Spectral Modeling and Numerical Simulation of Compressible Homogeneous Sheared Turbulence

1. Introduction

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, M_d, 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 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 M_g0 (Riahi & Lili, 2011).

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 M_g0 describing various regimes of flow. In fact, the study of the behavior of the non-dimensional turbulent kinetic energy q²(t)/q²(0) (withq2=uiui¯) allows to check relevance of an incompressible regime for low values of initial gradient Mach number, of an intermediate regime for moderate values of M_g0 and of a compressible regime for high values of M_g0 (Riahi et al., 2007). The pressure-released regime is related to infinite values of M_g0 (Riahi & Lili, 2011). The gradient Mach number M_g 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 τa=lafor a large eddy to the mean flow time scaleτd=1S: M_g =τaτd=Sla, where l is an integral lengthscale and a is the mean sound speed. The lengthscale of energetic structure is expressed byl=q3ε, where εis the rate of turbulent kinetic energy dissipation. Sarkar (1995) also used Sla(which he referred to as ‘gradient Mach number’ M_g), 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 M_t = 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 u_i and pressure p lead to general RDT equations:

where γis the ratio of specific heats,ρ¯is the mean density and νis the kinematic viscosity.

The dot superscript denotes a substantial derivative along the mean flow trajectories related to mean field velocityUi¯.

Equation (1) is derived from continuity equation

associated to isentropic fluctuations hypothesiss˙=0which is translated by

and leads to equation (1). So, the isentropic hypothesis s˙=0simplifies governing equations by keeping the variables u_i and p and eliminatingρ. We can obviously discard the hypothesis s˙=0by 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 hypothesiss˙=0, Blaisdell et al. (1993) introduced a polytropic coefficient n defined by,pP¯=nρρ¯n =γcorresponding 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 coefficientn=(p2¯/P¯2ρ2¯/ρ¯2)12.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 thanγ=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 " ^ ") 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:

where λijis the mean velocity gradient defined by:

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):

where φ^1(k→,t)and φ^2(k→,t)are the solenoidal contributions and φ^3(k→,t)is the dilatational contribution.

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

whereφ^4(k→,t)is related to the pressure and takes the following form:

We then write evolution equations of these doubles correlations:

where k'=k12+k32and k1,k2,k3are the components of the wave vectork→.

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

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

3.3. Results and discussion

All Figures 1-8 validate RDT approach and numerical code for low values of the non-dimensional 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.

In the compressible regime (M_t0 = 0.25 and M_g0 = 66.7), RDT and DNS equilibrium values of the non-dimensional production -2b₁₂ 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 -2b₁₂ (strongly compressible regime M_g0 = 66.7 >> 1 and r0>> 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 b₁₂ 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 b₁₂ (which is linked to production), b₁₁ and b₂₂ components anisotropy tensor in the incompressible (M_g0 = 2.2) and intermediate regimes (M_g0 = 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 b₁₂, b₁₁ and b₂₂ 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.

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 M_g0. This essential feature justifies the resort to RDT in order to critically study equilibrium states in the compressible regime.

4. Various regimes of flow

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 M_g0 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 M_g0 becomes more significant. It appears from this figure that the turbulent kinetic energy varies quasi-linearly in cases A₁ and A₂, where M_g0 takes respectively values 2.7 and 4. These two values of M_g0 correspond to the incompressible regime. A weak amplification of the turbulent kinetic energy shows that cases A₄ (M_g0 = 12) and A₁₀ (M_g0 = 66.7) correspond to the intermediate regime. From M_g0 = 16.5, the regime becomes compressible. Indeed, cases A₅,…, A₁₀ show a significant amplification of the total kinetic energy more and more marked when the initial gradient Mach number M_g0 increases.

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

Several explanations were proposed these last years to analyze causes of the stabilising effect of compressibility which are still not cleared up.

5. Compressibility effects on structure of homogeneous sheared turbulence

5.3. Results

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 Πdintervene significantly in this budget. In the compressible regime [Fig. 12(c)], the production is dominating and the pressure-dilatation Πdremains with relatively low values (practically null until St = 1). Figure 12(a) shows the results for M_g0 = 1 and M_t0 = 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 εdand the pressure-dilatation Πdare not negligible in spite of the small value of the initial gradient Mach number M_g0. Consequently, we can affirm that ‘‘the incompressible behavior’’ cannot be obtained in the borderline case of low values of M_g0; that’s why we preferred to give the results for M_g0 = 1 and not for M_g0 = 2.7 and 4 with an aim of better approaching the ‘‘incompressible limit’’.

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 u12¯and u1u2¯(Riahi et al., 2007).

In Figures 13(a), (b), (c), (d) are represented the different components of the anisotropy tensorbij. These figures show obviously a remarkable property concerning the anisotropy. In fact, it appears clearly that the anisotropy increases with M_g0 i.e. with compressibility and that it is responsible of the behavior of u12¯which becomes dominating compared to u22¯and u32¯in the compressible case (M_g0 = 66.7). As an indication, u12¯u22¯=12.62and u12¯u32¯=13.11for St = 3.5.

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 b₁₂ 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, b₁₂ 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 b₁₂ 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: (b11)∞= 0.66, (b22)∞=(b33)∞= - 0.33 and (b12)∞= 0. These values confirm the independence of the anisotropy tensor with M_t0 in the pressure-released 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):(u12¯)∞=q2, (u22¯)∞=0and(u32¯)∞=0.

Another interesting property of the pressure-released regime is the quadratic increase of u12¯as shown in Figure 15 related to M_g0 = 10⁶ and M_t0 = 0.25 (Riahi & Lili, 2011). This property has been established by Livescu et al. (2004) as analytical RDT solution in the pressure-released limit.

Table 2 lists equilibrium values of the normalized dissipation rate due to dilatation(εdεs+εd)∞, the normalized pressure-dilatation correlation (pd¯Sq2)∞and the normalized pressure variance (p2¯ρ¯2q4)∞for various values of initial turbulent Mach number M_t0 (Riahi & Lili, 2011). From this table, we deduce the dependence of all these parameters with M_t0.

Figure 16 shows variation of the equilibrium normalized pressure variance (p2¯ρ¯2q4)∞with various initial turbulent Mach number (M_t0 = 0.25, 0.4, 0.5 and 0.6 (M_t0 = 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 (p2¯ρ¯2q4)∞=(M_t0)αwhere α = 5.8.

7. Turbulence model to be tested

7.3. Results

Comparison between RDT and Fujiwara and Arakawa (1995) model results concerning the pressure-dilatation correlation term Πdare represented in Figures 17(a), (b), (c). As one can remark from these results, that the contribution of Πdobtained by this model is negligible. Except in the case of the intermediate regime (M_g0 = 12), the Fujiwara and Arakawa model does not represent an appreciable discrepancies with the RDT results.

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 M_g0 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 M_g0, of an intermediate regime for moderate values of M_g0 and of a compressible regime for high values of M_g0. 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 u12¯andu1u2¯. 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 M_g0 which allows to determine equilibrium states in the compressible regime. Evolution of the relevant component of the Reynolds stress anisotropy tensor b₁₂, for different values of initial gradient Mach number M_g0 (66.7, 200, 500 and 1000) and for large values of St, shows that b₁₂ is independent of the initial turbulent Mach number M_t0 (which is a parameter characterizing the effects of compressibility). In addition, b₁₂ becomes stationary all the more quickly as M_g0 is large (M_g0 = 1000). Considering equilibrium states associated to this value of M_g0 as representative of equilibrium states related to infinite M_g0 (pressure-released regime), equilibrium values of the Reynolds stress anisotropy components (b11)∞and (b22)∞are independent of the initial turbulent Mach number M_t0. 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 M_t0. Variation of the equilibrium normalized pressure variance with various initial turbulent Mach number (M_t0 = 0.25, 0.4, 0.5 and 0.6) can be written as (p2¯ρ¯2q4)∞=(M_t0)αwhere α = 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.