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NASA NTRS1989

Hypersonic blunt body computations including real gas effects

Various second-order explicit and implicit TVD shock-capturing methods, a generalization of Roe's approximate Riemann solver, and a generalized flux-vector splitting scheme are used to study two-dimensional hypersonic real-gas flows. Special attention is given to the identification of some of the elements and parameters which can affect the convergence rate for high Mach numbers or real gases, but have negligible effect for low Mach numbers, for cases involving steady-state inviscid blunt flows. Blunt body calculations at Mach numbers of greater than 15 are performed to treat real-gas effects, and impinging shock results are obtained to test the treatment of slip surfaces and complex structures. Even with the addition of improvements, the convergence rate of algorithms in the hypersonic flow regime is found to be generally slower for a real gas than for a perfect gas.

Montagne, J.-L.↗

NASA NTRS2005

Adaptive Numerical Dissipation Control in High Order Schemes for Multi-D Non-Ideal MHD

The required type and amount of numerical dissipation/filter to accurately resolve all relevant multiscales of complex MHD unsteady high-speed shock/shear/turbulence/combustion problems are not only physical problem dependent, but also vary from one flow region to another. In addition, proper and efficient control of the divergence of the magnetic field (Div(B)) numerical error for high order shock-capturing methods poses extra requirements for the considered type of CPU intensive computations. The goal is to extend our adaptive numerical dissipation control in high order filter schemes and our new divergence-free methods for ideal MHD to non-ideal MHD that include viscosity and resistivity. The key idea consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. These scheme-independent detectors are capable of distinguishing shocks/shears, flame sheets, turbulent fluctuations and spurious high-frequency oscillations. The detection algorithm is based on an artificial compression method (ACM) (for shocks/shears), and redundant multiresolution wavelets (WAV) (for the above types of flow feature). These filters also provide a natural and efficient way for the minimization of Div(B) numerical error.

Yee, H. C.↗

NASA NTRS2004

Nonlinear filtering and limiting in high order methods for ideal and non-ideal MHD

The various filtering mechanisms and base scheme options of the newly developed adaptive numerical dissipation control in spatially high order filter schemes for the ideal and non-ideal magnetohydrodynamics (MHD) equations are investigated. These filter schemes are applicable to complex unsteady MHD high-speed shock/shear/turbulence problems. They also provide a natural and efficient way for the minimization of Div(B) numerical error. The type of spatial base scheme to be used in conjunction with our filter idea is very general. For example, spectral, compact and non-compact spatially central finite difference schemes are possible candidates. The adaptive numerical dissipation mechanism consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and to leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is also very general. The objective of this paper is to investigate the performance of using compact and non-compact central base schemes in conjunction with three commonly used types of nonlinear numerical dissipation for both the ideal and non-ideal MHD. This extended abstract shows the performance of three nonlinear filters in conjunction with a sixth-order non-compact spatial central base scheme. In the final paper, the high order compact spatial central base scheme will be illustrated and compared with the non-compact base scheme. The reason for the investigation of the high order compact spatial central base scheme over the non-compact base scheme is to evaluate if additional accuracy can be gained in regions of fine scale turbulence that are away from shocks/shears.

Yee,H. C.↗

NASA NTRS1993

A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems

The finite-volume approach is presently used to obtain a 2D, high-order accurate and basically nonoscillatory shock-capture method whose high-order spatial accuracy is obtained by means of a piecewise polynomial approximation of the solution from cell averages. Attention is given to a high-order spatial operator that is able to both retain high-order accuracy in smooth regions and avoid the oscillations that are associated with interpolations across steep gradients. The operator is extended to hyperbolic systems of equations and curvilinear meshes.

Casper, Jay↗

NASA NTRS2004

High-Order Simulation of Non-Linear Oscillations and Shocks in the Solar Atmosphere

The solar atmosphere presents a rich source of highly non-linear magneto-hydrodynamic phenomena: strong gradients and forcing terms result in both large shocks and oscillations. The additional requirements of energy balance and initialization in hydrostatic equilibrium compound the challenge of this problem. A wealth of observational data allows us to check the results of our simulations. The problem of simulating the solar atmosphere provides, in addition to an interesting system in its own right, a challenging testbed for high-order shock-capturing methods. We discuss the challenge of simulating solar atmospheric phenomena, concentrating on various high-order central methods ranging from second to fourth order. Our method is based on the central-upwind scheme of Kurganov, Noelle and Petrova, which we extend to high order via various interpolants. We investigate various initial data for our simulations, corresponding to observed conditions in different regions of the solar surface: the normal quiet sun and sunspots. When non-oscillatory using second- and third-order methods, we are able to reproduce non-trivial observational results. In particular we find a correlation between initial data and both the shock speeds and particle oscillation spectra that match observations in the corresponding regions. When using fourth-order WENO interpolants, we find that while the individual shock profiles at any given time appear non-oscillatory, spurious oscillations appear in the fields after long time integrations. The issue of initialization in hydrostatic equilibrium raises difficult issues. Careful treatment of the gravitational source term can reduce violations of hydrostatic equilibrium, but difficulties remain, primarily due to discontinuities in the piecewise-polynomial reconstructions. In the case of the Euler equations in gravity, the use of high-order methods reduces the violation of hydrostatic equilibrium to a sufficiently low level for the above described results to be obtained. Including various energy flux terms in the equations, however, leads to severe instability when hydrostatic balance is violated. We discuss various strategies to address this problem, including adaptive grid methods.

Bryson, S.↗

NASA NTRS1986

Semi-implicit and fully implicit shock-capturing methods for hyperbolic conservation laws with stiff source terms

Some numerical aspects of finite-difference algorithms for nonlinear multidimensional hyperbolic conservation laws with stiff nonhomogenous (source) terms are discussed. If the stiffness is entirely dominated by the source term, a semi-implicit shock-capturing method is proposed provided that the Jacobian of the soruce terms possesses certain properties. The proposed semi-implicit method can be viewed as a variant of the Bussing and Murman point-implicit scheme with a more appropriate numerical dissipation for the computation of strong shock waves. However, if the stiffness is not solely dominated by the source terms, a fully implicit method would be a better choice. The situation is complicated by problems that are higher than one dimension, and the presence of stiff source terms further complicates the solution procedures for alternating direction implicit (ADI) methods. Several alternatives are discussed. The primary motivation for constructing these schemes was to address thermally and chemically nonequilibrium flows in the hypersonic regime. Due to the unique structure of the eigenvalues and eigenvectors for fluid flows of this type, the computation can be simplified, thus providing a more efficient solution procedure than one might have anticipated.

Yee, H. C.↗

NASA NTRS1987

Semi-implicit and fully implicit shock-capturing methods for hyperbolic conservation laws with stiff source terms

Some numerical aspects of finite-difference algorithms for nonlinear multidimensional hyperbolic conservation laws with stiff nonhomogeneous (source) terms are discussed. If the stiffness is entirely dominated by the source term, a semi-implicit shock-capturing method is proposed provided that the Jacobian of the source terms possesses certain properties. The proposed semi-implicit method can be viewed as a variant of the Bussing and Murman point-implicit scheme with a more appropriate numerical dissipation for the computation of strong shock waves. However, if the stiffness is not solely dominated by the source terms, a fully implicit method would be a better choice. The situation is complicated by problems that are higher than one dimension, and the presence of stiff source terms further complicates the solution procedures for alternating direction implicit (ADI) methods. Several alternatives are discussed. The primary motivation for constructing these schemes was to address thermally and chemically nonequilibrium flows in the hypersonic regime. Due to the unique structure of the eigenvalues and eigenvectors for fluid flows of this type, the computation can be simplified, thus providing a more efficient solution procedure than one might have anticipated.

Yee, H. C.↗

NASA NTRS1988

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Mostrel, M. M.↗

NASA NTRS1997

An Investigation of High-Order Shock-Capturing Methods for Computational Aeroacoustics

This project is motivated by the desire to develop numerical methods that will be useful in the study of compressible flows that exhibit aeroacoustic phenomena. Solutions to linear problems have been investigated through the development of a computer code based on the recent dispersion-relation-preserving (DRP) methodology. In regard to nonlinear problems, the class of essentially nonoscillatory (ENO) schemes have been considered as the primary candidates for solving aeroacoustic problems in which discontinuities are involved. Discontinuities in the solution itself (e.g. shocks) as well as in the geometry on which the problem is defined have been studied. Two-dimensional nonlinear problems were considered in order to determine if the one-dimensional results obtained in the first phase of this project were extendable to more realistic problems. Conclusions have been drawn in regard to the ability to numerically predict solutions of nonlinear problems with shocks to high-order accuracy.

Casper, Jay↗

NASA NTRS1991

Multigrid for hypersonic viscous two- and three-dimensional flows

The use of a multigrid method with central differencing to solve the Navier-Stokes equations for hypersonic flows is considered. The time-dependent form of the equations is integrated with an explicit Runge-Kutta scheme accelerated by local time stepping and implicit residual smoothing. Variable coefficients are developed for the implicit process that remove the diffusion limit on the time step, producing significant improvement in convergence. A numerical dissipation formulation that provides good shock-capturing capability for hypersonic flows is presented. This formulation is shown to be a crucial aspect of the multigrid method. Solutions are given for two-dimensional viscous flow over a NACA 0012 airfoil and three-dimensional viscous flow over a blunt biconic.

Turkel, E.↗

NASA NTRS1993

Conservative high-order-accurate finite-difference methods for curvilinear grids

Two fourth-order-accurate finite-difference methods for numerically solving hyperbolic systems of conservation equations on smooth curvilinear grids are presented. The first method uses the differential form of the conservation equations; the second method uses the integral form of the conservation equations. Modifications to these schemes, which are required near boundaries to maintain overall high-order accuracy, are discussed. An analysis that demonstrates the stability of the modified schemes is also provided. Modifications to one of the schemes to make it total variation diminishing (TVD) are also discussed. Results that demonstrate the high-order accuracy of both schemes are included in the paper. In particular, a Ringleb-flow computation demonstrates the high-order accuracy and the stability of the boundary and near-boundary procedures. A second computation of supersonic flow over a cylinder demonstrates the shock-capturing capability of the TVD methodology. An important contribution of this paper is the dear demonstration that higher order accuracy leads to increased computational efficiency.

Rai, Man M.↗

NASA NTRS1988

An improved flux-split algorithm applied to hypersonic flows in chemical equilibrium

An explicit, finite-difference, shock-capturing numerical algorithm is presented and applied to hypersonic flows assumed to be in thermochemical equilibrium. Real-gas chemistry is either loosely coupled to the gasdynamics by way of a Gibbs free energy minimization package or fully coupled using species mass conservation equations with finite-rate chemical reactions. A scheme is developed that maintains stability in the explicit, finite-rate formulation while allowing relatively high time steps. The codes use flux vector splitting to difference the inviscid fluxes and employ real-gas corrections to viscosity and thermal conductivity. Numerical results are compared against existing ballistic range and flight data. Flows about complex geometries are also computed.

Palmer, Grant↗

NASA NTRS1997

Computational Considerations for the Simulation of Discontinuous Flows

The numerical study of aeroacoustic problems places stringent demands on the choice of a computational algorithm, because it requires the ability to propagate disturbances of small amplitude and short wavelength. The demands are particularly high when shock waves are involved, because the chosen algorithm must also resolve discontinuities in the solution. In a previous work the capabilities and deficiencies of shock-capturing methods for aeroacoustic problems were demonstrated using a high-order essentially nonoscillatory (ENO) numerical method. It was shown that first-order results are obtained when simulating time-dependent flows with discontinuities. The present study reaffirms this conclusion by comparing the ENO results with those obtained using a conventional linear scheme. A sixth-order-accurate compact implicit finite difference scheme is used to investigate various discontinuous flows. The design order of accuracy is achieved in the smooth regions of a steady-state, quasi-one-dimensional Euler test case, as well as in the time-dependent Burgers' equation. However, in the unsteady Euler sound-shock interaction, first-order results are obtained downstream of the shock. A comparison is made between the linear and nonlinear results, noting the advantages of each method. A discontinuous linear model problem is then used to identify the cause of the first-order results. Here, the nature of the solution error is quantified as being predominantly a numerical phase shift, and a post-processing procedure is demonstrated which increases the solution accuracy downstream of the discontinuity to second-order.

Carpenter Mark H.↗

NASA NTRS1982

Iterative spectral methods and spectral solutions to compressible flows

A spectral multigrid scheme is described which can solve pseudospectral discretizations of self-adjoint elliptic problems in O(N log N) operations. An iterative technique for efficiently implementing semi-implicit time-stepping for pseudospectral discretizations of Navier-Stokes equations is discussed. This approach can handle variable coefficient terms in an effective manner. Pseudospectral solutions of compressible flow problems are presented. These include one dimensional problems and two dimensional Euler solutions. Results are given both for shock-capturing approaches and for shock-fitting ones.

Hussaini, M. Y.↗

NASA NTRS2004

On controlling nonlinear dissipation in high order filter methods for ideal and non-ideal MHD

The newly developed adaptive numerical dissipation control in spatially high order filter schemes for the compressible Euler and Navier-Stokes equations has been recently extended to the ideal and non-ideal magnetohydrodynamics (MHD) equations. These filter schemes are applicable to complex unsteady MHD high-speed shock/shear/turbulence problems. They also provide a natural and efficient way for the minimization of Div(B) numerical error. The adaptive numerical dissipation mechanism consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is very general. The objective of this paper is to investigate the performance of three commonly used types of nonlinear numerical dissipation for both the ideal and non-ideal MHD.

Yee, H. C.↗

NASA NTRS1976

Diffraction of a shock wave by a compression corner. I - Regular reflection

The unsteady, two-dimensional flowfield resulting from the interaction of a moving planar shock wave with a compression corner is determined using a second-order, discontinuity-fitting, finite-difference approach. The time-dependent Euler equations are transformed to normalize the distance between the body and peripheral shock and to include the existing self-similar property of the flow. The resulting set of partial differential equations in conservation-law form is then solved in a time-dependent fashion using MacCormack's scheme. The vortical singularity, which lies on the body surface, and the single reflected shock are both treated as discontinuities in the numerical procedure. The results of the numerical simulation compare quite favorably with existing experimental interferograms and yield better flowfield resolution than previous first-order, shock-capturing, numerical solutions.

Kutler, P.↗

NASA NTRS1976

Diffraction of a shock wave by a compression corner; regular and single Mach reflection

The two dimensional, time dependent Euler equations which govern the flow field resulting from the injection of a planar shock with a compression corner are solved with initial conditions that result in either regular reflection or single Mach reflection of the incident planar shock. The Euler equations which are hyperbolic are transformed to include the self similarity of the problem. A normalization procedure is employed to align the reflected shock and the Mach stem as computational boundaries to implement the shock fitting procedure. A special floating fitting scheme is developed in conjunction with the method of characteristics to fit the slip surface. The reflected shock, the Mach stem, and the slip surface are all treated as harp discontinuities, thus, resulting in a more accurate description of the inviscid flow field. The resulting numerical solutions are compared with available experimental data and existing first-order, shock-capturing numerical solutions.

Vijayashankar, V. S.↗

NASA NTRS1981

On various treatments of potential equations at shocks

The potential equation that governs transonic inviscid flowfields is elliptic in subsonic regions and hyperbolic in supersonic regions. The transition from supersonic to subsonic flow may occur either continuously or across a surface of discontinuity known as a shock. Because of the changing type of governing equation and the presence of the surfaces of discontinuity, special treatments of the governing equations are needed at shocks to preserve mass conservation and to ensure satisfaction of the governing equation. In the so-called shock-capturing techniques, shocks are captured by smearing the discontinuity over several mesh cells in the computational domain. Most existing conservative schemes conserve mass flux throughout the flowfield, including shock boundaries; however, they also introduce zero-order errors in the approximation to the governing equation. It is shown that the zero-order errors, which do not diminish as the mesh spacing approaches zero, can cause discrepancies in the prediction of shock strength and location. These parameters are extremely sensitive features of transonic inviscid flowfield calculations. It is also shown that the zero-order errors can be avoided by developing higher-order schemes that properly model the governing equation at shocks.

Chen, L. T.↗