Evaluation of a multigrid-based Navier-Stokes solver for aerothermodynamic computations

Abstract

A multigrid acceleration technique developed for solving the three-dimensional Navier-Stokes equations is used for computing high-Mach-number flows over configurations of practical interest. An explicit multistage RungeKutta time-stepping scheme is used as the basic algorithm. Solutions are presented for a spherically blunted cone at Mach 10 and a modified Shuttle orbiter at Mach 6. The computed surface heat-transfer distributions are shown to compare well with the experimental data. Effect of grid refinement on computed heat-transfer distributions is also examined to assess the numerical accuracy of the computed solutions. The rapid convergence rate associated with multigrid schemes in previous applications at transonic speeds is observed at the higher-Mach-number flows investigated here. NomenclatureCL = lift coefficient D = sum of dissipative and viscous fluxes d = artificial dissipation term E = energy per unit volume F, G, H = components of convective flux vector in stream, normal, and span directions G v = viscous flux component in normal direction i, j,k = coordinate indices in £, 77, and f directions / = Jacobian of transformation /, L = length of body, meters M = Mach number p = static pressure Q = sum of convective fluxes q = heat-transfer coefficient, W/cm 2 s = circumferential arc length, m £max = maximum arc length in circumferential direction, m TI , T 2 = parameters for shock detection switch t = time U = dependent-variable vector u,v,w = Cartesian velocity components, m/s W = a typical conserved variable x s = streamwise distance measured from apex of equivalent sharp cone, m a = angle of attack, deg 6 (2)= multiplicative coefficient for second-order dissipation 6 (4) = multiplicative coefficient for fourth-order dissipation /c (2) = scaling constant for second-order dissipation K (4) = scaling constant for fourth-order dissipation A.= eigenvalue used in dissipation scaling v = pressure-gradient parameter defined in Eq. (6) £, 77, ? = curvilinear coordinates along stream, normal, and span directions co = weight factor used in construction of shock detection switchPresented as Paper 92