We explore the motion of an axisymmetric gravity current in an anisotropic porous medium in which the horizontal permeability is larger than the vertical permeability. It is well known that the classical axisymmetric gravity current supplied by a constant point source of fluid has an unphysical singularity near the origin. We address this by considering a pressure-dominated region near the origin which allows for vertical flow from the source, such that the current remains of finite depth, whilst beyond this region the flow is gravity dominated. At early times the inner pressure-driven region controls the spreading of the current, but at late times the inner region occupies a progressively smaller fraction of the current such that the radius increases as ${\sim }t^{3/7}$, while the depth near the origin increases approximately as ${\sim }t^{1/7}$. The presence of anisotropy highlights this phenomenon, since the vertical permeability maintains an effect on the flow at late times through the pressure-driven flow near the origin. Using these results we provide some quantitative insights into the dominant dynamics which controls CO$_2$ migration through permeable aquifers, as occurs in the context of carbon capture and storage.

The wake of a long rectangular wall-mounted prism is investigated at Reynolds numbers of $Re=250\unicode{x2013}1200$ by directly solving the Navier–Stokes equations. The aim of this study was to examine the unsteady transition mechanism in the wake of a large-depth-ratio (streamwise length to width) prism as well as to characterize the unsteady wake evolution at low Reynolds numbers. The results highlighted that increasing Reynolds number significantly altered the dominance of upwash and downwash flows in the time-averaged flow and changed the characteristics of coherent structures, including their size, dominant frequency and interaction with other structures in the flow. The wake is, therefore, categorized into three regimes within the transition process: steady regime at $Re \leq 625$, regular unsteady regime at $625 < Re < 750$ and irregular unsteady regime at $Re \geq 750$. Particularly, the wake started to exhibit unsteady features at $Re=625\unicode{x2013}650$, which transitioned to an early irregular unsteady wake topology at $Re=750$. At $Re \geq 675$, horseshoe vortices transformed to vortex loops. There were hairpin-like structures formed on the upper and side faces of the long prism. The results further indicated that the transition to unsteadiness is attributed to separated leading-edge shear-layer instabilities and interactions of the shear layer with the horseshoe structures. The wake was more complex due to the interactions of multiple coherent structures in the flow, which resulted in a multiple-periodicity wake system. A skeleton model is proposed for a large-depth-ratio prism, to incorporate details of the unsteady flow features and specify various flow coherent structures at low Reynolds numbers.

Turbulent motions enhance the diffusion of large-scale flows and temperature gradients. Such diffusion is often parameterized by coefficients of turbulent viscosity ($\nu _{t}$) and turbulent thermal diffusivity ($\chi _{t}$) that are analogous to their microscopic counterparts. We compute the turbulent diffusion coefficients by imposing sinusoidal large-scale velocity and temperature gradients on a turbulent flow and measuring the response of the system. We also confirm our results using experiments where the imposed gradients are allowed to decay. To achieve this, we use weakly compressible three-dimensional hydrodynamic simulations of isotropically forced homogeneous turbulence. We find that the turbulent viscosity and thermal diffusion, as well as their ratio the turbulent Prandtl number, $\textit {Pr}_{t} = \nu _{t}/\chi _{t}$, approach asymptotic values at sufficiently high Reynolds and Péclet numbers. We also do not find a significant dependence of $\textit {Pr}_{t}$ on the microscopic Prandtl number $\textit {Pr} = \nu /\chi$. These findings are in stark contrast to results from the $k{-}\epsilon$ model, which suggests that $\textit {Pr}_{t}$ increases monotonically with decreasing $\textit {Pr}$. The current results are relevant for the ongoing debate on, for example, the nature of the turbulent flows in the very-low-$\textit {Pr}$ regimes of stellar convection zones.

The hydrodynamics of bubble columns in the heterogeneous regime is revisited. Focusing on air–water systems at large aspect ratio, we show from dimensional analysis that buoyancy equilibrates inertia, and that velocities scale as $(gD\varepsilon )^{1/2}$, where $D$ is the bubble column diameter, $\varepsilon$ the void fraction and $g$ the gravitational acceleration. From new experiments in a $0.4$ m diameter column with ${{O}}(10^3)$ particle Reynolds number bubbles and from a detailed analysis of published data, we confirm the self-organization prevailing in the heterogeneous regime, and that the liquid flow rate is only set by the column diameter $D$. Besides, direct liquid and gas velocity measurements demonstrate that the relative velocity increases above the terminal velocity $U_T$ in the heterogeneous regime, and that it tends to ${\sim }2.4 U_T$ at very large gas superficial velocities $V_{sg}$. The proposed velocity scaling is shown to hold for liquid and gas mean velocities and for their standard deviations. Furthermore, it is found to be valid over a wide range of conditions, corresponding to Froude numbers $Fr=V_{sg}/(gD)^{1/2}$ from 0.02 to 0.5. Then, the relevance of this scaling for coalescing media is discussed. Moreover, following the successful prediction of the void fraction with a Zuber & Findlay approach at the beginning of the heterogeneous regime, we show how the void fraction is correlated with $Fr$. Further investigations are finally suggested to connect the increase in relative velocity with meso-scale structures known to exist in the heterogeneous regime.

Slender fibres are ubiquitous in biology, physics and engineering, with prominent examples including bacterial flagella and cytoskeletal fibres. In this setting, slender body theories (SBTs), which give the resistance on the fibre asymptotically in its slenderness $\epsilon$, are useful tools for both analysis and computations. However, a difficulty arises when accounting for twist and cross-sectional rotation: because the angular velocity of a filament can vary depending on the order of magnitude of the applied torque, asymptotic theories must give accurate results for rotational dynamics over a range of angular velocities. In this paper, we first survey the challenges in applying existing SBTs, which are based on either singularity or full boundary integral representations, to rotating filaments, showing in particular that they fail to consistently treat rotation–translation coupling in curved filaments. We then provide an alternative approach which approximates the three-dimensional dynamics via a one-dimensional line integral of Rotne–Prager–Yamakawa regularized singularities. While unable to accurately resolve the flow field near the filament, this approach gives a grand mobility with symmetric rotation–translation and translation–rotation coupling, making it applicable to a broad range of angular velocities. To restore fidelity to the three-dimensional filament geometry, we use our regularized singularity model to inform a simple empirical equation which relates the mean force and torque along the filament centreline to the translational and rotational velocity of the cross-section. The single unknown coefficient in the model is estimated numerically from three-dimensional boundary integral calculations on a rotating, curved filament.

Convection within planetary bodies is often modelled using a temperature-dependent rheology which, when cooled from the surface, naturally leads to the formation of a so-called stagnant lid at the cold outer surface. However, for sufficiently large planets the phase diagram describing the partially molten system may depend significantly on pressure in addition to temperature, leading to significant variations in solid fraction. The aggregate rheology may therefore exhibit significant dependence on both the temperature and pressure, and hence may exhibit marked dependence on depth in addition to the dependence on the thermal structure due to convection. Here, we consider the growth and stability of a planetary stagnant lid. We first characterise the effect of a pressure- and temperature-dependent rheology on the evolution of a symmetric, planetary stagnant lid. This analysis further suggests that the pressure dependence of the rheology may lead to an instability of the growing stagnant lid which, importantly, may lead to asymmetric lid growth. We find that the most unstable mode is at the longest wavelengths, and discuss the implications for stagnant-lid convection and the growth of asymmetric surfaces of planetary bodies. In particular, we discuss the possibility that this instability has implications for the formation of the crustal dichotomy found on the Moon.

The interaction between a turbulent boundary layer flow and compliant surfaces is investigated experimentally. Three viscoelastic coatings with different material stiffnesses are used to identify the general surface response to the turbulent flow conditions. For the softest coating, the global force measurements show two obvious regimes of interaction with an indicated transition at $U_b/C_t\sim 3.5$, where $U_b$ is the bulk flow velocity and $C_t$ is the coating shear velocity. The one-way coupled regime shows friction values comparable to those of the rigid wall, while the two-way coupled regime indicate a significant increase in fluid friction. Within the one-way coupled regime for $U_b/C_t>1.2$, the flow measurements show a low level of two-way coupling represented by the change of the velocity profile as well as the increase in the Reynolds stresses in the near-wall region. This is supported by the surface deformation measurements. Initially, the turbulent flow structures induce only an imprint on the coating surface, while a change in surface response occurs when the surface wave propagation velocity $c_w$ equals the shear wave velocity of the coating $C_t$ (i.e. $c_w/C_t\sim 1$). Above $U_b/C_t>1.2$, a growth in wavelength is observed with increasing flow velocity, most probably due to the surface wave formation generated downstream the pressure features of the flow. The surface response is stable and correlates with the high-intensity turbulent pressure fluctuations in the turbulent boundary layer, with a wave propagation velocity $c_w\sim 0.7\unicode{x2013}0.8$ $U_b$. Within the two-way coupled regime, additional fluid motions and a downward shift in the logarithmic region of the velocity profile are observed due to substantial surface deformation and confirm the frictional drag increase. Another type of surface response is initiated by phase-lag instability in combination with surface undulations that start to protrude the viscous sublayer, where the propagation velocity of surface wave trains is $c_w\sim 0.17\unicode{x2013}0.18$ $U_b$.

An exact pairwise hydrodynamic theory is developed for the flow-induced spatial distribution of particles in dilute polydisperse suspensions undergoing two-dimensional unidirectional flows, including shear and planar Poiseuille flows. Coupled diffusive fluxes and a drift velocity are extracted from a Boltzmann-like master equation. A boundary layer is predicted in regions where the shear rate vanishes with thickness set by the radii of the upstream collision cross-sections for pair interactions. An analysis of this region yields linearly vanishing drift velocities and non-vanishing diffusivities where the shear rate vanishes, thus circumventing the source of the singular particle distribution predicted by the usual models. Outside of the boundary layer, a power-law particle distribution is predicted with exponent equal to minus half the exponent of the local shear rate. Trajectories for particles with symmetry-breaking contact interactions (e.g. rough particles, permeable particles, emulsion drops) are analytically integrated to yield particle displacements given by quadratures of hard-sphere (or spherical drop) mobility functions. Using this analysis, stationary particle distributions are obtained for suspensions in Poiseuille flow. The scale for the particle distribution in monodisperse suspensions is set by the collision cross-section of the particles but its shape is almost universal. Results for polydisperse suspensions show size segregation in the central boundary layer with enrichment of smaller particles. Particle densities at the centreline scale approximately with the inverse square root of particle size. A superposition approximation reliably predicts the exact results over a broad range of parameters. The predictions agree with experiments in suspensions up to approximately 20 % volume fraction without fitting parameters.

A principal topic of interest and importance in stably stratified flows is how turbulence irreversibly mixes the ambient density field. Because the density field is coupled dynamically to the velocity field through the stable stratification, this mixing affects the overall flow dynamics, and its accurate parameterisation has become a ‘grand challenge’ in environmental fluid mechanics (Dauxois et al., Phys. Rev. Fluids, vol. 6, issue 2, 2021, 020501). In order to better understand the detailed kinematics of mixing in a stably stratified fluid, Jiang et al. (2022) perform experiments using a unique laboratory facility, capable of generating controllable stratified shear flows, and providing almost instantaneous density and three-component velocity measurements on a high-resolution, three-dimensional grid. Using three-dimensional data sets from the experiments, they employ the rortex–shear decomposition to identify the morphology of instantaneous rortices in the flow fields, leading to the interpretation of the motion of the rortices and ultimately to how the rortices cause irreversible mixing of the density field. This marks one of the first studies where, in a laboratory setting, full use has been made of somewhat high-resolution, three-dimensional near-instantaneous measurements; it demonstrates what can be accomplished in the laboratory, setting a new standard for future experiments.

We consider nonlinear dynamical systems driven by stochastic forcing. It has been largely evidenced in the literature that the linear response of non-normal systems (e.g. fluid flows) may exhibit a large variance amplification, even in a linearly stable regime. This linear response, however, is relevant only in the limit of vanishing forcing intensity. As the intensity increases, the frequency distribution and the variance of the response may be strongly modified due to nonlinear effects. To go beyond this limitation, we propose a theoretical approach to derive an amplitude equation governing the weakly nonlinear evolution of these systems. This approach does not rely on the presence of an eigenvalue close to the neutral axis, applying instead to any sufficiently non-normal operator, and the Fourier components of the response are allowed to be arbitrarily different from any eigenmode. The methodology is outlined for a generic nonlinear dynamical system, and the application case highlights a common non-normal mechanism in hydrodynamics: convective non-normal amplification in the flow past a backward-facing step.

To date, very few careful and direct comparisons between experiments and direct numerical simulations (DNS) have been published on free-stream turbulence (FST) induced boundary layer transition, whilst there exist numerous published works on the comparison of canonical turbulent boundary layers. The primary reason is that the former comparison is vastly more difficult to carry out simply because all known transition scenarios have large energy gradients and are extremely sensitive to surrounding conditions. This paper presents a detailed comparison between new experiments and available DNS data of the complex FST transition scenario in a flat plate boundary layer at turbulence intensity level about $Tu = 3\,\%$ and FST Reynolds number about $Re_{{fst}} = 67$. The leading edge (LE) pressure gradient distribution and the full energy spectrum at the LE are identified as the two most important parameters for a satisfying comparison. Matching the LE characteristic FST parameters is not enough as previously thought, which is illustrated by setting up two experimental FST cases with about the same FST integral parameters at the LE but with different energy spectra. Finally, an FST boundary layer penetration depth (PD) measure is defined using DNS, which suggests that the PD grows with the downstream distance and stays around 20 % of the boundary layer thickness down to transition onset. With this result, one cannot rule out the significance of the continuous FST forcing along the boundary layer edge in this transition scenario, as indicated in previous studies.

Understanding the behaviour of scalar isosurfaces in a turbulent flow is of particular interest for many problems in turbulent mixing that contain sharp interfaces between regions of the flow. Common examples include combustion, where the chemical reactions occur in thin regions within the flow, and the turbulent/non-turbulent interface in shear flows, where a thin region separates the rotational, turbulent motions from the irrotational, non-turbulent background. Recent advances in computing technology allow for in-depth analysis of these interface problems that are difficult to quantify in a laboratory setting. In this paper, the results of a direct numerical simulation of a passive scalar $\varPhi$ evolving on a turbulent, temporally developing mixing layer are described. A novel approach has been taken to calculate the surface area of individual scalar isosurfaces, $A_{iso}$, throughout the simulation, as well as the mean isosurface area density, $\varSigma$, as a function of the cross-stream direction and time. A notable finding is that the profiles of $\varSigma$ develop in a self-similar manner when scaled by the Taylor scale of the scalar field, $\lambda _\phi$. Remarkably, the scaling appears to hold for a wide range of isovalues. A rough scaling argument based on the formal definition of $\varSigma$ and properties of a temporal mixing layer is presented which also exposes a dependence on $\lambda _\phi$. Based on these results, a possible scaling for the isosurface area is presented as $A_{iso}/A_0 \sim (Re \, Sc)^{1/2}$, where $Re$ and $Sc$ are local Reynolds and Schmidt numbers, respectively.

We conduct direct numerical simulations and study the evolution of a pair of counter-rotating vortices in a stratified and turbulent environment beyond the first vortex linking (which is due to the Crow instability). The initial position of the two vortices is perpendicular to the direction of thermal stratification. We vary the Froude number, the background turbulence intensity and the Reynolds number, covering strong, weak and neutral stratification; low, moderate and high background turbulence; and low and moderate Reynolds numbers. We observe a second vortex linking in a weakly stratified environment for the first time. The second vortex linking is followed by turbulence bursts that lead to the rapid decay of the residual vortices after the first vortex linking, leading to a short-living vortex pair. This challenges the conventional view that residual vortices have a long life span, which is true only in an unstratified environment. In addition to the second vortex linking, we observe the following. First, strong background turbulence quickly breaks the two vortices. Second, the two vortices induce secondary and tertiary vortices and lose energy to gravitational waves in a strongly stratified ($Fr\leqslant 0.7$) environment.

We investigate the Reynolds analogy over riblets, namely the analogy between the fractional increase in Stanton number $C_h$ and the fractional increase in the skin-friction coefficient $C_f$, relative to a smooth surface. We investigate the direct numerical simulation data of Endrikat et al. (Flow Turbul. Combust., vol. 107, 2021, pp. 1–29). The riblet groove shapes are isosceles triangles with tip angles $\alpha = {30}^{\circ }, {60}^{\circ }, {90}^{\circ }$, a trapezoid, a rectangle and a right triangle. The viscous-scaled riblet spacing varies between $s^+ \approx 10$ to $60$. The global Reynolds analogy is primarily influenced by Kelvin–Helmholtz rollers and secondary flows. Kelvin–Helmholtz rollers locally break the Reynolds analogy favourably, i.e. cause a locally larger fractional increase in $C_h$ than in $C_f$. These rollers induce negative wall shear stress patches which have no analogue in wall heat fluxes. Secondary flows at the riblets’ crests are associated with local unfavourable breaking of the Reynolds analogy, i.e. locally larger fractional increase in $C_f$ than in $C_h$. Only the triangular riblets with $\alpha = {30}^{\circ }$ trigger strong Kelvin–Helmholtz rollers without appreciable secondary flows. This riblet shape globally preserves the Reynolds analogy from $s^+ = 21$ to $33$. However, the other riblet shapes have weak or non-existent Kelvin–Helmholtz rollers, yet persistent secondary flows. These riblet shapes behave similarly to rough surfaces. They unfavourably break the global Reynolds analogy, and do so to a greater extent as $s^+$ increases.

Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.

Submerged flexible vegetation occupies the core of aquatic ecosystem research. Hydrodynamics of submerged flexible vegetation and its interaction with the flow in an open channel are of great significance in studying the mass and momentum transports in the flow. In this study, a numerical model for highly flexible vegetation based on large eddy simulation and the immersed boundary method was used to simulate the flow–vegetation interaction. It is recognised that alternate vortices with opposite sense rotations appear at the flow–vegetation interface. These vortices prompt the vegetation canopy to have wave-like coherent waving motion, commonly called the monami phenomena. The spatial scale and the spreading velocity in the streamwise direction of these vortices determine the wavelength, frequency and amplitude of the vegetation coherent waving motion. In this study, the fast Fourier transform method was applied to analyse the factors affecting the characteristics of the vegetation coherent waving motion. It is revealed that, as the flow velocity increases, the wavelength of the coherent waving motion decreases, while the frequency and amplitude increase. Besides, as the vegetation spacing increases, the wavelength and amplitude of the coherent waving motion increase, but the frequency decreases. Furthermore, an increase in the relative density of vegetation magnifies the amplitude of coherent waving motion without affecting the wavelength and frequency.

During the high velocity impact of an object on a solid covered with a thin fluid layer, a lubricated contact exists within the short time in which the liquid is squeezed out from the contact. This is important for e.g. the grip of shoes on wet surfaces. We experimentally study the squeeze flow of such layers and find that the amount of viscous dissipation determines how much fluid remains in the contact after the kinetic energy of the impacting object is absorbed. For impacts with sufficient amount of kinetic energy, it is possible to completely drain a Newtonian or shear thinning fluid from the contact on a short time scale. Viscoelastic liquids, however, cannot be drained by increasing impact velocity, because of the fluids’ elastic tendency to retract back into the contact after rapid squeezing. This explains why the presence of polymeric fluids can lead to extreme slipperiness of surfaces. Furthermore, we show that all our experimental results agree with the predictions given by hydrodynamic theory applied to the fluid flow in the gap.

By varying the oil volume fraction, the microscopic droplet size and the macroscopic rheology of emulsions are investigated in a Taylor–Couette turbulent shear flow. Although here oil and water in the emulsions have almost the same physical properties (density and viscosity), unexpectedly, we find that oil-in-water (O/W) and water-in-oil (W/O) emulsions have very distinct hydrodynamic behaviours, i.e. the system is clearly asymmetric. By looking at the micro-scales, the average droplet diameter hardly changes with the oil volume fraction for O/W or for W/O. However, for W/O it is about $50\,\%$ larger than that of O/W. At the macro-scales, the effective viscosity of O/W is higher when compared to that of W/O. These asymmetric behaviours are expected to be caused by the presence of surface-active contaminants from the walls of the system. By introducing an oil-soluble surfactant at high concentration, remarkably, we recover the symmetry (droplet size and effective viscosity) between O/W and W/O emulsions. Based on this, we suggest a possible mechanism responsible for the initial asymmetry and reach conclusions on emulsions where interfaces are fully covered by the surfactant. Next, we discuss what sets the droplet size in turbulent emulsions. We uncover a $-6/5$ scaling dependence of the droplet size on the Reynolds number of the flow. Combining the scaling dependence and the droplet Weber number, we conclude that the droplet fragmentation, which determines the droplet size, occurs within the boundary layer and is controlled by the dynamic pressure caused by the gradient of the mean flow, as proposed by Levich (Physicochemical Hydrodynamics, Prentice-Hall, 1962), instead of the dynamic pressure due to turbulent fluctuations, as proposed by Kolmogorov (Dokl. Akad. Nauk. SSSR, vol. 66, 1949, pp. 825–828). The present findings provide an understanding of both the microscopic droplet formation and the macroscopic rheological behaviours in dynamic emulsification, and connects them.

The unsteady hydrodynamics of two pitching foils arranged in a side-by-side (parallel) configuration is examined for a range of Strouhal numbers, phase differences, oscillation amplitudes and separation distances. Three distinct vortex patterns are identified in the wake maps, which include separated wake, merged wake and transitional-merged wake. Furthermore, a novel model is introduced based on fundamental flow variables including velocity, location and circulation of dipole structures to quantitatively distinguish vortex patterns in the wake. The physical mechanism of the wake merging process is also elucidated. When an oscillating foil experiences the jet deflection phenomenon, secondary structures separated from the primary street traverse in the other direction by making an angle with its parent vortex street. For in-phase pitching parallel foils, secondary structures from the vortex street of the lower foil interact with the primary vortex street of the upper foil under certain kinematic conditions. This interaction triggers the wake merging process by influencing circulation of coherent structures in the upper part of the wake. It is unveiled that merging of the wakes leads to enhancements in propulsive efficiency by increasing thrust generation without a significant alteration in power requirements. These are attributed to the formation of a high-momentum jet by the merged vortex street, which possesses significantly larger circulation due to the amalgamation of the vortices, and major alterations in the evolution of leading edge vortices. Thus, flow physics, which is thoroughly explored here, is crucial in providing novel insights for the future development of flow control techniques for efficient designs of bio-inspired underwater propulsors.