We develop a comprehensive model for the creeping Poiseuille Bingham flow in channels equipped with a patterned wall, i.e. decorated with grooves or stripes that may represent a superhydrophobic (SH) or a chemically patterned (CP) surface, respectively, with longitudinal, transverse and oblique groove (stripe) orientations with respect to the applied pressure gradient. We rely on the Navier slip law to model the boundary condition on the slippery grooves. We develop semi-analytical, explicit-form and complementary computational fluid dynamics models, with solutions that have reasonable agreement. In contrast to its Newtonian analogue, a distinct solution for the oblique configuration, with an a priori unknown transform matrix, must be developed due to the viscoplastic nonlinear rheology. Our focus is to systematically analyse the effects of the Bingham number ($B$), slip number ($b$), groove periodicity length ($\ell$), slip area fraction ($\varphi$) and groove orientation angle ($\theta$), on the slip velocities, effective slip length ($\chi$), slip angle difference ($\theta -s$), mixing index ($I_M$), flow anisotropy and flow regimes. In particular, we demonstrate that, as $B$ increases, the maximum values of the shear component of $\chi$, $\theta -s$ and $I_M$ occur progressively at smaller values of $\theta$, compared with their Newtonian counterparts.

Motivated by buoyancy-driven flows within geological formations, we study the evolution of a (dense) gravity current in a porous medium bisected by a thin interbed layer. The gravity current experiences distributed drainage along this low-permeability boundary. Our theoretical description of this flow takes into account dispersive mass exchange with the surrounding ambient fluid by considering the evolution of the bulk and dispersed phases of the gravity current. In turn, we model basal draining by considering two bookend limits, i.e. no mixing versus perfect mixing in the lower layer. Our formulations are assessed by comparing model predictions against the output of complementary numerical simulations run using COMSOL. Numerical output is essential both for determining the value of the entrainment coefficient used within our theory and for assessing the reasonableness of key modelling assumptions. Our results suggest that the degree of dispersion depends on the dip angle and the depth and permeability of the interbed layer. We further find that the nose position predictions made by our theoretical models are reasonably accurate up to the point where the no mixing model predicts a retraction of the gravity current front. Thereafter, the no mixing model significantly under-predicts, and the perfect mixing model moderately over-predicts, numerical data. Reasons for the failure of the no mixing model are provided, highlighting the importance of convective instabilities in the lower layer. A regime diagram is presented that defines the parametric region where our theoretical models do versus do not yield predictions in good agreement with numerical simulations.

The scaling universality of structure functions is studied for artificially thickened turbulent boundary-layer flows in over-tripped impacts by using hot-wire measurement datasets. The self-similarity behaviours in the inner and outer regions are examined from the viewpoint of different flow mechanisms. In the inner region, the relative ratios between structure functions for the energy-containing range of scales exhibit universality behaviour, in accordance with Townsend's attached eddy hypothesis. This universality of the energy-containing range of scales extends further away from the wall by increasing the tripping intensity. On the other hand, the impact of the external intermittency on the self-similarity of small-scale turbulence is examined through the intermittent zone in over-tripped conditions. Towards the boundary-layer edge, the structure functions exhibit a growing departure from self-similarity and analytical prediction, and it is demonstrated that the departure is primarily due to external intermittency. Moreover, based on the conditional statistics concentrated in the turbulent regimes, it is revealed that the small scales in the turbulence regime are homogenized in a self-similar behaviour, which is independent of the current tripping conditions.

An improved version of the non-equilibrium theory of non-homogeneous turbulence of Chen & Vassilicos (J. Fluid Mech., vol. 938, 2022, A7) predicts that an intermediate range of length scales exists where the interscale turbulence transfer rate, the two-point interspace turbulence transport rate and the two-point pressure gradient velocity correlation term in the two-point small-scale turbulent energy equation are all proportional to the turbulence dissipation rate and independent of length scale. Particle image velocimetry measurements in a field of view under the turbulence-generating impellers in a baffled water tank support these predictions and show that the measured small-scale turbulence is significantly non-homogeneous. The particle image velocimetry measurements also suggest that the rate with which large scales lose energy to the small scales in the two-point large-scale turbulent energy equation also appears to be approximately proportional to the turbulence dissipation rate and independent of length scale in the same intermediate range and that this rate may not balance the interscale turbulence transfer rate in the two-point small-scale turbulent energy equation because of turbulent energy transport caused by the non-homogeneity.

Heated supersonic rectangular twin jets (SRTJ) with a total temperature ratio of 2, using nozzles of design Mach number 1.5 and aspect ratio 2, were investigated in flow regimes from overexpanded to the design condition (Mj = 1.3–1.5). This work complements our recently published work in unheated SRTJ using the same experimental facility (Samimy et al., J. Fluid Mech, vol. 959, 2023, A13). Localized arc filament plasma actuators (LAFPAs) were used to excite the natural instabilities in the jets, thereby controlling the flow and acoustics. The results show that the jets were coupled primarily out-of-phase in overexpanded cases, that the coupling had significant effects on the near-field (NF) pressure fluctuations, and that these fluctuations were considerably higher for in-phase than for out-of-phase coupled cases. The results also revealed that the far-field (FF) overall sound pressure level is significantly higher on the minor axis plane of the SRTJ and that the onset of Mach wave radiation contributes to the increased acoustic radiation at the peak noise direction. The LAFPAs successfully controlled the coupling and were able to reduce the NF pressure fluctuations by 10 dB. However, only 1 to 2 dB FF noise reduction at the peak noise radiation direction was achieved. The overall trends of the baseline results and response of the flow to excitation are qualitatively similar in unheated and heated cases, but the details are significantly different.

This paper extends the work of Tamano & Morinishi (J. Fluid Mech., vol. 548, 2006, pp. 361–373) by simulating supersonic turbulent channel flow with asymmetric thermal walls using a larger computational domain and a finer mesh. Direct numerical simulation is carried out for four cases with different thermal wall boundaries at the top wall at fixed $Ma=1.5$, $Re=6000$ and $Pr=0.72$, while the bottom wall is maintained at a constant temperature of $T_L$ equal to the reference temperature. These cases are referred to as the adiabatic case TAd, where the top wall is adiabatic; the pseudo-adiabatic case T32, where the top wall is isothermal with temperature $T_{w,t}=T_A$; the sub-adiabatic case T25, with $T_{w,t}=0.77T_A$; and the super-adiabatic case T40, with $T_{w,t}=1.24T_A$. Here, $T_A=3.234$ is the mean temperature at the adiabatic wall in the TAd case. The objective of this study is to compare and contrast the TAd case with its corresponding T32 case, and to investigate the effect of the wall temperature difference between the two isothermal walls. Comparisons of the basic turbulent statistics, the heat transfer between the Favre-averaged mean-flow kinetic energy, the Favre-averaged turbulent kinetic energy and the Favre-averaged mean internal energy, as well as the wall heat transfer properties, indicate that the TAd case and its corresponding T32 case are generally equivalent. The only discernible difference is in the region very close to the top wall for the temperature-fluctuation-related quantities. The analysis reveals that the asymmetry of the thermal walls causes asymmetry in the flow and thermal fields. In addition, the transfer of the heat generated by the pressure dilatation and the viscous stress is facilitated by the turbulent heat flux term and the mean molecular heat flux term.

Shock-tube experiments and theoretical studies have been performed to highlight mode-coupling in an air–SF$_6$–air fluid layer. Initially, the two interfaces of the layer are designed as single mode with different basic modes. It is found that as the two perturbed interfaces become closer, interface coupling induces a different mode from the basic mode on each interface. Then mode coupling further generates new modes. Based on the linear model (Jacobs et al., J. Fluid Mech., vol. 295, 1995, pp. 23–42), a modified model is established by considering the different accelerations of two interfaces and the waves’ effects in the layer, and provides good predictions to the linear growth rates of the basic modes and the modes generated by interface coupling. It is observed that interface coupling behaves differently to the nonlinear growth of the basic modes, which can be characterized generally by the existing or modified nonlinear model. Moreover, a new modal model is established to quantify the mode-coupling effect in the layer. The mode-coupling effect on the amplitude growth is negligible for the basic modes, but is significant for the interface-coupling modes when the initial wavenumber of one interface is twice the wavenumber of the other interface. Finally, amplitude freeze-out of the second single-mode interface is achieved theoretically and experimentally through interface coupling. These findings may be helpful for designing the target in inertial confinement fusion to suppress the hydrodynamic instabilities.

Velocity gradient tensor, $A_{ij}\equiv \partial u_i/\partial x_j$, in a turbulence flow field is modelled by separating the treatment of intermittent magnitude ($A = \sqrt {A_{ij}A_{ij}}$) from that of the more universal normalised velocity gradient tensor, $b_{ij} \equiv A_{ij}/A$. The boundedness and compactness of the $b_{ij}$-space along with its universal dynamics allow for the development of models that are reasonably insensitive to Reynolds number. The near-lognormality of the magnitude $A$ is then exploited to derive a model based on a modified Ornstein–Uhlenbeck process. These models are developed using data-driven strategies employing high-fidelity forced isotropic turbulence data sets. A posteriori model results agree well with direct numerical simulation data over a wide range of velocity-gradient features and Reynolds numbers.

Gravitational sedimentation and the collision of particles play key roles in various natural and engineering processes. In practice, particles are often non-spherical in shape with non-uniform mass distribution. In this study we investigate how the mass eccentricity influences the settling and gravitational collision of non-spherical particles in a quiescent fluid. Firstly, we theoretically analyse the effect of mass centre offset on the settling motion of a single spheroid under the low-Reynolds-number assumption. We find that the competition of fluid-inertia torque and gravitational torque determines the terminal settling mode of the spheroid. As the mass centre offset increases from zero to a critical value, the orientation of a settling spheroid undergoes a transition from the broad-side-on to narrow-side-on alignment. With an intermediate mass centre offset, the settling spheroid prefers an oblique orientation with a horizontal drift. Secondly, we investigate the gravitational collision rate of settling spheroids. With the change of particle orientation, the collision kernel exhibits a non-monotonic variation with a maximum when particles settle with an intermediate oblique orientation. Therefore, adjusting the mass centre offset to alter particle orientation can indirectly affect the collision rate of settling spheroidal particles. In summary, our findings reveal the significance of the mass eccentricity on particle dynamics in fluid flows, and suggest a potential approach for manipulating the settling motion and collision rate of non-spherical particles by adjusting their mass centre position.

We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we uncover a rich parameter space featuring four distinct branches of stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two states composed of many similar SVs. Of the latter, the first is a dense vortex gas where all SVs have the same sign and diffuse across the domain. The second is a hexagonal vortex crystal forming from this gas when the linear instability is sufficiently weak. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. The vortex gas is continued in the forcing parameter, revealing a sharp transition towards the crystal state as the forcing strength decreases. This transition is analysed in terms of the diffusivity of individual vortices using ideas from statistical physics. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of the forcing details in two-dimensional turbulence and reveals the presence of non-trivial SV states in this system, specifically the emergence and melting of a vortex crystal.

Melting icebergs provide nearly half of the total freshwater flux from ice shelves to the ocean, but the availability of accurate, data-constrained melting rate parametrisations limits the correct representation of this process in ocean models. Here, we investigate the melting of a vertical ice face in a warm, salt-stratified environment in a laboratory setting. Observations of the depth-dependent melting rates ${m}$ and boundary layer flow speed $U$ are reported for a range of initially uniform far-field ambient temperatures $T_a$ above ${10}\,^{\circ }{\rm C}$. Ice scallops are characteristic features observed in all experiments, with the width of the scallops consistent with the theory of double-diffusive layers. The morphology of the scallops changes from symmetric about the scallop centre in the colder experiments to asymmetric in the warmer experiments. Observed melting rates are consistent with a melting rate scaling of the form ${m}\propto U\,\Delta T_a$ proposed by previous work in less extreme parameter regimes, where $\Delta T_a$ is the magnitude of thermal driving between the ambient and ice–fluid interface. Our results indicate that ice scalloping is closely linked to the naturally convecting flow of the ambient fluid. Depth-averaged melting rates depend on the buoyancy frequency in the ambient fluid, and double-diffusive convection promotes a turbulent-flux regime distinct from that explained previously in an unstratified regime. These findings have implications for parametrising melting rates of icebergs and glaciers in numerical models or potential freshwater harvesting operations, and provide insights into the interplay between stratification and ice melting.

We compute steady planar incompressible flows and wall shapes that maximize the rate of heat transfer ($Nu$) between hot and cold walls, for a given rate of viscous dissipation by the flow ($Pe^2$), with no-slip boundary conditions at the walls. In the case of no flow, we show theoretically that the optimal walls are flat and horizontal, at the minimum separation distance. We use a decoupled approximation to show that flat walls remain optimal up to a critical non-zero flow magnitude. Beyond this value, our computed optimal flows and wall shapes converge to a set of forms that are invariant except for a $Pe^{-1/3}$ scaling of horizontal lengths. The corresponding rate of heat transfer $Nu \sim Pe^{2/3}$. We show that these scalings result from flows at the interface between the diffusion-dominated and convection-dominated regimes. We also show that the separation distance of the walls remains at its minimum value at large $Pe$.

Dilatational motions in the shape of travelling wave packets have been identified recently to be dynamically significant in hypersonic turbulent boundary layers. The present study investigates the mechanisms of their generation and their association with the solenoidal motions, especially the well-recognized near-wall self-sustaining process of the regeneration cycle between the velocity streaks and quasi-streamwise vortices. By exploiting the direct numerical simulation databases and orchestrating numerical experiments, we explore systematically the near-wall flow dynamics in the processes of the formation and transient growth of low-speed streaks. We conclude via theoretical ansatz that the nonlinearity related to the parallel density and pressure gradients close to the wall due to the restriction of the isothermal boundary condition is the primary cause of the generation of the dilatational structures at small scales. In fully developed turbulence, the formation and the existence of healthy dilatational travelling wave packets require the participation of the turbulence at scales larger than those of the near-wall regeneration cycles, especially the occurrence of the bursting events that generate vortex clusters. This is proven by the less intensified dilatational motions in the numerical experiments in which the Orr mechanism is alleviated and the vortical structures and turbulent bursts are weakened.

The geostrophic turbulence in rapidly rotating thermal convection exhibits characteristics shared by many highly turbulent geophysical and astrophysical flows. In this regime, the convective length and velocity scales and heat flux are all diffusion-free, i.e. independent of the viscosity and thermal diffusivity. Our direct numerical simulations (DNS) of rotating Rayleigh–Bénard convection in domains with no-slip top and bottom and periodic lateral boundary conditions for a fluid with the Prandtl number $Pr=1$ and extreme buoyancy and rotation parameters (the Rayleigh number up to $Ra=3\times 10^{13}$ and the Ekman number down to $Ek=5\times 10^{-9}$) indeed demonstrate all these diffusion-free scaling relations, in particular, that the dimensionless convective heat transport scales with the supercriticality parameter $\widetilde {Ra}\equiv Ra\, Ek^{4/3}$ as $Nu-1\propto \widetilde {Ra}^{3/2}$, where $Nu$ is the Nusselt number. We further derive and verify in the DNS that with the decreasing $\widetilde {Ra}$, the geostrophic turbulence regime undergoes a transition into another geostrophic regime, the convective heat transport in this regime is characterized by a very steep $\widetilde {Ra}$-dependence, $Nu-1\propto \widetilde {Ra}^{3}$.

Recent studies have shown that superimposing rhythmic perturbations to oscillating tailbeats could simultaneously enhance both the thrust and efficiency (Lehn et al., Phys. Rev. Fluids, vol. 2, 2017, p. 023101; Chao et al., PNAS Nexus, vol. 3, 2024, p. 073). However, these investigations were conducted with a tethered flapping foil, overlooking the self-propulsion intrinsic to real swimming fish. Here, we investigate how the high-frequency, low-amplitude superimposed rhythmic perturbations impact the self-propelled pitching and heaving swimming of a rigid foil. The swimming-speed-based Reynolds number ranges from 1400 to 2700 in our study, depending on superimposed perturbations and swimming modes. Numerical results reveal that perturbations significantly increase swimming speeds in both pitching and heaving motions, while enhancing efficiency exclusively in the heaving motion. Further derived scaling laws elucidate the relationships of perturbations with speeds, power costs and efficiency, respectively. These findings not only hypothesise the potential advantages of perturbations in biological systems, but also inspire designs and controls in biomimetic propulsion and manoeuvring within aquatic environments.

We conducted a systematic numerical investigation of spherical, prolate and oblate particles in an inertial shear flow between two parallel walls, using smoothed particle hydrodynamics (SPH). It was previously shown that above a critical Reynolds number, spherical particles experience a supercritical pitchfork bifurcation of the equilibrium position in shear flow between two parallel walls, namely that the central equilibrium position becomes unstable, leading to the emergence of two new off-centre stable positions (Fox et al., J. Fluid Mech., vol. 915, 2021). This phenomenon was unexpected given the symmetry of the system. In addition to confirming this finding, we found, surprisingly, that ellipsoidal particles can also return to the centre position from the off-centre positions when the particle Reynolds number is further increased, while spherical particles become unstable under this increased Reynolds number. By utilizing both SPH and the finite element method for flow visualization, we explained the underlining mechanism of this reverse of bifurcation by altered streamwise vorticity and symmetry breaking of pressure. Furthermore, we expanded our investigation to include asymmetric particles, a novel aspect that had not been previously modelled, and we observed similar trends in particle dynamics for both symmetric and asymmetric ellipsoidal particles. While further validation through laboratory experiments is necessary, our research paves the road for development of new focusing and separation methods for shaped particles.

A variational principle is proposed to derive the governing equations for the problem of ocean wave interactions with a floating ice shelf, where the ice shelf is modelled by the full linear equations of elasticity and has an Archimedean draught. The variational principle is used to form a thin-plate approximation for the ice shelf, which includes water–ice coupling at the shelf front and extensional waves in the shelf, in contrast to the benchmark thin-plate approximation for ocean wave interactions with an ice shelf. The thin-plate approximation is combined with a single-mode approximation in the water, where the vertical motion is constrained to the eigenfunction that supports propagating waves. The new terms in the approximation are shown to have a major impact on predictions of ice shelf strains for wave periods in the swell regime.

This paper characterizes the refraction of a triple-shock configuration at planar fast–slow gas interfaces. The primary objective is to reveal the wave configurations and elucidate the mechanisms governing circulation deposition and velocity perturbation on the interface caused by triple-shock refraction. The incident triple-shock configuration is generated by diffracting a planar shock around a rigid cylinder, and four interfaces with various $Z_{{r}}$ (i.e. acoustic impedance ratio across the interface) are considered. An analytical model describing the triple-shock refraction is developed, which accurately predicts both the wave configurations as well as circulation deposition and velocity perturbation. Depending on $Z_{{r}}$, three distinct patterns of transmitted waves can be anticipated: a triple-shock configuration; a four-shock configuration; a four-wave configuration. The underlying mechanism for the formation of these wave configurations is elucidated through shock polar analysis. A novel physical insight into the contribution of triple-shock refraction to the interface perturbation growth is provided. The results indicate that the reflected shock in an incident triple-shock configuration makes significant negative contribution to both circulation deposition and velocity perturbation. This investigation elucidates the underlying mechanism responsible for the relatively insignificant contribution of baroclinic circulation to the Richtmyer–Meshkov-like instability induced by a non-uniform shock, and provides an explanation for the decrease in growth rate of interface perturbation amplitude with increasing Atwood number.

The interaction between a uniform current with a circular cylinder submerged in a fluid covered by a semi-infinite ice sheet is considered analytically. The ice sheet is modelled as an elastic thin plate, and the fluid flow is described by the linearised velocity potential theory. The Green function or the velocity potential due to a source is first obtained. As the water surface is divided into two semi-infinite parts with different boundary conditions, the Wiener–Hopf method (WHM) offers significant advantages over alternative approaches and is consequently adopted. To do that, the distribution of the roots of the dispersion equation for fluid fully covered by an ice sheet in the complex plane is first analysed systematically, which does not seem to have been done before. The variations of these roots with the Froude number are investigated, especially their effects or factorisation and decomposition required in the WHM. The result is verified by comparing with that obtained from the matched eigenfunction expansion method. Through differentiating the Green function with respect to the source position, the potentials due to multipoles are obtained, which are employed to construct the velocity potential for the circular cylinder. Extensive results are provided for hydrodynamic forces on the cylinder and wave profiles, and some unique features are discussed. In particular, it is found that the forces can be highly oscillatory with the Froude number when the body is below the ice sheet, whereas such an oscillation does not exist when the body is below the free surface.

When a cylinder is free to move along a transverse rectilinear path within a current, the vibrations developing with and without structural restoring force (SRF) noticeably deviate: if the elastic support is removed, their onset is delayed from a Reynolds number ($Re$, based on the body diameter and inflow velocity) value of approximately 20 to 30, and their peak amplitudes and frequency bandwidths are substantially reduced. The present study examines the influence of a curved path on this deviation by considering that the cylinder, mounted on an elastic support or not, is free to translate along a circular path whose radius is varied. The investigation is carried out numerically at $Re=25$ and $100$, i.e. subcritical and postcritical values relative to the threshold of $47$ that marks the onset of flow unsteadiness for a fixed body. The principal result of this work is that the behaviours of the flow–structure systems with and without SRF tend to converge under the effect of path curvature. Beyond a certain curvature magnitude, both systems explore the same vibration ranges and the presence or absence of SRF becomes indiscernible. This convergence is accompanied by an enhancement of the responses appearing without SRF. It is analysed in light of the evolution of the effective added mass which determines the subset of responses reached with SRF that remain accessible without SRF. The apparent continuity of the physical mechanisms between the subcritical- and postcritical-$Re$ values suggests that the convergence phenomenon uncovered here could persist at higher $Re$.