Exact versus average continuity equations in the Generalised Lagrangian Mean theory: compatibility equations

Abstract

Generalised Lagrangian Mean (GLM) theory aims to describe the joint evolution of the mean flow and its perturbations. This paper examines exact GLM forms of continuity equations (CEs) and clarifies the conditions of their validity. We do not consider equations of motion; therefore, we use only exact formulae and general notions of fluid dynamics and operate only with the general statements for exact CEs. The tools used are Lagrangian X, Eulerian x, averaged Eulerian x* coordinates of fluid particles, and ensemble-based averaging. The targeted forms of CEs operate on functions x(x*,t) and rho (x*, t), where rho(x*,t) is the density. The first step is to present the actual velocity divergence div u via div*u*, where u and u* are the actual and average fluid velocities. Then we derive three versions of exact CEs and demonstrate that each is controversial, prompting us to split each equation into average and oscillating parts. The third step is to resolve the contradiction by introducing compatibility equations that express the identical vanishing of the oscillating parts. For comparison, we consider two versions of the McIntyre-Andrews Transformation (MAT) for CEs. The original MAT introduces an auxiliary function that satisfies an auxiliary PDE and special initial conditions. Therefore, it works for a special class of fluid flows. The generalised version makes CEs applicable to arbitrary fluid motion. Both versions require compatibility equations, which we compare with ours. Finally, we consider examples of average flows with small perturbations, thus linking our exposition to the classical GLM theory.