Self-similar mixing in stratified plane Couette flow for varying Prandtl number


Type
Article
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Authors
Zhou, Q 
Taylor, JR 
Caulfield, CP 
Abstract

We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebio et al. (J. Fluid Mech., vol. 781, 2015, pp. 298-329) expanding the range of Prandtl number Pr examined by two orders of magnitude from 0.7 up to 70. Significant effects of Pr on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin-Obukhov (M-O) similarity theory and van Driest damping functions. We then employ M-O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number Rig is determined by the length scale ratio h/L, where h is the half-channel gap depth and L is the Obukhov length scale. As h/L approaches very large values, Rig asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number Reb ε/(νN2), where ε is the dissipation, ν is the kinematic viscosity and N is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio L+ L/δν, where δν is the near-wall viscous scale. The flux Richardson number Rif -B/P, where B is the buoyancy flux and P is the shear production, is found to be proportional to Rig. This then leads to a turbulent Prandtl number Prt νt/κt of order unity, where νt and κt are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number Frh ε/(NU2), where U is a turbulent horizontal velocity scale, is found to vary like Rig−1/2. All these scalings are consistent with our numerical data and appear to be independent of Pr. The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn, J. Phys. Oceanogr., vol. 10, 1980, pp. 83-89), together with M-O scalings, results in a parameterization of κt/ν ~ νt/ν ~ RebRig/(1-Rig). With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M-O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M-O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.

Description
Keywords
geophysical and geological flows, mixing, stratified turbulence
Journal Title
Journal of Fluid Mechanics
Conference Name
Journal ISSN
0022-1120
1469-7645
Volume Title
820
Publisher
Cambridge University Press
Sponsorship
Engineering and Physical Sciences Research Council (EP/K034529/1)
The EPSRC Programme grant EP/K034529/1 entitled ‘Mathematical Underpinnings of Stratified Turbulence’ is gratefully acknowledged for supporting the research presented here.
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