On turbulent mixing in stably stratified wall-bounded flows

Abstract

In this paper, we provide an analysis of turbulent mixing in stably stratified wall-bounded flows to highlight a number of important issues such as prediction of the turbulent viscosity, the turbulent diffusivity, and the irreversible flux Richardson number. By invoking the equilibrium assumption between the production rate of the turbulent kinetic energy (P), the dissipation rate of the turbulent kinetic energy (ϵ), and the turbulent potential energy dissipation rate (ϵPE) as P ≈ ϵ + ϵPE and also assuming equilibrium between the buoyancy flux (B) and ϵPE as −B ≈ ϵPE, we first propose that the irreversible flux Richardson number (Rf∗=ϵPE/(ϵ+ϵPE)) can be approximated with the flux Richardson number (Rf = − B/P), especially for low gradient Richardson numbers. Second, we propose that the turbulent viscosity νt≈(1−Rf∗)−1ϵ/S2, where S is the mean shear rate. We then extend our analysis to propose appropriate velocity and length scales. Tests using the direct numerical simulation (DNS) data of turbulent channel flow of García-Villalba and del Álamo [“Turbulence modification by stable stratification in channel flow,” Phys. Fluids 23, 045104 (2011)] are performed to evaluate our propositions. The comparisons between the exact turbulent viscosity, length and velocity scales, and the proposed formulations are excellent. Also, the agreement between Rf and Rf∗ is reasonable for the bulk of the flow depth with differences of about 20% on average, except at depths very close to the flow boundaries. Finally, by invoking the equilibrium assumption between the buoyancy flux (B) and the dissipation rate of the turbulent potential energy (ϵPE) as −B ≈ ϵPE, we infer the turbulent diffusivity as κt ≈ ϵPE/N2, where N is the buoyancy frequency. The comparison of the proposed turbulent diffusivity with the exact turbulent diffusivity computed from DNS data is good especially close to the wall but the agreement deteriorates far away from the wall, indicating the breakdown of assuming equilibrium as −B ≈ ϵPE which is attributed to the presence of linear internal wave motions in this far-wall region.