Large-Eddy and RANS Simulations of a Normal Shock Train in a Constant-Area Isolator

Abstract

Large-eddy simulations (LES) of a normal shock train in a constant-area isolator (M ∞ = 1.61, Re θ = 1660) are carried out with a high-order compact differencing scheme using localized artificial diffusivity (LAD) for shock capturing. We examine sensitivities of the solution to a variety of physical modeling assumptions. Simulations with spanwise periodic boundary conditions are first performed, the results of which are compared to experiment and validated with a three-level grid refinement study. Due to the computational cost associated with resolving near-wall structures, the LES is run at a Reynolds number lower than that in the comparison experiment; thus, the confinement effect of the turbulent boundary layers is not exactly duplicated. While this discrepancy affects the location of the first normal shock, the overall structure of the shock train and its interaction with the boundary layers matches the experiment quite closely. Observations of pertinent physical phenomena in the experiment such as a lack of reversed flow and the development of secondary shear layers are captured by the simulation. Next, a series of linear eddy viscosity based Reynolds Averaged Navier Stokes (RANS) simulations are performed to assess the ability of reduced-order models to accurately capture the complicated physics of multiple shock/boundary layer interactions. It is found that the RANS solutions exhibit a wide range of variations, indicating a strong sensitivity -particularly with respect to streamwise location of the initial shock --to model formulation. Finally, three-dimensional effects due to the side walls of the isolator are investigated by performing LES of the same shock train interaction in a three-dimensional, low-aspect-ratio rectangular duct geometry. A significant three-dimensional flow feature is observed in the region of the strong initial shock, which agrees well with experimental oil flow visualizations. NomenclatureC f = skin friction coefficient k = turbulence kinetic energy h = isolator half-height l = turbulent length scale L x , L y , L z = extent of the computational mesh M = Mach number N x , N y , N z = number of grid points in computational mesh p = static pressure Re = Reynolds number S ij = Strain-rate tensor s L = length of shock train t = time T = temperature u = velocity vector u = streamwise velocity component u τ = friction velocity w = value at wall v = wall-normal velocity component w = spanwise velocity component δ = boundary layer thickness δ ij = Dirac delta θ = momentum thickness μ t = eddy viscosity ρ = density τ = shear stress ω = specific dissipation rate Superscript + = wall unit quantity ′ = fluctuation quantity inn = inner boundary layer value out = outer boundary layer value 2 Subscript ∞ = freestream value in = value at isolator inlet out = value at isolator outlet r = reference value ahead of initial shock VD = van Driest-transformed quantity

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