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.

When a fluid-filled cube rotating rapidly about an axis passing through two opposite vertices is subjected to harmonic modulations of its rotation rate (librations) at a modulation frequency that is $2/\sqrt {3}$ times the mean rotation frequency, all walls of the cube have critical reflection slopes. As such, all inertial wave beams emitted from edges and vertices of the cube in response to the librations are trapped in thin oscillatory boundary layers for forcing amplitudes (Rossby numbers) below a critical value which depends on the Ekman number (ratio of rotation to viscous time scales). How the resulting oscillatory boundary layer flow, referred to as a boundary-confined wave, depends on Ekman and Rossby numbers is examined in detail over several decades. Of particular interest is how the mean flow grows with increasing forcing amplitude, leading to instability resulting from nonlinear interactions between the mean flow and waves in the oscillatory boundary layers, injecting intense small-scale structures throughout the cube.

A new two-equation model for gravity-driven liquid film flow based on the long-wave expansion has been derived. The novelty of the model consists in using a base velocity profile combining parabolic (Ruyer-Quil & Manneville, Eur. Phys. J. B, vol. 15, issue 2, 2000, pp. 357–369) and ellipse (Usha et al., Phys. Fluids, vol. 32, issue 1, 2020, 013603) profile functions in the wall-normal coordinate. The dependence on a free parameter $A$ related to the eccentricity of an ellipse serves as an adjustable parameter. The resulting models are consistent at $O(\varepsilon )$ for inertia terms and at $O(\varepsilon ^2)$ for viscous diffusion effects, and predict accurately the primary instability. Appropriate tuning of the adjustable parameter helps to recover accurate predictions for the asymptotic wave celerity of nonlinear solitary waves. Further, the model is shown to capture the closed separation vortices that can form underneath the troughs of precursory capillary ripples.

Instabilities of fluid flows often generate turbulence. Using extensive direct numerical simulations, we study two-dimensional turbulence driven by a wavenumber-localised instability superposed on stochastic forcing, in contrast to previous studies of state-independent forcing. As the contribution of the instability forcing, measured by a parameter $\gamma$, increases, the system undergoes two transitions. For $\gamma$ below a first threshold, a regular large-scale vortex condensate forms. Above this threshold, shielded vortices (SVs) emerge within the condensate. At a second, larger value of $\gamma$, the condensate breaks down, and a gas of weakly interacting vortices with broken symmetry spontaneously emerges, characterised by preponderance of vortices of one sign only and suppressed inverse energy cascade. The latter transition is shown to depend on the damping mechanism. The number density of SVs in the broken symmetry state slowly increases via a random nucleation process. Bistability is observed between the condensate and mixed SV-condensate states. Our findings provide new evidence for a strong dependence of two-dimensional turbulence phenomenology on the forcing.

The well-known self-propulsion, or swimming, of a deformable body in Stokes flow (i.e. at low Reynolds number) can be understood and modelled from the variation in the configuration-dependent hydrodynamic resistance tensor throughout the period of deformation. Remarkably, at the other extreme of high Reynolds number, a deformable body may also self-propel without doing any net work on the fluid in potential flow. As a body deforms, the mass of fluid displaced – the so-called added mass – depends on the instantaneous body configuration, and a net displacement is possible over a period of deformation. This potential swimming takes a form identical to that for Stokes swimmers with the configuration-dependent added mass replacing the hydrodynamic resistance tensor. Analytical insight into the swimming of a deformable body is obtained through an expansion of the nonlinear spatial dependence of the hydrodynamic interactions and connections between previous studies of swimming in Stokes flow to those in potential flow are made.

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.

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.

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.

Bifurcation analysis of confined salt-finger convection using single-mode equations obtained from a severely truncated Fourier expansion in the horizontal is performed. Strongly nonlinear staircase-like solutions having, respectively, one (S1), two (S2) and three (S3) regions of mixed salinity in the vertical direction are computed using numerical continuation, and their stability properties are determined. Near onset, the one-layer S1 solution is stable and corresponds to maximum salinity transport among the three solutions. The S2 and S3 solutions are unstable but exert an influence on the statistics observed in direct numerical simulations (DNS) in larger two-dimensional (2-D) domains. Secondary bifurcations of S1 lead either to tilted-finger (TF1) or to travelling wave (TW1) solutions, both accompanied by the spontaneous generation of large-scale shear, a process favoured for lower density ratios and Prandtl numbers ($Pr$). These states at low $Pr$ are associated, respectively, with two-layer and three-layer staircase-like salinity profiles in the mean. States breaking reflection symmetry in the midplane are also computed. In two dimensions and for low $Pr$, the DNS results favour direction-reversing tilted fingers resembling the pulsating wave state observed in other systems. Two-layer and three-layer mean salinity profiles corresponding to reversing tilted fingers and TW1 are observed in 2-D DNS averaged over time. The single-mode solutions close to the high wavenumber onset are in an excellent agreement with 2-D DNS in small horizontal domains and compare well with 3-D DNS.

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.

We consider a two-dimensional (2-D) model of an autophoretic particle. Beyond a certain emission/absorption rate (characterized by a dimensionless Péclet number, $Pe$) the particle is known to undergo a bifurcation from a non-motile to a motile state, with different trajectories, going from a straight to a chaotic motion by increasing $Pe$. From the full model, we derive a reduced closed model which involves only two time-dependent complex amplitudes $C_1(t)$ and $C_2(t)$ corresponding to the first two Fourier modes of the solute concentration field. It consists of two coupled nonlinear ordinary differential equations for $C_1$ and $C_2$ and presents several advantages: (i) the straight and circular motions can be handled fully analytically; (ii) complex motions such as chaos can be analysed numerically very efficiently in comparison with the numerically expensive full model involving partial differential equations; (iii) the reduced model has a universal form dictated only by symmetries (meaning that the form of the equations does not depend on a given phoretic model); (iv) the model can be extended to higher Fourier modes. The derivation method is exemplified for a 2-D model, for simplicity, but can easily be extended to three dimensions, not only for the presently selected phoretic model, but also for any model in which chemical activity triggers locomotion. A typical example can be found, for example, in the field of cell motility involving acto-myosin kinetics. This strategy offers an interesting way to cope with swimmers on the basis of ordinary differential equations, allowing for analytical tractability and efficient numerical treatment.

One novel trend in reducing aero-engine noise is to utilize the silent flight mechanism of owls by applying perforations on fan stator vanes. Consequently, the establishment of relevant theoretical models is of particular interest. The current efforts made in this regard are just targeting the features based on two-dimensional models without including the three-dimensionality. In this paper, we present a three-dimensional solution for acoustic scattering by annular perforated cascades, and the dipole source corresponding to the unsteady pressure loading on the vanes is identified as the dominant sound source. By the singularity method, the acoustic response is obtained with the soft boundary condition applied on the vane surfaces. It is found that considerable noise reduction can be achieved for rotor–stator interaction with a modest uniform porosity, and accordingly two mechanisms are proposed to understand the effect of porosity on propagating sound. The first is that the perforations allowing a normal velocity across the vane reduce the unsteady loading induced by the incident disturbances. The second is that the three-dimensional interactions among the dipole sources at different positions are also dampened by the soft boundaries, thus the distribution of the unsteady pressure loading on the vanes will also change significantly compared to hard-vane cases. Non-uniform distributions of porosity are investigated further, indicating that perforations in the vane upstream area are more effective in reducing propagating noise. Our method is fully three-dimensional and capable of investigating non-uniform porosity, and thus is able to provide useful guidance for future soft vane designs.

In Stokes flow, Purcell's scallop theorem forbids objects with time-reversible (reciprocal) swimming strokes from moving. In the presence of inertia, this restriction is eased and reciprocally deforming bodies can swim. A number of recent works have investigated dimer models that swim reciprocally at intermediate Reynolds numbers ${\textit Re} \approx 1$–1000. These show interesting results (e.g. switches of the swim direction as a function of inertia) but the results vary and seem to be case specific. Here, we introduce a general model and investigate the behaviour of an asymmetric spherical dimer of oscillating length for small-amplitude motion at intermediate ${\textit {Re}}$. In our analysis we make the important distinction between particle and fluid inertia, both of which need to be considered separately. We asymptotically expand the Navier–Stokes equations in the small-amplitude limit to obtain a system of linear partial differential equations. Using a combination of numerical (finite element) and analytical (reciprocal theorem, method of reflections) methods we solve the system to obtain the dimer's swim speed and show that there are two mechanisms that give rise to motion: boundary conditions (an effective slip velocity) and Reynolds stresses. Each mechanism is driven by two classes of sphere–sphere interactions, between one sphere's motion and (1) the oscillating background flow induced by the other's motion, and (2) a geometric asymmetry induced by the other's presence. We can thus unify and explain behaviours observed in other works. Our results show how sensitive, counterintuitive and rich motility is in the parameter space of finite inertia of particles and fluid.

In this work, theoretical modelling, quasi-three-dimensional (quasi-3D) simulations and micromodel experiments are conducted to study spontaneous imbibition with gravity in porous media micromodels. By establishing the force balance governing the spontaneous imbibition process, we develop a theoretical model for predicting the imbibition length against time in a rectangular capillary. The theoretical model is then extended to the prediction of a compact displacement process in a micromodel by using an equivalent width, which is derived by analogising the micromodel to a rectangular capillary. By simulating spontaneous imbibition in a rectangular capillary with various aspect ratios ($\varepsilon$), we show that the application condition of the quasi-3D method is $\varepsilon \leqslant 1/3$. Next, we simulate spontaneous imbibition in micromodels with various geometries and flow conditions. Fingering and compact displacement are identified for varying viscosity ratios and gravitational accelerations. At low (high) viscosity ratio of wetting to non-wetting fluids, an upward (downward) gravity can promote the stability of the wetting front, favouring the transition from fingering to compact displacement. In addition, we find that the depth-oriented interface curvature dominates the capillary effect during the imbibition, and such a mechanism is considered by introducing an equivalent contact angle into the theoretical model. With the help of equivalent width and contact angle, the theoretical model is shown to provide satisfactory prediction of the compact displacement process. Finally, a micromodel experiment is presented to further verify the developed theoretical model and the quasi-3D simulation.

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.

Particle-resolved direct numerical simulations are employed to investigate the particle–particle drag force in the bidisperse gas–particle suspensions where the particles are smooth and the single-particle velocity distribution function is Maxwellian. The particle Reynolds number ranges from 6.7 to 123.8, and in this range the particle inertia is high enough that the lubrication force is not essential. It is found that the relation derived by the kinetic theory of granular flow (KTGF) highly overestimates the particle–particle drag force. This is because the pre-collision velocities of colliding particles are not completely uncorrelated with each other. From the time sequence of collision events, it is observed that the particle pair that has just collided will probably collide again after a short time due to the restriction of the particle motion in dense suspensions. Since the post-collision velocities of the first collision cannot relax entirely in such a short time, the relative velocity before the subsequent collision is statistically smaller than the domain-averaged relative velocity. Consequently, the particle–particle drag force is over-predicted when the domain-averaged relative velocity is used. For this reason, this work assumes that the particle–particle drag force is determined by the relative velocity within a local region near large particles. When the local region is set to be the spherical shells centred on the centres of large particles and with an outer radius of a mean free path of small particles, the KTGF-based relation can reasonably predict the particle–particle drag force.

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.

We conducted a three-dimensional numerical simulation of bioconvection generated by oxygen-reactive chemotactic bacteria. This study investigated the bioconvection patterns, interference between plumes, and the wavelength of bioconvection patterns. In addition, we clarified the transport characteristics of cells and oxygen in the bioconvection. Multiple plumes occur in the suspension, and three-dimensional bioconvection is formed around the plumes by the cells with vortex rings arising around the plumes. Even if bioconvection at a high Rayleigh number is disturbed, the bioconvection is strongly stable with respect to disturbances, and the pattern does not change due to disturbances. Bioconvection changes depending on the physical properties of bacteria and oxygen, and in particular, the rate of oxygen consumption by bacteria significantly affects the strength of bioconvection. Bioconvection patterns with different plume arrangements and shapes are formed for different Rayleigh numbers or initial disturbances of the cell concentration. As a result, the wavelengths of the patterns also vary. As the Rayleigh number increases, interference between plumes is strengthened by the shortening of the pattern wavelength, so the velocities of both upward and downward flows increase. Many cells are located under the plumes, and a strong shear flow occurs in these regions. As the pattern wavelength decreases, the cells are affected by high shear stress. Then the convective transport of the entire suspension strengthens and the transport characteristics of cells and oxygen improve. When the chamber boundary is changed to side walls, bacteria adhere to the wall surface, and plumes are arranged regularly along the side walls.