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

Published online by Cambridge University Press:  04 May 2017

Qi Zhou Open the ORCID record for Qi Zhou [Opens in a new window] ,
John R. Taylor Open the ORCID record for John R. Taylor [Opens in a new window]  and
C. P. Caulfield Open the ORCID record for C. P. Caulfield [Opens in a new window]
Qi Zhou*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
John R. Taylor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
C. P. Caulfield
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: [email protected]
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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 $Ri_{g}$ 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, $Ri_{g}$ asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number $Re_{b}\equiv \unicode[STIX]{x1D700}/(\unicode[STIX]{x1D708}N^{2})$, where $\unicode[STIX]{x1D700}$ is the dissipation, $\unicode[STIX]{x1D708}$ 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 $L^{+}\equiv L/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$, where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ is the near-wall viscous scale. The flux Richardson number $Ri_{f}\equiv -B/P$, where $B$ is the buoyancy flux and $P$ is the shear production, is found to be proportional to $Ri_{g}$. This then leads to a turbulent Prandtl number $Pr_{t}\equiv \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D705}_{t}$ of order unity, where $\unicode[STIX]{x1D708}_{t}$ and $\unicode[STIX]{x1D705}_{t}$ are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number $Fr_{h}\equiv \unicode[STIX]{x1D700}/(NU^{\prime 2})$, where $U^{\prime }$ is a turbulent horizontal velocity scale, is found to vary like $Ri_{g}^{-1/2}$. 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 $\unicode[STIX]{x1D705}_{t}/\unicode[STIX]{x1D708}\sim \unicode[STIX]{x1D708}_{t}/\unicode[STIX]{x1D708}\sim Re_{b}Ri_{g}/(1-Ri_{g})$. 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.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Mixing: Mixing Turbulent Flows: Stratified turbulence
Type
Papers
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Journal of Fluid Mechanics , Volume 820 , 10 June 2017 , pp. 86 - 120
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© 2017 Cambridge University Press 

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