The approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction

Abstract

A formulation of the approximate deconvolution model (ADM) for the large-eddy simulation (LES) of compressible flows in complex geometries is detailed. The model is applied to supersonic compression ramp flow where shock-turbulence interaction occurs. With the ADM approach an approximation to the unfiltered solution is obtained from the filtered solution by a series expansion involving repeated filtering. Given a sufficiently good approximation of the unfiltered solution at a time instant, the flux terms of the underlying filtered transport equations can be computed directly, avoiding the need to explicitly compute subgrid-scale closures. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation and a dynamically estimated relaxation parameter. Results of the large-eddy simulation of the turbulent supersonic boundary layer along a compression ramp compare well with filtered DNS data. The filtered shock solution is correctly predicted by the ADM procedure, demonstrating that turbulent and nonturbulent subgrid-scales are properly modeled. We found that a computationally expensive shock-capturing technique was not necessary for stable integration. As a consequence, the computational effort for simulations with ADM is approximately as large as for a coarse-grid DNS with a hybrid compact-upwind-ENO scheme, since the additional computational cost for the subgrid-scale model is more than compensated due to the fact that in the LES flux-derivatives can be computed by linear central finite differences on the entire domain.

A Subgrid-Scale Deconvolution Approach for Shock Capturing

“…Also, further extensions of the present approach may include nonlinear limiters or filters. Since our approach shares a considerable degree of commonality with recent deconvolution models for subgrid-scale modeling for LES [45,47,48] we expect that it will prove useful for the LES of shock-turbulence interaction.…”

“…We construct a regularization based on a relaxation term which employs a secondary filter operation [47,48]. The advantage of relaxation regularizations is that they leave the underlying differential equation type unchanged since they constitute a lower order perturbation and do not affect its well-posedness [14].…”

The subgrid-scale scalar variance under supercritical pressure conditions

“…(25) is exact if both q and qZ fields are totally recovered and an infinite expansion is used. Otherwise, there is no proof that ratio between series (23) and (22) converges to the recoverable part of the quantity Z, which is Z * at a given truncation order N. Such an assumption, although not trivial, has already been used by Stolz et al 32 while reconstructing flux terms of the compressible NS equations and by Dubois et al 33 in the first of a two-step procedure for estimating the SGS stress tensor. The difference between the previous studies and our investigation is that ADM is here used in the context of a soft-deconvolution problem 27 in which no additional models for recovering the deficient SGS part are provided.…”

Convergence Analysis of a Fully Discrete Family of Iterated Deconvolution Methods for Turbulence Modeling with Time Relaxation

“…In fact, results show that only time relaxation regularization truncates scales sufficiently for practical computations. Indeed, it is shown that time relaxation term χ w − w for χ > 0 damps unresolved fluctuations over time 5,6 . Note that the choice of χ is an active area of research and that solutions are very sensitive to variations in χ. Deconvolution-based regularization is also an active area of research obtained, for example, by replacing w by D w in each 1.2 -1.5 for some deconvolution operator D. In 7 , Dunca proposed the general Leray-deconvolution problem D w instead of w as a more accurate extension to Leray's model 8 .…”