The influence of dense-gas effects on compressible wall-bounded turbulence is investigated by means of direct numerical simulations of supersonic turbulent channel flows. Results are obtained for PP11, a heavy fluorocarbon representative of dense gases, the thermophysics properties of which are described by using a fifth-order virial equation of state and advanced models for the transport properties. In the dense-gas regime, the speed of sound varies non-monotonically in small perturbations and the dependency of the transport properties on the fluid density (in addition to the temperature) is no longer negligible. A parametric study is carried out by varying the bulk Mach and Reynolds numbers, and results are compared to those obtained for a perfect gas, namely air. Dense-gas flow exhibits almost negligible friction heating effects, since the high specific heat of the fluids leads to a loose coupling between thermal and kinetic fields, even at high Mach numbers. Despite negligible temperature variations across the channel, the mean viscosity tends to decrease from the channel walls to the centreline (liquid-like behaviour), due to its complex dependency on fluid density. On the other hand, strong density fluctuations are present, but due to the non-standard sound speed variation (opposite to the mean density evolution across the channel), the amplitude is maximal close to the channel wall, i.e. in the viscous sublayer instead of the buffer layer like in perfect gases. As a consequence, these fluctuations do not alter the turbulence structure significantly, and Morkovin’s hypothesis is well respected at any Mach number considered in the study. The preceding features make high Mach wall-bounded flows of dense gases similar to incompressible flows with variable properties, despite the significant fluctuations of density and speed of sound. Indeed, the semi-local scaling of Patel et al. (Phys. Fluids, vol. 27 (9), 2015, 095101) or Trettel & Larsson (Phys. Fluids, vol. 28 (2), 2016, 026102) is shown to be well adapted to compare results from existing surveys and with the well-documented incompressible limit. Additionally, for a dense gas the isothermal channel flow is also almost adiabatic, and the Van Driest transformation also performs reasonably well. The present observations open the way to the development of suitable models for dense-gas turbulent flows.

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A detailed numerical study of the influence of dense gas effects on the large-scale dynamics of decaying homogeneous isotropic turbulence is carried out by using the van der Waals gas model. More specifically, we focus on dense gases of the Bethe–Zel’dovich–Thompson type, which may exhibit non-classical nonlinearities in the transonic and supersonic flow regimes, under suitable thermodynamic conditions. The simulations are based on the inviscid conservation equations, solved by means of a ninth-order numerical scheme. The simulations rely on the numerical viscosity of the scheme to dissipate energy at the finest scales, while leaving the larger scales mostly unaffected. The results are systematically compared with those obtained for a perfect gas. Dense gas effects are found to have a significant influence on the time evolution of the average and root mean square (r.m.s.) of the thermodynamic properties for flows characterized by sufficiently high initial turbulent Mach numbers (above 0.5), whereas the influence on kinematic properties, such as the kinetic energy and the vorticity, are smaller. However, the flow dilatational behaviour is very different, due to the non-classical variation of the speed of sound in flow regions where the dense gas is characterized by a value of the fundamental derivative of the gas dynamics (a measure of the variation of the speed of sound in isentropic compressions) smaller than one or even negative. The most significant differences between the perfect and the dense gas case are found for the repartition of dilatation levels in the flow field. For the perfect gas, strong compressions occupy a much larger volume fraction than expansion regions, leading to probability distributions of the velocity divergence highly skewed toward negative values. For the dense gas, the volume fractions occupied by strong expansion and compression regions are much more balanced; moreover, strong expansion regions are characterized by sheet-like structures, unlike the perfect gas which exhibits tubular structures. In strong compression regions, where compression shocklets may occur, both the dense and the perfect gas exhibit sheet-like structures. This suggests the possibility that expansion eddy shocklets may appear in the dense gas. This hypothesis is also supported by the fact that, in dense gas, vorticity is created with equal probability in strong compression and expansion regions, whereas for a perfect gas, vorticity is more likely to be created in the strong compression ones.

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A hypersonic, spatially evolving turbulent boundary layer at Mach 12.48 with a cooled wall is analysed by means of direct numerical simulations. At the selected conditions, massive kinetic-to-internal energy conversion triggers thermal and chemical non-equilibrium phenomena. Air is assumed to behave as a five-species reacting mixture, and a two-temperature model is adopted to account for vibrational non-equilibrium. Wall cooling partly counteracts the effects of friction heating, and the temperature rise in the boundary layer excites vibrational energy modes while inducing mild chemical dissociation of oxygen. Vibrational non-equilibrium is mostly driven by molecular nitrogen, characterized by slower relaxation rates than the other molecules in the mixture. The results reveal that thermal non-equilibrium is sustained by turbulent mixing: sweep and ejection events efficiently redistribute the gas, contributing to the generation of a vibrationally under-excited state close to the wall, and an over-excited state in the outer region of the boundary layer. The tight coupling between turbulence and thermal effects is quantified by defining an interaction indicator. A modelling strategy for the vibrational energy turbulent flux is proposed, based on the definition of a vibrational turbulent Prandtl number. The validity of the strong Reynolds analogy under thermal non-equilibrium is also evaluated. Strong compressibility effects promote the translational–vibrational energy exchange, but no preferential correlation was detected between expansions/compressions and vibrational over-/under-excitation, as opposed to what has been observed for unconfined turbulent configurations.

Mixing of several species in high-pressure (high-$p$) turbulent flows is investigated to understand the influence of the number of species on the flow characteristics. Direct numerical simulations are conducted in the temporal mixing layer configuration at approximately the same value of the momentum ratio for all realizations. The simulations are performed with mixtures of two, three, five and seven species to address various compositions at fixed number of species, at three values of initial vorticity-thickness-based Reynolds number, $Re_{0}$, and two values of the free-stream pressure, $p_{0}$, which is supercritical for each species except water. The major species are C7H16, O2 and N2, and the minor species are CO, CO2, H2 and H2O. The extensive database thus obtained allows the study of the influence not only of $Re_{0}$ and $p_{0}$, but also of the initial density ratio and of the initial density difference between streams, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$. The results show that the layer growth is practically insensitive to all of the above parameters; however, global vortical aspects increase with $Re_{0},p_{0}$ and the number of species; nevertheless, at the same $Re_{0},p_{0}$ and density ratio, vorticity aspects are not influenced by the number of species. Species mixing produces strong density gradients which increase with $p_{0}$ and otherwise scale with $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ but, when scaled by $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, are not affected by the number of species. Generalized Korteweg-type equations are developed for a multi-species mixture, and a priori estimates based on the largest density gradient show that the Korteweg stresses, which account for the influence of the density gradient, have negligible contribution in the momentum equation. The species-specific effective Schmidt number, $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$, is computed and it is found that negative values occur for all minor species – particularly for H2 – thus indicating uphill diffusion, while the major species experience only regular diffusion. The probability density function (p.d.f.) of $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$ shows strong variation with $p_{0}$ but weak dependence on the number of species; however, the p.d.f. substantially varies with the identity of the species. In contrast, the p.d.f. of the effective Prandtl number indicates dependence on both $p_{0}$ and the number of species. Similar to $Sc_{\unicode[STIX]{x1D6FC},\mathit{eff}}$, the species-specific effective Lewis-number p.d.f. depends on the species, and for all species the mean is smaller than unity, thus invalidating one of the most popular assumptions in combustion modelling. Simplifying the mixture composition by reducing the number of minor species does not affect the crucial species–temperature relationship of the major species that, for accuracy, must be retained in combustion simulations, but this relationship is affected for the minor species and in regions of uphill diffusion, indicating that the reduction is nonlinear in nature.

The present paper investigates the influence of dense gases governed by complex equations of state on the dynamics of homogeneous isotropic turbulence. In particular, we investigate how differences due to the complex thermodynamic behaviour and transport properties affect the small-scale structures, viscous dissipation and enstrophy generation. To this end, we carry out direct numerical simulations of the compressible Navier–Stokes equations supplemented by advanced dense gas constitutive models. The dense gas considered in the study is a heavy fluorocarbon (PP11) that is shown to exhibit an inversion zone (i.e. a region where the fundamental derivative of gas dynamics $\unicode[STIX]{x1D6E4}$ is negative) in its vapour phase, for pressures and temperatures of the order of magnitude of the critical ones. Simulations are carried out at various initial turbulent Mach numbers and for two different initial thermodynamic states, one immediately outside and the other inside the inversion zone. After investigating the influence of dense gas effects on the time evolution of mean turbulence properties, we focus on the statistical properties of turbulent structures. For that purpose we carry out an analysis in the plane of the second and third invariant of the deviatoric strain-rate tensor. The analysis shows a weakening of compressive structures and an enhancement of expanding ones. Strong expansion regions are found to be mostly populated by non-focal convergence structures typical of strong compression regions, in contrast with the perfect gas that is dominated by eddy-like structures. Additionally, the contribution of non-focal expanding structures to the dilatational dissipation is comparable to that of compressed structures. This is due to the occurrence of steep expansion fronts and possibly of expansion shocklets which contribute to enstrophy generation in strong expansion regions and that counterbalance enstrophy destruction by means of the eddy-like structures.

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