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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 NTRS1990

A comparison of optimization-based approaches for solving the aerodynamic design problem

Three optimization-based methods for solving aerodynamic design problems are compared. The Euler equations for one-dimensional duct flow was used as a model problem, and the three methods are compared for efficiency, robustness, and implementation difficulty. The smoothness of the design problem with respect to different shock-capturing finite difference schemes, and in the presence of grid refinement, is investigated.

Frank, Paul D.↗
NASA NTRS1993

A comparison of two formulations for high-order accurate essentially non-oscillatory schemes

The finite-volume and finite-difference implementations of high-order accurate essentially non-oscillatory shock-capturing schemes are discussed and compared. Results obtained with fourth-order accurate algorithms based on both formulations are examined for accuracy, sensitivity to grid irregularities, resolution of waves that are oblique to the mesh, and computational efficiency. Some algorithm modifications that may be required for a given application are suggested. Conclusions that pertain to the relative merits of both formulations are drawn, and some circumstances for which each might be useful are noted.

Casper, Jay↗
NASA NTRS1994

Numerical simulation of shock-induced combustion past blunt bodies using shock-fitting technique

Two-dimensional axisymmetric, reacting viscous flow over blunt projectiles is computed to study shock-induced combustion at Mach 5.11 and Mach 6.46 in hydrogen-air mixture. A finite-difference, shock-fitting method is used to solve the complete set of Navier-Stokes and species conservation equations. In this approach, the bow shock represents a boundary of the computational domain and is treated as a discontinuity across which Rankine-Hugoniot conditions are applied. All interior details of the flow such as compression waves, reaction front, and the wall boundary layer are captured automatically in the solution. Since shock-fitting approach reduces the amount of artificial dissipation, all the intricate details of the flow are captured much more clearly than has been possible with the shock-capturing approach. This has allowed an improved understanding of the physics of shock-induced combustion over blunt projectiles and the numerical results can now be explained more readily with one-dimensional wave-interaction model than before.

Ahuja, J. K.↗
NASA NTRS1994

On the dynamics of a shock-bubble interaction

We present a detailed numerical study of the interaction of a weak shock wave with an isolated cylindrical gas inhomogenity. Such interactions have been studied experimentally in an attempt to elucidate the mechanisms whereby shock waves propagating through random media enhance mixing. Our study concentrates on the early phases of the interaction process which are dominated by repeated refractions of acoustic fronts at the bubble interface. Specifically, we have reproduced two of the experiments performed by Haas and Sturtevant : M(sub s) = 1.22 planar shock wave, moving through air, impinges on a cylindrical bubble which contains either helium or Refrigerant 22. These flows are modelled using the two-dimensional, compressible Euler equations for a two component fluid (air-helium or air-Refrigerant 22). Although simulations of shock wave phenomena are now fairly commonplace, they are mostly restricted to single component flows. Unfortunately, multi-component extensions of successful single component schemes often suffer from spurious oscillations which are generated at material interfaces. Here we avoid such problems by employing a novel, nonconservative shock-capturing scheme. In addition, we have utilized a sophisticated adaptive mesh refinement algorithm which enables extremely high resolution simulations to be performed relatively cheaply. Thus we have been able to reproduce numerically all the intricate mechanisms that were observed experimentally (e.g., transitions from regular to irregular refraction, cusp formation and shock wave focusing, multi-shock and Mach shock structures, jet formation, etc.), and we can now present an updated description for the dynamics of a shock-bubble interaction.

Quirk, James J.↗
NASA NTRS1994

X-ray clusters in a cold dark matter + lambda universe: A direct, large-scale, high-resolution, hydrodynamic simulation

A new, three-dimensional, shock-capturing, hydrodynamic code is utilized to determine the distribution of hot gas in a cold dark matter (CDM) + lambda model universe. Periodic boundary conditions are assumed: a box with size 85/h Mpc, having cell size 0.31/h Mpc, is followed in a simulation with 270(exp 3) = 10(exp 7.3) cells. We adopt omega = 0.45, lambda = 0.55, h identically equal to H/100 km/s/Mpc = 0.6, and then, from the cosmic background explorer (COBE) and light element nucleosynthesis, sigma(sub 8) = 0.77, omega(sub b) = 0.043. We identify the X-ray emitting clusters in the simulation box, compute the luminosity function at several wavelength bands, the temperature function and estimated sizes, as well as the evolution of these quantities with redshift. This open model succeeds in matching local observations of clusters in contrast to the standard omega = 1, CDM model, which fails. It predicts an order of magnitude decline in the number density of bright (h nu = 2-10 keV) clusters from z = 0 to z = 2 in contrast to a slight increase in the number density for standard omega = 1, CDM model. This COBE-normalized CDM + lambda model produces approximately the same number of X-ray clusters having L(sub x) greater than 10(exp 43) erg/s as observed. The background radiation field at 1 keV due to clusters is approximately the observed background which, after correction for numerical effects, again indicates that the model is consistent with observations.

Cen, Renyue↗
NASA NTRS1994

Hot gas in the cold dark matter scenario: X-ray clusters from a high-resolution numerical simulation

A new, three-dimensional, shock-capturing hydrodynamic code is utilized to determine the distribution of hot gas in a standard cold dark matter (CDM) model of the universe. Periodic boundary conditions are assumed: a box with size 85 h(exp -1) Mpc having cell size 0.31 h(exp -1) Mpc is followed in a simulation with 270(exp 3) = 10(exp 7.3) cells. Adopting standard parameters determined from COBE and light-element nucleosynthesis, sigma(sub 8) = 1.05, omega(sub b) = 0.06, and assuming h = 0.5, we find the X-ray-emitting clusters and compute the luminosity function at several wavelengths, the temperature distribution, and estimated sizes, as well as the evolution of these quantities with redshift. We find that most of the total X-ray emissivity in our box originates in a relatively small number of identifiable clusters which occupy approximately 10(exp -3) of the box volume. This standard CDM model, normalized to COBE, produces approximately 5 times too much emission from clusters having L(sub x) is greater than 10(exp 43) ergs/s, a not-unexpected result. If all other parameters were unchanged, we would expect adequate agreement for sigma(sub 8) = 0.6. This provides a new and independent argument for lower small-scale power than standard CDM at the 8 h(exp -1) Mpc scale. The background radiation field at 1 keV due to clusters in this model is approximately one-third of the observed background, which, after correction for numerical effects, again indicates approximately 5 times too much emission and the appropriateness of sigma(sub 8) = 0.6. If we have used the observed ratio of gas to total mass in clusters, rather than basing the mean density on light-element nucleosynthesis, then the computed luminosity of each cluster would have increased still further, by a factor of approximately 10. The number density of clusters increases to z approximately 1, but the luminosity per typical cluster decreases, with the result that evolution in the number density of bright clusters is moderate in this redshift range, showing a broad peak near z = 0.7, and then a rapid decline above redshift z = 3. Detailed computations of the luminosity functions in the range L(sub x) = 10(exp 40) - 10(exp 44) ergs/s in various energy bands are presented for both cluster central regions and total luminosities to be used in comparison with ROSAT and other observational data sets. The quantitative results found disagree significantly with those found by other investigators using semianalytic techniques. We find little dependence of core radius on cluster luminosity and a dependence of temperature on luminosity given by log kT(sub x) = A + B log L(sub x), which is slightly steeper (B = 0.38) than is indicated by observations. Computed temperatures are somewhat higher than observed, as expected, in that COBE-normalized CDM has too much power on the relevant scales. A modest average temperature gradient is found, with temperatures dropping to 90% of central values at 0.4 h(exp -1) Mpc and 70% of central values at 0.9 h(exp -1) Mpc. Examining the ratio of gas to total mass in the clusters normalized to Omega(sub B) h(exp 2) = 0.015, and comparing with observations, we conclude, in agreement with White (1991), that the cluster observations argue for an open universe.

Kang, Hyesung↗
NASA NTRS1996

Interaction of a shock with a longitudinal vortex

In this paper we study the shock/longitudinal vortex interaction problem in axisymmetric geometry. Linearized analysis for small vortex strength is performed, and compared with results from a high order axisymmetric shock-fitted Euler solution obtained for this purpose. It is confirmed that for weak vortices, predictions from linear theory agree well with results from nonlinear numerical simulations at the shock location. To handle very strong longitudinal vortices, which may ultimately break the shock, we use an axisymmetric high order essentially non-oscillatory (ENO) shock capturing scheme. Comparison of shock-captured and shock-fitted results are performed in their regions of common validity. We also study the vortex breakdown as a function of Mach number ranging from 1.3 to 10, thus extending the range of existing results. For vortex strengths above a critical value. a triple point forms on the shock and a secondary shock forms to provide the necessary deceleration so that the fluid velocity can adjust to downstream conditions at the shock.

Erlebacher, Gordon↗
NASA NTRS1994

Numerical Simulation of Shock-Induced Combustion Past Blunt Bodies Using Shock-Fitting Technique

Two-dimensional axisymmetric, reacting viscous flow over blunt projectiles is computed to study shock induced combustion at Mach 5.11 and Mach 6.46 in hydrogen-air mixture. A finite-difference, shock-fitting method is used to solve the complete set of Navier Stokes and species conservation equations. In this approach, the bow shock represents a boundary of the computational domain and is treated as a discontinuity across which Rankine-Hugoniot conditions are applied. All interior details of the flow such as compression waves, reaction front, and the wall boundary layer are captured automatically in the solution. Since shock-fitting approach reduces the amount of artificial dissipation, all the intricate details of the flow are captured much more clearly than has been possible with the shock-capturing approach. This has allowed an improved understanding of the physics of shock-induced combustion over blunt projectiles and the numerical results can now be explained more readily with one dimensional wave-interaction model than before.

Ahuja, J. K.↗
NASA NTRS1994

Numerical Simulations of High-Speed Chemically Reacting Flow

The Essentially NonOscillatory (ENO) shock-capturing scheme for the solution of hyperbolic equations is extended to solve a system of coupled conservation equations governing two-dimensional, time-dependent, compressible chemically reacting flow with full chemistry. The thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are separated from the fluid motion by a fractional step algorithm. The methodology is used to study the concept of shock-induced mixing and combustion, a process by which the interaction of a shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity generation. Test cases with and without chemical reaction are explored here. Our results indicate that, in the temperature range examined, vorticity generation as well as the distribution of atomic species do not change significantly with the introduction of a chemical reaction and subsequent heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process. This suggests that the fluid mechanics of this problem may be successfully decoupled from the combustion processes, and that computation of the mixing problem (without combustion chemistry) can elucidate much of the important physical features of the flow.

Ton, V. T.↗
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 NTRS1994

Numerical Simulations of High-Speed Chemically Reacting Flow

The essentially nonoscillatory (ENO) shock-capturing scheme for the solution of hyperbolic equations is extended to solve a system of coupled conservation equations governing two-dimensional, time-dependent, compressible chemically reacting flow with full chemistry. The thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are separated from the fluid motion by a fractional step algorithm. The methodology is used to study the concept of shock-induced mixing and combustion, a process by which the interaction of a shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity generation. Test cases with and without chemical reaction are explored here. Our results indicate that, in the temperature range examined, vorticity generation as well as the distribution of atomic species do not change significantly with the introduction of a chemical reaction and subsequent heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process. This suggests that the fluid mechanics of this problem may be successfully decoupled from the combustion processes, and that computation of the mixing problem (without combustion chemistry) can elucidate much of the important physical features of the flow.

Ton, V. T.↗
NASA NTRS1999

Factorizable Schemes for the Equations of Fluid Flow

We present an upwind high-resolution factorizable (UHF) discrete scheme for the compressible Euler equations that allows to distinguish between full-potential and advection factors at the discrete level. The scheme approximates equations in their general conservative form and is related to the family of genuinely multidimensional upwind schemes developed previously and demonstrated to have good shock-capturing capabilities. A unique property of this scheme is that in addition to the aforementioned features it is also factorizable, i.e., it allows to distinguish between full-potential and advection factors at the discrete level. The latter property facilitates the construction of optimally efficient multigrid solvers. This is done through a relaxation procedure that utilizes the factorizability property.

Sidilkover, David↗
NASA NTRS1998

Ionospheric Instabilities and Shock Capturing Algorithms

We describe the basic physics of the gradient-drift instability and the collisional Rayleigh-Taylor instability, the two plasma instabilities which dominate the production of small-scale structure in the earth's ionosphere. We then focus our attention on the nonlinear evolution of these instabilities for two particular cases: 1) the recursive bifurcation and "freezing" of artificial ion clouds released in the ionosphere; and 2) equatorial spread F. Finally, we discuss the numerical challenges faced in attempting to numerically simulate these instabilities, and the role played by shock-capturing algorithms therein.

Zalesak, Steven T.↗
NASA NTRS2001

Adaptive Numerical Dissipation Controls for High Order Methods

A numerical scheme for direct numerical simulation of shock-turbulence interactions of high speed compressible flows would ideally not be significantly more expensive than the standard fourth or sixth-order compact or non-compact central differencing scheme. It should be possible to resolve all scales down to scales of order of the Kolmogorov scales of turbulence accurately and efficiently, while at the same time being able to capture steep gradients occurring at much smaller scales efficiently. The goal of this lecture is to review the progress and new development of the low dissipative high order shock-capturing schemes proposed by Yee et al. Comparison on the efficiency and accuracy of this class of schemes with spectral and the fifth-order WENO (weighted essentially nonoscillatory) scheme will be presented. A new approach to dynamically sense the appropriate amount of numerical dissipation to be added at each grid point using non-orthogonal wavelets will be discussed.

Yee, Helen C.↗
NASA NTRS2002

On Accuracy of Adaptive Grid Methods for Captured Shocks

The accuracy of two grid adaptation strategies, grid redistribution and local grid refinement, is examined by solving the 2-D Euler equations for the supersonic steady flow around a cylinder. Second- and fourth-order linear finite difference shock-capturing schemes, based on the Lax-Friedrichs flux splitting, are used to discretize the governing equations. The grid refinement study shows that for the second-order scheme, neither grid adaptation strategy improves the numerical solution accuracy compared to that calculated on a uniform grid with the same number of grid points. For the fourth-order scheme, the dominant first-order error component is reduced by the grid adaptation, while the design-order error component drastically increases because of the grid nonuniformity. As a result, both grid adaptation techniques improve the numerical solution accuracy only on the coarsest mesh or on very fine grids that are seldom found in practical applications because of the computational cost involved. Similar error behavior has been obtained for the pressure integral across the shock. A simple analysis shows that both grid adaptation strategies are not without penalties in the numerical solution accuracy. Based on these results, a new grid adaptation criterion for captured shocks is proposed.

Yamaleev, Nail K.↗
NASA NTRS2002

Analysis of High Order Difference Methods for Multiscale Complex Compressible Flows

Accurate numerical simulations of complex multiscale compressible viscous flows, especially high speed turbulence combustion and acoustics, demand high order schemes with adaptive numerical dissipation controls. Standard high resolution shock-capturing methods are too dissipative to capture the small scales and/or long-time wave propagations without extreme grid refinements and small time steps. An integrated approach for the control of numerical dissipation in high order schemes with incremental studies was initiated. Here we further refine the analysis on, and improve the understanding of the adaptive numerical dissipation control strategy. Basically, the development of these schemes focuses on high order nondissipative schemes and takes advantage of the progress that has been made for the last 30 years in numerical methods for conservation laws, such as techniques for imposing boundary conditions, techniques for stability at shock waves, and techniques for stable and accurate long-time integration. We concentrate on high order centered spatial discretizations and a fourth-order Runge-Kutta temporal discretizations as the base scheme. Near the bound-aries, the base scheme has stable boundary difference operators. To further enhance stability, the split form of the inviscid flux derivatives is frequently used for smooth flow problems. To enhance nonlinear stability, linear high order numerical dissipations are employed away from discontinuities, and nonlinear filters are employed after each time step in order to suppress spurious oscillations near discontinuities to minimize the smearing of turbulent fluctuations. Although these schemes are built from many components, each of which is well-known, it is not entirely obvious how the different components be best connected. For example, the nonlinear filter could instead have been built into the spatial discretization, so that it would have been activated at each stage in the Runge-Kutta time stepping. We could think of a mechanism that activates the split form of the equations only at some parts of the domain. Another issue is how to define good sensors for determining in which parts of the computational domain a certain feature should be filtered by the appropriate numerical dissipation. For the present study we employ a wavelet technique introduced in as sensors. Here, the method is briefly described with selected numerical experiments.

Sjoegreen, Bjoern↗
NASA NTRS2002

The Delta x B = 0 Constraint Versus Minimization of Numerical Errors in MHD Simulations

The MHD equations are a system of non-strictly hyperbolic conservation laws. The non-convexity of the inviscid flux vector resulted in corresponding Jacobian matrices with undesirable properties. It has previously been shown by Powell et al. (1995) that an 'almost' equivalent MHD system in non-conservative form can be derived. This non-conservative system has a better conditioned eigensystem. Aside from Powell et al., the MHD equations can be derived from basic principles in either conservative or non-conservative form. The Delta x B = 0 constraint of the MHD equations is only an initial condition constraint, it is very different from the incompressible Navier-Stokes equations in which the divergence condition is needed to close the system (i.e., to have the same number of equations and the same number of unknown). In the MHD formulations, if Delta x B = 0 initially, all one needs is to construct appropriate numerical schemes that preserve this constraint at later time evolutions. In other words, one does not need the Delta x B condition to close the MHD system. We formulate our new scheme together with the Cargo & Gallice (1997) form of the MHD approximate Riemann solver in curvilinear grids for both versions of the MHD equations. A novel feature of our new method is that the well-conditioned eigen-decomposition of the non-conservative MHD equations is used to solve the conservative equations. This new feature of the method provides well-conditioned eigenvectors for the conservative formulation, so that correct wave speeds for discontinuities are assured. The justification for using the non-conservative eigen-decomposition to solve the conservative equations is that our scheme has a better control of the numerical error associated with the divergence of the magnetic condition. Consequently, computing both forms of the equations with the same eigen-decomposition is almost equivalent. It will be shown that this approach, using the non-conservative eigensystem when solving the conservative equations, also works well in the context of standard shock-capturing schemes.

Yee, H. C.↗