**5.1. Introduction**

The study of compressibility effects on the turbulent homogeneous shear flow behavior made these last years the objective of several researches as mentioned in the works of Blaisdell et al. (1993 & 1996) and Sarkar et al. (1991 & 1992). DNS developed by Sarkar (1995) show that the temporal growth rate of the turbulent kinetic energy is extensively influenced by compressibility. Simone et al. (1997) identified the new coupling term in the quasiisentropic RDT equations, which was responsible for long-term stabilization, comparing incompressible and compressible RDT. They concluded that the modification of the compressible turbulence structure is due to linear processes.

The prediction of compressible turbulent flows by rapid distortion theory provides useful results which can be used to clarify the physics of the compressible turbulent flows and to study compressibility effects on structure of homogeneous sheared turbulence. This study is important for the analysis of various physical mechanisms which characteristic the turbulence. Thus, physical comprehension of compressibility effects on turbulence leads it 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 energy.

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 structure and evolution of a sheared homogeneous turbulence.

#### **5.2. The turbulent kinetic energy equation**

In the case of homogeneous turbulence, the turbulent kinetic energy is written (Simone, 1995) as:

$$\frac{d}{dt}(\frac{q^2}{2}) = P - \varepsilon\_s - \varepsilon\_d + \Pi\_{d'} \tag{22}$$

in which 1 2 *P Su u* is the rate of production by the mean flow and Π*d i,i pu* the pressuredilatation correlation. 4 and <sup>3</sup> *s i i d i,i i,i ε ν ωω ε ν u u* are respectively the solenoidal and the 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 (1990), Sarkar et al. (1991) and others.

#### **5.3. Results**

404 Numerical Simulation – From Theory to Industry

**4. Various regimes of flow** 

the initial gradient Mach number *Mg*0 increases.

effect of compressibility which are still not cleared up.

compressible turbulence structure is due to linear processes.

modification generated by compressibility.

**turbulence** 

**5.1. Introduction** 

compressible regime.

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

The various curves of Figure 1(b) permit to determine different regimes of the flow (Riahi & Lili, 2011). This figure shows that there is an increase in the turbulent kinetic energy when *Mg*0 increases for various cases considered A1,…,A10. In addition, when initial gradient Mach number increases, we observe a break of slope which is accentuated when the value of *Mg*<sup>0</sup> becomes more significant. It appears from this figure that the turbulent kinetic energy varies quasi-linearly in cases A1 and A2, where *Mg*0 takes respectively values 2.7 and 4. These two values of *Mg*0 correspond to the incompressible regime. A weak amplification of the turbulent kinetic energy shows that cases A4 (*Mg*0 = 12) and A10 (*Mg*0 = 66.7) correspond to the intermediate regime. From *Mg*0 = 16.5, the regime becomes compressible. Indeed, cases A5,…, A10 show a significant amplification of the total kinetic energy more and more marked when

These different regimes permit to better understanding compressibility effects on structure of homogeneous sheared turbulence and to analyze the causes of turbulence structure

Several explanations were proposed these last years to analyze causes of the stabilising

The study of compressibility effects on the turbulent homogeneous shear flow behavior made these last years the objective of several researches as mentioned in the works of Blaisdell et al. (1993 & 1996) and Sarkar et al. (1991 & 1992). DNS developed by Sarkar (1995) show that the temporal growth rate of the turbulent kinetic energy is extensively influenced by compressibility. Simone et al. (1997) identified the new coupling term in the quasiisentropic RDT equations, which was responsible for long-term stabilization, comparing incompressible and compressible RDT. They concluded that the modification of the

The prediction of compressible turbulent flows by rapid distortion theory provides useful results which can be used to clarify the physics of the compressible turbulent flows and to study compressibility effects on structure of homogeneous sheared turbulence. This study is important for the analysis of various physical mechanisms which characteristic the turbulence. Thus, physical comprehension of compressibility effects on turbulence leads it

**5. Compressibility effects on structure of homogeneous sheared** 

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 production *P* and the pressure-dilatation Π*d* intervene significantly in this budget. In the compressible regime [Fig. 12(c)], the production is dominating and the pressure-dilatation Π*<sup>d</sup>* remains with relatively low values (practically null until *St* = 1). Figure 12(a) shows the results for *Mg*0 = 1 and *Mt*0 = 0.1 (case A0). In this case, it is still the production which is dominating. The explicitly compressible terms which are the dilatational dissipation rate *<sup>d</sup> ε* and the pressure-dilatation Π*d* are not negligible in spite of the small value of the initial gradient Mach number *Mg*0. Consequently, we can affirm that ''the incompressible behavior'' cannot be obtained in the borderline case of low values of *Mg*0; that's why we preferred to give the results for *Mg*0 = 1 and not for *Mg*0 = 2.7 and 4 with an aim of better approaching the ''incompressible limit''.

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 407



(d)

*b*<sup>12</sup>

*b*<sup>33</sup>

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

*St*

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

*St*

(b)

**Figure 13.** Components evolution of the anisotropy tensor (a) *b*11, (b) *b*12, (c) *b*22 and (d) *b*33. : case A0 (*Mg*0 = 1), : case A4 (*Mg*0 = 12), : case A10 (*Mg*0 = 66.7).

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

*St*

(c)

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

*St*

(a)



*b*<sup>11</sup>

*b*<sup>22</sup>

**6. Equilibrium states of homogeneous compressible turbulence** 

in the direction of the shear (x1 direction): 2 2

released limit.

RDT is also used to determine equilibrium states in the compressible regime (Riahi & Lili, 2011) in particular for large values of *St* (*St* > 10) and *M*g0. Evolution of the relevant component of Reynolds stress anisotropy tensor *b*12 is presented in Figure 14 for different values of the initial gradient Mach number *M*g0 (66.7, 200, 500 and 1000). Numerical results show that, for large values of *St*, *b*12 is independent of the initial turbulent Mach number *M*t0 as shown in Figure 14(a) (*M*t0 = 0.25) and in Figure 14(b) (*M*t0 = 0.6). As one can remark also from these results that *b*12 reaches its stationary value all the more quickly as *M*g0 is large (*M*g0 = 1000). At this stage, we study equilibrium states corresponding to *M*g0 = 1000, value which we assimilate to *M*g0 arbitrarily large (*M*g0 infinity). We present now some properties relative to these equilibrium states corresponding to pressure-released regime. The case corresponding to pressure-released limit has been discussed by Cambon et al. (1993), Simone et al. (1997) and Livescu et al. (2004). We begin by presenting equilibrium values of Reynolds stress anisotropy components: *b*<sup>11</sup> = 0.66, *b*<sup>22</sup> = *b*<sup>33</sup> = - 0.33 and <sup>12</sup> *<sup>b</sup>* = 0. These values confirm the independence of the anisotropy tensor with *M*t0 in the pressurereleased regime; calculation gives effectively these values for *M*t0 = 0.25, 0.4, 0.5 and 0.6. Pressure-released state corresponds thus to a particular state of one-component turbulence

<sup>1</sup> *u q*

Another interesting property of the pressure-released regime is the quadratic increase of <sup>2</sup>

as shown in Figure 15 related to *Mg*0 = 106 and *Mt*0 = 0.25 (Riahi & Lili, 2011). This property has been established by Livescu et al. (2004) as analytical RDT solution in the pressure-

, <sup>2</sup>

<sup>2</sup> *<sup>u</sup>*<sup>0</sup>

and <sup>2</sup>

<sup>3</sup>*<sup>u</sup>* <sup>0</sup> .

1 *u*

**Figure 12.** The turbulent kinetic energy budget (a) case A0 (*Mg*0 = 1), (b) case A4 (*Mg*0 = 12) and (c) case A10 (*Mg*0 = 66.7). **——** : 2 ( ) Π 2 *sd d d q <sup>P</sup> ε ε dt* , -**··**- : rate of the turbulent production ( ) *P* , **···** : rate of the solenoidal dissipation ( )*<sup>s</sup> ε* , --- : rate of the dilatational dissipation ( ) *<sup>d</sup> ε* , -**·**- : pressuredilatation correlation term (Π ) *<sup>d</sup>* .

In conclusion, compressibility affects more the turbulent production term which represents the dominating parameter in the turbulent kinetic energy budget. In the compressible regime, the production becomes preponderant. This property is already observed in the analysis of the budget of the Reynolds stress equations related to <sup>2</sup> <sup>1</sup> *u* and 1 2 *u u* (Riahi et al., 2007).

In Figures 13(a), (b), (c), (d) are represented the different components of the anisotropy tensor *ij b* . These figures show obviously a remarkable property concerning the anisotropy. In fact, it appears clearly that the anisotropy increases with *Mg*0 i.e. with compressibility and that it is responsible of the behavior of <sup>2</sup> <sup>1</sup> *<sup>u</sup>* which becomes dominating compared to <sup>2</sup> <sup>2</sup> *u* and 2 <sup>3</sup> *u* in the compressible case (*Mg*0 = 66.7). As an indication, 2 1 2 2 12 62 *<sup>u</sup> . u* and 2 1 2 3 13 11 *<sup>u</sup> . u* for *St* = 3.5.

**Figure 13.** Components evolution of the anisotropy tensor (a) *b*11, (b) *b*12, (c) *b*22 and (d) *b*33. : case A0 (*Mg*0 = 1), : case A4 (*Mg*0 = 12), : case A10 (*Mg*0 = 66.7).

approaching the ''incompressible limit''.

(a)

0,0 0,2 0,4 0,6 0,8 1,0 1,2

*St*

2

( ) Π 2 *sd d*


analysis of the budget of the Reynolds stress equations related to <sup>2</sup>

<sup>3</sup> *u* in the compressible case (*Mg*0 = 66.7). As an indication,

*d q <sup>P</sup> ε ε dt*

A10 (*Mg*0 = 66.7). **——** :


2007).

2

*St* = 3.5.

dilatation correlation term (Π ) *<sup>d</sup>* .

that it is responsible of the behavior of <sup>2</sup>

production *P* and the pressure-dilatation Π*d* intervene significantly in this budget. In the compressible regime [Fig. 12(c)], the production is dominating and the pressure-dilatation Π*<sup>d</sup>* remains with relatively low values (practically null until *St* = 1). Figure 12(a) shows the results for *Mg*0 = 1 and *Mt*0 = 0.1 (case A0). In this case, it is still the production which is dominating. The explicitly compressible terms which are the dilatational dissipation rate *<sup>d</sup> ε* and the pressure-dilatation Π*d* are not negligible in spite of the small value of the initial gradient Mach number *Mg*0. Consequently, we can affirm that ''the incompressible behavior'' cannot be obtained in the borderline case of low values of *Mg*0; that's why we preferred to give the results for *Mg*0 = 1 and not for *Mg*0 = 2.7 and 4 with an aim of better

**Figure 12.** The turbulent kinetic energy budget (a) case A0 (*Mg*0 = 1), (b) case A4 (*Mg*0 = 12) and (c) case

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

(b)

*St*

In conclusion, compressibility affects more the turbulent production term which represents the dominating parameter in the turbulent kinetic energy budget. In the compressible regime, the production becomes preponderant. This property is already observed in the

In Figures 13(a), (b), (c), (d) are represented the different components of the anisotropy tensor *ij b* . These figures show obviously a remarkable property concerning the anisotropy. In fact, it appears clearly that the anisotropy increases with *Mg*0 i.e. with compressibility and

the solenoidal dissipation ( )*<sup>s</sup> ε* , --- : rate of the dilatational dissipation ( ) *<sup>d</sup> ε* , -**·**- : pressure-

, -**··**- : rate of the turbulent production ( ) *P* , **···** : rate of


<sup>1</sup> *<sup>u</sup>* which becomes dominating compared to <sup>2</sup>

12 62 *<sup>u</sup> .*

and

*u*

<sup>1</sup> *u* and 1 2 *u u* (Riahi et al.,

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

*St*

(c)

*u*

13 11 *<sup>u</sup> .*

<sup>2</sup> *u* and

for

#### **6. Equilibrium states of homogeneous compressible turbulence**

RDT is also used to determine equilibrium states in the compressible regime (Riahi & Lili, 2011) in particular for large values of *St* (*St* > 10) and *M*g0. Evolution of the relevant component of Reynolds stress anisotropy tensor *b*12 is presented in Figure 14 for different values of the initial gradient Mach number *M*g0 (66.7, 200, 500 and 1000). Numerical results show that, for large values of *St*, *b*12 is independent of the initial turbulent Mach number *M*t0 as shown in Figure 14(a) (*M*t0 = 0.25) and in Figure 14(b) (*M*t0 = 0.6). As one can remark also from these results that *b*12 reaches its stationary value all the more quickly as *M*g0 is large (*M*g0 = 1000). At this stage, we study equilibrium states corresponding to *M*g0 = 1000, value which we assimilate to *M*g0 arbitrarily large (*M*g0 infinity). We present now some properties relative to these equilibrium states corresponding to pressure-released regime. The case corresponding to pressure-released limit has been discussed by Cambon et al. (1993), Simone et al. (1997) and Livescu et al. (2004). We begin by presenting equilibrium values of Reynolds stress anisotropy components: *b*<sup>11</sup> = 0.66, *b*<sup>22</sup> = *b*<sup>33</sup> = - 0.33 and <sup>12</sup> *<sup>b</sup>* = 0. These values confirm the independence of the anisotropy tensor with *M*t0 in the pressurereleased regime; calculation gives effectively these values for *M*t0 = 0.25, 0.4, 0.5 and 0.6. Pressure-released state corresponds thus to a particular state of one-component turbulence in the direction of the shear (x1 direction): 2 2 <sup>1</sup> *u q* , <sup>2</sup> <sup>2</sup> *<sup>u</sup>*<sup>0</sup> and <sup>2</sup> <sup>3</sup>*<sup>u</sup>* <sup>0</sup> .

Another interesting property of the pressure-released regime is the quadratic increase of <sup>2</sup> 1 *u* as shown in Figure 15 related to *Mg*0 = 106 and *Mt*0 = 0.25 (Riahi & Lili, 2011). This property has been established by Livescu et al. (2004) as analytical RDT solution in the pressurereleased limit.

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 409

0.0410 0.0277 0.0230 0.0197

0.000387 0.000311 0.000280 0.000257

0.00937 0.00278 0.00156 0.000973

**Table 2.** Equilibrium values of the normalized dissipation rate due to dilatation, the normalized pressure-

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

*Mt*0

**Figure 16.** Variation of the equilibrium normalized pressure variance with various initial turbulent

RDT is an efficient and accurate method for testing linear turbulence models. The Fujiwara and Arakawa model (1995) has been proposed in literature for pressure-dilatation correlation studying compressible turbulence. In order to verify its capacity to describe the homogeneous compressible flow, since it was established to this objective, three cases of the initial gradient Mach number (*Mg*0 = 1, 12 and 66.7) are considered. The evaluation of this model, in the different regimes of flow, stays in the field of RDT validity (Riahi & Lili, 2011).

dilatation and the normalized pressure variance for various initial turbulent Mach number *Mt*0.

10-4

10-3

 

 *42 2 ρ q p*

Mach number (*Mt*0 = 0.25, 0.4, 0.5 and 0.6).

**7.1. Introduction** 

**7. Turbulence model to be tested** 

Its detailed linear form is provided below.

10-2

10-1

0.25 0.4 0.5 0.6

Equilibrium parameters

*d s d ε ε ε* 

> 2 *pd Sq*

2 2 4 *p <sup>ρ</sup> <sup>q</sup>*

 *Mt*<sup>0</sup>

**Figure 14.** Evolution of the relevant component of the Reynolds stress anisotropy tensor *b*12 for various values of initial gradient Mach number *Mg*0 with (a): *Mt*0 = 0.25 and (b): *Mt*0 = 0.6.

**Figure 15.** Quadratic evolution of <sup>2</sup> <sup>1</sup> *u* .

Table 2 lists equilibrium values of the normalized dissipation rate due to dilatation *d s d ε ε ε* , the normalized pressure-dilatation correlation 2 *pd Sq* and the normalized pressure variance 2 2 4 *p <sup>ρ</sup> <sup>q</sup>* for various values of initial turbulent Mach number *Mt*0 (Riahi & Lili, 2011). From this table, we deduce the dependence of all these parameters with *Mt*0. Figure 16 shows variation of the equilibrium normalized pressure variance 2 2 4 *p <sup>ρ</sup> <sup>q</sup>* with various initial turbulent Mach number (*Mt*0 = 0.25, 0.4, 0.5 and 0.6 (*Mt*0 = 0.6 is relatively high initial turbulent Mach number)) (Riahi & Lili, 2011). From this figure, it can be seen that this parameter can be written as 2 2 4 *p <sup>ρ</sup> <sup>q</sup>* (*Mt*0) *<sup>α</sup>* where *α* = 5.8.


**Table 2.** Equilibrium values of the normalized dissipation rate due to dilatation, the normalized pressuredilatation and the normalized pressure variance for various initial turbulent Mach number *Mt*0.

**Figure 16.** Variation of the equilibrium normalized pressure variance with various initial turbulent Mach number (*Mt*0 = 0.25, 0.4, 0.5 and 0.6).

## **7. Turbulence model to be tested**

#### **7.1. Introduction**

408 Numerical Simulation – From Theory to Industry

**Figure 14.** Evolution of the relevant component of the Reynolds stress anisotropy tensor *b*12 for various

2 4 6 8 10 20 30

*pd Sq*

for various values of initial turbulent Mach number *Mt*0 (Riahi &

and the normalized

2 2 4 *p <sup>ρ</sup> <sup>q</sup>*

with

 

 

*St*

Table 2 lists equilibrium values of the normalized dissipation rate due to dilatation

Lili, 2011). From this table, we deduce the dependence of all these parameters with *Mt*0.

various initial turbulent Mach number (*Mt*0 = 0.25, 0.4, 0.5 and 0.6 (*Mt*0 = 0.6 is relatively high initial turbulent Mach number)) (Riahi & Lili, 2011). From this figure, it can be seen that this

(*Mt*0) *<sup>α</sup>* where *α* = 5.8.

, the normalized pressure-dilatation correlation 2

Figure 16 shows variation of the equilibrium normalized pressure variance

2 2 4 *p <sup>ρ</sup> <sup>q</sup>*

 

values of initial gradient Mach number *Mg*0 with (a): *Mt*0 = 0.25 and (b): *Mt*0 = 0.6.

102

<sup>1</sup> *u* .

103

2 1 *u*

2 2 4 *p <sup>ρ</sup> <sup>q</sup>*

 

**Figure 15.** Quadratic evolution of <sup>2</sup>

*d s d ε ε ε* 

pressure variance

parameter can be written as

RDT is an efficient and accurate method for testing linear turbulence models. The Fujiwara and Arakawa model (1995) has been proposed in literature for pressure-dilatation correlation studying compressible turbulence. In order to verify its capacity to describe the homogeneous compressible flow, since it was established to this objective, three cases of the initial gradient Mach number (*Mg*0 = 1, 12 and 66.7) are considered. The evaluation of this model, in the different regimes of flow, stays in the field of RDT validity (Riahi & Lili, 2011). Its detailed linear form is provided below.

#### **7.2. Fujiwara and Arakawa model**

The model suggested by Fujiwara and Arakawa (1995) for the pressure-dilatation correlation takes into account the compressible part (dilatational) of the kinetic energy ( <sup>2</sup> *<sup>d</sup> q* ). Linear part of this model has the following form:

Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence 411

<sup>1</sup> *u* and 1 2 *u u* . Another important

(*Mt*0) *<sup>α</sup>* where *α* = 5.8.

non-dimensional turbulent kinetic energy *q2(t)/q2(0)* permit to determine various regimes of flow. This study allows to check relevance of an incompressible regime for low values of initial gradient Mach number *Mg*0, of an intermediate regime for moderate values of *Mg*0 and of a compressible regime for high values of *Mg*0. Agreement between RDT and DNS of Simone et al. (1997) and Sarkar (1995) is obtained for small values of the non-dimensional times *St* (*St* < 3.5). This agreement gives new insight into compressibility effects and reveals the extent to which linear processes are responsible for modifying the structure of compressible turbulence. The behavior analysis of the various terms presented in the turbulent kinetic energy budget shows that compressibility affects more the turbulent production which becomes preponderant in the compressible regime. This property is already observed in the analysis of

Agreement of RDT with DNS (Simone et al., 1997) found for large values of *St* (*St* > 10) in particular for large values of *Mg*0 which allows to determine equilibrium states in the compressible regime. Evolution of the relevant component of the Reynolds stress anisotropy tensor *b*12, for different values of initial gradient Mach number *Mg*0 (66.7, 200, 500 and 1000) and for large values of *St*, shows that *b*12 is independent of the initial turbulent Mach number *Mt*0 (which is a parameter characterizing the effects of compressibility).In addition, *b*12 becomes stationary all the more quickly as *Mg*<sup>0</sup> is large (*Mg*0 = 1000). Considering equilibrium states associated to this value of *Mg*0 as representative of equilibrium states related to infinite *Mg*<sup>0</sup> (pressure-released regime), equilibrium values of the Reynolds stress anisotropy components <sup>11</sup> *<sup>b</sup>* and <sup>22</sup> *<sup>b</sup>* are independent of the initial turbulent Mach number *Mt*0. In contrast, equilibrium values of the normalized dissipation rate due to dilatation, the normalized pressure-dilatation correlation and the normalized pressure variance are dependent of *Mt*0. Variation of the equilibrium normalized pressure variance with various initial turbulent Mach

In conclusion, after a critical analysis, we were able to justify that RDT permits to well identify compressibility effects in order to develop models taking them into account satisfactorily. The analysis of rapid distortion theory showed that it is possible to better understand the compressible turbulent flows. In addition, RDT is valid to predict asymptotic equilibrium states for compressible homogeneous sheared turbulence for large values of initial gradient Mach number (pressure-released regime). RDT is an efficient

*Faculté des Sciences de Tunis, Département de Physique, Laboratoire de Mécanique des Fluides,* 

2 2 4 *p <sup>ρ</sup> <sup>q</sup>* 

the budget of the Reynolds stress equations related to <sup>2</sup>

number (*Mt*0 = 0.25, 0.4, 0.5 and 0.6) can be written as

method for testing linear contribution of turbulence models.

**Author details** 

Mohamed Riahi and Taieb Lili

*Campus Universitaire, Manar II, Tunis, Tunisia* 

property reveals that anisotropy increases with compressibility.

$$
\Pi\_d = C\_1 \overline{\rho} q^2 b\_{ij} \sqrt{\frac{q\_d^2}{q^2}} \frac{\widehat{\odot} \mathcal{U}\_i}{\widehat{\odot} \mathcal{X}\_j},
$$

where <sup>1</sup> *C .*  0 3 and *ε* is the turbulent dissipation rate. *ij b* is the Reynolds anisotropy tensor.

#### **7.3. Results**

Comparison between RDT and Fujiwara and Arakawa (1995) model results concerning the pressure-dilatation correlation term Π*d* are represented in Figures 17(a), (b), (c). As one can remark from these results, that the contribution of Π*d* obtained by this model is negligible. Except in the case of the intermediate regime (*Mg*0 = 12), the Fujiwara and Arakawa model does not represent an appreciable discrepancies with the RDT results.

**Figure 17.** Evolution of the pressure dilatation correlation term Π*d* (a) case A0 (*Mg*0 = 1), (b) case A4 (*Mg*0 = 12) and (c) case A10 (*Mg*0 = 66.7). : RDT results, : Fujiwara and Arakawa model.

#### **8. Conclusion**

Rapid distortion theory (RDT) is a computationally viable option for examining linear compressible flow physics in the absence of inertial effects. Evolution of compressible homogeneous turbulence has been described completely by finding numerical solutions obtained by solving linear double correlations spectra evolution. Numerical integration of these equations has been carried out using a second-order simple and accurate scheme. This numerical method has proved more stable and faster and allows in particular to obtain accurately the asymptotic behavior of the turbulence parameters (for large values of *St*) characteristic of equilibrium states. In this chapter, RDT code developed by authors solves linearized equations for compressible homogeneous shear flows (Riahi & Lili, 2011). It has been validated by comparing RDT results with direct numerical simulation (DNS) of Simone et al. (1997) and Sarkar (1995) for various values of initial gradient Mach number *Mg*0 which is the key parameter controlling the level of compressibility. The study of the behavior of the non-dimensional turbulent kinetic energy *q2(t)/q2(0)* permit to determine various regimes of flow. This study allows to check relevance of an incompressible regime for low values of initial gradient Mach number *Mg*0, of an intermediate regime for moderate values of *Mg*0 and of a compressible regime for high values of *Mg*0. Agreement between RDT and DNS of Simone et al. (1997) and Sarkar (1995) is obtained for small values of the non-dimensional times *St* (*St* < 3.5). This agreement gives new insight into compressibility effects and reveals the extent to which linear processes are responsible for modifying the structure of compressible turbulence. The behavior analysis of the various terms presented in the turbulent kinetic energy budget shows that compressibility affects more the turbulent production which becomes preponderant in the compressible regime. This property is already observed in the analysis of

the budget of the Reynolds stress equations related to <sup>2</sup> <sup>1</sup> *u* and 1 2 *u u* . Another important property reveals that anisotropy increases with compressibility.

Agreement of RDT with DNS (Simone et al., 1997) found for large values of *St* (*St* > 10) in particular for large values of *Mg*0 which allows to determine equilibrium states in the compressible regime. Evolution of the relevant component of the Reynolds stress anisotropy tensor *b*12, for different values of initial gradient Mach number *Mg*0 (66.7, 200, 500 and 1000) and for large values of *St*, shows that *b*12 is independent of the initial turbulent Mach number *Mt*0 (which is a parameter characterizing the effects of compressibility).In addition, *b*12 becomes stationary all the more quickly as *Mg*<sup>0</sup> is large (*Mg*0 = 1000). Considering equilibrium states associated to this value of *Mg*0 as representative of equilibrium states related to infinite *Mg*<sup>0</sup> (pressure-released regime), equilibrium values of the Reynolds stress anisotropy components <sup>11</sup> *<sup>b</sup>* and <sup>22</sup> *<sup>b</sup>* are independent of the initial turbulent Mach number *Mt*0. In contrast, equilibrium values of the normalized dissipation rate due to dilatation, the normalized pressure-dilatation correlation and the normalized pressure variance are dependent of *Mt*0. Variation of the equilibrium normalized pressure variance with various initial turbulent Mach

$$\text{Number (M\uplus 0.25, 0.4, 0.5 and 0.6) can be written as } \left(\frac{p^2}{\overline{\rho}^2 q^4}\right)\_{\sigma} = \text{(M\uplus)}^{\alpha} \text{ where } \alpha = 5.8.$$

In conclusion, after a critical analysis, we were able to justify that RDT permits to well identify compressibility effects in order to develop models taking them into account satisfactorily. The analysis of rapid distortion theory showed that it is possible to better understand the compressible turbulent flows. In addition, RDT is valid to predict asymptotic equilibrium states for compressible homogeneous sheared turbulence for large values of initial gradient Mach number (pressure-released regime). RDT is an efficient method for testing linear contribution of turbulence models.
