A uniform representation of the mean turbulent velocity profile in the sum of a wall function and a wake function, already introduced for other parallel geometries, is applied to an open channel. The open channel with its wake function is thus found to coherently fit in to the same theoretical picture previously drawn for plane Couette, plane closed-channel and circular-pipe flow, and to share with them a universal law of the wall and a universal logarithmic law with a common value of von Kármán's constant.

The observed behaviour of passive objects in simple flows can be surprisingly intricate, and is complicated further by object activity. Inspired by the motility of bacterial swimmers, in this two-part study we examine the three-dimensional motion of rigid active particles in shear Stokes flow, focusing on bodies that induce rapid rotation as part of their activity. In Part 1 we develop a multiscale framework to investigate these emergent dynamics and apply it to simple spheroidal objects. In Part 2 (Dalwadi et al., J. Fluid Mech., vol. 979, 2024, A2) we apply our framework to understand the emergent dynamics of more complex shapes; helicoidal objects with chirality. Via a multiple scales asymptotic analysis for nonlinear systems, we systematically derive emergent equations of motion for long-term trajectories that explicitly account for the strong (leading-order) effects of fast spinning. Supported by numerical examples, we constructively link these effective dynamics to the well-known Jeffery's orbits for passive spheroids, deriving an explicit closed-form expression for the effective shape of the active particle, broadening the scope of Jeffery's seminal study to spinning spheroids.

In this two-part study, we investigate the motion of rigid, active objects in shear Stokes flow, focusing on bodies that induce rapid rotation as part of their activity. In Part 2, we derive and analyse governing equations for rapidly spinning complex-shaped particles – general helicoidal objects with chirality. Using the multiscale framework that we develop in Part 1 (Dalwadi et al., J. Fluid Mech., vol. 979, 2024, A1), we systematically derive emergent equations of motion for the angular and translational dynamics of these chiral spinning objects. We show that the emergent dynamics due to rapid rotation can be described by effective generalised Jeffery's equations, which differ from the classic versions via the inclusion of additional terms that account for chirality and other asymmetries. Furthermore, we use our analytic results to characterise and quantify the explicit effect of rotation on the effective hydrodynamic shape of the chiral objects, expanding significantly the scope of Jeffery's seminal study.

The growth of perturbations subject to the Crow instability along two vortex rings of equal and opposite circulation undergoing a head-on collision is examined. Unlike the planar case for semi-infinite line vortices, the zero-order geometry of the flow (i.e. the ring radius, core thickness and separation distance) and by extension the growth rates of perturbations vary in time. The governing equations are therefore temporally integrated to characterize the perturbation spectrum. The analysis, which considers the effects of ring curvature and the distribution of vorticity within the vortex cores, explains several key flow features observed in experiments. First, the zero-order motion of the rings is accurately reproduced. Next, the predicted emergent wavenumber, which sets the number of secondary vortex structures emerging after the cores come into contact, agrees with experiments, including the observed increase in the number of secondary structures with increasing Reynolds number. Finally, the analysis predicts an abrupt transition at a critical Reynolds number to a regime dominated by a higher-frequency, faster-growing instability mode that may be consistent with the experimentally observed rapid generation of a turbulent puff following the collision of rings at high Reynolds numbers.

An efficient numerical method for calculation of the sea surface skewness and kurtosis for arbitrary wave spectra, taking up to third-order nonlinear effects into account, is developed. The skewness and kurtosis are calculated for a large number of sea states, covering a wide range of sea-state parameters in terms of wave steepness, water depth, directional spreading and frequency bandwidth. The results are used to propose new accurate expressions for skewness and kurtosis, valid over a wide range of sea states. Existing expressions for skewness and kurtosis reported in the literature are reviewed, and their accuracy is evaluated. Comparison to model-test results and phase-resolved numerical simulation are presented. It is suggested that the new improved parametrizations for skewness and kurtosis, which include dependence on wave steepness, water depth, spectral bandwidth and directional spreading, represent a convenient way to include these covariates into higher-order distributions for crest heights, wave heights and surface elevation.

Natural flyers and swimmers employ flexible wings or fins to propel. While the complex interaction between the foil with deformation and the surrounding non-steady fluid environment defines the propulsion performance of the propellers, elucidating the interaction mechanism through theoretical models earns much challenge. Based on elastokinetics and linear potential flow theory, this study proposes a simplified analytical model to clarify the kinematics and the propulsion performance of a flexible thin foil pitching in flow. The dynamical forces, including the inertial force of the foil and the non-steady fluid pressure, are used to determine the averaged deformation angle of the foil. Combining the averaged deformation angle and the prescribed driving pitching motion, the kinematics of the foil is resolved analytically. Based on the analytical expressions for the corresponding pitching motion, analytical relations among the physical parameters of the stiffness and the mass of the foil and the driving frequency are given to these critical conditions, including resonance of the flow–structure system, equal pitching amplitude between the flexible foil and the rigid counterpart, phase angle transition from ${\rm \pi}/2$ to $- {\rm \pi}/2$. Subsequently, the performance of the foil, including the thrust, the power and the propulsive efficiency, as a function of the flexibility of the foil are derived, together with the introduction of a bluff body type offset drag to the thrust. The formulated analytical theory, which matches nicely with previous reports, will help to interpret the effect of the flexibility and regulate the propulsive performance of the flexible foil when pitching in fluid.

Many of the cell membrane's vital functions are achieved by the self-organization of the proteins and biopolymers embedded in it. The protein dynamics is in part determined by its drag. A large number of these proteins can polymerize to form filaments. In vitro studies of protein–membrane interactions often involve using rigid beads coated with lipid bilayers, as a model for the cell membrane. Motivated by this, we use slender-body theory to compute the translational and rotational resistance of a single filamentous protein embedded in the outer layer of a supported bilayer membrane and surrounded on the exterior by a Newtonian fluid. We first consider the regime where the two layers are strongly coupled through their inter-leaflet friction. We find that the drag along the parallel direction grows linearly with the filament's length and quadratically with the length for the perpendicular and rotational drag coefficients. These findings are explained using scaling arguments and by analysing the velocity fields around the moving filament. We then present and discuss the qualitative differences between the drag of a filament moving in a freely suspended bilayer and a supported membrane as a function of the membrane's inter-leaflet friction. Finally, we briefly discuss how these findings can be used in experiments to determine membrane rheology. In summary, we present a formulation that allows computation of the effects of membrane properties (its curvature, viscosity and inter-leaflet friction), and the exterior and interior three-dimensional fluids’ depth and viscosity on the drag of a rod-like/filamentous protein, all in a unified theoretical framework.

Viscous streaming is an efficient rectification mechanism to exploit flow inertia at small scales for fluid and particle manipulation. It typically entails a fluid vibrating around an immersed solid feature that, by concentrating stresses, modulates the emergence of steady flows of useful topology. Motivated by its relevance in biological and artificial settings characterized by soft materials, recent studies have theoretically elucidated, in two dimensions, the impact of body elasticity on streaming flows. Here, we generalize those findings to three dimensions, via the minimal case of an immersed soft sphere. We first improve existing solutions for the rigid-sphere limit, by considering previously unaccounted terms. We then enable body compliance, exposing a three-dimensional, elastic streaming process available even in Stokes flows. Such effect, consistent with two-dimensional analyses but analytically distinct, is validated against direct numerical simulations and shown to translate to bodies of complex geometry and topology, paving the way for advanced forms of flow control.

The presence of a mean magnetic field aligned with the direction of the acceleration greatly modifies the development of the Rayleigh–Taylor instability (RTI). High resolution direct numerical simulations of the Boussinesq–Navier–Stokes equations under the magnetohydrodynamics approximation reveal that, after an initial damping of the perturbations at the interface between the two miscible fluids, a rapid increase of the mixing layer is observed. Structures are significantly stretched in the vertical direction because magnetic tension prevents small-scale shear instabilities. When the vertical turbulent velocity exceeds the Alfvén velocity, the flow transitions to turbulence, structures break and an enhanced mixing occurs with strong dissipation. Afterwards, the mixing zone slows down and its growth rate is decreased compared to the hydrodynamic case. For larger magnitudes of the mean magnetic field, a strong anisotropy persists, and an increased fraction of potential energy injected into the system is lost into turbulent magnetic energy: as a consequence, the mixing zone growth rate is decreased even more. This phenomenology is embedded in a general buoyancy-drag equation, derived from simplified equations that reflect the large-scale dynamics, in which the drag coefficient is increased by the presence of turbulent magnetic energy.

A computational study was performed of the transport of both non-adhesive and adhesive particles in a porous bed with a body-centred cubic (BCC) structure. Pore-scale simulation of the flow within the porous bed was achieved through combining the immersed boundary method and the lattice Boltzmann method. Particle transport is computed using an adhesive discrete-element method based on a multi-time-scale soft-sphere model. The fluid flow results are validated by comparison with experimental data for dimensionless permeability of flow in a porous bed of spheres. For computations with non-adhesive particles, the particles are observed to drift to the centre of ‘channels’ in the BCC array, within which most of the fluid flow occurs. The mechanism of this inward drift was found to be related to the phenomenon of oscillatory clustering, which is an inertial drift mechanism observed for particles in a corrugated channel. A measure for particle drift into these channels was developed, and the time rate of change of this measure was found to compare closely with an approximate theoretical prediction based on oscillatory clustering theory. The drift measure was observed to be limited at long time by hold-up of outlier particles caught in long-duration collisions with the fixed bed particles in regions of low fluid velocity magnitude. Simulations with adhesive particles exhibited marked increase in collision duration, as well as inhibition of the tendency to drift toward the flow channels due to adhesive hold-up.

The finitely extensible nonlinear elastic with Peterlin closure (FENE-P) and simplified Phan-Thien–Tanner (sPTT) viscoelastic models are used widely for modelling of complex fluids. Although they are derived from distinct micro-structural theories, these models can become mathematically identical in steady and homogeneous flows with a particular choice of the values of the model parameters. However, even with this choice of parameter values, the model responses are known to differ from each other in transient flows. In this work, we investigate the responses of the FENE-P and sPTT constitutive models in large-amplitude oscillatory shear (LAOS). In steady shear, the shear stress scales with the non-dimensional group $Wi/(aL)$ ($Wi\,\sqrt {\epsilon }$) for the FENE-P (sPTT) model, where $Wi$ is the Weissenberg number, $L^2$ is the limit of extensibility in the FENE-P model ($a$ being $L^2/(L^2-3)$), and $\epsilon$ is the extensibility parameter in the sPTT model. Our numerical and analytical results show that in LAOS, the FENE-P model shows this universality only for large values of $L^2$, whereas the sPTT model shows it for all values of $\epsilon$. In the strongly nonlinear region, there is a drastic difference between the responses of the two models, with the FENE-P model exhibiting strong shear stress overshoots that manifest as self-intersecting secondary loops in the viscous Lissajous curves. We quantify the nonlinearity exhibited by each constitutive model using the sequence of physical processes framework. Despite the high degree of nonlinearity exhibited by the FENE-P model, we also show using fully nonlinear one-dimensional simulations that it does not shear band in LAOS within the range of conditions studied.

A computational model is developed to study the time-averaged mean dynamics of red blood cells (RBCs) driven by the time-averaged mean stress generated by two phase-shifted orthogonal ultrasonic standing waves in a viscous fluid. The cell is modelled as an ellipsoidal viscoelastic membrane enclosing the viscous fluid cytoplasm, the motion of which is described by the inclination angle of the ellipsoidal cell shape and the phase angle of the potential membrane cycle. Based on the acoustic perturbation method, the acoustic field and acoustic streaming field are solved to obtain the time-averaged mean stress, and then the temporal evolution equations of the inclination and phase angles of the cell are determined considering the torque balance and energy conservation. At a small acoustic pressure amplitude, this model reproduces the experimentally observed features of cell motion in orthogonal standing waves: the transition from steady stationary orientation to unsteady tumbling with the increase of the phase difference between the two standing waves. By turning up the acoustic pressure amplitude above a critical value, it is further predicted that the previously observed motions can be accompanied by the membrane tank-treading rotation. Observations of these motions, combined with the present computational model, can help to evaluate the mechanical properties of RBC membranes in an automated and high-throughput manner by acoustic methods.

The statistical properties of uniform momentum zones (UMZs) are extracted from laboratory and field measurements in rough wall turbulent boundary layers to formulate a set of stochastic models for the simulation of instantaneous velocity profiles. A spatiotemporally resolved velocity dataset, covering a field of view of $8 \times 9\,{\rm m}^2$, was obtained in the atmospheric surface layer using super-large-scale particle image velocimetry (SLPIV), as part of the Grand-scale Atmospheric Imaging Apparatus (GAIA). Wind tunnel data from a previous study are included for comparison (Heisel et al., J. Fluid Mech., vol. 887, 2020, R1). The probability density function of UMZ attributes such as their thickness, modal velocity and averaged vertical velocity are built at varying elevations and modelled using log-normal and Gaussian distributions. Inverse transform sampling of the distributions is used to generate synthetic step-like velocity profiles that are spatially and temporally uncorrelated. Results show that in the wide range of wall-normal distances and $Re_\tau$ up to $\sim O(10^6)$ investigated here, shear velocity scaling is manifested in the velocity jump across shear interfaces between adjacent UMZs, and attached eddy behaviour is observed in the linear proportionality between UMZ thickness and their wall normal location. These very same characteristics are recovered in the generated instantaneous profiles, using both fully stochastic and data-driven hybrid stochastic (DHS) models, which address, in different ways, the coupling between modal velocities and UMZ thickness. Our method provides a stochastic approach for generating an ensemble of instantaneous velocity profiles, consistent with the structural organisation of UMZs, where the ensemble reproduces the logarithmic mean velocity profile and recovers significant portions of the Reynolds stresses and, thus, of the streamwise and vertical velocity variability.

Based on the multiple-scale expansion technique, a new set of extended nonlinear Schrödinger (ENLS) equations up to the third order is derived to account for the additional high-order bottom and dispersion effects as well as the nonlinear wave interaction on wave transformation over periodic sandbars of sinusoidal geometry. By employing the small-amplitude wave assumption, a closed-form analytical solution for Bragg scattering is obtained from the linearised ENLS equations, which demonstrates that a downshift of wave frequency of the maximum reflection is mainly due to the inclusion of the high-order bottom effect. The factors that affect the downshift of the resonant frequency are identified and a theoretical expression in parabolic form is derived to quantify the downshift magnitude. The fully ENLS equations are further analysed to reveal the additional wave nonlinear effects on Bragg scattering characteristics. Under the condition of infinitesimal sandbar amplitude, the ENLS equations render a theoretical expression of the critical value of $kh$ when the nonlinear wave self-modulation effect and the nonlinear wave cross-modulation effect are equal, whereas the former effect is responsible for wavenumber upshifting and the latter downshifting. When $kh$ is larger than the critical value, the increase of wave nonlinearity will enhance the downshift magnitude of the Bragg resonance, and vice versa. For finite amplitude of the bottom sandbar, the ENLS equations are solved numerically to examine the influence of both wave nonlinearity and sandbar amplitude on the characteristics of Bragg resonance. The results reveal that as the increase of sandbar amplitude, the critical $kh$ increases monotonically.

In this paper, we study non-isochoric models for mixtures of solid particles, at high volume concentration, and a gas. One of the motivations for this work concerns geophysics and more particularly the pyroclastic density currents, whose dense basal parts are precisely mixtures of pyroclasts and lithic fragments and air. They are extremely destructive phenomena, capable of devastating urbanised areas, and are known to propagate over long distances, even over almost flat topography. Fluidisation of these dense granular flows by pore gas pressure is one response that could explain this behaviour and must therefore be taken into account in the models. Starting from a solid–gas mixing model and invoking the compressibility of the gas through a law of state, we rewrite the conservation of mass equation of the gas phase into an equation of the pore gas pressure whose net effect is to reduce the friction between the particles. The momentum equation of the solid phase is completed by generic constitutive laws, specified as in Schaeffer et al. (J. Fluid Mech., vol. 874, 2019, pp. 926–951) by a yield function and a dilatancy function. Therefore, the divergence of the velocity field, which reflects the ability of the granular flow to expand or compress, depends on the volume fraction, pressure, strain rate and inertial number. In addition, we require the dilatancy function to describe the rate of volume change of the granular material near an isochoric equilibrium state, i.e. at constant volume. This property ensures that the volume fraction, which is the solution to the conservation of mass equation, is positive and finite at all times. We also require that the non-isochoric fluidised model be linearly stable and dissipate energy (over time). In this theoretical framework, we derive the dilatancy models corresponding to classical rheologies such as Drucker–Prager and $\mu (I)$ (with or without expansion effects). The main result of this work is to show that it is possible to obtain non-isochoric and fluidised granular models satisfying all the properties necessary to correctly account for the physics of granular flows and which are well posed, at least linearly stable.

Flow-induced vibration (FIV) of a flexible cylinder with a triangular cross-section, allowed to oscillate in the cross-flow, inline and torsional direction, is studied experimentally through water tunnel tests. The dynamic response of the cylinder was studied for three different angles of attack ($0^\circ, 30^\circ, 60^\circ$), at reduced velocities of 0.9–16.27, corresponding to Reynolds numbers of 364–3600. At the angle of attack of $0^\circ$, vortex-induced vibration at low reduced velocity was observed, which transitioned to galloping at higher reduced velocities. At the angles of attack of $30^\circ$ and $60^\circ$, galloping-type response was observed over the range of the reduced velocities tested. Our results show that the cylinder's torsional oscillation breaks the system's symmetry and affects the structural response at higher reduced velocities regardless of the angle of attack. The FIV response of the flexible triangular cylinder is distinct from that of a rigid flexibly mounted triangular cylinder due to torsional oscillation, spanwise flexibility and the two fixed boundary conditions limiting the amplitude of oscillation. Flow field analysis in the wake of the cylinder was done qualitatively and quantitatively using hydrogen bubble flow visualisation and time-resolved volumetric particle tracking velocimetry techniques, respectively. Our results show the existence of highly three-dimensional vortex structures in the wake of the cylinder. We studied the coupling between the vortex shedding modes in the wake of the cylinder and the structural vibration modes through the spatiotemporal mode analysis using the proper orthogonal decomposition technique to distinguish between different types of the FIV response observed.

A computational analysis is performed to study the three-dimensional response of rectangular shear layers to plasma actuator-based control, in the context of sound mitigation of supersonic non-axisymmetric jets. A Mach $1.5$ rectangular jet with an aspect ratio $2:1$ is controlled using experimentally informed actuation patterns, referred to as M0, M1, M2, M3, M${\rm \pi}$ and M+/$-$1. While the first five progressively increase the phase difference between successive actuators thus enhancing three-dimensionality of the shear layer structures, the latter corresponds to the flapping mode of the jet. A preliminary linear analysis identifies that the frequency, $St\sim 1$, has a relatively high overall amplification within the baseline shear layer, and is hence utilized for control in the subsequent nonlinear simulations. Each actuation reveals unique near-field vortical and acoustic responses that have a profound impact on far-field noise levels. The M0 actuation induces circumferentially interconnected strong streamwise vortices, while M1 actuation enhances the circumferential variability in the coherent structures. The M2 actuation encompasses both these effects, and along with a very low tonal impact of forcing, produces the most desirable far-field noise mitigation (${\sim }2.6\,{\rm dB}$), contributed by a broadband reduction around the column-mode peak of the baseline jet. Beyond M2 actuation, effectiveness of control saturates, particularly along the direction of peak noise radiation. Through a near-field analysis of the acoustic component, the efficacy of M2 actuation is attributed to the attenuation of the radiative efficiency of the jet, including reduced energy in the supersonic phase speeds, and redistribution of energy into the higher helical modes. Further, it curtails the nonlinear difference interactions in the plume that energize column-mode frequencies, which often appear as strong intermittent sound-producing events. While the shear layer turbulent kinetic energy decreases with actuation, the controlled jets show minimal variations in mean flow properties, particularly under M2 actuation, suggesting this to be a promising small-perturbation-based noise control strategy.

This article concerns the time-evolution, spectral structure and scaling of weak turbulence subject to rotation and stable stratification. The flow is expressed as a combination of particular solutions, referred to as modes, of the linearised governing equations without viscosity or diffusion. Modes are of two types: oscillatory ones which represent inertial-gravity waves and time-independent ones that express a non-propagating (NP) component of the flow. The presence of the NP component, which plays an active role in the dynamics apart from in the case of pure rotation, renders wave-turbulence analysis problematic because the NP mode is non-dispersive. Equations are derived for the time evolution of the modal amplitudes, evolution which is due to nonlinearity and visco-diffusion. Subsequent analysis assumes that one or other (or both) of the Rossby and Froude numbers is small (weak turbulence). Given this assumption, the NP component is found to evolve independently of the wave one and a numerical scheme, similar to, though significantly different from classical direct numerical simulation, is used to determine its time evolution. The treatment of the wave component assumes its amplitude is large compared with the NP one, otherwise there are seemingly intractable difficulties of closure in the analysis. Given this further assumption, the wave component decouples from the NP one. Evolution equations for the wave spectra are derived using wave-turbulence analysis and are integrated numerically. As might be expected, these equations indicate that nonlinear coupling of wave modes is dominated by resonances. Results are given for both the NP and wave components.

Jetting of collapsing bubbles is a key aspect in cavitation-driven fluid–solid interactions as it shapes the bubble dynamics and additionally due to its direct interaction with the wall. We study experimentally and numerically the near-wall collapse and jetting of a single bubble seeded into the stagnation flow of a wall jet, i.e. a jet that impinges perpendicular onto a solid wall. High-speed imaging shows rich and rather distinct bubble dynamics for different wall jet flow velocities and bubble-to-wall stand-off distances. The simulations use a volume-of-fluid method that allows us to numerically determine the microscopic and transient pressures and shear stresses on the wall. It is shown that a wall jet at moderate flow velocities of a few metres per second already shapes the bubble ellipsoidally inducing a planar and convergent jet flow. The distinct bubble dynamics allow us to tailor the wall interaction. In particular, the shear stresses can be increased by orders of magnitude without increasing impact pressures the same way. Interestingly, at small seeding stand-offs, the bubble during the final collapse stage can lift off the wall and migrate against the flow direction of the wall jet such that the violent collapse occurs away from the wall.

This work studies two-dimensional fixed-flux Rayleigh–Bénard convection with periodic boundary conditions in both horizontal and vertical directions and analyses its dynamics using numerical continuation, secondary instability analysis and direct numerical simulation. The fixed-flux constraint leads to time-independent elevator modes with a well-defined amplitude. Secondary instability of these modes leads to tilted elevator modes accompanied by horizontal shear flow. For $Pr=1$, where $Pr$ is the Prandtl number, a subsequent subcritical Hopf bifurcation leads to hysteresis behaviour between this state and a time-dependent direction-reversing state, followed by a global bifurcation leading to modulated travelling waves without flow reversal. Single-mode equations reproduce this moderate Rayleigh number behaviour well. At high Rayleigh numbers, chaotic behaviour dominated by modulated travelling waves appears. These transitions are characteristic of high wavenumber elevator modes since the vertical wavenumber of the secondary instability is linearly proportional to the horizontal wavenumber of the elevator mode. At a low $Pr$, relaxation oscillations between the conduction state and the elevator mode appear, followed by quasi-periodic and chaotic behaviour as the Rayleigh number increases. In the high $Pr$ regime, the large-scale shear weakens, and the flow shows bursting behaviour that can lead to significantly increased heat transport or even intermittent stable stratification.