Supersonic flux-split procedure for second moments of turbulence

A finite volume procedure which combines flux-difference splitting and flux-vector splitting is presented in this paper. The diffusive fluxes are treated explicitly to preserve the upwinding structure of the split Euler fluxes. This procedure is extended in this paper from its essentially laminar or eddy viscosity form to include the equations for the six components of the Reynolds stress tensor and an additional equation for the solenoidal dissipation. The models used for the unclosed terms are presented as are the extensions of the numerical procedures to cover the turbulence equations. To validate the proposed procedures, a compressible turbulent flow over a flat plate at Mach number M_∞ = 2.87 and Reynolds number per unit length Re/m of 6.5× 10⁷ was calculated and compared with experimental measurements. Also simulated was the case when the foregoing boundary-layer flow was made to pass over a ramp. A strong adverse pressure gradient case in which boundary-layer thickness-to-curvature ratio (δ/Rc) is 0.1 was considered. In this case, the supersonic turbulent boundary layer experiences the combined effects of an adverse pressure gradient, bulk compression, and concave streamline curvature. For the various tests, the results obtained from our calculations are in agreement with the experimental measurements.