Implicit WENO Schemes with Anti-Diffusive Flux for Compressible Flows

Implicit variant WENO schemes with anti-diffusive flux for compressible flow computations are presented. These variant WENO schemes include the antis for the preconditioned compressible Euler/Navier-Stokes equations are also considered. The numerical flux of the variant WENO scdiffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator. Implicit WENO schemeheme is consisted of a first-order entropy-satisfying part and a high-order modulated flux part and allows for a more flexible choice of low order schemes. The lower-upper symmetric-Gauss-Seidel (LU-SGS) method is adopted for the implicit operator. Various aerodynamic flows ranging from low subsonic to supersonic Mach numbers are presented to illustrate the methods. Comparisons of present solutions with available experimental and computational results are made. Nomenclature a = speed of sound C B A ˆ , ˆ , ˆ = Jacobian matrices d = distance from the wall e = energy per unity volume G F E , , = Cartesian components of the convective flux vector v v v G F E , , = Cartesian components of the viscous flux vector Ê = physical flux E ~ = numerical flux L E ~ = first order numerical flux HW E ~ = higher order numerical flux H = source state vector IS = smoothness indicators J = Jacobian of coordinates transformation 19th AIAA Computational Fluid Dynamics 22 25 June 2009, San Antonio, Texas AIAA 2009-3661 Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 2 , l t K K = laminar and turbulent thermal conductivity s l = left eigenvectors of Jacobian matrices L _ = characteristic length n  = unit normal vector p = pressure Pr = Prandtl number t l Pr , Pr = laminar and turbulent Prandtl number q = heat flux Q = state vector of conserved variables p Q = state vector of primitive variables s r = right eigenvectors of Jacobian matrices Re = Reynolds number  S  = surface area vector in  direction t = time w v u , , = Cartesian components of velocity vector   = non-dimensional eddy viscosity  = pre-conditioned matrix 1   = inverse of pre-conditioned matrix  = specific heat ratio    , , = body-fitted curvilinear coordinates       , , = difference operators with respect to    , ,  = molecular viscosity , l t   = laminar and turbulent viscosity  = density ij  = viscous stress tensor i  = discontinuity indicator superscripts k j i , , = grid point index of variables  = viscosity variable  = free stream value