We consider the efficiency of turbulence, a dimensionless parameter that characterises the fraction of the input energy stored in a turbulent flow field. We first show that the inverse of the efficiency provides an upper bound for the dimensionless energy injection in a turbulent flow. We analyse the efficiency of turbulence for different flows using numerical and experimental data. Our analysis suggests that efficiency is bounded from above, and, in some cases, saturates following a power law reminiscent of phase transitions and bifurcations. We show that for the von Kármán flow the efficiency saturation is insensitive to the details of the forcing impellers. In the case of Rayleigh–Bénard convection, we show that within the Grossmann and Lohse model, the efficiency saturates in the inviscid limit, while the dimensionless kinetic energy injection/dissipation goes to zero. In the case of pipe flow, we show that saturation of the efficiency cannot be excluded, but would be incompatible with the Prandtl law of the drag friction coefficient. Furthermore, if the power-law behaviour holds for the efficiency saturation, it can explain the kinetic energy and the energy dissipation defect laws proposed for shear flows. Efficiency saturation is an interesting empirical property of turbulence that may help in evaluating the ‘closeness’ of experimental and numerical data to the true turbulent regime, wherein the kinetic energy saturates to its inviscid limit.

The interface shape near a moving contact line is described by the Cox–Voinov theory, which contains a constant term that is not trivially obtained. In this work, an approximate expression of this term in explicit form is derived under the condition of a Navier slip. Introducing the approximation of a local slippery wedge flow, we first propose a novel form of the generalised lubrication equation. A matched asymptotic analysis of this equation yields the Cox–Voinov relation with the constant term expressed in elementary functions. For various viscosity ratios and contact angles, the theoretical predictions are rigorously validated against full numerical solutions of the Stokes equations and available asymptotic results.

The dynamics of self-propelled colloidal particles is strongly influenced by their environment through hydrodynamic and, in many cases, chemical interactions. We develop a theoretical framework to describe the motion of confined active particles by combining the Lorentz reciprocal theorem with a Galerkin discretisation of surface fields, yielding an equation of motion that efficiently captures self-propulsion without requiring an explicit solution for the bulk fluid flow. Applying this framework, we identify and characterise the long-time behaviours of a Janus particle near rigid, permeable and fluid–fluid interfaces, revealing distinct motility regimes, including surface-bound skating, stable hovering and chemo-hydrodynamic reflection. Our results demonstrate how the solute permeability and the viscosity contrast of the surface influence a particle’s dynamics, providing valuable insights into experimentally relevant guidance mechanisms for autophoretic particles. The computational efficiency of our method makes it particularly well suited for systematic parameter sweeps, offering a powerful tool for mapping the phase space of confined active particles and informing high-fidelity numerical simulations.

Interactions of turbulent boundary layers with a compliant surface are investigated experimentally at Reτ = 3300–8900. Integrating tomographic particle tracking with Mach–Zehnder interferometry enables simultaneous mapping of the compliant wall deformation and the three-dimensional velocity and pressure fields. Our initial study (J. Fluid. Mech. vol. 980, R2) shows that the flow–deformation correlations decrease with increasing Reτ, despite an order of magnitude increase in deformation amplitude. To elucidate the mechanisms involved, the same velocity, pressure and kinetic energy fields are decomposed to ‘wave-coherent’ and ‘stochastic’ parts using a Hilbert projection method. The phase dependent coherent variables, especially the pressure, are highly correlated with the wave, but decrease with increasing Reτ. While the coherent energy is 6 %–10 % of the stochastic level, the pressure root mean square is comparable near the wall. The energy flux between the coherent and stochastic parts and the pressure diffusion reverse sign at the critical layer. To explain the Reτ dependence, the characteristic deformation wavelength (three times the thickness) is compared with the scales of the energy-containing eddies in the boundary layer represented by the k−1 range in the energy spectrum. When the deformation wavelength is matched with the kxEuu peak at the present lowest Reτ, the flow–deformation correlations and coherent pressure become strong, even for submicron deformations. In this case, the flow and wall motion become phase locked, suggesting resonant behaviours. As Reτ increases, the wall wavelengths and spectral range of attached eddies are no longer matched, resulting in reduced correlations and lower coherent energy and pressure, despite larger deformation.

Understanding the interplay between buoyancy and fluid motions within stably stratified shear layers is crucial for unravelling the contribution of flow structures to turbulent mixing. In this study, we examine statistically the local relationship between stratification and fluid deformation rate in wave and turbulent regimes, using experimental datasets obtained from a stratified inclined duct (SID) containing fluids of different densities that form an exchange flow. We introduce rotational and shear components of varying strength within the vorticity and a family of coherent gradient Richardson numbers ($Ri_C$), ratios related to the buoyancy frequency and the strength of either the rotational or shearing motion. Conditional statistical analysis reveals that both shear and stratification intensity affect the probability distribution of the $Ri_C$, with extreme events occurring more frequently in areas of weak stratification. In the wave regime, we identify the persistence of fast-spin vortices within the strongly stratified density interface. However, scouring of the density interface is primarily driven by shearing motions, with baroclinic torque making a notable contribution to enstrophy transport. In the turbulent regime, rigid-body rotations occur at significantly shorter time scales than that associated with the local buoyancy frequency, making them more disruptive to stratification than shear. Additionally, correlation analysis reveals that irrotational strain distorts stable stratification similarly to shearing motions, but is weaker than both shearing and rotational motions and less likely to have a time scale longer than that related to the buoyancy frequency. Moreover, we observed that the interplay between rotational and shearing motions intensifies as stratification increases. Finally, a comparison of length scales along the shear layers highlights the $Ri_C$ as a valuable measure of the relative sizes of different motions compared with the Ozmidov scale and shows that stratification can influence sub-Ozmidov scales through baroclinic torque. This study highlights the critical impact of the type, strength and location of fluid deformations on localised mixing, providing new insights into the role of rotational motions in shear-driven stratified flows.

Steady flow at low Reynolds (Re) number through a planar channel with converging or diverging width is investigated in this study. Along the primary direction of flow, the small dimension of the channel cross-section remains constant while the sidewalls bounding the larger dimension are oriented at a constant angle. Due in part to ease of manufacturing, parallel-plate geometries such as this have found widespread use in microfluidic devices for mixing, heat exchange, flow control and flow patterning at small length scales. Previous analytical solutions for flows of this nature have required the converging or diverging aspect of the channel to be gradual. In this work, we derive a matched asymptotic solution, validated against numerical modelling results, that is valid for any sidewall angle, without requiring the channel width to vary gradually. To accomplish this, a cylindrical coordinate system defined by the angle of convergence between the channel sidewalls is considered. From the mathematical form of the composite expansion, a delineation between two secondary flow components emerges naturally. The results of this work show how one of these two components, originating from viscous shear near the channel sidewalls, corresponds to convective mixing, whereas the other component impresses the sidewall geometry on streamlines in the outer flow.

The Lamb–Oseen vortex is a model for practical vortical flows with a finite vortex core. Vortices with a Lamb–Oseen vortex velocity profile are stable according to the Rayleigh criterion in an infinite domain. Practical situations introduce boundary conditions over finite domains. Direct numerical simulations are performed on the evolution of perturbations to a viscous Lamb–Oseen vortex with uniform inlet axial velocity in a pipe of finite length. Linear stability boundaries are determined in the $(\textit{Re},\omega )$ plane. For a given swirl ratio $\omega$, the flow is found to become linearly unstable when the Reynolds number $\textit{Re}$ is above a critical value. The complete evolution history of the flow is followed until it reaches its final state. For small swirl ratios, the axisymmetric mode is linearly unstable and evolves to a final steady axisymmetric but non-columnar accelerated flow state after nonlinear saturation. For large swirl ratios, the spiral mode is linearly unstable. The spiral mode is found to force growth of an axisymmetric component due to nonlinear interaction. The flow evolves to a final unsteady spiral vortex breakdown state after it undergoes nonlinear saturation. The energy transfer between the mean flow and perturbations is studied by the Reynolds–Orr equation. The pressure work at the exit of the finite pipe is a major source of energy production. Finite-domain boundary conditions also modify the perturbation mode shapes, which can render the vortex core from absorbing energy to producing energy, and thus lead to instabilities. As the pipe length increases, the stability behaviour of the flow is found to approach that predicted by the classical Rayleigh criterion.

We consider the vortex–wedge interaction problem, taking as a departure point Howe’s model of a point vortex interacting with a semi-infinite half-plane, where the vortex path is influenced by its image and a closed-form analytical solution is obtained for the sound field. We generalise Howe’s model to consider wedges of arbitrary angles and explore the influence of vortex circulation, distance from the edge and the wedge half-angle. The effect of wedge angle on sound emission involves a reduced amplitude of the latter as the former is increased. An extension of the model is proposed to account for convection effects by a non-zero ambient flow. We identify a non-dimensional parameter that characterises the vortex kinematics close to the edge and the associated acoustic effect: high and low values of the parameter correspond, respectively, to high- and low-amplitude sound emission of high and low frequency.

We report our finding from direct numerical simulations that polygonal cell structures are formed by inertial particles in turbulent Rayleigh–Bénard convection in a large aspect ratio channel at Rayleigh numbers of $10^6, 10^7$ and $10^8$, and Prandtl number of 0.7. The settling of small particles modified the flow structures only through momentum interactions. From the simulations performed for various sizes and mass loadings of particles, we discovered that for small- and intermediate-sized particles, cell structures such as square, pentagonal or hexagonal cells were observed, whereas a roll structure was formed by large particles. As the mass loading increased, the sizes of the cells or rolls decreased for all particle sizes. The Nusselt number increased with the mass loading of intermediate and large particles, whereas it decreased with the mass loading of small particles compared with the value for particle-free convection. A detailed investigation of the effective feedback forces of the settling particles inside the hot and cold plumes near the walls revealed that the feedback forces break the up–down symmetry between the hot and cold plumes near the surfaces. This enhances the hot plume ascent while not affecting the cold plume, which leads to the preferred formation of cellular structures. The energy budget analysis provides a detailed interaction between particles and fluid, revealing that the net energy is transferred from the fluid to particles when the particles are small, while settling intermediate and large particles drag the fluid so strongly that energy is transferred from particles to fluid.

This study suggests that partial changes in adverse pressure gradient (APG) turbulent boundary layers (TBLs) relative to zero pressure gradient (ZPG) conditions can be obtained quantitatively by the wall-normal integral, while clarifying the partial influence of non-equilibrium effects. Specifically, the term $u_{\tau }^{2}/ ( {U_{e}V_{e}} )$, which is found to describe the degree of scale separation under non-equilibrium conditions, is decomposed into three terms. Here, $u_{\tau }$ is the frictional velocity, $U_{e}$ is the streamwise velocity at the boundary layer edge, and $V_{e}$ is the normal velocity at the boundary layer edge. This equation includes a ZPG term, a pressure gradient term and a streamwise variation term, indicating that the pressure gradient promotes scale separation. The equation can be applied to ZPG TBLs and equilibrium APG TBLs by separately ignoring the pressure gradient term and the streamwise variation term. By using this equation to simplify the integral of the inertia term of the mean momentum equation, an expression for the Reynolds shear stress in the outer region can be obtained, which indicates how APG affects the Reynolds shear stress through the mean velocity. The above quantitative results support further study of non-equilibrium APG TBLs.

The interaction between the flow in a channel with multiple obstructions on the bottom and an elastic ice sheet covering the liquid is studied for the case of steady flow. The mathematical model employs velocity potential theory and fully accounts for the nonlinear boundary conditions at the ice/liquid interface and on the channel bottom. The integral hodograph method is used to derive the complex velocity potential of the flow, explicitly containing the velocity magnitude at the interface. This allows the boundary-value problem to be reduced to a system of nonlinear equations for the unknown velocity magnitude at the ice/liquid interface, which is solved using the collocation method. Case studies are carried out for a widened rectangular obstruction, whose width exceeds the wavelength of the interface, and for arrays of triangular ripples forming the undulating bottom shape. The influence of the bottom shape on the interface is investigated for three flow regimes: the subcritical regime, $F \lt F_{{cr}}$, for which the depth-based Froude number is less than the critical Froude number, and the interface perturbation decays upstream and downstream of the obstruction; the ice-supercritical and channel-subcritical regime, $F_{cr} \lt F \lt 1$, for which two waves of different wavelengths extend upstream and downstream to infinity; and the channel-supercritical regime, $F \gt 1$, for which the hydroelastic wave extends downstream to infinity. The results revealed a trapped interface wave above the rectangular obstruction and the ripple patch. The resonance behaviour of the interface over the undulating bottom occurs when the period of ripples approaches the wavelength of the ice/liquid interface.