes code was used to obtain solutions for axisymmetric jets with on-design operating conditions at Mach numbers ranging from 0.6 to 3.0, supersonic jets containing weak shocks and Mach disks, and supersonic jets with nonaxisymmetric nozzle exit geometries. This report discusses computational methods, code implementation, computed results, and comparisons with available experimental data. Very good agreement is shown between the numerical solutions and available experimental data over a wide range of operating conditions. The Navier-Stokes method using the standard Jones-Launder two-equation kappa-epsilon turbulence model can accurately predict jet flow, and such predictions are made without any modification to the published constants for the turbulence model.
An, Hongli; Yuen, Manwai
2014-05-15
In this paper, we investigate the analytical solutions of the compressible Navier-Stokes equations with dependent-density viscosity. By using the characteristic method, we successfully obtain a class of drifting solutions with elliptic symmetry for the Navier-Stokes model wherein the velocity components are governed by a generalized Emden dynamical system. In particular, when the viscosity variables are taken the same as Yuen [M. W. Yuen, “Analytical solutions to the Navier-Stokes equations,” J. Math. Phys. 49, 113102 (2008)], our solutions constitute a generalization of that obtained by Yuen. Interestingly, numerical simulations show that the analytical solutions can be used to explain the drifting phenomena of the propagation wave like Tsunamis in oceans.
NASA Astrophysics Data System (ADS)
Wang, Yong
2016-09-01
In the present paper, we study the uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system whose viscosity and heat conductivity are allowed to vanish at different orders. The problem is studied in a three dimensional bounded domain with Navier-slip type boundary conditions. It is shown that there exists a unique strong solution to the full compressible Navier-Stokes system with the boundary conditions in a finite time interval which is independent of the viscosity and heat conductivity. The solution is uniformly bounded in {W^{1,infty}} and is a conormal Sobolev space. Based on such uniform estimates, we prove the convergence of the solutions of the full compressible Navier-Stokes to the corresponding solutions of the full compressible Euler system in {L^infty(0,T; L^2)}, {L^infty(0,T; H1)} and {L^infty([0,T]×Ω)} with a rate of convergence.
NASA Astrophysics Data System (ADS)
Peng, NaiFu; Guan, Hui; Wu, ChuiJie
2016-04-01
In this paper, the theory of constructing optimal dynamical systems based on weighted residual presented by Wu & Sha is applied to three-dimensional Navier-Stokes equations, and the optimal dynamical system modeling equations are derived. Then the multiscale global optimization method based on coarse graining analysis is presented, by which a set of approximate global optimal bases is directly obtained from Navier-Stokes equations and the construction of optimal dynamical systems is realized. The optimal bases show good properties, such as showing the physical properties of complex flows and the turbulent vortex structures, being intrinsic to real physical problem and dynamical systems, and having scaling symmetry in mathematics, etc.. In conclusion, using fewer terms of optimal bases will approach the exact solutions of Navier-Stokes equations, and the dynamical systems based on them show the most optimal behavior.
A high-order gas-kinetic Navier-Stokes flow solver
Li Qibing; Xu Kun; Fu Song
2010-09-20
The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge-Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge-Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier-Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier-Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier-Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such
Parallel performance investigations of an unstructured mesh Navier-Stokes solver
NASA Technical Reports Server (NTRS)
Mavriplis, Dimitri J.
2000-01-01
A Reynolds-averaged Navier-Stokes solver based on unstructured mesh techniques for analysis of high-lift configurations is described. The method makes use of an agglomeration multigrid solver for convergence acceleration. Implicit line-smoothing is employed to relieve the stiffness associated with highly stretched meshes. A GMRES technique is also implemented to speed convergence at the expense of additional memory usage. The solver is cache efficient and fully vectorizable, and is parallelized using a two-level hybrid MPI-OpenMP implementation suitable for shared and/or distributed memory architectures, as well as clusters of shared memory machines. Convergence and scalability results are illustrated for various high-lift cases.
A concurrent hybrid Navier-Stokes/Euler approach to fluid dynamic computations
NASA Technical Reports Server (NTRS)
Tavella, Domingo A.; Djomehri, M. J.; Kislitzin, Katherine T.; Blake, Matthew W.; Erickson, Larry L.
1993-01-01
We present a methodology for the numerical simulation of flow fields by the simultaneous application of two distinct approaches to computational aerodynamics. We compute the three dimensional flow field of a missile at moderate angle of attack by dividing the flow field into two regions: a region near the surface where we use a structured grid and a Navier Stokes solver, and a region farther away from the surface where we utilize an unstructured grid and an Euler solver. The two solvers execute as independent UNIX processes either on the same machine or on two machines. The solvers communicate data across their common interfaces within the same machine or over the network. The computations indicate that extensively separated flow fields can be computed without significant distortion by combining viscous and inviscid solvers.
Navier-Stokes computations about complex configurations including a complete F-16 aircraft
NASA Technical Reports Server (NTRS)
Holst, Terry L.; Flores, Jolen; Chaderjian, Neal M.; Kaynak, Unver
1990-01-01
Transonic Navier-Stokes (TNS) code solutions gathered from the literature for three-dimensional geometries, including two different wings and a complete F-16A aircraft, are presently discussed. The TNS codes use a zonal grid approach whose number of zones vary from four to 54. The Euler equations are solved in zones away from no-slip surfaces, and the thin-layer TNS equations are solved in all zones immediately adjacent to no-slip surfaces. In the case of 'corner' zones possessing no-slip boundary conditions on two different surfaces, a thin-layer formulation along both directions is employed. Employing these features, a zonal construction with the requisite set of boundary conditions can be devised for almost any application.
A vertex-based finite-volume algorithm for the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Chakrabartty, S. K.; Dhanalakshmi, K.
1993-07-01
A vertex-based, finite-volume algorithm has been developed to solve the Reynolds-averaged Navier-Stokes equations without thin-layer approximation. An explicit, five-stage Runge-Kutta, time-stepping scheme has been used for time integration along with different acceleration techniques to reach the steady state. A code employing multi-block grid structure has been developed. This code can accept any type of grid topology. As test cases, the turbulent flow past RAE-2822 and NACA-0012 airfoils, and the laminar flow past a cropped delta wing at ten degrees angle of attack have been computed and the results compared with available numerical and experimental results. The Baldwin-Lomax turbulence model has been used in the case of turbulent flows.
Parallel computation of 3-D Navier-Stokes flowfields for supersonic vehicles
NASA Technical Reports Server (NTRS)
Ryan, James S.; Weeratunga, Sisira
1993-01-01
Multidisciplinary design optimization of aircraft will require unprecedented capabilities of both analysis software and computer hardware. The speed and accuracy of the analysis will depend heavily on the computational fluid dynamics (CFD) module which is used. A new CFD module has been developed to combine the robust accuracy of conventional codes with the ability to run on parallel architectures. This is achieved by parallelizing the ARC3D algorithm, a central-differenced Navier-Stokes method, on the Intel iPSC/860. The computed solutions are identical to those from conventional machines. Computational speed on 64 processors is comparable to the rate on one Cray Y-MP processor and will increase as new generations of parallel computers become available.
NASA Technical Reports Server (NTRS)
Cook, C. H.
1977-01-01
The results of a comprehensive numerical investigation of the basic capabilities of the finite element method (FEM) for numerical solution of compressible flow problems governed by the two-dimensional and axis-symmetric Navier-Stokes equations in primitive variables are presented. The strong and weak points of the method as a tool for computational fluid dynamics are considered. The relation of the linear element finite element method to finite difference methods (FDM) is explored. The calculation of free shear layer and separated flows over aircraft boattail afterbodies with plume simulators indicate the strongest assets of the method are its capabilities for reliable and accurate calculation employing variable grids which readily approximate complex geometry and capably adapt to the presence of diverse regions of large solution gradients without the necessity of domain transformation.
Efficiency and Accuracy of Time-Accurate Turbulent Navier-Stokes Computations
NASA Technical Reports Server (NTRS)
Rumsey, Christopher L.; Sanetrik, Mark D.; Biedron, Robert T.; Melson, N. Duane; Parlette, Edward B.
1995-01-01
The accuracy and efficiency of two types of subiterations in both explicit and implicit Navier-Stokes codes are explored for unsteady laminar circular-cylinder flow and unsteady turbulent flow over an 18-percent-thick circular-arc (biconvex) airfoil. Grid and time-step studies are used to assess the numerical accuracy of the methods. Nonsubiterative time-stepping schemes and schemes with physical time subiterations are subject to time-step limitations in practice that are removed by pseudo time sub-iterations. Computations for the circular-arc airfoil indicate that a one-equation turbulence model predicts the unsteady separated flow better than an algebraic turbulence model; also, the hysteresis with Mach number of the self-excited unsteadiness due to shock and boundary-layer separation is well predicted.
Calculation of Turbulent Subsonic Diffuser Flows Using the NPARC Navier-Stokes Code
NASA Technical Reports Server (NTRS)
Dudek, J. C.; Georgiadis, N. J.; Yoder, D. A.
1996-01-01
Axisymmetric subsonic diffuser flows were calculated with the NPARC Navier-Stokes code in order to determine the effects various code features have on the flow solutions. The code features examined in this work were turbulence models and boundary conditions. Four turbulence models available in NPARC were used: the Baldwin-Lomax algebraic model, the Baldwin-Barth one-equation model, and the Chien kappa-epsilon and Wilcox kappa-omega two-equation models. The three boundary conditions examined were the free boundary, the mass flux boundary and the subsonic outflow with variable static pressure. In addition to boundary condition type, the geometry downstream of the diffuser was varied to see if upstream influences were present. The NPARC results are compared with experimental data and recommendations are given for using NPARC to compute similar flows.
Recent advances in Runge-Kutta schemes for solving 3-D Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Vatsa, Veer N.; Wedan, Bruce W.; Abid, Ridha
1989-01-01
A thin-layer Navier-Stokes has been developed for solving high Reynolds number, turbulent flows past aircraft components under transonic flow conditions. The computer code has been validated through data comparisons for flow past isolated wings, wing-body configurations, prolate spheroids and wings mounted inside wind-tunnels. The basic code employs an explicit Runge-Kutta time-stepping scheme to obtain steady state solution to the unsteady governing equations. Significant gain in the efficiency of the code has been obtained by implementing a multigrid acceleration technique to achieve steady-state solutions. The improved efficiency of the code has made it feasible to conduct grid-refinement and turbulence model studies in a reasonable amount of computer time. The non-equilibrium turbulence model of Johnson and King has been extended to three-dimensional flows and excellent agreement with pressure data has been obtained for transonic separated flow over a transport type of wing.
Navier-Stokes simulation of the crossflow instability in swept-wing flows
NASA Technical Reports Server (NTRS)
Reed, Helen L.
1989-01-01
The computational modeling of the transition process characteristic of flows over swept wings are described. Specifically, the crossflow instability and crossflow/T-S wave interactions are analyzed through the numerical solution of the full three-dimensional Navier-Stokes equations including unsteadiness, curvature, and sweep. This approach is chosen because of the complexity of the problem and because it appears that linear stability theory is insufficient to explain the discrepancies between different experiments and between theory and experiments. The leading edge region of a swept wing is considered in a three-dimensional spatial simulation with random disturbances as the initial conditions. The work has been closely coordinated with the experimental program of Professor William Saric, examining the same problem. Comparisons with NASA flight test data and the experiments at Arizona State University were a necessary and an important integral part of this work.
NASA Technical Reports Server (NTRS)
Chen, Shu-Cheng; Liu, Nan-Suey; Kim, Hyun Dae
1991-01-01
Presented here is an algorithm for solving the multidimensional unsteady Navier-Stokes equations for compressible flows. It is based on a diagonally-dominant approximate factorization procedure. The factorization error and the timewise linearization error associated with this procedure are reduced by performing Newton-type inner iterations at each time step. The inviscid fluxes are evaluated by the fourth-order central differencing scheme amended with a numerical dissipation directly proportional to the entire dissipative part of the truncation error intrinsic to the third order biased upwind scheme. The important features of the proposed solution are elucidated by the numerical results of the convection of a vortex and the backward-facing step flows.
Two-dimensional Navier-Stokes heat transfer analysis for rough turbine blades
NASA Technical Reports Server (NTRS)
Boyle, R. J.; Civinskas, K. C.
1991-01-01
A quasi-three-dimensional thin-layer Navier-Stokes analysis was used to predict heat transfer to rough surfaces. Comparisons are made between predicted and experimental heat transfer for turbine blades and flat plates of known roughness. The effect of surface roughness on heat transfer was modeled using a mixing length approach. The effect of near-wall grid spacing and convergence criteria on the accuracy of the heat transfer predictions are examined. An eddy viscosity mixing length model having an inner and outer layer was used. A discussion of the appropriate model for the crossover between the inner and outer layers is included. The analytic results are compared with experimental data for both flat plates and turbine blade geometries. Comparisons between predicted and experimental heat transfer showed that a modeling roughness effects using a modified mixing length approach results in good predictions of the trends in heat transfer due to roughness.
NASA Technical Reports Server (NTRS)
Vatsa, Veer N.; Turkel, Eli L.
2004-01-01
We report research experience in applying an Unsteady Reynolds-Averaged Navier-Stokes (URANS) solver for the prediction of time-dependent flows in the presence of an active flow control device. The configuration under consideration is a synthetic jet created by a single diaphragm piezoelectric actuator in quiescent air. Time-averaged and instantaneous data for this case were obtained at Langley Research Center, using multiple measurement techniques. Computational results for this case using one-equation Spalart-Allmaras and two-equation Menter s turbulence models are presented here along with comparisons with the experimental data. The effect of grid refinement, preconditioning and time-step variation are also examined.
Navier-Stokes simulation of rotor-body flowfield in hover using overset grids
NASA Technical Reports Server (NTRS)
Srinivasan, G. R.; Ahmad, J. U.
1993-01-01
A free-wake Navier-Stokes numerical scheme and multiple Chimera overset grids have been utilized for calculating the quasi-steady hovering flowfield of a Boeing-360 rotor mounted on an axisymmetric whirl-tower. The entire geometry of this rotor-body configuration is gridded-up with eleven different overset grids. The composite grid has 1.3 million grid points for the entire flow domain. The numerical results, obtained using coarse grids and a rigid rotor assumption, show a thrust value that is within 5% of the experimental value at a flow condition of M(sub tip) = 0.63, Theta(sub c) = 8 deg, and Re = 2.5 x 10(exp 6). The numerical method thus demonstrates the feasibility of using a multi-block scheme for calculating the flowfields of complex configurations consisting of rotating and non-rotating components.
Analytic solutions for the three-dimensional compressible Navier-Stokes equation
NASA Astrophysics Data System (ADS)
Barna, I. F.; Mátyás, L.
2014-10-01
We investigate the three-dimensional compressible Navier-Stokes (NS) and the continuity equations in Cartesian coordinates for Newtonian fluids. The problem has an importance in different fields of science and engineering like fluid, aerospace dynamics or transfer processes. Finding an analytic solution may bring considerable progress in understanding the transport phenomena and in the design of different equipments where the NS equation is applicable. For solving the equation the polytropic equation of state is used as a closing condition. The key idea is the three-dimensional generalization of the well-known self-similar ansatz which was already used for non-compressible viscous flow in our former study. The geometrical interpretations of the trial function is also discussed. We compared our recent results to the former non-compressible ones.
Analysis of the Scramjet inlet flow field using two-dimensional Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Kumar, A.; Tiwari, S. N.
1982-01-01
A computer code was developed to solve the full two dimensional Navier-Stokes equations in a scramjet inlet. The analysis uses a numerical coordinate transformation which generates a set of boundary-fitted curvilinear coordinates. The explicit finite difference algorithm of MacCormack is used to solve the governing equations. A two-layer eddy viscosity model is used for the turbulent flow. The code can analyze both inviscid and viscous flows with multiple struts in the flow field. Detailed results are presented for two model problems and two scramjet inlets with one and two struts. The application of the two dimensional analysis in the preliminary design of the actual scramjet inlet is briefly discussed.
Development of a three-dimensional Navier-Stokes code on CDC star-100 computer
NASA Technical Reports Server (NTRS)
Vatsa, V. N.; Goglia, G. L.
1978-01-01
A three-dimensional code in body-fitted coordinates was developed using MacCormack's algorithm. The code is structured to be compatible with any general configuration, provided that the metric coefficients for the transformation are available. The governing equations are developed in primitive variables in order to facilitate the incorporation of physical boundary conditions and turbulence-closure models. MacCormack's two-step, unsplit, time-marching algorithm is used to solve the unsteady Navier-Stokes equations until steady-state solution is achieved. Cases discussed include (1) flat plate in supersonic free stream; (2) supersonic flow along an axial corner; (3) subsonic flow in an axial corner at M infinity = 0.95; and (4) supersonic flow in an axial corner at M infinity 1.5.
Prediction of airfoil stall using Navier-Stokes equations in streamline coordinates
NASA Technical Reports Server (NTRS)
Choi, D. H.; Sohn, C. H.; Oh, C. S.
1992-01-01
A Navier-Stokes procedure to calculate the flow about an airfoil at incidence was developed. The parabolized equations are solved in the streamline coordinates generated for an arbitrary airfoil shape using conformal mapping. A modified k-epsilon turbulence model is applied in the entire domain, but the eddy viscosity in the laminar region is suppressed artificially to simulate the region correctly. The procedure was applied to airfoils at various angles of attack, and the results are quite satisfactory for both laminar and turbulent flows. It is shown that the present choice of the coordinate system reduces the error due to numerical diffusion, and that the lift is accurately predicted for a wide range of incidence.
Entropy Stable Wall Boundary Conditions for the Compressible Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Parsani, Matteo; Carpenter, Mark H.; Nielsen, Eric J.
2014-01-01
Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite volume, finite difference, discontinuous Galerkin, and flux reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.
A Navier-Stokes Solution of Hull-Ring Wing-Thruster Interaction
NASA Technical Reports Server (NTRS)
Yang, C.-I.; Hartwich, P.; Sundaram, P.
1991-01-01
Navier-Stokes simulations of high Reynolds number flow around an axisymmetric body supported in a water tunnel were made. The numerical method is based on a finite-differencing high resolution second-order accurate implicit upwind scheme. Four different configurations were investigated, these are: (1) barebody; (2) body with an operating propeller; (3) body with a ring wing; and (4) body with a ring wing and an operating propeller. Pressure and velocity components near the stern region were obtained computationally and are shown to compare favorably with the experimental data. The method correctly predicts the existence and extent of stern flow separation for the barebody and the absence of flow separation for the three other configurations with ring wing and/or propeller.
Grid induced errors in stream function solutions of the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Ho, P. Y.; Decher, R.; Hermanson, J. C.
1991-06-01
This paper examines the effects of the flow Reynolds number, grid stretching, and velocity gradients on the accuracy of computed solutions to the Navier-Stokes stream function equations. It is shown that second order truncation errors become small only as the flow Reynolds number approaches zero. For sufficiently high values of Reynolds number, the highly nonlinear flow solutions that result from the large velocity gradients can result in large truncation error, even for the case of nonuniform grids. These errors behave as momentum sources and sinks within the computed flow domain, causing significant error in the predictions of boundary layer separation and reattachment. The streamfunction-vorticity method is shown to be unsuitable to the modeling of viscid-inviscid interactive flows. Limitations on the use of higher order finite difference approximations are discussed.
Navier-Stokes solutions for two-dimensional subsonic base flow
NASA Technical Reports Server (NTRS)
Rudy, D. H.
1984-01-01
Methods for determining the effects of mass injection from the trailing edge of a bluff body at low speeds and in transonic flow were numerically studied along with an unmodified blunt-based body to gain insight into the effects of vortex shedding on the base drag. The methodology used to obtain finite-difference solutions to the Navier-Stokes equations for subsonic compressible two-dimensional near-wake flows is presented. The effectiveness of an introduced outflow boundary condition which minimizes reflections back into the computational domain was demonstrated with the solution of a model vortex problem. Calculations of the near-wake flow past a circular cylinder were in excellent agreement with experimental data. Laminar-flow solutions for a blunt-based model with and without a base cavity and with mass injection into the wake agreed qualitatively with experimental observations. The drag reduction capability provided by such base modifications was demonstrated.
A third-order-accurate upwind scheme for Navier-Stokes solutions at high Reynolds numbers
NASA Astrophysics Data System (ADS)
Agarwal, R. K.
1981-01-01
A third-order-accurate upwind scheme is presented for solution of the steady two-dimensional Navier-Stokes equations in stream-function/vorticity form. The scheme is found to be accurate and stable at high Reynolds numbers. A series of test computations is performed on flows with large recirculating regions. In particular, highly accurate solutions are obtained for flow in a driven square cavity up to Reynolds numbers of 10,000. These computations are used to critically evaluate the accuracy of other existing first- and second-order-accurate upwind schemes. In addition, computations are carried out for flow in a channel with symmetric sudden expansion, flow in a channel with a symmetrically placed blunt base, and the flowfield of an impinging jet. Good agreement is obtained with the computations of other investigators as well as with the available experimental data.
Progress in development of a Navier-Stokes solver for evaluation of iced airfoil performance
NASA Technical Reports Server (NTRS)
Potapczuk, M. G.; Gerhart, P. M.
1985-01-01
A method is being developed for evaluation of the flow field behavior about an airfoil with significant ice accretion on the leading edge. The computer code, being evaluated for this purpose, solves the Navier-Stokes equations in a body-fitted curvilinear coordinate system. This requires the use of a grid generation code to transform the x-y coordinates of the physical space into xi-eta coordinates of the computational space. Evaluation of the suitability of these two codes for predicting iced airfoil performance is presently being carried out in anticipation of use in an overall icing analysis effort. Results of this evaluation to date indicate good correlation with known information on clean airfoils. Preliminary results for rime and glaze, iced airfoil shapes are also presented.
Navier-Stokes and Comprehensive Analysis Performance Predictions of the NREL Phase VI Experiment
NASA Technical Reports Server (NTRS)
Duque, Earl P. N.; Burklund, Michael D.; Johnson, Wayne
2003-01-01
A vortex lattice code, CAMRAD II, and a Reynolds-Averaged Navier-Stoke code, OVERFLOW-D2, were used to predict the aerodynamic performance of a two-bladed horizontal axis wind turbine. All computations were compared with experimental data that was collected at the NASA Ames Research Center 80- by 120-Foot Wind Tunnel. Computations were performed for both axial as well as yawed operating conditions. Various stall delay models and dynamics stall models were used by the CAMRAD II code. Comparisons between the experimental data and computed aerodynamic loads show that the OVERFLOW-D2 code can accurately predict the power and spanwise loading of a wind turbine rotor.
Transonic Navier-Stokes calculations about a 65 deg delta wing
NASA Technical Reports Server (NTRS)
Londenberg, W. Kelly
1994-01-01
A computational study has been conducted in which the CFL3D Navier-Stokes solver coupled with an algebraic and a one-equation nonequilibrium turbulence model has been used to predict the flow over a 65 degree delta wing at transonic conditions for Reynolds numbers ranging from 6 x 10(exp 6) to 120 x 10(exp 6) based on mean aerodynamic chord. Solutions obtained indicated that the computational method when used with the one-equation turbulence model predicts results that compare well with experiment for attached flow conditions. Comparisons with experimental pressure at separated conditions show that the computational method, even though primary flow-field features are predicted well, does not predict secondary flow features.
An Evaluation of Parameters Influencing Jet Mixing Using the WIND Navier-stokes Code
NASA Technical Reports Server (NTRS)
Dembowski, Mary Ann; Georgiadis, Nicholas J.
2002-01-01
The WIND code, a Reynolds-averaged Navier-Stokes solver used for a variety of aerospace flow simulations, was investigated for a Mach 2 nozzle at a series of nozzle stagnation temperatures. Comparisons of WIND calculations are made to experimental measurements of axial velocity, Mach number, and stagnation temperature along the jet centerline. The primary objective was to investigate the capabilities of the two-equation turbulence models available in WIND, version 4.0, for the analysis of heated supersonic nozzle flows. The models examined were the Menter Shear Stress Transport (SST) model and the Chien k-epsilon model, with and without the compressibility correction due to Sarkar. It was observed that all of the turbulence models investigated produced solutions that did not agree well with the experimental measurements. The effects of freestream Mach number and turbulent Prandtl number specifications were also investigated.
On velocity errors due to irrotational forces in the Navier-Stokes momentum balance
NASA Astrophysics Data System (ADS)
Linke, A.; Merdon, C.
2016-05-01
This contribution studies the influence of the pressure on the velocity error in finite element discretisations of the Navier-Stokes equations. Four simple benchmark problems that are all close to real-world applications convey that the pressure can be comparably large and is not to be underestimated. In fact, the velocity error can be arbitrarily large in such situations. Only pressure-robust mixed finite element methods, whose velocity error is pressure-independent, can avoid this influence. Indeed, the presented examples show that the pressure-dependent component in velocity error estimates for classical mixed finite element methods is sharp. In consequence, classical mixed finite element methods are not able to simulate some classes of real-world flows, even in cases where dominant convection and turbulence do not play a role.
Dynamic stall computations using a zonal Navier-Stokes model. Master's thesis
Conroyd, J.H.
1988-06-01
A zonal Navier-Stokes model is installed and verified on the NASA Ames Cray X/MP-48 computer and is used to calculate the flow field about a NACA 0012 airfoil oscillating in pitch. Surface-pressure distributions and integrated lift, pitching moment, and drag coefficient versus angle of attack are compared to existing experimental data for four cases and existing computational data for one case. These cases involve deep dynamic stall and fully detached flow at and below a free-stream Mach number of .184. The flow field about the oscillating airfoil is investigated through the study of pressure, vorticity, local velocity, and stream function. Finally, the effects of pitch rate on dynamic stall are investigated.
Labyrinth seal rotordynamic forces using a three-dimensional Navier-Stokes code
NASA Astrophysics Data System (ADS)
Rhode, D. L.; Hensel, S. J.; Guidry, M. J.
1992-10-01
A finite difference method for determining rotordynamic forces on an eccentric whirling labyrinth cavity has been developed. A coordinate-transformation was applied to the Reynolds time-averaged Navier-Stokes equations in order to use the modified bipolar coordinate system. The SIMPLER algorithm with QUICK differencing and the high Reynolds number k-epsilon turbulence model are used to compute the complex turbulent flowfield. A circular whirl orbit about the geometric center of the housing was specified for simplicity. The new model was tested against the rotordynamic force measurements, and close agreement was found. For the cases considered, the radial and tangential force components become rotordynamically less desirable with increasing inlet swirl. Also, circumferential pressure variations are included for enhanced insight into the flowfield.
Discrete sensitivity derivatives of the Navier-Stokes equations with a parallel Krylov solver
NASA Technical Reports Server (NTRS)
Ajmani, Kumud; Taylor, Arthur C., III
1994-01-01
This paper solves an 'incremental' form of the sensitivity equations derived by differentiating the discretized thin-layer Navier Stokes equations with respect to certain design variables of interest. The equations are solved with a parallel, preconditioned Generalized Minimal RESidual (GMRES) solver on a distributed-memory architecture. The 'serial' sensitivity analysis code is parallelized by using the Single Program Multiple Data (SPMD) programming model, domain decomposition techniques, and message-passing tools. Sensitivity derivatives are computed for low and high Reynolds number flows over a NACA 1406 airfoil on a 32-processor Intel Hypercube, and found to be identical to those computed on a single-processor Cray Y-MP. It is estimated that the parallel sensitivity analysis code has to be run on 40-50 processors of the Intel Hypercube in order to match the single-processor processing time of a Cray Y-MP.
Navier-Stokes Computations on Full Wing-Body Configuration with Oscillating Control Surfaces
NASA Technical Reports Server (NTRS)
Obayashi, Shigeru; Chiu, Ing-Tsau; Guruswamy, Guru P.
1995-01-01
Unsteady Navier-Stokes simulations have been performed for vortical flows over an "arrow-wing" configuration of a supersonic transport in the transonic regime. Computed steady pressures and integrated force coefficients with and without control surface deflection at a moderate angle of attack are compared with experiment. For unsteady cases, oscillating trailing-edge control surfaces are modeled by using moving grids. Response characteristics between symmetric and antisymmetric oscillatory motions of the control surfaces on the left and right wings are studied. The antisymmetric case produces higher lift than the steady case with no deflection and the unsteady symmetric case produces higher lift than the antisymmetric case. The detailed analysis of the wake structure revealed a strong interaction between the primary vortex and the wake vortex sheet from the flap region when the flap is deflected up.
A split finite element algorithm for the compressible Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Baker, A. J.
1979-01-01
An accurate and efficient numerical solution algorithm is established for solution of the high Reynolds number limit of the Navier-Stokes equations governing the multidimensional flow of a compressible essentially inviscid fluid. Finite element interpolation theory is used within a dissipative formulation established using Galerkin criteria within the Method of Weighted Residuals. An implicit iterative solution algorithm is developed, employing tensor product bases within a fractional steps integration procedure, that significantly enhances solution economy concurrent with sharply reduced computer hardware demands. The algorithm is evaluated for resolution of steep field gradients and coarse grid accuracy using both linear and quadratic tensor product interpolation bases. Numerical solutions for linear and nonlinear, one, two and three dimensional examples confirm and extend the linearized theoretical analyses, and results are compared to competitive finite difference derived algorithms.
Calculation of steady and unsteady airfoil flow fields via the Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Shamroth, S. J.
1985-01-01
A compressible time-dependent procedure for the two-dimensional ensemble averaged Navier-Stokes equations has been applied to the isolated airfoil problem in steady and unsteady flows. The procedure solves the governing equations via the linearized block implicit technique. Turbulence is modeled either via a mixing length or turbulence energy approach. The equations are solved in general non-orthogonal form with no-slip boundary conditions applied at the airfoil surface. Results are presented for airfoils at constant incidence, an airfoil in ramp motion and an airfoil oscillating through a dynamic stall loop. In general, steady converged solutions are obtained within 70 time steps over the range of Mach numbers considered. Comparisons with measured data show good agreement between computation and measurement.
Numerical Simulations of Homogeneous Turbulence Using Lagrangian-Averaged Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Mohseni, Kamran; Shkoller, Steve; Kosovic, Branko; Marsden, Jerrold E.; Carati, Daniele; Wray, Alan; Rogallo, Robert
2000-01-01
The Lagrangian-averaged Navier-Stokes (LANS) equations are numerically evaluated as a turbulence closure. They are derived from a novel Lagrangian averaging procedure on the space of all volume-preserving maps and can be viewed as a numerical algorithm which removes the energy content from the small scales (smaller than some a priori fixed spatial scale alpha) using a dispersive rather than dissipative mechanism, thus maintaining the crucial features of the large scale flow. We examine the modeling capabilities of the LANS equations for decaying homogeneous turbulence, ascertain their ability to track the energy spectrum of fully resolved direct numerical simulations (DNS), compare the relative energy decay rates, and compare LANS with well-accepted large eddy simulation (LES) models.
LDV measurement and Navier-Stokes computation of parallel jet mixing in a rectangular confinement
Kunz, R.F.; D`Amico, S.W.; Vassallo, P.F.; Zaccaria, M.A.; Aksoy, H.; So, R.M.C.
1995-06-01
Laser Doppler Velocimetry (LDV) measurements were taken in a rectangular confinement into which issues a row of parallel jets. Two-component measurements were taken with two optics orientations yielding three mean velocity components and four Reynolds stress components. As observed in isolated three dimensional wall bounded jets, the transverse diffusion of the jets is quite large. The data indicates that this rapid mixing process is due to strong secondary flows, transport of large inlet intensities and Reynolds stress anisotropy effects. Navier-Stokes analyses of this configuration underpredict the rate of transverse jet diffusion. Detailed numerical accuracy studies show that this is attributed to shortcomings in low-Reynolds number two-equation turbulence modelling. A low-Reynolds number full-Reynolds stress model is shown to provide improvement.
On the Helicity in 3D-Periodic Navier-Stokes Equations II: The Statistical Case
NASA Astrophysics Data System (ADS)
Foias, Ciprian; Hoang, Luan; Nicolaenko, Basil
2009-09-01
We study the asymptotic behavior of the statistical solutions to the Navier-Stokes equations using the normalization map [9]. It is then applied to the study of mean energy, mean dissipation rate of energy, and mean helicity of the spatial periodic flows driven by potential body forces. The statistical distribution of the asymptotic Beltrami flows are also investigated. We connect our mathematical analysis with the empirical theory of decaying turbulence. With appropriate mathematically defined ensemble averages, the Kolmogorov universal features are shown to be transient in time. We provide an estimate for the time interval in which those features may still be present. Our collaborator and friend Basil Nicolaenko passed away in September of 2007, after this work was completed. Honoring his contribution and friendship, we dedicate this article to him.
An Evaluation of Architectural Platforms for Parallel Navier-Stokes Computations
NASA Technical Reports Server (NTRS)
Jayasimha, D. N.; Hayder, M. E.; Pillay, S. K.
1996-01-01
We study the computational, communication, and scalability characteristics of a computational fluid dynamics application, which solves the time accurate flow field of a jet using the compressible Navier-Stokes equations, on a variety of parallel architecture platforms. The platforms chosen for this study are a cluster of workstations (the LACE experimental testbed at NASA Lewis), a shared memory multiprocessor (the Cray YMP), and distributed memory multiprocessors with different topologies - the IBM SP and the Cray T3D. We investigate the impact of various networks connecting the cluster of workstations on the performance of the application and the overheads induced by popular message passing libraries used for parallelization. The work also highlights the importance of matching the memory bandwidth to the processor speed for good single processor performance. By studying the performance of an application on a variety of architectures, we are able to point out the strengths and weaknesses of each of the example computing platforms.
Parallelizing Navier-Stokes Computations on a Variety of Architectural Platforms
NASA Technical Reports Server (NTRS)
Jayasimha, D. N.; Hayder, M. E.; Pillay, S. K.
1997-01-01
We study the computational, communication, and scalability characteristics of a Computational Fluid Dynamics application, which solves the time accurate flow field of a jet using the compressible Navier-Stokes equations, on a variety of parallel architectural platforms. The platforms chosen for this study are a cluster of workstations (the LACE experimental testbed at NASA Lewis), a shared memory multiprocessor (the Cray YMP), distributed memory multiprocessors with different topologies-the IBM SP and the Cray T3D. We investigate the impact of various networks, connecting the cluster of workstations, on the performance of the application and the overheads induced by popular message passing libraries used for parallelization. The work also highlights the importance of matching the memory bandwidth to the processor speed for good single processor performance. By studying the performance of an application on a variety of architectures, we are able to point out the strengths and weaknesses of each of the example computing platforms.
An Empirical Model for Vane-Type Vortex Generators in a Navier-Stokes Code
NASA Technical Reports Server (NTRS)
Dudek, Julianne C.
2005-01-01
An empirical model which simulates the effects of vane-type vortex generators in ducts was incorporated into the Wind-US Navier-Stokes computational fluid dynamics code. The model enables the effects of the vortex generators to be simulated without defining the details of the geometry within the grid, and makes it practical for researchers to evaluate multiple combinations of vortex generator arrangements. The model determines the strength of each vortex based on the generator geometry and the local flow conditions. Validation results are presented for flow in a straight pipe with a counter-rotating vortex generator arrangement, and the results are compared with experimental data and computational simulations using a gridded vane generator. Results are also presented for vortex generator arrays in two S-duct diffusers, along with accompanying experimental data. The effects of grid resolution and turbulence model are also examined.
A Priori Bound on the Velocity in Axially Symmetric Navier-Stokes Equations
NASA Astrophysics Data System (ADS)
Lei, Zhen; Navas, Esteban A.; Zhang, Qi S.
2016-01-01
Let v be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound |v(x, t)| ≤ C |ln r|^{1/2}/r^2, qquad 0 < r ≤ 1/2, where r is the distance from x to the z axis, and C is a constant depending only on the initial value. This provides a pointwise upper bound (worst case scenario) for possible singularities, while the recent papers (Chiun-Chuan et al., Commun PDE 34(1-3):203-232, 2009; Koch et al., Acta Math 203(1):83-105, 2009) gave a lower bound. The gap is polynomial order 1 modulo a half log term.
NASA Technical Reports Server (NTRS)
Bart, Timothy J.; Kutler, Paul (Technical Monitor)
1998-01-01
Chapter 1 briefly reviews several related topics associated with the symmetrization of systems of conservation laws and quasi-conservation laws: (1) Basic Entropy Symmetrization Theory; (2) Symmetrization and eigenvector scaling; (3) Symmetrization of the compressible Navier-Stokes equations; and (4) Symmetrization of the quasi-conservative form of the magnetohydrodynamic (MHD) equations. Chapter 2 describes one of the best known tools employed in the study of differential equations, the maximum principle: any function f(x) which satisfies the inequality f(double prime)>0 on the interval [a,b] attains its maximum value at one of the endpoints on the interval. Chapter three examines the upwind finite volume schemes for scalar and system conservation laws. The basic tasks in the upwind finite volume approach have already been presented: reconstruction, flux evaluation, and evolution. By far, the most difficult task in this process is the reconstruction step.
Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas.
Kolvin, Itamar; Livne, Eli; Meerson, Baruch
2010-08-01
We show that, in dimension higher than one, heat diffusion and viscosity cannot arrest thermal collapse in a freely evolving dilute granular gas, even in the absence of gravity. Thermal collapse involves a finite-time blowup of the gas density. It was predicted earlier in ideal, Euler hydrodynamics of dilute granular gases in the absence of gravity, and in nonideal, Navier-Stokes granular hydrodynamics in the presence of gravity. We determine, analytically and numerically, the dynamic scaling laws that characterize the gas flow close to collapse. We also investigate bifurcations of a freely evolving dilute granular gas in circular and wedge-shaped containers. Our results imply that, in general, thermal collapse can only be arrested when the gas density becomes comparable with the close-packing density of grains. This provides a natural explanation to the formation of densely packed clusters of particles in a variety of initially dilute granular flows. PMID:20866801
A three dimensional multigrid Reynolds-averaged Navier-Stokes solver for unstructured meshes
NASA Technical Reports Server (NTRS)
Mavriplis, D. J.
1994-01-01
A three-dimensional unstructured mesh Reynolds averaged Navier-Stokes solver is described. Turbulence is simulated using a single field-equation model. Computational overheads are minimized through the use of a single edge-based data-structure, and efficient multigrid solution technique, and the use of multi-tasking on shared memory multi-processors. The accuracy and efficiency of the code are evaluated by computing two-dimensional flows in three dimensions and comparing with results from a previously validated two-dimensional code which employs the same solution algorithm. The feasibility of computing three-dimensional flows on grids of several million points in less than two hours of wall clock time is demonstrated.
Navier-Stokes Entropy Controlled Combustion Instability Analysis for Liquid Propellants
NASA Technical Reports Server (NTRS)
Chung, T. J.; Yoon, W. S.
1990-01-01
Navier-Stokes solutions are used to calculate oscillatory components of pressure, velocity, and density, which in turn provide necessary data to compute energy growth factors to determine combustion instability. It is shown that wave instabilities are associated with changes in entropy and the space and time averages of oscillatory components of pressure, velocity and density, together with the mean flow field in the energy equation. Compressible laminar and turbulent flows and reacting flows with hydrogen/oxygen combustion are considered. The SSME combustion/thrust chamber is used for illustration of the theory. The analysis shows that the increase of mean pressure and disturbances consistently results in the increase of instability. It is shown that adequate combustion instability analysis requires at least third order nonlinearity in energy growth or decay.
NASA Technical Reports Server (NTRS)
Parsani, Matteo; Carpenter, Mark H.; Nielsen, Eric J.
2015-01-01
Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.
Recent Analytical and Numerical Results for The Navier-Stokes-Voigt Model and Related Models
NASA Astrophysics Data System (ADS)
Larios, Adam; Titi, Edriss; Petersen, Mark; Wingate, Beth
2010-11-01
The equations which govern the motions of fluids are notoriously difficult to handle both mathematically and computationally. Recently, a new approach to these equations, known as the Voigt-regularization, has been investigated as both a numerical and analytical regularization for the 3D Navier-Stokes equations, the Euler equations, and related fluid models. This inviscid regularization is related to the alpha-models of turbulent flow; however, it overcomes many of the problems present in those models. I will discuss recent work on the Voigt-regularization, as well as a new criterion for the finite-time blow-up of the Euler equations based on their Voigt-regularization. Time permitting, I will discuss some numerical results, as well as applications of this technique to the Magnetohydrodynamic (MHD) equations and various equations of ocean dynamics.
PARC Navier-Stokes code upgrade and validation for high speed aeroheating predictions
NASA Technical Reports Server (NTRS)
Liver, Peter A.; Praharaj, Sarat C.; Seaford, C. Mark
1990-01-01
Applications of the PARC full Navier-Stokes code for hypersonic flowfield and aeroheating predictions around blunt bodies such as the Aeroassist Flight Experiment (AFE) and Aeroassisted Orbital Transfer Vehicle (AOTV) are evaluated. Two-dimensional/axisymmetric and three-dimensional perfect gas versions of the code were upgraded and tested against benchmark wind tunnel cases of hemisphere-cylinder, three-dimensional AFE forebody, and axisymmetric AFE and AOTV aerobrake/wake flowfields. PARC calculations are in good agreement with experimental data and results of similar computer codes. Difficulties encountered in flowfield and heat transfer predictions due to effects of grid density, boundary conditions such as singular stagnation line axis and artificial dissipation terms are presented together with subsequent improvements made to the code. The experience gained with the perfect gas code is being currently utilized in applications of an equilibrium air real gas PARC version developed at REMTECH.
Large Eddy And Reynolds-Averaged Navier-Stokes Simulations Of An Underexpanded Sonic Jet
NASA Astrophysics Data System (ADS)
Gorle, Catherine; Iaccarino, Gianluca
2011-05-01
The complex phenomena involved in scramjet propulsion can be investigated using Reynolds-averaged Navier- Stokes (RANS) simulations, but the computations can not fully represent all the physics involved. The mixing of fuel and air inside supersonic combustion chambers is one of the critical processes for successful operation of scramjet engines that requires modeling. The goal of the present work is the identification and characterization of the uncertainties in RANS mixing models, to support the development of an error model. In the first part of this paper simulations of an underexpanded hydrogen jet in quiescent air are described and compared to Schlieren images for verification of the jet injection modeling. In the second part RANS simulations of an underexpanded ethylene jet in a supersonic cross flow of Mach 2 are presented to identify the uncertainties in the mixing model, and quantify the sensitivity of the model outcome to these uncertainties.
NASA Astrophysics Data System (ADS)
Parsani, Matteo; Carpenter, Mark H.; Nielsen, Eric J.
2015-07-01
Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.
On the Regularity Set and Angular Integrability for the Navier-Stokes Equation
NASA Astrophysics Data System (ADS)
D'Ancona, Piero; Lucà, Renato
2016-09-01
We investigate the size of the regular set for suitable weak solutions of the Navier-Stokes equation, in the sense of Caffarelli-Kohn-Nirenberg (Commun Pure Appl Math 35:771-831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space {\\{t > 0\\}} in an appropriate limit. In particular, we obtain that if the {L2} norm with weight {|x|^{-frac12}} of the data tends to 0, the regular set invades {\\{t > 0\\}}; this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771-831, 1982).
Application of an unsteady Navier-Stokes solver to transonic turbine design
NASA Technical Reports Server (NTRS)
Rangwalla, Akil A.; Madavan, Nateri K.; Johnson, Paul D.
1991-01-01
This study presents a numerical evaluation of the performance of the first stage of a new-generation turbine design. The numerical method solves the two-dimensional Navier-Stokes equations using a system of patched grids. Three-dimensional effects of stream-tube contraction are also modeled. The study focuses on the effects of axial gap variation on the unsteady rotor-stator interactions and on stage performance. Results are presented for three different axial gaps. The results indicate that the unsteady interactions can be very large in this design. These interactions affect not only the stage efficiency but also substantially alter the time-averaged features of the flow. In particular, for the case of the smallest axial gap, it was found that there was an unsteady shock on the stator suction surface which spanned the gap region and impinged upon the moving rotor airfoils.
Three-dimensional Navier-Stokes analysis of turbine passage heat transfer
NASA Technical Reports Server (NTRS)
Ameri, Ali A.; Arnone, Andrea
1991-01-01
The three-dimensional Reynolds-averaged Navier-Stokes equations are numerically solved to obtain the pressure distribution and heat transfer rates on the endwalls and the blades of two linear turbine cascades. The TRAF3D code which has recently been developed in a joint project between researchers from the University of Florence and NASA Lewis Research Center is used. The effect of turbulence is taken into account by using the eddy viscosity hypothesis and the two-layer mixing length model of Baldwin and Lomax. Predictions of surface heat transfer are made for Langston's cascade and compared with the data obtained for that cascade by Graziani. The comparison was found to be favorable. The code is also applied to a linear transonic rotor cascade to predict the pressure distributions and heat transfer rates.
Nonperturbative renormalization group study of the stochastic Navier-Stokes equation.
Mejía-Monasterio, Carlos; Muratore-Ginanneschi, Paolo
2012-07-01
We study the renormalization group flow of the average action of the stochastic Navier-Stokes equation with power-law forcing. Using Galilean invariance, we introduce a nonperturbative approximation adapted to the zero-frequency sector of the theory in the parametric range of the Hölder exponent 4-2ε of the forcing where real-space local interactions are relevant. In any spatial dimension d, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for ε = 2 as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a saturation in the ε dependence of the scaling dimension of the eddy diffusivity at ε = 3/2 when, according to perturbative renormalization, the velocity field becomes infrared relevant. PMID:23005533