Comparison of Several Numerical Methods for Simulation of Compressible Shear Layers

An investigation is conducted on several numerical schemes for use in the computation of two-dimensional, spatially evolving, laminar variable-density compressible shear layers. Schemes with various temporal accuracies and arbitrary spatial accuracy for both inviscid and viscous terms are presented and analyzed. All integration schemes use explicit or compact finite-difference derivative operators. Three classes of schemes are considered: an extension of MacCormack''s original second-order temporally accurate method, a new third-order variant of the schemes proposed by Rusanov and by Kutler, Lomax, and Warming (RKLW), and third- and fourth-order Runge-Kutta schemes. In each scheme, stability and formal accuracy are considered for the interior operators on the convection-diffusion equation Ut + aUx = aUxx. Accuracy is also verified on the nonlinear problem, Ut + Fx = 0. Numerical treatments of various orders of accuracy are chosen and evaluated for asymptotic stability. Formally accurate boundary conditions are derived for several sixth- and eighth-order central-difference schemes. Damping of high wave-number data is accomplished with explicit filters of arbitrary order. Several schemes are used to compute variable-density compressible shear layers, where regions of large gradients exist.

[1]  D. Anderson,et al.  A comparison of numerical solutions to the inviscid equations of fluid motion , 1974 .

[2]  Sanjiva K. Lele,et al.  Direct numerical simulation of compressible free shear flows , 1989 .

[3]  A. Bayliss,et al.  A fourth-order scheme for the unsteady compressible Navier-Stokes equations , 1985 .

[4]  Marcel Lesieur,et al.  Direct and large-eddy simulations of transition in the compressible boundary layer , 1992 .

[5]  D. Gottlieb,et al.  Stable and accurate boundary treatments for compact, high-order finite-difference schemes , 1993 .

[6]  W. Ng,et al.  A concentration probe for the study of mixing in supersonic shear flows , 1989 .

[7]  V. Rusanov,et al.  On difference schemes of third order accuracy for nonlinear hyperbolic systems , 1970 .

[8]  F. Yuan,et al.  SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) , 1999 .

[9]  Garrett Birkhoff,et al.  Numerical Fluid Dynamics. , 1983 .

[10]  Wen-Yih Sun.  Comments on “A Comparative Study of Numerical Advection Schemes Featuring a One-Step Modified WKL Algorithm” , 1993 .

[11]  M. H. Carpenter,et al.  A high-order compact numerical algorithm for supersonic flows , 1990 .

[12]  D. Anderson,et al.  A comparison of numerical solutions of the advective equation , 1974 .

[13]  R. S. Amando.  Numerical computation of internal and external flows, volume 1: Fundamentals of numerical discretization , 1989 .

[14]  David H. Rudy,et al.  Secondary frequencies in the wake of a circular cylinder with vortex shedding , 1991, Journal of Fluid Mechanics.

[15]  D. Gottlieb,et al.  The theoretical accuracy of Runge-Kutta . . . , 1997 .

[16]  B. Fornberg.  On a Fourier method for the integration of hyperbolic equations , 1975 .

[17]  J. Sowa.  Stability of a Runge-Kutta method for the Navier-Stokes equation , 1990 .

[18]  S. Raman,et al.  A comparative study of numerical advection schemes featuring a one-step modified WKL algorithm , 1991 .

[19]  C. Kennedy,et al.  Self‐similar supersonic variable‐density shear layers in binary systems , 1994 .

[20]  Neil D. Sandham,et al.  Three-dimensional simulations of large eddies in the compressible mixing layer , 1991, Journal of Fluid Mechanics.

[21]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[22]  R. F. Warming,et al.  The modified equation approach to the stability and accuracy analysis of finite-difference methods , 1974 .

[23]  Gordon Erlebacher,et al.  High-order ENO schemes applied to two- and three-dimensional compressible flow , 1992 .

[24]  D. Gottlieb,et al.  Difference Schemes with Fourth Order Accuracy for Hyperbolic Equations , 1974 .

[25]  D. Gottlieb,et al.  Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes , 1994 .

[26]  J. Beckers,et al.  Analytical linear numerical stability conditions for an anisotropic three-dimensional advection-diffusion equation , 1992 .

[27]  David I. Gottlieb,et al.  The Theoretical Accuracy of Runge-Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error , 1995, SIAM J. Sci. Comput..

[28]  Satoru Yamamoto,et al.  Higher-order-accurate upwind schemes for solving the compressible Euler and Navier-Stokes equations , 1993 .

[29]  Paul Kutler,et al.  Computation of Space Shuttle Flowfields Using Noncentered Finite-Difference Schemes , 1973 .

[30]  Arthur Rizzi,et al.  Computer-aided analysis of the convergence to steady state of discrete approximations to the euler equations , 1985 .

[31]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[32]  Krishnan Mahesh,et al.  Direct numerical simulation , 1998 .

[33]  Hervé Guillard,et al.  Adaptive spectral methods with application to mixing layer computations , 1992 .

[34]  J. Philip Drummond,et al.  A two-dimensional numerical simulation of a supersonic, chemically reacting mixing layer , 1988 .

[35]  T. Poinsot.  Boundary conditions for direct simulations of compressible viscous flows , 1992 .

[36]  J. Riley,et al.  A study of inviscid, supersonic mixing layers using a second-order TVD scheme , 1988 .

[37]  Direct numerical simulation of laminar breakdown in high-speed, axisymmetric boundary layers , 1992 .

[38]  M. Carpenter,et al.  Direct Simulation of High-Speed Mixing Layers , 1992 .

[39]  Thomas A. Zang,et al.  Direct numerical simulation of laminar breakdown in high-speed, axisymmetric boundary layers , 1992 .

[40]  J. Williamson.  Low-storage Runge-Kutta schemes , 1980 .

[41]  Thomas H. Pulliam,et al.  Artificial Dissipation Models for the Euler Equations , 1985 .

[42]  Eli Turkel,et al.  Multidimensional Difference Schemes with Fourth-Order Accuracy , 1976 .

[43]  Gordon Erlebacher,et al.  The analysis and simulation of compressible turbulence , 1990, Theoretical and Computational Fluid Dynamics.