**1. Introduction**

18 Will-be-set-by-IN-TECH

[33] K. Trulsen and K. B. Dysthe. Action of windstress and breaking on the evolution of a wavetrain. In M. L. Banner and R. H. J. Grimshaw, editors, *Breaking Waves*, pages

[34] T. Hara and C. C. Mei. Frequency downshift in narrowbanded surface waves under the

[35] C. Skandrani, C. Kharif, and J. Poitevin. On benjamin feir instability and evolution of a nonlinear wave with finite amplitude sidebands. *Cont. Math.*, 200:157–171, 1996. [36] E. Lo and C. C. Mei. A numerical study of water wave modulation based on a higher

order nonlinear schrödinger equation. *J. Fluid Mech.*, 150:395–416, 1985.

243–249. Springer Verlag, 1992.

influence of wind. *J. Fluid Mech.*, 230:429–477, 1991.

This chapter is mainly in the area of the use of Rapid Distortion Theory (RDT) to clarify and to well better increase our understanding of the physics of the compressible turbulent flows. This theory is a computationally viable option for examining linear compressible flow physics in the absence of inertial effects. In this linear limit, the statistical evolution of incompressible homogeneous turbulence can be described completely in terms of closed spectral covariance equations (see Refs. (Hunt, 1990 & Savill, 1987) and references therein). Many papers in literature deal with homogeneous compressible turbulence and RDT solution (Cambon et al., 1993; Coleman & Mansour, 1991; Blaisdell et al., 1993, 1996; Durbin & Zeman, 1992; Jacquin et al., 1993; Livescu & Madnia, 2004; Riahi et al., 2007; Riahi, 2008; Riahi & Lili, 2011; Sarkar, 1995; Simone, 1995; Simone et al., 1997). These studies have yielded very valuable physical insight and closure model suggestions. In all the above works, the fluctuation equations are solved directly to infer turbulence physics. For the case of viscous compressible homogeneous shear flow in the RDT limit no analytical solutions are known. Simone et al. (1997) performed RDT simulations of homogeneous shear flow and showed that the role of the distortion Mach number, *Md*, on the time variation of the turbulent kinetic energy is consistent with that found in the direct numerical simulation (DNS) results. In this chapter, numerical solutions to the RDT equations for the special case of mean shear is described completely by finding numerical solutions obtained by solving linear double point correlations equations. Numerical integration of these equations is carried out using a second-order simple and accurate scheme (Riahi & Lili, 2011). Indeed, this numerical method is proved more stable and faster than the previous one which use linear transfer matrix (Riahi et al., 2007 & Riahi, 2008) and allows in particular to obtain accurately the asymptotic behavior of the turbulence parameters (for large values of the non-dimensional times *St*) characteristic of equilibrium states. To perform this work, RDT code solving

© 2012 Riahi and Lili, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

linearized equations for compressible homogeneous shear flows is validated by comparing RDT results to those of direct numerical simulation (DNS) of Simone et al. (1997) and Sarkar (1995) for various values of initial gradient Mach number *Mg*0 (Riahi & Lili, 2011).

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 395

2 2

(1)

) (<sup>3</sup>

(4)

*<sup>P</sup> <sup>ρ</sup> <sup>n</sup>*<sup>=</sup> *<sup>γ</sup>*

2

*j*

*x*

 , (2)

, *i,i*

<sup>1</sup> , 3 3 *j i i*

*<sup>p</sup> <sup>u</sup> γ P* 

*j i ij*

where *γ* is the ratio of specific heats, *ρ* is the mean density and *ν* is the kinematic

The dot superscript denotes a substantial derivative along the mean flow trajectories related

associated to isentropic fluctuations hypothesis *s* 0 which is translated by

*i,i <sup>ρ</sup> <sup>u</sup> ρ* 

*γ P ρ* 

and leads to equation (1). So, the isentropic hypothesis *s* 0 simplifies governing equations by keeping the variables *ui* and *p* and eliminating *ρ* . We can obviously discard the hypothesis *s* 0 by writing the (exact) energy equation. The linearized derived equation within the framework of RDT can be written as pressure fluctuation equation which contains some viscous terms (Livescu et al., 2004; see equation (8)). Such an equation is compatible with equation (2) which contains also viscous terms and represents the linearized equation of momentum. Concerning now the validity of isentropic hypothesis

*<sup>s</sup>* <sup>0</sup> , Blaisdell et al. (1993) introduced a polytropic coefficient *n* defined by, *<sup>p</sup> <sup>ρ</sup> <sup>n</sup>*

corresponding obviously to isentropic fluctuations. These authors performed direct numerical simulations (DNS) related to homogeneous compressible sheared turbulence with various initial conditions. They calculated temporal variation of an average polytropic

times), the evolution of *n* with *St* depends on initial conditions (and in particular of initial entropy fluctuations). However, for all of their simulations and for large values of *St*, *n* tends

. During the period of the turbulence establishment (for early

*x ρ x xx*

*U u p ν ν u*

 

*i j*

*u u*

Equation (1) is derived from continuity equation

*<sup>p</sup> <sup>ρ</sup>*

viscosity.

coefficient

*n*

*p / P*

2 2 2

1

*ρ / ρ*

2 2

to mean field velocity *Ui* .

A study of compressibility effects on structure and evolution of a sheared homogeneous turbulent flow is carried out using this theory (RDT). An analysis of the behavior of different terms appearing in the turbulent kinetic energy and the Reynolds stress equations permit to well identify compressibility effects which allow us to analyze performance of the compressible model of Fujiwara and Arakawa concerning the pressure-dilatation correlation (Riahi et al., 2007). The evaluation of this model stays in the field of RDT validity (Riahi & Lili, 2011).

Equilibrium states of homogeneous compressible turbulence subjected to rapid shear can be studied using rapid distortion theory (RDT) for large values of *St* (*St* > 10) in particular for large values of the initial gradient Mach number *Mg*0 describing various regimes of flow. In fact, the study of the behavior of the non-dimensional turbulent kinetic energy *q2(t)/q2(0)* (with <sup>2</sup> *<sup>q</sup> i i u u* ) allows to check relevance of an incompressible regime for low values of initial gradient Mach number, of an intermediate regime for moderate values of *Mg*0 and of a compressible regime for high values of *Mg*<sup>0</sup> (Riahi et al., 2007). The pressure-released regime is related to infinite values of *Mg*0 (Riahi & Lili, 2011). The gradient Mach number *Mg* appears naturally when scaling the (linearized) RDT equations for homogeneous compressible turbulence (see equations (A.10), (A.11), (A.12) and (A.13) in appendix (Riahi et al., 2007)). This parameter can be viewed as the ratio of an acoustic time *<sup>a</sup> l τ <sup>a</sup>* for a large eddy to the mean

flow time scale <sup>1</sup> *d τ <sup>S</sup>* : *Mg*<sup>=</sup> *<sup>a</sup> d τ Sl <sup>τ</sup> <sup>a</sup>* , where *l* is an integral lengthscale and *a* is the mean sound

speed. The lengthscale of energetic structure is expressed by <sup>3</sup> *<sup>q</sup> <sup>l</sup> <sup>ε</sup>* , where *ε* is the rate of turbulent kinetic energy dissipation. Sarkar (1995) also used *Sl a* (which he referred to as 'gradient Mach number' *Mg*), to quantify compressibility effects for homogeneous shear flow. In the case of shear flow, *l* is chosen to be the integral lengthscale of the streamwise fluctuating velocity in the shearing direction x2. Another Mach number relevant to homogeneous shear flow and that characterize the effects of compressibility on turbulence is the turbulent Mach number *Mt* = *u/a* based on a characteristic fluctuation velocity *u* and mean sound speed *a*.

## **2. RDT equations for compressible homogeneous turbulence**

The flow to be considered is a homogeneous, compressible turbulent shear flow where we retain the same RDT equations adopted by Simone (1995), Simone et al. (1997), Riahi et al. (2007) and Riahi (2008). The linearized equations of continuity, momentum and entropy controlling the fluctuating of velocity *ui* and pressure *p* lead to general RDT equations:

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 395

$$\left(\frac{\dot{p}}{\gamma'\overline{P}}\right) = -u\_{i,i'} \tag{1}$$

$$\dot{u}\_i + u\_j \frac{\partial \overline{II}\_i}{\partial \mathbf{x}\_j} = -\frac{1}{\overline{\rho}} \frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\nu}{3} \frac{\partial^2 u\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} + \frac{\nu}{3} \frac{\partial^2 u\_i}{\partial \mathbf{x}\_j^2} \,\_{\mathbf{x}'} \tag{2}$$

where *γ* is the ratio of specific heats, *ρ* is the mean density and *ν* is the kinematic viscosity.

The dot superscript denotes a substantial derivative along the mean flow trajectories related to mean field velocity *Ui* .

Equation (1) is derived from continuity equation

394 Numerical Simulation – From Theory to Industry

Lili, 2011).

flow time scale <sup>1</sup>

*d τ*

linearized equations for compressible homogeneous shear flows is validated by comparing RDT results to those of direct numerical simulation (DNS) of Simone et al. (1997) and Sarkar

A study of compressibility effects on structure and evolution of a sheared homogeneous turbulent flow is carried out using this theory (RDT). An analysis of the behavior of different terms appearing in the turbulent kinetic energy and the Reynolds stress equations permit to well identify compressibility effects which allow us to analyze performance of the compressible model of Fujiwara and Arakawa concerning the pressure-dilatation correlation (Riahi et al., 2007). The evaluation of this model stays in the field of RDT validity (Riahi &

Equilibrium states of homogeneous compressible turbulence subjected to rapid shear can be studied using rapid distortion theory (RDT) for large values of *St* (*St* > 10) in particular for large values of the initial gradient Mach number *Mg*0 describing various regimes of flow. In fact, the study of the behavior of the non-dimensional turbulent kinetic energy *q2(t)/q2(0)* (with <sup>2</sup> *<sup>q</sup> i i u u* ) allows to check relevance of an incompressible regime for low values of initial gradient Mach number, of an intermediate regime for moderate values of *Mg*0 and of a compressible regime for high values of *Mg*<sup>0</sup> (Riahi et al., 2007). The pressure-released regime is related to infinite values of *Mg*0 (Riahi & Lili, 2011). The gradient Mach number *Mg* appears naturally when scaling the (linearized) RDT equations for homogeneous compressible turbulence (see equations (A.10), (A.11), (A.12) and (A.13) in appendix (Riahi et al., 2007)). This

'gradient Mach number' *Mg*), to quantify compressibility effects for homogeneous shear flow. In the case of shear flow, *l* is chosen to be the integral lengthscale of the streamwise fluctuating velocity in the shearing direction x2. Another Mach number relevant to homogeneous shear flow and that characterize the effects of compressibility on turbulence is the turbulent Mach number *Mt* = *u/a* based on a characteristic fluctuation velocity *u* and mean sound speed *a*.

The flow to be considered is a homogeneous, compressible turbulent shear flow where we retain the same RDT equations adopted by Simone (1995), Simone et al. (1997), Riahi et al. (2007) and Riahi (2008). The linearized equations of continuity, momentum and entropy

controlling the fluctuating of velocity *ui* and pressure *p* lead to general RDT equations:

*l τ*

*<sup>τ</sup> <sup>a</sup>* , where *l* is an integral lengthscale and *a* is the mean sound

<sup>3</sup> *<sup>q</sup> <sup>l</sup>*

*a*

*<sup>a</sup>* for a large eddy to the mean

*<sup>ε</sup>* , where *ε* is the rate of

(which he referred to as

(1995) for various values of initial gradient Mach number *Mg*0 (Riahi & Lili, 2011).

parameter can be viewed as the ratio of an acoustic time *<sup>a</sup>*

*d τ Sl*

speed. The lengthscale of energetic structure is expressed by

turbulent kinetic energy dissipation. Sarkar (1995) also used *Sl*

**2. RDT equations for compressible homogeneous turbulence** 

*<sup>S</sup>* : *Mg*<sup>=</sup> *<sup>a</sup>*

$$
\left(\frac{\dot{\rho}}{\overline{\rho}}\right) = -\mu\_{i,i} \tag{3}
$$

associated to isentropic fluctuations hypothesis *s* 0 which is translated by

$$
\left(\frac{\dot{p}}{\overline{P}}\right) = \mathcal{V}\left(\frac{\dot{\rho}}{\overline{\rho}}\right) \tag{4}
$$

and leads to equation (1). So, the isentropic hypothesis *s* 0 simplifies governing equations by keeping the variables *ui* and *p* and eliminating *ρ* . We can obviously discard the hypothesis *s* 0 by writing the (exact) energy equation. The linearized derived equation within the framework of RDT can be written as pressure fluctuation equation which contains some viscous terms (Livescu et al., 2004; see equation (8)). Such an equation is compatible with equation (2) which contains also viscous terms and represents the linearized equation of momentum. Concerning now the validity of isentropic hypothesis

$$\text{As } \dot{s} = 0 \text{, Blaisdell et al. (1993) introduced a polytropic coefficient } n \text{ defined by, } \frac{p}{\overline{P}} = n \frac{\rho}{\overline{\rho}} \quad n = \overline{\rho} - \overline{\rho}$$

corresponding obviously to isentropic fluctuations. These authors performed direct numerical simulations (DNS) related to homogeneous compressible sheared turbulence with various initial conditions. They calculated temporal variation of an average polytropic

coefficient 1 2 2 2 2 2 *p / P n ρ / ρ* . During the period of the turbulence establishment (for early

times), the evolution of *n* with *St* depends on initial conditions (and in particular of initial entropy fluctuations). However, for all of their simulations and for large values of *St*, *n* tends toward an equilibrium value independent of initial conditions and slightly less than *γ* 1 4*.* . Blaisdell et al. (1993) concluded that thermodynamic variables "follow a nearly isentropic process" for large values of *St* and for compressible sheared turbulence. As a conclusion, we retain that with isentropic fluctuations hypothesis, we can obtained a good prediction of equilibrium states *(St )* for compressible homogeneous sheared turbulence.

After these remarks, we consider the Fourier transform (denoted here by the symbol *" "* ˆ ) of various terms in equations (1) and (2) which leads to the following equations expressed in the spectral space and in the case of turbulent shear flow:

$$\frac{\dot{\overline{p}}}{\gamma \overline{P}} = -\hat{u}\_{i,i'} \tag{5}$$

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 397

<sup>12</sup> <sup>2</sup> 13 14 <sup>23</sup> <sup>33</sup> <sup>2</sup> , <sup>7</sup> 2( ) <sup>3</sup> *<sup>d</sup><sup>Φ</sup> Sk k' Sk k Sk Sk k Φ ν<sup>k</sup> <sup>Φ</sup> akΦΦ Φ dt k k kk' k*

(12)

(14)

(15)

*dt <sup>k</sup> k'* (16)

(17)

(18)

(19)

 *f* (*t +Δt*) *= f* (*t*) *+ Δt f'*(*t*) *+ Δt2 f''*(*t*)*/2*, (21)

*dt* (20)

 .

(13)

<sup>2</sup> <sup>12</sup> <sup>13</sup> <sup>22</sup> <sup>23</sup> (2 ) , *<sup>d</sup><sup>Φ</sup> Sk k Sk Sk Sk k νk ΦΦΦ Φ*

*dt k k' k kk'*

14 2 3 2 3 13 14 <sup>24</sup> <sup>34</sup> , *<sup>d</sup><sup>Φ</sup> Sk Sk k akΦ ν<sup>k</sup> ΦΦ Φ dt k kk'*

<sup>2</sup> <sup>22</sup> <sup>23</sup> (2 2 ) 2 , *<sup>d</sup><sup>Φ</sup> Sk k Sk νk Φ Φ*

<sup>2</sup> <sup>22</sup> 23 24 <sup>33</sup> , <sup>7</sup> <sup>2</sup>

2 2 <sup>23</sup> 33 34 , <sup>4</sup> 4 2( ) 2 <sup>3</sup>

2 2 24 33 34 44 , <sup>4</sup> 2 () <sup>3</sup>

*dt k k'*

3 *<sup>d</sup><sup>Φ</sup> Sk k' Sk Φ ν<sup>k</sup> <sup>Φ</sup> akΦ Φ*

24 2 1 2 1 <sup>23</sup> <sup>2</sup> <sup>24</sup> <sup>34</sup> ( ) , *<sup>d</sup><sup>Φ</sup> Sk k Sk akΦ ν<sup>k</sup> Φ Φ dt k k'*

*<sup>d</sup><sup>Φ</sup> Sk k' Sk k Φ ν<sup>k</sup> <sup>Φ</sup> ak<sup>Φ</sup>*

*<sup>d</sup><sup>Φ</sup> Sk k' Sk k <sup>Φ</sup> akΦ ν<sup>k</sup> <sup>Φ</sup> ak<sup>Φ</sup>*

<sup>34</sup> <sup>2</sup> , *<sup>d</sup><sup>Φ</sup> ak<sup>Φ</sup>*

Numerical integration of these equations ((11)-(20)) is carried out using a simple second-

where the derivatives *f'*(*t*)and *f''*(*t*) are expressed exactly from evolution equations (11)-

In this section, results are presented and used to verify the validity of the RDT code. For this, tests of this code have been performed by comparing RDT results with those of direct numerical simulation (DNS) of Simone et al. (1997) for various values of initial gradient Mach number *Mg*0 which describe different regimes of the flow (see below). Comparisons

33 1 2 1 2

34 1 2 1 2

1 3 *k' k k* and 123 *k , k , k* are the components of the wave vector *k*

*dt k k*

44

where 2 2

order accurate scheme:

(20) and *Δt* is the time-step size.

**3. RDTcode validation** 

**3.1. Introduction** 

*dt k k*

<sup>12</sup> <sup>2</sup> 1 2 <sup>1</sup> 3 23

<sup>22</sup> <sup>2</sup> 1 2 <sup>1</sup>

<sup>23</sup> 1 1 <sup>2</sup>

<sup>13</sup> <sup>1</sup> <sup>2</sup> 1 2 <sup>3</sup> 2 3

$$
\dot{\hat{u}}\_i + \lambda\_{ij}\hat{u}\_j + \frac{\nu}{3}k\_i \ k\_j \hat{u}\_j + \frac{\nu}{3}k^2 \ \hat{u}\_i = -\text{I}k\_i \frac{\hat{p}}{\overline{\rho}}\tag{6}
$$

where *ij λ* is the mean velocity gradient defined by:

$$
\lambda\_{\vec{\eta}} = \mathcal{S} \delta\_{i1} \delta\_{j1} \tag{7}
$$

and *I2* = - 1.

In the Fourier space, velocity field is decomposed on solenoidal and dilatational contributions by adopting a local reference mark of Craya (Cambon et al., 1993):

$$
\hat{\boldsymbol{\mu}}\_{\text{i}}(\vec{\boldsymbol{k}},t) = \hat{\boldsymbol{\phi}}^{1}(\vec{\boldsymbol{k}},t) \, \boldsymbol{e}\_{\text{i}}^{1}(\vec{\boldsymbol{k}}) + \hat{\boldsymbol{\phi}}^{2}(\vec{\boldsymbol{k}},t) \, \boldsymbol{e}\_{\text{i}}^{2}(\vec{\boldsymbol{k}}) + \hat{\boldsymbol{\phi}}^{3}(\vec{\boldsymbol{k}},t) \, \boldsymbol{e}\_{\text{i}}^{3}(\vec{\boldsymbol{k}}) \, \tag{8}
$$

where 1 2 ˆ ˆ *(k,t) (k,t)* and are the solenoidal contributions and <sup>3</sup> ˆ *(k,t)* is the dilatational contribution.

By separating solenoidal and dilatational contributions, we introduce the spectrums of doubles correlations:

$$\mathcal{O}\_{\vec{\boldsymbol{\eta}}}(\vec{k},t) = \frac{1}{2} \left[ \hat{\boldsymbol{\phi}}^{\boldsymbol{i}^{\ast}}(\vec{k},t) \hat{\boldsymbol{\phi}}^{\boldsymbol{j}}(\vec{k},t) + \hat{\boldsymbol{\phi}}^{\boldsymbol{i}}(\vec{k},t) \hat{\boldsymbol{\phi}}^{\boldsymbol{j}^{\ast}}(\vec{k},t) \right] \quad \boldsymbol{i},\boldsymbol{j} = 1..4,\tag{9}$$

where <sup>4</sup> *k,t* ˆ ( ) is related to the pressure and takes the following form:

$$
\hat{\phi}^4(\vec{k}, t) = I \frac{\hat{p}}{\overline{\rho}a} \tag{10}
$$

We then write evolution equations of these doubles correlations:

$$\frac{d\Phi\_{11}}{dt} = -2\nu k^2 \Phi\_{11} - 2\frac{Sk\_3}{k} \Phi\_{12} + 2\frac{Sk\_2k\_3}{kk'} \Phi\_{13'} \tag{11}$$

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 397

$$\frac{d\Phi\_{12}}{dt} = (-2\nu k^2 + \frac{Sk\_1k\_2}{k^2})\Phi\_{12} - \frac{Sk\_1}{k'}\Phi\_{13} - \frac{Sk\_3}{k}\Phi\_{22} + \frac{Sk\_2k\_3}{kk'}\Phi\_{23},\tag{12}$$

$$\frac{d\varPhi\_{13}}{dt} = 2\frac{Sk\_1k'}{k^2}\varPhi\_{12} - (\frac{7}{3}\nu k^2 + \frac{Sk\_1k\_2}{k^2})\varPhi\_{13} - ak\varPhi\_{14} - \frac{Sk\_3}{k}\varPhi\_{23} + \frac{Sk\_2k\_3}{kk'}\varPhi\_{33}.\tag{13}$$

$$\frac{d\Phi\_{14}}{dt} = ak\Phi\_{13} - \nu k^2 \Phi\_{14} - \frac{Sk\_3}{k} \Phi\_{24} + \frac{Sk\_2k\_3}{kk'} \Phi\_{34},\tag{14}$$

$$\frac{d\Phi\_{22}}{dt} = (-2\nu k^2 + 2\frac{\mathrm{Sk}\_1 \mathrm{k}\_2}{k^2})\Phi\_{22} - 2\frac{\mathrm{Sk}\_1}{k'}\Phi\_{23'} \tag{15}$$

$$\frac{d\Phi\_{23}}{dt} = 2\frac{Sk\_1k'}{k^2}\Phi\_{22} - \frac{7}{3}\nu k^2\Phi\_{23} - ak\Phi\_{24} - \frac{Sk\_1}{k'}\Phi\_{33},\tag{16}$$

$$\frac{d\Phi\_{24}}{dt} = ak\Phi\_{23} + (-\nu k^2 + \frac{Sk\_1k\_2}{k^2})\Phi\_{24} - \frac{Sk\_1}{k'}\Phi\_{34}.\tag{17}$$

$$\frac{d\varPhi\_{33}}{dt} = 4\frac{Sk\_1k'}{k^2}\varPhi\_{23} - 2(\frac{4}{3}\nu k^2 + \frac{Sk\_1k\_2}{k^2})\varPhi\_{33} - 2ak\varPhi\_{34},\tag{18}$$

$$\frac{d\varPhi\_{34}}{dt} = 2\frac{Sk\_1k'}{k^2}\varPhi\_{24} + ak\varPhi\_{33} - (\frac{4}{3}\nu k^2 + \frac{Sk\_1k\_2}{k^2})\varPhi\_{34} - ak\varPhi\_{44}.\tag{19}$$

$$\frac{d\Phi\_{44}}{dt} = 2ak\Phi\_{34} \,\text{\AA} \tag{20}$$

where 2 2 1 3 *k' k k* and 123 *k , k , k* are the components of the wave vector *k* .

Numerical integration of these equations ((11)-(20)) is carried out using a simple secondorder accurate scheme:

$$f\left(t+\Delta t\right) = f\left(t\right) + \Delta t \, f'(t) + \Delta t^2 \, f'(t)/2,\tag{21}$$

where the derivatives *f'*(*t*)and *f''*(*t*) are expressed exactly from evolution equations (11)- (20) and *Δt* is the time-step size.

### **3. RDTcode validation**

#### **3.1. Introduction**

396 Numerical Simulation – From Theory to Industry

toward an equilibrium value independent of initial conditions and slightly less than *γ* 1 4*.* . Blaisdell et al. (1993) concluded that thermodynamic variables "follow a nearly isentropic process" for large values of *St* and for compressible sheared turbulence. As a conclusion, we retain that with isentropic fluctuations hypothesis, we can obtained a good prediction of

After these remarks, we consider the Fourier transform (denoted here by the symbol *" "* ˆ ) of various terms in equations (1) and (2) which leads to the following equations expressed in

> , <sup>ˆ</sup> <sup>ˆ</sup>*i,i <sup>p</sup> <sup>u</sup> <sup>γ</sup><sup>P</sup>*

*ν ν <sup>p</sup> <sup>u</sup> <sup>λ</sup> u k k u k u Ik*

*ij i j* 1 2 *λ Sδ δ* (7)

In the Fourier space, velocity field is decomposed on solenoidal and dilatational

11 22 33 ˆˆ ˆ ˆ ( ) ( ) ( ) ( ) ( ) ( ) ( ), *i iii u k,t k,t e k k,t e k k,t e k*

By separating solenoidal and dilatational contributions, we introduce the spectrums of

( ) ( ) ( ) ( ) ( ) 1 4, ˆ ˆ ˆˆ

*ρa*

 

 

<sup>11</sup> <sup>12</sup> <sup>13</sup> 22 2 , *<sup>d</sup><sup>Φ</sup> Sk Sk k νk ΦΦ Φ*

*dt k kk'*

*i j ij <sup>Φ</sup>ij k,t k,t k,t k,t k,t i, j ..*

 (8)

(9)

ˆ *(k,t)*

(10)

(11)

is the dilatational

contributions by adopting a local reference mark of Craya (Cambon et al., 1993):

ˆ ˆ *(k,t) (k,t)* and are the solenoidal contributions and <sup>3</sup>

 

is related to the pressure and takes the following form:

1

We then write evolution equations of these doubles correlations:

11 <sup>2</sup> <sup>3</sup> 2 3

2

<sup>4</sup> <sup>ˆ</sup> <sup>ˆ</sup> ( ) *<sup>p</sup> k,t I*

(5)

*ρ* (6)

equilibrium states *(St )* for compressible homogeneous sheared turbulence.

the spectral space and in the case of turbulent shear flow:

where *ij λ* is the mean velocity gradient defined by:

and *I2* = - 1.

where 1 2 

doubles correlations:

contribution.

where <sup>4</sup> *k,t* ˆ ( )

 

<sup>2</sup> <sup>ˆ</sup> ˆˆ ˆ ˆ , 3 3 *i ij j i j j i i*

In this section, results are presented and used to verify the validity of the RDT code. For this, tests of this code have been performed by comparing RDT results with those of direct numerical simulation (DNS) of Simone et al. (1997) for various values of initial gradient Mach number *Mg*0 which describe different regimes of the flow (see below). Comparisons concern the turbulent kinetic energy *q2(t)/q2(0)* [Figs. 1, 2], the non-dimensional production term 2 <sup>12</sup> <sup>2</sup> <sup>2</sup> *<sup>P</sup> <sup>b</sup> Sq* [Figs. 3, 4] and the turbulent kinetic energy growth rate 2 2 <sup>1</sup> *dq <sup>Λ</sup> Sq dt* [Figs. 5, 6]. In the same way, comparisons deal with the solenoidal and the dilatational *b*<sup>12</sup> anisotropy tensor components [Figs. 7, 8] (Riahi & Lili, 2011). We note that 1 2 12 2 *u u <sup>b</sup> <sup>q</sup>* is the relevant component of the Reynolds stress anisotropy tensor. Tavoularis & Corrsin (1981) Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 399

Case *Mt*<sup>0</sup> *Mg*<sup>0</sup> *r*<sup>0</sup> *Ret*<sup>0</sup>

A1 0.25 2.7 10.7 296 A2 0.25 4 16 296 A3 0.25 8.3 33.1 296 A4 0.25 12 48 296 A5 0.25 16.5 66.2 296 A6 0.25 24 96.1 296 A7 0.25 32 128.1 296 A8 0.25 42.7 170.8 296 A9 0.25 53.4 213.5 296 A10 0.25 66.7 266.9 296

All Figures 1-8 validate RDT approach and numerical code for low values of the nondimensional times *St* (until 3.5). Indeed, comparisons carried out between our RDT results (Riahi & Lili, 2011) and DNS results of Simone et al. (1997) show that the linear analysis predicted correctly – as well qualitatively as quantitatively – the behavior of turbulence in

**Figure 1.** Histories of the turbulent kinetic energy for pure-shear flow: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for both DNS

**Table 1.** Initial parameters.

**3.3. Results and discussion** 

particular for small values of *St*.

and RDT; arrows show trend with increasing *Mg*0.

have shown that in the incompressible homogeneous shear flow 12 *b* tends toward an equilibrium value which is independent of initial conditions.

Concerning the existence of the evolution zones where shocks develop, we neither observe any anomaly in physic parameters (no problem of realisability), nor any instability in the calculation course. In addition, it is important to specify that our code was not conceived to take explicitly into account the existence of shocks. However, we think that in the applications presented, the flow is not probably contaminated by the appearance of shocks.

#### **3.2. Initial parameters**

Intrinsic parameters that characterize the flow include the initial turbulent Mach number *Mt*0 <sup>0</sup> *<sup>q</sup> <sup>a</sup>* (recall that <sup>2</sup> 0 1 2 *q* is the initial turbulent kinetic energy and *a* is the mean speed of sound), the initial gradient Mach number *Mg*0 = *Mt*<sup>0</sup> 2 0 0 *q S ε* (*S* is the constant mean shear and <sup>0</sup> *ε* the initial total rate of turbulent kinetic energy dissipation) and the initial turbulent Reynolds number 4 0 0 0 *et <sup>q</sup> <sup>R</sup> νε* . The ratio of the gradient Mach number to the turbulent Mach number 2 0 0 *<sup>q</sup> r S <sup>ε</sup>* characterizes the rapidity of the shear.

Table 1 lists ten simulations cases labelled A1, A2 … A10 corresponding to different values of the initial gradient Mach number *Mg*0. In these simulations, *Mg*0 increases respectively in cases A1 (*Mg*0 = 2.7) to A10 (*Mg*0 = 66.7) by keeping the same initial value of the turbulent Mach number *Mt*0 and the turbulent Reynolds number *Ret*0.

In these cases, the initial turbulent kinetic energy spectrum is similar to that used by Simone (1995),

$$E(\mathbf{K}, t\_0) = K^4 \exp(-2\frac{K^2}{\kappa\_{pic}^2}),$$

where *κpic* denotes the wave number corresponding to the peak of the power spectrum and *K* is the initial wave number.


**Table 1.** Initial parameters.

398 Numerical Simulation – From Theory to Industry

term 2 <sup>12</sup> <sup>2</sup> <sup>2</sup> *<sup>P</sup> <sup>b</sup>*

**3.2. Initial parameters** 

*<sup>a</sup>* (recall that <sup>2</sup>

2 0 0

*K* is the initial wave number.

*<sup>q</sup> r S*

0 1 2

sound), the initial gradient Mach number *Mg*0 = *Mt*<sup>0</sup>

4 0

0

number *Mt*0 and the turbulent Reynolds number *Ret*0.

*<sup>ε</sup>* characterizes the rapidity of the shear.

0

*et <sup>q</sup> <sup>R</sup>*

*Mt*0 <sup>0</sup> *<sup>q</sup>*

number

(1995),

Reynolds number

concern the turbulent kinetic energy *q2(t)/q2(0)* [Figs. 1, 2], the non-dimensional production

5, 6]. In the same way, comparisons deal with the solenoidal and the dilatational *b*<sup>12</sup>

relevant component of the Reynolds stress anisotropy tensor. Tavoularis & Corrsin (1981) have shown that in the incompressible homogeneous shear flow 12 *b* tends toward an

Concerning the existence of the evolution zones where shocks develop, we neither observe any anomaly in physic parameters (no problem of realisability), nor any instability in the calculation course. In addition, it is important to specify that our code was not conceived to take explicitly into account the existence of shocks. However, we think that in the applications presented, the flow is not probably contaminated by the appearance of shocks.

Intrinsic parameters that characterize the flow include the initial turbulent Mach number

the initial total rate of turbulent kinetic energy dissipation) and the initial turbulent

Table 1 lists ten simulations cases labelled A1, A2 … A10 corresponding to different values of the initial gradient Mach number *Mg*0. In these simulations, *Mg*0 increases respectively in cases A1 (*Mg*0 = 2.7) to A10 (*Mg*0 = 66.7) by keeping the same initial value of the turbulent Mach

In these cases, the initial turbulent kinetic energy spectrum is similar to that used by Simone

4 <sup>0</sup> <sup>2</sup> exp 2 ,

*<sup>K</sup> E(K,t ) K (- )*

where *κpic* denotes the wave number corresponding to the peak of the power spectrum and

*q* is the initial turbulent kinetic energy and *a* is the mean speed of

*νε* . The ratio of the gradient Mach number to the turbulent Mach

2

*pic*

*κ*

2 0 0 *q S ε*

2

*Sq dt* [Figs.

*<sup>q</sup>* is the

2 <sup>1</sup> *dq <sup>Λ</sup>*

> 12 2 *u u <sup>b</sup>*

(*S* is the constant mean shear and <sup>0</sup> *ε*

*Sq* [Figs. 3, 4] and the turbulent kinetic energy growth rate

equilibrium value which is independent of initial conditions.

anisotropy tensor components [Figs. 7, 8] (Riahi & Lili, 2011). We note that 1 2

#### **3.3. Results and discussion**

All Figures 1-8 validate RDT approach and numerical code for low values of the nondimensional times *St* (until 3.5). Indeed, comparisons carried out between our RDT results (Riahi & Lili, 2011) and DNS results of Simone et al. (1997) show that the linear analysis predicted correctly – as well qualitatively as quantitatively – the behavior of turbulence in particular for small values of *St*.

**Figure 1.** Histories of the turbulent kinetic energy for pure-shear flow: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for both DNS and RDT; arrows show trend with increasing *Mg*0.

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 401

**Figure 5.** Histories of the temporal energy growth rate for pure-shear flow: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for both DNS

**Figure 6.** Evolution of the turbulent kinetic energy growth rate (a) case A2 (*Mg*0 = 4), (b) case A4 (*Mg*0 =

5 10 15 *St*

5 10 15 *St*

(c)

(b)

**Figure 7.** Histories of the solenoidal *b*12 component anisotropy tensor for pure-shear: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for

both DNS and RDT; arrows show trend with increasing *Mg*0.

12) and (c) case A10 (*Mg*0 = 66.7). : RDT results, o : DNS results of Simone et al. (1997).

and RDT; arrows show trend with increasing *Mg*0.

0 5 10 15

*St*

(a)

0,0

0,2

0,4

0,6

**Figure 2.** Evolution of the turbulent kinetic energy (a) case A2 (*Mg*0 = 4), (b) case A4 (*Mg*0 = 12) and (c) case A10 (*Mg*0 = 66.7). : RDT results, o : DNS results of Simone et al. (1997).

**Figure 3.** Histories of the non-dimensional production term -2*b*12 for pure-shear flow: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for both DNS and RDT; arrows show trend with increasing *Mg*0.

**Figure 4.** Evolution of the non-dimensional production term (a) case A2 (*Mg*0 = 4), (b) case A4 (*Mg*0 = 12) and (c) case A10 (*Mg*0 = 66.7).: RDT results, o : DNS results of Simone et al. (1997).

0 5 10 15

*St*

(a)

*q2(t)/q2(0)* 

**Figure 2.** Evolution of the turbulent kinetic energy (a) case A2 (*Mg*0 = 4), (b) case A4 (*Mg*0 = 12) and (c)

*St*

5 10 15

(b)

**Figure 3.** Histories of the non-dimensional production term -2*b*12 for pure-shear flow: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for

**Figure 4.** Evolution of the non-dimensional production term (a) case A2 (*Mg*0 = 4), (b) case A4 (*Mg*0 = 12)

*St*

5 10 15

(b)

5 10 15

(c)

5 10 15

*St*

(c)

*St*

and (c) case A10 (*Mg*0 = 66.7).: RDT results, o : DNS results of Simone et al. (1997).

case A10 (*Mg*0 = 66.7). : RDT results, o : DNS results of Simone et al. (1997).

both DNS and RDT; arrows show trend with increasing *Mg*0.

(a)

0 510 15

*St*

0,0

0,2


*b*12

0,4

0,6

**Figure 5.** Histories of the temporal energy growth rate for pure-shear flow: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for both DNS and RDT; arrows show trend with increasing *Mg*0.

**Figure 6.** Evolution of the turbulent kinetic energy growth rate (a) case A2 (*Mg*0 = 4), (b) case A4 (*Mg*0 = 12) and (c) case A10 (*Mg*0 = 66.7). : RDT results, o : DNS results of Simone et al. (1997).

**Figure 7.** Histories of the solenoidal *b*12 component anisotropy tensor for pure-shear: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for both DNS and RDT; arrows show trend with increasing *Mg*0.

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 403

**Figure 9.** Evolution of the non-dimensional production term with (a): *Mg*0 = 2.2 and

**Figure 10.** Evolution of *b*11 component anisotropy tensor with (a): *Mg*0 = 2.2 and (b): *Mg*0 = 13.2.

**Figure 11.** Evolution of *b*22 component anisotropy tensor with (a): *Mg*0 = 2.2 and (b): *Mg*0 = 13.2.




*b*22


0,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

0,0 0,1 0,2 0,3 0,4 0,5 0,6


*b*<sup>11</sup>

0 5 10 15 20 25

*St*

(b)

0 5 10 15 20 25

0 5 10 15 20 25

*St*

(b)

*St*

(b)

(b): *Mg*0 = 13.2. : RDT results, o : DNS results of Sarkar (1995).

0 5 10 15 20 25

*St*

(a)

0 5 10 15 20 25

*St*

: RDT results, o : DNS results of Sarkar (1995).

(a)

: RDT results, o : DNS results of Sarkar (1995).




*b*22


0,0

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

0,0 0,1 0,2 0,3 0,4 0,5 0,6


*b*<sup>11</sup>

0 5 10 15 20 25

*St*

(a)

**Figure 8.** Histories of the dilatational *b*12 component anisotropy tensor for pure-shear flow: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7 to 67 for both DNS and RDT; arrows show trend with increasing *Mg*0.

In the compressible regime (*Mt*0 = 0.25 and *Mg*0 = 66.7), RDT and DNS equilibrium values of the non-dimensional production -2*b*12 of Simone et al. (1997) are very close [Fig. 4(c)]. As one can remark from this figure that for large values of *St* (*St* > 10), there is a small difference between RDT and DNS results of Simone et al. (1997). Thus, in this compressible regime, RDT predicts correctly equilibrium behavior of the turbulence (asymptotic behavior ( ) *St* ) relatively to the non-dimensional production -2*b*12 (strongly compressible regime *Mg*0 = 66.7 >> 1 and 0 *r* >> 1) which is a rapid distortion regime compatible with the framework of RDT. The intermediate regime (case A4, Mg0 = 12) and especially the incompressible regime, for which non-linear effects are preponderant, are poorly estimated by RDT for large values of *St*. Moreover, Simone et al. (1997) indicate, as general conclusion that compressibility effects on homogeneous sheared turbulence, concerning particularly *b*<sup>12</sup> and the non-dimensional production, which are generally associated with non linear phenomena, can be explained in terms of RDT. So, non linear and dissipative effects (through the non linear cascade) are not significant for predicting equilibrium states for strongly compressible regime (Simone et al., 1997). Consequently, we can justify the resort to RDT in order to determine equilibrium states in the compressible regime.

In Figures 9-11, we present evolutions of *b*12 (which is linked to production), *b*11 and *b*<sup>22</sup> components anisotropy tensor in the incompressible (*Mg*0 = 2.2) and intermediate regimes (*Mg*0 = 13.2). In the incompressible regime, as illustrated in Figures 9(a), 10(a), 11(a), we confirm that RDT validity is limited for small values of *St*; considerable differences appear between RDT and DNS results of Sarkar (1995) beyond. With RDT, it is not possible to predict evolution of sheared turbulence for large values of *St*. Consequently, we can't predict, by means of RDT, equilibrium states of sheared turbulence in the incompressible regime. We note that in the intermediate regime [Figs. 9(b), 10(b), 11(b)], the behavior of DNS results of Sarkar (1995) concerning *b*12, *b*11 and *b*22 seems to be compatible with RDT results for large values of *St*. In the intermediate zone of *St* (between small and large values of *St*) differences are appreciable.

**Figure 9.** Evolution of the non-dimensional production term with (a): *Mg*0 = 2.2 and (b): *Mg*0 = 13.2. : RDT results, o : DNS results of Sarkar (1995).

*Mg*0 = 66.7 >> 1 and 0

of *St*) differences are appreciable.

**Figure 8.** Histories of the dilatational *b*12 component anisotropy tensor for pure-shear flow: (a) DNS results of Simone et al. (1997) and (b) our RDT results. Initial gradient Mach number *Mg*0 ranges from 2.7

In the compressible regime (*Mt*0 = 0.25 and *Mg*0 = 66.7), RDT and DNS equilibrium values of the non-dimensional production -2*b*12 of Simone et al. (1997) are very close [Fig. 4(c)]. As one can remark from this figure that for large values of *St* (*St* > 10), there is a small difference between RDT and DNS results of Simone et al. (1997). Thus, in this compressible regime, RDT predicts correctly equilibrium behavior of the turbulence (asymptotic behavior ( ) *St* ) relatively to the non-dimensional production -2*b*12 (strongly compressible regime

framework of RDT. The intermediate regime (case A4, Mg0 = 12) and especially the incompressible regime, for which non-linear effects are preponderant, are poorly estimated by RDT for large values of *St*. Moreover, Simone et al. (1997) indicate, as general conclusion that compressibility effects on homogeneous sheared turbulence, concerning particularly *b*<sup>12</sup> and the non-dimensional production, which are generally associated with non linear phenomena, can be explained in terms of RDT. So, non linear and dissipative effects (through the non linear cascade) are not significant for predicting equilibrium states for strongly compressible regime (Simone et al., 1997). Consequently, we can justify the resort

In Figures 9-11, we present evolutions of *b*12 (which is linked to production), *b*11 and *b*<sup>22</sup> components anisotropy tensor in the incompressible (*Mg*0 = 2.2) and intermediate regimes (*Mg*0 = 13.2). In the incompressible regime, as illustrated in Figures 9(a), 10(a), 11(a), we confirm that RDT validity is limited for small values of *St*; considerable differences appear between RDT and DNS results of Sarkar (1995) beyond. With RDT, it is not possible to predict evolution of sheared turbulence for large values of *St*. Consequently, we can't predict, by means of RDT, equilibrium states of sheared turbulence in the incompressible regime. We note that in the intermediate regime [Figs. 9(b), 10(b), 11(b)], the behavior of DNS results of Sarkar (1995) concerning *b*12, *b*11 and *b*22 seems to be compatible with RDT results for large values of *St*. In the intermediate zone of *St* (between small and large values

to RDT in order to determine equilibrium states in the compressible regime.

*r* >> 1) which is a rapid distortion regime compatible with the

to 67 for both DNS and RDT; arrows show trend with increasing *Mg*0.

**Figure 10.** Evolution of *b*11 component anisotropy tensor with (a): *Mg*0 = 2.2 and (b): *Mg*0 = 13.2. : RDT results, o : DNS results of Sarkar (1995).

**Figure 11.** Evolution of *b*22 component anisotropy tensor with (a): *Mg*0 = 2.2 and (b): *Mg*0 = 13.2. : RDT results, o : DNS results of Sarkar (1995).

In conclusion, RDT is valid for small values of the non-dimensional times *St* (*St* < 3.5). RDT is also valid for large values of *St* (*St* > 10) in particular for large values of *Mg*0. This essential feature justifies the resort to RDT in order to critically study equilibrium states in the compressible regime.

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 405

possible to better predict compressible turbulent flows and to improve existing turbulence models. The Helmoltz decomposition of the velocity field in solenoidal and dilatational parts reveals two additional terms in the turbulent kinetic energy budget. Several studies of the behavior of the different terms present in this budget and the Reynolds stress equations show the role of the explicit compressible terms. From Simone et al. (1997) and Sarkar (1995), for the case in which the rate of shear is much larger than the rate of non-linear interactions of the turbulence, amplification of turbulent kinetic energy by the mean shear caused by compressibility is due to the implicit pressure-strain correlations effects and to the anisotropy of the Reynolds stress tensor. These authors also recommend to take into account correctly the explicit dilatational terms such as the pressure-dilatation correlation and the dilatational dissipation. In contrast, the role of explicit terms was over-estimated by Zeman (1990) and Sarkar et al. (1991). These last authors show that both those terms have a dissipative contribution in shear flow, leading to the reduced growth of the turbulent kinetic

The study of the budget behaviors of the turbulent kinetic energy and the Reynolds stress anisotropy by RDT enables to better understand and explain compressibility effects on

In the case of homogeneous turbulence, the turbulent kinetic energy is written (Simone,

2 *sd d*

in which 1 2 *P Su u* is the rate of production by the mean flow and Π*d i,i pu* the pressure-

dilatational parts of the turbulent dissipation rate given that *ωi* is the fluctuating vorticity and *i,i u* denotes the fluctuating divergence of velocity. *<sup>s</sup> ε* represents the turbulent dissipation arising from the traditional energy cascade which is solenoidal, *<sup>d</sup> ε* represents the turbulent dissipation arising from dilatational regimes. Note that the last two terms on the right-hand side in equation (22) do not appear when the flow is incompressible. The explicit/energetic approach is embodied in the modeling of *<sup>d</sup> ε* and Π*d* done by Zeman

Figures 12(a), (b), (c) show the budget of the turbulent kinetic energy for three values of initial gradient Mach number (respectively 1, 12 and 66.7) which describe the various regimes of the flow (Riahi & Lili, 2011). It will be shown from Figures 12(b), (c) that the

*( )P ε ε*

Π ,

<sup>3</sup> *s i i d i,i i,i ε ν ωω ε ν u u* are respectively the solenoidal and the

(22)

2

*d q*

*dt*

structure and evolution of a sheared homogeneous turbulence.

**5.2. The turbulent kinetic energy equation** 

dilatation correlation. 4 and

(1990), Sarkar et al. (1991) and others.

energy.

1995) as:

**5.3. Results** 
