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NASA NTRS2000

Solution of Third Computational Workshop Internal Propagation Problems using Low Order Schemes

The problems solved are propagation of sound waves through a transonic nozzle and shock-sound interaction. The first problem is solved using a MacCormack scheme. The second problem is solved by perturbation of a conservative Euler equation solution to the steady state. The problems are solved on evenly spaced grids. While solutions were found, the methods selected and not using stretched grids lead to solutions which do not compare well with those found using more accurate schemes. However, the perturbation of the mean flow scheme used to solve the second problem shows promise.

Miles, Jeffrey Hilton↗

NASA NTRS1973

Survey of computational methods for three-dimensional supersonic inviscid flows with shocks

Hyperbolic systems of partial differential equations governing supersonic inviscid flows are discussed and analyzed. Finite-difference analogues for integrating these systems in the interior of fluid domains are described from two points of view: a differential form approach and an integral form approach. The algorithms presented are analyzed for stability and accuracy. The concept of time splitting is discussed and supplied to these methods to achieve increased numerical efficiency. Techniques for treating conditions at the boundaries of the fluid domain and shock-wave discontinuities at surfaces within the domain are described.

Maccormack, R. W.↗

NASA NTRS1976

A rapid solver for hyperbolic systems of equations

A numerical method is proposed for solving the time-dependent compressible Navier-Stokes equations in two dimensions. The equations are time-split into a hyperbolic part and a parabolic part. This paper describes an explicit numerical method for solving the hyperbolic (inviscid) part. The hyperbolic operator is explicit, conservative, uses characteristic relations to predict convection and pressure fields, and is stable under a specified condition.

Maccormack, R. W.↗

NASA NTRS1984

Analysis of inviscid and viscous flows in cascades with an explicit multiple-grid algorithm

A rapid technique is used for calculating inviscid and viscous flows in turbomachinery cascades. The Euler and thin-layer Navier-Stokes equations are solved using the original explicit MacCormack algorithm. The Baldwin-Lomax eddy viscosity model is used for turbulent flows. Convergence to a steady state is accelerated by use of a variable time-step and a multiple-grid scheme. Computer time is reduced through vectorization. Details of the numerical method are presented along with computed results for two low-speed wind tunnel turning vanes, a space shuttle fuel pump turbine rotor, and a supersonic inflow compressor rotor. The method can predict subtle viscous flow phenomena in cascades and is fast enough to be used as a design tool.

Chima, R. V.↗

NASA NTRS1984

Analysis of inviscid and viscous flows in cascades with an explicit multiple-grid algorithm

Chima, R. V.↗

NASA NTRS1988

A numerical method for predicting hypersonic flowfields

The flow about a body traveling at hypersonic speed is energetic enough to cause the atmospheric gases to react chemically and reach states in thermal nonequilibrium. In this paper, a new procedure based on Gauss-Seidel line relaxation is shown to solve the equations of hypersonic flow fields containing finite reaction rate chemistry and thermal nonequilibrium. The method requires a few hundred time steps and small computer times for axisymmetric flows about simple body shapes. The extension to more complex two-dimensional body geometries appears straightforward.

Maccormack, Robert W.↗

NASA NTRS2001

CAA for Jet Noise Physics: Issues and Recent Progress

Dr. Mankbadi summarized recent CAA results. Examples of the effect of various boundary condition schemes on the computed acoustic field, for a point source in a uniform flow, were shown. Solutions showing the impact of inflow excitations on the result were also shown. Results from a large eddy simulation, using a fourth-order MacCormack scheme with a Smagorinsky sub-grid turbulence model, were shown for a Mach 2.1 unheated jet. The results showed that the results were free from spurious modes. Results were shown for a Mach 1.4 jet using LES in the near field and the Kirchhoff method for the far field. Predicted flow field characteristics were shown to be in good agreement with data and predicted far field directivities were shown to be in qualitative agree with experimental measurements.

Mankbadi, Reda↗

NASA NTRS2001

CAA for Jet Noise Physics

Mankbadi, Reda↗

NASA NTRS1978

Status and future prospects of using numerical methods to study complex flows at High Reynolds numbers

The calculation of flow fields past aircraft configuration at flight Reynolds numbers is considered. Progress in devising accurate and efficient numerical methods, in understanding and modeling the physics of turbulence, and in developing reliable and powerful computer hardware is discussed. Emphasis is placed on efficient solutions to the Navier-Stokes equations.

Maccormack, R. W.↗

NASA NTRS1982

Computation of three-dimensional inviscid flow over hypersonic missile configurations using the GIM code

A three-dimensional computational technique was used to obtain flowfield solutions to the Euler equations over selected hypersonic missile configurations. The General Interpolants Method (GIM) computer code was used with interpolation functions in an algebraic approach to generate a discrete computational grid for each configuration. The spatial marching version of the GIM code, which treats the parabolized Navier-Stokes (PNS) equations or the Euler equations with a shock capturing, 'MacCormack-like' scheme, was used to advance the solution hyperbolically over each configuration. The inviscid flowfield solutions over the two three-dimensional missile configurations, calculated using the GIM hyperbolic scheme, are presented here. The flow field over a wing/body configuration at zero degree angle of attack is presented. Flow over the fuselage of a tactical missile, termed the TAME 10, at both zero degree and 7.5 degree angles of attack is presented. In addition, an inviscid, two-dimensional analysis of an inlet configuration designed to mount on the TAME 10 is included. Contour maps of velocity and pressure are included for each configuration. Comparison of calculation and data show good agreement.

Xiques, K. E.↗

NASA NTRS1976

An efficient numerical method for solving the time-dependent compressible Navier-Stokes equations at high Reynolds number

A fine-mesh method incorporating two new operators, which drastically reduces the computation time, has been developed for solving the time-dependent Navier-Stokes equations at flight Reynolds numbers. The approach time-splits the equations into a hyperbolic part and a parabolic part, solves the hyperbolic part by a new explicit numerical method based on characteristics theory, and solves the parabolic part by a new efficient implicit parabolic method. The method has reduced the computation time by one and two orders of magnitude from that required previously to solve for the interaction of a shock wave with a boundary layer on a flat plate.

Maccormack, R. W.↗

NASA NTRS1989

A numerical method for predicting hypersonic flowfields

The flow about a body traveling at hypersonic speed is energetic enough to cause the atmospheric gases to chemically react and reach states in thermal nonequilibrium. The prediction of hypersonic flowfields requires a numerical method capable of solving the conservation equations of fluid flow, the chemical rate equations for specie formation and dissociation, and the transfer of energy relations between translational and vibrational temperature states. Because the number of equations to be solved is large, the numerical method should also be as efficient as possible. The proposed paper presents a fully implicit method that fully couples the solution of the fluid flow equations with the gas physics and chemistry relations. The method flux splits the inviscid flow terms, central differences of the viscous terms, preserves element conservation in the strong chemistry source terms, and solves the resulting block matrix equation by Gauss Seidel line relaxation.

Maccormack, Robert W.↗

NASA NTRS1991

Non equilibrium flow in an arc-jet

A method for simulating the flow of a nitrogen plasma in thermodynamic nonequilibrium is described. The method relies on the modeling of the relevant physical and chemical phenomena and on a fully coupled implicit finite volume upwind numerical approximation of the governing equations. The results agree with experimental data.

Rostand, P.↗

NASA NTRS1987

Supercomputing of supersonic flows using upwind relaxation and MacCormack schemes

The performance of two numerical solution schemes, (1) an implicit upwind relaxation with a finite-volume discretization (Thomas and Walters, 1985) and (2) an explicit-implicit MacCormack (1981) scheme with a finite-difference discretization, is compared in two-dimensional simulations of supersonic flow past a flat plate with leading edge, a rearward-facing step, a 10-deg compression corner, a NACA 0012 airfoil at high angle of attack, and a cavity. The algebraic turbulence model, the solution methods, and the boundary conditions and SIMD coding are explained, and the results are presented in tables and graphs and characterized with reference to published experimental data. Scheme (1) is found to converge more rapidly and to give more accurate results than (2) in a wide range of problem types.

Baysal, O.↗

NASA NTRS1995

Category 5 problem solution using an unstructured finite volume algorithm

For the simulation of flows with complex geometries, unstructured finite volume methods have proven to be very popular, and simulations of a large number of flows have been done with good results using this approach. Since most of the simulations to date were done for steady flows, it is not clear that present unstructured finite volume algorithms can accurately track the unsteady propagation of acoustic waves in a computation. Therefore, there is a need to assess the accuracy of these methods for acoustic calculations. In this paper, we perform the numerical simulation of a very small amplitude acoustic wave incident on the non-uniform steady flow in a quasi- 1 D convergent-divergent nozzle using an unstructured finite volume algorithm with piece-wise linear, least square reconstruction, Roe flux difference splitting, and second-order MacCormack time marching. First, the spatial accuracy of the algorithm is evaluated for the steady flow by running the simulation with a sequence of successively finer meshes. Then the unsteady numerical solution with the acoustic perturbation is presented.

Bui, Trong T.↗

NASA NTRS1995

A computational fluid dynamics simulation of the hypersonic flight of the Pegasus(TM) vehicle using an artificial viscosity model and a nonlinear filtering method

The computational fluid dynamics code, PARC3D, is tested to see if its use of non-physical artificial dissipation affects the accuracy of its results. This is accomplished by simulating a shock-laminar boundary layer interaction and several hypersonic flight conditions of the Pegasus(TM) launch vehicle using full artificial dissipation, low artificial dissipation, and the Engquist filter. Before the filter is applied to the PARC3D code, it is validated in one-dimensional and two-dimensional form in a MacCormack scheme against the Riemann and convergent duct problem. For this explicit scheme, the filter shows great improvements in accuracy and computational time as opposed to the nonfiltered solutions. However, for the implicit PARC3D code it is found that the best estimate of the Pegasus experimental heat fluxes and surface pressures is the simulation utilizing low artificial dissipation and no filter. The filter does improve accuracy over the artificially dissipative case but at a computational expense greater than that achieved by the low artificial dissipation case which has no computational time penalty and shows better results. For the shock-boundary layer simulation, the filter does well in terms of accuracy for a strong impingement shock but not as well for weaker shock strengths. Furthermore, for the latter problem the filter reduces the required computational time to convergence by 18.7 percent.

Mendoza, John Cadiz↗

NASA NTRS1983

A time-split finite-volume algorithm for three-dimensional flow-field simulation

A general finite-volume algorithm is developed for solving three-dimensional, time-dependent, compressible Navier-Stokes equations for high Reynolds number flows over an arbitrary geometry. This algorithm adapts MacCormack's (1982) explicit-implicit scheme to a time-split, three-dimensional finite-volume concept in a general coordinate system. It is shown that the thin-layer approximation in all three spatial directions significantly reduces the evaluation of viscous terms and allows the algorithm to solve more complicated geometries with all boundaries in two or all three directions. The calculated results using this method are found to be in good agreement with the experimental measurements of a blunt-fin induced shock wave and boundary-layer interaction problems. Observations of the existence of peak pressure, primary horseshoe and secondary vortices, and reversed supersonic zones show that computational fluid dynamics can effectively supplement the wind tunnel tests for aerodynamic design as well as for understanding basic fluid dynamics.

Hung, C. M.↗

NASA NTRS1981

A numerical method for solving the equations of compressible viscous flow

Although much progress has already been made in solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A method for solving these equations has been devised that: (1) is second-order accurate in space and time; (2) is unconditionally stable; (3) preserves conservation form; (4) requires no block or scalar tridiagonal inversions; (5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs); (6) is more efficient than present methods; and (7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 300,000 to 3 x 10 to the 7th and at CFL numbers as high as 1000 are compared with theory and experiment.

Maccormack, R. W.↗