On the turbulent Prandtl number in homogeneous stably stratified turbulence

Abstract

In this paper, we derive a general relationship for the turbulent Prandtl number Pr t for homogeneous stably stratified turbulence from the turbulent kinetic energy and scalar variance equations. A formulation for the turbulent Prandtl number, Pr t , is developed in terms of a mixing length scale L M and an overturning length scale L E , the ratio of the mechanical (turbulent kinetic energy) decay time scale T L to scalar decay time scale T ρ and the gradient Richardson number Ri . We show that our formulation for Pr t 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 L M /L E ≈ 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, T L /T ρ . 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 Pr t in terms of the gradient Richardson number Ri . The formulation presented here provides a general framework for calculating Pr t that will be useful for turbulence closure schemes in numerical models.

Comments on the Numerical Simulation of Homogeneous Stably-Stratified Turbulence

“…4 of Venayagamoorthy and Stretch (2010), are consistent with these conditions. This is not surprising: any monotonically increasing function linking the neutral stability limit: Pr T | Ri=0 = Pr 0 ≈ 0.7−0.8 with the large Ri limit: Pr T = R −1 ∞ Ri (e.g., the simple interpolation: Pr T = Pr 0 + R −1 ∞ Ri) would also look reasonable.…”

On the Velocity Gradient in Stably Stratified Sheared Flows. Part 1: Asymptotic Analysis and Applications

“…The maximum value of the flux Richardson number: R ∞ ≈ 0.2 has been determined from various experimental, large-eddy simulation (LES) and direct numerical simulation (DNS) data for extremely strong static stabilities, namely, for the values of Ri between 1 and 10 2 , at which Ri f definitely levels off (e.g., Yamada 1975;Stroscio 1982;Zilitinkevich et al , 2008Zilitinkevich et al , 2009Stretch et al 2009;Venayagamoorthy and Stretch 2010). It is remarkable that the coefficient 1/R ∞ ≈ 5 in Eq.…”

RANS Modeling of Stably Stratified Turbulent Boundary Layer Flows in OpenFOAM®

“…Venayagamoorthy & Stretch [9] found that Γ ∞ = 1/3 fits the data corresponding to Ri f ∞ = 0.25 [7]. Pr t0 is taken to be 0.7 from the previous discussion.…”

“…Additionally, the analytical Pr t model of Venayagamoorthy & Stretch [9] (hereafter VS10) is considered. The authors provided a theoretical modification to the empirical formulation of Schumann & Gerz [8] based on DNS data [5,7].…”

“…Pr t is found to be on the order of unity for simple turbulent flows [1]. While Pr t is generally assumed to be a constant in the range 0.5 − 1.0 in numerical modeling of stably stratified turbulent flows, observational studies [2][3][4], direct numerical simulation (DNS) [5][6][7], and analytical models [8][9][10] have shown that the value of Pr t is strongly linked to stability. The research presented in this paper seeks to investigate several Pr t propositions for efficacy in numerical simulations of stably stratified turbulent flows.…”