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Efficient upwind algorithms for solution of the Euler and Navier-stokes equations

By C. Y. McNeil

Abstract

An efficient three-dimensionasl tructured solver for the Euler and Navier-Stokese quations is developed based on a finite volume upwind algorithm using Roe fluxes. Multigrid and optimal smoothing multi-stage time stepping accelerate convergence. The accuracy of the new solver is demonstrated for inviscid flows in the range 0.675 :5M :5 25. A comparative grid convergence study for transonic turbulent flow about a wing is conducted with the present solver and a scalar dissipation central difference industrial design solver. The upwind solver demonstrates faster grid convergence than the central scheme, producing more consistent estimates of lift, drag and boundary layer parameters. In transonic viscous computations, the upwind scheme with convergence acceleration is over 20 times more efficient than without it. The ability of the upwind solver to compute viscous flows of comparable accuracy to scalar dissipation central schemes on grids of one-quarter the density make it a more accurate, cost effective alternative. In addition, an original convergencea cceleration method termed shock acceleration is proposed. The method is designed to reduce the errors caused by the shock wave singularity M -+ 1, based on a localized treatment of discontinuities. Acceleration models are formulated for an inhomogeneous PDE in one variable. Results for the Roe and Engquist-Osher schemes demonstrate an order of magnitude improvement in the rate of convergence. One of the acceleration models is extended to the quasi one-dimensiona Euler equations for duct flow. Results for this case d monstrate a marked increase in convergence with negligible loss in accuracy when the acceleration procedure is applied after the shock has settled in its final cell. Typically, the method saves up to 60% in computational expense. Significantly, the performance gain is entirely at the expense of the error modes associated with discrete shock structure. In view of the success achieved, further development of the method is proposed

Publisher: Cranfield University
Year: 1995
OAI identifier: oai:dspace.lib.cranfield.ac.uk:1826/4147
Provided by: Cranfield CERES

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Citations

  1. (1989). A class of high-resolution explicit and implicit shock-capturing methods. doi
  2. (1992). A comparative study of advanced shockcapturing applied to Burgers' equation. doi
  3. (1977). A comparative study of computational methods in cosmic gas dynamics. doi
  4. A comparison of numerical flux formulas for the Euler and Navier-Stokes equations. AIAA Paper 87-1104,
  5. (1993). A comparison of several implicit smoothing methods. doi
  6. A contribution to numerical prediction of unsteady flows. doi
  7. (1992). A contribution to the great Riernarm solver debate. doi
  8. (1977). A finite-volume method for transonic potential flow calculations. doi
  9. (1992). A flux splitting scheme with high-resolution and robustness for discontinuities. doi
  10. (1993). A generalised Riemann problem for quasi-one-dimensional gas flows. doi
  11. (1982). A multiple grid scheme for solving the euler equations. doi
  12. (1989). A new multigrid approach to convection problems. doi
  13. A systematic comparative study of several high resolution schemes for complex problems in high speed flows. doi
  14. (1995). A unified multigrid solver for the Navier-Stokes equations on mixed element meshes. doi
  15. a-GMRES: A new parallelizable iterative solver for large sparse non-symmetric linear systems arising from CFD. doi
  16. Accurate solution of the 2-D Euler equations with and efficient cell-vertex upwind scheme. doi
  17. (1987). Algebraic multigrid. doi
  18. (1991). An adaptive grid algorithm for computational shock dynamics. doi
  19. An adaptively-refined cartesian mesh solver for the Euler equations. doi
  20. (1991). Appendix 5 to GARTEUR AD(AG-15) report -Itesults premsented at the third meeting held at Aerospatiale,
  21. (1948). Application of a new class of high accuracy TVD schemes to the NavierStokes equations. doi
  22. (1987). Application of Wynn's c-algorithm to transonic flow calculations. doi
  23. (1981). Approximate IUemann solvers, parameter vectors, and difference schemes. doi
  24. (1992). Basins of attraction and the time-dependent approach to obtaining steady-state numerical solutions. Preprint to appear in doi
  25. (1990). Blunt-body flow simulations. AIAA Paper 88-2904,1988. [1111
  26. (1979). Boundary-Layer Theory. McGraw-Hill, seventh edition edition, doi
  27. Calculation of plane steady transonic flow. doi
  28. (1993). Canonical forms of multidimensional steady inviscid flows.
  29. (1994). Canonical variables multigrid method for steady-state Euler equations. doi
  30. Characteristic decompostion methods for the multidimensional Euler equations.
  31. Characteristic time-stepping or local preconditioning of the Euler equations. doi
  32. Characteristic-based schemes for the Euler equations. doi
  33. (1986). Comparison of finite volume flux vector splittings for the Euler equations. doi
  34. (1991). Comparison of solution of various Euler solvers and one Navier-Stokes solver for the flow about a sharp-edged cropped delta wing.
  35. (1984). Compressible Fluid Flow and Systems of Conservation Laws doi
  36. Computation of turbulent supersonic flows around pointed bodies having cross-flow separation. doi
  37. Computational efficiency achieved by time splitting of finite difference operators. doi
  38. Computer-aided analysis of the convergence to steady state of discrete approximations to the Euler equations. doi
  39. Condition improvement for point relaxation in multigrid, subsonic Euler-flow computations. doi
  40. (1992). Contamination of laminar boundary layers by artifical dissipation in Navier-Stokes solutions.
  41. (1965). Convergence acceleration methods: The past decade. doi
  42. Convergence acceleration through the use of time inclining. doi
  43. (1980). Convergence to steadyĆ½-state solutions of the Euler equations on unstructured grids with limiters. doi
  44. (1991). Damped direction dependant multigrid for hypersonic flow computations. doi
  45. (1995). de Zeeuw. Developments of semi-coarsening and sparse-grid techniques.
  46. Design of optimally smoothing multi-stage schemes for the Euler equations. doi
  47. (1987). Dirichlet boundary conditions for the random-choice method for solving one-dimensional unsteady flows of a perfect gas.
  48. (1986). Discrete models for the numerical analysis of time-dependent multi-dimensional gas-dynamics. doi
  49. Dissipation additions to flux-difference splitting. doi
  50. (1991). Dynamical approach study of spurious steady-state numerical s' olutions of nonlinear differential equations, Part I: The ODE connection and its implications for algorithm development in computational fluid dynamics. doi
  51. (1991). Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations, Part III: The effects of nonlinear source terms and boundary conditions in reaction-convection equations. doi
  52. (1992). Dynamical approach study of spurious steadystate numerical solutions of nonlinear differential equations, Part II: The dynamics of numerics of systems of 2x2 ODEs and its connection to finite discretizations of nonlinear PDEs.
  53. (1990). Efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations. doi
  54. (1987). Efficient implementation of essentially nonoscillatory shock capturing schemes. doi
  55. (1957). Elements of Gas Dynamics. doi
  56. (1987). Euler solver for three-dimensional supersonic flows with subsonic pockets. doi
  57. (1992). Euler solvers for transonic applications. In NASA 71-ansonic Symposium: Theory, Application and Experiment,
  58. (1991). Extension of multigrid methodology to supersonic/hypersonic 3-D viscous flows. doi
  59. (1987). Extrapolation methods for vector sequences. doi
  60. (1993). Finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics.
  61. (1995). Flux difference splitting techniques for the Euler equations in non-cartesian geometry.
  62. GMRES acceleration of computational fluid dynamic codes. doi
  63. (1986). GMRES: A general minimum residual algorithm for solving nonsymmetric linear systems. doi
  64. (1975). High lift aerodynamics. doi
  65. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. doi
  66. Improved convergence to the steady state of the Euler equations by enhanced wave propagation. doi
  67. Modification of quasi-Newton method for nonlinear equa, tions with a sparse jacobian. doi
  68. (1985). Multi-Grid Methods and Applications. doi
  69. Multi-level adaptive solutions to boundary value problems. doi
  70. (1973). Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems. doi
  71. Multi-stage schemes for the Euler and NavierStokes equations with optimal smoothing. doi
  72. (1993). Multigrid calculations of threedimensional viscous cascade flows. doi
  73. (1991). Multigrid for hypersonic viscous two- and three-dimensional flows. doi
  74. (1982). Multilevel adaptive computations in fluid dynamics. doi
  75. (1982). Multiple steady states for 1-D transonic flow. Pre-print, Center for Pure and Applied Mathematics,
  76. (1991). New light on numerical boundary conditions. doi
  77. Newton-like methods for fast high resolution simulation of hypersonic viscous flows. doi
  78. Non-reflecting boundary conditions for Euler equation calculations. doi
  79. (1993). Nonparametized entropy fix for Roe's method. doi
  80. Numerical computation of two-dimensional viscous blunt body flows with an impinging shock. doi
  81. Numerical solutions for the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. doi
  82. Numerical wave propagation in an advection equation with a nonlinear source term. doi
  83. (1991). On Godunov-type methods for gas dynamics. doi
  84. On the estimation of sparse jacobian matrices. doi
  85. On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe. doi
  86. (1995). Optimal multi-stage schemes for Euler equations with residual smoothing. doi
  87. (1991). Preconditioned conjugate gradient methods for the compressible Navier-Stokes equations. doi
  88. Preconditioned methods for solving the incompressible and low speed compressible equations. doi
  89. (1979). Pressure distribution on the ONERA N16 wing at transonic mach numbers. In AGARD Advisory Report 138. Experimental database for computer program assessment,
  90. (1992). Private communication,
  91. (1991). Progress with multigrid schemes for bypersonic flow problems. doi
  92. (1990). Recent progress in multidimensional upwinding. doi
  93. (1992). Review of preconditioning methods for fluid dynamics. doi
  94. (1986). Roe's method in gas dynamical problems associated with the combustion of high-energy solids in a closed tube. College of Aeronautics Report NFP86/18,
  95. Simple improvements of an upwind TVD scheme for hypersonic flow. doi
  96. (1976). Skew upstream differencing schemes for problems involving fluid flow. doi
  97. (1985). Some contributions to the modelling of discontinuous flows.
  98. (1992). Sonic flux formulae. doi
  99. Sparse quasi-Newton method for high resolution schemes. doi
  100. Sparse quasi-Newton method for Navier-Stokes solutions. doi
  101. Stiffness and deflection analysis of complex structures.
  102. (1994). Surface boundary conditions for the numerical solution of the euler equations. doi
  103. (1992). Systems of conservation laws. doi
  104. (1988). The behaviour of flux difference splitting schemes near slowly moving shock waves. doi
  105. (1983). The evolution of computational methods in aerodynamics. doi
  106. The speed of convergence of one iterative process. doi
  107. (1980). The use of the Riemann problem in finite difference schemes. In doi
  108. Thin layer approximation and algebraic model for separated turbulent flows. doi
  109. (1990). Topics in Numerical Computation of Compressible Flow.
  110. (1979). Towards the ultimate conservative difference scheme V, A second-order sequel to Godunov's method. doi
  111. (1993). Turbulence Modelling for CFD. DCW Industries, doi
  112. Upwind implicit residual smoothing method for multi-stage schemes. doi
  113. (1985). Upwind-difference methods for aerodynamic problems governed by the Euler equations.
  114. Validation of a multigrid method for the Reynolds averaged equations. doi
  115. (1992). Validation of central and upwind 3-D compressible flow solvers.
  116. Variable coefficient implicit residual smoothing. doi

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