NSFLEX — An Implicit Relaxation Method for the Navier-Stokes Equations for a Wide Range of Mach Numbers

Discussed is the well-proven NSFLEX method, which was applied to various sub- and transonic flow cases in the past. Here the extension to hypersonic flows is discribed. Applications are given for simple two-dimensional as well as for complex three-dimensional configurations. Real gas effects are incorporated in the solution procedure. For turbulent flows the Reynolds-averaged Navier-Stokes equations are employed using an algebraic turbulence model. To evaluate the inviscid fluxes a Riemann problem is solved at the finite-volume faces. A third-order accurate local characteristic flux extrapolation scheme (MUSCL type flux difference splitting) is utilized, using van Albada sensors to detect non-monotonous behaviour of the flow variables, where the scheme degrades to a first-order one. At very strong shocks, which occur at hypersonic flow conditions, a hybrid StegerWarming (flux vector splitting) local characteristic flux is used to avoid negative pressures in the transient phase. For the present application the Steger Warming flux is modified to overcome some disadvantages of the original one. Up to third-order accuracy is employed to reduce the inherent numerical viscosity of the inviscid flux discretisation. The viscous fluxes are constructed with central differences at each cell face. The unfactored implicit equations are solved in time-dependent form by a point Gauss-Seidel relaxation technique with red-black strategy. Thereby the code is perfectly vectorized. Because it is a finite-volume scheme complex geometries can be handled. Highly accurate results are given in the present paper for very low and very high Mach number applications.