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NASA NTRS2007

URANS Application with CFL3D

This case was run using CFL3D, a multi-zone Reynolds-averaged Navier-Stokes code developed at NASA Langley [1]. It solves the thin-layer form of the Navier-Stokes equations in each of the (selected) coordinate directions. It can use 1-to-1, patched, or overset grids, and employs local time step scaling, grid sequencing, and multigrid to accelerate convergence to steady state. In time-accurate mode, CFL3D has the option to employ dual-time stepping with subiterations and multigrid, and it achieves second order temporal accuracy. CFL3D is a finite volume method. It uses third-order upwind-biased spatial differencing on the convective and pressure terms, and second-order differencing on the viscous terms; it is globally second-order spatially accurate. The flux difference-splitting (FDS) method of Roe is employed to obtain fluxes at the cell faces. It is advanced in time with an implicit three-factor approximate factorization method.

Rumsey, C. L.↗
NASA NTRS2007

RANS and URANS Application with CFL3D

This case was run using CFL3D, a multi-zone Reynolds-averaged Navier-Stokes code developed at NASA Langley [1]. It solves the thin-layer form of the Navier-Stokes equations in each of the (selected) coordinate directions. It can use 1-to-1, patched, or overset grids, and employs local time step scaling, grid sequencing, and multigrid to accelerate convergence to steady state. In time-accurate mode, CFL3D has the option to employ dual-time stepping with subiterations and multigrid, and it achieves second order temporal accuracy. CFL3D is a finite volume method. It uses third-order upwind-biased spatial differencing on the convective and pressure terms, and second-order differencing on the viscous terms; it is globally second-order spatially accurate. The flux difference-splitting (FDS) method of Roe is employed to obtain fluxes at the cell faces. It is advanced in time with an implicit three-factor approximate factorization method.

Rumsey, C. L.↗
NASA NTRS1989

Multigrid calculation of three-dimensional turbomachinery flows

Research was performed in the general area of computational aerodynamics, with particular emphasis on the development of efficient techniques for the solution of the Euler and Navier-Stokes equations for transonic flows through the complex blade passages associated with turbomachines. In particular, multigrid methods were developed, using both explicit and implicit time-stepping schemes as smoothing algorithms. The specific accomplishments of the research have included: (1) the development of an explicit multigrid method to solve the Euler equations for three-dimensional turbomachinery flows based upon the multigrid implementation of Jameson's explicit Runge-Kutta scheme (Jameson 1983); (2) the development of an implicit multigrid scheme for the three-dimensional Euler equations based upon lower-upper factorization; (3) the development of a multigrid scheme using a diagonalized alternating direction implicit (ADI) algorithm; (4) the extension of the diagonalized ADI multigrid method to solve the Euler equations of inviscid flow for three-dimensional turbomachinery flows; and also (5) the extension of the diagonalized ADI multigrid scheme to solve the Reynolds-averaged Navier-Stokes equations for two-dimensional turbomachinery flows.

Caughey, David A.↗
NASA NTRS1988

Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations

Parameter studies are conducted using the Euler and potential flow equation models for steady and unsteady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux vector splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flow results are made, as well as with experimental data where available.

Anderson, W. Kyle↗
NASA NTRS1988

Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations

Parameter studies are conducted using the Euler and potential flow equation models for unsteady and steady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux-vector-splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flows results are made as well as with experimental data where available.

Anderson, W. Kyle↗
NASA NTRS1993

Multigrid properties of upwind-biased data reconstructions

The multigrid properties of two data reconstruction methods used for achieving second-order spatial accuracy when solving the two-dimensional Euler equations are examined. The data reconstruction methods are used with an implicit upwind algorithm which uses linearized backward-Euler time-differencing. The solution of the resulting linear system is performed by an iterative procedure. In the present study only regular quadrilateral grids are considered, so a red-black Gauss-Seidel iteration is used. Although the Jacobian is approximated by first-order upwind extrapolation, two alternative data reconstruction techniques for the flux integral that yield higher-order spatial accuracy at steady state are examined. The first method, probably most popular for structured quadrilateral grids, is based on estimating the cell gradients using one-dimensional reconstruction along curvilinear coordinates. The second method is based on Green's theorem. Analysis and numerical results for the two dimensional Euler equations show that data reconstruction based on Green's theorem has superior multigrid properties as compared to the one-dimensional data reconstruction method.

Warren, Gary P.↗
NASA NTRS1974

Hydrodynamic models of a cepheid atmosphere

A method for including the solution of the transfer equation in a standard Henyey type hydrodynamic code was developed. This modified Henyey method was used in an implicit hydrodynamic code to compute deep envelope models of a classical Cepheid with a period of 12(d) including radiative transfer effects in the optically thin zones. It was found that the velocity gradients in the atmosphere are not responsible for the large microturbulent velocities observed in Cepheids but may be responsible for the occurrence of supersonic microturbulence. It was found that the splitting of the cores of the strong lines is due to shock induced temperature inversions in the line forming region. The adopted light, color, and velocity curves were used to study three methods frequently used to determine the mean radii of Cepheids. It is concluded that an accuracy of 10% is possible only if high quality observations are used.

Karp, A. H.↗
NASA NTRS1982

Application of the Hughes-LIU algorithm to the 2-dimensional heat equation

An implicit explicit algorithm for the solution of transient problems in structural dynamics is described. The method involved dividing the finite elements into implicit and explicit groups while automatically satisfying the conditions. This algorithm is applied to the solution of the linear, transient, two dimensional heat equation subject to an initial condition derived from the soluton of a steady state problem over an L-shaped region made up of a good conductor and an insulating material. Using the IIT/PRIME computer with virtual memory, a FORTRAN computer program code was developed to make accuracy, stability, and cost comparisons among the fully explicit Euler, the Hughes-Liu, and the fully implicit Crank-Nicholson algorithms. The Hughes-Liu claim that the explicit group governs the stability of the entire region while maintaining the unconditional stability of the implicit group is illustrated.

Malkus, D. S.↗
NASA NTRS1989

Parabolized Navier-Stokes algorithm for chemically reacting flows

A second-order parabolized Navier-Stokes algorithm based on the MacCormack (1969) explicit scheme is used to study three-dimensional chemically reacting flows with finite-rate chemistry. The method can treat the chemical source term implicitly, and it accounts for the multicomponent diffusion and convection of the chemical species. The method is demonstrated with the nonreacting case of a Mach-3 flow over a double-wedge compression corner and the case of streamwise hydrogen injection at sonic velocity in a Mach-2.44 vitiated air stream.

Kamath, H.↗
NASA NTRS1995

Adaptive unstructured triangular mesh generation and flow solvers for the Navier-Stokes equations at high Reynolds number

A method for generating high quality unstructured triangular grids for high Reynolds number Navier-Stokes calculations about complex geometries is described. Careful attention is paid in the mesh generation process to resolving efficiently the disparate length scales which arise in these flows. First the surface mesh is constructed in a way which ensures that the geometry is faithfully represented. The volume mesh generation then proceeds in two phases thus allowing the viscous and inviscid regions of the flow to be meshed optimally. A solution-adaptive remeshing procedure which allows the mesh to adapt itself to flow features is also described. The procedure for tracking wakes and refinement criteria appropriate for shock detection are described. Although at present it has only been implemented in two dimensions, the grid generation process has been designed with the extension to three dimensions in mind. An implicit, higher-order, upwind method is also presented for computing compressible turbulent flows on these meshes. Two recently developed one-equation turbulence models have been implemented to simulate the effects of the fluid turbulence. Results for flow about a RAE 2822 airfoil and a Douglas three-element airfoil are presented which clearly show the improved resolution obtainable.

Ashford, Gregory A.↗
NASA NTRS1993

A comparison of three-dimensional nonequilibrium solution algorithms applied to hypersonic flows with stiff chemical source terms

Three solution algorithms, explicit underrelaxation, point implicit, and lower upper symmetric Gauss-Seidel (LUSGS), are used to compute nonequilibrium flow around the Apollo 4 return capsule at 62 km altitude. By varying the Mach number, the efficiency and robustness of the solution algorithms were tested for different levels of chemical stiffness. The performance of the solution algorithms degraded as the Mach number and stiffness of the flow increased. At Mach 15, 23, and 30, the LUSGS method produces an eight order of magnitude drop in the L2 norm of the energy residual in 1/3 to 1/2 the Cray C-90 computer time as compared to the point implicit and explicit under-relaxation methods. The explicit under-relaxation algorithm experienced convergence difficulties at Mach 23 and above. At Mach 40 the performance of the LUSGS algorithm deteriorates to the point it is out-performed by the point implicit method. The effects of the viscous terms are investigated. Grid dependency questions are explored.

Palmer, Grant↗
NASA NTRS1993

Reliability enhancement of Navier-Stokes codes through convergence enhancement

Reduction of total computing time required by an iterative algorithm for solving Navier-Stokes equations is an important aspect of making the existing and future analysis codes more cost effective. Several attempts have been made to accelerate the convergence of an explicit Runge-Kutta time-stepping algorithm. These acceleration methods are based on local time stepping, implicit residual smoothing, enthalpy damping, and multigrid techniques. Also, an extrapolation procedure based on the power method and the Minimal Residual Method (MRM) were applied to the Jameson's multigrid algorithm. The MRM uses same values of optimal weights for the corrections to every equation in a system and has not been shown to accelerate the scheme without multigriding. Our Distributed Minimal Residual (DMR) method based on our General Nonlinear Minimal Residual (GNLMR) method allows each component of the solution vector in a system of equations to have its own convergence speed. The DMR method was found capable of reducing the computation time by 10-75 percent depending on the test case and grid used. Recently, we have developed and tested a new method termed Sensitivity Based DMR or SBMR method that is easier to implement in different codes and is even more robust and computationally efficient than our DMR method.

Choi, K.-Y.↗
NASA NTRS1987

Numerical tests of a modified full implicit continuous Eulerian (FICE) scheme with projected normal characteristic boundary conditions for MHD flows

A numerical method has been developed based on a modified full implicit continuous Eulerian (FICE) scheme and projected normal characteristic boundary conditions for simulating MHD flows which undergo a long process of evolution. An astrophysical flow is chosen for illustration of this procedure, and numerical tests are made to verify the computational stability and physically realistic solution. Three computational tests have been accomplished; they are tests of solving methods, characteristic boundary condition, and time steps. The tests show that the program from the modified FICE scheme with proper boundary conditions and time steps can be made numerically stable for a time long enough to obtain physically plausible solutions.

Wu, S. T.↗
NASA NTRS1986

An LU implicit scheme for high speed inlet analysis

A numerical method is developed to analyze the inviscid flowfield of a high speed inlet by the solution of the Euler equations. The LU implicit scheme in conjunction with adaptive dissipation proves to be an efficient and robust nonoscillatory shock capturing technique for high Mach number flows as well as for transonic flows.

Yoon, S.↗
NASA NTRS1995

Newton-Krylov-Schwarz: An implicit solver for CFD

Newton-Krylov methods and Krylov-Schwarz (domain decomposition) methods have begun to become established in computational fluid dynamics (CFD) over the past decade. The former employ a Krylov method inside of Newton's method in a Jacobian-free manner, through directional differencing. The latter employ an overlapping Schwarz domain decomposition to derive a preconditioner for the Krylov accelerator that relies primarily on local information, for data-parallel concurrency. They may be composed as Newton-Krylov-Schwarz (NKS) methods, which seem particularly well suited for solving nonlinear elliptic systems in high-latency, distributed-memory environments. We give a brief description of this family of algorithms, with an emphasis on domain decomposition iterative aspects. We then describe numerical simulations with Newton-Krylov-Schwarz methods on aerodynamics applications emphasizing comparisons with a standard defect-correction approach, subdomain preconditioner consistency, subdomain preconditioner quality, and the effect of a coarse grid.

Cai, Xiao-Chuan↗
NASA NTRS1995

Comparison of Nonequilibrium Solution Algorithms Applied to Chemically Stiff Hypersonic Flows

Three solution algorithms, explicit under-relaxation, point implicit, and lower-upper symmetric Gauss-Seidel, are used to compute nonequilibrium flow around the Apollo 4 return capsule at the 62-km altitude point in its descent trajectory. By varying the Mach number, the efficiency and robustness of the solution algorithms were tested for different levels of chemical stiffness.The performance of the solution algorithms degraded as the Mach number and stiffness of the flow increased. At Mach 15 and 30, the lower-upper symmetric Gauss-Seidel method produces an eight order of magnitude drop in the energy residual in one-third to one-half the Cray C-90 computer time as compared to the point implicit and explicit under-relaxation methods. The explicit under-relaxation algorithm experienced convergence difficulties at Mach 30 and above. At Mach 40 the performance of the lower-upper symmetric Gauss-Seidel algorithm deteriorates to the point that it is out performed by the point implicit method. The effects of the viscous terms are investigated. Grid dependency questions are explored.

Palmer, Grant↗
NASA NTRS1990

Multigrid solution of unsteady Navier-Stokes equations using a pressure method

A multigrid relaxation method is applied to a pressure-based implicit procedure to solve unseady, incompressible Navier-Stokes equations. The present multigrid method is a Correction Scheme according to Brandt. This method is used to solve the scalar matrices resulting from the finite-volume formulation and uses flux averaging as the restriction operator. The accuracy and computational efficiency are demonstrated with a steady state driven cavity flow and an unsteady flow over a circular cylinder case. The results are compared with single grid results using the OrthoMin conjugate gradient method and experimental data.

Jiang, Y.↗