Investigation of the Local Equilibrium Approximation in a Planar Momentumless Turbulent Wake in a Passively Stratified Fluid

Abstract

The flow in the far planar momentumless turbulent wake in a passively stratified medium is investigated with the use of a mathematical model that includes, along with the averaged equation for transfer of the longitudinal component of velocity, the differential equations for balance of the turbulence energy \(e\), transfer of its dissipation rate \(\varepsilon\), fluid density defect \(\langle\rho_{1}\rangle\), shear turbulent stress \(\langle{u}'{v}'\rangle\), and vertical component of the mass flux vector \(\langle v'\rho'\rangle\) in the far wake approximation. Algebraic truncation of the last two equations leads to the known gradient relations for the shear turbulent stress and the vertical component of the mass flux vector. It has been established that with a certain limitation on the values of the empirical constants of the mathematical model and with a time scale growth law consistent with the mathematical model, these relations are joint differential constraints of the model. It has been found that the local equilibrium approximation for the shear turbulent stress is equivalent to the equality to zero of the Poisson bracket for dimensionless values of the turbulent viscosity coefficient and the defect of the longitudinal velocity component. It has been obtained that the local equilibrium approximation for the vertical component of the mass flux vector is equivalent to the equality to zero of the Poisson bracket for dimensionless values of the turbulent diffusion coefficient and mean density. Results of numerical experiments illustrating the theoretical results are presented.