Subgrid scale analysis of turbulence after the shock-turbulence interaction

Abstract eng:
Turbulence structure and statistics of the subgrid scales (SGS) in the context of Large Eddy Simulations (LES) are studied after the interaction with a shock wave. Recent high resolution shock-resolved Direct Numerical Simulations (DNS) [1] show that, when there is a large separation in scale between turbulence and the shock width, the interaction can be described by the Linear Interaction Approximation (LIA). By using LIA to alleviate the need to resolve the shock, DNS post-shock data can be generated at much higher Reynolds numbers than previously possible. Here, results with Reλ ~180 are used to investigate the structure of post-shock turbulence. In particular, the interaction with the shock leads to a local axisymmetric flow state. In turn, this induces an Ms dependent symmetrization of the SGS dissipation PDF and a large increase in its variance. This corresponds to significant enhancement in size of the regions and magnitude of backscatter. The interaction of shock waves with turbulence is an important aspect in many types of flows, from hypersonic flight, to supersonic combustion, to astrophysics and Inertial Confinement Fusion (ICF). In general, in practical applications, the shock width is much smaller than the turbulence scales, even at low shock Mach numbers, Ms, and it becomes comparable to the molecular mean free path at high Ms values. When there is a large-scale separation between the shock and turbulence, viscous effects become negligible during the interaction. If, in addition, the turbulent Mach number, Mt, of the upstream turbulence is small, the nonlinear effects can also be neglected during the interaction. In this case, the interaction can be treated analytically using the linearized Euler equations and Rankine-Hugoniot jump conditions. This is known as the Linear Interaction Approximation (LIA) [2]. However, due to the high cost of simulations for the parameter space close to practical applications and difficulties with accurate measurements close to the shock, previous studies have demonstrated only limited agreement with LIA. Recently, Ryu and Livescu [1], using high resolution fully resolved DNS extensively covering the parameter range, have shown that the DNS results converge to the LIA solutions as the ratio δ/η where δ is the shock width and η is the Kolmogorov microscale of the upstream turbulence, becomes small. The results reconcile a long time open question about the role of the LIA theory and establish LIA as a reliable prediction tool for low Mt turbulence-shock interaction problems. Furthermore, when there is a large separation in scale between the shock and the turbulence, the exact shock profile is no longer important for the interaction, so that LIA can be used to predict arbitrarily high Ms interaction problems, when the Navier-Stokes equations are no longer valid and fully resolved DNS are not feasible. The shock-turbulence interaction has been traditionally studied in an open-ended domain, with the turbulence fed through the inlet plane encountering a stationary shock at some distance from the inlet. This approach is very expensive even when a shockcapturing scheme is used and limited to low Reynolds numbers. However, the range of the achievable Re values can be significantly increased if, instead, one uses the LIA theory to generate the post-shock fields. In order to generate full 3-D postshock fields, Refs. [1,3] have extended the classical LIA formulas, which traditionally have been used to calculate second order moments only. Using this procedure, Refs. [1,3] have shown profound changes in the structure of post-shock turbulence, with significant potential implications on turbulence modelling. High Re post-shock DNS data are generated by first performing triply periodic forced compressible isotropic turbulence (IT) simulations using the linear forcing method [4]. This forcing method has the advantage of specifying the Kolmogorov microscale and ratio of dilatational to solenoidal kinetic energies, χ, at the outset. Here, we present results from simulations with Reλ~180, χ<0.01 (quasi-vortical turbulence) and Mt=0.05. The resulting turbulence fields are passed through the generalized LIA formulas [2,3] to obtain the post-shock turbulence data. In order to examine the properties of the subgrid scales, the post-shock data is filtered using a Gaussian filter. The usual picture of an energy cascade typically holds in a statistically-averaged sense, it does not always describe the local behaviour of a turbulent flow. The turbulent dissipation is actually the difference between two energy fluxes, a "forwardscatter", corresponding to the classical energy cascade, and the "backscatter", a reversal of this process in which energy is transferred from the small scales back to the large scales. In LES approaches, the SGS backscatter acts as a source term in the kinetic energy equation and poses significant difficulties in maintaining stable computations [5]. Many of the simple SGS models do not account for backscatter and properly describing this phenomenon is an active area of research [5].

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