Centered and Upwind Multigrid Turbulent Flow Simulations of Launch Vehicle Configurations

a = speed of sound C = convective operator Cp = pressure coefficient D = artificial dissipation operator d = minimum distance to the wall e = total energy per unit volume ei = internal energy f = source term of the multigrid method k = specific turbulent kinetic energy p = static pressure Pe = inviscid flux vector Pv = viscous flux vector Q = vector of conserved properties q = heat flux vector RHS = right-hand side operator S = absolute value of the mean strain-rate tensor S = area vector Sij = mean strain-rate tensor component T = static temperature u, v, w = Cartesian velocity components V = viscous operator v = Cartesian velocity vector x, y, z = Cartesian coordinates = angle of attack 1 . . . 5 = Runge–Kutta control parameters = ratio of specific heats t = time step = von Karman constant = dynamic viscosity coefficient = kinematic viscosity coefficient ~ = modified Spalart–Allmaras eddy-viscosity coefficient = density = viscous stress tensor = gradient ratio for limiter computation = control volume limiter ! = turbulent dissipation = absolute value of the mean rotation tensor ij = mean rotation tensor component