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All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Prediction of vortical flows on wings using incompressible Navier-Stokes equations

Numerical solutions to the incompressible Navier-Stokes equations are in greater demand than ever before as the field of computational fluid dynamics (CFD) increases its impact as an engineering tool. An implicit upwind-relation finite-difference scheme is developed for solving unsteady incompressible Navier-Stokes equations incorporating an artificial time derivative of the pressure to the continuity equation. The essential features observed in experiments for complicated vortical flows past three round-edged low-aspect-ratio wings at high angles of attack are reproduced both qualitatively and quantitatively.

Hsu, C.-H.↗
NASA NTRS1989

A dual potential formulation of the Navier-Stokes equations

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.

Gegg, S. G.↗
NASA NTRS1988

An upwind-differencing scheme for the incompressible Navier-Stokes equations

The steady state incompressible Navier-Stokes equations in 2-D are solved numerically using the artificial compressibility formulation. The convective terms are upwind-differenced using a flux difference split approach that has uniformly high accuracy throughout the interior grid points. The viscous fluxes are differenced using second order accurate central differences. The numerical system of equations is solved using an implicit line relaxation scheme. Although the current study is limited to steady state problems, it is shown that this entire formulation can be used for solving unsteady problems. Characteristic boundary conditions are formulated and used in the solution procedure. The overall scheme is capable of being run at extremely large pseudotime steps, leading to fast convergence. Three test cases are presented to demonstrate the accuracy and robustness of the code. These are the flow in a square-driven cavity, flow over a backward facing step, and flow around a 2-D circular cylinder.

Rogers, Stuart E.↗
NASA NTRS1990

Implicit methods for the Navier-Stokes equations

Numerical solutions of the Navier-Stokes equations using explicit schemes can be obtained at the expense of efficiency. Conventional implicit methods which often achieve fast convergence rates suffer high cost per iteration. A new implicit scheme based on lower-upper factorization and symmetric Gauss-Seidel relaxation offers very low cost per iteration as well as fast convergence. High efficiency is achieved by accomplishing the complete vectorizability of the algorithm on oblique planes of sweep in three dimensions.

Yoon, S.↗
NASA NTRS1989

Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations

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.

Hafez, Mohammed↗
NASA NTRS1987

Analysis of unsteady wake of a circular cylinder using Navier-Stokes equations

An unsteady Navier-Stokes analysis of the separated flow in the wake of a circular cylinder is presented in order to study the wake structure and the various interactions between its scales at low Re. The quasi-periodic and chaotic flows noted by Sreenivasan (1985) are searched for. The analysis is used to characterize the secondary vortex formation in the early stages of the flow development and its interaction with the primary vortex.

Ghia, K. N.↗
NASA NTRS1991

A study of three dimensional turbulent boundary layer separation and vortex flow control using the reduced Navier Stokes equations

A reduced Navier-Stokes (RNS) initial value space marching solution technique was applied to vortex generator and separated flow problems and demonstrated good predictions of the engine face flow field. This RNS solution technique using FLARE approximations can adequately describe the topological and topographical structure flow separation associated with vortex liftoff, and this conclusion led to the concept of a subclass of separations which can be called vorticity separations: separations dominated by the transport of vorticity. Adequate near wall resolution of vorticity separations appears necessary for good predictions of these flows.

Anderson, Bernhard H.↗
NASA NTRS1978

An efficient explicit-implicit-characteristic method for solving the compressible Navier-Stokes equations

Explicit, implicit, and characteristic finite-difference methods are applied to solve model equations representative of the compressible Navier-Stokes equations. An approach is then formulated for solving the Navier-Stokes equation at high Reynolds numbers. The approach has drastically reduced the computation time required to obtain viscous flow solutions. Computational results for shock wave separated flows are presented.

Maccormack, R. W.↗
NASA NTRS1981

Two-boundary grid generation for the solution of the three dimensional compressible Navier-Stokes equations

A grid generation technique called the two boundary technique is developed and applied for the solution of the three dimensional Navier-Stokes equations. The Navier-Stokes equations are transformed from a cartesian coordinate system to a computational coordinate system, and the grid generation technique provides the Jacobian matrix describing the transformation. The two boundary technique is based on algebraically defining two distinct boundaries of a flow domain and the distribution of the grid is achieved by applying functions to the uniform computational grid which redistribute the computational independent variables and consequently concentrate or disperse the grid points in the physical domain. The Navier-Stokes equations are solved using a MacCormack time-split technique. Grids and supersonic laminar flow solutions are obtained for a family of three dimensional corners and two spike-nosed bodies.

Smith, R. E.↗
NASA NTRS1980

Restricted invariance of the Navier-Stokes equation

It is well known that the classical Navier-Stokes equation for three-dimensional incompressible viscous flow is invariant under Galilean transformations. In the present paper, it is shown that, although the general invariance group is no larger than Galilean-scale dilitation, the Navier-Stokes equation for incompressible viscous flow admits time-dependent rotational invariance for solutions of special types.

Rosen, G.↗
DOE OSTI2021

Modeling and simulation of transitional Rayleigh–Taylor flow with partially averaged Navier–Stokes equations

In this work, the partially averaged Navier–Stokes (PANS) equations are used to predict the variable-density Rayleigh–Taylor (RT) flow at Atwood number 0.5 and maximum Reynolds number 500. This is a prototypical problem of material mixing, featuring laminar, transitional, and turbulent flow, instabilities and coherent structures, density fluctuations, and production of turbulence kinetic energy by both shear and buoyancy mechanisms. These features pose numerous challenges to modeling and simulation, making the RT flow ideal to develop the validation space of the recently proposed PANS Besnard–Harlow–Rauenzahn-linear eddy viscosity model closure. The numerical simulations are conducted at different levels of physical resolution and test three approaches to set the parameters $f_\phi$ defining the range of physically resolved scales. The computations demonstrate the efficiency (accuracy vs cost) of the PANS model predicting the spatiotemporal development of the RT flow. Results comparable to large-eddy simulations and direct numerical simulations are obtained at significantly lower physical resolution without the limitations of the Reynolds-averaged Navier–Stokes equations in these transitional flows. The data also illustrate the importance of appropriate selection of the physical resolution and the resolved fraction of each dependent quantity $\phi$ of the turbulent closure, $f_\phi$. These two aspects determine the ability of the model to resolve the flow phenomena not amenable to modeling by the closure and, as such, the computations’ fidelity.

Navier Stokes equations↗
NASA NTRS1994

Preconditioned conjugate gradient methods for the Navier-Stokes equations

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.

Ajmani, Kumud↗
NASA NTRS2004

Evaluation of a Multigrid Scheme for the Incompressible Navier-Stokes Equations

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.

Swanson, R. C.↗
NASA NTRS1991

Analysis of regularized Navier-Stokes equations. I, II

A regularized form of the conventional Navier-Stokes equations is analyzed. The global existence and uniqueness are established for two classes of generalized solutions. It is shown that the solution of this regularized system converges to the solution of the conventional Navier-Stokes equations for low Reynolds numbers. Particular attention is given to the structure of attractors characterizing the solutions. Both local and global invariant manifolds are found, and the regularity properties of these manifolds are analyzed.

Ou, Yuh-Roung↗
NASA NTRS1982

Error norms for the adaptive solution of the Navier-Stokes equations

The adaptive solution of the Navier-Stokes equations depends upon the successful interaction of three key elements: (1) the ability to flexibly select grid length scales in composite grids, (2) the ability to efficiently control residual error in composite grids, and (3) the ability to define reliable, convenient error norms to guide the grid adjustment and optimize the residual levels relative to the local truncation errors. An initial investigation was conducted to explore how to approach developing these key elements. Conventional error assessment methods were defined and defect and deferred correction methods were surveyed. The one dimensional potential equation was used as a multigrid test bed to investigate how to achieve successful interaction of these three key elements.

Forester, C. K.↗
NASA NTRS1988

An upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner in using the method of pseudocompressibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme based on flux-difference splitting is used to compute the convective terms. The upwind differencing is biased based on the sign of the local eigenvalue of the Jacobian matrix. Third-order or fifth-order spatial accuracy is maintained throughout the interior grid points. The equations are solved using an implicit line-relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudotime, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at -90 deg angle-of-attack is also computed.

Rogers, Stuart E.↗
NASA NTRS1989

Time dependent viscous incompressible Navier-Stokes equations

Information on time dependent incompressible Navier-Stokes equations is given in viewgraph form. Information is given on streamfunction equations for unsteady incompressible flow, the streamfunction algorithm for unsteady incompressible flow, and a multigrid solver for the laminar implicit equations.

Goodrich, John W.↗