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On the turbulent Prandtl number in homogeneous stably stratified turbulence

Published online by Cambridge University Press:  11 February 2010

SUBHAS K. VENAYAGAMOORTHY  and
DEREK D. STRETCH
SUBHAS K. VENAYAGAMOORTHY*
Affiliation:
Department of Civil and Environmental Engineering, Colorado State University, Fort Collins, CO 80523-1372, USA School of Civil Engineering, University of KwaZulu-Natal, Durban 4041, South Africa
DEREK D. STRETCH
Affiliation:
School of Civil Engineering, University of KwaZulu-Natal, Durban 4041, South Africa
*
†Email address for correspondence: vskaran@colostate.edu
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Abstract

In this paper, we derive a general relationship for the turbulent Prandtl number Prt for homogeneous stably stratified turbulence from the turbulent kinetic energy and scalar variance equations. A formulation for the turbulent Prandtl number, Prt, is developed in terms of a mixing length scale LM and an overturning length scale LE, the ratio of the mechanical (turbulent kinetic energy) decay time scale TL to scalar decay time scale Tρ and the gradient Richardson number Ri. We show that our formulation for Prt is appropriate even for non-stationary (developing) stratified flows, since it does not include the reversible contributions in both the turbulent kinetic energy production and buoyancy fluxes that drive the time variations in the flow. Our analysis of direct numerical simulation (DNS) data of homogeneous sheared turbulence shows that the ratio LM/LE ≈ 1 for weakly stratified flows. We show that in the limit of zero stratification, the turbulent Prandtl number is equal to the inverse of the ratio of the mechanical time scale to the scalar time scale, TL/Tρ. We use the stably stratified DNS data of Shih et al. (J. Fluid Mech., vol. 412, 2000, pp. 1–20; J. Fluid Mech., vol. 525, 2005, pp. 193–214) to propose a new parameterization for Prt in terms of the gradient Richardson number Ri. The formulation presented here provides a general framework for calculating Prt that will be useful for turbulence closure schemes in numerical models.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 644 , 10 February 2010 , pp. 359 - 369
Copyright
Copyright © Cambridge University Press 2010

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References

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