A stochastic extension of the approximate deconvolution model

Abstract

The approximate deconvolution model (ADM) for large-eddy simulation exploits a range of represented but non-resolved scales as buffer region for emulating the subgrid-scale energy transfer. ADM can be related to Langevin models for turbulence when filter operators are interpreted as stochastic kernel estimators. The main conceptual difference between ADM and Langevin models for turbulence is that the former is formulated with respect to an Eulerian reference frame whereas the latter are formulated with respect to a Lagrangian reference frame. This difference can be resolved by transforming the Langevin models to the Eulerian reference frame. However, the presence of a stochastic force prevents the classical convective transformation from being applicable. It is shown that for the transformation a stochastic number-density field can be introduced that essentially represents the Lagrangian particle distribution of the original model. Unlike previous derivations, the number-density field is derived by invoking the δ-function calculus, and for the resulting stochastic-momentum-field transport equation implies the necessity of a repulsive force in order to maintain a unique mapping between Lagrangian and Eulerian frame. Based on the number-density field and the stochastic-momentum field, a stochastic modification of ADM is possible by an approximate reconstruction of the small-scale field on the above-mentioned range of buffer scales. The objective of this paper is to introduce the concept of the Eulerian formulation of the Langevin model in a consistent form, allowing for stable numerical integration and to show how this model can be used for a modified way of subfilter-scale estimation. It should be noted that the overall concept can be applied more generally to any situation where a Lagrangian Langevin model is used. For an initial verification of the concept, which is within the scope of this paper, we consider the example of compressible isotropic turbulence and that of the three-dimensional Taylor-Green-Vortex.