Continuity equations in the Generalised Lagrangian Mean theory

Generalised Lagrangian Mean (GLM) theory aims to describe the joint evolution of the mean flow and its perturbations based on the exact system of governing equations.

This paper analyses the formulations of exact GLM continuity equations (GLM-CEs) and clarifies the conditions of their applicability.

We restrict ourselves only to general statements on CEs and do not consider the equations of motion.

The tools used are Lagrangian X, Eulerian x, averaged Eulerian x* coordinates of fluid particles, and ensemble-based averaging over the parameter alpha.

The targeted forms of GLM-CE operate with functions x(x*, t, alpha) and rho(x*, t, alpha), where rho is density.

We 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 GLM-CEs and divide each into average and alpha-dependent (or tilde) parts.

The latter can be seen as compatibility equations that define the class of admissible fluid motions to be considered in the averaged equations.

The main result of the paper is that each original CE is explicitly presented as a self-consistent system of GLM-CEs consisting of average and tilde equations.

For comparison, we consider two versions of the McIntyre-Andrews Transformation (MAT) for CEs.

The original MAT introduces an auxiliary function that satisfies a PDE with special initial conditions.

It also requires introducing specific compatibility equations.

Hence, the original MAT is applicable to a special class of fluid flows.

The generalised MAT yields results similar to those described above, but uses a more complex derivation.

Finally, we consider examples of average flows with small perturbations, thus linking our exposition to the classical GLM theory.