S. Hickel, C.P. Egerer, J. Larsson (2014)
Physics of Fluids 26: 106101. doi: 10.1063/1.4898641

We derive and analyze a model for implicit Large Eddy Simulation (LES) of compressible flows that is applicable to a broad range of Mach numbers and particularly efficient for LESof shock-turbulence interaction. Following a holistic modeling philosophy, physically sound turbulence modeling and numerical modeling of unresolved subgrid scales (SGS) are fully merged, in a manner quite different from that of traditional implicit LES approaches. The implicit subgrid model is designed in such a way that asymptotic consistency with incompressible turbulence theory is maintained in the low Mach number limit. Compressibility effects are properly accounted for by a novel numerical flux function, which can capture strong shock waves in supersonic flows and also ensures an accurate representation of smooth waves and turbulence without excessive numerical dissipation. Simulations of shock-tube problems, Noh's three-dimensional implosion problem, large-scale forced and decaying three-dimensional homogeneous isotropic turbulence, supersonic turbulent boundary layer flows, and a Mach = 2.88 compression-expansion ramp flow demonstrate the good performance of the SGS model; across this range of flows, predictions are in excellent agreement with theory, direct numerical simulations, and experimental reference data. Results for implicit LES of canonical shock-turbulence interaction are compared with results of explicit LES using the dynamic Smagorinsky model. The analysis shows that details of the numerical method used for shock capturing clearly outweigh the effect of different turbulence modeling strategies in explicit and implicit LES. The implicit LES model recovers the ideal 2nd-order grid convergence of shock-capturing errors that has been predicted using Rapid Distortion Theory. The dynamic Smagorinsky model in conjunction with a hybrid method that combines sixth-order central differences with a seventh-order weighted essentially non-oscillatory scheme yields turbulence statistics that are very similar to the implicit LES results. However, while the explicit LES requires a tailored high-order low-dissipative numerical method that applies numerical dissipation only in shock normal direction, no such ad hoc adjustments are necessary with the proposed implicit LES method.


Modified wavenumber analysis for (a) centered differences of second, fourth, and sixth order, (b) present scheme for implicit LES.


Validation for decaying isotropic turbulence at (a) weakly compressible conditions at turbulent Mach number Mat = 0.3, and (b) strongly compressible conditions including eddy shocklets at Mat = 0.6: normalized —– ◯ turbulence kinetic energy, and variances of – - – ▢   density, – - - – △ pressure, and - - - - ▽ temperature, for implicit LES with the present scheme (lines) and DNS (symbols). The time is given in large-eddy turnover times.


Validation for compression-expansion ramp flow at Ma = 2.88 and Reδ = 132 840. The upper panel shows computational domain and grid topology, the lower panel shows time averaged (a) wall pressure and (b) friction coefficient. Lines denote LES results and symbols denote experimental data by Zheltovodov et al. (1990), see also Grilli et al. (2012) and Grilli et al. (2013) for a detailed analysis of these LES results.