The role of turbulent large-scale streaks or large-scale motions in forming subaqueous sediment ridges on an initially flat sediment bed is investigated with the aid of particle resolved direct numerical simulations of open channel flow at bulk Reynolds numbers up to 9500. The regular arrangement of quasi-streamwise ridges and troughs at a characteristic spanwise spacing between 1 and 1.5 times the mean fluid height is found to be a consequence of the spanwise organisation of turbulence in large-scale streamwise velocity streaks. Ridges predominantly appear in regions of weaker erosion below large-scale low-speed streaks and vice versa for troughs. The interaction between the dynamics of the large-scale streaks in the bulk flow and the evolution of sediment ridges on the sediment bed is best described as ‘top-down’ process, as the arrangement of the sediment bedforms is seen to adapt to changes in the outer flow with a time delay of several bulk time units. The observed ‘top-down’ interaction between the outer flow and the bed agrees fairly well with the conceptual model on causality in canonical channel flows proposed by Jiménez (J. Fluid Mech., vol. 842, 2018, P1, § 5.6). Mean secondary currents of Prandtl's second kind of comparable intensity and lateral spacing are found over developed sediment ridges and in single-phase smooth-wall channels alike in averages over ${O}(10)$ bulk time units. This indicates that the secondary flow commonly observed together with sediment ridges is the statistical footprint of the regularly organised large-scale streaks.

We present laboratory measurements of the interaction between thermoelectric currents and turbulent magnetoconvection. In a cylindrical volume of liquid gallium heated from below and cooled from above and subject to a vertical magnetic field, it is found that the large-scale circulation (LSC) can undergo a slow axial precession. Our experiments demonstrate that this LSC precession occurs only when electrically conducting boundary conditions are employed, and that the precession direction reverses when the axial magnetic field direction is flipped. A thermoelectric magnetoconvection (TEMC) model is developed that successfully predicts the zeroth-order magnetoprecession dynamics. Our TEMC magnetoprecession model hinges on thermoelectric current loops at the top and bottom boundaries, which create Lorentz forces that generate horizontal torques on the overturning large-scale circulatory flow. The thermoelectric torques in our model act to drive a precessional motion of the LSC. This model yields precession frequency predictions that are in good agreement with the experimental observations. We postulate that thermoelectric effects in convective flows, long argued to be relevant in liquid metal heat transfer and mixing processes, may also have applications in planetary interior magnetohydrodynamics.

Inhomogeneous rough surfaces in which strips of roughness alternate with smooth-wall strips are known to generate large-scale secondary motions. Those secondary motions are strongest if the strip width is of the order of the half-channel height and they generate a spatial wall shear stress distribution whose mean value can significantly exceed the area-averaged mean value of a homogeneously smooth and rough surface. In the present paper it is shown that a parametric forcing approach (Busse & Sandham, J. Fluid Mech., vol. 712, 2012, pp. 169–202; Forooghi et al., Intl J. Heat Fluid Flow, vol. 71, 2018, pp. 200–209), calibrated with data from turbulent channel flows over homogeneous roughness, can capture the topological features of the secondary motion over protruding and recessed roughness strips (Stroh et al., J. Fluid Mech., vol. 885, 2020, R5). However, the results suggest that the parametric forcing approach roughness model induces a slightly larger wall offset when applied to the present heterogeneous rough-wall conditions. Contrary to roughness-resolving simulations, where a significantly higher resolution is required to capture roughness geometry, the parametric forcing approach can be applied with usual smooth-wall direct numerical simulation resolution resulting in less computationally expensive simulations for the study of localized roughness effects. Such roughness model simulations are employed to systematically investigate the effect of the relative roughness protrusion on the physical mechanism of secondary flow formation and the related drag increase. It is found that strong secondary motions present over spanwise heterogeneous roughness with geometrical height difference generally lead to a drag increase. However, the physical mechanism guiding the secondary flow formation, and the resulting secondary flow topology, is different for protruding roughness strips and recessed roughness strips separated by protruding smooth surface strips.

Anelastic convection at high Rayleigh number in a plane parallel layer with no slip boundaries is considered. Energy and entropy balance equations are derived, and they are used to develop scaling laws for the heat transport and the Reynolds number. The appearance of an entropy structure consisting of a well-mixed uniform interior, bounded by thin layers with entropy jumps across them, makes it possible to derive explicit forms for these scaling laws. These are given in terms of the Rayleigh number, the Prandtl number and the bottom to top temperature ratio, which also measures how much the density varies across the layer. The top and bottom boundary layers are examined and they are found to be very different, unlike in the Boussinesq case. Elucidating the structure of these boundary layers plays a crucial part in determining the scaling laws. Physical arguments governing these boundary layers are presented, concentrating on the case in which the boundary layers are so thin that temperature and density vary little across them, even though there may be substantial temperature and density variations across the whole layer. Different scaling laws are found, depending on whether the viscous dissipation is primarily in the boundary layers or in the bulk. The cases of both high and low Prandtl number are considered. Numerical simulations of no-slip anelastic convection up to a Rayleigh number of $10^7$ have been performed and our theoretical predictions are compared with the numerical results.

This paper presents a comprehensive analysis of the momentum and energy transfer in compressible turbulent boundary layers based on integral identities. By considering data obtained from direct numerical simulations for a wide parameter range, the superordinate influences of compressibility, wall heat transfer and pressure gradient on the terms of the governing equations are identified and visualized. This allows us both to determine to what degree cases corresponding to different Mach number, heat transfer and pressure-gradient conditions have physically comparable behaviour and to design turbulent boundary-layer cases with specific sought-after behaviour. To this end, newly formulated identities for the skin-friction coefficient $c_f$ and the specific heat-transfer coefficient $c_h$ from wall-normal integrals based on the non-dimensional compressible momentum and total-enthalpy equations are derived and evaluated. As the individual terms of the resulting identities stay formally close to the terms of the governing equations, the integral analysis further allows the evaluation of common arguments derived from the ‘strong’ Reynolds analogy from an integral perspective. A particular formulation of the Eckert number $Ec$ is proposed as a similarity parameter, mapping cases with different Mach numbers and wall heat transfer conditions.

A series of experiments on stratified Taylor–Couette flow in short aspect ratio wide-gap annuli found an intriguing and not-well-understood dynamics: nonlinear coherent structures appearing and disappearing periodically, along with density layering reminiscent of staircase profiles. A detailed numerical study is presented of the nonlinear dynamics near onset of instability in this setting, which explains most of the characteristics found in the experiments. The simulations show that centrifugal instability of the boundary layer on the inner rotating cylinder produces jets of angular momentum forming Taylor cells that are compressed axially due to the strong stratification. These cells are not axisymmetric from the onset, but are in fact two sets of Taylor cells displaced axially that meet in localized azimuthal defect regions where the cells are patched together; the whole structure is a rotating wave with azimuthal wavenumber $m=1$. The presence of endwalls in this short aspect ratio annulus is critical for the understanding of the dynamics. Their impact cannot be accounted for in idealized axially periodic models. Another key ingredient is the role played by the symmetries of the system. Although the axial reflection symmetry is weakly broken by centrifugal buoyancy effects, following instability there are various branches of solutions corresponding to the different ways the system's symmetries may be broken.

High-resolution simulations are performed for gravity currents propagating on a no-slip boundary to study the merging and splitting processes in the lobe-and-cleft structure at a gravity current head. The simulations reproduce the morphological features observed in the laboratory and provide more detailed flow information to elucidate the merging and splitting processes. Our mean lobe width $\tilde {b}$ and mean maximum lobe width $\tilde {b}_{max}$ satisfy the empirical relationships $\tilde {b}/\tilde {d}=7.4 Re^{-0.39}_f$ and $\tilde {b}_{max}/\tilde {d}=12.6 Re^{-0.38}_f$, respectively, over the front Reynolds number in the range $383 \le Re_f \le 3267$, where the front Reynolds number is defined as $Re_f= \tilde {u}_f \tilde {d} / \tilde {\nu }, \tilde {u}_f$ is the front velocity, $\tilde {d}$ is the height of the gravity current head and $\tilde {\nu }$ is the fluid kinematic viscosity. When measured in terms of the viscous length scale $\tilde {\delta }_\nu = \tilde {\nu }/\tilde {u}^*$, where $\tilde {u}^*$ is the shear velocity at the gravity current head, the mean lobe width and the mean maximum lobe width increase with increasing front Reynolds number and asymptotically approach $126 \tilde {\delta }_\nu$ and $230 \tilde {\delta }_\nu$ at $Re_f=3267$, respectively. The vortical structure inside a lobe has an elongated tooth-like shape and a pair of counter-rotating streamwise vortices are positioned on the left- and right-hand sides of each cleft. For the merging process, it requires the interaction of three tooth-like vortices and the middle tooth-like vortex breaks up and reconnects with the two neighbouring tooth-like vortices. Therefore, a cleft may continually merge with another neighbouring cleft but may never disappear. For the splitting process, even before the new cleft appears, a new born streamwise vortex is created by the parent vortex of opposite orientation and the parent vortex can be either the left part or the right part of the existing tooth-like vortex inside the splitting lobe. The new born streamwise vortex then induces the other counter-rotating streamwise vortex as the new cleft develops. The initiation of the splitting process can be attributed to the Brooke–Hanratty mechanism reinforced by the baroclinic production of vorticity. Depending on the orientation of the parent vortex, the resulting new cleft after the splitting process can shift laterally in the positive or negative spanwise direction along the leading edge of the gravity currents as the lobe-and-cleft structure moves forward in the streamwise direction. For gravity currents propagating on a no-slip boundary, the lobe-and-cleft structure is self-sustaining and the manifestations of the merging and splitting processes are in accord with reported laboratory observations.

We study the stability of unsteady particle-laden flows in long, tilted water columns in batch settling mode, where the quasi-steady assumption of base flow no longer holds for the fast settling of particles. For this purpose, we introduce a settling time scale in the momentum and transport equations to solve the unsteady base flow, and utilise non-modal analysis to examine the stability of the disturbance flow field. The base flow increases in magnitude as the settling speed decreases and attains its maximum value when the settling speed becomes infinitesimal. The time evolution of the disturbance flow energy experiences an algebraic growth caused by the lift-up mechanism of the wall-normal disturbance, followed by an exponential growth owing to the shear instability of the base flow. The streamwise and spanwise wavenumbers corresponding to the peak energy gain are identified for both stages. In particular, the flow instability is enhanced as the Prandtl number increases, which is attributed to the sharpening of the particle-laden interface. On the other hand, the flow instability is suppressed by the increase in settling speed, because less disturbance energy can be extracted from the base flow. There exists an optimal tilted angle for efficient sedimentation, where the particle-laden flow is relatively stable and is accompanied by a smaller energy gain of the disturbance.

Air filaments and cavities in plunging breaking waves, generically cylinders, produce bubbles through an interface instability. The effects of gravity, surface tension and surface curvature on cylinder breakup are explored. A generalized dispersion relation is obtained that spans the Rayleigh–Taylor and Plateau–Rayleigh instabilities as cylinder radius varies. The analysis provides insight into the role of surface tension in the formation of bubbles from filaments and cavities. Small filaments break up into bubbles through a Plateau–Rayleigh instability driven through the action of surface tension. Large air cavities produce bubbles through a Rayleigh–Taylor instability driven by gravity and moderated by surface tension, which has a stabilizing effect. Surface tension, interface curvature and gravity are all important for cases between these two extremes. Predicted unstable mode wavenumber and bubble size show good agreement with direct numerical simulations of plunging breaking waves and air cylinders.

Electrophoresis of a tightly fitting sphere of radius $a$ along the centreline of a liquid-filled circular cylinder of radius $R$ is studied for a gap width $h_0=R-a\ll a$. We assume a Debye length $\kappa ^{-1}\ll h_0$, so that surface conductivity is negligible for zeta potentials typically seen in experiments, and the Smoluchowski slip velocity is imposed as a boundary condition at the solid surfaces. The pressure difference between the front and rear of the sphere is determined. If the cylinder has finite length $L$, this pressure difference causes an additional volumetric flow of liquid along the cylinder, increasing the electrophoretic velocity of the sphere, and an analytic prediction for this increase is found when $L\gg R$. If $N$ identical, well-spaced spheres are present, the electrophoretic velocity of the spheres increases with $N$, in agreement with the experiments of Misiunas & Keyser (Phys. Rev. Lett., vol. 122, 2019, 214501).

The main goal of the present study is to thoroughly test the recently derived OBurnett equations for the normal shock wave flow problem for a wide range of Mach number ($3 \leq Ma \leq 9$). A dilute gas system composed of hard-sphere molecules is considered and the numerical results of the OBurnett equations are validated against in-house results from the direct simulation Monte Carlo method. The primary focus is to study the orbital structures in the phase space (velocity–temperature plane) and the variation of hydrodynamic fields across the shock. From the orbital structures, we observe that the heteroclinic trajectory exists for the OBurnett equations for all the Mach numbers considered, unlike the conventional Burnett equations. The thermodynamic consistency of the equations is also established by showing positive entropy generation across the shock. Further, the equations give smooth shock structures at all Mach numbers and significantly improve upon the results of the Navier–Stokes equations. With no tweaking of the equations in any way, the present work makes two important contributions by putting forward an improved theory of shock waves and establishing the validity of the OBurnett equations for solving complex flow problems.

Drop deformation in fluid flows is investigated as an exchange between the kinetic energy of the fluid and the surface energy of the drop. We show analytically that this energetic exchange is controlled only by the stretching (or compression) of the drop surface by the rate-of-strain tensor. This mechanism is analogous to the stretching of the vorticity field in turbulence. By leveraging the non-local nature of turbulence dynamics, we introduce a new decomposition that isolates the energetic exchange arising from local drop-induced surface effects from the non-local action of turbulent fluctuations. We perform direct numerical simulations of single inertial drops in isotropic turbulence and show that an important contribution to the increments of the surface energy arises from the non-local stretching of the fluid–fluid interface by eddies far from the drop surface (outer eddies). We report that this mechanism is dominant and independent of surface dynamics in a range of Weber numbers in which drop breakup occurs. These findings shed new light on drop deformation and breakup in turbulent flows, and opens the possibility for the improvement and simplification of breakup models.

A quantitative assessment of the acoustic source field produced by a laboratory-scale heated jet with a gas dynamic Mach number of 1.55 and an acoustic Mach number of 2.41 is performed using arrays of microphones that are traversed across the axial and radial plane of the jet's acoustic field. The nozzle contour comprises a method of characteristics shape so that shock-related noise is minimal and the dominant sound production mechanism is from Mach waves. The spatial topography of the overall sound pressure level is shown to be dominated by a distinct lobe residing on the principal acoustic emission path, which is expected from flows of this kind with supersonic convective acoustic Mach numbers. The sound field is then analysed on a per-frequency basis in order to identify the location, strength, convection velocity and propagation angle of the various axially distributed noise sources. The analysis reveals a collection of unique data-informed polar patterns of the sound intensity for each frequency. It is shown how these polar patterns can be propagated to any point in the far field with extreme accuracy using the inverse square law. Doing so allows one to gauge the kinds of errors that are encountered using a nozzle-centred source to calculate sound pressure spectrum levels and acoustic power. It is proposed that the measurement strategy described here be used for situations where measurements are being used to compare different facilities, for extrapolating measurements to different geometric scales, for model validation or for developing noise control strategies.

A computational study is presented on cross-stream migration and focusing of deformable capsules in curved microchannels of square and rectangular sections under inertial and non-inertial regimes. The numerical methodology is based on immersed boundary methods for fluid–structure coupling, a finite-volume-based flow solver and finite-element method for capsule deformation. Different focusing behaviours in the two regimes are predicted that arise due to the interplay of inertia, deformation, altered shear gradient, streamline curvature effect and secondary flow. In the non-inertial regime, a single-point focusing occurs on the central plane, and at a radial location between the interior face (i.e. face with highest curvature) of the channel and the location of zero shear. The focusing position is nearly independent of capsule deformability (represented by the capillary number, $Ca$). A two-step migration is observed that is comprised of a faster radial migration, followed by a slower migration toward the centre plane. The focusing location progressively moves further toward the interior face with increasing curvature and width, but decreasing height. In the inertial regime, single-point focusing is observed near the interior face for channel Reynolds number $Re_{C}\sim {O}(1)$, that is also highly sensitive to $Re_{C}$ and $Ca$, and moves progressively toward the exterior face with increasing $Re_{C}$ but decreasing $Ca$. As $Re_{C}$ increases by an order, secondary flow becomes stronger, and two focusing locations appear close to the centres of the Dean vortices. This location becomes practically independent of $Ca$ at even higher inertia. The inertial focusing positions move progressively toward the exterior face with increasing channel width and decreasing height. For wider channels, the equilibrium location is further toward the exterior face than the vortex centre.

Long and slender liquid filaments are produced during inkjet printing, which can subsequently either retract to form a single droplet, or break up to form a primary droplet and one or more satellite droplets. These satellite droplets are undesirable since they degrade the quality and reproducibility of the print, and lead to contamination within the enclosure of the print device. Existing strategies for the suppression of satellite droplet formation include, among others, adding viscoelasticity to the ink. In the present work, we aim to improve the understanding of the role of viscoelasticity in suppressing satellite droplets in inkjet printing. We demonstrate that very dilute viscoelastic aqueous solutions ($\text {concentrations} \sim 0.003\,\%$ wt. polyethylene oxide, corresponding to nozzle Deborah number $De_{n}\sim 3$) can suppress satellite droplet formation. Furthermore, we show that, for a given driving condition, upper and lower bounds of polymer concentration exist, within which satellite droplets are suppressed. Satellite droplets are formed at concentrations below the lower bound, while jetting ceases for concentrations above the upper bound (for fixed driving conditions). Moreover, we observe that, with concentrations in between the two bounds, the filaments retract at velocities larger than the corresponding Taylor–Culick velocity for the Newtonian case. We show that this enhanced retraction velocity can be attributed to the elastic tension due to polymer stretching, which builds up during the initial jetting phase. These results shed some light on the complex interplay between inertia, capillarity and viscoelasticity for retracting liquid filaments, which is important for the stability and quality of inkjet printing of polymer solutions.

The movements of some massive (${O}(100)\ \textrm {t}$) clifftop boulders, once thought to have been caused by tsunami, have been reattributed to storm waves in several recent papers. However, the precise wave-impact modes and transport mechanisms are unknown. We present preliminary linear acceleration, pressure and displacement data recorded by a $1\,{:}\,30$ scale clifftop boulder impacted by a focused breaking wave in a laboratory flume. The 8 kg boulder was placed atop a 0.25 m high platform and struck with a breaking wave of 0.34 m amplitude. Wave focus position was varied from 0.8 m fore of the platform to 0.27 m aft of the platform to alter the breaking crest shape and wave impact type while maintaining total wave spectral energy. Pressure and acceleration time series measurements from within the boulder show distinct impact types across focus positions. All impacts produced boulder displacement, ranging from 5 mm to 42 mm (0.15 m to 1.3 m at full scale, assuming Froude scaling). The largest boulder pressures were recorded when the wave crest and trough struck the boulder at the same position (flip-through). The largest boulder displacements were measured when high pressures and long impact durations occurred simultaneously and wave focusing was close to flip-through.

The dynamics of suspensions plays a crucial role in the evolution of geophysical systems such as lava lakes, magma chambers and magma oceans. During their cooling and solidification, these magmatic bodies involve convective viscous fluids and dispersed solid crystals that can form either a cumulate or a floating lid by sedimentation. We study such systems based on internal heating convection experiments in high Prandtl fluids bearing plastic beads. We aim to determine the conditions required to produce a floating lid or a sedimented deposit. We show that, although the sign of particles buoyancy is the key parameter, it is not sufficient to predict the particles fate. To complement the model we introduce the Shields formalism and couple it with scaling laws describing convection. We propose a generalized Shields number that enables a self-consistent description of the fate of particles in the system, especially the possibility to segregate from the convective bulk. We provide a quantification of the partition of the mass of particles in the different potential reservoirs (bulk suspension, floating lid, settled cumulate) through reconciling the suspension stability framework with the Shields formalism. We illustrate the geophysical implications of the model by revisiting the problem of the stability of flotation crusts on solidifying rocky bodies.

We present an investigation of the fundamental physical processes involved in deep-water gravity wave breaking. Our motivation is to identify the underlying reason causing the deficiency of the eddy viscosity breaking model (EVBM) in predicting surface elevation for strongly nonlinear waves. Owing to the limitation of experimental methods in the provision of high-resolution flow information, we propose a numerical methodology by developing an EVBM enclosed standalone fully nonlinear quasi-potential (FNP) flow model and a coupled FNP plus Navier–Stokes flow model. The numerical models were firstly verified with a wave train subject to modulational instability, then used to simulate a series of broad-banded focusing wave trains under non-, moderate- and strong-breaking conditions. A systematic analysis was carried out to investigate the discrepancies of numerical solutions produced by the two models in surface elevation and other important physical properties. It is found that EVBM predicts accurately the energy dissipated by breaking and the amplitude spectrum of free waves in terms of magnitude, but fails to capture accurately breaking induced phase shifting. The shift of phase grows with breaking intensity and is especially strong for high-wavenumber components. This is identified as a cause of the upshift of the wave dispersion relation, which increases the frequencies of large-wavenumber components. Such a variation drives large-wavenumber components to propagate at nearly the same speed, which is significantly higher than the linear dispersion levels. This suppresses the instant dispersive spreading of harmonics after the focal point, prolonging the lifespan of focused waves and expanding their propagation space.

The multiscale dynamics of a shock–droplet interaction is crucial in understanding the atomisation of droplets due to external airflow. The interaction phenomena are classified into wave dynamics (stage I) and droplet breakup dynamics (stage II). Stage I involves the formation of different wave structures after an incident shock impacts the droplet surface. These waves momentarily change the droplet's ambient conditions, while in later times they are mainly influenced by shock-induced airflow. Stage II involves induced airflow interaction with the droplet that leads to its deformation and breakup. Primarily, two modes of droplet breakup, i.e. shear-induced entrainment and Rayleigh–Taylor piercing (RTP) (based on the modes of surface instabilities) were observed for the studied range of Weber numbers $(We\sim 30\text{--}15\,000)$. A criterion for the transition between two breakup modes is obtained, which successfully explains the observation of RTP mode of droplet breakup at high Weber numbers $(We\sim 800)$. For $We > 1000$, the breakup dynamics is governed by the shear-induced surface waves. After formation, the Kelvin–Helmholtz waves travel on the droplet surface and merge to form a liquid sheet near the droplet equator. Henceforth, the liquid sheet undergoes breakup processes via nucleation of several holes. The breakup process is recurrent until the complete droplet disintegrates or external drag acting on the droplet is insufficient for further disintegration. At lower Weber numbers, the droplet undergoes complete deformation like a flattened disk, and a multibag mode of breakup based on RTP is observed.

Using a dynamical scaling analysis of the flow variables and their evolution due to bubble bursting, here we predict the size and speed of ejected droplets for the whole range of experimental Ohnesorge and Bond numbers where ejection occurs. The transient ejection, which requires the backfire of a vortex ring inside the liquid to preserve physical symmetry, shows a delicate balance between inertia, surface tension and viscous forces around a critical Ohnesorge number, akin to an apparent singularity. Like in other natural phenomena, this balance makes the process extremely sensitive to initial conditions. Our model generalizes or displaces other recently proposed ones, impacting on, for instance, the statistical description of sea spray.