The complex behaviours of supersonic turbulent boundary-layer flows interacting with combustion is explored through the use of direct numerical simulations. The chosen flow model is a non-premixed hydrogen–air flame ignited within a three-dimensional supersonic turbulent flat-plate boundary layer operating at a Mach number of 2.33 and a friction Reynolds number of approximately 1000. The simulation involves a finite-rate model for the hydrogen–air reaction including 9 species and 19 steps with adiabatic non-catalytic wall conditions. The inlet flow consists of preheated air in the main stream and a hydrogen film injected in proximity to the cold wall, inducing mixing and ignition in the outer layer. The specific inlet configuration leads to two successive transition stages of distinct mechanisms, with the first stage related to the mixing-layer instability and the second one to the boundary-layer instability. The excess Reynolds stresses resulting from the transition exhibit a downstream decrease, with values being lower than canonical adiabatic profiles, particularly in the outer layer. This difference is attributed to the reduced local friction Reynolds numbers, associated with the non-classical wall-normal distributions of density and viscosity. The effect of combustion on the recovery process and skin friction is analysed in comparison with a supplemented chemistry-frozen setting. Velocity–temperature and velocity–species correlations are further examined. The wall-normal profiles of turbulent Prandtl number tend to classical non-reacting values and the turbulent Schmidt number is only slightly affected by the reaction. The strong Reynolds analogy linking velocity and temperature fluctuations is found to be invalid in the outer layer due to the presence of large-scale temperature fluctuations and the inhomogeneity of hydrogen gas distribution, as revealed by a spectrum analysis. A statistical analysis of elementary reactions at varying wall-normal distances is provided, highlighting the dominance of hydrogen atom depletion in the inner region and the prevalence of water vapour production in the outer region, together contributing to the chemical heat release across the boundary layer. Turbulence–chemistry interaction is assessed through a comparison of mean turbulent and laminar chemical heat release rates, showing significant differences of up to an order of magnitude. Using a simple diagnostic tool, the modelling of fluctuations associated with hydrogen gas and two intermediate products is found to be crucial for turbulent chemistry closure.

The unresolved scalar variance in large-eddy simulations of turbulent flows is a fundamental physical and modelling parameter. Despite its importance, relatively few algebraic models have been developed for this important variable with the most prominent models to date being the classic scale-similarity and gradient models. In this work a new generalized modelling framework based on reconstruction has been developed, which in contrast to classic modelling approaches allows the construction of base static variance models of arbitrary accuracy. It is demonstrated that higher-order reconstructions naturally lead to base static variance models of increased accuracy, and that the classic scale-similarity and gradient models are subsets of more general and higher-order models. The classic scale-similarity assumption for developing dynamic models is also revisited, and it is demonstrated that this can essentially be reinterpreted as a two-level reconstruction approach. Based on this result, a new general methodology is proposed that allows the construction of dynamic models for any given base static model, and a corresponding general reconstruction operator, algebraic or iterative. Consequently, improved static and dynamic models for the scalar variance are developed. The newly developed models are then thoroughly tested a priori using two high-fidelity direct numerical simulation databases corresponding to two substantially different flame and flow configurations, and are shown to outperform classic algebraic models for the variance.

The tangential strain rate in premixed flames impacts significantly the flame surface area generation and thus the combustion process. Studies on incompressible isotropic turbulence have revealed that the mean tangential strain rate at material and iso-scalar surfaces is positive and exhibits a universal value when normalized by the Kolmogorov time. This is associated with the preferential alignment of the surface normal with the most compressive principal strain rate. The present study investigates such effects in premixed hydrogen and iso-octane flame kernels using direct numerical simulations. It is shown that the normalized mean tangential strain rate of the investigated flames has a very similar value compared with the incompressible flows. However, in the reaction zone, the flame surface normal aligns preferentially with the most extensive principal strain rate. Furthermore, this alignment depends on the reaction progress variable and the Lewis number, while the tangential strain rate remains independent of these parameters. Such counter-intuitive behaviour is systematically investigated by decomposing the effects of dilatation and residual solenoidal turbulence. It is found that the solenoidal turbulence influences significantly the tangential strain rate. A general effect of turbulence on the tangential strain rate is identified, which is consistent with incompressible flows and independent of the Lewis number and the reaction progress variable. This is a remarkable finding indicating that models of the tangential strain rate developed based on incompressible flows apply also to premixed flames with different Lewis numbers, and, for the modelling, only the solenoidal turbulence should be considered.

Lean hydrogen/air premixed flame flashback in a turbulent boundary layer over a flat plate is investigated using three-dimensional direct numerical simulation with detailed chemical kinetics. The upstream propagation of the flame takes place in near-wall turbulence and the interaction between the flame and the approaching reactant flow is studied. It is found that backflow regions are always present immediately upstream of flame bulges that are convex towards the reactants, confirming earlier observations. A flashback speed, including the effects of flame displacement speed and flow velocity, is introduced to quantify the flame flashback behaviour. This analysis indicates that the flashback speed is overall positive and it is considerably affected by the presence of the backflow regions. A budget analysis of the pressure transport equation is performed to explain the presence of the backflow regions. It is suggested that the positive dilatation and thermal diffusion terms near the leading edge of flame bulges are the main reasons for the pressure increase, leading to an adverse pressure gradient. The effects of the flame-induced adverse pressure gradient on the structures of the turbulent boundary layer are also investigated. It is revealed that the near-wall mean velocity and skin-friction coefficient are reduced due to the adverse pressure gradient. The coherent vortical structures of the boundary layer turbulence are lifted by the adverse pressure gradient. The analysis of the Reynolds stress component showed that the ejection event is augmented by combustion while the sweep event is attenuated, which facilitates the occurrence of flame flashback.

Thermoacoustic systems are complex systems where the interactions between the hydrodynamic, acoustic and heat release rate fluctuations lead to diverse dynamics such as chaos, intermittency and ordered dynamics. Such complex interactions cause catastrophically high-amplitude acoustic pressure oscillations and the emergence of order in the spatiotemporal dynamics, referred to as thermoacoustic instability. In this work, we use multilayer networks to study the spatial pattern of inter-subsystem interactions between the vorticity dynamics and thermoacoustic power generated due to acoustically coupled combustion in a bluff-body-stabilised turbulent dump combustor. We construct a two-layered network where the layers represent the thermoacoustic power and vorticity fields. The inter-layer links are determined using cross-variable short-window correlations between vorticity and thermoacoustic power fluctuations at any two locations in the flow field. Analysing the topology of inter-layer networks, using network properties such as degree correlations and link-rank distributions, helps us infer the spatial inhomogeneities in inter-subsystem interactions and unravel the fluid mechanical processes involved during different dynamical states. We show that, during chaotic dynamics, interactions between subsystems are non-localised and spread throughout the flow field of the combustor. During the state of thermoacoustic instability (order), we find that intense interactions occur in between regions of coherent vortex shedding and thermoacoustic power generation and we understand that these processes are strongly and locally coupled. Moreover, we discover that such dense inter-layer connections emerge in spatial pockets in the dump plane of the combustor during the state of intermittency much prior to the onset of order.

The presence of large-scale coherent structures can significantly impact the dynamics of a turbulent flow field and the behaviour of a flame stabilized in that flow. The goal of this study is to analyse how increasing free-stream turbulence can change the response of the flow to longitudinal acoustic excitation of varying amplitudes. We study the flow in the wake of a cylindrical bluff body at both non-reacting and reacting conditions, as the presence of a flame can significantly alter the global stability of the flow. The frequency of longitudinal acoustic excitation is set to match the natural frequency of anti-symmetric vortex shedding for this configuration and we vary the free-stream turbulence using perforated plates upstream of the bluff body. The results show that varying the level of free-stream turbulence can influence not only the amplitude of the coherent flow response, but also the symmetry of vortex shedding in the presence of longitudinal acoustic excitation. Increasing the turbulence intensity can fundamentally change the structure of the time-averaged flow and can directly impact the coherent flow response in two ways. First, increasing turbulence intensity can enhance the amplitude of the natural anti-symmetric vortex shedding mode in the wake. Second, increasing turbulence intensity weakens the symmetric response of the flow to the longitudinal acoustic excitation. In the non-reacting and reacting conditions, both symmetric and anti-symmetric modes are present and are characterized using a spectral proper orthogonal decomposition. We see evidence of interaction between the symmetric and anti-symmetric modes, which leads to an interference pattern in the coherent vorticity response in the shear layers. We conclude by presenting a conceptual model for the influence that turbulence has on these flows.

Direct numerical simulation data obtained from four pairs of turbulent, lean hydrogen–air, complex-chemistry flames are analysed to explore the influence of molecular diffusion on flame surface density, displacement speed $S_d$ and the flame surface density transport equation terms. Each pair involves (i) a flame where mixture-averaged molecular diffusivities are adopted and Lewis number $Le$ is significantly less than unity and (ii) an equidiffusive flame where all molecular diffusivities are set equal to molecular heat diffusivity of the mixture and $Le=1$, with other things being equal. Reported results show that significantly higher turbulent burning rates simulated in the former flames result mainly from an increase in the local fuel consumption rate, whereas an increase in flame surface area plays a secondary role, especially in more intense turbulence. The rate increase stems from (i) an increase in the peak local fuel consumption rate and (ii) an increase in a width of a zone where the rate is significant. The latter phenomenon is of more importance in richer flames and both phenomena are most pronounced in the vicinity of the flame leading edges, thus indicating a crucial role played by the leading edge of a premixed turbulent flame in its propagation. Moreover, mean displacement speed differs significantly from the laminar flame speed even in the equidiffusive flames, varies substantially across flame brush and may be negative at the leading edges of highly turbulent flames.

This study analyses the stability characteristics of the shear layer vortices (SLV) in a reacting jet in crossflow, analysing effects of flame position, momentum flux ratio ($J$) and density ratio ($S$). It utilizes 40 kHz particle image velocimetry to characterize the dominant SLV frequencies, streamwise evolution and convective/global stability characteristics for three different canonical configurations, one non-reacting and two reacting (‘R1’ and ‘R2’). In the non-reacting case, both convective and global instability is observed, depending upon $S$ and $J$. Qualitatively similar $S$ dependencies occur for the R1 reacting case where the radial flame position lies outside the jet shear layer, albeit with slower SLV growth rates. When the flame lies inside the jet shear layer, the R2 reacting case, a qualitatively different behaviour is observed, as vorticity concentration in the shear layers is suppressed almost completely. Finally, we show that frequency and stability characteristics of the non-reacting and R1 cases can be scaled in a unified manner using a counter-current shear layer model. This model relates these SLV behaviours to a vorticity layer thickness, a velocity scale and an effective density ratio (noting that there are three distinct densities associated with the jet, the crossflow and the burned gases). These parameters were extracted from the data and used to collapse the frequency scaling, and to explain the transition to self-excited oscillatory behaviour.

To explore the direction of inter-scale transfer of scalar variance between subgrid scale (SGS) and resolved scalar fields, direct numerical simulation data obtained earlier from two complex-chemistry lean hydrogen–air flames are analysed by applying Helmholtz–Hodge decomposition (HHD) to the simulated velocity fields. Computed results show backscatter of scalar (combustion progress variable $c$) variance, i.e. its transfer from SGS to resolved scales, even in a highly turbulent flame characterized by a unity-order Damköhler number and a ratio of Kolmogorov length scale to thermal laminar flame thickness as low as 0.05. Analysis of scalar fluxes associated with the solenoidal and potential velocity fields yielded by HHD shows that the documented backscatter stems primarily from the potential velocity perturbations generated due to dilatation in instantaneous local flames, with the backscatter being substantially promoted by a close alignment of the spatial gradient of mean scalar progress variable and the potential-velocity contribution to the local SGS scalar flux. The alignment is associated with the fact that combustion-induced thermal expansion increases local velocity in the direction of $\boldsymbol {\nabla } c$. These results call for development of SGS models capable of predicting backscatter of scalar variance in turbulent flames in large eddy simulations.

We study the spatio-temporal dynamics of high-frequency combustion instability in a model single-element rocket combustor using an acoustic energy flux-based spatial network. The acoustic energy source collapses by the formation of small communities with weak connection when the flame edge is attached to the injector rim. In contrast, large communities with strong connection are formed in the shear layer between the oxygen and hydrogen jets when the flame edge is detached from the injector rim, which has a significant impact on driving combustion instability. The switching between the attachment and detachment of the flame edge during combustion instability can be explained by the spectral-clustering-based transition network constructed from the pressure and flow velocity of the hydrogen jet at the injector exit, and the temperature near the injector rim.

In the present work, direct numerical simulation of a laboratory-scale lean premixed reacting jet in cross-flow was performed to explore the flow–flame structures and turbulence–flame interactions. A jet of lean premixed ethylene–air mixtures (equivalence ratio $\phi = 0.6$) was injected into a hot vitiated cross-flow. Both non-reacting and reacting cases were simulated. It was found that the reacting jet penetrates deeper in the cross-flow with a weaker shear layer compared with the non-reacting one. The wake of the non-reacting and reacting jet is characterized by vertical vortices and recirculation zones, respectively. As for the flame structure of the reacting case, the reaction intensity varies considerably in different flame zones. The heat release rate on the leeward side is higher than that on the windward side, but lower than that of the corresponding laminar flame. The analysis of the turbulence–flame interactions of the reacting case showed that the large local Damköhler number ($Da$) related to reaction-induced dilatations results in an increased tendency of the scalar gradient to align with the most extensive strain rate, which is more evident in the regions with high heat release rate on the leeward side. Negative dilatation regions with positive tangential strain rate and negative normal strain rate are observed on the windward side. High positive dilatations appear on the flame front of the leeward side. The tangential strain rate is negatively correlated with the normal strain rate and curvature. Regions with a high local $Da$ on the windward side correspond with high positive curvature regions.

Premixed turbulent flames, encountered in power generation and propulsion engines, are an archetype of a randomly advected, self-propagating surface. While such a flame is known to exhibit large-scale intermittent flapping, the possible intermittency of its small-scale fluctuations has been largely disregarded. Here, we experimentally reveal the inner intermittency of a premixed turbulent V-flame, while clearly distinguishing this small-scale feature from large-scale outer intermittency. From temporal measurements of the fluctuations of the flame, we find a frequency spectrum that has a power-law subrange with an exponent close to $-2$, which is shown to follow from Kolmogorov phenomenology. Crucially, however, the moments of the temporal increment of the flame position are found to scale anomalously, with exponents that saturate at higher orders. This signature of small-scale inner intermittency is shown to originate from high-curvature, cusp-like structures on the flame surface, which have significance for modelling the heat release rate and other key properties of premixed turbulent flames.

A high-fidelity large-eddy simulation and unsteady flamelet combustion model construct is deployed to numerically investigate the effects of crosswind magnitude and pool fire shape on large-scale pool fire attributes. These include general flame dynamics, flame shape and radiative flux magnitude in and around the fire. Three pool fire shapes at a nominal length scale of 10 m are subjected to four crosswind magnitudes between 0 and 20 m s$^{-1}$. The pool shapes studied are circular, square and rectangular. The study includes the sensitivity of parameters to mesh and time step refinement. Results demonstrate that the rectangular shape, under crosswind, has low-levels of vertical velocity induction, resulting in a plume that is closer to the ground. In the quiescent regime, under-resolved meshes provide a higher radiative heat flux prediction compared with the most refined mesh. However, as crosswind increases, low mesh resolutions underpredicted radiative flux. This is due to the coarse mesh resolution not capturing small-scale vortical features that increased mixing and combustion efficiency. A transition of peak radiative flux with respect to crosswind occurs from the leeward- to windward-side of the pool, while sharp pool features result in larger radiative heat fluxes concentrated in regions of high scalar dissipation rate.

In the present study, turbulent particle-laden boundary layer combustion over a flat plate is investigated using direct numerical simulation (DNS). A two-way coupled Eulerian–Lagrangian point particle method is used for the solid phase. The effects of particle Stokes numbers, mass loadings and chemical reactions on the interactions between particles and boundary layer turbulence in the near-wall region are explored. It was found that particle heat transfer is dominant over wall heat transfer in the reacting case with heavy particles and large mass loadings, resulting in a lower fluid temperature. Particle accumulation due to the turbophoresis effect in the near-wall region is observed, which is more prominent in the cases with a large Stokes number. The turbophoresis effect is examined via the magnitude of streamwise vorticity $\varOmega _x$. It is shown that $\varOmega _x$ is attenuated by heavy particles, and the attenuation increases with increasing mass loadings. Therefore, particle wall–accumulation is less prominent in the cases with large mass loadings. Compared with the non-reacting cases, the distribution of particles is more inhomogeneous for the reacting cases, where the particles move faster due to intense reactions with increasing wall-normal distance. Finally, the flow topologies and the Reynolds stress anisotropy are examined to understand turbulence modulation by combustion and particles. It was suggested that light particles augment the vortex-dominant topologies, whereas heavy particles have an opposite effect. The anisotropy mapping of the Reynolds stress shows that the flows become more one-dimensional in the near-wall region for the cases with combustion and/or large mass loadings.

Global instabilities in swirling flows can significantly alter the flame and flow dynamics of swirl-stabilized flames, such as those in modern gas turbine engines. In this study, we characterize the interaction between the precessing vortex core (PVC), which is the consequence of a global hydrodynamic instability, and thermoacoustic instabilities, which are the result of a coupling between combustor acoustics and the unsteady heat release rate. This study is performed using experimental data obtained from a model gas turbine combustor employing two concentric swirling nozzles of air, separated by a ring of fuel injectors, operating at five bar pressure. The flow split between the two streams is systematically varied to observe the impact of flow structure variation on the system dynamics at both non-reacting and reacting conditions. High-speed stereoscopic particle image velocimetry, OH planar laser-induced fluorescence and acetone planar laser-induced fluorescence are used to obtain information about the velocity fields, flame and fuel flow behaviour, respectively. Spectral proper orthogonal decomposition and a complex network analysis are used to identify and characterize the dominant oscillation mechanisms driving the system. In the non-reacting data, a PVC is present in most cases and the amplitude of the oscillation increases with increasing flow through the centre nozzle. In the reacting data, three dominant modes are seen: two thermoacoustic modes and the PVC. Our results show that in the cases where the frequency of the PVC overlaps with either of the thermoacoustic modes, the thermoacoustic modes are suppressed. The complex network analysis coupled with a weakly nonlinear theoretical analysis suggests the mechanisms by which this coupling and suppression of the thermoacoustic mode occur.

A new flamelet model is developed to be used for subgrid modelling and coupled with the resolved flow description for turbulent combustion. The model differs from current models in several critical ways. (i) Non-premixed flames, premixed flames or multi-branched flame structures are determined rather than prescribed. (ii) The effects of shear strain and vorticity on the flames are determined. (iii) The strain rates and vorticity applied at the subgrid level are directly determined from the resolved-scale strain rates and vorticity without the use of a contrived progress variable. (iv) The flamelet model is three-dimensional without need for assuming axisymmetry or planar geometry. (v) The effect of variable density is addressed in the flamelet model. Solutions to the Navier–Stokes equations and the associated scalar equations governing the flamelet model are obtained without the boundary-layer approximation. By appropriate coordinate transformation, a similar solution is found for the rotational flamelet model, reducing it to a system of ordinary differential equations. Vorticity is shown to create a centrifugal force on the subgrid counterflow that modifies the molecular transport rates and burning rate. Sample computations of the rotational flamelet model without coupling to the resolved flow are presented first to demonstrate the importance of the new features of the model. Scaling laws are presented for relating strain rates and vorticity at the subgrid level to those quantities at the resolved-flow level for coupling with large-eddy simulations or the time-averaged mean-flow level for Reynolds-averaged flows. The time-averaged behaviour of a simple turbulent flow is resolved with coupling to the rotational flamelet model. Specifically, a two-dimensional, multicomponent, time-averaged planar shear layer with variable density and energy release is employed using a mixing-length description for the eddy viscosity. Needs for future study are identified.

The objective of this work is to investigate the interaction of turbulence with the nonlinear processes of particle nucleation and growth that occur in reaction crystallisation, also known as precipitation. A validated methodology for coupling the population balance equation with direct numerical simulation of turbulent flows is employed for simulating an experiment conducted by Schwarzer et al. (Chem. Engng Sci., vol. 61, no. 1, 2006, pp. 167–181), where barium sulphate nanoparticles are formed by mixing and reaction of barium chloride and sulphate acid in a T-mixer, with the spatial resolution resolved down to the Kolmogorov scale. A unity Schmidt number is assumed, since at present it is not possible to resolve the Batchelor scale for realistic Schmidt numbers (order of 1000 or more). The probability density function, filtered averages and spatial distribution of time and length scales are all examined in order to shed light on the interplay of turbulence and precipitation. Separate Damköhler numbers are defined for nucleation and growth and both are found to be close to unity, indicating that the process is neither mixing nor kinetics controlled. The nucleation length scales are also evaluated and compared with the Kolmogorov scale to show the importance of resolving nucleation bursts. In addition, zones of different rate-determining mechanisms are identified. The ultimate aim of precipitation is to obtain control over the product particle size distribution, and the present study elucidates the synergistic or competing roles of mixing, nucleation and growth on the process outcome and discusses the implications for modelling.

This study experimentally investigates reacting jets in cross-flow (RJICF), considering flame/shear layer offset, momentum flux ratio ($J$) and density ratio ($S$) effects. These results demonstrate that non-reacting JICF and RJICF can exhibit very similar or completely different dynamics and controlling physics, depending upon streamwise and radial flame location. Consistent with prior measurements of Getsinger et al. (Exp. Fluids, vol. 53 (3), 2012, pp. 783–801), spatial amplification rates of shear layer vortex (SLV) structures increased with decreasing $S$ for non-reacting cases. Similar $S$ dependencies exist for reacting cases in which the flame lay outside the jet shear layer, whose flow topology is also quite similar to non-reacting cases, albeit with reduced SLV growth rates. However, although the reacting cases have lower growth rates, these SLV structures ultimately reach approximately the same peak strength as in the non-reacting cases. Finally, the SLV decay rate in both non-reacting and reacting cases was found to similarly scale inversely with the initial SLV growth rate. As such, primarily inertial mechanisms govern SLV growth and decay for reacting cases where the flame lies outside the shear layer. In contrast, very different behaviour is exhibited by reacting cases where the flame lies inside the shear layer, where the locally increased viscosity exerts significant influences. In these reacting cases, SLV roll-up is completely suppressed and the entire jet column undulates over a long length scale relative to the jet diameter. As such, the relative roles of inertial and viscous mechanisms in controlling combustion influences on the SLV dynamics, changes markedly with shear layer–flame offset.

Direct numerical simulations of a three-dimensional cylindrical Richtmyer–Meshkov instability with and without chemical reactions are carried out to explore the chemical reaction effects on the statistical characteristics of transition and turbulent mixing. We adopt 9-species and 19-reaction models of non-premixed hydrogen and oxygen separated by a multimode perturbed cylindrical interface. A new definition of mixing width suitable for a chemical reaction is introduced, and we investigate the spatio-temporal evolution of typical flow parameters within the mixing regions. After reshock with a fuller mixing of fuels and oxygen, the chemical reaction becomes sufficiently apparent at affecting the evolution of the flow fields. Because of the generation of a combustion wave within the combustion regions and propagation, the growth of the mixing width with a chemical reaction is accelerated, especially around the outer radius with large temperature gradient profiles. However, the viscous dissipation rate in the early stage of the chemical reaction is greater because of heat release, which results in weakened turbulent mixing within the mixing regions. We confirm that small-scale structures begin to develop after reshock and then decay over time. During the developing process, helicity also begins to develop, in addition to kinetic energy, viscous dissipation rate, enstrophy, etc. In the present numerical simulations with cylindrical geometry, the fluctuating flow fields evolve from quasi-two-dimensional perturbations, and the generations of helicity can capture this transition process. The weakened fluctuations during shock compression can be explained as the inverse energy cascade, and the chemical reaction can promote this inverse energy cascade process.

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.