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An explicit algebraic Reynolds-stress and scalar-flux model for stably stratified flows

Published online by Cambridge University Press:  16 April 2013

W. M. J. Lazeroms ,
G. Brethouwer ,
S. Wallin  and
A. V. Johansson
W. M. J. Lazeroms*
Affiliation:
Linné FLOW Centre, Department of Mechanics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden
G. Brethouwer
Affiliation:
Linné FLOW Centre, Department of Mechanics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden
S. Wallin
Affiliation:
Linné FLOW Centre, Department of Mechanics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden Swedish Defence Research Agency (FOI), SE-164 90, Stockholm, Sweden
A. V. Johansson
Affiliation:
Linné FLOW Centre, Department of Mechanics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden
*
†Email address for correspondence: [email protected]
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Abstract

This work describes the derivation of an algebraic model for the Reynolds stresses and turbulent heat flux in stably stratified turbulent flows, which are mutually coupled for this type of flow. For general two-dimensional mean flows, we present a correct way of expressing the Reynolds-stress anisotropy and the (normalized) turbulent heat flux as tensorial combinations of the mean strain rate, the mean rotation rate, the mean temperature gradient and gravity. A system of linear equations is derived for the coefficients in these expansions, which can easily be solved with computer algebra software for a specific choice of the model constants. The general model is simplified in the case of parallel mean shear flows where the temperature gradient is aligned with gravity. For this case, fully explicit and coupled expressions for the Reynolds-stress tensor and heat-flux vector are given. A self-consistent derivation of this model would, however, require finding a root of a polynomial equation of sixth-order, for which no simple analytical expression exists. Therefore, the nonlinear part of the algebraic equations is modelled through an approximation that is close to the consistent formulation. By using the framework of a $K\text{{\ndash}} \omega $ model (where $K$ is turbulent kinetic energy and $\omega $ an inverse time scale) and, where needed, near-wall corrections, the model is applied to homogeneous shear flow and turbulent channel flow, both with stable stratification. For the case of homogeneous shear flow, the model predicts a critical Richardson number of 0.25 above which the turbulent kinetic energy decays to zero. The channel-flow results agree well with DNS data. Furthermore, the model is shown to be robust and approximately self-consistent. It also fulfils the requirements of realizability.

JFM classification

Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 723 , 25 May 2013 , pp. 91 - 125
Copyright
©2013 Cambridge University Press 

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