2.1 Governing equations and approximation schemes

In the present study we consider flows of gases in the single-phase regime, governed by the compressible Navier–Stokes equations that are written in differential form

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}u_{j})}{\unicode[STIX]{x2202}x_{j}}=0 & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}u_{i}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}u_{i}u_{j})}{\unicode[STIX]{x2202}x_{j}}=-\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{ij}}{\unicode[STIX]{x2202}x_{j}} & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}E}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}((\unicode[STIX]{x1D70C}E+p)u_{j})}{\unicode[STIX]{x2202}x_{j}}=\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70E}_{ij}u_{i})}{\unicode[STIX]{x2202}x_{j}}-\frac{\unicode[STIX]{x2202}q_{j}}{\unicode[STIX]{x2202}x_{j}}, & \displaystyle\end{eqnarray}$$

with $u_{i}$ the velocity vector components, $\unicode[STIX]{x1D70C}$ the density and $p$ the pressure. The specific total energy $E$ and the viscous stress tensor $\unicode[STIX]{x1D70E}_{ij}$ are given by

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle E\equiv e+{\textstyle \frac{1}{2}}u_{j}u_{j} & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}_{ij}\equiv \unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}u_{j}}{\unicode[STIX]{x2202}x_{i}}\right)-\frac{2}{3}\unicode[STIX]{x1D707}\unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}_{ij}, & \displaystyle\end{eqnarray}$$

where $e$ is the specific internal energy, $\unicode[STIX]{x1D707}$ the dynamic viscosity, $\unicode[STIX]{x1D703}\equiv \unicode[STIX]{x2202}u_{k}/\unicode[STIX]{x2202}x_{k}$ the dilatation, $\unicode[STIX]{x1D6FF}_{ij}$ the Kronecker delta and where we have used the well-known Stokes’ hypothesis in neglecting the contribution of the bulk viscosity, which is a widely accepted approximation for most flows and namely for air flows. The study of Cramer (Reference Cramer2012) provides some numerical estimates for the bulk viscosity of ideal gases. It shows that the latter can vary significantly with the temperature and can be as large as hundreds or thousands of times the shear viscosity, even for some common diatomic gases. Nevertheless, its effect, additive to that of the thermodynamic pressure, is generally several orders of magnitude smaller. Its contribution is then expected to be negligible except, possibly, in very high dilatation regions such as the locations of shocklets where it may introduce an additional damping. For more complex gases, bulk viscosity data are even scarcer. Based on the preceding arguments, we will neglect its contribution also for dense gas flows.

Finally, $q_{j}$ represents the heat flux, modelled by means of Fourier’s law:

(2.6) $$\begin{eqnarray}q_{j}=-\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{j}},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ is the thermal conductivity. The system is supplemented by thermal and caloric equations of state, respectively:

(2.7a,b ) $$\begin{eqnarray}p=p(\unicode[STIX]{x1D70C},T)\quad \text{and}\quad e=e(\unicode[STIX]{x1D70C},T).\end{eqnarray}$$

These equations satisfy the compatibility relation:

(2.8) $$\begin{eqnarray}e=e_{\mathit{ref}}+\int _{T_{\mathit{ref}}}^{T}c_{v,\infty }(T)\,\text{d}T-\int _{\unicode[STIX]{x1D70C}_{\mathit{ref}}}^{\unicode[STIX]{x1D70C}}\left[T\left.\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}T}\right|_{\unicode[STIX]{x1D70C}}-p\right]\frac{\text{d}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}^{2}},\end{eqnarray}$$

where the subscript $(\cdot )_{\mathit{ref}}$ indicates a reference state, and $c_{v,\infty }(T)$ is the isochoric specific heat in the ideal gas limit.

In the present work, the gas behaviour is modelled through the Martin–Hou thermal equation of state (Martin & Hou Reference Martin and Hou1955). Such a model involves five virial terms and satisfies ten thermodynamic constraints, and it ensures high accuracy with a minimum of experimental information, thus providing a realistic description of the gas behaviour and of the inversion zone size. The equation reads

(2.9) $$\begin{eqnarray}p=\frac{RT}{v-b}+\mathop{\sum }_{i=2}^{5}\frac{f_{i}(T)}{(v-b)^{i}},\end{eqnarray}$$

where $v\equiv 1/\unicode[STIX]{x1D70C}$ is the specific volume, $R={\mathcal{R}}/{\mathcal{M}}$ is the gas constant (with ${\mathcal{R}}$ the universal gas constant and ${\mathcal{M}}$ the molecular weight), $b$ is the co-volume occupied by the fluid molecules and the functions $f_{i}(T)$ are of the form

(2.10) $$\begin{eqnarray}f_{i}(T)=A_{i}+B_{i}T+C_{i}\text{e}^{-k(T/T_{c})},\end{eqnarray}$$

with $T_{c}$ the critical temperature and $k=5.475$ . The gas-dependent coefficients $A_{i}$ , $B_{i}$ , $C_{i}$ can be expressed in terms of the critical temperature and pressure, the critical compressibility factor, the Boyle temperature and one point on the vapour pressure curve. A power law of the form

(2.11) $$\begin{eqnarray}c_{v_{\infty }}=c_{v_{\infty }}(T_{c})\left(\frac{T}{T_{c}}\right)^{n}\end{eqnarray}$$

is used to model variations of the low-density specific heat with temperature, where $n$ is a parameter that depends on the gas used (for more details, see Cramer Reference Cramer1989a ). The viscosity and thermal conductivity have been modelled according to the accurate dense gas formulation of Chung, Lee & Starling (Reference Chung, Lee and Starling1984), Chung et al. (Reference Chung, Ajlan, Lee and Starling1988), also described in the book of Poling et al. (Reference Poling, Prausnitz, O’Connell and Reid2001).

In the following, simulations are carried out considering PP11 as the working fluid, whose thermodynamic properties are provided in table 1. This gas has been often used in the dense gas literature because it is predicted to exhibit a relatively wide inversion zone and, as a consequence, BZT effects.

Table 1. Thermodynamic properties of PP11: molecular weight ( ${\mathcal{M}}$ ), critical temperature ( $T_{c}$ ), critical density ( $\unicode[STIX]{x1D70C}_{c}$ ), critical pressure ( $p_{c}$ ), critical compressibility factor ( $Z_{c}$ ), acentric factor ( $\unicode[STIX]{x1D714}_{ac}$ ), dipole moment ( $\unicode[STIX]{x1D709}$ ) of the gas phase, boiling temperature ( $T_{b}$ ), ratio of ideal gas specific heat at constant volume over the gas constant ( $c_{v}(T_{c})/R$ ) at the critical point and exponent of the power law for the low-density specific heat ( $n$ ).

The numerical strategy relies on a fourth-order discretization, whereby the convective terms are approximated by optimized finite differences with an eleven-point stencil in each mesh direction (Bogey & Bailly Reference Bogey and Bailly2004) and the viscous terms are approximated by fourth-order central differences. To damp grid-to-grid oscillations in smooth flow regions, an optimized selective sixth-order filter based on an eleven-point stencil is applied in each direction. In order to ensure computational robustness across flow discontinuities, an adaptive nonlinear selective filtering strategy similar to that of Bogey, De Cacqueray & Bailly (Reference Bogey, De Cacqueray and Bailly2009) is employed. In particular, to avoid the appearance of spurious oscillations, a low-order filter based on a Ducros-type shock sensor (Ducros et al. Reference Ducros, Ferrand, Nicoud, Weber, Darracq, Gacherieu and Poinsot1999) is introduced. The sensor triggers the required additional damping in regions characterized by high dilatation levels compared with the local vorticity magnitude, while it is not active in regions where the flow is smooth or dominated by vortical structures (see Bogey et al. Reference Bogey, De Cacqueray and Bailly2009 for details). A low-storage six-stage Runge–Kutta scheme optimized in the wavenumber space (Bogey & Bailly Reference Bogey and Bailly2004) is used for time integration. The adaptive filter is applied at the last stage of the Runge–Kutta integration. The preceding numerical strategy has been widely demonstrated for direct and large eddy simulations of compressible flows with and without shocks (see, e.g., Bogey & Bailly Reference Bogey and Bailly2009; Bogey, Marsden & Bailly Reference Bogey, Marsden and Bailly2012; Aubard, Gloerfelt & Robinet Reference Aubard, Gloerfelt and Robinet2013; Gloerfelt & Berland Reference Gloerfelt and Berland2013). Validations of the present DNS code against results in the literature are reported in appendix A.

2.2 Initial conditions

The CHIT decay problem is solved in the cubic domain $[0,2\unicode[STIX]{x03C0}]^{3}$ , and periodic boundary conditions are applied in the three directions.

The issue of the initial conditions for compressible isotropic turbulence has been addressed by various authors (Blaisdell, Mansour & Reynolds Reference Blaisdell, Mansour and Reynolds1993; Ristorcelli & Blaisdell Reference Ristorcelli and Blaisdell1997; Samtaney et al. Reference Samtaney, Pullin and Kosovic2001). In general, the shape of the initial three-dimensional spectrum and the root-mean-square (r.m.s.) level of each flow variable must be provided. Furthermore, different spectra can be assigned to the solenoidal and dilatational components. The main difference between an incompressible and compressible initialization is the presence of an intrinsic velocity scale for the latter, which is related to the speed of sound. The initial r.m.s. of the velocity ( $u_{\mathit{rms}}$ ) is defined by prescribing the turbulent Mach number ( $u_{\mathit{rms}}=M_{t}\langle c\rangle$ , where $\langle c\rangle$ is the prescribed average speed of sound). Temperature and pressure fluctuations are specified in accordance with the velocity fluctuations. Several simplifying hypotheses are made in the compressible case (Blaisdell et al. Reference Blaisdell, Mansour and Reynolds1993). Normally, the same spectrum shape is imposed for all the fluctuating fields, while fluctuations of the thermodynamic quantities are neglected. A review of initialization procedures for CHIT can be found in Samtaney et al. (Reference Samtaney, Pullin and Kosovic2001), as well as an analysis of their influence on the evolution of turbulent quantities.

For a PFG, the initialization used in this work is similar to that described in Lee et al. (Reference Lee, Lele and Moin1991), Sarkar et al. (Reference Sarkar, Erlebacher, Hussaini and Kreiss1991), Samtaney et al. (Reference Samtaney, Pullin and Kosovic2001), Pirozzoli & Grasso (Reference Pirozzoli and Grasso2004) and also used in Sciacovelli et al. (Reference Sciacovelli, Cinnella, Content and Grasso2016). In the case of dense gases that use a complex EoS, additional assumptions are required. Since the thermodynamic variables are nonlinearly related, in contrast to the PFG case it is not possible to assume a direct proportionality between density (or temperature) and dilatational velocity fluctuations. In the present work, in order to ensure fair comparisons between perfect and dense gas simulations, fluctuations of the thermodynamic quantities are set to zero, and the turbulent velocity field is assumed to be purely solenoidal in all of the following simulations. Specifically, the turbulent spectrum for the initial velocity field is of the form (Passot & Pouquet Reference Passot and Pouquet1987)

(2.12) $$\begin{eqnarray}E(k)=Ak^{4}\exp \left[-2\left(\frac{k}{k_{0}}\right)^{2}\right],\end{eqnarray}$$

where $k_{0}$ is the peak wavenumber and $A$ is a constant used to set the initial amount of kinetic energy, and thus the initial turbulent Mach number. For small values of $k_{0}$ , this distribution associates most of the energy with the largest scales and practically none to the smallest ones.

The expression for the initial turbulent kinetic energy spectrum can be used to compute analytically the initial turbulent kinetic energy ( $K_{0}$ ), the initial enstrophy ( $\unicode[STIX]{x1D6FA}_{0}$ ) and the initial large eddy turnover time ( $\unicode[STIX]{x1D70F}_{0}$ ) by means of the following relations (rigorously valid for an incompressible velocity field):

(2.13a-c ) $$\begin{eqnarray}K_{0}=\frac{3A}{64}\sqrt{2\unicode[STIX]{x03C0}}k_{0}^{5},\quad \unicode[STIX]{x1D6FA}_{0}=\frac{15A}{256}\sqrt{2\unicode[STIX]{x03C0}}k_{0}^{7},\quad \unicode[STIX]{x1D70F}_{0}=\sqrt{\frac{32}{A}}(2\unicode[STIX]{x03C0})^{1/4}k_{0}^{-7/2}.\end{eqnarray}$$

All simulations reported in the following were carried out for $k_{0}=2$ , whereas the value of $A$ is such that the initial turbulent kinetic energy is $K_{0}=0.5M_{t_{0}}^{2}c_{0}^{2}$ .

The Taylor Reynolds number, taken equal to 200 in all simulations, is based on the initial Taylor microscale, r.m.s. velocity and kinematic viscosity. The latter depends on the initial thermodynamic conditions both for air and PP11, and has to be rescaled (by a factor of the order of $10^{5}$ for both fluids) to match simultaneously the prescribed Mach and Reynolds number conditions.

4.1 Local flow topology

In turbulence, it is especially interesting to study the behaviour of critical points, i.e. points where the velocity gradient is degenerated, such as nodes, saddles, foci, etc. (Chong, Perry & Cantwell Reference Chong, Perry and Cantwell1990) and their contribution to the flow dynamics. The notion of flow topology refers to the spatial characterization of the flow field through its partitioning into simpler geometrical units, and is related to the presence of coherent turbulent structures (Wang & Peters Reference Wang and Peters2013). To elucidate the influence of dense gas effects on the statistical properties of turbulence structures, we consider the topological classification proposed by Perry & Chong (Reference Perry and Chong1987), Chong et al. (Reference Chong, Perry and Cantwell1990) and Kevlahan, Mahesh & Lee (Reference Kevlahan, Mahesh and Lee1992) in the framework of incompressible turbulence, and also employed by Pirozzoli & Grasso (Reference Pirozzoli and Grasso2004) and by Wang et al. (Reference Wang, Shi, Wang, Xiao, He and Chen2012) to analyse the small-scale behaviour of CHIT.

Let $\unicode[STIX]{x1D608}_{ij}\equiv \unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}$ be the velocity gradient tensor, and $P$ , $Q$ and $R$ be, respectively, its first, second and third invariant defined as

(4.1) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}P=-\unicode[STIX]{x1D703}=-\unicode[STIX]{x1D61A}_{ii}=-(\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D706}_{2}+\unicode[STIX]{x1D706}_{3})\\ Q={\textstyle \frac{1}{2}}(P^{2}-\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{ij}+\unicode[STIX]{x1D61E}_{ij}\unicode[STIX]{x1D61E}_{ij})=\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}+\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{3}+\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{3}\\ R={\textstyle \frac{1}{3}}(-P^{3}+3PQ-\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{jk}\unicode[STIX]{x1D61A}_{ki}-3\unicode[STIX]{x1D61E}_{ij}\unicode[STIX]{x1D61E}_{jk}\unicode[STIX]{x1D61A}_{ki})=-\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}\unicode[STIX]{x1D706}_{3},\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D61A}_{ij}=(\unicode[STIX]{x1D608}_{ij}+\unicode[STIX]{x1D608}_{ji})/2$ and $\unicode[STIX]{x1D61E}_{ij}=(\unicode[STIX]{x1D608}_{ij}-\unicode[STIX]{x1D608}_{ji})/2$ are the strain- and rotation-rate tensor components, and $\unicode[STIX]{x1D706}_{i}$ the three eigenvalues of $\unicode[STIX]{x1D608}_{ij}$ .

The nature of turbulent structures is classified according to the sign of the discriminant of $A$ :

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}={\textstyle \frac{27}{4}}R^{2}+\left(P^{3}-{\textstyle \frac{9}{2}}PQ\right)R+\left(Q^{3}-{\textstyle \frac{1}{4}}P^{2}Q^{2}\right).\end{eqnarray}$$

When $\unicode[STIX]{x1D6E5}$ is positive, the velocity gradient tensor has one real eigenvalue and two complex-conjugate ones, and focal regions are present; in contrast, when $\unicode[STIX]{x1D6E5}$ is negative, the eigenvalues of $\unicode[STIX]{x1D608}_{ij}$ are all real and turbulent regions are non-focal. Moreover, in the case of incompressible turbulence ( $P=0$ ) flow regions are further classified according to the sign of $R$ , leading to the following families of structures:

(4.3) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\unicode[STIX]{x1D6E5}>0 & \left\{\begin{array}{@{}ll@{}}R<0 & \quad \text{stable focus-stretching}\\ R>0 & \quad \text{unstable focus-compressing}\end{array}\right.\\ \\ \unicode[STIX]{x1D6E5}<0 & \left\{\begin{array}{@{}ll@{}}R<0 & \quad \text{stable node-saddle-saddle}\\ R>0 & \quad \text{unstable node-saddle-saddle.}\end{array}\right.\end{array}\right\}\end{eqnarray}$$

In the case of compressible turbulence, according to the sign of $P$ additional topological structures can be identified, referred to as stable-focus compressing, unstable focus stretching, stable node-stable node-stable node, and unstable node-unstable node-unstable node regions (Chong et al. Reference Chong, Perry and Cantwell1990). Focal regions are representative of vortical structures (Chong et al. Reference Chong, Soria, Perry, Chacin, Cantwell and Na1998), whereas non-focal ones are more related to expansion/compression phenomena. In shock/turbulence interaction Kevlahan et al. (Reference Kevlahan, Mahesh and Lee1992) have analysed the evolution of turbulent structures in terms of the deviatoric strain-rate tensor ( $\unicode[STIX]{x1D61A}_{ij}^{\ast }=\unicode[STIX]{x1D61A}_{ij}-(1/3)\unicode[STIX]{x1D61A}_{kk}\unicode[STIX]{x1D6FF}_{ij}$ ) and the rotation rate tensor $\unicode[STIX]{x1D61E}_{ij}$ , and have classified the flow into four structures

(4.4) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}W^{2}<S^{\ast 2}/2 & \text{convergence zones (essentially irrotational)}\\ S^{\ast 2}/2\leqslant W^{2}\leqslant 2S^{\ast 2} & \text{shear zones (strain dominated)}\\ W^{2}>2S^{\ast 2} & \text{eddy-dominated zones (highly rotational).}\end{array}\right\}\end{eqnarray}$$

Furthermore, they also classified turbulent structures as compressible or incompressible depending on the value of the parameter $\unicode[STIX]{x1D701}$ :

(4.5) $$\begin{eqnarray}\unicode[STIX]{x1D701}=\frac{\unicode[STIX]{x1D703}^{2}}{S^{\ast 2}+W^{2}};\quad \left.\begin{array}{@{}ll@{}}\unicode[STIX]{x1D701}\leqslant 0.05 & \text{incompressible structures}\\ \unicode[STIX]{x1D701}>0.05 & \text{compressible structures,}\end{array}\right\}\end{eqnarray}$$

where the threshold value of 0.05 (rather arbitrary), has been often used in the literature (Lee et al. Reference Lee, Lele and Moin1991; Kevlahan et al. Reference Kevlahan, Mahesh and Lee1992) to identify flow regions where dilatational dissipation represents more than 5 % of the total kinetic energy dissipation. All possible combinations of turbulent structures are summarized in table 3.

Table 3. Classification of turbulent regions in three-dimensional compressible flow according to Kevlahan et al. (Reference Kevlahan, Mahesh and Lee1992).

To systematically quantify the effects of local compressibility, we classify the overall flow volume into six subregions, according to the local dilatation scaled by the initial turbulent Mach number, as done in Wang et al. (Reference Wang, Shi, Wang, Xiao, He and Chen2012) and Sciacovelli et al. (Reference Sciacovelli, Cinnella, Content and Grasso2016):

1. (i) strong compressions, $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}\in [-\infty ,-2]$ ;

2. (ii) moderate compressions, $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}\in [-2,-1]$ ;

3. (iii) weak compressions, $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}\in [-1,0]$ ;

4. (iv) weak expansions, $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}\in [0,1]$ ;

5. (v) moderate expansions, $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}\in [1,2]$ ;

6. (vi) strong expansions, $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}\in [2,\infty ]$ .

In the discussion that follows, the statistical data are plotted at time $t/\unicode[STIX]{x1D70F}_{0}=2$ .

The distribution of the volume fractions occupied by the different structures conditioned on the above-defined dilatation levels is presented in figure 9 as a function of $M_{t_{0}}$ , for a perfect and dense gas. Up to $M_{t_{0}}=0.5$ , moderate and strong dilatation regions are close to zero, and the flow volume is nearly symmetrically occupied by weak expansion and compression regions. At the higher $M_{t_{0}}$ , this symmetry is rapidly lost in PFGs, due to the development of moderate and strong compressions and the progressive reduction of weak expansion regions, whereas a more symmetric behaviour is observed for PP11, as previously remarked for the p.d.f. of the local dilatation. At $M_{t_{0}}=1$ , strong compressions occupy approximately 3.5 % of the total volume for air, while they represent about 3 % for PP11. The remaining 50 % of the flow volume is mostly occupied by weak expansions for air, while strong expansions occupy only ${\approx}0.5\,\%$ of the total volume, in agreement with results of Wang et al. (Reference Wang, Shi, Wang, Xiao, He and Chen2012). In dense gas, strong compressions tend to be inhibited and strong expansions enhanced, leading to a nearly symmetric repartition of the volume fractions, as already observed in inviscid CHIT of BZT Van der Waals gas (Sciacovelli et al. Reference Sciacovelli, Cinnella, Content and Grasso2016). At $M_{t_{0}}=1$ , weak dilatation regions occupy ${\approx}40\,\%$ only, while moderate and strong expansion occupy, respectively, approximately 10 % and 2 % of the flow volume. The present results show that strong expansions are highly enhanced in the dense gas even when the operating conditions (IC1) do not allow BZT effects like expansion eddy shocklets. However, BZT effects contribute to further enhance expansions at the expense of compressions. Table 4 summarizes the percentage of focal and non-focal structures conditioned on the different dilatation levels as a function of the initial turbulent Mach number. The results are rather insensitive to the type of gas up to $M_{t_{0}}=0.5$ . The repartition changes more and more as compressibility effects increase. At $M_{t_{0}}=1$ , focal structures occupy 63 % of the total volume for air, whereas it is 58 % for PP11-IC1 and even lower (57 %) for PP11-IC2. The largest deviations are observed in moderate and strong expansion regions. Interestingly, these regions are much more populated by non-focal structures in the dense gas than air.

Figure 9. Volume fractions (%) occupied by flow regions characterized by different normalized dilatation intervals as a function of $M_{t_{0}}$ ( $t\unicode[STIX]{x1D70F}_{0}=2$ ): ——, air; – – –, PP11-IC1; — ⋅ ⋅ —, PP11-IC2; (a) strong compressions; (b) moderate compressions; (c) weak compressions; (d) weak expansions; (e) moderate expansions; (f) strong expansions.

Table 4. Percentage of focal/non-focal structures according to dilatation levels (at non-dimensional time $t/\unicode[STIX]{x1D70F}_{0}=2$ ).

For a variety of incompressible flows, including forced isotropic turbulence (Ooi et al. Reference Ooi, Martin, Soria and Chong1999), plane mixing layers (Soria & Cantwell Reference Soria and Cantwell1994), channel flows (Blackburn, Mansour & Cantwell Reference Blackburn, Mansour and Cantwell1996) and turbulent boundary layers (Chong et al. Reference Chong, Soria, Perry, Chacin, Cantwell and Na1998), a universal behaviour has been observed when analysing the flow topology in the $(Q-R)$ plane. Specifically, the joint p.d.f. $(Q,R)$ is found to exhibit a teardrop shape. In order to analyse the influence of dense gas effects on the dynamics of CHIT, we follow the approach of Pirozzoli & Grasso (Reference Pirozzoli and Grasso2004) developed for PFG flows. We have then formulated the problem in terms of the second and third invariants of the anisotropic part of the velocity gradient tensor ( $\unicode[STIX]{x1D608}_{ij}^{\ast }=\unicode[STIX]{x1D608}_{ij}-\unicode[STIX]{x1D703}\unicode[STIX]{x1D6FF}_{ij}/3$ ):

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle Q^{\ast }=-{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D61A}_{ij}^{\ast }\unicode[STIX]{x1D61A}_{ij}^{\ast }-\unicode[STIX]{x1D61E}_{ij}\unicode[STIX]{x1D61E}_{ij})=Q-{\textstyle \frac{1}{3}}P^{2} & \displaystyle\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle R^{\ast }=-{\textstyle \frac{1}{3}}(\unicode[STIX]{x1D61A}_{ij}^{\ast }\unicode[STIX]{x1D61A}_{jk}^{\ast }\unicode[STIX]{x1D61A}_{ki}^{\ast }+3\unicode[STIX]{x1D61E}_{ij}\unicode[STIX]{x1D61E}_{jk}\unicode[STIX]{x1D61A}_{ki}^{\ast })=R-{\textstyle \frac{1}{3}}PQ+{\textstyle \frac{2}{27}}P^{3}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{i}^{\ast }=\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D703}/3$ (note that, by construction, $\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D706}_{i}^{\ast }=0$ and $\unicode[STIX]{x1D6E5}^{\ast }=27/4R^{\ast 2}+Q^{\ast 3}$ ). A detailed analysis in the invariant plane has been carried out, the objective being twofold: (i) verify if the universal behaviour of the solenoidal component of the flow is recovered in dense gas flows, despite the peculiar thermophysical properties of these gases; (ii) investigate the influence of dense gas effects on the various flow structures.

Even though the analysis has been conducted at various $M_{t_{0}}$ , in the following only results corresponding to $M_{t_{0}}=1$ are discussed. In figure 10 we report the $\log$ of the total (a,d,g) and conditioned (b,e,h and c,f,i) joint p.d.f. of $Q^{\ast }$ and $R^{\ast }$ for air (a–c) and dense gas (PP11-IC1, d–f; PP11-IC2, g–i) for $M_{t_{0}}=1$ (strong compressions, b,e,h; strong expansions, c,f,i). The joint p.d.f. is computed from the conditional averages of the data by dividing the $(Q^{\ast },R^{\ast })$ plane into a number of equally sized partitions. For a good compromise between accuracy and statistical convergence each axis has been divided in $(2n)^{1/3}$ subdivisions, where $n=512^{3}$ is the number of samples. As done in Chong et al. (Reference Chong, Perry and Cantwell1990) and in Pirozzoli & Grasso (Reference Pirozzoli and Grasso2004) the invariants are normalized by the average of the second invariant of the rotation-rate tensor ( $\langle Q_{W}\rangle =\langle \unicode[STIX]{x1D61E}_{ij}\unicode[STIX]{x1D61E}_{ij}/2\rangle$ ). The thick solid line represents points where the discriminant of the anisotropic part of the deformation-rate tensor is zero (so as to discriminate between focal and non-focal regions). Independently of the type of gas, the joint p.d.f. $(Q^{\ast },R^{\ast })$ exhibits a teardrop shape as in the incompressible limit (Ooi et al. Reference Ooi, Martin, Soria and Chong1999), with a statistical preference in the second and in the fourth quadrant, where points are aligned with the right branch of the zero-discriminant curve (Cantwell Reference Cantwell1993; Martín et al. Reference Martín, Ooi, Chong and Soria1998; Pirozzoli & Grasso Reference Pirozzoli and Grasso2004). The kinematics of critical points is thus not affected by the gas type. It is interesting to observe that conditioning on strong compressions reveals an even stronger alignment with the right branch, as in Wang et al. (Reference Wang, Shi, Wang, Xiao, He and Chen2012). However, due to the weakening/suppression of compression structures, the tail of the teardrop is shorter in the dense gas cases, especially for the BZT condition IC2. For instance the isocontour $10^{-3}$ (corresponding to the bold solid line in figure 10), intersects the zero-discriminant curve at $(2.2,-3.2)$ in the case of air, against $(1.93,-2.93)$ and $(1.69,-2.69)$ for PP11-IC1 and PP11-IC2, respectively. The joint p.d.f.s conditioned on strong expansions exhibits more striking differences. For the dense gas case PP11-IC2 the p.d.f. tends to be skewed toward the left branch of the zero-discriminant curve (in particular for probability levels of $10^{-3}$ , or higher), in contrast with air and PP11-IC1 that exhibit a more symmetric shape. This suggests that different flow structures may appear in the strong expansion regions of BZT flows.

Figure 10. Isocontours ( $-4$ to 0, spacing equal to 1) of the $\log _{10}$ of the joint p.d.f. of the scaled second and third invariants of the anisotropic part of the velocity gradient at $M_{t_{0}}=1$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ) for air (a–c), PP11-IC1 (d–f) and PP11-IC2 (g–i). (a,d,g) Total joint p.d.f. (b,e,h) The p.d.f. conditioned on strong compressions (regions with $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}\in [-\infty ,-2]$ ). (c,f,i) The p.d.f. conditioned on strong expansions (regions with $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}\in [2,\infty ]$ ). The bold solid isoline denotes the isocontour  $-3$ .

4.2 Contribution of flow structures to dissipation

To further elucidate the influence of dense gases on focal and non-focal structures, we analyse the second invariant of the rotation-rate tensor ( $Q_{W}=\unicode[STIX]{x1D61E}_{ij}\unicode[STIX]{x1D61E}_{ij}/2$ , a measure of the rotational energy), of the anisotropic part of the strain-rate tensor ( $-Q_{S^{\ast }}=\unicode[STIX]{x1D61A}_{ij}^{\ast }\unicode[STIX]{x1D61A}_{ij}^{\ast }/2$ ) and of the variance of the dilatation ( $P^{2}$ ). Specifically, we consider the volume fractions occupied by focal and non-focal structures, as well as their contributions to $Q_{W}$ and $Q_{S^{\ast }}$ , as a function of the initial turbulent Mach number.

We recall that the total dissipation ( $-2\overline{\unicode[STIX]{x1D707}}\langle Q_{S^{\ast }}\rangle$ ) is the sum of the solenoidal ( $2\overline{\unicode[STIX]{x1D707}}\langle Q_{W}\rangle$ ) and dilatational ( $2/3\overline{\unicode[STIX]{x1D707}}\langle P^{2}\rangle$ ) dissipation.

The distributions of $\langle V\rangle ^{\pm }$ (where $+$ and $-$ indicate conditioning on focal and non-focal structures, respectively) are reported in figure 11 as a function of $M_{t_{0}}$ (a,d, incompressible structures; b,e, compressed structures; c,f, expanding structures) for air and PP11-IC1. At low $M_{t_{0}}$ we observe that, in the PFG case: (i) structures of an incompressible nature occupy a larger volume fraction than compressible ones; (ii) shear-like structures are the most frequent and occupy ${\approx}42\,\%$ of the flow volume; and (iii) eddy and convergence zones represent ${\approx}19\,\%$ and ${\approx}38\,\%$ of the total flow volume, respectively. Up to $M_{t_{0}}=0.5$ , PP11 exhibits similar statistical volume fraction distributions. As the initial turbulent Mach number increases, the volume fraction occupied by incompressible structures is reduced significantly due to the development of compressible structures, whose volume fractions nearly triple when $M_{t_{0}}$ is varied from 0.5 to 1. As compressibility effects become significant, the volume percentage associated with the different flow structures is most affected by the working fluid. Specifically, the volume fraction of compressible structures is 58 % for PP11-IC1, whereas it is only 44 % for air, and convergence and shear compressed regions are ${\approx}30\,\%$ larger than in air. The choice of the initial conditions (IC1 versus IC2) has a small influence on the computed volume fractions.

Figure 11. Fractional contribution of the various flow structures to $\langle V\rangle ^{+}$ (a–c) and $\langle V\rangle ^{-}$ (d–f) at various $M_{t_{0}}$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ): (a,d) incompressible structures; (b,e) compressed structures; (c,f) expanding structures; ——, air; – – –, PP11-IC1; $+$ , EI; $\times$ , CI; ♢, SI; ▫, EC; ○, CC; ▵, SC; ▪, EE; ●, CE; ▴, SE.

An interesting feature is that, for dense gas, expanding convergence regions (associated with strong expansion phenomena and, possibly, shocklets for the IC2 case) are almost as frequent as expanding shear zones, unlike air for which shear regions dominate. This behaviour is likely to be related to the non-convex nature of the inviscid fluxes and to the degeneracy of the characteristic fields in the proximity of the inversion zone ( $\unicode[STIX]{x1D6E4}\approx 0$ ), leading to the formation of steep expansion fronts, as discussed in greater detail in a later section.

The isosurfaces of the strong compression regions $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}=-3$ (a,c,e) and of the strong expansion regions $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}=3$ (b,d,f) are reported in figure 12 both for air (a,b) and for the dense gas cases (c–f) at $M_{t_{0}}=1$ . For air, strong compression regions are populated by random sheet-like structures of convergence-type topology over a wide range of scales, contrary to strong expansions that are characterized by sparse blob or tubular structures mainly of the eddy and shear type. The dense gas also exhibits compressed sheet-like shocklets and an increased volume percentage of expanding structures having tubular as well as sheet-like convergence topology. The continuous convergence structures steepen in the case of IC2 (figure 12 e,f), leading to expansion shocklets that are admissible since the initial thermodynamic state falls in the inversion region.

Figure 12. Strong compressions $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}=-3$ (a,c,e) and strong expansions $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}=3$ (b,d,f) for air (a,b), PP11-IC1 (c,d) and PP11-IC2 (e,f) coloured with the local type of structure: blue, eddy; white, shear; red, convergence.

Next, we consider the fractional contributions of the different flow structures to the solenoidal, total and dilatational dissipation. For both perfect and dense gas cases, shear regions account approximately for 50 % of the solenoidal dissipation, whereas compressible regions contribute less than 20 % of the total, the contribution being slightly higher for the dense gas. In all cases, the contribution of eddy-like regions is almost negligible. Furthermore, focal and non-focal structures contribute to total dissipation by the same amount (results not reported for brevity), whereas at high $M_{t_{0}}$ the contribution of non-focal compressible convergence structures is significantly increased, especially in the dense gas case.

The most interesting differences between the perfect and the dense gas concern the dilatational dissipation, reported in figure 13. The contribution of incompressible structures is almost the same both for air and PP11; however, the contribution of compressible structures changes significantly. We observe in particular that, for the dense gas at high $M_{t_{0}}$ , non-focal expanding structures play a major role, and contribute to the dilatational dissipation by an amount that is comparable to that of compressed focal structures ( ${\approx}20\,\%$ versus ${\approx}30\,\%$ ). For condition IC2, for which the occurrence of expansion and compression shocklets is almost equally probable, this effect is even more accentuated (a comparison of the fractional contributions to $\langle Q_{P}\rangle ^{\pm }$ for IC1 and IC2 is reported in figure 14). In contrast, in the PFG compressed non-focal structures (associated with the formation of eddy shocklets) contribute more significantly to dilatational dissipation than expanding ones. The amount of dilatational dissipation increases with the initial turbulent Mach number and, at $M_{t_{0}}=1$ , it represents more than 50 % of the total, whereas expanding ones contribute for less than 10 % and the dependence on $M_{t_{0}}$ is weak.

Figure 13. Fractional contribution of the various flow structures to the dilatational dissipation $\langle Q_{P}\rangle ^{+}$ (a–c) and $\langle Q_{P}\rangle ^{-}$ (d–f) at various $M_{t_{0}}$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ): (a,d) incompressible structures; (b,e) compressed structures; (c,f) expanding structures; ——, air; – – –, PP11-IC1; $+$ , EI; $\times$ , CI; ♢, SI; ▫, EC; ○, CC; ▵, SC; ▪, EE; ●, CE; ▴, SE.

Figure 14. Influence of the initial thermodynamic state on the fractional contribution of the various flow structures to the dilatational dissipation $\langle Q_{P}\rangle ^{+}$ (a–c) and $\langle Q_{P}\rangle ^{-}$ (d–f) at various $M_{t_{0}}$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ): (a,d) incompressible structures; (b,e) compressed structures; (c,f) expanding structures; – – –, PP11-IC1; — ⋅ ⋅ —, PP11-IC2; $+$ , EI; $\times$ , CI; ♢, SI; ▫, EC; ○, CC; ▵, SC; ▪, EE; ●, CE; ▴, SE.

To further investigate the correlation between the various flow structures and the occurrence of strong compression/expansion phenomena, in figure 15 we report the volume fractions of the different flow structures conditioned with respect to strong compression ( $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}<-2$ ) and strong expansion zones ( $\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{rms}M_{t_{0}}^{2}>2$ ). The former are populated by similar structures both in the perfect and dense gas, with convergence compressed zones representing more than 80 % of the total volume and eddy zones being negligible. In the case of dense gas, convergence and shear zones represent, respectively, 75–80 % (depending on the initial thermodynamic state) and 15–20 % (eddy-like being negligible) of the volume occupied by expanding structures, whereas in air the different flow structures are nearly equally probable (45 % shear, 30 % convergence and 25 % eddy-like).

Figure 15. Volume fractions of the various flow structures conditioned with respect to strong compression regions (a) and strong expansion regions (b) at various $M_{t_{0}}$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ): ——, air; – – –, PP11-IC1; — ⋅ ⋅ —, PP11-IC2; ▫, EC; ○, CC; ▵, SC; ▪, EE; ●, CE; ▴, SE.

The preceding analysis supports the conclusion that, in dense gases, strong expansion regions are populated by a significant number of convergence structures that contribute considerably to the dilatational dissipation. In this sense, these regions tend to exhibit a behaviour similar to compression regions. This phenomenon is observed not only for BZT flow conditions (IC2) but also for non-BZT conditions characterized by $\unicode[STIX]{x1D6E4}\approx 0$ (IC1), although to a lesser extent. A possible mechanism enhancing convergence structure is the formation of steep expansion fronts (for IC1) or possibly expansion shocklets (for IC2).

4.3 Enstrophy generation mechanisms

In the preceding section we highlighted that dense gas effects modify the contribution of the various flow structures to the dilatational dissipation, whereas the rotational energy (related to enstrophy) is less affected. On the other hand, in compressible flows enstrophy is contributed by shocks in addition to shear layers and vortical motions. It is then interesting to analyse how dense gas effects modify sources of enstrophy generation. We recall that enstrophy is governed by the following transport equation (Erlebacher & Sarkar Reference Erlebacher and Sarkar1993):

(4.8) $$\begin{eqnarray}\frac{\text{D}}{\text{D}t}\left(\frac{\unicode[STIX]{x1D714}^{2}}{2}\right)=\unicode[STIX]{x1D714}_{i}V_{i}^{T}+\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D716}_{ijk}\frac{1}{\unicode[STIX]{x1D70C}^{2}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{j}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{k}}+\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D716}_{ijk}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}\left(\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{mk}}{\unicode[STIX]{x2202}x_{m}}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D714}^{2}/2=-4Q_{W}=\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/2$ is the enstrophy density, $\unicode[STIX]{x1D714}_{i}$ is the vorticity vector and $\unicode[STIX]{x1D716}_{ijk}$ is the Levi-Civita symbol. The three terms on the right-hand side represent, respectively, enstrophy generation due to vortex stretching, the baroclinic term and the viscous diffusion. The vortex stretching vector $V_{i}^{T}$ is the sum of two contributions due to the deviatoric strain rate ( $V_{i}^{S}=\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D61A}_{ij}^{\ast }$ ) and to dilatation ( $V_{i}^{D}=-2/3\unicode[STIX]{x1D703}\unicode[STIX]{x1D714}_{i}$ ).

Integration of (4.8) over the entire volume yields

(4.9) $$\begin{eqnarray}\frac{\text{D}\langle \unicode[STIX]{x1D714}^{2}/2\rangle }{\text{D}t}=\langle \unicode[STIX]{x1D714}_{i}V_{i}^{S}\rangle +\langle \unicode[STIX]{x1D714}_{i}V_{i}^{D}\rangle +\left\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D716}_{ijk}\frac{1}{\unicode[STIX]{x1D70C}^{2}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}x_{j}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{k}}\right\rangle +\left\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D716}_{ijk}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}\left(\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}_{mk}}{\unicode[STIX]{x2202}x_{m}}\right)\right\rangle .\end{eqnarray}$$

Kida & Orszag (Reference Kida and Orszag1989), Pirozzoli & Grasso (Reference Pirozzoli and Grasso2004) and Wang et al. (Reference Wang, Shi, Wang, Xiao, He and Chen2012) showed that the baroclinic effects are negative and rather negligible, while the viscous dissipation acts as a sink.

We focus our analysis on the action of the strain rate $\langle \unicode[STIX]{x1D714}_{i}V_{i}^{S}\rangle$ and dilatation $\langle \unicode[STIX]{x1D714}_{i}V_{i}^{D}\rangle$ terms. To relate enstrophy generation with the occurrence of shocklets, as in Wang et al. (Reference Wang, Shi, Wang, Xiao, He and Chen2012), we project the vorticity and the vortex stretching vectors (both the one associated with the deviatoric strain rate and the dilatation rate) in the normal and tangential directions relative to the local density isosurfaces, i.e. the direction aligned with the density gradient $\boldsymbol{n}=\text{grad}(\unicode[STIX]{x1D70C})/\Vert \text{grad}(\unicode[STIX]{x1D70C})\Vert$ and a direction $\boldsymbol{t}$ orthogonal to $\boldsymbol{n}$ .

Figure 16. Distribution of the averages of the vortex stretching vector $\unicode[STIX]{x1D714}_{i}V_{i}^{T}$ (a,d), $\unicode[STIX]{x1D714}_{i}V_{i}^{D}$ (b,e) and $\unicode[STIX]{x1D714}_{i}V_{i}^{S}$ (c,f) normalized by $\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D61A}_{ij}^{\ast }\rangle$ and conditioned on the dilatation for air (a–c) and PP11-IC1 (d–f) at $M_{t_{0}}=1$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ). In each panel the contributions to enstrophy generation in the direction parallel (-–○-–  $r=n$ ) and tangential (-- --▵-- --  $r=t$ ) to the density gradient and the total contribution (--.--▿--.--) are reported.

In figure 16 we report the volume average of the rate of total enstrophy generation $\langle \unicode[STIX]{x1D714}_{i}V_{i}^{T}\rangle$ (a,d), the rate of dilatational enstrophy generation $\langle \unicode[STIX]{x1D714}_{i}V_{i}^{D}\rangle$ (b,e) and the deviatoric strain-rate enstrophy generation $\langle \unicode[STIX]{x1D714}_{i}V_{i}^{S}\rangle$ (c,f) conditioned on dilatation, both for air (a–c) and dense gas (d–f). The vorticity vector having a tendency to be orthogonal to the density gradient, the main contribution to enstrophy generation is in the tangential direction. Enstrophy is produced due to vortex stretching across eddy shocklets (compressed structures). Expanding PFG structures destroy enstrophy and both $\langle \unicode[STIX]{x1D714}_{i}V_{i}^{S}\rangle$ and $\langle \unicode[STIX]{x1D714}_{i}V_{i}^{D}\rangle$ increase rapidly in magnitude with $\unicode[STIX]{x1D703}$ . On the contrary, dense gases have a tendency to strongly inhibit enstrophy destruction across moderate and strong expansions. Furthermore, dilatational enstrophy generation is of comparable order both in the directions normal and tangential to the expanding structures. The attenuation of enstrophy destruction in expansion regions is likely due to two competing mechanisms: expansion processes associated with thermodynamic states having $\unicode[STIX]{x1D6E4}=O(1)$ that contribute to enstrophy destruction as in PFGs, and (strong) expansions associated with thermodynamic states having $\unicode[STIX]{x1D6E4}$ close to or less than zero that have a tendency to behave similar to compressions (thus producing enstrophy and counteracting its destruction).

To further elucidate into the enstrophy generation mechanism and its relation with the flow topology, we have also analysed the contribution due to vortex stretching in the eigendirections of the deviatoric strain rate. We recall that in incompressible flows (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Cantwell Reference Cantwell1993), vorticity has a tendency to align with the eigendirection associated with the intermediate eigenvalue of the strain rate, which is found to be positive in the average (i.e. more often positive than negative). For compressible flows, we consider the deviatoric strain rate $\unicode[STIX]{x1D61A}_{ij}^{\ast }$ and project the vortex stretching vector in the eigendirections $\unicode[STIX]{x1D726}_{k}$ associated with the eigenvalues $\unicode[STIX]{x1D706}_{k}^{\ast }$ of $\unicode[STIX]{x1D61A}_{ij}^{\ast }$ . The vortex stretching term then reduces to

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D61A}_{ij}^{\ast }=\unicode[STIX]{x1D714}^{2}\mathop{\sum }_{k}\unicode[STIX]{x1D706}_{k}^{\ast }\cos ^{2}\unicode[STIX]{x1D719}_{k},\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{k}=\cos ^{-1}(\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\unicode[STIX]{x1D726}_{k}/\Vert \unicode[STIX]{x1D74E}\Vert \,\Vert \unicode[STIX]{x1D726}_{k}\Vert )$ is the angle between the vorticity vector and the $k$ th eigendirection.

Figure 17. Conditional averages of squares of cosines of angles between the density gradient direction $\boldsymbol{n}$ and the strain-rate eigenvectors $\unicode[STIX]{x1D726}_{k}$ for air (a), PP11-IC1 (b) and PP11-IC2 (c) at $M_{t_{0}}=1$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ): -–○-–, $k=1$ ; -- --▵-- --, $k=2$ ; --.--▿--.--, $k=3$ .

The p.d.f.s and the conditional p.d.f.s of the eigenvalue ratios $\unicode[STIX]{x1D706}_{2}^{\ast }/\unicode[STIX]{x1D706}_{1}^{\ast }$ and $\unicode[STIX]{x1D706}_{2}^{\ast }/\unicode[STIX]{x1D706}_{3}^{\ast }$ of the deviatoric strain rate for air and PP11 (not reported for brevity), as well as the conditional p.d.f.s of the normalized eigenvalues of the strain-rate tensor $\unicode[STIX]{x1D6FD}_{k}=\unicode[STIX]{x1D706}_{k}/\sqrt{\sum _{i}\unicode[STIX]{x1D706}_{i}^{2}}$ , follow the classical behaviour observed both in incompressible and compressible turbulence (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Erlebacher & Sarkar Reference Erlebacher and Sarkar1993; Pirozzoli & Grasso Reference Pirozzoli and Grasso2004; Lee, Girimaji & Kerimo Reference Lee, Girimaji and Kerimo2009; Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2012). Specifically, the most probable values of $\unicode[STIX]{x1D706}_{k}^{\ast }$ are approximately in the ratio $[-4:1:3]$ . Furthermore, as compression increases, the p.d.f. of $\unicode[STIX]{x1D6FD}_{1}$ peaks at lower values, $\unicode[STIX]{x1D6FD}_{2}$ and $\unicode[STIX]{x1D6FD}_{3}$ become much smaller than $\unicode[STIX]{x1D6FD}_{1}$ and the eigenvalues are approximately in the ratio $[-1:0:0]$ . In strongly expanding regions the p.d.f. of $\unicode[STIX]{x1D6FD}_{3}$ peaks at approximately 1 and the eigenvalues are in the ratio $[0:0:1]$ . Hence compression is dominant in the first eigendirection, while expansion is dominant in the third one.

Reported in figure 17 are the conditional averages of the squared cosines of the angles between the density gradient direction and the principal strain ones ( $\unicode[STIX]{x1D713}_{k}=\cos ^{-1}(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D726}_{k})/(\Vert \unicode[STIX]{x1D726}_{k}\Vert )$ ) for air (a), PP11-IC1 (b) and PP11-IC2 (c) at $M_{t_{0}}=1$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ). In strong compressions, the eigendirection associated with the most negative eigenvalue of $\unicode[STIX]{x1D61A}_{ij}$ has a tendency to align with the density gradient (i.e. it tends to be perpendicular to shocklets) both for the perfect and the dense gas. In expansion regions, the first eigenvector is nearly orthogonal to the density gradient, while the third strain-rate eigendirection has a tendency to somewhat align with the density gradient. This tendency is strongly marked in dense gas, supporting the possibility of the occurrence of steep expansion fronts for condition IC1 and eventually expansion shocklets for IC2.

From (4.10) we observe that the amount of enstrophy production/destruction depends on the alignment between vorticity and the deviatoric strain-rate eigendirections, and on the sign and magnitude of the associated eigenvalues. An inspection of the p.d.f.s and the conditional p.d.f.s of the angles between vorticity and $\unicode[STIX]{x1D726}_{k}$ (not reported), shows that the vorticity is preferentially parallel or antiparallel to the intermediate eigenvector, and has a tendency to be orthogonal to the first eigendirection. The angle between vorticity and the eigendirection associated with the most positive eigenvector has a tendency to be more uniformly distributed as expansion strengthens (i.e. any angle between vorticity and $\unicode[STIX]{x1D726}_{3}$ is equally probable). This behaviour, which is in agreement with the results of Wang et al. (Reference Wang, Shi, Wang, Xiao, He and Chen2012), does not depend on the type of gas and is rather similar to that observed in incompressible flows (Erlebacher & Sarkar Reference Erlebacher and Sarkar1993).

The conditional averages of enstrophy production by vortex stretching in the principal strain-rate directions is reported in figure 18 for air (a), PP11-IC1 (b) and PP11-IC2 (c). For air, the main contributions to enstrophy production in strong compression are associated with the second and third eigenvectors, whereas in regions where the gas undergoes strong expansions the small positive contribution in the direction associated with the most positive eigenvalue ( $\unicode[STIX]{x1D706}_{3}^{\ast }$ ) is overwhelmed by the negative contributions in the other directions, as also found by Wang et al. (Reference Wang, Shi, Wang, Xiao, He and Chen2012). For dense gas, a similar behaviour is observed in compression regions. However, in expansion zones the large negative contributions in the $\unicode[STIX]{x1D726}_{1}$ and $\unicode[STIX]{x1D726}_{2}$ eigendirections are reduced by one order of magnitude, yielding striking differences with the perfect gas case. This can be interpreted by examining the vorticity distribution whose conditional average is reported in figure 19 as a function of dilatation. In general, dense gas effects reduce vorticity both in compression and expansion zones. In strong compressions, perfect and dense gases generate a comparable average vorticity. On the contrary, in strong expansions dense gas exhibits a stronger decrease in vorticity. The p.d.f. of vorticity magnitude conditioned on dilatation is reported in figure 20, which shows indeed that high vorticity magnitudes are somewhat less probable in dense gas (the effect being stronger for case PP11-IC2) than in PFG.

Figure 18. Conditional averages of enstrophy production associated with the three strain-rate eigenvectors at $M_{t_{0}}=1$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ) normalized by $\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x1D61A}_{ij}^{\ast }\rangle$ : -–○-–, $k=1$ ; -- --▵-- --, $k=2$ ; --.--▿--.--., $k=3$ .

Figure 19. Conditional average of the normalized vorticity magnitude at $M_{t_{0}}=1$ ( $t/\unicode[STIX]{x1D70F}_{0}=2$ ): ——, air; – – –, PP11-IC1; — ⋅ ⋅ —, PP11-IC2.

Figure 20. Total and conditional p.d.f. of the normalized vorticity magnitude: (a) air; (b) PP11-IC1; (c) PP11-IC2; ——, $[-\infty ,+\infty ]$ ; – – –, $[-\infty ,-2]$ ; — ⋅ ⋅ —, $[2,\infty ]$ .