Evaluation SSG-LRR Model on an Homogeneous Turbulence under Inclined Shear for High and Low Stratification: Shear Number Effect

Abstract

This chapter develops proposals for an evaluation coupled second order model of SSG-LRR on an homogeneous turbulence submitted to an inclined shear for high and low stratification. The effect of Shear number on thermal and dynamic turbulent fields of the problem is performed for Shear number fixed at 2, 6, 14 and 20. Two values of Froude number equal to 0.35 and 1.29 are adopted for all numerical simulations corresponding to high and low stratification, respectively. For all simulations, value of angle theta is fixed at θ = π/4 corresponding to the angle between the shear and the vertical gradient of stratification. SSG-LRR model is adopted to compute turbulent parameters of principal component of anisotropy b12, normalized turbulence dissipation ε/KS and the density flux ρu1¯. A good agreement is detected by comparison of findings via model of SSG-LRR with the reported results in the literature by Direct Numerical Simulation of Jacobitz (DNSJ). It is found that the variation of Shear number predict a very strong influence on thermal and dynamic turbulent characteristics. Hence, findings with SSG-LRR model prove the existence of an asymptotic equilibrium states for various thermal and dynamic parameters in particularly for a low stratification.

1. Introduction

Homogeneous sheared and stably stratified turbulence is considered as a fundamental flow relevant to the study of geophysical turbulence and, generally, anisotropic turbulence. Homogeneous and stratified turbulence has received considerable attention in the past few decades. This interest stems from its importance in many industrial and engineering applications. Such applications include geophysical flows, oceanic and atmospheric mixed layers, turbulent thermal plumes, atmospheric circulation, pipes, gas turbines, airplanes, etc. Complex dynamical turbulent processes associated to the stratified turbulence under shear are observed in atmosphere and in industry. This kind of turbulence problem presents an attractive physical field as it plays a key role to improve our understanding of behaviors of geophysical flows [1, 2, 3, 4]. Dynamical turbulent processes associated to the stratified turbulence submitted to an inclined shear are encountered in industry and atmosphere. This problem of the turbulence is relevant in many thermal engineering applications in both nature and industry owing to it playing a key role to better describe turbulent characteristics of geophysical flows. Furthermore, the understanding of behaviors of homogeneous and stratified turbulence related to geophysical flows presents a considerable role to develop turbulence theories and performed a comparison of results via different models of LES, SSG (Speziale, Sarkar, and Gatski), k-ε, SL, and DNS [5, 6, 7, 8, 9]. So, the exam of the turbulence leads to engineering applications with benefits for society such as the development of high-resolution climate and space weather models.

The available review illustrates numerous investigations on the characterization of thermal and dynamic fields of the turbulence flow using different methods [10, 11, 12, 13]. Komori et al. [14] approve that an experimental analysis on a stratified turbulent via vertical shear flow. Experimental studies are performed for a horizontal shear by various authors [15, 16]. Comprehensive studies of homogeneous and stratified turbulence submitted to inclined shear include the experimentally work by Rohr et al. [17] and Piccirillo et al. [18]; they confirmed the importance of the Froude number and reported an influence of additional dimensional parameter such as shear number on evolution of turbulence. Development of LRR (Launder, Reece, and Rodi) model of turbulence in which the Reynolds stresses is studied by Launder et al. [19, 20], they conclude that the numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows. After that, Jacobitz et al. [21] performed the direct numerical simulations (DNS) to investigate the evolution of turbulence in a uniformly sheared and stably stratified flow. They conclude that the shear number has a strong non-monotone influence on the turbulence evolution. Jacobitz el al. [5] used the direct numerical simulations in order to investigate the evolution of turbulence in a stably stratified fluid forced by nonvertical shear. The flux Richardson number Ri_f depends on the gradient Richardson number Ri but not on the shear inclination angle θ. It is interesting that the normalized turbulence production is strongly influenced by the angle θ, but that the flux Richardson number remains unaffected.

The anisotropy of fluctuating motion in a stably stratified medium with uniform mean shear is studied by Sarkar [22] using the numerical simulation of uniform shear flow. He concludes that the vertical of streamwise velocity is found to dominate the components of turbulent dissipation in both horizontal and vertical shear flows. After that, Bouzaiane et al. [23] studied the evolution of homogeneous stably stratified turbulence submitted to a nonvertical shear using second-order closure models (Craft and Launder CL and Shih and Lumley SL). They improve that a good agreement between the predictions of second-order models and values of DNSJ is generally observed for the principal component of anisotropy b ₁₂ and a qualitative agreement is observed for the ratios K/E and K _ρ/E of the kinetic and potential energies to the total energy E. Thamri et al. [24] applied three coupled second-order models (of SSG-CL, SSG-SL, and SSG-LRR) in order to simulate the influence of nonvertical shear on homogeneous turbulence and for a low stratification Ri = 0.2. They showed a good accord between results of three second-order models and those of DNSJ.

Basing on the literature, it is found that no combination between Speziale, Sarkar, and Gatski (SSG) model [25] and Launder, Reece, Rodi (LRR) model [19, 20] is improved. The present work is the first systematic investigation related to the shear number effects on thermal and dynamics parameters of a homogeneous turbulence submitted to an inclined shear using second-order coupled models of SSG-LRR, with low and high stratification. For the case of high stratification of the turbulence, the Richardson number is equal to Ri = 2 associated to Froude number equal to 0.35. For low stratification, the Richardson number is fixed at Ri = 0.15 corresponding to Froude number equal to 1.29. In addition, a comparison between our work (SSG-LRR model) and the results of DNSJ [21, 26, 27] is carried out. The paper is organized as follows: The problem is defined in Section 2, which also presents the governing equations and parameters. The methodology is presented in Section 3.1, which describes the numerical method, model setup, boundary conditions, and validation. Results and discussion follow in Section 3.2, and finally, conclusions are drawn in Section 4.

2. Governing equations and numerical approach

All numerical computations are based on the continuity equation of an incompressible fluid, three-dimensional unsteady Navier–Stokes equation with the Boussinesq approximation and the transport equation for the density. The mean velocity $\overline{U}=\left(\overline{U_1},0,0\right)$ has a constant inclined shear rate $S=S.sin\left(\theta \right)$ where θ = π/4 is the angle between the shear and the vertical gradient of stratification. The mean density has a constant vertical stratification gradient $S_{\rho }=\frac{\partial \overline{\rho }}{\partial x_3}$. Using the decomposition of Reynolds, the dependent variables U_i (i-th component of the total velocity), ρ (total density), and P (total pressure) are decomposed into a mean part (denoted by an overbar) and a fluctuating part (denoted by small letters):

$$U_i=\overline{U_i}+u_i,\rho =\overline{\rho }+\rho \text{and}P=\overline{P}+p$$E1

The decomposition of dependent variables is introduced into equations of motion and the following evolution equations for the fluctuating parts are obtained:

$$\frac{\partial u_i}{\partial x_i}=0$$E2

$$\begin{array}{l}\frac{\partial u_i}{\partial t}+u_k\frac{\partial u_i}{\partial x_k}+\left(S\frac{\sqrt{2}}{2}{x}_2+S\frac{\sqrt{2}}{2}{x}_3\right)\frac{\partial u_i}{\partial x_1}+\left(S\frac{\sqrt{2}}{2}{u}_2+S\frac{\sqrt{2}}{2}{u}_3\right){\delta }_{i1}=\\ -\frac{1}{{\rho }_0}\frac{\partial p}{\partial x_i}+\nu \frac{{\partial }^2{u}_i}{\partial x_k\partial x_k}-\frac{g}{{\rho }_0}{\mathit{\rho \delta }}_{i3}\end{array}$$E3

$$\frac{\partial \rho }{\partial t}+u_k\frac{\partial \rho }{\partial x_k}+\left(S\frac{\sqrt{2}}{2}{x}_2+S\frac{\sqrt{2}}{2}{x}_3\right)\frac{\partial \rho }{\partial x_1}+S_{\rho }{u}_3=\alpha \frac{{\partial }^2\rho }{\partial x_k\partial x_k}$$E4

g is the acceleration due to gravity, ν the kinematic viscosity, α the scalar diffusivity, δ_ij is the Kronecker symbol, ρ is the fluctuation of the scalar, and ρ₀ is the density of reference.

$S_{\rho }=\frac{\partial \overline{\rho }}{\partial x_3}$ is the stable density stratification, and S is the shear rate.

The dimensionless shear number SK/ε is the ratio of the turbulence time scale K/ε to the shear time scale 1/S. Dimensionless Richardson number is the ratio of the turbulence stratification by the turbulence shear. Richardson number is defined as follows:

$$\mathrm{R}i=\frac{N^2}{S^2}$$

The Froude number, which depends on Richardson number, is given as:

$$\mathit{Fr}=\sqrt{\frac{{0.5}^2}{\mathit{Ri}}}$$

where S is the uniform mean shear, $N^2=-g\left(\partial \rho /\partial z\right)/{\rho }_0$ is the Brunt–Väisäla frequency, g is the acceleration due to gravity, ρ is the fluctuation of the scalar, ρ₀ is the density of reference, and z is the vertical coordinate.

The non-dimensional time related to temporal evolution of physical turbulent parameters, which characterize the considered problem, is defined as: τ = S.t.

The adopted calculation algorithm related to the present problem consists the following steps:

1. Initialization of thermal and dynamic fields of the flow.

2. Declaration of initial conditions of isotropy of findings DNSJ [21, 26, 27]:

   $${\left(b_{\mathit{ij}}\right)}_0=0,0,{\left(\frac{\epsilon }{\mathit{KS}}\right)}_0=0,5,{\left(F_i\right)}_0=0,0,{\left(\eta \right)}_0=0,0$$

3. Integration of dimensionless equations of SSG-LRR model.

4. A fourth-order Runge–Kutta method is adopted to resolve various equations of the problem.

5. Calculation of thermal and dynamic turbulent parameters (b₁₂, $\overline{\rho u_1}$, ε/KS).

The set of differential equations governing the transport of the velocity correlation$\overline{u_i{u}_j}$, transport equation for the density flux $\overline{u_i\rho }$, transport equation for the turbulent kinetic energy $K=\frac{\overline{u_i{u}_i}}{2}$, and transport equation for the density fluctuation $\overline{{\rho }^2}$ may be written in the form:

$$\begin{array}{l}\frac{d\overline{u_i{u}_j}}{\mathit{dt}}=-S\frac{\sqrt{2}}{2}\overline{u_j{u}_2}{\delta }_{i1}-S\frac{\sqrt{2}}{2}\overline{u_j{u}_3}{\delta }_{i1}-S\frac{\sqrt{2}}{2}\overline{u_i{u}_2}{\delta }_{j1}-S\frac{\sqrt{2}}{2}\overline{u_i{u}_3}{\delta }_{j1}-\\ \frac{g}{{\rho }_0}\left(\overline{u_i\rho }{\delta }_{j3}+\overline{u_j\rho }{\delta }_{i3}\right)+\frac{1}{{\rho }_0}\overline{p\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)}-2\nu \overline{\frac{\partial u_i}{\partial x_k}\frac{\partial u_j}{\partial x_k}}\end{array}$$E5

$$\begin{array}{l}\frac{d\overline{u_i\rho }}{\mathit{dt}}=-S\frac{\sqrt{2}}{2}\overline{u_2\rho }{\delta }_{i1}-S\frac{\sqrt{2}}{2}\overline{u_3\rho }{\delta }_{i1}-S_{\rho }\overline{u_i{u}_3}-\frac{g}{{\rho }_0}\overline{{\rho }^2}{\delta }_{i3}+\frac{1}{{\rho }_0}\overline{p\frac{\partial \rho }{\partial x_i}}-\\ \left(\alpha +\nu \right)\overline{\frac{\partial \rho }{\partial x_k}\frac{\partial u_i}{\partial x_k}}\end{array}$$E6

$$\frac{\mathit{dK}}{\mathit{dt}}=-S\frac{\sqrt{2}}{2}\overline{u_1{u}_2}-S\frac{\sqrt{2}}{2}\overline{u_1{u}_3}-\frac{g}{{\rho }_0}\overline{u_3\rho }-\nu \overline{\frac{\partial u_i}{\partial x_k}\frac{\partial \mathit{ui}}{\partial x_k}}$$E7

$$\frac{d\overline{{\rho }^2}}{\mathit{dt}}=-2S_{\rho }\overline{\rho u_3}+2\nu \overline{\frac{\partial \rho }{\partial x_k}\frac{\partial \rho }{\partial x_k}}$$E8

g is the acceleration due to gravity, $\nu$ the kinematic viscosity, and $\alpha$ the scalar diffusivity.

$S_{\rho }=\frac{\partial \overline{\rho }}{\partial x_3}$ is the stable density stratification, S is the shear rate, and δ_ij is the Kronecker symbol.ρ is the fluctuation of the scalar, and ρ₀ is the density of reference.where component denoted by an over bar and component denoted by small letters denote mean and fluctuating components of velocity, p is the fluctuation of static pressure about its mean value, and x_i (i = 1, 2, 3) denote Cartesian space coordinates.

$$U_i=\overline{U_i}+u_i,,P=\overline{P}+p\rho =\overline{\rho }+\rho$$

U_i the total values of velocity components, P the total pressure, and ρ the total density.

Nearly every worker obtained a closed system of differential equations. The turbulence modeling remains an important approach retained by several authors [28]. The coupled between second-order modeling (SSG-LRR) is retained here and consists of modeling the pressure-strain ${\phi }_{\mathit{ij}}$and pressure–temperature gradient ${\phi }_{\mathit{i\rho }}$ correlations.

$${\phi }_{\mathit{ij}}=\frac{1}{{\rho }_0}\overline{p\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)}={\phi }_{\mathit{ij}}^1+{\phi }_{\mathit{ij}}^2+{\phi }_{\mathit{ij}}^3\text{and}{\phi }_{\mathit{i\rho }}=\frac{1}{{\rho }_0}\overline{p\frac{\partial \rho }{\partial x_i}}={\phi }_{\mathit{i\rho }}^1+{\phi }_{\mathit{i\rho }}^2+{\phi }_{\mathit{i\rho }}^3$$

The SSG model [25] has concerned only kinematic turbulence, and it is retained here for these terms, which are written in the following forms:

$${\phi }_{\mathit{ij}}={\phi }_{\mathit{ij}}^1+{\phi }_{\mathit{ij}}^2$$E9

$${{\phi }_{\mathit{ij}}}^1=-C_1{b}_{\mathit{ij}}+3\left(C_1-2\right)\left(b_{\mathit{ij}}^2+{\mathit{II}}_b\frac{{\delta }_{\mathit{ij}}}{3}\right)$$E10

$${\mathit{II}}_b=b_{\mathit{kl}}{b}_{\mathit{lk}},C_1=3,4$$

$$\begin{array}{l}{{\phi }_{\mathit{ij}}}^2=C_2{b}_{\mathit{mn}}{S}_{\mathit{mn}}{b}_{\mathit{ij}}+\frac{1}{2}\left(C_3-\sqrt{b_{\mathit{mn}}{b}_{\mathit{mn}}}{C}_{33}\right)S_{\mathit{ij}}+\frac{1}{2}{C}_4\left(b_{\mathit{ik}}{S}_{\mathit{kj}}+b_{\mathit{jk}}{S}_{\mathit{ki}}-\frac{2}{3}{b}_{\mathit{mn}}{S}_{\mathit{mn}}{\delta }_{\mathit{ij}}\right)\\ +\frac{1}{2}{C}_5\left(b_{\mathit{ik}}{w}_{\mathit{jk}}+b_{\mathit{jk}}{w}_{\mathit{ik}}\right)\end{array}$$

$$C_2=1,8C_3=0,8C_4=1,25C_5=0,4C_{33}=1,3$$

The LRR model [19, 20] is retained for scalars field:

$${\phi }_{i\rho }^1=-C_1^{\text{'}}\frac{\epsilon }{k}\overline{\rho u_i}$$E11

$$C_1^{\text{'}}=3.2$$

$${\phi }_{i\rho }^2=0,8\overline{\rho u_k}{U}_{i,k}-0,2\overline{\rho u_k}{U}_{k,i}$$E12

The contributions 3, ${\phi }_{\mathit{ij}}^3$, and ${\phi }_{\mathit{i\rho }}^3$ are terms due to buoyancy. Lumley et al. [29] and Zeman et al. [30] retained only the classical model for this contribution.

$${\phi }_{\mathit{ij}}^3=-C_3\left({\beta }_j\overline{u_i\rho }+{\beta }_i\overline{u_j\rho }-\frac{2}{3}{\beta }_l\overline{u_l\rho }{\delta }_{\mathit{ij}}\right)$$E13

$${\phi }_{\mathit{i\rho }}^3=-C_{3\rho }{\beta }_i\overline{{\rho }^2}$$E14

$$C_3=0,5,C_{3\rho }=0,5$$

With the reason of acquisition, the dimensionless equations, basic Eqs. (5)–(8) are deposited in dimensionless form by laying on the dimensionless time St = τ, the kinematic components $b_{\mathit{ij}}=\frac{\overline{u_i{u}_j}}{2K}-\frac{1}{3}{\delta }_{\mathit{ij}}$, ε/KS and the scalar component $\eta =\frac{1}{2}\frac{g}{{\rho }_0{S}_{\rho }}\frac{\overline{{\rho }^2}}{k}=\frac{k_{\rho }}{k}$

$$\frac{{\mathit{db}}_{12}}{\mathit{d\tau }}=-\frac{\sqrt{2}}{2}\left(b_{22}+\frac{1}{3}\right)-\frac{\sqrt{2}}{2}{b}_{23}+\frac{{\phi }_{12}}{2\mathit{kS}}+2\frac{\sqrt{2}}{2}{b}_{12}^2+2\frac{\sqrt{2}}{2}{b}_{12}{b}_{13}+F_3{b}_{12}+\left(\frac{\epsilon }{\mathit{kS}}\right)b_{12}$$E15

$$\begin{array}{l}\frac{{\mathit{db}}_{22}}{\mathit{d\tau }}=\frac{{\phi }_{22}}{2\mathit{kS}}-\frac{\epsilon }{3\mathit{kS}}+\left(b_{22}+\frac{1}{3}\right)\left(2\frac{\sqrt{2}}{2}{b}_{12}+2\frac{\sqrt{2}}{2}{b}_{13}+F_3+\frac{\epsilon }{\mathit{kS}}\right)\\ \end{array}$$E16

$$\frac{\mathit{d\eta }}{\mathit{d\tau }}=-F_3+\eta \left(2\frac{\sqrt{2}}{2}{b}_{12}+2\frac{\sqrt{2}}{2}{b}_{13}+F_3-\frac{\epsilon }{\mathit{kS}}\right)$$E17

$$\begin{array}{l}\frac{d}{\mathit{d\tau }}\left(\frac{\epsilon }{\mathit{kS}}\right)=-2C_{\epsilon 1}\left(\frac{\epsilon }{\mathit{kS}}\right)b_{13}+\left(1-C_{\epsilon 2}\right){\left(\frac{\epsilon }{\mathit{kS}}\right)}^2-C_{\epsilon 1}\left(1-C_{\epsilon 3}\right)\left(\frac{\epsilon }{\mathit{kS}}\right)F_3\\ +\left(\frac{\epsilon }{\mathit{kS}}\right)\left(2\frac{\sqrt{2}}{2}{b}_{12}+2\frac{\sqrt{2}}{2}{b}_{13}+F_3\right)\end{array}$$E18

where $F_i=\frac{g}{{\rho }_0}\frac{\overline{u_i\rho }}{\mathit{kS}}$ is the non-dimensional component of the turbulent scalar flow.

A numerical integration of the differential equations is done. Obtained results are discussed in the following sections.

3. Numerical integration and results

3.1 Validation of the present code based on coupled model of SSG-LRR

In order to make sure on the precision of coupled model of SSG-LRR employed for the computations of the considered problem, the present code taped with Fortran is validated with the published findings of Jacobitz et al. [21, 26, 27] related to direct numerical simulations of the stratified turbulence under shear. However, a comparative analysis of predictions via SSG-LRR model and results of DNSJ is performed (Figures 1–3).

Figure 1.

Time evolution of the principal component of anisotropy b₁₂ for low stratification with Fr = 1.11, θ = π/4, SK/ε = 2.

Figure 2.

Time evolution of the normalized turbulence dissipation ε/KS for low stratification with Fr = 1.11, θ = π/4, SK/ε = 2.

Figure 3.

Time evolution of the normalized turbulence dissipation ε/KS for high stratification with Fr = 0.35, θ = π/4, SK/ε = 2.

A fourth-order Runge–Kutta method is used for integrating a closed system of nonlinear dimensionless differential equations. The initial conditions are taken from the same initial conditions of result of DNSJ [21, 26, 27].

$${\left(b_{\mathit{ij}}\right)}_0=0,0,{\left(\frac{\epsilon }{\mathit{KS}}\right)}_0=0,5,{\left(F_i\right)}_0=0,0,{\left(\eta \right)}_0=0,0$$

An inclined θ = π/4 is studied for different values of Shear number SK/ε ranging from 2 to 20 and with a constant Froude number Fr equal to 0.35 for high stratification and equal to 1.11 for low stratification. Here θ is the inclination angle between vertical stratification and shear.

A primary focus of research has been the parameterization of the Froude number Fr associated with stationary, where Fr is defined by: $\mathit{Fr}=\sqrt{\frac{{0.5}^2}{\mathit{Ri}}}$.

The gradient Richardson number Ri is defined by: $R_i=\frac{g}{{\rho }_0}\frac{S_{\rho }}{S^2}=\frac{N^2}{S^2}$.

$N^2$ is the frequency scale relevant to density stratification, and S is the mean vertical shear.

The Shear number SK/ε is the ratio of the turbulence time scale K/ε to the shear time scale 1/S.

However, to our knowledge, no previous work is interested in the prediction of equilibrium states by the coupled between SSG and LRR second-order models. This creates the aim of this part of our work. The numerical study is to carried out a comparison between results related to the case of an homogeneous and a stratified (Fr = 0.35 for high stratification and Fr = 1.11 for low stratification) turbulence under inclined shear θ = π/4 for various Shear number (2 ≤ SK/ε ≤ 20) via SSG-LRR model and findings of those of direct numerical simulation of Jacobitz (DNSJ) [21, 26, 27]. Obtained results via SSG-LRR model and DNSJ [21, 26, 27] are presented in Figures 1–3. Figure 1 shows time evolution of principal component of anisotropy b₁₂ according to the SSG-LRR model for a low stratification related to Fr = 1.11 for an intermediate band of τ, in the range [0, 50]. The SSG-LRR model shows a good agreement with the asymptotic value of the component of anisotropy (b₁₂)_∞ of DNSJ [21, 26, 27] when a non-dimensional time τ is greater than 20. Using the same model, results of the problem of comparative analysis of coupled second-order models on shear and Richardson numbers effects on homogeneous and stratified turbulence [31] are compared to find numerical approaches with RANS [6]. It has shown a good agreement between different predictions of components of anisotropy b₁₂. Consequently, it can be asserted that these favorable comparisons corroborate the coupled model of SSG-LRR, which can produce reliable numerical findings of turbulent flows.

Figures 2 and 3 show time evolution of the normalized turbulence dissipation ε/KS according to SSG-LRR model compared with values of DNSJ for initial value of shear number SK/ε = 2 and for a constant Froude number equal to 1.11 (low stratification) and Fr = 0.35 (high stratification), respectively. The normalized dissipation ε/KS is affected only slightly by the direction of shear. For high and low stratification, the normalized dissipation ε/KS has a tendency to predict the asymptotic state at long-time evolution for non-dimensional time τ ≥ 20. For the case of inclined shear (θ = π/4), an excellent agreement between predictions of the SSG-LRR model and values of DNSJ [21, 26, 27] is detected for higher time (τ ≥ 30).

3.2 Effect of shear number

In this section, the influence of shear number (SK/ε) on the evolution of homogeneous and stratified turbulence is addressed. Two cases of stratified turbulence have been performed; a low stratification Fr = 1.29 and a high stratification Fr = 0.35.

3.2.1 Case of low stratification (Fr = 1.29)

Numerical study is presented for various values of shear number SK/ε equal to 2, 6, 14, and 20 in order to simulate the problem of homogeneous and stratified turbulence (Fr = 1.29). A numerical code is taped with Fortran language using a fourth-order Range-Kutta method. All simulations of the present study are approved for initial conditions of Jacobitz [21, 26, 27].

$${\left(b_{\mathit{ij}}\right)}_0=0,0,{\left(\frac{\epsilon }{\mathit{KS}}\right)}_0=0,5,{\left(F_i\right)}_0=0,0,{\left(\eta \right)}_0=0,0$$

In Figure 4, the evolution of the principal component of anisotropy b₁₂ is shown for a non-dimensional time 0 ≤ τ ≤ 40 with different values of shear number (SK/ε = 2 to 20) for a fixed Froude number at 1.29. The simulations of b₁₂ as initial spectra show a large initial decay of b₁₂ during the transient phase, and finally, the coupled model SSG-LRR confirms the existence of an asymptotic equilibrium states beyond a non-dimensional time τ = 30 for various values of shear number.

Figure 4.

Time evolution of the principal component of anisotropy b₁₂ for several values of SK/ε with F r = 1.29 and θ = π/4.

The evolution of the normalized turbulence dissipation ε/KS is shown in Figure 5. It appears that the magnitude of ε/KS decreases with increasing shear number (from 2 to 20). Again, an asymptotically constant value of ε/KS is observed. SSG-LRR model confirms the existence of an asymptotic equilibrium states for a long time τ. The time evolution of the thermal parameter (density flux) $\overline{\rho u_1}$as a function of the shear number is illustrated in Figure 6. It is found that the $\overline{\rho u_1}$ is affected by the variation of shear number. The SSG-LRR model confirms the existence of asymptotic equilibrium states for the component $\overline{\rho u_1}$ beyond non-dimensional time τ = 25. Furthermore, the figure shows that the parameter ${\left(\overline{\rho u_1}\right)}_{\infty }$ increases as increasing _SK/ε from 2 to 20.

Figure 5.

Time evolution of the normalized turbulence dissipation ε/KS for several values of SK/ε with Fr = 1.29 and θ = π/4.

Figure 6.

Time evolution of the density flux $\overline{\rho u_1}$ for several values of SK/ε with Fr = 1.29 and θ = π/4.

3.2.2 Case of high stratification (Fr = 0.35)

In this section, the results of the principal component of anisotropy b₁₂, the normalized turbulence dissipation ε/KS, and the density flux $\overline{\rho u_1}$with different Shear number SK/ε equal to 2, 6, 14, and 20 and for a constant Froude number equal to 0.35 (high stratification) are presented in Figures 7, 8, and 9, respectively. All simulation demonstrates that the b₁₂ and ε/KS show an initial decay due to the isotropic initial conditions. After this initial phase, the principal component of anisotropy and the normalized turbulence dissipation grow for small and decay for large values of the shear number. Contrary, the density flux decays for small and grows for large values of the shear number. Figures 7 and 8 depict the evolution of the principal component of anisotropy b₁₂ and the normalized turbulence dissipation ε/KS, respectively. Obtained by SSG-LRR as a function of the normalized time τ for inclined shear associated to θ = π/4. SSG-LRR confirms the existence of an asymptotic equilibrium states for b₁₂ and for ε/KS beyond a non-dimensional time τ = 25 for diverse values of shear number. Also, SSG-LRR indicates that asymptotic equilibrium state (b₁₂)_∞ and (ε/KS)_∞ increase as decreasing shear number from 20 to 2. In Figure 9 is shown evolution of the density flux $\overline{u_1\rho }$ versus the shear number using SSG-LRR model with Fr = 0.35. For SSG-LRR model, the density flux $\overline{u_1\rho }$ presents a slight decrease with the decrease of shear number from 20 to 2.

Figure 7.

Time evolution of the principal component of anisotropy b₁₂ for several values of SK/ε with Fr = 0.35 and θ = π/4.

Figure 8.

Time evolution of the normalized turbulence dissipation ε/KS for several values of SK/ε with Fr = 0.35 and θ = π/4.

Figure 9.

Time evolution of the density flux $\overline{\rho u_1}$ for several values of SK/ε with Fr = 0.35 and θ = π/4.

4. Conclusion

The effect of shear number on dynamic and thermal components of an homogeneous and stratified turbulence under inclined shear is investigated using coupled second-order model (SSG-LRR). Firstly, a comparison between results obtained with SSG-LRR model and those of DNSJ [21, 26, 27] is performed. A good agreement between predictions of second-order model of SSG-LRR and those of DNSJ is detected for a higher dimensionless time τ. Beyond τ ≥ 25, an excellent agreement between results of the SSG-CL model and those of DNSJ is observed for the principal component of anisotropy b₁₂ and the normalized turbulence dissipation ε/KS for high and low stratification. Secondly, for high and low stratification Fr = 0.35 and 1.29, respectively, it is found that coupled model of SSG-LRR confirms the existence of an asymptotic equilibrium states of different parameters (b₁₂, ε/KS, and $\overline{\rho u_1}$) for various shear number (2, 6, 14, and 20). Hence, the coupling between the SSG model for the kinematic field and the LRR model for the scalars field proves a significant contribution to predict asymptotic equilibrium states of turbulent thermal and dynamic characteristics. This comparative investigation proves that the coupling between second-order models of SSG and LRR model can present a great solution to modelize homogeneous and stratified turbulence flows under inclined shear. Finally, it can be finished by that coupled second-order models of SSG-LRR present an important contribution for the modeling of turbulence for stratified shear flows and can help to better understand and realized a complex turbulent phenomena of geophysical flows, which propagate in the atmosphere.

Conflict of interest

The authors declare that they have no conflict of interest.

Nomenclature

b	Anisotropic tensor of Reynolds
Cp	Specific heat at constant pressure
g	Acceleration due to gravity
P	Pressure
p	Fluctuation of the pressure
Fr	Froude number
Ri	Richardson number
SK/ε	Shear number
S	Uniform mean shear
Sρ	Uniform stable stratification
K	Turbulent kinetic energy
T,i¯	Gradient of the scalar
t	Time
τ	Non-dimensional time
ui	i-th component of the fluctuating velocity
Ui	i-th component of mean velocity
uiuj¯	Reynolds stress tensor
uiρ¯	Turbulent flux of the scalar
Up,q¯	Gradient of mean speed
xi	Component of an orthonormal Cartesian coordinate system
μ	Dynamic viscosity
λ	Viscosity ratio
ν	Kinematic viscosity
α	Thermal diffusivity
τ	Non-dimensional time
δij	Kronecker symbol
ρ	Fluctuation of the scalar
ρ0	Density of reference
ρ2¯	Density variance
ε	Terms of dissipation of turbulent kinetic energy
ερρ	Terms of dissipation of variance of the scalar
εij	Terms of dissipation of tensor of Reynolds
εiρ	Terms of dissipation of the scalar flux turbulent
φij	Terms of pressure-strain correlation
φiρ	Terms of pressure- scalar gradient correlation
θ	Angle between the shear and the vertical gradient of stratification
SSG	Speziale, Sarkar, and Gatski model
LRR	Launder, Reece, and Rodi model
DNSJ	Direct numerical simulation of Jacobitz