While it has long been recognized that Lagrangian drift at the ocean surface plays a critical role in the kinematics and dynamics of upper ocean processes, only recently has the contribution of wave breaking to this drift begun to be investigated through direct numerical simulations (Deike et al., J. Fluid Mech., vol. 829, 2017, pp. 364–391; Pizzo et al., J. Phys. Oceanogr., vol. 49(4), 2019, pp. 983–992). In this work, laboratory measurements of the surface Lagrangian transport due to focusing deep-water non-breaking and breaking waves are presented. It is found that wave breaking greatly enhances mass transport, compared to non-breaking focusing wave packets. These results are in agreement with the direct numerical simulations of Deike et al. (J. Fluid Mech., vol. 829, 2017, pp. 364–391), and the increased transport due to breaking agrees with their scaling argument. In particular, the transport at the surface scales with $S$, the linear prediction of the maximum slope at focusing, while the surface transport due to non-breaking waves scales with $S^{2}$, in agreement with the classical Stokes prediction.

We develop a novel ‘moving-capacitor’ dynamic network model to simulate immiscible fluid–fluid displacement in porous media. Traditional network models approximate the pore geometry as a network of fixed resistors, directly analogous to an electrical circuit. Our model additionally captures the motion of individual fluid–fluid interfaces through the pore geometry by completing this analogy, representing interfaces as a set of moving capacitors. By incorporating pore-scale invasion events, the model reproduces, for the first time, both the displacement pattern and the injection-pressure signal under a wide range of capillary numbers and substrate wettabilities. We show that at high capillary numbers the invading patterns advance symmetrically through viscous fingers. In contrast, at low capillary numbers the flow is governed by the wettability-dependent fluid–fluid interactions with the pore structure. The signature of the transition between the two regimes manifests itself in the fluctuations of the injection-pressure signal.

The dynamics of ideally polarizable spherical particles in concentrated suspensions under the effects of nonlinear electrokinetic phenomena is analysed using large-scale numerical simulations. Particles are assumed to carry no net charge and considered to undergo the combination of dielectrophoresis and induced-charge electrophoresis termed dipolophoresis. Chaotic motion and resulting hydrodynamic diffusion are known to be driven by the induced-charge electrophoresis, which dominates the dielectrophoresis. Up to a volume fraction $\unicode[STIX]{x1D719}\approx 35\,\%$, the particle dynamics seems to be hindered by the increase in the magnitude of excluded volume interactions with concentration. However, a non-trivial suspension behaviour is observed in concentrated regimes, where the hydrodynamic diffusivity starts to increase with the volume fraction at $\unicode[STIX]{x1D719}\approx 35\,\%$, before reaching a local maximum, and then drastically decreases on approaching random close packing. Similar non-trivial behaviours are observed in the particle velocity and number-density fluctuations around volume fractions at which the non-trivial behaviour of the hydrodynamic diffusion is observed. We explain these non-trivial behaviours as a consequence of particle contacts, which are related to the dominant mechanism of particle pairings. The particle contacts are classified into attractive and repulsive classes by the nature of contacts, and in particular, the strong repulsive contact becomes predominant at $\unicode[STIX]{x1D719}>20\,\%$. Moreover, this transition is visible in the pair distribution functions, which also reveal the change in the suspension microstructure in concentrated regimes. It appears that strong and massive repulsive contacts along the direction perpendicular to an electric field promote the non-trivial suspension behaviours observed in concentrated regimes.

We consider a three-dimensional acoustic field of an ideal gas in which all entropy production is confined to weak shocks and show that similar scaling relations hold for such a field as for forced Burgers turbulence, where the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$ and the $p$th-order structure function scales as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, $\unicode[STIX]{x1D716}$ being the mean energy dissipation per unit mass, $d$ the mean distance between the shocks and $r$ the separation distance. However, for the acoustic field, $\unicode[STIX]{x1D716}$ should be replaced by $\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712}$, where $\unicode[STIX]{x1D712}$ is associated with entropy production due to heat conduction. In particular, the third-order longitudinal structure function scales as $\langle \unicode[STIX]{x1D6FF}u_{r}^{3}\rangle =-C(\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712})r$, where $C$ takes the value $12/5(\unicode[STIX]{x1D6FE}+1)$ in the weak shock limit, $\unicode[STIX]{x1D6FE}=c_{p}/c_{v}$ being the ratio between the specific heats at constant pressure and constant volume.

Online in situ prediction of 3-D flame evolution has been long desired and is considered to be the Holy Grail for the combustion community. Recent advances in computational power have facilitated the development of computational fluid dynamics (CFD), which can be used to predict flame behaviours. However, the most advanced CFD techniques are still incapable of realizing online in situ prediction of practical flames due to the enormous computational costs involved. In this work, we aim to combine the state-of-the-art experimental technique (that is, time-resolved volumetric tomography) with deep learning algorithms for rapid prediction of 3-D flame evolution. Proof-of-concept experiments conducted suggest that the evolution of both a laminar diffusion flame and a typical non-premixed turbulent swirl-stabilized flame can be predicted faithfully in a time scale on the order of milliseconds, which can be further reduced by simply using a few more GPUs. We believe this is the first time that online in situ prediction of 3-D flame evolution has become feasible, and we expect this method to be extremely useful, as for most application scenarios the online in situ prediction of even the large-scale flame features are already useful for an effective flame control.

Experiments and computations are presented for a foil pitching about its leading edge near a planar, solid boundary. The foil is examined when it is constrained in space and when it is unconstrained or freely swimming in the cross-stream direction. It was found that the foil has stable equilibrium altitudes: the time-averaged lift is zero at certain altitudes and acts to return the foil to these equilibria. These stable equilibrium altitudes exist for both constrained and freely swimming foils and are independent of the initial conditions of the foil. In all cases, the equilibrium altitudes move farther from the ground when the Strouhal number is increased or the reduced frequency is decreased. Potential flow simulations predict the equilibrium altitudes to within 3 %–11 %, indicating that the equilibrium altitudes are primarily due to inviscid mechanisms. In fact, it is determined that stable equilibrium altitudes arise from an interplay among three time-averaged forces: a negative jet deflection circulatory force, a positive quasistatic circulatory force and a negative added mass force. At equilibrium, the foil exhibits a deflected wake and experiences a thrust enhancement of 4 %–17 % with no penalty in efficiency as compared to a pitching foil far from the ground. These newfound lateral stability characteristics suggest that unsteady ground effect may play a role in the control strategies of near-boundary fish and fish-inspired robots.

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$, where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

Reversed von Kármán streets are responsible for a velocity surplus in the wake of flapping foils, indicating the onset of thrust generation. However, the wake pattern cannot be predicted based solely on the flapping peak-to-peak amplitude $A$ and frequency $f$ because the transition also depends sensitively on other details of the kinematics. In this work we replace $A$ with the cycle-averaged swept trajectory ${\mathcal{T}}$ of the foil chordline. Two-dimensional simulations are performed for pure heave, pure pitch and a variety of heave-to-pitch coupling. In a phase space of dimensionless ${\mathcal{T}}-f$ we show that the drag-to-thrust wake transition of all tested modes occurs for a modified Strouhal $St_{{\mathcal{T}}}\rightarrow 1$. Physically, the product ${\mathcal{T}}f$ expresses the induced velocity of the foil and indicates that propulsive jets occur when this velocity exceeds $U_{\infty }$. The new metric offers a unique insight into the thrust-producing strategies of biological swimmers and flyers alike, as it directly connects the wake development to the chosen kinematics, enabling a self-similar characterisation of flapping foil propulsion.

As a counterpart of energy cascade, turbulent momentum cascade (TMC) in the wall-normal direction is important for understanding wall turbulence. Here, we report an analytic prediction of non-universal Reynolds number ($Re_{\unicode[STIX]{x1D70F}}$) scaling transition of the maximum TMC located at $y_{p}$. We show that in viscous units, $y_{p}^{+}$ (and $1+\overline{u^{\prime }v^{\prime }}_{p}^{+}$) displays a scaling transition from $Re_{\unicode[STIX]{x1D70F}}^{3/7}$ ($Re_{\unicode[STIX]{x1D70F}}^{-6/7}$) to $Re_{\unicode[STIX]{x1D70F}}^{3/5}$ ($Re_{\unicode[STIX]{x1D70F}}^{-3/5}$) in turbulent boundary layer, in sharp contrast to that from $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ ($Re_{\unicode[STIX]{x1D70F}}^{-2/3}$) to $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ ($Re_{\unicode[STIX]{x1D70F}}^{-1/2}$) in a channel/pipe, countering the prevailing view of a single universal near-wall scaling. This scaling transition reflects different near-wall motions in the buffer layer for small $Re_{\unicode[STIX]{x1D70F}}$ and log layer for large $Re_{\unicode[STIX]{x1D70F}}$, with the non-universality being ascribed to the presence/absence of mean wall-normal velocity $V$. Our predictions are validated by a large set of data, and a probable flow state with a full coupling between momentum and energy cascades beyond a critical $Re_{\unicode[STIX]{x1D70F}}$ is envisaged.

When a drop of a low-viscosity liquid of radius $R$ impacts against an inclined smooth solid substrate at a velocity $V$, a liquid sheet of thickness $H_{t}\ll R$ is expelled at a velocity $V_{t}\gg V$. If the impact velocity is such that $V>V^{\ast }$, with $V^{\ast }$ the critical velocity for splashing, the edge of the expanding liquid sheet lifts off from the wall as a consequence of the gas lubrication force at the wedge region created between the advancing liquid front and the substrate. Here we show that the magnitude of the gas lubrication force is limited by the values of the slip length $\ell _{\unicode[STIX]{x1D707}}$ at the gas–liquid interface and of the slip length $\ell _{g}\propto \unicode[STIX]{x1D706}$ at the solid, with $\unicode[STIX]{x1D706}$ the mean free path of gas molecules. We demonstrate that the splashing regime changes depending on the value of the ratio $\ell _{\unicode[STIX]{x1D707}}/\ell _{g}$ – a fact explaining the spreading–splashing–spreading–splashing transition for a fixed (low) value of the gas pressure as the drop impact velocity increases (Xu et al., Phys. Rev. Lett., vol. 94, 2005, 184505; Hao et al., Phys. Rev. Lett., vol. 122, 2019, 054501). We also provide an expression for $V^{\ast }$ as a function of the inclination angle of the substrate, the drop radius $R$, the material properties of the liquid and the gas, and the mean free path $\unicode[STIX]{x1D706}$, in very good agreement with experiments.

The low-frequency unsteadiness of oblique shock wave/boundary layer interactions (SBLIs) has been investigated using large-eddy simulation (LES) and high-frequency pressure measurements from experiments. Particular attention has been paid to off-centreline behaviour: the LES dataset was generated including sidewalls, and experimental pressure measurements were acquired across the entire span of the reflected shock foot. The datasets constitute the first maps of low-frequency unsteadiness in both streamwise and spanwise directions. The results reveal that significant low-frequency shock motion (with $St\approx 0.03$) occurs away from the centreline, along most of the central separation shock and in the corner regions. The most powerful low-frequency unsteadiness occurs off-centre, likely due to the separation shock being strengthened by shocks arising from the swept interactions on the sidewalls. Both simulation and experimental results exhibit asymmetry about the spanwise centre. In simulations, this may be attributed to a lack of statistical convergence; however, the fact that this is also seen in experiments is indicative that some SBLIs may exhibit some inherent asymmetry across the two spanwise halves of the separation bubble. There is also significant low-frequency power in the corner separations. The relation of the unsteadiness in the corner regions to that in the centre is investigated by means of two-point correlations: a key observation is that significant correlation does not extend across the attached flow channel between the central and corner separations.

An examination is undertaken of the validity and limitations of the quasi-steady hypothesis of near-wall turbulence. This hypothesis is based on the supposition that the statistics of the turbulent fluctuations are universal if scaled by the local, instantaneous, wall shear when its variations are determined from footprints of large-scale, energetic, structures that reside in the outer part of the logarithmic layer. The examination is performed with the aid of direct numerical simulation data for a single Reynolds number, which are processed in a manner that brings out the variability of locally scaled statistics when conditioned on the local value of the wall friction. The key question is to what extent this variability is insignificant, thus reflecting universality. It is shown that the validity of the quasi-steady hypothesis is confined, at best, to a thin layer above the viscous sublayer. Beyond this layer, substantial variations in the conditioned shear-induced production rate of large-scale turbulence cause substantial departures from the hypothesis. Even within the wall-proximate layer, moderate departures are provoked by large-scale distortions in the conditioned strain rate that result in variations in small-scale production of turbulence down to the viscous sublayer.

Vortex-induced vibration of a circular cylinder that is free to move in the transverse ($Y$) and streamwise ($X$) directions is investigated at subcritical Reynolds numbers ($1500\lesssim Re\lesssim 9000$) via three-dimensional (3-D) numerical simulations. The mass ratio of the system for all the simulations is $m^{\ast }=10$. It is observed that while some of the characteristics associated with the $XY$-oscillation are similar to those of the $Y$-only oscillation (in line with the observations made by Jauvtis & Williamson (J. Fluid Mech., vol. 509, 2004, pp. 23–62)), notable differences exist between the two systems with respect to the transition between the branches of the cylinder response in the lock-in regime. The flow regime between the initial and lower branch is characterized by intermittent switching in the cylinder response, aerodynamic coefficients and modes of vortex shedding. Similar to the regime of laminar flow, the system exhibits a hysteretic response near the lower- and higher-$Re$ end of the lock-in regime. The frequency spectrum of time history of the cylinder response shows that the most dominant frequency in the streamwise oscillation on the initial branch is the same as that of the transverse oscillation.

The changes in spectra and intensities of the streamwise component of turbulent velocity are calculated in the inflow of a turbine rotor. The flow is initially calculated in the limit when the turbulence is of small scale compared with the rotor diameter. Rapid distortion theory (RDT), Batchelor & Proudman (Q. J. Mech. Appl. Maths, vol. 7 (1), 1954, pp. 83–103) (BP), for small-scale turbulence is combined with the effect of the fluctuating potential flow field on the turbulence caused by the direct interaction of the incident turbulence with the rotor as a sheet of resistance. A second computation is then carried out for turbulence of larger length scale. The results of the calculations are compared with velocity measurements in the inflow of both a commercial wind turbine and a tidal turbine rotor.

In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier–Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute the maximum vorticity $\unicode[STIX]{x1D714}_{max}$ as a function of vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}$ in the range $2000\leqslant R_{\unicode[STIX]{x1D6E4}}\leqslant 3400$, and deduce a compatible behaviour $\unicode[STIX]{x1D714}_{max}\sim \unicode[STIX]{x1D714}_{0}\exp [1+220(\log [R_{\unicode[STIX]{x1D6E4}}/2000])^{2}]$ as $R_{\unicode[STIX]{x1D6E4}}\rightarrow \infty$. This may be described as a physical (although not strictly mathematical) singularity, for all $R_{\unicode[STIX]{x1D6E4}}\gtrsim 4000$.

The scattering of inertia-gravity waves by large-scale geostrophic turbulence in a rapidly rotating, strongly stratified fluid leads to the diffusion of wave energy on the constant-frequency cone in wavenumber space. We derive the corresponding diffusion equation and relate its diffusivity to the wave characteristics and the energy spectrum of the turbulent flow. We check the predictions of this equation against numerical simulations of the three-dimensional Boussinesq equations in initial-value and forced scenarios with horizontally isotropic wave and flow fields. In the forced case, wavenumber diffusion results in a $k^{-2}$ wave energy spectrum consistent with as-yet-unexplained features of observed atmospheric and oceanic spectra.

Fractal features of the turbulent/non-turbulent interface (TNTI) in shock wave/turbulent boundary-layer interaction (SWBLI) flows are essential in understanding the physics of the SWBLI and the supersonic turbulent boundary layer, yet have received almost no attention previously. Accordingly, this study utilises a high spatiotemporal resolution visualisation technique, ice-cluster-based planar laser scattering (IC-PLS), to acquire the TNTI downstream of the reattachment in a SWBLI flow. Evolution of the fractal features of the TNTI in this SWBLI flow is analysed by comparing the parameters of the TNTI acquired in this study with those from a previous result (Zhuang et al.J. Fluid Mech., vol. 843, 2018a).

We report an experimental study of turbulent Rayleigh–Bénard (RB) convection in an annular cell of water (Prandtl number $Pr=4.3$) with a radius ratio $\unicode[STIX]{x1D702}\simeq 0.5$. Global quantities, such as the Nusselt number $Nu$ and the Reynolds number $Re$, and local temperatures were measured over the Rayleigh range $4.2\times 10^{9}\leqslant Ra\leqslant 4.5\times 10^{10}$. It is found that the scaling behaviours of $Nu(Ra)$, $Re(Ra)$ and the temperature fluctuations remain the same as those in the traditional cylindrical cells; both the global and local properties of turbulent RB convection are insensitive to the change of cell geometry. A visualization study, as well as local temperature measurements, shows that in spite of the lack of the cylindrical core, there also exists a large-scale circulation (LSC) in the annular system: thermal plumes organize themselves with the ascending hot plumes on one side and the descending cold plumes on the opposite side. Near the upper and lower plates, the mean flow moves along the two circular branches. Our results further reveal that the dynamics of the LSC in this annular geometry is different from that in the traditional cylindrical cell, i.e. the orientation of the LSC oscillates in a narrow azimuthal angle range, and no cessations, reversals or net rotation were detected.

We present experiments and theory for viscous fingering of a suspension of non-colloidal particles undergoing radial flow in a Hele-Shaw cell. As the suspension displaces air, shear-induced migration causes particles to move faster than the average suspension velocity and to accumulate on the suspension–air interface. The resultant particle accumulation generates a pattern in which low-concentration, low-viscosity suspension displaces high-concentration, high-viscosity suspension and is unstable due to the classic Saffman–Taylor instability mechanism. While the destabilising mechanism is well-understood, what remains unknown is the stabilising mechanism that suppresses fine fingers characteristic of miscible fingering. In this work, we demonstrate how the stable suspension–air interface interacts with the unstable miscible interface to set the critical wavelength. We present a linear stability analysis for the time-dependent radial flow and show that the wavenumber predicted by the analysis is in good agreement with parametric experiments investigating the effect of suspension concentration and gap thickness of the Hele-Shaw cell.

The coating of a plate withdrawn from a bath of a suspension of non-Brownian, monodisperse and neutrally buoyant spherical particles suspended in a Newtonian liquid has been studied. Using laser profilometry, particle tracking and local sample weighing we have quantified the thickness $h$ and the particle content of the film for various particle diameters $d$ and volume fractions ($0.10\leqslant \unicode[STIX]{x1D719}\leqslant 0.50$). Three coating regimes have been observed as the withdrawal velocity is increased: (i) no particle entrainment ($h\lesssim d$), (ii) a monolayer of particles ($h\sim d$), and (iii) a thick film ($h\gtrsim d$), where the suspension behaves as an effective viscous fluid following the Landau–Levich–Derjaguin law. We discuss the boundaries between these regimes, as well as the evolution of the liquid and solid content of the coating over the whole range of withdrawal capillary number and volume fractions.