Turbulence–chemistry interaction in a Mach-7 hypersonic boundary layer with significant production of radical species is characterised using direct numerical simulation. Overriding a non-catalytic surface maintained as isothermal at 3000 K, the boundary layer is subject to finite-rate chemical effects, comprising both dissociation/recombination processes as well as the production of nitric oxide as mediated by the Zel’dovich mechanism. With kinetic-energy dissipation giving rise to temperatures exceeding 5300 K, molecular oxygen is almost entirely depleted within the aerodynamic heating layer, producing significant densities of atomic oxygen and nitric oxide. Owing to the coupling between turbulence-induced thermodynamic fluctuations and the chemical-kinetic processes, the Reynolds-averaged production rates ultimately depart significantly from their mean-field approximations. To better characterise this turbulence–chemistry interaction, which arises primarily from the exchange reactions in the Zel’dovich mechanism, a decomposition for the mean distortion of finite-rate chemical processes with respect to thermodynamic fluctuations is presented. Both thermal and partial-density fluctuations, as well as the impact of their statistical co-moments, are shown to contribute significantly to the net chemical production rate of each species. Dissociation/recombination processes are confirmed to be primarily affected by temperature fluctuations alone, which yield an augmentation of the molecular dissociation rates and reduction of the recombination layer’s off-wall extent. While the effect of pressure perturbations proves largely negligible for the mean chemical production rates, fluctuations in the species mass fractions are shown to be the primary source of turbulence–chemistry interaction for the second Zel’dovich reaction, significantly modulating the production of all major species apart from molecular nitrogen.

We derive the scale-by-scale uncertainty energy budget equation and demonstrate theoretically and computationally the presence of a self-similar equilibrium cascade of decorrelation in an inertial range of scales during the time range of power-law growth of uncertainty in statistically stationary homogeneous turbulence. This cascade is predominantly inverse and driven by compressions of the reference field’s relative deformation tensor and their alignments with the uncertainty velocity field. Three other subdominant cascade mechanisms are also present, two of which are forward and also dominated by compressions and one of which, the weakest and the only nonlinear one of the four, is inverse. The uncertainty production and dissipation scalings which may follow from the self-similar equilibrium cascade of decorrelation lead to power-law growths of the uncertainty integral scale and the average uncertainty energy which are also investigated. Compressions are key not only to chaoticity, as previously shown, but also to stochasticity.

We study experimentally, numerically and theoretically the gravitational instability induced by dissolution of carbon dioxide with a forced lateral flow. The study is restricted to the model case of a vertical Hele-Shaw cell filled with water. While a transverse (horizontal) flow is continuously forced through the whole cell, the carbon dioxide is introduced above the liquid–gas interface so that a $\textrm {CO}_2$-enriched diffusive layer gradually forms on top of the liquid phase. The diffusive layer destabilises through a convective process which entrains the $\textrm {CO}_2$–water mixture towards the bottom of the cell. The concentration fields are measured quantitatively by means of a pH-sensitive dye (bromocresol green) that reveals a classic fingering pattern. We observe that the transverse background flow has a stabilising effect on the gravitational instability. At low velocity (i.e. for small thickness-based Péclet numbers), the behaviour of the system is hardly altered by the background flow. Beyond a threshold value of the Péclet number ($\textit{Pe} \sim 15$), the emergence of the fingering instability is delayed (i.e. the growth rate becomes smaller), while the most unstable wavelength is increased. These trends can be explained by the stabilising role of the Taylor–Aris dispersion in the horizontal direction and a model is proposed, based on previous works, which justifies the scalings observed in the limit of large Péclet number for the growth rate ($\sigma ^\star \sim \textit{Pe}^{-4}$) and the most unstable wavelength ($\lambda ^\star \sim \textit{Pe}^{\,5/2}$). The flux (rate mass transfer) of $\textrm {CO}_2$ in the nonlinear regime is also weakly decreased by the background transverse flow.

In the present study, we investigate the relation between temperature ($T^{\prime}$) and streamwise velocity ($u^{\prime}$) fluctuations by assessing the state-of-the-art Reynolds analogy models. These analyses are conducted on three levels: in the statistical sense, in spectral space and via the distribution characteristics of temperature fluctuations. It is observed that the model proposed by Huang et al. (HSRA) (1995 J. Fluid Mech. 305, 185–218), is the only model that works well for both channel flows and turbulent boundary layers in the statistical sense. In spectral space, the intensities of $T^{\prime}$ at small scales are discovered to be larger than the predictions of these models, whereas those at scales corresponding to the energy-containing eddies and the large-scale motions are approximately equal to and smaller than the predictions of the HSRA, respectively. The success of the HSRA arises from this combined effect. In compressible turbulent boundary layers, the relationship between the intensities of positive temperature and negative velocity fluctuations is found to be well described by a model proposed by Gaviglio (1987 Intl J. Heat Mass Transfer, 30, 911–926), whereas that between negative temperature and positive velocity fluctuations is accurately depicted by the HSRA. The streamwise length scale, rather than the spanwise length scale, is found to be more suitable for characterising the scale characteristics of the $u^{\prime}-T^{\prime}$ relation in spectral space. Combining these observations and a newly proposed modified generalised Reynolds analogy (Cheng & Fu 2024 J. Fluid Mech. 999, A20), models regarding the relations in spectral space for both compressible channel flows and turbulent boundary layers are developed, and a strategy for generating more reliable temperature fluctuations as the inlet boundary condition for simulations of compressible boundary layers is also suggested.

The momentum dispersion model for flows in isotropic porous media has been validated and successfully applied by Rao & Jin (2022, J. Fluid Mech., vol. 937, A17). However, the anisotropic coupled models concerning heat–fluid–solid interactions in turbulent forced convection requires further development. This research proposes various anisotropic physical coefficient tensors to model the total drag ${R}_{i}$, interphase energy resistance $H$, momentum dispersion and thermal dispersion accounting for both anisotropic and isotropic scenarios. The effective physical coefficients of the Darcy–Forchheimer equation regarding ${R}_{i}$ are adapted to accommodate anisotropy. The heat transfer coefficient $h$ between the solid and fluid, despite being a scalar, is also required to depend on the local flow direction in anisotropic cases. Two scaling laws of $h$ with respect to a local Reynolds number ${\textit{Re}_{K}}$ are found: $h\sim \textit{Re}_K^2$ for the Darcy regime, and $h\sim \textit{Re}_{K}^{1/2}$ for the Forchheimer regime, with a transition at ${\textit{Re}_{K}}\sim 1$. The influence of momentum and thermal dispersions, along with the modelling errors of ${R}_{i}$ and $H$ originating from heterogeneity, are approximated using a second-order pseudo-stress tensor and a pseudo-flux vector, respectively. The effective viscosity and thermal diffusivity tensors are simplified into longitudinal and transverse components using tensor symmetries, and are assumed to rely mainly on another local Reynolds number ${\textit{Re}_{d}}$. Both components of the effective viscosity are positive in isotropic cases, whereas the longitudinal component may be negative in anisotropic cases, mainly serving as a compensation of overestimated drag. The coupled models are applied to simulate turbulent forced thermal convection in porous media with one or two length scales across a wide range of Reynolds numbers. The comparisons with direct numerical simulations results imply that the coupled macroscopic models can accurately predict not only statistically stationary distributions but also real-time changes in velocity and temperature.

Standing acoustic waves in a channel generate time-mean Eulerian flows. In homogeneous fluids, these streaming flows have been shown by Rayleigh to result from viscous attenuation of the waves in oscillatory boundary (i.e. Stokes) layers. However, the strength and structure of the mean flow significantly depart from the predictions of Rayleigh when inhomogeneities in fluid compressibility or density are present. This change in mean flow behaviour is of particular interest in thermal management, as streaming flows can be used to enhance cooling. In this work, we consider standing acoustic wave oscillations of an ideal gas in a differentially heated channel with hot- and cold-wall temperatures respectively set to $T_* + \Delta \varTheta _*$ and $T_*$. An asymptotic analysis for a normalised temperature differential $\Delta \varTheta _*/T_*$ comparable to the small acoustic Mach number is performed to capture the transition between the two documented regimes of Rayleigh streaming ($\Delta \varTheta _*\,{=}\,0$) and baroclinic streaming ($\Delta \varTheta _* =O(T_*)$). Our analytical solution accounts for existing experimental and numerical results and elucidates the separate contributions of viscous torques in Stokes layers and baroclinic forcing in the interior to driving the streaming flow. The analysis yields a scaling estimate for the temperature difference $\Delta \varTheta _{c_*}$ at which baroclinic driving is comparable to viscous forcing, signalling the smooth transition from Rayleigh to baroclinic acoustic streaming.

When a liquid film on a horizontal plate is driven in motion by a shear stress, surface waves are easily generated. This paper studies such flow at moderate Reynolds numbers, where the surface tension and inertial force are equally important. The governing equations for two-dimensional flows are derived using the long-wave approximation along with the integral boundary-layer theory. For small disturbances, the dispersion relation and neutral curves are determined by the linear stability analysis. For finite-amplitude perturbations, the numerical simulation suggests that the oscillations generated by the perturbation in a certain place continuously spread to the surrounding areas. When the effects of surface tension and gravity reach equilibrium, steady-state solutions will emerge, which include two cases: solitary waves and periodic waves. The former have heteroclinic trajectories between two stationary points, while the latter include five patterns at different parameters. In addition, there are also periodic waves that do not converge after a long period of time. During these evolution processes, strange attractors appear in the phase space. By examining the Poincaré section and the sensitivity to initial values, we demonstrate that these waves can be divided into two types: quasi-periodic and chaotic solutions. The specific type depends on parameters and initial conditions.