Bridging Filtering and Point-Splitting Approaches for Variable-Density Flows

TL;DR

This paper develops a general framework linking filtering and correlation-based descriptions of energy transfer in variable-density turbulent flows, validated through numerical simulations of buoyancy-driven bubbly flows.

Contribution

It introduces a novel framework to connect filtered energy definitions with correlation functions in compressible turbulence, unifying different approaches.

Findings

Framework successfully relates Favre filtered energy to multi-point correlations.

Numerical validation confirms the framework's applicability to buoyancy-driven bubbly flows.

Provides a basis for analyzing energy transfer in variable-density turbulence.

Abstract

Energy transfer in turbulent flows is typically described either through correlation functions, via the K\'arm\'an-Howarth-Monin relation, or through a scale-by-scale budget of filtered energy (Frisch 1995). For constant-density turbulence, the equivalence between these two descriptions is well understood. In compressible turbulence, however, several definitions of filtered energy exist, and for most of them the associated formulation in terms of correlation functions is unclear. We develop a general framework to determine the multi-point correlation functions corresponding to any specified filtered energy. As a demonstration, we show that the Favre filtered energy--defined as the ratio of the squared filtered momentum to the filtered density--and the terms in its budget can be written as an infinite series of multi-point correlation functions. We validate the framework numerically using three-dimensional buoyancy-driven bubbly flows.