Turbulent shear flows are abundant in geophysical and astrophysical systems and in engineering-technology applications. They are often riddled with large-scale secondary flows that drastically modify the characteristics of the primary stream, preventing or enhancing mixing, mass and heat transfer. Using experiments and numerical simulations, we study the possibility of modifying these secondary flows by using superhydrophobic surface treatments that reduce the local shear. We focus on the canonical problem of Taylor–Couette flow, the flow between two coaxial and independently rotating cylinders, which has robust secondary structures called Taylor rolls that persist even at significant levels of turbulence. We generate these structures by rotating only the inner cylinder of the system, and show that an axially spaced superhydrophobic treatment can weaken the rolls through a mismatching surface heterogeneity, as long as the roll size can be fixed. The minimum hydrophobicity of the treatment required for this flow control is rationalized, and its effectiveness beyond the Reynolds numbers studied here is also discussed.

Jumps in granular chute flow are obtained as continuous solutions of the properly adapted Saint–Venant equations. We elucidate their internal structure via a dynamical systems approach and show that the jumps in phase space manifest themselves as trajectories organized around the stable/unstable manifold of the fixed point representing the uniform flow. Based on this, we derive an analytic approximate expression for the jump length. The paper concludes with a numerical experiment confirming the stability of the jumps.

Design of isolated microparticles which stabilize at the centreline of a channel flow is examined numerically, for moderate Reynolds numbers ($10\leqslant Re\leqslant 80$). This problem is motivated by the need for the design of shaped particle carriers for use in next generation microfluidic cell analysis devices. Stability metrics for particles with arbitrary shapes are formulated based on linear stability theory. Particle shape is parametrized by a compact, non-uniform rational B-spline-based representation. Shape design is posed as an optimization problem and solved using adaptive Bayesian optimization. We focus on designing particles for stability at the channel centreline robust to perturbations. Our results indicate that centreline focusing particles are families of characteristic ‘fish’/‘bottle’/‘dumbbell’-like shapes, exhibiting fore–aft asymmetry. A parametric exploration is then performed to identify stable particle designs at different $k$ values (particle chord-to-channel width ratio) and $Re$ values ($0.1\leqslant k\leqslant 0.4,10\leqslant Re\leqslant 80$). Particles at high $k$ values and $Re$ values are highly stabilized when compared to those at low $k$ values and $Re$ values. We validate the performance of designed particles to perturbations in flow using detailed fluid–structure interaction simulations over different $k$ values and $Re$ values. We identify a basin of attraction around the centreline, within which any arbitrary release results in rotationally stable centreline focusing. We find that this basin spans larger release angle ranges and lateral locations (tending to the channel width) for narrower channels. This effectively standardizes the notion of global focusing using the current stability paradigm in narrow channels, which eliminates the need for an independent design for global focusing in such configurations. The framework detailed in this work is illustrated for two-dimensional cases and is generalizable to stability in three-dimensional flow fields. The current formulation is agnostic to $Re$ and particle/channel geometry, which indicates substantial potential for integration with imaging flow cytometry tools and microfluidic biosensing assays.

Experiments to study the effect of edge undulation on vortex formation have been conducted on impulsively accelerated plates. Abstractions of propulsors found in nature are produced by imprinting undulatory features with varying wavelengths onto the circumferential vortex-forming edge of circular plates. The effects of the small-scale disturbances introduced by these modifications are accessed by means of force measurements and time-resolved particle image velocimetry. Investigations of four different geometries at two different Reynolds numbers reveal an insensitivity of the flow towards length scales smaller than or similar to the thickness of the feeding shear layer. However, the instabilities in the shear layer and the coherence of the vortex wake are influenced when the wavelength of the undulation exceeds the shear-layer thickness by a significant margin. This results in a force augmentation due to enhanced entrainment into the turbulent vortex core, and thus an associated faster vortex growth rate. Yet, contrary to prior expectations, the time of vortex pinch-off remains constant for all edge modifications. The cause–effect relationship behind the stability of the vortex wake is further investigated. While for small edge undulations a turbulent transition of the vortex core results in vortex pinch-off, for larger edge undulations the turbulent vortex core is found to be fed constantly with additional circulation from the shear layer.

We investigate the role of linear mechanisms in the emergence of nonlinear horizontal self-propelled states of a heaving foil in a quiescent fluid. Two states are analysed: a periodic state of unidirectional motion and a quasi-periodic state of slow back and forth motion around a mean horizontal position. The states emergence is explained through a fluid–solid Floquet stability analysis of the non-propulsive symmetric base solution. Unlike a purely hydrodynamic analysis, our analysis accurately determines the locomotion states onset. An unstable synchronous mode is found when the unidirectional propulsive solution is observed. The obtained mode has a propulsive character, featuring a mean horizontal velocity and an asymmetric flow that generates a horizontal force accelerating the foil. An unstable asynchronous mode, also featuring flow asymmetry and a non-zero velocity, is found when the back and forth state is observed. Its associated complex multiplier introduces a slow modulation of the flapping period, agreeing with the quasi-periodic nature of the back and forth regime. The temporal evolution of this perturbation shows how the horizontal force exerted by the flow is alternatively propulsive or resistive over a slow period. For both modes, an analysis of the velocity and force perturbation time-averaged over the flapping period is used to establish physical instability criteria. The behaviour for a large solid-to-fluid density ratio of the modes is thus analysed. The asynchronous fluid–solid mode converges towards the purely hydrodynamic one, whereas the synchronous mode becomes marginally unstable in our analysis not converging to the purely hydrodynamic analysis where it is never destabilised.

The changing three-dimensional vortex shedding topology of a heavy, tethered sphere exposed to uniform flow crossing the onset of vortex induced vibrations (VIV) is reported together with simultaneously tracked sphere positions. Transient upstream flow conditions were imposed by increasing the reduced velocity from $U^{*} = 2.2$ to 4.5 (‘lock-in’, mode I). Three-dimensional vortex shedding under these transient conditions strongly resembled those at steady conditions at the same $U^{*}$. Changes in the wake accompanying the onset of VIV indicated several well-defined stages associated with the changing instantaneous location of the velocity deficit centroid and the wake's symmetry plane's orientation. Before the onset of VIV, the appearance of induced vortices led to lock-in of streamwise sphere oscillations prior to lock-in in the transverse direction. Shortly after ($U^{*} \approx 3.6$), preferential vortex shedding (wake symmetry plane aligned with tether) was lost. Upon reaching $U^{*} = 4.5$, flow–structure interaction reorganised vortex shedding and sphere motion, resulting in steady state conditions after some delay. At this stage, the symmetry plane was aligned perpendicular to the tether and pairs of alternately shed, single hairpins exerted transverse forcing on the sphere. At $U^{*} = 7.2$ (mode II, steady upstream flow), pairs of double hairpins were shed per oscillation period with maximum instantaneous vortex force coefficients that were higher than at $U^{*} = 4.5$. While the present Reynolds numbers ($Re$) were chosen low, the different identified stages in the onset of VIV are also expected to be relevant at a higher $Re$ range.

The Rayleigh–Taylor instability of a linear viscoelastic fluid overlying a passive gas is considered, where, under neutral conditions, the key dimensionless groups are the Bond number and the Weissenberg number. The branching behaviour upon instability to sinusoidal disturbances is determined by weak nonlinear analysis with the Bond number advanced from its critical value at neutral stability. It is shown that the solutions emanating from the critical state either branch out supercritically to steady waves at predictable wavelengths or break up subcritically with a wavelength having a single node. The nonlinear analysis leads to the counterintuitive observation that Rayleigh–Taylor instability of a viscoelastic fluid in a laterally unbounded layer must always result in saturated steady waves. The analysis also shows that the subcritical breakup in a viscoelastic fluid can only occur if the layer is laterally bounded below a critical horizontal width. If the special case of an infinitely deep viscoelastic layer is considered, a simple expression is obtained from which the transition between steady saturated waves and subcritical behaviour can be determined in terms of the leading dimensionless groups. This expression reveals that the supercritical saturation of the free surface is due to the influence of the normal elastic stresses, while the subcritical rupture of the free surface is attributed to the influence of capillary effects. In short, depending upon the magnitude of the scaled shear modulus, there exists a wavenumber at which a transition from saturated waves to subcritical breakup occurs.

When an object is accelerated in a fluid, a primary vortex is formed through the roll-up of a shear layer. This primary vortex does not grow indefinitely and will reach a limiting size and strength. Additional vorticity beyond the critical limit will end up in a trailing shear layer and accumulate into secondary vortices. The secondary vortices are typically considerably smaller than the primary vortex. In this paper, we focus on the formation, shedding and trajectory of secondary vortices generated by a rotating rectangular plate in a quiescent fluid using time-resolved particle image velocimetry. The Reynolds number $\textit {Re}$ based on the maximum rotational velocity of the plate and the distance between the centre of rotation and the tip of the plate is varied from 840 to 11 150. At low $\textit {Re}$, the shear layer is a continuous uninterrupted layer of vorticity that rolls up into a single coherent primary vortex. At $\textit {Re} = 1955$, the shear layer becomes unstable and secondary vortices emerge and subsequently move away from the tip of the plate. For $\textit {Re} > 4000$, secondary vortices are discretely released from the plate tip and are not generated from the stretching of an unstable shear layer. First, we demonstrate that the roll-up of the shear layer, the trajectory of the primary vortex and the path of secondary vortices can be predicted by a modified Kaden spiral for the entire $\textit {Re}$ range considered. Second, the timing of the secondary vortex shedding is analysed using the swirling strength criterion. The separation time of each secondary vortex is identified as a local maximum in the temporal evolution of the average swirling strength close to the plate tip. The time interval between the release of successive secondary vortices is not constant during the rotation but increases the more vortices have been shed. The shedding time interval also increases with decreasing Reynolds number. The increased time interval under both conditions is due to a reduced circulation feeding rate.

A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

Orbits tracked by ellipsoids immersed in inviscid and viscous environments are studied by means of Kirchhoff's equations and high resolution numerical simulations using a variant of the immersed boundary method. We explore the consequences of Kozlov and Onishchenko's theorem of non-integrability of Kirchhoff's equations to show how the fraction of phase space in chaotic orbits is sensitively determined by the body shape, fluid/solid density ratio and the fraction of initial energy in rotational motion. We show how the added mass tensor of the system is an important player in both viscous and inviscid flow, in causing chaos in a triaxial ellipsoid while acting to suppress it in a spheroid. We identify a new integral of motion for a spheroid in inviscid fluid: one component of the generalised angular momentum. A spheroid, which can never execute chaotic dynamics in inviscid flow, is shown to display chaos in viscous flow due to irregular vortex shedding. But the dynamics of the spheroid is restricted whether in viscous or in inviscid flow, unlike in the triaxial ellipsoid, due to our extra integral of motion.

An equation for the evolution of mean kinetic energy, $E$, in a two-dimensional (2-D) or 3-D Rayleigh–Bénard system with domain height $L$ is derived. Assuming classical Nusselt-number scaling, $Nu \sim Ra^{1/3}$, and that mean enstrophy, in the absence of a downscale energy cascade, scales as $\sim E/L^2$, we find that the Reynolds number scales as $Re \sim Pr^{-1}Ra^{2/3}$ in the 2-D system, where $Ra$ is the Rayleigh number and $Pr$ the Prandtl number. Using the evolution equation and the Reynolds-number scaling, it is shown that $\tilde {\tau } \gtrsim Pr^{-1/2}Ra^{1/2}$, where $\tilde {\tau }$ is the non-dimensional convergence time scale. For the 3-D system, we make the estimate $\tilde {\tau } \gtrsim Ra^{1/6}$ for $Pr = 1$. It is estimated that the total computational cost of reaching the high $Ra$ limit in a simulation is comparable between two and three dimensions. The predictions are compared with data from direct numerical simulations.

We study the inverse flux of waves in one of the simplest geophysical fluid dynamics models: one-dimensional rotating shallow water equations. Based on direct numerical integration of the governing equations, we find that waves injected at small scales get transferred upscale predominantly via resonant quartic interactions between wave modes. The waves’ upscale transfer is non-local and involves turbulent transfer between disparate scales of the flow. Our analysis reveals that the upscale transfer of waves is extremely intermittent and is a result of localized-in-time bursts in wave action flux. These intermittent events of flux bursts lead to shallower waves’ spectrum and relatively higher amplitude wave fields in physical space. On examining statistics of the flow fields, we find that low-energy high wavenumbers more or less comply with the assumptions used in wave turbulence theory, such as uniformly distributed wave phases and the Gaussian distribution of fields, while non-uniform distribution of wave phases and non-Gaussian statistics dominate at large scales or low wavenumbers that contain a major share of the flow energy. Our findings point out that the one-dimensional rotating shallow water equations, despite being a simple geophysical fluid dynamic model, harbour complex and intricate features associated with the upscale transfer of waves that have not been recognized in the past.

Body-force modelling in the lattice Boltzmann method (LBM) has been studied extensively in the incompressible limit but rarely discussed for thermal compressible flows. Here we present a systematic approach of incorporating body force in the LBM which is valid for thermal compressible and non-equilibrium flows. In particular, a LBM forcing scheme accurate for the energy equation with second-order time accuracy is given. New and essential in this scheme is the third-moment contribution of the force term. It is shown via Chapman–Enskog analysis that the absence of this contribution causes an erroneous heat flux quadratic in Mach number and linear in temperature variation. The theoretical findings are verified and the necessity of the third-moment contribution is demonstrated by numerical simulations.

Immiscible and incompressible liquid–liquid flows are considered in a Taylor–Couette geometry and analysed by direct numerical simulations coupled with the volume-of-fluid method and a continuum surface force model. The system Reynolds number $Re \equiv r_i \omega _i d / \nu$ is fixed to $960$, where the single-phase flow is in the steady Taylor vortex regime, whereas the secondary-phase volume fraction $\varphi$ and the system Weber number $We \equiv \rho r_i^2 \omega _i^2 d / \sigma$ are varied to study the interactions between the interface and the Taylor vortices. We show that different Weber numbers lead to two distinctive flow regimes, namely an advection-dominated regime and an interface-dominated regime. When $We$ is high, the interface is easily deformed because of its low surface tension. The flow patterns are then similar to the single-phase flow, and the system is dominated mainly by advection (advection-dominated regime). However, when $We$ is low, the surface tension is so large that stable interfacial structures with sizes comparable to the cylinder gap can exist. The background velocity field is modulated largely by these persistent structures, thus the overall flow dynamics is governed by the interface (interface-dominated regime). The effect of the interface on the global system response is assessed by evaluating the Nusselt number $Nu_{\omega }$ based on the non-dimensional angular velocity transport. It shows non-monotonic trends as functions of the volume fraction $\varphi$ for both low and high $We$. We explain how these dependencies are closely linked to the velocity and interfacial structures.

We demonstrate a novel shear-induced mechanism for growth of concentration fluctuations in a bacterial suspension. Using a linear stability analysis, a homogeneous bacterial suspension, subject to a simple shear flow, is shown to be susceptible to exponentially growing layering perturbations in the shear rate and bacterial concentration. A semi-analytical expression for the growth rate of concentration perturbations is first obtained using the method of multiple scales, in the limit where the time scales characterizing the positional and orientation degrees of freedom are well separated. Next, the eigenspectrum obtained numerically from a full linear stability analysis is used to validate and extend the multiple scales result, and draw a contrast with the known orientation-shear instability. Finally, fully nonlinear simulations, but restricted to one-dimensional variations of the relevant fields (velocity, concentration and swimmer orientation distribution) show that the initial instability leads to gradient-banded velocity profiles, with a local depletion of bacteria at the interface between the homogeneous shear bands. Our results demonstrate that long-ranged hydrodynamic interactions serve as an alternate explanation for recent observations of shear bands in bacterial suspensions.

We report a systematic experimental study of the mean temperature profile $\theta (\delta z)$ and temperature variance profile $\eta (\delta z)$ across a stable and immiscible liquid–liquid (water–FC770) interface formed in two-layer turbulent Rayleigh–Bénard convection. The measured $\theta (\delta z)$ and $\eta (\delta z)$ as a function of distance $\delta z$ away from the interface for different Rayleigh numbers are found to have the scaling forms $\theta (\delta z/\lambda )$ and $\eta (\delta z/\lambda )$, respectively, with varying thermal boundary layer (BL) thickness $\lambda$. By a careful comparison with the simultaneously measured BL profiles near a solid conducting surface, we find that the measured $\theta (\delta z)$ and $\eta (\delta z)$ near the liquid interface can be well described by the BL equations for a solid wall, so long as a thermal slip length $\ell _T$ is introduced to account for the convective heat flux passing through the liquid interface. Direct numerical simulation results further confirm that the turbulent thermal diffusivity $\kappa _t$ near a stable liquid interface has a complete cubic form, $\kappa _t(\xi )/\kappa \sim (\xi +\xi _0)^3$, where $\kappa$ is the molecular thermal diffusivity of the convecting fluid, $\xi =\delta z/\lambda$ is the normalized distance away from the liquid interface and $\xi _0$ is the normalized slip length associated with $\ell _T$.

Free drops of uncharged and charged inviscid, conducting fluids subjected to small-amplitude perturbations undergo linear oscillations (Rayleigh, Proc. R. Soc. London, vol. 29, no. 196–199, 1879, pp. 71–97; Rayleigh, Philos. Mag., vol. 14, no. 87, 1882, pp. 184–186). There exist a countably infinite number of oscillation modes, $n=2, 3, \ldots$, each of which has a characteristic frequency and mode shape. Presence of charge ($Q$) lowers modal frequencies and leads to instability when $Q>Q_R$ (Rayleigh limit). The $n=0$ and $n=1$ modes are disallowed because they violate volume conservation and cause centre of mass (COM) motion. Thus, the first mode to become unstable is the $n=2$ prolate–oblate mode. For free drops, there is a one-to-one correspondence between mode number and shape (Legendre polynomial $P_n$). Recent research has shifted to studying oscillations of spherical drops constrained by solid rings. Pinning the drop introduces a new low-frequency mode of oscillation ($n=1$), one associated primarily with COM translation of the constrained drop. We analyse theoretically the effect of charge on oscillations of constrained drops. Using normal modes and solving a linear operator eigenvalue problem, we determine the frequency of each oscillation mode. Results demonstrate that for ring-constrained charged drops (RCCDs), the association between mode number and shape is lost. For certain pinning locations, oscillations exhibit eigenvalue veering as $Q$ increases. While slightly charged RCCDs pinned at zeros of $P_2$ have a first mode that involves COM motion and a second mode that entails prolate–oblate oscillations, the modes flip as $Q$ increases. Thereafter, prolate–oblate oscillations of RCCDs adopt the role of being the first mode because they exhibit the lowest vibration frequency. At the Rayleigh limit, the first eigenmode – prolate–oblate oscillations – loses stability while the second – involving COM motion – remains stable.

Turbulent flows above a solid surface are characterised by a hydrodynamic roughness that represents, for the far velocity field, the typical length scale at which momentum mixing occurs close to the surface. Here, we are theoretically interested in the hydrodynamic roughness induced by a two-dimensional modulated surface, the elevation profile of which is decomposed in Fourier modes. We describe the flow for a sinusoidal mode of given wavelength and amplitude with Reynolds-averaged Navier–Stokes equations closed by means of a mixing-length approach that takes into account a possible surface geometrical roughness as well as the presence of a viscous sublayer. It also incorporates spatial transient effects at the laminar–turbulent transition. Performing a weekly nonlinear expansion in the bedform aspect ratio, we predict the effective hydrodynamic roughness when the surface wavelength is varied and we show that it presents a non-monotonic behaviour at the laminar–turbulent transition when the surface is hydrodynamically smooth. Further, with a self-consistent looped calculation, we are able to recover the smooth–rough transition of a flat surface, for which the hydrodynamic roughness changes from a regime where it is dominated by the viscous length to another one where it scales with the surface corrugation. We finally apply the results to natural patterns resulting from hydrodynamic instabilities such as those associated with dissolution or sediment transport. We discuss in particular the aspect ratio selection of dissolution bedforms and roughness hierarchy in superimposed ripples and dunes.