A theoretical model is developed to predict the maximum spreading of liquid metal drops when impacting onto dry surfaces under the influence of a vertical magnetic field. This model, which is constructed based on the energy conversion principle, agrees very well with the numerical results, covering a wide range of impact speeds, contact angles and magnetic strengths. When there is no magnetic field, we found that the maximum spreading factor can be predicted well by an interpolating scheme between the viscous and capillary effects, as proposed by Laan et al. (Phys. Rev. Appl., vol. 2 (4), 2014, 044018). However, when gradually increasing the magnetic field strength, the induced Lorentz forces are dominant over the viscous and capillary forces, taking the spreading behaviour into the ‘Joule regime’, where the Joule dissipation is significant. For most situations of practical interest, namely when the strength of the magnetic field is less than 3 T, all three energy conversion routes are important. Therefore, we determine the correct scaling behaviours for the magnetic influence by first equating the loss of kinetic energy to the Joule dissipation in the Joule regime, then by interpolating it with the viscous dissipation and the capillary effects, which allows for a universal rescaling. By plotting the numerical results against the theoretical model, all the results can be rescaled onto a single curve regardless of the materials of the liquid metals or the contact angles of the surfaces, proving that our theoretical model is correct in predicting the maximum spreading factor by constructing a balanced formula between kinetic energy, capillary energy, viscous dissipation energy and Joule dissipation energy.

We analyse velocity fluctuations inside coherent vortices generated as a result of the inverse cascade in the two-dimensional (2-D) turbulence in a finite box. As we demonstrated in Kolokolov & Lebedev (Phys. Rev. E, vol. 93, 2016, 033104), the universal velocity profile, established in Laurie et al. (Phys. Rev. Lett., vol. 113, 2014, 254503), corresponds to the passive regime of the flow fluctuations. This property enables one to calculate correlation functions of the velocity fluctuations in the universal region. We present the results of the calculations that demonstrate a non-trivial scaling of the structure function. In addition the calculations reveal strong anisotropy of the structure function.

Simultaneous measurements of the phase-averaged velocity distribution and the underlying discharge quantities of a dielectric barrier discharge plasma actuator (PA) are performed at $10~\text{k}\text{Hz}$ discharge frequency to investigate the interplay of the discharge and the surrounding flow. The underlying velocity information for the force estimation is obtained by means of phase-averaged particle image velocimetry; the discharge quantities are determined from a Lissajous-figure analysis. The results uncover a clear cause–effect relation between the phase-dependent effective discharge capacitance of the PA and the resulting spatiotemporal volume-force distributions. From this novel insight, it must be concluded that the instantaneous effective discharge intensity dominates the momentum-transfer rate rather than the formerly assumed operating voltage.

We consider the slow motion generated when a body is set into motion relative to an incompressible, inviscid, non-diffusive rotating stratified fluid, showing that there is generated in general a topographic Rossby wave which leads to non-decaying fluctuations in the lift on the obstacle and a fluctuating non-zero drag. The problem is relevant to the flow patterns and forces excited when slow oceanic flows cross bottom topography, and suggests a mechanism for slow fluctuations observed in laboratory experiments.

We investigate numerically the rotational behaviour of a circular porous particle suspended in a two-dimensional (2D) simple shear flow with fluid inertia at particle shear Reynolds number up to 108. We use the volume-averaged macroscopic momentum equation to formulate the flow field inside and outside the moving porous particle, which is solved by a modified single relaxation time lattice Boltzmann method. The effects of fluid inertia, confinement of the bounding walls, and permeability of the particle are studied. Our two-dimensional simulation results confirm that the permeability has little effect on the rotation of a porous particle in unbounded shear flow without fluid inertia (Masoud, Stone & Shelley, J. Fluid Mech., vol. 733, 2013, R6), but also suggest that the role of permeability cannot be neglected when the confinement effect is significant, or the fluid inertia is not negligible. The fluid inertia and the confined walls have similar effects on the rotation of a porous particle as that on a solid impermeable particle. The angular velocity decays with an increase in fluid inertia, and the confinement effect suppresses the angular velocity to a shear rate ratio below 0.5. A simple scaling argument based on the balance of torque exerted by fluid flows adjacent to the two bounding walls and that due to the flow recirculation can explain our results.

An asymptotic theory is developed for the linear stability curve of rapidly rotating Taylor–Couette flow. The analytic curve obtained by the theory excellently explains the limiting Navier–Stokes stability result for general disturbances. When the cylinders are corotating, the asymptotic theory describes the gap between the neutral curve and the Rayleigh stability criterion. For the case when the cylinders are counter-rotating, it is found that, along the stability boundary, the Reynolds number based on the inner cylinder speed is proportional to that based on the outer cylinder speed to the power of $3/5$ .

We present a new adaptive control strategy to isolate and stabilize turbulent states in transitional, stably stratified plane Couette flow in which the gravitational acceleration (non-dimensionalized as the bulk Richardson number $Ri$ ) is adjusted in time to maintain the turbulent kinetic energy (TKE) of the flow. We demonstrate that applying this method at various stages of decaying stratified turbulence halts the decay process and allows a succession of intermediate turbulent states of decreasing energy to be isolated and stabilized. Once the energy of the initial flow becomes small enough, we identify a single minimal turbulent spot, and lower-energy states decay to laminar flow. Interestingly, the turbulent states which emerge from this process have very similar time-averaged $Ri$ , but TKE levels different by an order of magnitude. The more energetic states consist of several turbulent spots, each qualitatively similar to the minimal turbulent spot. This suggests that the minimal turbulent spot may well be the lowest-energy turbulent state which forms a basic building block of stratified plane Couette flow. The fact that a minimal spot of turbulence can be stabilized, so that it neither decays nor grows, opens up exciting opportunities for further study of spatiotemporally intermittent stratified turbulence.

Linear stability of a laminar boundary-layer flow in a streamwise corner can only be treated with an ansatz that considers two-dimensional eigenfunctions with inhomogeneous boundary conditions in cross-flow directions. It is common practice to use Sommerfeld’s radiation condition with a certain wavenumber $\unicode[STIX]{x1D6FD}$ at the lateral domains of the integration domain which are at the same time the far-field domains for each wall. So far, this radiation condition has been exclusively used in a ‘symmetrical’ way, i.e. with the same $\unicode[STIX]{x1D6FD}$ on either far-field boundary plane. This has led to wave patterns that either enter or leave the corner region from the lateral sides for $\unicode[STIX]{x1D6FD}<0$ or $\unicode[STIX]{x1D6FD}>0$ respectively. Here, an ‘asymmetric’ use of Sommerfeld’s radiation condition is suggested, i.e.  $\unicode[STIX]{x1D6FD}<0$ on one far side of the corner and $\unicode[STIX]{x1D6FD}>0$ on the other. With this modification, waves enter the corner area from one side and leave it through the other, i.e. they travel obliquely through the corner. In contrast to before, their amplification rate is always symmetric with respect to $\unicode[STIX]{x1D6FD}=0$ and there is no amplification-rate increase or decrease due to information that either continuously enters the corner from both sides or continuously leaves it through the far sides. The present analysis also shows that the inviscid corner modes are unaffected by the parameters of the far-field radiation boundary conditions. Nevertheless, superposition of two oppositely running single waves obtained by the modified application of the radiation condition leads to a similar wave pattern to that in the case with $\unicode[STIX]{x1D6FD}<0$ on both sides; however, with a slightly smaller amplification rate and a strictly streamwise propagation direction.

We present a new robust method for identifying three dynamically distinct regions in a stratified turbulent flow, which we characterise as quiescent flow, intermittent layers and turbulent patches. The method uses the cumulative filtered distribution function of the local density gradient to identify each region. We apply it to data from direct numerical simulations of homogeneous stratified turbulence, with unity Prandtl number, resolved on up to $8192\times 8192\times 4096$ grid points. In addition to classifying regions consistently with contour plots of potential enstrophy, our method identifies quiescent regions as regions where $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(1)$ , layers as regions where $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(10)$ and patches as regions where $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(100)$ . Here, $\unicode[STIX]{x1D716}$ is the dissipation rate of turbulence kinetic energy, $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $N$ is the (overall) buoyancy frequency. By far the highest local dissipation and mixing rates, and the majority of dissipation and mixing, occur in patch regions, even when patch regions occupy only 5 % of the flow volume. We conjecture that treating stratified turbulence as an instantaneous assemblage of these different regions in varying proportions may explain some of the apparently highly scattered flow dynamics and statistics previously reported in the literature.

In order to better understand the mechanism governing segregation in dense granular flows, the force experienced by a large particle embedded in a granular flow made of small particles is studied using discrete numerical simulations. Accurate force measurements have been obtained in a large range of flow parameters by trapping the large particle in a harmonic potential well to mimic an optical tweezer. Results show that positive or negative segregation lift forces (perpendicular to the shear) exist depending on the stress inhomogeneity. An empirical expression of the segregation force is proposed as a sum of a term proportional to the gradient of pressure and a term proportional to the gradient of shear stress, which both depend on the local friction and particle size ratio.

The nonlinear stability of two-fluid Couette flows is studied using a novel evolution equation whose dynamics is validated by direct numerical simulation (DNS). The evolution equation incorporates inertial effects at arbitrary Reynolds numbers through a non-local term arising from the coupling between the two fluid regions, and is valid when one of the layers is thin. The equation predicts asymmetric solutions and exhibits bistability, features that are essential observations in the experiments of Barthelet et al. (J. Fluid Mech., vol. 303, 1995, pp. 23–53). Related low-inertia models have been used in qualitative predictions rather than the direct comparisons carried out here, and ad hoc modifications appear to be necessary in order to predict asymmetry and bistability. Comparisons between model solutions and DNS show excellent agreement at Reynolds numbers of $O(10^{3})$ found in the experiments. Direct comparisons are also made with the available experimental results of Barthelet et al. (J. Fluid Mech., vol. 303, 1995, pp. 23–53) when the thin layer occupies $1/5$ of the channel height. Pointwise comparisons of the travelling wave shapes are carried out, and once again the agreement is very good.

We study the effect of severe geometrical confinement in Rayleigh–Bénard convection with a wide range of width-to-height aspect ratio $\unicode[STIX]{x1D6E4}$ , $1/128\leqslant \unicode[STIX]{x1D6E4}\leqslant 1$ , and Rayleigh number $Ra$ , $3\times 10^{4}\leqslant Ra\leqslant 1\times 10^{11}$ , at a fixed Prandtl number of $Pr=4.38$ by means of direct numerical simulations in Cartesian geometry with no-slip walls. For convection under geometrical confinement (decreasing $\unicode[STIX]{x1D6E4}$ from 1), three regimes can be recognized (Chong et al., Phys. Rev. Lett., vol. 115, 2015, 264503) based on the global and local properties in terms of heat transport, plume morphology and flow structures. These are Regime I: classical boundary-layer-controlled regime; Regime II: plume-controlled regime; and Regime III: severely confined regime. The study reveals that the transition into Regime III leads to totally different heat and momentum transport scalings and flow topology from the classical regime. The convective heat transfer scaling, in terms of the Nusselt number $Nu$ , exhibits the scaling $Nu-1\sim Ra^{0.61}$ over three decades of $Ra$ at $\unicode[STIX]{x1D6E4}=1/128$ , which contrasts sharply with the classical scaling $Nu-1\sim Ra^{0.31}$ found at $\unicode[STIX]{x1D6E4}=1$ . The flow in Regime III is found to be dominated by finger-like, long-lived plume columns, again in sharp contrast with the mushroom-like, fragmented thermal plumes typically observed in the classical regime. Moreover, we identify a Rayleigh number for regime transition, $Ra^{\ast }=(29.37/\unicode[STIX]{x1D6E4})^{3.23}$ , such that the scaling transition in $Nu$ and $Re$ can be clearly demonstrated when plotted against $Ra/Ra^{\ast }$ .

We experimentally study resonant interactions of oblique surface gravity waves in a large basin. Our results strongly extend previous experimental results performed mainly for perpendicular or collinear wave trains. We generate two oblique waves crossing at an acute angle, while we control their frequency ratio, steepnesses and directions. These mother waves mutually interact and give birth to a resonant wave whose properties (growth rate, resonant response curve and phase locking) are fully characterized. All our experimental results are found in good quantitative agreement with four-wave interaction theory with no fitting parameter. Off-resonance experiments are also reported and the relevant theoretical analysis is conducted and validated.

We consider a rapidly rotating, Boussinesq fluid stirred by buoyant anomalies. In such a system it is known that, in the absence of a magnetic field, inertial waves whose wave vectors lie normal to the rotation axis play a key role in establishing quasi-geostrophic motion. In particular, buoyant anomalies radiate low-frequency inertial wave packets which disperse along the rotation axis, leading to axially elongated columnar vortices. Here we focus on the influence of an ambient magnetic field on this process, motivated by the dynamics of planetary cores. We find that, once again, the waves responsible for establishing quasi-geostrophic structures have wave vectors normal to the rotation axis; however, these are not conventional inertial waves, but rather hybrid ‘inertial–Alfvén waves’. Their frequency equals that of an Alfvén wave but their axial group velocity is half that of the equivalent inertial wave. They have maximal kinetic, magnetic and cross-helicity, carry magnetic and kinetic energy in equal amounts, and are particularly potent in establishing columnar, helical vortices through the spontaneous emission of axially elongated wave packets. Although our hybrid inertial–Alfvén waves have been overlooked in dynamo literature to date, we speculate that they in fact play a central role in planetary dynamos.

Multi-particle dispersion is studied using direct numerical simulations of temporally evolving mixing layers and planar jets for tetrahedra consisting of four fluid particles which are seeded in the turbulent regions or in the non-turbulent regions near the turbulent/non-turbulent interface (TNTI). The modified Richardson law for decaying turbulence is observed for particle pairs. The size dependence of the mean and relative motions of the entrained tetrahedra indicates that the characteristic length scale of the entrained lumps of fluid is approximately 10 times the Kolmogorov microscale. When the tetrahedra move within the TNTI layer they are flattened and elongated by vortex stretching at a deformation rate that is characterized by the Kolmogorov time scale. The shape evolutions of the tetrahedra show that in free-shear flows, thin-slab structures of advected scalars are generated within the TNTI layers.

In their bipolar-coordinate analysis of circular-cylinder electrophoresis near a dielectric wall, Keh et al. (J. Fluid Mech., vol. 231, 1991, pp. 211–228) found that, when an electric field is applied parallel to the wall, the translational and rotational electrophoretic mobilities increase monotonically as the ratio $\unicode[STIX]{x1D6FF}$ of the cylinder–wall separation to the cylinder radius decreases, eventually diverging as $\unicode[STIX]{x1D6FF}^{-1/2}$ when $\unicode[STIX]{x1D6FF}\rightarrow 0$ . Considering the singular limit $\unicode[STIX]{x1D6FF}\ll 1$ from the outset, we conduct here an asymptotic analysis of that electrokinetic problem, providing insight to the manner by which the intense electric field in the narrow gap is transformed into $O(\unicode[STIX]{x1D6FF}^{-3/2})$ shear stresses; these stresses, in turn, overcome the large Stokes resistance so as to provide the large electrophoretic mobilities. In a companion problem, where the cylinder is exposed to a uniform current emanating from a nearby reactive electrode, the intense gap-scale electric field results in an $O(\unicode[STIX]{x1D6FF}^{-2})$ pressure, giving rise in turn to a large repulsive force. In that problem we find that the cylinder velocity perpendicular to the wall approaches a finite limit as $\unicode[STIX]{x1D6FF}\rightarrow 0$ . We also discuss the role of ‘dielectrophoretic’ forces which are inevitable in the above semi-bounded configurations.

A novel method to estimate unsteady aerodynamic force coefficients from pointwise velocity measurements is presented. As opposed to other existing methodologies, time-resolved full velocity fields are not required. The methodology is based on a resolvent-based reduced-order model which requires the mean flow to obtain physical flow structures and pointwise measurement to calibrate their amplitudes. A computationally affordable time-stepping methodology to obtain resolvent modes in non-trivial flow domains is introduced and compared with previous existing matrix-free and matrix-forming strategies. The technique is applied to the unsteady flow around an inclined square cylinder at low Reynolds number. The potential of the methodology is demonstrated through good agreement between the fluctuating pressure distribution on the cylinder and the temporal evolution of the unsteady lift and drag coefficients predicted by the model and those computed by direct numerical simulation.

Since their discovery by Yves Couder and Emmanuel Fort, droplets walking on a vibrating liquid bath have attracted considerable attention because they unexpectedly exhibit certain features reminiscent of quantum particles. While the behaviour of walking droplets in unbounded geometries has to a large extent been rationalized theoretically, no such rationale exists for their behaviour in the presence of boundaries, as arises in a number of key quantum analogue systems. We here present the results of a combined experimental and theoretical study of the interaction of walking droplets with a submerged planar barrier. Droplets exhibit non-specular reflection, with a small range of reflection angles that is only weakly dependent on the system parameters, including the angle of incidence. The observed behaviour is captured by simulations based on a theoretical model that treats the boundaries as regions of reduced wave speed, and rationalized in terms of momentum considerations.

Direct numerical simulations (DNS) are performed to study the behaviour of flow past a sphere in the regime of high stratification (low Froude number $Fr$ ). In contrast to previous results at lower Reynolds numbers, which suggest monotone suppression of turbulence with increasing stratification in flow past a sphere, it is found that, below a critical $Fr$ , increasing the stratification induces unsteady vortical motion and turbulent fluctuations in the near wake. The near wake is quantified by computing the energy spectra, the turbulence energy equation, the partition of energy into horizontal and vertical components, and the buoyancy Reynolds number. These diagnostics show that the stabilizing effect of buoyancy changes flow over the sphere to flow around the sphere. This qualitative change in the flow leads to a new regime of unsteady vortex shedding in the horizontal planes and intensified horizontal shear which result in turbulence regeneration.

A laboratory investigation was performed to study distinctive features of the laminar-to-turbulent transition over distributed roughness characterized by two-dimensional (2D) and three-dimensional (3D) periodic, low-order topographies at roughness Reynolds number $Re_{k}\approx 300$ . Systematic experiments were performed using high-spatial-resolution planar particle image velocimetry (PIV) in a refractive-index-matching (RIM) channel, where the roughness covered the entire length of the test section. The results show that the flow over the 2D roughness becomes turbulent much sooner than its 3D counterpart ( $Re_{x}=50\,000$ versus 120 000). This is attributed to the presence of a velocity inflection point resulting from flow separation within the troughs of the 2D roughness. In the transitional region, unsteady disturbances above the two roughnesses appear upstream of near-roughness disturbances. The above-roughness disturbances are associated with the inflection point in the vertically displaced boundary layer for the 2D case, and with the mean velocity deficit resulting from the interaction of the wakes of upstream elements for the 3D case. The near-roughness fluctuations are associated with the shear layer present behind the crests of both roughnesses. The transitional region is characterized by the interaction between above- and near-roughness disturbances, which merge, leading to a rapid vertical growth of the turbulent fluctuations.