This study explores the turbulent breakdown of high-enthalpy hypersonic boundary layers under adiabatic wall conditions using direct numerical simulations, with a focus on finite-rate chemistry effects. By subjecting a Mach 10 boundary layer to controlled perturbations via suction and blowing, the investigation shows the evolution from laminar to fully turbulent regime, via second-mode transition. In contrast to conventional understanding, this work identifies subharmonic breakdown scenario as significant contributor to turbulent transition, with energy transfer to secondary disturbances driving the process. The influence of finite-rate chemistry on growth rates is analyzed, revealing that chemical non-equilibrium has a stabilizing effect on secondary instabilities. Dynamic mode decomposition analysis further elucidates the modes predominantly involved in turbulent breakdown.

As shown by Wenzel et al. (J. Fluid Mech., vol. 930, 2022, A1), the Eckert number Ec defined using the difference between recovery temperature (T) over bar (r) and wall temperature (T) over bar (w) can be understood as a meaningful quantity to compare heat-transfer effects inside compressible turbulent boundary layers (for a calorically perfect gas), no matter whether these are caused by different Mach-number or wall-temperature conditions. While the named study deduced this comparative behaviour of Ec from an integral perspective in a strict sense, Cogo et al. (J. Fluid Mech., vol. 974, 2023, A10) performed a systematic parameter study based on the previous findings to look at wall-normal profiles. They have shown that the diabatic parameter Theta, being equivalent to Ec, is capable of categorizing heat-transfer effects for cases at different Mach numbers, even to some extent for some of the wall-normal profiles. Building on this progress, the present paper provides a comprehensive classification of both existing and newly computed super- and hypersonic direct numerical simulation data at various wall temperature conditions into heated cases, adiabatic cases or weakly/moderately/strongly/quasi-incompressibly cooled cases. Hereby, the classification is largely based on the wall-normal position of the temperature peak occurring in cooled boundary-layer cases, which is one of the determining factors for the topological characteristics of diabatic boundary-layer profiles. Integrating high-enthalpy data into the analysis allowed us to confirm the reliability of the proposed classification also in more complex scenarios, where the calorically perfect gas assumption no longer applies and additional heat-transfer mechanisms come into play. While the Eckert number is shown to well characterize heat-transfer effects on most important temperature-related quantities for a wide range of Mach numbers and (T) over bar (w)/(T) over bar (r) conditions, also the local Reynolds number Ret is shown to notably affect the strength of heat-transfer effects. Since both Ec and Ret can be determined in advance - or estimated to a reasonable extent - a key advantage of the classification scheme is to allow for an effective a priori estimation of the extent to which heat-transfer effects are to be expected for a given compressible turbulent boundary-layer configuration.

A high-order shock-capturing central finite-difference scheme is evaluated for numerical simulations of hypersonic high-enthalpy flows out of thermochemical equilibrium. The scheme is an extension to thermochemical out-of-equilibrium flows of the technique presented in Sciacovelli et al. (2021) for high-speed flows in chemical nonequilibrium. It relies on a standard tenth-order accurate central-difference approximation of the inviscid fluxes, supplemented with a high-order accurate nonlinear artificial dissipation term of ninth-order accuracy in smooth flow regions. Close to flow discontinuities, a shock-capturing low-order term is activated based on a highly selective shock sensor. To enable robust and non oscillatory solutions in regions of strong discontinuities of the thermodynamic variables, including the vibrational temperature, a shock detector consisting of a combination of a pressure-based term and Ducros’ vorticity/dilatation sensor is applied to all conservation equations except that of vibrational energy. For the latter, a shock sensor based on the vibrational temperature itself is adopted instead, to account for the loose coupling between vibrational energy and pressure. The accuracy and robustness of the proposed approach is demonstrated for a variety of thermochemical non-equilibrium configurations, ranging from one-dimensional benchmarks to three-dimensional turbulent flows, for which the ILES capabilities of the selective high-order numerical dissipation are also showcased.

PurposeThe purpose of the paper is to analyse the performances of closures and compressibility corrections classically used in turbulence models when applied to highly-compressible turbulent boundary layers (TBLs) over flat plates.Design/methodology/approachA direct numerical simulation (DNS) database of TBLs, covering a wide range of thermodynamic conditions, is presented and exploited to perform a priori analyses of classical and recent closures for turbulent models. The results are systematically compared to the "exact" terms computed from DNS.FindingsThe few compressibility corrections available in the literature are not found to capture DNS data much better than the uncorrected original models, especially at the highest Mach numbers. Turbulent mass and heat fluxes are shown not to follow the classical gradient diffusion model, which was shown instead to provide acceptable results for modelling the vibrational turbulent heat flux.Originality/valueThe main originality of the present paper resides in the DNS database on which the a priori tests are conducted. The database contains some high-enthalpy simulations at large Mach numbers, allowing to test the performances of the turbulence models in the presence of both chemical dissociation and vibrational relaxation processes.

The dynamics of a shock wave impinging on a transitional high-enthalpy boundary layer out of thermochemical equilibrium is investigated for the first time by means of a direct numerical simulation. The freestream Mach number is equal to 9 and the oblique shock impinges with a cooled flat-plate boundary layer with an angle of 10{\deg}, generating a reversal flow region. In conjunction with freestream disturbances, the shock impingement triggers a transition to a fully turbulent regime shortly downstream of the interaction region. Accordingly, wall properties emphasize the presence of a laminar region, a recirculation bubble, a transitional zone and fully turbulent region. The breakdown to turbulence is characterized by an anomalous increase of skin friction and wall heat flux, due to the particular shock pattern. At the considered thermodynamic conditions the flow is found to be in a state of thermal non-equilibrium throughout, with non-equilibrium effects enhanced and sustained by the shock-induced laminar/turbulent transition, while chemical activity is almost negligible due to wall cooling. In the interaction region, relaxation towards thermal equilibrium is delayed and the fluctuating values of the rototranslational and the vibrational temperatures strongly differ, despite the strong wall-cooling. The fully turbulent portion exhibits evolutions of streamwise velocity, Reynolds stresses and turbulent Mach number in good accordance with previous results for highly-compressible cooled-wall boundary layers in thermal nonequilibrium, with turbulent motions sustaining thermal nonequilibrium. Nevertheless, the vibrational energy is found to contribute little to the total wall heat flux.

A priori tests of turbulence models for the compressible Reynolds-Averaged Navier--Stokes (RANS) are performed by using Direct Numerical Simulations (DNS) data of zero-pressure-gradient flat-plate turbulent boundary layers. The DNS database covers a wide range of operating conditions, ranging from supersonic ($M_\infty = 2.25$) up to the high-enthalpy hypersonic regime ($M_\infty= 12.48$). Several RANS closures and compressibility corrections in the literature are assessed against the exact terms from the DNS. Particular attention is paid to closure models for the turbulent heat fluxes and the dilatational dissipation, as well as to the analysis of turbulence/chemistry and turbulence/vibrational relaxation interactions for the high-enthalpy simulations.

: A high-order shock-capturing finite-difference scheme for scale-resolving numerical simulations of hypersonic high-enthalpy flows, involving thermal non-equilibrium effects, is presented. The suitability of the numerical strategy for such challenging configurations is assessed in terms of accuracy and robustness, with special focus on shock-capturing capabilities. The approach is demonstrated for a variety of thermochemical non-equilibrium configurations.

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.

The influence of high-enthalpy effects on hypersonic turbulent boundary layers is investigated by means of direct numerical simulations (DNS). A quasi-adiabatic flat-plate air flow at free-stream Mach number equal to 10 is simulated up to fully-developed turbulent conditions using a five-species, chemically-reacting model. A companion DNS based on a frozen-chemistry assumption is also carried out, in order to isolate the effect of finite-rate chemical reactions and assess their influence on turbulent quantities. In order to reduce uncertainties associated with turbulence generation at the inlet of the computational domain, both simulations are initiated in the laminar flow region and the flow is let to evolve up to the fully turbulent regime. Modal forcing by means of localized suction and blowing is used to trigger laminar-to-turbulent transition. The high temperatures reached in the near wall region including the viscous and buffer sublayers activate significant dissociation of both oxygen and nitrogen. This modifies in turn the thermodynamic and transport properties of the reacting mixture, affecting the first-order statistics of thermodynamic quantities. Due to the endothermic nature of the chemical reactions in the forward direction, temperature and density fluctuations in the reacting layer are smaller than in the frozen-chemistry flow. However, the first- and second-order statistics of the velocity field are found to be little affected by the chemical reactions under a scaling that accounts for the modified fluid properties. We also observed that the Strong Reynolds Analogy (SRA) remains well respected despite the severe hypersonic conditions and that the computed skin friction coefficient distributions match well the results of the Renard-Deck decomposition extended to compressible flows.

High-speed turbulent flows are encountered in most space-related applications (including exploration, tourism and defense fields) and represent a subject of growing interest in the last decades. A major challenge in performing high-fidelity simulations of such flows resides in the stringent requirements for the numerical schemes to be used. These must be robust enough to handle strong, unsteady discontinuities, while ensuring low amounts of intrinsic dissipation in smooth flow regions. Furthermore, the wide range of temporal and spatial active scales leads to concurrent needs for numerical stabilization and accurate representation of the smallest resolved flow scales in cases of under-resolved configurations. In this paper, we present a finite-difference high-order shock-capturing technique based on Jameson's artificial diffusivity methodology. The resulting scheme is ninth-order-accurate far from discontinuities and relies on the addition of artificial dissipation close to large gradients. The shock detector is slightly revised to enhance its selectivity and avoid spurious activations of the shock-capturing term. A suite of test cases ranging from 1D to 3D configurations (namely, shock tubes, Shu-Osher problem, isentropic vortex advection, under-expanded jet, compressible Taylor-Green Vortex, supersonic and hypersonic turbulent boundary layers) is analysed in order to test the capability of the proposed numerical strategy to handle a large variety of problems, ranging from calorically-perfect air to multi-species reactive flows. Results obtained on under-resolved grids are also considered to test the applicability of the proposed strategy in the context of implicit Large-Eddy Simulations.

A study of dense-gas effects on the stability of compressible boundary-layer flows is conducted. From the laminar similarity solution, the temperature variations are small due to the high specific heat of dense gases, leading to velocity profiles close to the incompressible ones. Concurrently, the complex thermodynamic properties of dense gases can lead to unconventional compressibility effects. In the subsonic regime, the Tollmien-Schlichting viscous mode is attenuated by compressibility effects and becomes preferentially skewed in line with the results based on the ideal-gas assumption. However, the absence of a generalized inflection point precludes the sustainability of the first mode by inviscid mechanisms. On the contrary, the viscous mode can be completely stable at supersonic speeds. At very high speeds, we have found instances of radiating supersonic instabilities with substantial amplification rates, i.e. waves that travel supersonically relative to the free-stream velocity. This acoustic mode has qualitatively similar features for various thermodynamic conditions and for different working fluids. This shows that the leading parameters governing the boundary-layer behaviour for the dense gas are the constant-pressure specific heat and, to a minor extent, the density-dependent viscosity. A satisfactory scaling of the mode characteristics is found to be proportional to the height of the layer near the wall that acts as a waveguide where acoustic waves may become trapped. This means that the supersonic mode has the same nature as Mack's modes, even if its frequency for maximal amplification is greater. Direct numerical simulation accurately reproduces the development of the supersonic mode and emphasizes the radiation of the instability waves.

High-speed flows of gases with complex thermodynamic behavior (often referred to as “real gases”) have been paid growing interest from the scientific community due to the manifold applications in aerospace and power generation systems. In this work, we focus on so-called dense-gas flows found in several engineering applications, ranging from energy production to high-speed wind tunnels. Dense gases are single-phase fluids of complex molecules, at pressure and temperature conditions of the same order of their thermodynamic critical point. These fluids, in particular those belonging to the Bethe–Zel’dovich–Thompson family, may exhibit nonclassical phenomena in the transonic and supersonic regimes such as expansion shocks or mixed waves; these nonclassical effects may be characterized by means of the fundamental derivative of gas dynamics [8].

A study of dense-gas effects on the laminar, transitional and turbulent characteristics of boundary layer flows is conducted. The laminar similarity solution shows that temperature variations are small due to the high specific heats of dense gases, leading to velocity profiles close to the incompressible ones. Nevertheless, the complex thermodynamics of the base flow has a major impact on unstable modes, which bear similarities with those obtained for a strongly cooled wall. Numerical simulations of spatially developing boundary layers yield turbulent statistics for the dense gas flow that remain closer to the incompressible regime than perfect gas ones despite the presence of strongly compressible structures.

High-speed turbulent boundary layers of a dense gas (PP11) and a perfect gas (air) over flat plates are investigated by means of direct numerical simulations and large eddy simulations. The thermodynamic conditions of the incoming flow are chosen to highlight dense gas effects, and laminar-to-turbulent transition is triggered by suction and blowing. In the paper, the behavior of the fully developed turbulent flow region is investigated. Due to the low characteristic Eckert number of dense gas flows ( $$\hbox {Ec}=U_\infty ^2/c_{p,\infty }T_\infty$$ ), the mean velocity profiles are largely insensitive to the Mach number and very close to the incompressible case even at high speeds. Second-order velocity statistics are also weakly affected by the flow Mach number and the velocity spectra are characterized by a secondary peak in the outer region of the boundary layer because of the higher local friction Reynolds number. Despite the incompressible-like velocity and Reynolds-stress profiles, the strongly non-ideal thermodynamic and transport-property behavior of the dense gas results in unconventional distributions of the fluctuating thermo-physical quantities. Specifically, density and viscosity fluctuations reach a peak close to the wall, instead of vanishing as in perfect gas flows. Additionally, dense gas boundary layers exhibit higher values of the fluctuating Mach number and velocity divergence and a larger dilatational-to-solenoidal dissipation ratio in the near-wall region, which represents a major deviation from high-Mach-number perfect gas boundary layers. Other significant deviations are represented by the more symmetric probability distributions of fluctuating quantities such as the density and velocity divergence, due to the more balanced occurrence of strong expansion and compression events.

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.

In this work we investigate the influence of dense gas effects on compressible wall-bounded turbulence. Turbulent flows of dense gases represent a research field of great importance for a wide range of applications in engineering. Dense gases are single-phase fluids with a molecular complexity such that the fundamental derivative of gas dynamics (Thompson, Phys Fluids, 14:1843–1849, 1979, [1]) \(\varGamma := 1 + \frac{\rho }{c} \left. \frac{\partial c}{\partial \rho }\right| _s\) (where \(\rho \) is the density, p the pressure, s the entropy, and c the sound speed), which measures the rate of change of the sound speed in isentropic transformations, is less than one in a range of thermodynamic conditions close to the saturation curve. In such conditions, the speed of sound increases in isentropic expansions and decreases in isentropic compressions, unlike the case of perfect gases. For dense gases, the perfect gas model is no longer valid, and more complex equations of state must be used to account for their peculiar thermodynamic behavior. Moreover, in the dense gas regime, the dynamic viscosity \(\mu \) and the thermal conductivity \(\lambda \) depend on temperature and pressure through complex relationships. Similarly, the approximation of nearly constant Prandtl number Pr=\(\mu c_p/\lambda \) is no longer valid. Numerical simulations of turbulent dense gas flows of engineering interest are based on the (Reynolds-Averaged Navier–Stokes) RANS equations, which need to be supplemented by a model for the Reynolds stress tensor and turbulent heat flux. The accuracy of RANS models for dense-gas flows has not been properly assessed up to date, due to the lack of both experimental and numerical reference data.

Transonic flows through axial, multi-stage, transcritical ORC turbines, are investigated by using a numerical solver including advanced multiparameter equations of sta te and a high-order discretization scheme. The working fluids in use are the refrigerants R134a and R245fa, classifie d as dense gases due to their complex molecules and relatively high molecular weight. Both inviscid and viscou s numerical simulations are carried out to quantify the impact of dense gas effects and viscous effects on turbine pe rformance. Both supercritical and subcritical inlet conditions are studied for the considered working fluids. In the former case, flow across the turbine is transcritical, since turbine output pressure is subcritical. Numerical re sults show that, due to dense gas effects characterizing the flow at supercritical inlet conditions, supercritical O RC turbines enable, for a given pressure ratio, a higher isentropic efficiency than subcritical turbines using the s ame working fluid. Moreover, for the selected operating conditions, R134a provides a better performance than R245f a. Nomenclature a Speed of sound Cf Friction coefficient cp Specific heat capacity at constant pressure cv Specific heat capacity at constat volume e Internal energy E Total internal energy f Flux GTP-14-1031 Cinnella 1 H Total enthalpy ̄̄ I Unit tensor p Pressure Pr Prandtl number q Actual order of convergence q Heat flux vector R Universal gas constant r Grid refinement ratio s Entropy T Temperature v Specific volume v Flow velocity vector V Control Volume ∂V Boundary of the control volume w Conservative variables vector Z Compressibility factor γ Specific heat ratio Γ Fundamental Derivative of Gas Dynamics δ Reduced density η Isentropic efficiency κ Thermal Conductivity μ Dinamic Viscosity ρ Density σ Dipole moment σr Dimensionless dipole moment τ Inverse reduced temperature ̄̄ τ Viscous stress tensor Φ Helmholtz free energy ω Acentric factor Ω̄ Angular velocity Subscript ∞ ideal-gas state c critical value GTP-14-1031 Cinnella 2

Dense gas effects, encountered in many engineering applications, lead to unconventional variations of the thermodynamic and transport properties in the supersonic flow regime, which in turn are responsible for considerable modifications of turbulent flow behavior with respect to perfect gases. The most striking differences for wall-bounded turbulence are the decoupling of dynamic and thermal effects for gases with high specific heats, the liquid-like behavior of the viscosity and thermal conductivity, which tend to decrease away from the wall, and the increase of density fluctuations in the near wall region. The present work represents a first attempt of quantifying the influence of such dense gas effects on modeling assumptions employed for the closure of the Reynolds-averaged Navier–Stokes equations, with focus on the eddy viscosity and turbulent Prandtl number models. For that purpose, we use recent direct numerical simulation results for supersonic turbulent channel flows of PP11 (a heavy fluorocarbon representative of dense gases) at various bulk Mach and Reynolds numbers to carry out a priori tests of the validity of some currently-used models for the turbulent stresses and heat flux. More specifically, we examine the behavior of the modeled eddy viscosity for some low-Reynolds variants of the \(k-\varepsilon \) model and compare the results with those found for a perfect gas at similar conditions. We also investigate the behavior of the turbulent Prandtl number in dense gas flow and compare the results with the predictions of two well-established turbulent Prandtl number models.