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NASA NTRS1986

A coupled marching procedure for the partially parabolized Navier-Stokes equations

A coupled finite-difference formulation is described for solving a reduced form of the compressible Navier-Stokes equations by a multiple space marching procedure. The properties of the equations are discussed and the solution algorithm presented. The scheme is used to compute incompressible flows by taking the incompressible limit (M about 0.1) of the compressible formulation. A separate Poisson equation is not required for the pressure. Results are compared with experimental data and other numerical predictions for low Reynolds number channel inlet flow, flow over a rearward-facing step in a channel, and flow near the trailing edge of a flat plate.

Liu, Xuezong↗

NASA NTRS1987

An adaptive finite element scheme for the Euler and Navier-Stokes equations

The implementation of two explicit finite element schemes for the solution of the compressible Euler and Navier-Stokes equations is presented. The schemes can be employed with general unstructured triangular meshes in two dimensions. Either scheme can therefore be used as the basic solver in a solution adaptive mesh procedure in a direct manner. The particular adaptive approach which is advocated here is intended for the solution of steady state problems only and involves an adaptive regeneration of the grid at prescribed stages during the false transient. The grid regeneration is accomplished by a mesh generator which has the capability of generating triangular grids over computational domains of arbitrary shape. The procedure is illustrated by solving transonic flows over multi-airfoil configurations and high speed flows, involving shock interactions, past circular cylinders.

Morgan, K.↗

NASA NTRS1989

A time accurate finite volume high resolution scheme for three dimensional Navier-Stokes equations

A time accurate, three-dimensional, finite volume, high resolution scheme for solving the compressible full Navier-Stokes equations is presented. The present derivation is based on the upwind split formulas, specifically with the application of Roe's (1981) flux difference splitting. A high-order accurate (up to the third order) upwind interpolation formula for the inviscid terms is derived to account for nonuniform meshes. For the viscous terms, discretizations consistent with the finite volume concept are described. A variant of second-order time accurate method is proposed that utilizes identical procedures in both the predictor and corrector steps. Avoiding the definition of midpoint gives a consistent and easy procedure, in the framework of finite volume discretization, for treating viscous transport terms in the curvilinear coordinates. For the boundary cells, a new treatment is introduced that not only avoids the use of 'ghost cells' and the associated problems, but also satisfies the tangency conditions exactly and allows easy definition of viscous transport terms at the first interface next to the boundary cells. Numerical tests of steady and unsteady high speed flows show that the present scheme gives accurate solutions.

Liou, Meng-Sing↗

NASA NTRS1991

Relative efficiency and accuracy of two Navier-Stokes codes for simulating attached transonic flow over wings

Two codes which solve the 3-D Thin Layer Navier-Stokes (TLNS) equations are used to compute the steady state flow for two test cases representing typical finite wings at transonic conditions. Several grids of C-O topology and varying point densities are used to determine the effects of grid refinement. After a description of each code and test case, standards for determining code efficiency and accuracy are defined and applied to determine the relative performance of the two codes in predicting turbulent transonic wing flows. Comparisons of computed surface pressure distributions with experimental data are made.

Bonhaus, Daryl L.↗

NASA NTRS1991

An Explicit Upwind Algorithm for Solving the Parabolized Navier-Stokes Equations

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 for the upwind PNS code are approximately equal to an explicit PNS MacCormack's code and existing implicit PNS solvers.

Korte, John J.↗

NASA NTRS1991

Addition of equilibrium air to an upwind Navier-Stokes code and other first steps toward a more generalized flow solver

An upwind three-dimensional volume Navier-Stokes code is modified to facilitate modeling of complex geometries and flow fields represented by proposed National Aerospace Plane concepts. Code enhancements include an equilibrium air model, a generalized equilibrium gas model and several schemes to simplify treatment of complex geometric configurations. The code is also restructured for inclusion of an arbitrary number of independent and dependent variables. This latter capability is intended for eventual use to incorporate nonequilibrium/chemistry gas models, more sophisticated turbulence and transition models, or other physical phenomena which will require inclusion of additional variables and/or governing equations. Comparisons of computed results with experimental data and results obtained using other methods are presented for code validation purposes. Good correlation is obtained for all of the test cases considered, indicating the success of the current effort.

Rosen, Bruce S.↗

NASA NTRS1991

Parabolized Navier-Stokes methods for hypersonic flows

A representative sampling of the techniques used in the integration of the Parabolized Navier-Stokes (PNS) equations is presented. Special atention is given to recent algorithms developed specifically for application to high speed flows, characterized by the presence of strong embedded shock waves and real gas effects. It is shown that PNS solvers are being used in the analysis of sonic boom signatures. Methods for modeling physical effects are discussed, including an overview of commonly used turbulence models and a more detailed discussion of techniques for including equilibrium and finite rate real gas effects.

Lawrence, Scott L.↗

NASA NTRS1991

Implementation of wall boundary conditions for transpiration in F3D thin-layer Navier-Stokes code

Numerical boundary conditions for mass injection/suction at the wall are incorporated in the thin-layer Navier-Stokes code, F3D. The accuracy of the boundary conditions and the code is assessed by a detailed comparison of the predictions of velocity distributions and skin-friction coefficients with exact similarity solutions for laminar flow over a flat plate with variable blowing/suction, and measurements for turbulent flow past a flat plate with uniform blowing. In laminar flow, F3D predictions for friction coefficient compare well with exact similarity solution with and without suction, but produces large errors at moderate-to-large values of blowing. A slight Mach number dependence of skin-friction coefficient due to blowing in turbulent flow is computed by F3D code. Predicted surface pressures for turbulent flow past an airfoil with mass injection are in qualitative agreement with measurements for a flat plate.

Kandula, M.↗

NASA NTRS1993

Navier-Stokes prediction of a delta wing in roll with vortex breakdown

The three-dimensional, Reynolds-averaged, Navier-Stokes (RANS) equations are used to numerically simulate vortical flow about a 65 degree sweep delta wing. Subsonic turbulent flow computations are presented for this delta wing at 30 degrees angle of attack and static roll angles up to 42 degrees. This work is part of an on going effort to validate the RANS approach for predicting high-incidence vortical flows, with the eventual application to wing rock. The flow is unsteady and includes spiral-type vortex breakdown. The breakdown positions, mean surface pressures, rolling moments, normal forces, and streamwise center-of-pressure locations compare reasonably well with experiment. In some cases, the primary vortex suction peaks are significantly underpredicted due to grid coarseness. Nevertheless, the computations are able to predict the same nonlinear variation of rolling moment with roll angle that appeared in the experiment. This nonlinearity includes regions of local static roll instability, which is attributed to vortex breakdown.

Chaderjian, Neal M.↗

NASA NTRS1993

Multigrid solution of the Navier-Stokes equations on highly stretched grids with defect correction

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.

Sockol, Peter M.↗

NASA NTRS1994

Transonic Navier-Stokes calculations about a 65 deg delta wing

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.

Londenberg, W. Kelly↗

NASA NTRS1995

Application of Navier-Stokes code PAB3D with kappa-epsilon turbulence model to attached and separated flows

A three-dimensional Navier-Stokes solver was used to determine how accurately computations can predict local and average skin friction coefficients for attached and separated flows for simple experimental geometries. Algebraic and transport equation closures were used to model turbulence. To simulate anisotropic turbulence, the standard two-equation turbulence model was modified by adding nonlinear terms. The effects of both grid density and the turbulence model on the computed flow fields were also investigated and compared with available experimental data for subsonic and supersonic free-stream conditions.

Abdol-Hamid, Khaled S.↗

NASA NTRS1995

Navier-Stokes analysis of a delta wing in static and dynamic roll

The three-dimensional, Reynolds-averaged, Navier-Stokes equations are used to numerically simulate non-steady high-incidence vortical flow about a 65 degree sweep delta wing under static roll and forced periodic roll motions. These computations have been previously reported in the literature, where emphasis was placed on validating the computational results with nonsteady experimental surface pressures, forces and moments. The emphasis of this research is to revisit these earlier computations and use nonsteady numerical flow visualization to analyze the nonlinear surface flow and vortex breakdown dynamics. Most notable are the periodic formation of surface-flow separation lines, and the periodic formation of vortex breakdown near the delta wing trailing edge.

Chaderjian, N.↗

NASA NTRS1995

A fast Poisson solver for unsteady incompressible Navier-Stokes equations on the half-staggered grid

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.

Golub, G. H.↗

NASA NTRS1999

Distributed Relaxation Multigrid and Defect Correction Applied to the Compressible Navier-Stokes Equations

The distributed-relaxation multigrid and defect- correction methods are applied to the two- dimensional compressible Navier-Stokes equations. The formulation is intended for high Reynolds number applications and several applications are made at a laminar Reynolds number of 10,000. A staggered- grid arrangement of variables is used; the coupled pressure and internal energy equations are solved together with multigrid, requiring a block 2x2 matrix solution. Textbook multigrid efficiencies are attained for incompressible and slightly compressible simulations of the boundary layer on a flat plate. Textbook efficiencies are obtained for compressible simulations up to Mach numbers of 0.7 for a viscous wake simulation.

Thomas, J. L.↗

NASA NTRS1975

Finite element computation of a viscous compressible free shear flow governed by the time dependent Navier-Stokes equations

A finite element algorithm for solution of fluid flow problems characterized by the two-dimensional compressible Navier-Stokes equations was developed. The program is intended for viscous compressible high speed flow; hence, primitive variables are utilized. The physical solution was approximated by trial functions which at a fixed time are piecewise cubic on triangular elements. The Galerkin technique was employed to determine the finite-element model equations. A leapfrog time integration is used for marching asymptotically from initial to steady state, with iterated integrals evaluated by numerical quadratures. The nonsymmetric linear systems of equations governing time transition from step-to-step are solved using a rather economical block iterative triangular decomposition scheme. The concept was applied to the numerical computation of a free shear flow. Numerical results of the finite-element method are in excellent agreement with those obtained from a finite difference solution of the same problem.

Cooke, C. H.↗

NASA NTRS1981

Reduction of the two dimensional stationary Navier-Stokes problem to a sequence of Fredholm integral equations of the second kind

Present approaches to solving the stationary Navier-Stokes equations are of limited value; however, there does exist an equivalent representation of the problem that has significant potential in solving such problems. This is due to the fact that the equivalent representation consists of a sequence of Fredholm integral equations of the second kind, and the solving of this type of problem is very well developed. For the problem in this form, there is an excellent chance to also determine explicit error estimates, since bounded, rather than unbounded, linear operators are dealt with.

Gabrielsen, R. E.↗

NASA NTRS1986

Efficient solution methods for the Navier-Stokes equations

Implicit finite difference schemes for solving two-dimensional and three-dimensional Euler and thin layer Navier-Stokes equations are addressed. The methods are demonstrated in fully vectorized codes for a Cray type architecture. The Beam and Warming implicit approximate factorization algorithm in generalized coordinates is used. The methods are either time accurate or accelerated non-time accurate steady state schemes. Acceleration and efficiency modifications such as matrix reduction, diagonalization, and flux split schemes are presented. Two dimensional inviscid and viscous calculations (e.g., airfoils with a deflected spoiler, circulation control airfoils, and unsteady buffeting) and of three dimensional viscous elliptical bodies, exhausting boattails, and generic oblique wing computations are discussed.

Pulliam, T. H.↗