Turbulence, mixing and Prandtl number effects in stratified plane Couette flows

Turbulence, mixing and Prandtl number effects in stratified plane Couette flows Qi Zhou 1 , John Taylor 1 and Colm Caulfield 2,1 Department of Applied Mathematics and Theoretical Physics BP Institute, University of Cambridge Abstract We report direct numerical simulations (DNS) of stratified plane Couette (SPC) flows in both continuous and layered stratifications for a wide range of Prandtl number Pr from 0.7, 7 to 70. In the continuously stratified set-up, Pr has a significant impact on the momentum and heat fluxes across the flow. Variations of these fluxes then modifies the turbulence characteristics in the way that is consistent with Monin–Okukhov (M-O) similarity theory. The mixing properties in the interior of the flow, however, appear to be independent of Pr . We employ M-O theory and the Osborn (1980) model to formulate scalings for the turbulent diffusivities. These scalings are then verified by DNS and discussed with respect to existing scalings in the literature. In the layered configuration, we investigate the transient mixing of a density interface, where the variation of Pr indeed becomes important. Introduction In this paper, we present a computational study of stratified plane Couette (SPC) flows. An SPC flow is bounded by two horizontal walls located at y = ±h respectively, and the walls move in opposite directions with a constant speed U w . The temperature θ at the upper and lower walls is held at ±T w respectively, and the fluid density ρ relates to θ via a linear equation of state ρ = ρ 0 (1 − α V θ), which results in a stably stratified system. The three external parameters are Reynolds number Re ≡ U w h/ν, Richardson number Ri ≡ α V T w gh/U w 2 and Prandtl number Pr ≡ ν/κ, where α V is thermal expansion coefficient, g is gravity, ν is viscosity and κ is diffusivity. The direct numerical simulations (DNS) reported here are all for Re = 4250 and for varying Ri and Pr values. The DNS algorithm can be found in Taylor (2008), and the configurations of the SPC simulations are reported in Deusebio et al. (2015) and Zhou et al. (2016) (the latter is hereinafter referred to as ‘ZTC’). Two configurations of SPC flows are examined in this paper, one being the fully developed statistically stationary turbulent SPC (§2), the other being a ‘layered’ set-up featuring a density interface introduced as an initial condition (§3). One common thread through these two configurations is the role of Prandtl number Pr . While most existing SPC studies have focused on Pr values of order unity (e.g., Garc´ia-Villalba et al. (2011); Deusebio et al. (2015)), there has been growing evidence indicating that Pr can indeed have some first-order effects on stratified shear flows. For example, the effects of Pr on the characteristics of secondary instabilities and diapycnal mixing were reported by Salehipour et al. (2015) through simulations of growing Kelvin-Helmholtz instabilities. Motivated by these observations, we aim to investigate the effects of Pr systematically in SPC flows by varying Pr from 0.7, 7 to 70 (note the first two values correspond to the scenarios of heat in air and heat in water, respectively, and the third value is investigated VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016