École Nationale Supérieure de
Techniques Avancées

Résumé

In this report we study a turbulent, stratified plane Couette flow with Reynolds number of Re = 4250 and Prandtl number of Pr = 0.7 using three-dimensional numerical simulations. The goal is to examine mixing efficiency that relates turbulent buoyancy fluxes to turbulent viscous dissipation. The Navier-Stokes equation is solved in the Boussinesq approximation with an advection-diffusion equation for density. Because of a lower computational cost, two-dimensional simulations are first run to analyze the transition to turbulence from a laminar initial flow state. We show how the destabilization into a turbulent flow and its development with time are really dependent on two things: shear and stratification. A fully turbulent three-dimensional simulation without stratification is then conducted with a highly unstable sharp initial velocity profile in order to produce a statistically steady turbulent flow. This velocity field is then used as the initial condition for full simulations including the density transport. A linear density stratification with varying gradient Richardson numbers Ri is added. The evolution of mixing efficiency is studied using a local flux Richardson number and different behaviors are observed for high and low Ri. Mixing efficiency is related to turbulent diffusivity and our data obtained from numerical simulations are compared to theoretical models that assume a linear relation between density diffusivity and turbulence intensity. The study of turbulent mixing in vertically-inhomogeneous shear such as that of turbulent Couette flows is particularly important for direct applications in oceanic environments. Turbulent mixing in the presence of stratification is often considered in periodic, idealized frameworks, and the present study aims at better understanding local processes involved in turbulent energy transfers in complex flows.