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.

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.

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.

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 this study, we present a fractal dimension analysis of high Schmidt number passive scalar mixing in experiments of turbulent pipe flow. By using the high-resolution planar laser-induced fluorescence technique, the scalar concentration fields are measured at Reynolds numbers $10\,000$, $15\,000$ and $20\,000$. In the inertial–convective range, the iso-scalar surface exhibits self-similar fractal characteristics, giving fractal dimension $1.67 \pm 0.05$ from the two-dimensional measurements over a range of length scales. This fractal dimension is approximately independent of the criteria of extracting the iso-scalar surfaces, the corresponding thresholds and the Reynolds numbers examined in this study. The crossover length scale, beyond which the $1.67 \pm 0.05$ fractal dimension is exhibited, is about ten times the Kolmogorov length scale, in agreement with previous studies. As the length scales decrease to be smaller than this crossover length scale, the fractal dimension, calculated from the one-dimensional signals, increases and approaches a saturation at approximately 2 (with the additive law) in the viscous–convective range, manifesting the space-filling characteristics, as theoretically predicted by Grossmann & Lohse (1994, Europhys. Lett., vol. 27, 347). This observation presents first-time experimental evidence for the fractal characteristics predicted by Grossmann and Lohse for the high Schmidt number passive scalar mixing.

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.

We analyse the pressure-driven radial flow of a shear-thinning fluid between two parallel plates. Complex fluid rheology may significantly affect the hydrodynamic features of such non-Newtonian flows, which remain not fully understood, compared with Newtonian flows. We describe the shear-thinning rheology using the Ellis model and present a theoretical framework for calculating the pressure distribution and the flow rate–pressure drop relation. We first derive a closed-form expression for the pressure gradient, which allows us to obtain semi-analytical expressions for the pressure, velocity and flow rate–pressure drop relation. Specifically, we provide the corresponding asymptotic solutions for small and large values of the dimensionless flow rates. We further elucidate the entrance length required for the radial velocity of a shear-thinning fluid to reach its fully developed form, showing that this length approximates the Newtonian low-Reynolds-number value at low shear rates, but may strongly depend on the fluid’s shear-thinning rheology and exceed the Newtonian value at high shear rates. We validate our theoretical results with finite-element numerical simulations and find excellent agreement. Furthermore, we compare our semi-analytical, asymptotic and finite-element simulation results for the pressure distribution with the experimental measurements of Laurencena & Williams (Trans. Soc. Rheol. vol. 18, 1974, pp. 331–355), showing good agreement. Our theoretical results using the Ellis model capture the interplay between the shear-thinning and the zero-shear-rate effects on the pressure drop, which cannot be explained using a simple power-law model, highlighting the importance of using an adequate constitutive model to accurately describe non-Newtonian flows of shear-thinning fluids.

Particle suspensions in confined geometries can become clogged, which can have a catastrophic effect on function in biological and industrial systems. Here, we investigate the macroscopic dynamics of dense suspensions in constricted geometries. We develop a minimal continuum two-phase model that allows for variation in particle volume fraction. The model comprises a ‘wet solid’ phase with material properties dependent on the particle volume fraction, and a seepage Darcy flow of fluid through the particles. We find that spatially varying geometry (or material properties) can induce emergent heterogeneity in the particle fraction and trigger the abrupt transition to a high-particle-fraction ‘clogged’ state.

Numerical studies on the statistical properties of irregular waves in finite depth have to date been based on models founded on weak nonlinearity; as a consequence, only lower-order (usually third-order) nonlinear interactions have thus far been investigated. The present study performs numerical simulations with a fully nonlinear, spectrally accurate model to investigate the statistics of irregular, unidirectional wave fields in finite water depth initially given by a Texel, Marsen and Arsloe spectrum. A series of random unidirectional wave fields are considered, covering a wide range of water depth. The wave spectrum and statistical properties, including the probability density function of the surface elevation, exceedance probability of wave crests and occurrence probability of extreme (rogue) waves, are investigated. The importance of full nonlinearity in comparison with third-order results is likewise evaluated. The results show that full nonlinearity increases kurtosis and enhances the occurrence probability of large wave crests and rogue waves substantially, in both deep water and finite water depth. Therefore, we propose that full nonlinearity may contribute significantly to the formation of rogue waves. Furthermore, to account for the effects of higher-order nonlinearity on modulational instability, we analyse the relationship between the Benjamin–Feir index (BFI) and maximal excess kurtosis. Our results show a strong linear relationship i.e. $({\mathcal{K}}_{max}-3)\propto {\textrm{BFI}}$, in contrast to $({\mathcal{K}}_{max}-3)\propto {\textrm{BFI}}^2$ based on the assumptions of weak nonlinearity, a narrow-banded spectrum and deep-water conditions. Above, $\mathcal{K}_{max}$ is the maximal kurtosis.

Waves transport particles in the direction of wave propagation with the Stokes drift. When the Earth’s rotation is accounted for, waves induce an additional (Eulerian-mean) current that reduces drift and is known as the anti-Stokes drift. This effect is often ignored in oceanic particle-tracking simulations, despite being important. Although different theoretical models exist, they have not been validated by experiments. We conduct laboratory experiments studying the surface drift induced by deep-water waves in a purpose-built rotating wave flume. With rotation, the Lagrangian-mean drift deflects to the right (counterclockwise rotation) and reduces in magnitude. Compared with two existing steady theoretical models, measured drift speed follows a similar trend with wave Ekman number but is larger. The difference is largely explained by unsteadiness on inertial time scales. Our results emphasise the importance of considering unsteadiness when predicting and analysing the transport of floating material by waves.

We theoretically investigate the small-amplitude broadside oscillations of an annular disk within an unbounded fluid domain. Specifically, we formulate a semi-analytical framework to examine the effects of the oscillation frequency and pore radius on the disk’s added mass and damping coefficients. By leveraging the superposition principle, we decompose the complex original problem into two simpler ones. The force exerted on the disk by the fluid is linked to the solutions of these sub-problems through the reciprocal theorem; the first solution is readily available, while the second is derived asymptotically, assuming a small inner radius. Both solutions are evaluated by transforming dual integral equations into systems of algebraic equations, which are then solved numerically. Building on these solutions, we extract asymptotic expressions for the variations of the quantities of interest in the limits of low and high oscillatory Reynolds numbers. Notably, at high frequencies, we uncover a previously overlooked logarithmic term in the force coefficient expansions, absent in prior scaling analyses of oscillating solid (impermeable) disks. Our findings indicate that, when viscosity plays a dominant role, an annular (porous) disk behaves similarly to a solid one, with reductions in the force coefficients scaling with the cube of the pore radius. We also discover, perhaps surprisingly, that, as inertial effects intensify, the damping coefficient initially increases with the pore radius, reaches a maximum and subsequently declines as the disk’s inner hole enlarges further; at its peak, it can exceed the value for an equivalent solid disk by up to approximately 62 % in the asymptotic limit of extremely high oscillatory Reynolds number. Conversely, the added mass coefficient decreases monotonically with increasing porosity. The decay rate of the added mass in the inertial regime initially scales with the cube of the pore radius before transitioning to linear scaling when the pore radius is no longer extremely small. Although our analysis assumes a small pore radius, direct numerical simulations confirm that our asymptotic formulation remains accurate for inner-to-outer radius ratios up to at least $1/2$.

Cilia exist ubiquitously in nature, and they are very effective in generating flow in a low Reynolds number environment. Inspired by nature, various artificial cilia have been invented for microfluidic applications, and a nature-mimicking tilted conical motion was often used for flow generation due to its simplicity and effectiveness. However, the current theoretical model for predicting the net flow rate generated by the tilted conical motion fails when the cilia are in close confinement, i.e. when the tips of the cilia are close to the ceiling of their channel or chamber, which is, in reality, the most practical way to enhance flow rate generation. Moreover, numerical simulations are very expensive for optimisation of such designs. In this study, we derive a new theoretical model, taking into account the tilting and opening angles of the cone, the height of the chamber and the length of the cilia. The results differ significantly from when the ceiling is not considered, and counter-intuitively in some cases the flow can even reverse. These unexpected results have important implications for artificial cilium design and applications. We validate the model with both numerical simulations and experiments using magnetic artificial cilia, and show that the flow optimisation based on tilted conical cilium motion can now be performed accurately in a realistic and practical manner. This study not only offers a simple tool for optimising designs of artificial cilium-based systems for microfluidic applications, but it also provides fresh insights for understanding natural cilium-driven flows.

The dynamics of self-excited shock train oscillations in a back pressured axisymmetric duct was investigated to deepen the understanding of the isolator/combustor coupling in high-speed propulsion systems. The test article consisted of an internal compression inlet followed by a constant area isolator, both having a circular cross-section. A systematic back pressure variation was implemented by using a combination of aerodynamic and physical blockages at the isolator exit. High bandwidth two-dimensional pressure field imaging was performed at $8\,{\rm kHz}$ repetition rate within the isolator for different back pressure settings. The acquisition rate was considerably higher than the dominant frequency of the shock train oscillations across the different back pressure settings. The power spectral density of the pressure fluctuations beneath the leading shock foot exhibited broadband low frequency oscillations across all back pressures that resembled the motions of canonical shock–boundary layer interaction units. A node in the vicinity of reattachment location that originated the pressure perturbations within the separation shock was also identified, which further ascertained that the leading shock low frequency motions were driven by the separation bubble pulsations. Above a threshold back pressure, additional peaks appeared at distinct higher frequencies that resembled the acoustic modes within the duct. However, none of the earlier expressions of the resonance acoustic frequency within a straight duct agreed with the experimentally observed value. Cross-spectral analyses suggested that these modes were caused by the shock interactions with upstream propagating acoustic waves that emanate from the reattachment location, originally proposed for transonic diffusers by Robinet & Casalis (2001) Phys. Fluids 13, 1047–1059. Feedback interactions described using one-dimensional stability analysis of the shock perturbations by obliquely travelling acoustic waves (Robinet & Casalis 2001 Phys. Fluids 13, 1047–1059) made favourable comparisons on the back pressure threshold that emanated the acoustic modes as well as the acoustic mode frequencies.

A deep reinforcement learning method for training a jellyfish-like swimmer to effectively track a moving target in a two-dimensional flow was developed. This swimmer is a flexible object equipped with a muscle model based on torsional springs. We employed a deep Q-network (DQN) that takes the swimmer’s geometry and dynamic parameters as inputs, and outputs actions that are the forces applied to the swimmer. In particular, an action regulation was introduced to mitigate the interference from complex fluid–structure interactions. The goal of these actions is to navigate the swimmer to a target point in the shortest possible time. In the DQN training, the data on the swimmer’s motions were obtained from simulations using the immersed boundary method. During tracking a moving target, there is an inherent delay between the application of forces and the corresponding response of the swimmer’s body due to hydrodynamic interactions between the shedding vortices and the swimmer’s own locomotion. Our tests demonstrate that the swimmer, with the DQN agent and action regulation, is able to dynamically adjust its course based on its instantaneous state. This work extends the application scope of machine learning in controlling flexible objects within fluid environments.

In this study, the statistical properties and formation mechanisms of particle clusters that consider the influence of particle–wall interactions in particle-laden wall turbulence are systematically investigated through wind tunnel experiments. In the experiments, two particle release modes, including particle top-releasing mode (Case 1) and particle locally laying mode (Case 2), were adopted to establish varying conditions with different particle–wall interaction strengths. The Voronoï diagram method was employed to identify the particle clusters, and the impact of particle–wall interactions on the characteristics of the clusters was analysed. The results indicate that particle–wall interaction is the predominant factor in the formation of particle clusters in the near-wall region. Under Case 1 and Case 2, the maximum concentration of particles in the clusters could reach nearly five times the average particle concentration; however, the clusters with large particle numbers ($N_C\gt 5$) in Case 1 tended to form near the wall and the vertical velocities of these clusters were greater than the average velocities of all particles. In contrast, under Case 2, clusters with large particle numbers exhibited a higher probability of occurrence further from the wall and the vertical velocities of these clusters were lower than the average velocity of all particles. Furthermore, this study found that the presence of particle clusters in these flows significantly alters the flow field properties surrounding them, implying that a region of high strain and low vorticity constitutes an essential but non-sufficient condition for the generation of particle clusters in wall turbulence.

Drops in a shear flow experience shear-induced diffusion due to drop–drop interactions. Here, the effects of medium viscoelasticity on shear-induced collective diffusivity are numerically investigated. A layer of viscous drops suspended in a viscoelastic fluid was simulated, fully resolving each deforming drop using a front-tracking method. The collective diffusivity is computed from the spreading of the drop layer with time, specifically a one-third scaling, as well as using an exponentially decaying dynamic structure factor of the system of drops. Both methods led to matching results. The surrounding viscoelasticity was shown to linearly reduce the diffusion-led spreading of the drop layer, the effect being stronger for less deformable drops (low capillary number). Because of the competition between the increasing effect with capillary number (Ca) and the decreasing effect with Weissenberg number (Wi), collective diffusivity vanishes at very low Ca and high enough Wi. The physics behind the hindering effects of viscoelasticity on shear-induced diffusion is explained with the help of drop–drop interactions in a viscoelastic fluid, where shear-induced interaction leads to trapping of drops into tumbling trajectories at lower Ca and higher Wi due to viscoelastic stresses. Using the simulated values, phenomenological correlations relating the shear-induced gradient diffusivity with Wi and Ca were found.

In recent years, integrating physical constraints within deep neural networks has emerged as an effective approach for expediting direct numerical simulations in two-phase flow. This paper introduces physics-informed neural networks (PINNs) that utilise the phase-field method to model three-dimensional two-phase flows. We present a fully connected neural network architecture with residual blocks and spatial parallel training using the overlapping domain decomposition method across multiple graphics processing units to enhance the accuracy and computational efficiency of PINNs for the phase-field method (PF-PINNs). The proposed PINNs framework is applied to a bubble rising scenario in a three-dimensional infinite water tank to quantitatively assess the performance of PF-PINNs. Furthermore, the computational cost and parallel efficiency of the proposed method was evaluated, demonstrating its potential for widespread application in complex training environments.

The shape of a free-surface slump of viscoplastic material supported by an oblique barrier on an inclined plane is investigated theoretically and experimentally. The barrier is sufficiently tall that it is not surmounted by the viscoplastic fluid, and a focus of this study is the largest volume of rigid viscoplastic fluid that can be supported upstream of it. A lubrication model is integrated numerically to determine the transient flow as the maximal rigid shape is approached. Away from the region supported by the barrier, the viscoplastic layer attains a uniform thickness in which the gravitational stresses are in balance with the yield stress of the material. However, closer to the barrier, the layer thickens and the barrier bears the additional gravitational loading. An exact solution for the rigid shape of the viscoplastic material is constructed from the steady force balance and computed by integrating Charpit’s equations along characteristics that emanate from the barrier wall. The characteristics represent the late-time streamlines of the flow as it approaches the rigid shape. The exact solution depends on a single dimensionless group, which incorporates the slope inclination, the barrier width and the fluid’s yield stress. It is shown that the shape is insensitive to the transient flow from which it originates. The force exerted by the slump is calculated for different barrier shapes. The results of new laboratory experiments are reported; these show that although convergence to the final rigid state is slow, there is good agreement with the experimental measurements at long times.