Supersonic Corner Flow Predictions Using the Quadratic Constitutive Relation

Abstract

A series of Reynolds-averaged Navier-Stokes simulations is performed for a supersonic wall-bounded turbulent corner flow. These simulations are compared to a high-order implicit large-eddy simulation for the same geometry and flow conditions. Inclusion of the quadratic constitutive relation in the Reynolds-averaged Navier-Stokes simulation results in significant improvement in qualitative agreement with the large-eddy simulation: specifically, the presence of secondary flow (a counter-rotating vortex pair). The range of valid values for the constant in the quadratic constitutive relation formulation is explored. Additionally, the effects on this range from different turbulence models and momentum thickness Reynolds numbers for the corner are also examined. These effects of the quadratic constitutive relation are explored using both the finite difference fluid solver OVERFLOW and the finite volume fluid solver US3D. The results indicate that the quadratic constitutive relation term directly affects the strength of the vortex pair in the secondary flow and that its influence appears directly dependent on all the aforementioned parameters. Nomenclature Cf = skin-friction coefficient C cr = quadratic constitutive relation constant k = turbulent kinetic energy L = distance of vortex core from corner l = reference length M = Mach number O ij = antisymmetric rotation tensor p = pressure Re=m = unit Reynolds number Re θ = momentum thickness Reynolds number S = strain-rate tensor T = temperature u = streamwise velocity v = normal velocity w = spanwise velocity x = streamwise distance y = wall-normal/spanwise distance z = spanwise distance α = angle of vortex core from nearest wall δ = boundary-layer thickness, 0.99u ∞ δ ij = Kronecker delta θ = boundary-layer momentum thickness μ = dynamic viscosity ρ = density τ = Reynolds-stress tensor Subscripts t = turbulent w = wall 1 = first cell off the wall ∞ = freestream Superscripts QCR = quadratic constitutive relation t = turbulent = inner coordinates − = time mean 0 = fluctuations

Turbulence anisotropy effects on corner-flow separation: physics and turbulence modelling

“…0.6). However, Spalart, Garbaruk & Strelets (2014) and Leger, Bisek & Poggie (2016) pointed out that increasing C cr1 of QCR2000, which is equivalent to C q1 /2 in this study, may cause non-physical vortices. The proposed model avoids this by introducing C q2 as explained in the following.…”