We perform direct numerical simulations of wall sheared Rayleigh–Bénard convection for Rayleigh numbers up to $Ra=10^{8}$, Prandtl number unity and wall shear Reynolds numbers up to $Re_{w}=10\,000$. Using the Monin–Obukhov length $L_{MO}$ we observe the presence of three different flow states, a buoyancy dominated regime ($L_{MO}\lesssim \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$; with $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ the thermal boundary layer thickness), a transitional regime ($0.5H\gtrsim L_{MO}\gtrsim \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$; with $H$ the height of the domain) and a shear dominated regime ($L_{MO}\gtrsim 0.5H$). In the buoyancy dominated regime, the flow dynamics is similar to that of turbulent thermal convection. The transitional regime is characterized by rolls that are increasingly elongated with increasing shear. The flow in the shear dominated regime consists of very large-scale meandering rolls, similar to the ones found in conventional Couette flow. As a consequence of these different flow regimes, for fixed $Ra$ and with increasing shear, the heat transfer first decreases, due to the breakup of the thermal rolls, and then increases at the beginning of the shear dominated regime. In the shear dominated regime the Nusselt number $Nu$ effectively scales as $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}\ll 1/3$, while we find $\unicode[STIX]{x1D6FC}\simeq 0.30$ in the buoyancy dominated regime. In the transitional regime, the effective scaling exponent is $\unicode[STIX]{x1D6FC}>1/3$, but the temperature and velocity profiles in this regime are not logarithmic yet, thus indicating transient dynamics and not the ultimate regime of thermal convection.

An artificial intelligence (AI) control system is developed to maximize the mixing rate of a turbulent jet. This system comprises of six independently operated unsteady minijet actuators, two hot-wire sensors placed in the jet and genetic programming for the unsupervised learning of a near-optimal control law. The ansatz of this law includes multi-frequency open-loop forcing, sensor feedback and nonlinear combinations thereof. Mixing performance is quantified by the decay rate of the centreline mean velocity of the jet. Intriguingly, the learning process of AI control discovers the classical forcings, i.e. axisymmetric, helical and flapping achievable from conventional control techniques, one by one in the order of increased performance, and finally converges to a hitherto unexplored forcing. Careful examination of the control landscape unveils typical control laws, generated in the learning process, and their evolutions. The best AI forcing produces a complex turbulent flow structure that is characterized by periodically generated mushroom structures, helical motion and an oscillating jet column, all enhancing the mixing rate and vastly outperforming others. Being never reported before, this flow structure is examined in various aspects, including the velocity spectra, mean and fluctuating velocity fields and their downstream evolution, and flow visualization images in three orthogonal planes, all compared with other classical flow structures. Along with the knowledge of the minijet-produced flow and its effect on the initial condition of the main jet, these aspects cast valuable insight into the physics behind the highly effective mixing of this newly found flow structure. The results point to the great potential of AI in conquering the vast opportunity space of control laws for many actuators and sensors and in optimizing turbulence.

Enhancing solute dispersion in electrically actuated flows has always been a challenging proposition, as attributed to the inherent uniformity of the flow field in the absence of surface patterns. Over the years, researchers have focused their attention towards circumventing this limitation, by employing several fluidic and geometric modulations. However, the corresponding improvements in solute dispersion often turn out to be inconsequential. Here we reveal that by exploiting the interplay between an externally imposed temperature gradient, subsequent electrical charge redistribution and ionic motion, coupled with the rheological complexities of the fluid, one can achieve enhancement of up to one order of magnitude of solute dispersion in a pressure-driven flow of an electrolyte solution. Our results demonstrate that the complex coupling between thermal, electrical, hydrodynamic and rheological parameters over small scales, responsible for such exclusive phenomenon, can be utilized in designing novel thermally actuated microfluidic and bio-microfluidic devices with favourable solute separation and dispersion characteristics.

We study the formation, longevity and breakdown of convective rings during impulsive spin up in square and cylindrical containers using direct numerical simulations. The rings, which are axisymmetric alternating regions of up- and downwelling flow that can last for $O(100)$ rotation times, were first demonstrated experimentally and arise due to a balance between Coriolis and viscous effects. We study the formation of these rings in the context of the Greenspan–Howard spin-up process, the disruption of which modifies ring formation and evolution. We show that, unless imprinted by boundary geometry, convective rings can only form when the surface providing buoyancy forcing is a free-slip surface, thereby explaining an apparent disagreement between experimental results in the literature. For Prandtl numbers from 1–5 we find that the longest-lived rings occur for intermediate Prandtl numbers, with a Rossby number dependence. Finally, we find that the constant evaporative heat-flux conditions imposed in the experiments are essential in sustaining the rings and in maintaining the vortices that form in consequence of the ring breakdown.

The motion of a slightly rarefied gas in a long straight two-dimensional channel caused by a discontinuous surface temperature is investigated on the basis of kinetic theory with a special interest in the fluid-dynamic description. More precisely, the channel is longitudinally divided into two parts and each part is kept at a uniform temperature different from each other, so that the surface temperature of the whole channel has a jump discontinuity at the junction. Under the assumption that the amount of jump in the surface temperature is small, the steady behaviour of the gas induced in the channel is studied on the basis of the linearized Boltzmann equation and the diffuse reflection boundary condition in the case where the Knudsen number, defined by the ratio of the molecular mean free path and the width of the channel, is small. Using a matched asymptotic expansion method combined with Sone’s asymptotics, a Stokes system describing the overall macroscopic behaviour of the gas inside the channel is derived, with a new feature of the ‘slip boundary condition’ for the flow velocity due to the jump discontinuity in the surface temperature of the channel. This condition takes the form of a diverging singularity with source and sink located at the point of discontinuity, with a multiplicative factor determined through the analysis of a spatially two-dimensional Knudsen-layer (or a Knudsen-zone) problem. Some numerical demonstrations based on the Bhatnagar–Gross–Krook equation are also presented.

With decreasing Reynolds number, $Re$, turbulence in channel flow becomes spatio-temporally intermittent and self-organises into solitary stripes oblique to the mean flow direction. We report here the existence of localised nonlinear travelling wave solutions of the Navier–Stokes equations possessing this obliqueness property. Such solutions are identified numerically using edge tracking coupled with arclength continuation. All solutions emerge in saddle-node bifurcations at values of $Re$ lower than the non-localised solutions. Relative periodic orbit solutions bifurcating from branches of travelling waves have also been computed. A complete parametric study is performed, including their stability, the investigation of their large-scale flow, and the robustness to changes of the numerical domain.

The dynamics of a hyperelastic splitter plate interacting with the laminar wake flow of a circular cylinder is investigated numerically at a Reynolds number of 80. By decreasing the plate’s stiffness, four regimes of flow-induced vibrations are identified: two regimes of periodic oscillation about a symmetric position, separated by a regime of periodic oscillation about asymmetric positions, and finally a regime of quasi-periodic oscillation occurring at very low stiffness and characterized by two fundamental (high and low) frequencies. A linear fully coupled fluid–solid analysis is then performed and reveals the destabilization of a steady symmetry-breaking mode, two high-frequency unsteady modes and one low-frequency unsteady mode, when varying the plate’s stiffness. These unstable eigenmodes explain the emergence of the nonlinear self-sustained oscillating states and provide a good prediction of the oscillation frequencies. A comparison with nonlinear calculations is provided to show the limits of the linear approach. Finally, two simplified analyses, based on the quiescent-fluid or quasi-static assumption, are proposed to further identify the linear mechanisms at play in the destabilization of the fully coupled modes. The quasi-static static analysis allows an understanding of the behaviour of the symmetry-breaking and low-frequency modes. The quiescent-fluid stability analysis provides a good prediction of the high-frequency vibrations, unlike the bending modes of the splitter plate in vacuum, as a result of the fluid added-mass correction. The emergence of the high-frequency periodic oscillations can thus be predicted based on a resonance condition between the frequencies of the hydrodynamic vortex-shedding mode and of the quiescent-fluid solid modes.

In this paper we re-examine the flow produced by the normal impact of a laminar liquid jet onto an infinite plane when the flow is dominated by surface tension. Over the range of parameters we consider, which are typical of water from a tap over a kitchen sink, it is observed experimentally that after impact the liquid spreads radially over the plane away from the point of impact in a thin film. It is also observed that, at a finite radius, there is an abrupt increase in thickness of the film which has been identified as a hydraulic jump. Once the jump is formed this radius remains constant in time and, further, is independent of the orientation of the surface showing that gravity is unimportant (Bhagat et al., J. Fluid Mech., vol. 851, 2018, R5). We show that the application of conservation of momentum in the film, subject only to viscosity and surface tension and ignoring gravity completely, predicts a singularity in the curvature of the liquid film and consequently a jump in the depth of the film at a finite radius. This location is almost identical to the radius of the jump predicted by conservation of energy and agrees with experimental observations. We also provide the correct boundary condition to be applied at an interface, where there is a change in interfacial area as a result of the fluid flow, that accounts for the energy change associated with fluid molecules’ exchange between the interface and the bulk.

The stability and transition in the bottom boundary layer under a solitary wave are analysed in the presence of finite-amplitude disturbances. First, the receptivity of the boundary layer is investigated using a linear input-output analysis, in which the environment noise is modelled as distributed body forces. The most ‘dangerous’ perturbations in a time frame until flow reversal are found to be arranged as counter-rotating streamwise-constant vortices. One of these vortex configurations is then selected and deployed to nonlinear equations, and streaks of various amplitudes are generated via the lift-up mechanism. By means of secondary stability analysis and direct numerical simulations, the dual role of streaks in the boundary-layer transition is shown. When the amplitude of streaks remains moderate, these elongated features remain stable until the adverse-pressure-gradient stage and have a dampening effect on the instabilities developing thereafter. In contrast, when the low-speed streaks reach high amplitudes exceeding 15 % of the free stream velocity at the respective phase, they become highly unstable to secondary sinuous modes in the outer shear layers. Consequently, a subcritical transition to turbulence, i.e. bypass transition, can be initiated already in the favourable-pressure-gradient region ahead of the wave crest.

We investigate the turbulence statistics in a multiphase plume made of heavy particles (particle Reynolds number at terminal velocity is 450). Using refractive-index-matched stereoscopic particle image velocimetry, we measure the locations of particles whose buoyancy drives the formation of a multiphase plume, together with the local velocity of the induced flow in the ambient salt–water. Measurements of the mean axial flow in the plume centreplane follow Gaussian profiles and that of the mean radial flow is consistent with integral plume theory. The turbulence characteristics resemble those measured in a bubble plume, including strong anisotropy in the normal Reynolds stresses. However, we observe structural differences between the two multiphase plumes. First, the skewness of the probability density function of the axial velocity fluctuations is not that which would be predicted by simply reversing the direction of a bubble plume. Second, in contrast to a bubble plume, the particle plume has a non-negligible fluid-shear production term in the turbulent kinetic energy (TKE) budget. Third, the radial decay of all measured terms in the TKE budget is slower than those in a bubble plume. Despite these dissimilarities, a bigger picture emerges that applies to both flows. The TKE production by particles (or bubbles) roughly balances the viscous dissipation, except near the plume centreline. The one-dimensional power spectra of the velocity fluctuations show a $-3$ power law that puts both the particle and bubble plume in a category different from single-phase shear-flow turbulence.

The Holmboe wave instability is one of the classic examples of a stratified shear instability, usually explained as the result of a resonance between a gravity wave and a vorticity wave. Historically, it has been studied by linear stability analyses at infinite Reynolds number, $Re$, and by direct numerical simulations at relatively low $Re$ in the regions known to be unstable from the inviscid linear stability results. In this paper, we perform linear stability analyses of the classical ‘Hazel model’ of a stratified shear layer (where the background velocity and density distributions are assumed to take the functional form of hyperbolic tangents with different characteristic vertical scales) over a range of different parameters at finite $Re$, finding new unstable regions of parameter space. In particular, we find instability when the Richardson number is everywhere greater than $1/4$, where the flow would be stable at infinite $Re$ by the Miles–Howard theorem. We find unstable modes with no critical layer, and show that, despite the necessity of viscosity for the new instability, the growth rate relative to diffusion of the background profile is maximised at large $Re$. We use these results to shed new light on the wave-resonance and over-reflection interpretations of stratified shear instability. We argue for a definition of Holmboe instability as being characterised by propagating vortices above or below the shear layer, as opposed to any reference to sharp density interfaces.

We present lattice Boltzmann pore-scale numerical simulations of solute transport and reaction in porous electrodes at a high Schmidt number, $Sc=10^{2}$. The three-dimensional geometry of real materials is reconstructed via X-ray computed tomography. We apply a volume-averaging upscaling procedure to characterise the microstructural terms contributing to the homogenised description of the macroscopic advection–reaction–dispersion equation. We firstly focus our analysis on its asymptotic solution, while varying the rate of reaction. The results confirm the presence of two working states of the electrodes: a reaction-limited regime, governed by advective transport, and a mass-transfer-limited regime, where dispersive mechanisms play a pivotal role. For all materials, these regimes depend on a single parameter, the product of the Damköhler number and a microstructural aspect ratio. The macroscopic dispersion is determined by the spatial correlation between solute concentration and flow velocity at the pore scale. This mechanism sustains reaction in the mass-transfer-limited regime due to the spatial rearrangement of the solute transport from low-velocity to high-velocity pores. We then compare the results of pre-asymptotic transport with a macroscopic model based on effective dispersion parameters. Interestingly, the model correctly represents the transport at short characteristic times. At longer times, high reaction rates mitigate the mechanisms of heterogeneous solute transport. In the mass-transfer-limited regime, the significant yet homogeneous dispersion can thus be modelled via an effective dispersion. Finally, we formulate guidelines for the design of porous electrodes based on the microstructural aspect ratio.

The turbulent boundary layer developing under a turbulence-laden free stream is numerically investigated using the temporal boundary layer framework. This study focuses on the interaction between the fully turbulent boundary layer and decaying free-stream turbulence. Previous experiments and simulations of this physical problem have considered a spatially evolving boundary layer beset by free-stream turbulence. The state of the boundary layer at any given downstream position in fact reflects the accumulated history of the co-evolution of boundary layer and free-stream turbulence. The central aim of the present work is to isolate the effect of local free-stream disturbances existing at the same time as the ‘downstream’ boundary layer. The temporal framework used here helps expose when and how disturbances directly above the boundary layer actively impart change upon it. The bulk of our simulations were completed by seeding the free stream above boundary layers that were ‘pre-grown’ to a desired thickness with homogeneous isotropic turbulence from a precursor simulation. Moreover, this strategy allowed us to test various combinations of the turbulence intensity and large-eddy length scale of the free-stream turbulence with respect to the corresponding scales of the boundary layer. The relative large-eddy turnover time scale between the free-stream turbulence and the boundary layer emerges as an important parameter in predicting if the free-stream turbulence and boundary layer interaction will be ‘strong’ or ‘weak’ before the free-stream turbulence eventually fades to a negligible level. If the large-eddy turnover time scale of the free-stream turbulence is much smaller than that of the boundary layer, the interaction will be ‘weak’, as the free-stream disturbances will markedly decay before the boundary layer is able be altered significantly as a result of the free-stream disturbances. For a ‘strong’ interaction, the injected free-stream turbulence causes increased spreading of the boundary layer away from the wall, permitting large incursions of free-stream fluid deep within it.

An exact set of equations describing deep-water irrotational surface gravity waves, originally proposed by Balk (Phys. Fluids, vol. 8 (2), 1996, pp. 416–420), and studied in the case of standing waves by Longuet-Higgins (J. Fluid Mech., vol. 423, 2000, pp. 275–291) and Longuet-Higgins (Proc. R. Soc. Lond. A, vol. 457 (2006), 2001, pp. 495–510), are analytically examined and put in a form more suitable for practical applications. The utility of this approach is its simplicity. The Lagrangian is a low-order polynomial in the Fourier coefficients, leading to equations of motion that are correspondingly of low degree. The structure of these equations is examined, and the existence of solutions is considered. To gain intuition about the system of equations, a truncated model is first examined. Linear stability analysis is performed, and the evolution of the fully nonlinear system is discussed. The theory is then applied to fully resolved permanent progressive deep-water waves and a simple method for finding the eigenvalues and eigenvectors of the normal modes of this system is presented. Potential applications of these results are then discussed.

Grain-scale discrete element simulations of bidisperse mixtures during bedload transport are used to understand, and model, bedload transport and particle-size segregation in granular media. For an initial distribution of fine particles on top of a coarse granular bed, this paper investigates the gravity driven percolation/segregation of the fine particles down into the quasi-static part of the bed. The segregation is observed to be driven by the inertial number at the bottom of the fine particle layer, and is independent of the number of fine particles. A novel travelling wave solution for the evolving concentration distribution is constructed using the continuum particle-size segregation model of Thornton, Gray & Hogg (J. Fluid Mech., vol. 550, 2006, pp. 1–25) and Gray & Chugunov (J. Fluid Mech., vol. 569, 2006, pp. 365–398). The observed behaviour is shown to be related to a local equilibrium between the influence of the concentration and of the inertial number. The existence of the exact solution relies on the segregation flux and the diffusion coefficient having the same dependency on the inertial number. This functional dependence allows the continuum model to quantitatively reproduce the discrete simulations. These results significantly improve on our understanding of the size segregation dynamics and represent a step forward in the up-scaling process to polydisperse granular flows in the context of turbulent bedload transport.

A delayed detached eddy simulation of an overexpanded nozzle flow with shock-induced separation is carried out at a Reynolds number of $1.7\times 10^{7}$, based on nozzle throat diameter and stagnation chamber properties. In this flow, self-sustained shock oscillations induce local unsteady loads on the nozzle wall as well as global off-axis forces. Despite several studies in the last few decades, a clear physical understanding of the factors driving this unsteadiness is still lacking. The geometry under investigation is a subscale truncated ideal contour nozzle, which was experimentally tested at the University of Texas at Austin at a nozzle pressure ratio of 30. Under these conditions, the nozzle operates in a highly overexpanded state and comprises a conical separation shock that merges to form a Mach disk at the nozzle centre. The delayed detached eddy simulation model agrees well with the experimental results in terms of mean and fluctuating wall-pressure statistics. Wall-pressure spectra reveal a large bump at low frequencies associated with an axisymmetric (piston-like) motion of the shock system, followed by a broad and high-amplitude peak at higher frequencies associated with the Mach waves produced by turbulent eddies convecting through the detached shear layer. Moreover, a distinct peak at an intermediate frequency (${\sim}1~\text{kHz}$) persists in the wall-pressure spectra downstream of the separation shock. A Fourier-based analysis performed in both time and space (azimuthal wavenumber) reveals that this intermediate-frequency peak is associated with the $m=1$ (non-symmetric) pressure mode and is thus related to the generation of aerodynamic side loads. It is then shown how the unsteady Mach disk motion is characterized by an intense vortex shedding activity that, together with the vortical structures of the annular shear layer, contributes to the sustainment of an aeroacoustic feedback loop occurring within the nozzle.

Experiments on the receptivity of two-dimensional boundary layers to acoustic disturbances from two-dimensional roughness strips were performed in a low-turbulence wind tunnel on a flat plate model. The free stream was subjected to a plane acoustic wave so that a Stokes layer (SL) was created on the plate, thus generating a Tollmien–Schlichting (T–S) wave through the receptivity process. An improved technique to measure the T–S component is described based on a retracting two-dimensional roughness, which allowed for phase-locked measurements at the acoustic wave frequency to be made. This improved technique enables both protuberances and cavities to be explored in the range $30~\unicode[STIX]{x03BC}\text{m}<|h|<750~\unicode[STIX]{x03BC}\text{m}$ (equivalent to $0.025<|h|/\unicode[STIX]{x1D6FF}_{B}^{\ast }<0.630$ in relative roughness height to the local unperturbed Blasius boundary layer displacement thickness). These depths are designed to cover both the predicted linear and nonlinear response of the T–S excitation. Experimentally, cavities had not previously been explored. Results show that a linear regime is identifiable for both positive and negative roughness heights up to ${\approx}150~\unicode[STIX]{x03BC}\text{m}$ ($|h|/\unicode[STIX]{x1D6FF}_{B}^{\ast }\approx 0.126$). The departure from the linear behaviour is, however, dependent on the geometry of the surface imperfection. For cavities of significant depth, the nonlinear behaviour is found to be milder than in the case of protuberances – this is attributed to the flow physics in the near field of the surface features. Nonetheless, results for positive heights agree well with previous theoretical work which predicted a linear disturbance response for small-height perturbations.

Turbulent mixing of passive scalars is studied in the canonical shock–turbulence interaction configuration via shock-capturing direct numerical simulations, varying the shock Mach number ($M=1.28{-}5$), turbulence Mach number ($M_{t}=0.1{-}0.4$), Taylor microscale Reynolds number ($Re_{\unicode[STIX]{x1D706}}\approx 40,70$) and Schmidt number ($Sc=0.5$, 1, 2). The shock-normal evolution of scalar variance and dissipation transport equations, spectra and probability density functions (PDFs) are examined. Scalar dissipation, its production and destruction increase across the shock with higher $M$, lower $M_{t}$ and lower $Re_{\unicode[STIX]{x1D706}}$. Mixing enhancement for different flow topologies across the shock is studied from changes in the PDFs of velocity gradient tensor invariants and conditional distributions of scalar dissipation. The proportion of the stable-focus-stretching flow topology is the highest among all the topologies in the flow both before and after the shock. Unstable-node/saddle/saddle topology is the most dissipative throughout the flow domain, despite variations across the shock. Preshock and postshock distributions of the alignment between the strain-rate tensor eigenvectors and the scalar gradient, vorticity and the mean streamwise vector conditioned on flow topology are studied. A novel barycentric map representation is introduced for a more direct visualization of the alignments and conditioned scalar dissipation distributions. Interaction with the shock increases alignment of the scalar gradient with the most extensive eigenvector, decreasing it with the most compressive, which is still dominant. The barycentric map of the passive scalar gradient also reveals that, across the shock, the most probable alignment between scalar gradient and strain eigendirections converges towards the alignment that provides the most dissipation. This also leads to an enhancement of scalar dissipation immediately downstream of the shock.

Double emulsion formation in a hierarchical flow-focusing channel is systematically investigated using a free-energy ternary lattice Boltzmann model. A three-dimensional formation regime diagram is constructed based on the capillary numbers of the inner ($Ca_{i}$), middle ($Ca_{m}$) and outer ($Ca_{o}$) phase fluids. The results show that the formation diagram can be classified into periodic two-step region, periodic one-step region, and non-periodic region. By varying $Ca_{i}$ and $Ca_{m}$ in the two-step formation region, different morphologies are obtained, including the regular double emulsions, decussate regimes with one or two alternate empty droplets, and structures with multiple inner droplets contained in the continuous middle phase thread. Bidisperse behaviours are also frequently encountered in the two-step formation region. In the periodic one-step formation region, scaling laws are proposed for the double emulsion size and for the size ratio between the inner droplet and the overall double emulsion. Furthermore, we show that the interfacial tension ratio can greatly change the morphologies of the obtained emulsion droplets, and the channel geometry plays an important role in changing the formation regimes and the double emulsion sizes. In particular, narrowing the side inlets or the distance between the two side inlets promotes the conversion from the two-step formation regime to the one-step formation regime.

The air–sea momentum exchanges in the presence of surface waves play an integral role in coupling the atmosphere and the ocean. In the current study, we present a detailed laboratory investigation of the momentum fluxes over wind-generated waves. Experiments were performed in the large wind-wave facility at the Air–Sea Interaction Laboratory of the University of Delaware. Airflow velocity measurements were acquired above wind waves using a combination of particle image velocimetry and laser-induced fluorescence techniques. The momentum budget is examined using a wave-following orthogonal curvilinear coordinate system. In the wave boundary layer, the phase-averaged turbulent stress is intense (weak) and positive downwind (upwind) of the crests. The wave-induced stress is also positive on the windward and leeward sides of wave crests but with asymmetric intensities. These regions of positive wave stress are intertwined with regions of negative wave stress just above wave crests and downwind of wave troughs. Likewise, at the interface, the viscous stress exhibits along-wave phase-locked variations with maxima upwind of the wave crests. As a general trend, the mean profiles of the wave-induced stress decrease to a negative minimum from a near-zero value far from the surface and then increase rapidly to a positive value near the interface where the turbulent stress is reduced. Far away from the surface, however, the turbulent stress is nearly equal to the total stress. Very close to the surface, in the viscous sublayer, the wave and turbulent stresses vanish, and therefore the stress is supported by the viscosity.