ON THE TURBULENT PRANDTL NUMBER IN STABLY STRATIFIED TURBULENCE BY SECOND ORDER MODELS

The aim of this work is to investigate the behavior of the Turbulent Prandtl number by second order modeling of a stably stratified homogeneous sheared turbulence. By analytic solutions, we have confirmed the asymptotic equilibrium behavior of the turbulent Prandtl number. Then two between the most second order models of turbulence; the Classic Launder-Reece-Model and the sophisticated Craft Launder model are retained. A non dimensional form of transport equations have been obtained when non dimensional parameters are introduced to substitute second order moments. A numerical integration using the fourth order Runge kutta method has been conducted for different values of the gradient Richardson number Ri. In comparison with direct numerical simulation result’s of Shih et al. the obtained results by the Craft Launder model has shown for the turbulent Prandtl number the best agreement at moderate values of gradient Richardson number 0.15 < Ri < 0.28. The classic model has shown a great default for the different values of Ri. No any concordance with retained results of DNS has been obtained by this model. We show also that prediction of this model can be improved by introducing variation and optimization of model constants.


INTRODUCTION
The problem of turbulent mixing in a stably stratified turbulence is a problem of often importance since stably stratified are present in the atmospheric boundary layer, oceans, lakes and many engineering flows. Homogenous turbulence in stratified shear flow has been investigated essentially through direct numerical simulations by several authors during the last three decades. Gerz et al. [1] and Holt et al. [2] have analyzed the behavior of a stratified homogeneous sheared turbulence as a function of the important parameter i R . The gradient Richardson number i R indicates the importance of stratification effect to shear effect.Shih et al. [3][4] have accorded much attention to the turbulent Prandtl number than the other works and their results are retained in this work. This number is a widely used parameter in stably stratified homogeneous turbulence. More recently, Subhas and Derek [5] have argued these results and derived a new relationship for the turbulent Prandtl number in terms of the gradient Richardson number i R , and between the turbulent kinetic energy and scalar variance. The second order modeling of stratified homogeneous turbulence remains to our sense an important approach of studying homogeneous turbulence since its application to the great number of engineering flows and industrial applications [6][7]. In a previous work, we are interested to the prediction of equilibrium states in stratified homogenous turbulence with vertical or inclined shear [8][9]. We were focused essentially to the asymptotic behavior of dimensionless kinematic parameters such as the component of the anisotropic tensor and the non dimensional shear number  KS .
However to our best knowledge not previous results have been dedicated to the prediction of the turbulent Prandtl number behavior using second order models. This simple and important goal constitutes the motivation of this paper which is organized as follows. In section 2 we present the mathematical formulation of a stratified homogeneous turbulence by the transport equation of second order moments of turbulence. Some analytical comments are than proposed for the turbulent Prandtl number t Pr and the ratio E L M L . In section 3 the second order modeling of transport equations is described and the retained second order models of turbulence are introduced. A non dimensional form of the transport equation is also obtained, parameterized by

MATHEMATICAL FORMULATION GENERAL EQUATIONS
In the case of a stably stratified homogenous sheared turbulence, the basic transport equation for the components j i u u of the Reynolds stress, the turbulent kinetic energy, the turbulent scalar flux and scalar variance may be obtained by standard method [2]: In these equations terms noted P are production terms due to mean velocity and mean scalar gradients: ij  and   i are the pressure strain correlation and pressure scalar gradient correlation terms: ij B and B are buoyancy terms: Journal of Thermal Engineering, Technical Note, Vol.6, No.3, pp.369-380, April, 2020 (13) and  are dissipation due to molecular effects terms: Now, analytical comments are developed for transport equations (1) - (4). The behavior of the turbulent Prandtl number t Pr and the ratio E M L L will be investigated. In their direct numerical simulations, Holt et al. [2] have confirmed that the non linear and viscosity effects are very small comparing to productions terms at high shear ( >>). If we take into account of this hypothesis in the transport equations (1) -(4), a linear set of differential equations are obtained and written in the following form: Journal of Thermal Engineering, Technical Note, Vol.6, No.3, pp.369-380, April, 2020

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The Laplace transform is retained for integrating the linear differential system, the principal solutions are written as follows: Where coefficients A, B, D are functions of initial values of turbulent parameters and the gradient Richardson number i R . These solutions will now be used to study the asymptotic behavior (when . Firstly we deduce the expressions of these parameters: And at high shear (  >>), we obtain respectively: We conclude that at high shear ( >>), the two dimensionless scalar parameters tend to equilibrium states functions of the initial values of turbulent parameters and the gradient Richardson number i R . This analytic solution is important since it confirms that at equilibrium, buoyancy terms equilibrate production terms. It confirms also the result of direct numerical simulation of Shih et al. [4] and the study of Subhas and Derek [5]. This approach is only a qualitative one and a more quantitative study of the evolution of the turbulent Prandtl number t Pr and the ratio E M L L will be addressed in the following section when second order modeling of transport equations is retained.

SECOND ORDER MODELING
At this step of our work and with the aim of obtaining a closed set of equations, the second order modeling is the approach retained here. Second order models are retained for nonlinear terms of pressure strain correlation, pressure scalar gradient correlation and transport equation of dissipation.
In a stratified shear flow, the correlations ij  and   i are classically [11] decomposed on three contributions: Where the contributions 1 are the return-to-the isotropy terms, the terms 2 represent the interaction between mean and turbulent flows. The terms 3 are terms due to buoyancy.
Two between the most known second order models are retained for the pressure strain correlation and the pressure scalar gradient correlation and the evolution equation of the dissipation rate  . The classic second order model of Launder, Reece and Rodi (LRR) [12] in one hand and the sophisticated model of Craft and Launder (CL) [13] have been retained. For its success in many different applications of stratified homogeneous shear flow [8][9][10], the Zeman Lumley model is retained for the third contributions of the pressure strain correlation and pressure scalar gradient correlation: At this step, a numerical integration of the differential equations is started. Obtained results will be discussed in the following sections.

NUMERICAL INTEGRATION AND RESULTS
Two non linear systems of seven differential equations are obtained corresponding to the two retained models. The numerical integration is advanced in time using the fourth order Runge-Kutta method. Numerical integration has been advanced to long time evolution Another optimization which we propose here is for the coefficient 1 It is essentially here to precise that coefficients 1 C and 1 C  of return to isotropy terms are generally the only coefficients which we can modify. The values of the others coefficients of the linear terms of pressure strain correlation and pressure scalar gradient correlations are exact. They cannot be modified since they are submitted in addition to kinematic constraints (continuity and symmetry...) to the strong form of realisability condition [13].

CONCLUDING REMARKS
In this work, we are interested to the study of the evolution of scalar parameters in stably stratified homogeneous sheared turbulence using second order models of turbulence. The turbulent Prandtl number is the principal parameter concerned. The ratio E L / M L of mechanical length scale to scalar length scale is also investigated. At a first step, we have confirmed by analytical linear solutions, deduced from Laplace transform and available only at high shear that the two mentioned dimensionless parameters have a general tendency to asymptotic equilibrium states, observed by the DNS of Shih et al. In the second step, two second order models have been retained for pressure strain correlation and pressure scalar gradient correlation to model transport equations. The fourth order Runge-Kutta method has been retained for integrating two non linear systems of seven differential equations.