3.1. Computational details

Turbulence is governed by the Navier–Stokes equations representing the conservation of mass, momentum and energy,

Here, ${\it\rho}$ is density, $u_{i}$ is the $i$ th component of velocity, $p$ is pressure, $e$ is internal energy per unit mass, $T$ is temperature, ${\it\kappa}$ is thermal conductivity and $f_{i}$ is the external forcing. The viscous stress tensor, ${\it\sigma}_{ij}$ , and strain rate tensor, $\unicode[STIX]{x1D61A}_{ij}$ , are given by, Due to the continuous input of energy through $\boldsymbol{f}$ in the momentum equations, in order to maintain a stationary state energy needs to be removed from the system. This is accomplished through an additional term, ${\it\Lambda}$ , in the energy equation (3.3). This can be done in several ways as discussed in § 3.2. In order to close the set of equations, we consider an ideal gas that follows the equation of state,

where $R$ is the gas constant of the fluid.

The equations are solved in conservative form using compact schemes (Lele Reference Lele1992) with periodic boundary conditions in all three spatial directions. In order to control aliasing errors, nonlinear terms are computed using a skew-symmetric form (Blaisdell, Mansour & Reynolds Reference Blaisdell, Mansour and Reynolds1993). Compact schemes have a very high resolving efficiency compared to conventional finite difference schemes and, unlike spectral schemes, are more amenable to more complex geometries involving different types of boundary conditions. Formally, the derivative $g^{\prime }$ of a function $g$ is approximated by, $\sum _{j=-m}^{m}{\it\alpha}_{j}g_{i+j}^{\prime }=\sum _{j=-n}^{n}a_{j}g_{i+j}$ , where $m$ , $n$ define the stencil size and ${\it\alpha}_{j},a_{j}$ are the coefficients of the stencil for the derivative and function. This leads to a system of linear equations of the form $\unicode[STIX]{x1D648}\boldsymbol{g}^{\prime }=\unicode[STIX]{x1D63C}\boldsymbol{g}$ , where $\unicode[STIX]{x1D648}$ is a banded matrix. A family of schemes is obtained by varying $m$ and $n$ . Specifically, for $m=2$ , $n=3$ a tenth-order scheme is obtained, which requires solving a penta-diagonal system. This can be solved efficiently using highly optimized mathematical libraries in $\mathit{O}(59N)$ operations (Lv & Le Reference Lv and Le2008). The code, which leverages the spectral-like accuracy of compact schemes, is parallelized using a hybrid MPI-OpenMP technique. The domain is decomposed in two directions which enables the utilization of up-to $N^{2}$ processors for a problem size of $N^{3}$ . For problems of current interest where $N=2048$ , a maximum of $2048^{2}$ processing cores can be used which is beyond the largest supercomputing facilities available currently, but may be available in so-called exascale computing.

The derivatives can be efficiently computed when one processor has all the data along the direction in which the derivative is sought. This ‘pencil’ of data is initially aligned, say, with $x$ . All the terms involving derivatives in the $x$ direction are then computed. However, in order to compute the entire right-hand side of the Navier–Stokes equations, derivatives along the other two directions are required. There are two options for computing these derivatives: (i) by halo exchanges along the processor boundaries, (Cook et al. Reference Cook, Cabot, Williams, Miller, Supinski, de Yates and Welcome2005) or (ii) by transposition of data from $x$ to $y$ and subsequently $y$ to $z$ pencils (Jagannathan & Donzis Reference Jagannathan and Donzis2012). While both methods have been shown to scale well at core counts of $\mathit{O}(10^{4})$ , the latter is independent of the order of accuracy and facilitates overlapping of communication and computation and has been used for this work. Once the derivatives are computed, the variables are marched in time using a third-order low-storage Runge–Kutta scheme. For the following time step, the order of transposition is inverted to save one transpose per variable every other step.

The code has shown very good scaling up to 524, 288 cores on grid resolutions up to $4096^{3}$ . Small departures from ideal scaling are seen only at very large core counts, where the memory footprint is of the order of cache size and hence latency effects, which can be attributed to communication, tend to overwhelm the computational time. Employing an additional level of parallelism based on shared memory alongside message passing techniques has decreased the communication time by as much as 30 %. The implementation, benchmarks, and other diagnostic methods are described in Jagannathan & Donzis (Reference Jagannathan and Donzis2012).

3.2. Simulation database

Direct numerical simulations (DNS) at different Reynolds and Mach number are required, for instance, to predict asymptotes at high Reynolds number and assess effects of compressibility. We use a large database of homogeneous forced isotropic compressible turbulence (Donzis & Jagannathan Reference Donzis and Jagannathan2013b ) to investigate the Reynolds and Mach number scaling of solenoidal and dilatational components of several quantities, effects of dilatation on thermodynamic quantities and the dynamics of energy transfer between internal and kinetic modes. Some of the simulations in the database have been extended for a longer duration to decrease statistical variability.

In order to maintain a stationary state, energy is injected at the large scales through the momentum equations, a technique that has been used extensively in incompressible (Eswaran & Pope Reference Eswaran and Pope1988; Yeung & Brasseur Reference Yeung and Brasseur1991; Overholt & Pope Reference Overholt and Pope1998; Rosales & Meneveau Reference Rosales and Meneveau2005) and in compressible flows (Kida & Orszag Reference Kida and Orszag1990; Petersen & Livescu Reference Petersen and Livescu2010; Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2012). Forcing in compressible turbulence, however, has a much wider and interdependent parameter space because the forcing parameters (mean temperature $T_{0}$ , number of forced wavenumbers ${\it\kappa}_{f}$ , amplitude of forcing, extent of solenoidal or dilatational forcing and time scale of forcing) are used to control not only the Reynolds number but also the turbulent Mach number which in turns depends on e.g. the mean temperature. Each of these parameters may affect the flow variables simultaneously. For instance, if $T_{0}$ is decreased, we might expect $M_{t}$ to increase ( $M_{t}\propto u/\sqrt{T}$ ), but it could also change ${\it\mu}$ thus affecting the Reynolds number as well. This interplay of parameters compounded by constraints of achieving (i) well-resolved simulations ( $k_{max}{\it\eta}\approx 1.5$ , to be shown later), (ii) the integral length scale being a fraction of the domain size, and (iii) the maximum $R_{{\it\lambda}}$ for a given grid resolution, makes it a difficult problem to conduct a parametric variation of $R_{{\it\lambda}}$ and $M_{t}$ . This may account for why investigations of forced compressible turbulence have been scarce compared to incompressible turbulence.

We have implemented a low-wavenumber stochastic forcing based on Eswaran & Pope (Reference Eswaran and Pope1988), that acts on wavenumbers ( ${\it\kappa}$ ) inside a spherical shell of radius ${\it\kappa}_{f}$ ( ${\it\kappa}\leqslant {\it\kappa}_{f},{\it\kappa}_{f}=3$ ). Stochastic forcing for compressible flows have been used before, though it has been limited to, e.g., a single wavenumber (Kida & Orszag Reference Kida and Orszag1990). In this work, the forcing term in (3.2) is given by the vector

where $\hat{\boldsymbol{f}}_{\bot }(\boldsymbol{k})$ is the projection onto the plane perpendicular to the wavenumber vector (which assures purely solenoidal forcing) of a random vector constructed using six independent integrated Uhlenbeck–Ornstein random processes. The injected energy cascades by nonlinear interactions and is dissipated at the small scales. Since this increases the total energy of the system, one has to devise a mechanism to remove energy to maintain a stationary state. We have implemented this in two different ways: by removing energy (i) such that the mean internal energy is constant and (ii) homogeneously and with a value equal to the average energy input at every step. Consistent with previous studies (Wang et al. Reference Wang, Wang, Xiao, Shi and Chen2010) results are virtually unaffected by the specific means in which energy is removed from the system. The temperature dependence of viscosity and diffusivity (for scalars) follow a power-law dependence with an exponent of 0.5, and the fluid is assumed to be an ideal gas with a Prandtl number, $Pr$ of $0.72$ . The relative difference between different temperatures dependencies of viscosity used in the literature is only a few percentage points (Wang et al. Reference Wang, Wang, Xiao, Shi and Chen2010) and the conclusions here are, thus, unaffected by the specific choice of the exponent as our own tests confirm.

While it is possible to force independently both the solenoidal and dilatational mode of velocity, in this work, our focus will be in simulations where only the solenoidal mode is forced. The main reason for this choice is to study dilatational motions as they emerge from Navier–Stokes dynamics as opposed to those imposed by external factors such as dilatational forcing. The DNS database, summarized in table 1, consists of grid resolutions from $64^{3}$ to $2048^{3}$ with Taylor Reynolds number ranging from 38 to 450 at different Mach numbers, $M_{t}=0.1,0.2,0.3,0.4$ and $0.6$ . The table includes basic information about integral and Taylor length scales as well as the ratio of dilatational to solenoidal kinetic energy, dissipation rate, and pressure.

Table 1.

Simulation database: number of grid points ( $N^{3}$ ), Reynolds and Mach numbers ( $R_{{\it\lambda}}$ and $M_{t}$ ), normalized integral and Taylor length scales, ratio of dilatational to solenoidal kinetic energy, dissipation and r.m.s. of pressure, ratio of mean pressure-dilatation correlation to mean dissipation, as well as the solenoidal and dilatational pressure-dilatation correlation to mean dissipation.

To ensure accurate small-scale resolution, we have done convergence studies. We first note that while strong compressions (so-called ‘shocklets’) may in principle impose further constraints in resolution, it has been suggested that the most probable shock thickness is of the order of the mean Kolmogorov scale (Samtaney et al. Reference Samtaney, Pullin and Kosovic2001). Furthermore, even the strongest shocklets have been suggested to be comparable to the strongest velocity gradients found in incompressible turbulence due to intermittency (Donzis & Jagannathan Reference Donzis and Jagannathan2013b ). This suggests that the resolution criterion may not be much stricter than in well-resolved incompressible DNS. Still, since these are theoretical results based on strong assumptions, it is important to assess accuracy based on numerical data.

Figure 1.

P.d.f. of pressure fluctuations normalized at $R_{{\it\lambda}}\approx 60$ for (a) $M_{t}\approx 0.1$ (b) $M_{t}\approx 0.6$ . Symbols represent ${\it\eta}/{\rm\Delta}x\approx 0.125$ (○), $0.25$ (▫), $0.50$ (▵), $1.0$ (♢).

It is also known that the resolution requirement typically depends on the quantity of interest (Watanabe & Gotoh Reference Watanabe and Gotoh2007; Donzis, Yeung & Pekurovsky Reference Donzis, Yeung and Pekurovsky2008a ). Since our main interest is in the effect of thermodynamic variables and dilatation of the flow, we consider pressure and higher-order moments of velocity gradients as our convergence criteria. The p.d.f. of pressure is shown in figure 1 at $R_{{\it\lambda}}\approx 60$ and $M_{t}~\approx ~0.1$ and 0.6 for different grid resolutions. While the shape of the p.d.f. differs for different $M_{t}$ (see § 6 for detailed analysis), the p.d.f.s themselves are converged even at ${\it\eta}/{\rm\Delta}x\approx 0.25$ showing little sensitivity to small-scale resolution. High-order moments of velocity gradients, on the other hand, are more sensitive than other quantities such as correlations and spectra (Donzis, Yeung & Sreenivasan Reference Donzis, Yeung and Sreenivasan2008b ). In figure 2 we show these moments up to order $4$ . At low $M_{t}$ ( ${\approx}0.1$ ), both third and fourth-order moments are converged at ${\it\eta}/{\rm\Delta}x\approx 0.5$ . Indeed, we have verified that moments up to order three are converged when ${\it\eta}/{\rm\Delta}x\approx 0.5$ for all Mach numbers in the database. We note, however, that the fourth-order moment at the highest $M_{t}$ ( ${\approx}0.6$ ) shows a slight increase at ${\it\eta}/{\rm\Delta}x\approx 1$ , which may suggest a more stringent resolution requirement. For all other Mach numbers, however, fourth-order moment are converged. Thus, for the results shown in this paper, a resolution of ${\it\eta}/{\rm\Delta}x\approx 0.5$ is found to be sufficient. This seems consistent with the results of Samtaney et al. (Reference Samtaney, Pullin and Kosovic2001) where it was estimated that the most probable shocklet thickness is a few times larger than the Kolmogorov scale. Thus, resolving Kolmogorov scales would also provide enough resolution for shocklets. The convergence seen in figure 2 and other conditions in our database seems to support this conclusion.

Figure 2.

Normalized moments of longitudinal velocity gradients, $X=(\partial u_{i}/\partial x_{i})/3$ , at $R_{{\it\lambda}}\approx 60$ for $M_{t}\approx 0.1$ (a,c) and $M_{t}\approx 0.6$ (b,d) for different ${\it\eta}/{\rm\Delta}x$ . The error bars correspond to a 95 % confidence interval.

3.3. Solenoidal and dilatational components

As mentioned before, the velocity field is decomposed into solenoidal and dilatational field, for which we will use superscripts $s$ and $d$ respectively. This decomposition is unique for homogeneous flows. Since $\boldsymbol{u}^{\boldsymbol{s}}$ and $\boldsymbol{u}^{\boldsymbol{d}}$ are orthogonal to each other on the average $(\langle \boldsymbol{u}^{\boldsymbol{s}}\boldsymbol{\cdot }\boldsymbol{u}^{\boldsymbol{d}}\rangle =0)$ , respective kinetic energies are related as $\langle \mathscr{K}\rangle =\langle \mathscr{K}^{s}\rangle +\langle \mathscr{K}^{d}\rangle$ (Kida & Orszag Reference Kida and Orszag1990; Lele Reference Lele1994), where the solenoidal and dilatational components of mean kinetic energy is given by $\langle \mathscr{K}^{s}\rangle =\langle {\it\rho}\boldsymbol{u}_{i}^{\boldsymbol{s}}\boldsymbol{u}_{i}^{\boldsymbol{s}}/2\rangle$ and $\langle \mathscr{K}^{d}\rangle =\langle {\it\rho}\boldsymbol{u}_{i}^{\boldsymbol{d}}\boldsymbol{u}_{i}^{\boldsymbol{d}}/2\rangle$ , with summation implied on $i$ . Using Reynolds decomposition, we decompose a random variable $X$ into mean $\langle X\rangle$ and fluctuations $X^{\prime }$ , such that $X=\langle X\rangle +X^{\prime }$ . For quantities with zero mean, the primes are dropped for simplicity in notation. Two examples are the velocity field $u$ and forcing term $f$ .

Sarkar et al. (Reference Sarkar, Erlebacher, Hussaini and Kreiss1991) proposed a way to decompose $\langle {\it\epsilon}\rangle$ into a solenoidal ( $\langle {\it\epsilon}^{s}\rangle$ ) and dilatational ( $\langle {\it\epsilon}^{d}\rangle$ ) component by recasting them in terms of fluctuating vorticity ( ${\it\omega}_{i}$ ) and dilatation ( ${\it\theta}$ ). Here we use

Note that if the fluctuations in ${\it\mu}$ are small, one can write (Sarkar et al. Reference Sarkar, Erlebacher, Hussaini and Kreiss1991) $\langle {\it\mu}{\it\omega}_{i}{\it\omega}_{i}\rangle \approx \langle {\it\mu}\rangle \langle {\it\omega}_{i}{\it\omega}_{i}\rangle$ and $\langle {\it\mu}{\it\theta}^{2}\rangle \approx \langle {\it\mu}\rangle \langle {\it\theta}^{2}\rangle$ .

Similarly, the fluctuating pressure can be decomposed into a solenoidal pressure satisfying the incompressible pressure Poisson equation,

and the dilatational pressure, $p^{d}=p^{\prime }-p^{s}$ , defined as the difference between the total pressure and solenoidal pressure fluctuations (Sagaut & Cambon Reference Sagaut and Cambon2008). To avoid confusion, it is important to note that we use here the term total pressure fluctuations to denote $p^{\prime }$ (the sum of solenoidal and dilatational components) and not the stagnation pressure, which is sometimes also called total pressure.

4.1. Dissipative anomaly

A particular topic in incompressible turbulence that has accrued several decades of research is the phenomenon of dissipative anomaly, the concept that at sufficiently high Reynolds number and away from walls, the mean energy dissipation rate becomes independent of the fluid viscosity (Sreenivasan Reference Sreenivasan1984, Reference Sreenivasan1998; Kaneda et al. Reference Kaneda, Ishihara, Yokokawa, Itakura and Uno2003; Donzis et al. Reference Donzis, Sreenivasan and Yeung2005). Following the original work of Taylor (Reference Taylor1935), the dissipation rate per unit volume $\langle {\it\epsilon}\rangle$ can be estimated from large scale characteristics of the flow as:

where $L$ is the integral length scale and $u$ the r.m.s. velocity ( $u=(\langle u_{i}u_{i}/3\rangle )^{1/2}$ ). Since density is constant in incompressible flows, one can instead use the dissipation rate per unit mass $\langle {\it\varepsilon}\rangle =\langle {\it\epsilon}/{\it\rho}\rangle =\langle {\it\epsilon}\rangle /\langle {\it\rho}\rangle$ , a relation clearly not satisfied in compressible flows. The brackets around ${\it\rho}$ are here unnecessary since ${\it\rho}$ is constant but are kept for consistency in notation when used for compressible flows. Dissipative anomaly is then assessed by plotting the normalized energy dissipation rate $D=\langle {\it\varepsilon}\rangle L/u^{3}$ against the Reynolds numbers. At high enough Reynolds numbers, $D$ approaches an asymptotic value supporting thus the independence of the dissipation rate on viscosity.

We now consider (4.1) for compressible flows. The limited results that are available in literature suggest that at low $M_{t}$ the influence of compressibility on the asymptotic value of the normalized dissipation is small though results are for isothermal flows (Pearson et al. Reference Pearson, Yousef, Haugen, Brandenburg and Krogstad2004), or flows with artificial dissipation added by the numerical scheme and an equation of state independent of temperature (Schmidt, Hillebrandt & Niemeyer Reference Schmidt, Hillebrandt and Niemeyer2006).

Figure 3(a) shows $D=\langle {\it\epsilon}\rangle L/u^{3}\langle {\it\rho}\rangle$ versus $R_{{\it\lambda}}$ from our database. Qualitatively, the figure is very similar to that in incompressible flows (Sreenivasan Reference Sreenivasan1984; Donzis et al. Reference Donzis, Sreenivasan and Yeung2005). In particular, there is a decrease of $D$ with $R_{{\it\lambda}}$ for low values of $R_{{\it\lambda}}$ , but approaches an asymptotic value at high $R_{{\it\lambda}}$ . The asymptotic state appears to be reached at $R_{{\it\lambda}}\approx 100$ consistent with previous incompressible results (Sreenivasan Reference Sreenivasan1984; Donzis et al. Reference Donzis, Sreenivasan and Yeung2005). For $M_{t}\approx 0.3$ , the asymptotic value of $D$ is around $0.43$ . For higher $M_{t}$ , the achievable $R_{{\it\lambda}}$ is not as high making it difficult to assess with certainty the value at the asymptotic state.

The phenomenological underpinning of dissipative anomaly is that large scales set the rate at which energy is transferred through increasingly smaller scales until viscosity is effective and dissipates this energy. A straightforward extension to compressible flows would indicate that $D$ approach a constant too at high $R_{{\it\lambda}}$ , which is approximately, what is seen in figure 3. However, in compressible flows, the mean dissipation could be decomposed into solenoidal and dilatational component, and hence, a similar asymptotic constant can be established for each ( $D_{s},D_{d}$ ). Thus, if dissipative anomaly holds, both components should approach an asymptotic state at high $R_{{\it\lambda}}$ . Since $D_{s}$ is just the difference between $D$ and $D_{d}$ , we show $D_{d}$ in figure 3(b) as a function of $M_{t}$ . Though the overall contribution from $D_{d}$ is less than 5 % for the cases considered, there is a dramatic increase in its relative contribution for $M_{t}>0.3$ . Thus, one can expect the asymptotic value of $D$ to depend on $M_{t}$ though it may be noticeable only at much higher $M_{t}$ . For the range of $M_{t}$ considered in this paper, $D$ is only weakly affected by compressibility. Simulations at increasingly higher Reynolds and Mach numbers are needed to reach an unambiguous conclusion about the dependence of the asymptotic constant on $M_{t}$ .

Figure 3.

Variation of (a) normalized mean energy dissipation rate ( $D$ ), and (b) its dilatational component ( $D_{d}$ ) with $M_{t}$ . Symbols for different Mach numbers in (a) correspond to $M_{t}\approx 0.1$ (○), $0.2$ (▫), $0.3$ (▵), $0.4$ (♢), and $0.6$ (▿), and for different Reynolds numbers in (b) correspond to $R_{{\it\lambda}}\approx 38$ (○), $60$ (▫), $100$ (▵), $160$ (♢), $270$ (▿), and $450$ (◃).

4.2. Velocity and length scales

Kolmogorov (Reference Kolmogorov1941) defined characteristic scales for the smallest scales of motion. The well-known Kolmogorov length, velocity and time scales are, respectively,

Since this applies to incompressible flows, the kinematic viscosity ${\it\nu}$ is constant (and thus brackets are omitted) and ${\it\varepsilon}$ is the mean energy dissipation rate per unit mass. These relations lead to a simple scaling law if the normalized mean energy dissipation rate asymptotes to a constant ( ${\it\varepsilon}L/u^{3}=D$ , as shown in § 4.1). Substituting ${\it\varepsilon}\propto u^{3}/L$ in (4.2) and after some algebra, we obtain

where $L$ is the integral length scale.

Figure 4.

Scaling of (a) Taylor micro scale ( ${\it\lambda}$ ) and integral length scale ( $L$ ) and (b) Kolmogorov velocity scale ( $u_{{\it\eta}}$ ). Dashed lines indicate classical Kolmogorov scaling for incompressible turbulence: ${\it\lambda}/{\it\eta}\sim R_{{\it\lambda}}^{1/2}$ , $L/{\it\eta}\sim R_{{\it\lambda}}^{3/2}$ , $u/u_{{\it\eta}}\sim R_{{\it\lambda}}^{1/2}$ . Symbols for different $M_{t}$ as in figure 3(a).

The extension of (4.2) to compressible flows, though typically straightforward, is not unique. Since ${\it\nu}$ varies in space and time, one could devise dimensionally consistent alternatives. For example, one could use $\langle {\it\mu}\rangle /\langle {\it\rho}\rangle$ or $\langle {\it\mu}/{\it\rho}\rangle$ . Similarly, the energy dissipation rate per unit mass in a compressible flow can be taken as $\langle {\it\epsilon}\rangle /\langle {\it\rho}\rangle$ or $\langle {\it\epsilon}/{\it\rho}\rangle$ . Indeed, one can find different definitions using these alternatives in the literature (Shivamoggi Reference Shivamoggi1995; Samtaney et al. Reference Samtaney, Pullin and Kosovic2001; Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2011). Although we have found that, for the range of parameters studied here, there is no appreciable difference between the different forms, we prefer to proceed by extending Kolmogorov (Reference Kolmogorov1941) reasoning to compressible flows. If we assert that small scales are independent of large scales and will only depend on the rate at which kinetic energy is converted into heat $\langle {\it\epsilon}\rangle$ , the mean viscosity $\langle {\it\mu}\rangle$ and the mean density $\langle {\it\rho}\rangle$ , then dimensional analysis yields:

With these definitions and $\langle {\it\epsilon}\rangle \propto \langle {\it\rho}\rangle u^{3}/L$ , we obtain again (4.3).

In figure 4(a) we plot the non-dimensionalized Taylor micro-scales and integral length scale. While the scaling of Taylor micro-scales conform very well to that of the proposed scaling ( ${\it\lambda}/{\it\eta}\sim R_{{\it\lambda}}^{1/2}$ ) without any $M_{t}$ dependence, the scaling of integral length scales deviate slightly from $R_{{\it\lambda}}^{3/2}$ at low Reynolds number. This is not surprising given that Kolmogorov scaling has been proposed only when the Reynolds number is high enough. However, even at high Reynolds numbers, a close inspection of the data seems to suggest a slight correction with the data presenting a shallower power law. Results at higher Reynolds numbers are needed to unambiguously determine this departure. In any case, for turbulent Mach numbers up to 0.6, the effect of compressibility on low order quantities tend to be weak as is also seen in figure 4(b) for Kolmogorov velocity scale.

5.1. Consequences of equipartition: scaling of dilatational components

One of the results from the asymptotic analysis of Erlebacher et al. (Reference Erlebacher, Hussaini, Kreiss and Sarkar1990) and Sarkar et al. (Reference Sarkar, Erlebacher, Hussaini and Kreiss1991) is the model for the dilatational dissipation rate in the low Mach-number regime ( $\langle {\it\epsilon}^{d}\rangle \propto M_{t}^{2}\langle {\it\epsilon}^{s}\rangle$ ). This result was derived by assuming equipartition in the form $F_{w}\approx 1$ . In addition, this model assumed the ratio of solenoidal and dilatational Taylor scales to be a constant, which as we show below, follows a power law in $M_{t}$ instead. Note that based on the definition of Taylor scales and assuming that density is weakly correlated with the velocity field, we can show that the ratio of solenoidal to dilatational Taylor scales varies as, ${\it\lambda}^{s}/{\it\lambda}^{d}\sim \sqrt{(\langle \mathscr{K}^{s}\rangle /\langle \mathscr{K}^{d}\rangle )(\langle {\it\epsilon}^{d}\rangle /\langle {\it\epsilon}^{s}\rangle )}$ . It can be clearly seen that this would become independent of Reynolds and Mach number only if $\langle \mathscr{K}^{d}\rangle /\langle \mathscr{K}^{s}\rangle$ and $\langle {\it\epsilon}^{d}\rangle /\langle {\it\epsilon}^{s}\rangle$ scale similarly. As will be shown momentarily, they scale differently and a correction factor has to be incorporated while modelling ${\it\lambda}^{s}/{\it\lambda}^{d}$ .

Available results on the scaling of dissipation appear not to be in agreement with each other. Bertoglio, Bataille & Marion (Reference Bertoglio, Bataille and Marion2001) suggested that $\langle {\it\epsilon}^{d}\rangle /\langle {\it\epsilon}^{s}\rangle$ also scales as $M_{t}^{2}$ based on an EDQNM analysis at low Mach numbers. Ristorcelli (Reference Ristorcelli1997) and Fauchet & Bertoglio (Reference Fauchet, Bertoglio and Frisch1998), on the other hand, proposed $M_{t}^{4}$ . It is worth noting here that the scaling exponents predicted by Erlebacher et al. (Reference Erlebacher, Hussaini, Kreiss and Sarkar1990) and Ristorcelli (Reference Ristorcelli1997) are based on different assumptions. While the former decomposed the compressible problem based on convective and acoustic time scales, the latter assumed that the length scales (inner and outer scales) are disparate. However, both these analyses were made for low $M_{t}$ . An even steeper dependence, as $M_{t}^{5}$ , at high turbulent Mach numbers has also been proposed using EDQNM (Sagaut & Cambon Reference Sagaut and Cambon2008). Other relations have also been put forth in the literature as $\langle {\it\epsilon}^{d}\rangle \propto \langle {\it\epsilon}^{s}\rangle F(M_{t})$ , where different relations are possible based on the definition of $F(M_{t})$ (Zeman Reference Zeman1990; Blaisdell & Zeman Reference Blaisdell, Zeman and Spinks1992). Some of these results have been supported by simulations of decaying compressible turbulence at low $R_{{\it\lambda}}$ and $M_{t}$ though conclusions may be dependent on specific features of the particular initial conditions used (Vreman, Sandham & Luo Reference Vreman, Sandham and Luo1996; Ristorcelli Reference Ristorcelli1997). One way of removing the dependence on initial condition is to force the simulation by adding energy to the large scales so that a statistically stationary state is achieved. This is the case in the present study.

Figure 7.

Ratio of dilatational to solenoidal (a) dissipation (b) turbulent kinetic energy. Symbols for different Reynolds numbers as in figure 3(b). Solid line in (a) corresponds to best-fit curve $M_{t}^{4.1}$ and dash-dotted line in (b) corresponds to a representative $M_{t}^{2}$ and $M_{t}^{4}$ curve.

Consider first the ratio of solenoidal to dilatational dissipation in figure 7. The DNS appears to suggest two different scaling ranges depending on $M_{t}$ with the transition occurring around $M_{t}\approx 0.3$ . While the contribution of dilatational dissipation is negligibly small at low $M_{t}$ , it increases sharply beyond $M_{t}\geqslant 0.3$ , following a steep power law, $\langle {\it\epsilon}^{d}\rangle /\langle {\it\epsilon}^{s}\rangle \sim M_{t}^{4.1}$ for the range of $M_{t}$ investigated. This indicates that the contribution of dilatational motions to dissipation grows quickly as $M_{t}$ increases, as has been noted in figure 3(b). Stronger fluctuations of local values of dilatational dissipation at high $M_{t}$ may also indicate stronger intermittency, a phenomenon with important implications in shock-turbulence interactions (Donzis & Jagannathan Reference Donzis and Jagannathan2013b ). The Reynolds number effect on the ratio of the different components of dissipation is seen to be very weak.

Scaling laws for the ratio of dilatational and solenoidal kinetic energy, $\langle \mathscr{K}^{d}\rangle /\langle \mathscr{K}^{s}\rangle$ , have also been proposed in the literature. For example, Sarkar et al. (Reference Sarkar, Erlebacher, Hussaini and Kreiss1991) suggested a $M_{t}^{2}$ dependence at low $M_{t}$ based on an asymptotic analysis while Fauchet & Bertoglio (Reference Fauchet, Bertoglio and Frisch1998) proposed $M_{t}^{4}$ and $M_{t}^{2}\mathit{Re}_{L}$ scaling using an EDQNM approximation with different decorrelation functions where $\mathit{Re}_{L}$ is the integral scale Reynolds number. Figure 7(b) shows this ratio from our DNS database. Similar to the scaling of ratio of dissipation rates, we can distinguish different scaling regimes at low and high $M_{t}$ that vary as $M_{t}^{4}$ and $M_{t}^{2}$ respectively. Their dependence on Reynolds number becomes weak at high $M_{t}$ .

We can now use the scaling of dissipation and kinetic energy to obtain the ratio of Taylor scales. Since both $\langle {\it\epsilon}^{d}\rangle /\langle {\it\epsilon}^{s}\rangle$ and $\langle \mathscr{K}^{d}\rangle /\langle \mathscr{K}^{s}\rangle$ appear to be independent of $R_{{\it\lambda}}$ for $M_{t}\geqslant 0.3$ , the ratio of solenoidal and dilatational Taylor scales should also be independent of $R_{{\it\lambda}}$ at high $M_{t}$ . Using the best fits for $\langle {\it\epsilon}^{d}\rangle /\langle {\it\epsilon}^{s}\rangle$ and $\langle \mathscr{K}^{d}\rangle /\langle \mathscr{K}^{s}\rangle$ at high $M_{t}$ , we obtain ${\it\lambda}^{s}/{\it\lambda}^{d}\sim \sqrt{M_{t}^{4.1}/M_{t}^{2.2}}\sim M_{t}^{0.9}$ . This result, along with DNS data is shown in figure 8. Again, two qualitatively different regimes can be identified with a transition around $M_{t}\approx 0.3$ . For low Mach numbers we see a strong Reynolds number dependence and a decreasing trend with $M_{t}$ . This is due to the Reynolds number dependence seen at low $M_{t}$ in the scaling of $\langle \mathscr{K}^{d}\rangle /\langle \mathscr{K}^{s}\rangle$ . This dependence weakens beyond $M_{t}\approx 0.3$ eventually becoming almost independent of $R_{{\it\lambda}}$ at $M_{t}\approx 0.6$ . A best-fit power-law for $M_{t}\geqslant 0.3$ shows a $M_{t}^{1.2}$ scaling which is close to the prediction.

Figure 8.

Ratio of solenoidal and dilatational Taylor scales. Symbols for different Reynolds numbers as in figure 3(b). Dash-dotted lines correspond to a representative $M_{t}^{-2}$ and $M_{t}^{0.9}$ .

From figure 8 we conclude that the DNS data do not support a constant ${\it\lambda}^{s}/{\it\lambda}^{d}$ as assumed by Sarkar et al. (Reference Sarkar, Erlebacher, Hussaini and Kreiss1991). However, the latter considered only decaying flows and, thus, does not account for the effect of forcing. As we have already shown above, statistically steady forced flows present significant differences, for example in equilibrium states (equipartition), compared to decaying flows.

6.1. Solenoidal and dilatational pressure scaling

Considering that equipartition appears to depend on $R_{{\it\lambda}}$ and $M_{t}$ , it seems important to understand not only total pressure, but in particular the solenoidal and dilatational components as mentioned in § 3.3. Though different decompositions are possible (Sarkar Reference Sarkar1992), (3.9) is widely used in turbulence modelling. Since by definition the solenoidal pressure satisfies the incompressible pressure Poisson equation, we can expect the relation, $p_{rms}^{s}\sim \langle {\it\rho}\rangle \boldsymbol{u}_{rms}^{\boldsymbol{s}}\boldsymbol{u}_{rms}^{\boldsymbol{s}}$ , to be only weakly affected by Mach number effects (Erlebacher et al. Reference Erlebacher, Hussaini, Kreiss and Sarkar1990).

Figure 9(a) shows that indeed $p_{rms}^{s}$ scales as $\langle {\it\rho}\rangle \langle \boldsymbol{u}_{i}^{\boldsymbol{s}}\boldsymbol{u}_{i}^{\boldsymbol{s}}\rangle$ with a weak Mach number dependence. For $M_{t}\approx 0.1$ the high Reynolds number asymptotic constant is around 0.88 close to the incompressible case at 0.91 (Donzis, Sreenivasan & Yeung Reference Donzis, Sreenivasan and Yeung2012). However, at the highest Reynolds number available for $M_{t}\approx 0.3$ ( $R_{{\it\lambda}}\approx 450$ ), the asymptotic constant decreases to around $0.82$ . While clearly data at higher Reynolds number are needed to establish asymptotes conclusively, small trends with Mach number appear to be present.

One could extend the incompressible scaling to total pressure fluctuations, $p_{rms}\sim \langle {\it\rho}\rangle u^{2}$ , and normalize by the mean pressure to obtain $p_{rms}/\langle p\rangle \sim \langle {\it\rho}\rangle u^{2}/\langle {\it\rho}\rangle R\langle T\rangle \sim u^{2}/c^{2}\sim M_{t}^{2}$ . Recent numerical simulations suggest that $p_{rms}/\langle p\rangle$ grows slightly faster with $M_{t}$ , as $M_{t}^{2.2}$ (Donzis & Jagannathan Reference Donzis and Jagannathan2013a ). This departure can be ascertained by exploring the scaling of the individual components of pressure. When $p_{rms}^{s}$ is scaled by the mean pressure, as shown in figure 9(b), we see excellent agreement with $M_{t}^{2}$ and almost no Reynolds number dependence. However, when the dilatational pressure is scaled similarly (in figure 9(c)), a steeper exponent is observed at low $M_{t}$ (close to 4.0) while an emerging $M_{t}^{2}$ scaling is seen for $M_{t}\geqslant 0.3$ . The higher exponent at low $M_{t}$ may explain the minor deviation seen in Donzis & Jagannathan (Reference Donzis and Jagannathan2013a ) only if dilatational pressure provides a non-negligible contribution to total pressure at low $M_{t}$ . As we show momentarily, however, this is not the case.

Figure 9.

Scaling of solenoidal pressure (a,b) and dilatational pressure (c). Symbols for different Mach numbers in (a) and Reynolds numbers in (b,c) as in figure 3. Dashed line in (a) corresponds to $p_{rms}^{s}/\langle {\it\rho}\rangle \boldsymbol{u}_{\mathit{rms}}^{\boldsymbol{s}}\boldsymbol{u}_{\mathit{rms}}^{\boldsymbol{s}}=1$ for reference. Dash-dotted lines in (b,c) indicate a slope of 2.

In order to assess the relative contributions of the dilatational and solenoidal pressure, the ratio of their r.m.s. values is shown in figure 10(a). While at low $M_{t}$ , $p_{rms}^{d}$ is only approximately 10 % of $p_{rms}^{s}$ , there is a sharp increase around $M_{t}\approx 0.3$ . Beyond $M_{t}\approx 0.3$ , the ratio remains close to $1.0$ , with a weaker Reynolds number dependence than at lower $M_{t}$ . Here, we note again the qualitative change in behaviour seen at $M_{t}\approx 0.3$ .

Figure 10.

(a) Ratio of dilatation to solenoidal pressure fluctuations. (b) Correlation coefficient, $C_{p}$ between solenoidal and dilatational pressure respectively. Dashed line indicates a best-fit curve, $C_{p}=-0.334M_{t}^{2}$ . Symbols for different Reynolds numbers as in figure 3(b).

While a linear analysis of the (inviscid) governing equation may suggest some independence between the components of pressure, it is expected that in real flows, solenoidal and dilatational pressure may in fact be correlated to a degree that would depend on the Reynolds and Mach numbers. Note that this is different from the velocity field whose solenoidal and dilatational components are by construction uncorrelated. The correlation coefficient for pressure components from the DNS data is plotted in figure 10(b). While at low Mach numbers the solenoidal and dilatational pressure behave independently, at higher Mach numbers they become more negatively correlated. One can also see a decreasing dependence on Reynolds number as $M_{t}$ increases. If by lack of any theoretical guidance, one assumes a power-law scaling in $M_{t}$ , the data is seen to be consistent with $C_{p}=-0.334M_{t}^{2}$ . Since by construction, correlation coefficients can take values in the interval $[-1,1]$ , this expression is clearly inadequate for higher $M_{t}$ .

From a theoretical perspective, one could now argue that this particular decomposition of $p$ has merit only at low $M_{t}$ where the two components are uncorrelated, just like the Helmholtz decomposition for the velocity field. At high $M_{t}$ , on the other hand, as dilatational pressure becomes more correlated with solenoidal pressure, its ability to isolate compressibility effects seems to weaken. Yet, from a practical perspective, this correlation might be useful for turbulence modelling. As we have seen, solenoidal pressure is virtually unaffected by compressibility for the range of Reynolds and Mach numbers investigated. This suggests that the p.d.f. of $p^{s}$ may be modelled according to theories from incompressible turbulence and dilatational pressure obtained from this model. For instance, in incompressible turbulence, Vedula & Yeung (Reference Vedula and Yeung1999) proposed a stretched exponential fit for the negative tails of pressure, and a Gaussian distribution for positive tails. Using the correlation between solenoidal and dilatational pressure it is possible to model both components of pressure, and thus the total pressure fluctuations.

We can now go back to the departures from $M_{t}^{2}$ observed for total pressure fluctuations in Donzis & Jagannathan (Reference Donzis and Jagannathan2013a ). By considering both figures 9, and 10 together we can see that at low $M_{t}$ , where the $M_{t}$ dependence is stronger for dilatational pressure, its contribution to total pressure is small. However, as $M_{t}$ increases dilatational pressure provides an increasingly large contribution. By using $p^{\prime }=p^{s}+p^{d}$ we can easily obtain $\langle p^{\prime 2}\rangle =\langle p^{s}p^{s}\rangle +\langle p^{d}p^{d}\rangle +2\langle p^{s}p^{d}\rangle$ or

Using figures 9(a) and 10(b), we can see that the first term on the right-hand side scales as $M_{t}^{4}$ which is expected to be the dominant term at low Mach numbers. In the other limit, at high $M_{t}$ , the variance of dilatational pressure also scales as $M_{t}^{4}$ and since $C_{p}$ cannot continue to grow with $M_{t}$ due to its boundedness, it is plausible to assume that it will approach a constant. In this case, the last term on the right-hand side will also scale as $M_{t}^{4}$ . Thus, $\langle p^{\prime 2}\rangle$ will scale as $M_{t}^{4}$ at low and high $M_{t}$ but the prefactors at both limits may be different.

Therefore, we conclude that the apparent departure from $M_{t}^{4}$ scaling for the variance of pressure when considering the entire range of $M_{t}$ in Donzis & Jagannathan (Reference Donzis and Jagannathan2013a ) appears to reflect the transition of dilatational pressure from low to high Mach numbers at $M_{t}\approx 0.3$ .

6.2. P.d.f. of pressure

Large pressure fluctuations or asymmetries in these fluctuations cannot be captured by its first or second-order moments. Instead one needs to study high-order moments or even the entire p.d.f. The skewness of pressure $S_{p}=\langle p^{\prime 3}\rangle /\langle p^{\prime 2}\rangle ^{3/2}$ , for example, which measures the asymmetry of the p.d.f., has been shown to change from negative to positive as $M_{t}$ is increased (Donzis & Jagannathan Reference Donzis and Jagannathan2013a ). This transition occurs around $M_{t}\approx 0.3$ , similar to other flow statistics presented in previous sections. Negative and positive skewness represents an increased likelihood of moderately low and high-pressure regions in the flow respectively and indicates a qualitative difference in the role of pressure at low and high $M_{t}$ . In figure 11(a) we show this change in the p.d.f. of pressure at $R_{{\it\lambda}}\approx 160$ for different $M_{t}$ .

Figure 11.

P.d.f. of (a) pressure fluctuations and its (b) solenoidal and (c) dilatational components at $R_{{\it\lambda}}\approx 160$ for different $M_{t}$ . Lines in grey correspond to a Gaussian distribution. Symbols for different Mach numbers as in figure 3(a). Arrows are in the direction of increasing $M_{t}$ .

It has been suggested (Blaisdell et al. Reference Blaisdell, Mansour and Reynolds1993) that this change is a result of pressure being a positive quantity and therefore bounded from the negative side ( $p^{\prime }>-\langle p\rangle$ ). However, this does not provide a mechanism that can explain why or how pressure fluctuations increase on the positive side. One could explore the p.d.f. of solenoidal and dilatational pressure individually, for example, to elucidate the contribution from each component.

The p.d.f. of $p^{s}$ ( $f_{ps}$ ) shown in figure 11(b), has a negligible dependence on $M_{t}$ and remains negatively skewed for the range of $M_{t}$ investigated, with positive fluctuations following a Gaussian distribution. This is consistent with results in incompressible turbulence (Pumir Reference Pumir1994; Vedula & Yeung Reference Vedula and Yeung1999). On the other hand, the p.d.f. of $p^{d}$ ( $f_{pd}$ ), shown in figure 11(c), remains close to Gaussian for negative fluctuations, but becomes positively skewed beyond $M_{t}\approx 0.3$ . Thus the positive skewness of pressure at high $M_{t}$ appears to be a direct result of the large contributions of dilatational pressure. Note that the appearance of dilatational pressure is a consequence of Navier–Stokes dynamics as the forcing scheme provides only solenoidal modes.

This conclusion is strengthened if we consider the following observations. First, since $p^{d}$ becomes anti-correlated with $p^{s}$ at high $M_{t}$ (see figure 10 b) and $f_{ps}$ is skewed to the left, we can expect the tails of $f_{pd}$ to be wider on the right. As long as this anti-correlation becomes stronger with $M_{t}$ and $p^{s}$ remains skewed to the left, $p^{d}$ would continue to show stronger and stronger positive fluctuations mirroring more and more the negative tails of solenoidal pressure. Second, in figure 10(a), we already observed that as $M_{t}$ is increased beyond $0.3$ , the dilatational pressure fluctuations tend to be of the same order as that of its solenoidal counterpart which suggests that strong positive fluctuations are related to dilatational effects. It then seems natural to investigate the differences in regions of low and high dilatational motions. This is the focus of the next sections.

6.3. P.d.f. of dilatation

The behaviour of dilatation is particularly relevant in understanding the effects of compressions and expansions in the flow. Though somewhat arbitrary, shocklets are commonly defined as regions with instantaneous dilatation being less than a threshold, ${\it\theta}^{\prime }<-3{\it\theta}_{rms}$ (Samtaney et al. Reference Samtaney, Pullin and Kosovic2001) and have been suggested to have different effects on the flow. For instance, shocklets can increase the rate of dilatational dissipation. While considerable attention has been given to the role of shocklets, the effect of expansions has not been investigated in any detail. A strong local expansion, for instance, can result in a large value of pressure-dilatation correlation, which could bring about a strong energy transfer between kinetic and internal energy.

Figure 12.

P.d.f. of dilatation at $M_{t}\approx$ (a) 0.1 (b) 0.3 (c) 0.6. Symbols for different Reynolds numbers as in figure 3. Arrows are in the direction of increasing $R_{{\it\lambda}}$ .

We consider the Reynolds and Mach number variation of the p.d.f. of normalized dilatation ( ${\it\theta}^{\prime }/{\it\theta}_{rms}$ ) for $M_{t}\approx 0.1,0.3,0.6$ in figure 12. At low $M_{t}$ , the tails of the p.d.f. become wider with $R_{{\it\lambda}}$ , but remains symmetric, indicating that regions of large positive and negative dilatation are equally likely to appear intermittently. This $R_{{\it\lambda}}$ effect is typical of all velocity gradients in incompressible turbulence (Donzis et al. Reference Donzis, Yeung and Sreenivasan2008b ). A similar behaviour is observed at $M_{t}\approx 0.3$ , though tails appear to be narrower at low $R_{{\it\lambda}}$ . It is conceivable that this decline in the appearance of large gradients in the flow is related to the change in several flow statistics at $M_{t}\approx 0.3$ that has been investigated so far. For higher $R_{{\it\lambda}}$ , the tails grow wider but retain the symmetry. Since high $R_{{\it\lambda}}$ simulations are available only for $M_{t}\approx 0.3$ and $0.4$ , it is not possible to comment on their Mach number variation. Finally, at $M_{t}\approx 0.6$ , the p.d.f. becomes asymmetric with wider tails on the left, that grow with $R_{{\it\lambda}}$ .

More quantitative information about large fluctuations can be obtained by computing high-order moments, in particular skewness, $S_{{\it\theta}}$ , and flatness factors, $F_{{\it\theta}}=\langle {\it\theta}^{\prime 4}\rangle /\langle {\it\theta}^{\prime 2}\rangle ^{2}$ , which quantifies the asymmetry and the wideness of the tails. The variation of skewness and flatness of dilatation with $M_{t}$ is seen in figure 13(a,b). Beyond $M_{t}\approx 0.3$ , the negative skewness increases (in magnitude) with $M_{t}$ , a result also reported in the literature (Lee et al. Reference Lee, Lele and Moin1991; Pirozzoli & Grasso Reference Pirozzoli and Grasso2004; Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2012).

Figure 13.

Variation of (a) skewness and (b) flatness of dilatation with $M_{t}$ . Arrows are in the direction of increasing $R_{{\it\lambda}}$ . Symbols for different Reynolds number as in figure 3. Dash-dotted line in (b) corresponds to a flatness value of 3 for a Gaussian distribution. Dashed lines in (a,b) connect data at fixed Reynolds number for clarity except for the highest Reynolds numbers where not enough data at different $M_{t}$ are available to see trends (cf. table 1).

Table 2.

Percentage volume of normalized dilatation ( ${\it\theta}^{\prime }/{\it\theta}_{rms}$ ) in different bins.

While the $R_{{\it\lambda}}$ dependence of skewness is negligible at low $M_{t}$ , there is a substantial change with $M_{t}$ beyond $M_{t}\approx 0.3{-}0.4$ . The flatness values, on the other hand, show a decrease with $M_{t}$ until $M_{t}\approx 0.3{-}0.4$ when it reaches a value around $3.0$ (Gaussian) for $R_{{\it\lambda}}\lesssim 160$ , and then increase sharply. This behaviour is consistent with the narrowing of the p.d.f. at intermediate $M_{t}$ and at low $R_{{\it\lambda}}$ . The data at higher Reynolds numbers ( $R_{{\it\lambda}}\approx 275,450$ ) appear to have higher flatness value at $M_{t}\approx 0.3$ which may indicate stronger intermittency as $R_{{\it\lambda}}$ increases consistent with velocity gradients in incompressible flows. In order to assess the precise Reynolds and Mach number scaling of high order statistics, however, simulations at higher $R_{{\it\lambda}}$ and $M_{t}$ areneeded.

Large flatness value such as seen in figure 13, typically indicates that regions of large positive and negative dilatation appear intermittently in the flow. The percentage volume of their occurrence is given in table 2. While regions of strong compressions ( ${\it\theta}^{\prime }/{\it\theta}_{rms}<-3$ ) and expansions ( ${\it\theta}^{\prime }/{\it\theta}_{rms}>3$ ) are equally likely to occur at low $M_{t}$ , both have an increased propensity to appear at higher Reynolds numbers. At the highest Reynolds number for $M_{t}\approx 0.1$ , the volume occupied by these strong dilatations is close to 1 %. At high $M_{t}$ , this Reynolds number dependence weakens and compressions are about four times more likely to occur than expansions at any given Reynolds number. Also note that for all Reynolds and Mach numbers, between 70 and 90 % of the flow has dilatation smaller than its r.m.s., i.e. $|{\it\theta}^{\prime }|/{\it\theta}_{rms}\leqslant 1$ .

6.4. Relation between thermodynamic variables

It is expected that dilatation, through the coupling of energy equations by the pressure-dilatation correlation, also affects the fluctuations of thermodynamic variables. Consider the Reynolds decomposition of the equation of state for an ideal gas,

which can be used to examine the relative contributions of each of these terms for different $M_{t}$ at different levels of dilatation. The conditional expectation of each term given dilatation is shown in figure 14. In (a) we see that at $M_{t}\approx 0.1$ , the conditional means are symmetric with respect to dilatation, supporting our earlier observation that at low $M_{t}$ , the thermodynamic fluctuations tend to be the same regardless of whether the flow experiences compressions or expansions, contingent on the magnitude of dilatation remaining the same. It is also clear that the dominant contributor to pressure fluctuations is density while the temperature and density–temperature correlation are negligibly small. Note that this does not imply an isothermal flow, for example. Instead, it implies that positive and negative temperature fluctuations are approximately equally likely in regions of same dilatation values which results then in strong cancellations and a very small conditional mean. At high $M_{t}$ , on the other hand, a qualitatively different picture emerges. First, the symmetry between positive and negative values of dilatation is no longer present indicating that expansions and compressions have different effects on the flow. Second, it can be seen that strong expansions and large positive pressure fluctuations (high pressure) are more likely to appear together. Intuitively this may not be unexpected as one can argue that high pressure regions act as precursors for strong expansions. Third, in regions of large expansions, temperature fluctuations and the density–temperature correlation are significantly higher than in corresponding regions of compressions.

Figure 14.

Conditional expectation $\langle X|{\it\theta}^{\prime }/{\it\theta}_{rms}\rangle$ given dilatation with $X=p^{\prime }/\langle {\it\rho}\rangle R\langle T\rangle$ (○), $X={\it\rho}^{\prime }/\langle {\it\rho}\rangle$ (▫), $X=T^{\prime }/\langle T\rangle$ (▵), $X={\it\rho}^{\prime }T^{\prime }/\langle {\it\rho}\rangle \langle T\rangle$ (♢) at $M_{t}\approx$ 0.1 (a) and, $0.6$ (b) at $R_{{\it\lambda}}\approx 160$ .

The asymmetry observed for thermodynamic variables may have an impact on how the relation between them is modelled. Consider for example the so-called isentropic assumption,

By linearizing the above equation, the fluctuations of thermodynamic quantities can be related as, $p^{\prime }/\langle p\rangle \approx {\it\gamma}{\it\rho}^{\prime }/\langle {\it\rho}\rangle \approx ({\it\gamma}/({\it\gamma}-1))T^{\prime }/\langle T\rangle$ . Substituting these relations in (6.2) yields

where the quadratic term in density fluctuations is due to the density–temperature correlation. It is now possible to use with DNS data to assess the validity of this approximation by investigating the conditional expectation of pressure given density. A typical result from such comparison is shown in figure 15 for $R_{{\it\lambda}}\approx 100$ and $M_{t}\approx 0.1$ and $0.6$ . The right-hand side of (6.2) is also included with and without the density–temperature correlation. While the fluctuations in density are very small for $M_{t}\approx 0.1$ , as seen from the small excursions around zero in figure 15, they become significant at high $M_{t}$ , and more specifically, the positive fluctuations are larger. As a result, the fluctuations in pressure are positive and larger in magnitude. Since density fluctuations are normalized by their mean and cannot be less than $-1.0$ , it seems that these large values of pressure, which cause the positive skewness at high $M_{t}$ , are likely to be from density–temperature correlation which is the quadratic term in (6.4). For negative fluctuations of density, the linearized isentropic assumption seems to be a very good approximation when the density–temperature correlation is included, while minor deviations are seen for large positive density fluctuations. Finally, since compressions and expansions behave differently at high $M_{t}$ , they may have profound effect on the flow, for instance, in energy exchanges, through the pressure-dilatation correlation as shown in § 6.6.

Figure 15.

Conditional expectation of $p^{\prime }/\langle {\it\rho}\rangle R\langle T\rangle$ given ${\it\rho}^{\prime }/\langle {\it\rho}\rangle$ at $R_{{\it\lambda}}\approx 100$ and $M_{t}\approx 0.1$ (grey solid circle) and $0.6$ (black solid circle). The linearized isentropic assumption with and without density–temperature correlation are shown as dash-dotted and dashed lines. The inset shows the zoomed in view of the conditional expectation at $M_{t}\approx 0.1$ which follows the linearized isentropic assumption.

6.5. Enstrophy and pressure

Here, we investigate the behaviour of enstrophy ( ${\it\Omega}\equiv \langle {\bf\omega}^{2}\rangle$ , where ${\bf\omega}$ is the vorticity vector), and its relation to pressure. The Reynolds and Mach number dependence of the p.d.f. of enstrophy is shown in figure 16 at $R_{{\it\lambda}}\approx 100$ and $160$ . While fluctuations close to the mean show a weak dependence on $M_{t}$ , stronger fluctuations show a significant monotonic dependence on the Mach number. The tails of the p.d.f. recede with increasing $M_{t}$ regardless of the Reynolds number indicating that with increasing levels of compressibility, high-enstrophy regions become less prominent. The predominance of dilatational field over its solenoidal component at high $M_{t}$ may be thought as one of the reasons for the overall decrease in enstrophy since, by construction, only the solenoidal component of velocity has non-zero vorticity. However, the additional dilatational motions may bring about more shocklets which may also act as source of more enstrophy. A detailed analysis of this effect will be reported elsewhere. Our focus here is on the relation of enstrophy and pressure.

Since high-pressure regions are more probable to occur at high $M_{t}$ (due to the positive skewness of the $f_{p}$ ), and the tails of enstrophy become narrower with increasing $M_{t}$ , it is possible that enstrophy and high-pressure regions may not be as well correlated as low-pressure regions are.

Figure 16.

Mach number dependence of p.d.f. of enstrophy at (a) $R_{{\it\lambda}}\approx 100$ and (b) $R_{{\it\lambda}}\approx 160$ . Symbols for different $M_{t}$ according to figure 3(a).

This can be directly tested by studying the conditional p.d.f. of pressure given enstrophy as shown in figure 17. At low $M_{t}$ (a), consistent with our previous observations, the p.d.f. is seen to be negatively skewed. The skewness, however, tends to increase (in magnitude) with increasing enstrophy, showing that at low $M_{t}$ , low-pressure and high-enstrophy regions tend to co-exist, similar to incompressible turbulence (Pumir Reference Pumir1994).

Figure 17.

Conditional p.d.f. of pressure given enstrophy ( $X={\it\Omega}/\langle {\it\Omega}\rangle$ ) at $R_{{\it\lambda}}\approx 160$ for (a) $M_{t}\approx 0.1$ and (b) $M_{t}\approx 0.6$ . Symbols correspond to $X\approx 0.2$ (○), $0.7$ (▫), $2.7$ (▵), $8$ (♢), and $16$ (▿). Lines in grey correspond to a Gaussian distribution. Arrow in (a) is in the direction of increasing $X$ .

At high $M_{t}$ (figure 17 b), two important observations can be made. First, the conditional p.d.f.s are positively skewed just as the unconditional p.d.f. Second, we see a very weak dependence on the value of the conditioning variable ${\it\Omega}$ , with all p.d.f.s collapsing to a significant degree especially for positive fluctuations. The negative fluctuations stay close to Gaussian. This insensitivity to enstrophy levels is qualitatively different from data at low $M_{t}$ (figure 17 a) and suggests an increasingly weak correlation between pressure and enstrophy with compressibility. Thus, it may not be appropriate, for instance, to visualize vortical structures by identifying regions of low-pressure.

This de-correlation between enstrophy and low/high pressure at high $M_{t}$ can be interpreted in light of the results in figure 10. At low $M_{t}$ , the pressure fluctuations are mainly solenoidal, and thus we see low pressure and high-enstrophy regions to be correlated, similar to incompressible turbulence. As $M_{t}$ increases, dilatational pressure fluctuations have an increasingly dominant effect on the overall behaviour of pressure. Also, at high $M_{t}$ dilatational pressure and dilatational kinetic energy tend to be in equipartition. Since, dilatational motions are irrotational, then it is not unexpected that pressure and enstrophy become less correlated. This suggests, for example, that the occurrence of high-pressure regions at high $M_{t}$ , are not primarily caused by solenoidal effects.

6.6. Correlation between pressure and dilatation

Conditional statistics of pressure given dilatation can also provide information about the correlation between them, which represents a reversible exchange mechanism between kinetic and internal energy ((1.1) and (1.2)). In decaying simulations, Lee et al. (Reference Lee, Lele and Moin1991) found the role of pressure-dilatation correlation confined to the early acoustic transient time period. In many realistic applications however, systems are stationary and hence pressure-dilatation correlation may continue to play a role in the energy dynamics.

Figure 18.

Conditional p.d.f. of pressure given dilatation ( $X={\it\theta}^{\prime }/{\it\theta}_{rms}$ ) for different Reynolds and Mach numbers. (a,c,e), (b,d,f) are at $R_{{\it\lambda}}\approx 100,\,160$ , respectively. (a,b), (c,d), (e,f) are at $M_{t}\approx 0.1,\,0.3,\,0.6$ respectively. Arrows indicate increasing magnitude of dilatation. Symbols correspond to $X\approx 0.47$ (○), $1.1$ (▫), $3.0$ (▵), $4.8$ (♢). Solid and dashed lines stand for negative and positive dilatation values. Lines in grey correspond to a Gaussian distribution.

The conditional p.d.f. of pressure for different values of dilatation is shown for higher Reynolds numbers in figure 18. At low Mach numbers ( $M_{t}\approx 0.1$ ), the tails of the conditional p.d.f. become wider to the left with increasing magnitude of dilatation, generally indicating that regions of large negative pressure fluctuations could co-exist with regions of strong compressions or expansions. On the other hand, the positive pressure fluctuations stay closer to Gaussian and are only slightly affected by the value of dilatation. The p.d.f. is also insensitive to the sign of dilatation and Reynolds number which is seen as curves for positive dilatation (dashed lines) and negative dilatation (solid lines) coincide in figure 18. This means that the instantaneous value of pressure is insensitive to locally expanding or compressing fluid elements that have the same magnitude of dilatation. This has wide implications in the dynamics of energy transfer between the internal and kinetic energy modes. When the dilatation is large and negative, for example, pressure fluctuations are more likely to be negative than positive (see figure 18 a). Thus, one expects a positive correlation between them which corresponds to transfer from internal to kinetic energy. This, indeed, will be further supported momentarily with data on conditional expectations. A similar argument for expanding regions (positive dilatation) indicates that the energy transfer is likely to be from kinetic to internal energy.

As $M_{t}$ increases ( $M_{t}\approx 0.3$ ), the only change is the receding of the tails from the left that tends to approach Gaussian for small dilatation. But at high Mach numbers ( $M_{t}\approx 0.6$ ), the tails of the p.d.f., as also seen in figure 11(a), become wider on the right. While for small dilatation the p.d.f. is still similar for expansions and compressions, substantial differences can be seen for large dilatations. Due to the change in the asymmetry of the p.d.f., the energy dynamics changes for $M_{t}\approx 0.6$ . Applying the same analysis as before, but with pressure fluctuations more likely to be positive for large compressions, we can argue that at high Reynolds numbers, the net energy transfer is likely to be from kinetic to internal. For expansions, on the other hand, the net energy transfer will depend on the relative magnitudes of $p^{\prime }{\it\theta}^{\prime }$ and ${\it\epsilon}$ . Table 3 summarizes the conclusion on energy exchanges for strong compressions and expansions at low and high $M_{t}$ for high Reynolds numbers.

Figure 19.

Conditional expectation of pressure-dilatation correlation given dilatation at $M_{t}\approx$ (a) $0.1$ , (b) $0.6$ . Symbols for different Reynolds numbers as in figure 3(b). Arrows are in the direction of increasing Reynolds numbers.

Table 3.

Summary of net energy exchanges for strong compressions (C) or expansions (E).

6.7. Energy exchanges

A quantitative way to test the above argument on energy exchanges is to compute the conditional expectation of pressure-dilatation correlation given dilatation at different $M_{t}$ , and is shown in figure 19 for different $R_{{\it\lambda}}$ as well. When $M_{t}$ is small and close to the incompressible limit, the expected value of $p^{\prime }{\it\theta}^{\prime }$ is seen to increase in magnitude with $R_{{\it\lambda}}$ for both positive and negative fluctuations of dilatation, with a change in sign occurring around zero dilatation. We also observe a narrow range of fluctuations ( $-0.5<{\it\theta}^{\prime }/{\it\theta}_{rms}<0.5$ ) for which the expected value of pressure-dilatation correlation is approximately zero. As $M_{t}$ is increased to $0.6$ , in addition to the reduction in magnitude of expected value for negative dilatation, there is an extended range of dilatation values ( $-1.0<{\it\theta}^{\prime }/{\it\theta}_{rms}<0.6$ ) where the expected value of $p^{\prime }{\it\theta}^{\prime }$ is close to zero. This shows that for small dilatation fluctuations, up to $M_{t}\approx 0.6$ , pressure-dilatation correlation is very small suggesting that the energy transfer is dominated by dissipation. This is consistent with the view that at high $M_{t}$ , pressure responds to changes in the flow slower than at low $M_{t}$ and hence an increasing range of fluctuations in dilatation is sustained without significant changes to pressure, and hence pressure-dilatation correlation. As seen at low $M_{t}$ , there is a sign change in the expected value of $p^{\prime }{\it\theta}^{\prime }$ that occurs around zero dilatation. Qualitatively, the direction of energy transfer is the opposite of that at low $M_{t}$ . For expansions, however, we observe instantaneous bursts of energy transferred from internal to kinetic mode with a weakening Reynolds number dependence.

To facilitate a comparison between the hypothesis established in § 6.6 and the conditional expectation, we compare the expected values for larger dilatation values at different $M_{t}$ for high Reynolds numbers. The direction of energy transfer changes only when the expected value changes sign. Since the expected value is positive for large compressions at low $M_{t}$ , the energy transferred due to $p^{\prime }{\it\theta}^{\prime }$ is from internal to kinetic, while at high $M_{t}$ , this direction of transfer is opposite and agrees well with table 3. Similarly for expansions at low and high $M_{t}$ , we find the expected value to be in line with table 3 as long as the Reynolds number is high.

An assessment of the dependence of fluctuations of pressure-dilatation correlation on $R_{{\it\lambda}}$ and $M_{t}$ can be obtained by studying its normalized p.d.f. ( $f_{pdil}$ ) and is shown in figure 20. At low $M_{t}$ , the p.d.f. is symmetric, and becomes wider with Reynolds number consistent with the behaviour of other intermittent quantities in incompressible turbulence (Sreenivasan & Antonia Reference Sreenivasan and Antonia1997; Donzis et al. Reference Donzis, Yeung and Sreenivasan2008b ). At $M_{t}\approx 0.3$ , they recede becoming narrower for low Reynolds numbers, following the behaviour of dilatation (figure 12). For higher $M_{t}$ , the Reynolds number dependence of the p.d.f. weakens for a wide range of fluctuations (at least up to 30 times that of the r.m.s. fluctuations). This result suggests that modelling pressure-dilatation correlation at high $M_{t}$ may be in some respect easier than at low $M_{t}$ .

Figure 20.

P.d.f. of pressure-dilatation correlation normalized by r.m.s. ( $X=p^{\prime }{\it\theta}^{\prime }/(p^{\prime }{\it\theta}^{\prime })_{rms}$ ) at $M_{t}\approx$ (a) 0.1, (b) 0.3, (c) 0.6. Symbols for different Reynolds numbers as in figure 3(b). Arrows in (a) are in the direction of increasing Reynolds numbers. The wide tails in (b) are for $R_{{\it\lambda}}\approx 275,430$ .

Table 4.

Percentage volume of ${\it\alpha}=p^{\prime }{\it\theta}^{\prime }/{\it\epsilon}$ for different $R_{{\it\lambda}}$ and $M_{t}$ .

One could also explore directly the relative importance of pressure-dilatation correlation due to compressions and expansions, and dissipation by investigating the ratio ${\it\alpha}=p^{\prime }{\it\theta}^{\prime }/{\it\epsilon}$ . A value of ${\it\alpha}$ greater (less) than $1.0$ would indicate a net energy transfer from internal to kinetic modes (kinetic to internal). The regime where pressure-dilatation correlation dominates the energy transfer can be identified as regions with $|{\it\alpha}|\geqslant 1$ . We identify the percentage volume of occurrence of different value of ${\it\alpha}$ in eight different bins ranging from $-5$ to $5$ and beyond, as shown in table 4. At low $M_{t}$ , regardless of the $R_{{\it\lambda}}$ , ${\it\alpha}$ tends to be in the interval $[-1,1]$ in more than 90 % of the domain. Small values of ${\it\alpha}$ could be the result of very high dissipation or a negligible pressure-dilatation correlation. As we show at the end of this section, we anticipate the result that the latter is true. This means that the energy transfer is primarily uni-directional for $M_{t}\approx 0.1$ and governed by dissipation for small dilatational motions. However, as $M_{t}$ is increased to $0.3$ , for a large fraction of volume the energy transfer tends to be dominated by $p^{\prime }{\it\theta}^{\prime }$ (regions where ${\it\alpha}<-1$ or ${\it\alpha}>1$ ), which in the case of $R_{{\it\lambda}}\approx 163$ and $M_{t}\approx 0.3$ is more than 50 %. As $M_{t}$ is further increased to $0.6$ , very large transfers dominated by pressure-dilatation correlation are very frequent (see, for e.g. $R_{{\it\lambda}}\approx 158$ and $M_{t}\approx 0.6$ and bins where ${\it\alpha}<-5$ or ${\it\alpha}>5$ ), which is not seen at low $M_{t}$ . We find that with increasing $M_{t}$ , the regions where $p^{\prime }{\it\theta}^{\prime }$ tend to play a dominant role in energy transfer, also increases. Intuitively one may be able to attribute this to regions with strong dilatations that occurs more frequently as $M_{t}$ is increased (cf. table 3).

Figure 21.

Ratio of (a) ${\it\beta}=\langle p^{\prime }{\it\theta}^{\prime }\rangle /\langle {\it\epsilon}\rangle$ , (b) ${\it\beta}_{s}=\langle p^{s}{\it\theta}^{\prime }\rangle /\langle {\it\epsilon}\rangle$ , (c) ${\it\beta}_{d}=\langle p^{d}{\it\theta}^{\prime }\rangle /\langle {\it\epsilon}\rangle$ versus $M_{t}$ for different Reynolds numbers. Symbols for different Reynolds numbers as in figure 3(b).

It is also of interest to understand how the energy is transferred in the mean sense and is often studied in the literature (Kida & Orszag Reference Kida and Orszag1990; Sarkar Reference Sarkar1992). For this purpose, we show the ratio of mean pressure-dilatation correlation to mean dissipation in figure 21 including solenoidal and dilatational contributions ( ${\it\beta}=\langle p^{\prime }{\it\theta}^{\prime }\rangle /\langle {\it\epsilon}\rangle$ , ${\it\beta}_{s}$ , ${\it\beta}_{d}$ ). We observe that ${\it\beta}$ is close to zero at low $M_{t}$ indicating negligible contribution from pressure dilatation to the net energy transfer. While there is a minor scatter around zero for $M_{t}\approx 0.4$ as seen in other statistics, we see that for higher $M_{t}$ , ${\it\beta}$ is negative and close to zero, an observation also noted in Kida & Orszag (Reference Kida and Orszag1990). This means that at high $M_{t}$ , the net transfer of energy is from kinetic to internal mode.

From figure 21(b,c), we see that, unlike at low $M_{t}$ where the contributions of ${\it\beta}_{s}$ and ${\it\beta}_{d}$ are close to zero, at high $M_{t}$ their contributions oppose each other with the net transfer being dominated by the contributions from solenoidal pressure-dilatation correlation. However, it is important to consider that these transfers might actually come from different processes. As observed in figure 11(b), the solenoidal pressure is insensitive to changes in $M_{t}$ and $R_{{\it\lambda}}$ , and remains negatively skewed. But, from figure 12 we have seen that the p.d.f. of dilatation becomes negatively skewed as $M_{t}$ increases. Hence, the change in ${\it\beta}_{s}$ occurs, in part, due to the changes in the behaviour of dilatation with $M_{t}$ , and not directly because of solenoidal pressure. Since, ${\it\beta}_{s}$ is negative, both dissipation and solenoidal pressure-dilatation correlation work together in converting kinetic energy to internal energy. The processes associated with ${\it\beta}_{d}$ , on the other hand, is more difficult to ascertain since the p.d.f.s of both dilatational pressure and dilatation change beyond the threshold of $M_{t}\approx 0.3$ . One could thus state that the change in energy transfer due to ${\it\beta}_{d}$ is in part due to the change in p.d.f. of dilatational pressure.