Some properties of a Lagrangian Wiener–Hermite expansion

Abstract

Wiener proposed that the turbulent velocity be expanded in Hermite functionals of a Gaussian white noise random function advected by the fluid. This paper describes the mechanics of converting his suggestion into a computable model, and assesses its range of validity as an approximation for incompressible, homogeneous, and isotropic turbulence. The terms retained are a linear term, v¹, representing an arbitrary Gaussian velocity, and a quadratic term, v², representing a non-Gaussian contribution to the velocity needed for energy transfer. The requirement that advection by the dependent velocity v = v¹ + v² does not alter the statistics of the base necessitates a further truncation of the base to antisymmetric quadratic basis elements. Realizability of any statistics of v is common to all Wiener–Hermite expansions. The projected equations for the Lagrangian expansion conserve energy by non-linear interaction, preserve the inviscid Gaussian equipartition ensemble, and are invariant to random Galilean transformations. Numerical calculations with an approximate form of these equations reveal that irreversible relaxation to the inviscid equipartition solution is not a property of the Lagrangian model, and that the rapid convergence advanced as the original motivation for studying Wiener–Hermite expansions does not survive closure by truncation. The dynamics of the model is not inconsistent with the existence of an inertial range. A simple numerical search routine failed to produce a solution corresponding to such an equilibrium ensemble.