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.

An asymptotic model is developed for the parametric subharmonic instability (PSI) of finite-width nearly monochromatic internal gravity wave beams in the presence of a background constant horizontal mean flow. The subharmonic perturbations are taken to be short-scale wavepackets that may extract energy via resonant triad interactions while in contact with the underlying beam, and the mean flow is assumed to be small so that its advection effect on the perturbations is as important as dispersion, triad nonlinearity and viscous dissipation. In this ‘distinguished limit’, the perturbation dynamics are governed by the same evolution equations as those derived in Karimi & Akylas (J. Fluid Mech., vol. 757, 2014, pp. 381–402), except for a mean flow term that affects the group velocity of the perturbations and imposes an additional necessary condition for PSI, which stabilizes very short-scale perturbations. As a result, it is possible for a small amount of mean flow to weaken PSI dramatically.

In turbulent Rayleigh–Bénard (RB) convection with regular, mono-scale, surface roughness, the scaling exponent $\unicode[STIX]{x1D6FD}$ in the relationship between the Nusselt number $Nu$ and the Rayleigh number $Ra$, $Nu\sim Ra^{\unicode[STIX]{x1D6FD}}$ can be ${\approx}1/2$ locally, provided that $Ra$ is large enough to ensure that the thermal boundary layer thickness $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ is comparable to the roughness height. However, at even larger $Ra$, $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ becomes thin enough to follow the irregular surface and $\unicode[STIX]{x1D6FD}$ saturates back to the value for smooth walls (Zhu et al., Phys. Rev. Lett., vol. 119, 2017, 154501). In this paper, we prevent this saturation by employing multiscale roughness. We perform direct numerical simulations of two-dimensional RB convection using an immersed boundary method to capture the rough plates. We find that, for rough boundaries that contain three distinct length scales, a scaling exponent of $\unicode[STIX]{x1D6FD}=0.49\pm 0.02$ can be sustained for at least three decades of $Ra$. The physical reason is that the threshold $Ra$ at which the scaling exponent $\unicode[STIX]{x1D6FD}$ saturates back to the smooth wall value is pushed to larger $Ra$, when the smaller roughness elements fully protrude through the thermal boundary layer. The multiscale roughness employed here may better resemble the irregular surfaces that are encountered in geophysical flows and in some industrial applications.

Several metres below the coastal ocean surface there are areas of high ecological activity that contain thin layers of concentrated motile phytoplankton. Gyrotactic trapping has been proposed as a potential mechanism for layer formation of bottom-heavy swimming algae cells, especially in flows where the vorticity varies linearly with depth (Durham et al., Science, vol. 323(5917), 2009, pp. 1067–1070). Using a continuum model for dilute microswimmer suspensions, we report that an instability of a gyrotactically trapped cell layer can arise in a pressure-driven plane channel flow. The linear stability analysis reveals that the equilibrium cell-layer solution is hydrodynamically unstable due to negative microswimmer buoyancy (i.e. a gravitational instability) over a range of biologically relevant parameter values. The critical cell concentration for this instability is found to be $N_{c}\simeq 10^{4}~\text{cells}~\text{cm}^{-3}$, a value comparable to the typical maximum cell concentration observed in thin layers. This result indicates that the instability may be a potential mechanism for limiting the layer’s maximum cell concentration, especially in regions where turbulence is weak, and motivates the study of its nonlinear evolution, perhaps, in the presence of turbulence.

An exact solution of a baroclinic three-dimensional vortex dipole in geophysical flows with constant background rotation and constant background stratification is provided under the quasi-geostrophic (QG) approximation. The motion of the dipole is unsteady but the potential vorticity contours move rigidly. The vortex comprises three potential vorticity anomaly modes, with a radial dependence given by the spherical Bessel functions and with azimuthal and polar dependences given by the spherical harmonics. The first mode, or spherical mode, accounts for the horizontal asymmetry of the vortex dipole and curvature of the dipole’s horizontal trajectory. The second mode, or dipolar mode, accounts for the speed of displacement of the vortex dipole. A third mode, or vertical tilting mode, accounts for the dipole’s vertical asymmetry. The QG vertical velocity field has two contributions: the first one is octupolar and depends entirely on the dipolar mode, and the second one is dipolar and depends on the nonlinear interaction between dipolar and vertical tilting modes.

A monochromatic, small amplitude, normally incident standing wave on a sloping beach is unstable to perturbation by subharmonic (half the frequency) edge waves. At equilibrium, edge wave shoreline amplitudes can exceed incident wave amplitudes. Here, the effect of incident wave randomness on subharmonic edge wave excitation is explored following a weakly nonlinear stability analysis under the assumption of narrow-band incident random waves. Edge waves respond to variations in both incident wave phase and amplitude, and the edge wave amplitudes and incident wave groups vary on similar time scales. When bottom friction is included, intermittent subharmonic edge wave excitation is predicted due to the combination of bottom friction and wave phase. Edge wave amplitude can be near zero for long times, but for short periods reaches relatively large values, similar to amplitudes with monochromatic incident waves and no friction.

We determine the smallest instantaneous increase in the strength of an opposing wind that is necessary to permanently reverse the forward displacement flow that is driven by a two-layer thermal stratification. With an interpretation in terms of the flow’s energetics, the results clarify why the ventilation of a confined space with a stably stratified buoyancy field is less susceptible to being permanently reversed by the wind than the ventilation of a space with a uniform buoyancy field. For large opposing wind strengths we derive analytical upper and lower bounds for the system’s marginal stability, which exhibit a good agreement with the exact solution, even for modest opposing wind strengths. The work extends a previous formulation of the problem (Lishman & Woods, Build. Environ., vol. 44 (4), 2009, pp. 666–673) by accounting for the transient dynamics and energetics associated with the homogenisation of the interior, which prove to play a significant role in buffering temporal variations in the wind.

Topology is introducing new tools for the study of fluid waves. The existence of unidirectional Yanai and Kelvin equatorial waves has been related to a topological invariant, the Chern number, that describes the winding of $f$-plane shallow water eigenmodes around band-crossing points in parameter space. In this previous study, the topological invariant was a property of the interface between two hemispheres. Here we ask whether a topological index can be assigned to each hemisphere. We show that this can be done if the shallow water model in the $f$-plane geometry is regularized by an additional odd-viscosity term. We then compute the spectrum of a shallow water model with a sharp equator separating two flat hemispheres, and recover the Kelvin and Yanai waves as two exponentially trapped waves along the equator, with all the other modes delocalized into the bulk. This model provides an exactly solvable example of bulk-interface correspondence in a flow with a sharp interface, and offers a topological interpretation for some of the transition modes described by Iga (J. Fluid Mech., vol. 294, 1995, pp. 367–390). It also paves the way towards a topological interpretation of coastal Kelvin waves along a boundary and, more generally, to an understanding of bulk-boundary correspondence in continuous media.

We present a conditional space–time proper orthogonal decomposition (POD) formulation that is tailored to the eduction of the average, rare or intermittent events from an ensemble of realizations of a fluid process. By construction, the resulting spatio-temporal modes are coherent in space and over a predefined finite time horizon, and optimally capture the variance, or energy of the ensemble. For the example of intermittent acoustic radiation from a turbulent jet, we introduce a conditional expectation operator that focuses on the loudest events, as measured by a pressure probe in the far field and contained in the tail of the pressure signal’s probability distribution. Applied to high-fidelity simulation data, the method identifies a statistically significant ‘prototype’, or average acoustic burst event that is tracked over time. Most notably, the burst event can be traced back to its precursor, which opens up the possibility of prediction of an imminent burst. We furthermore investigate the mechanism underlying the prototypical burst event using linear stability theory and find that its structure and evolution are accurately predicted by optimal transient growth theory. The jet-noise problem demonstrates that the conditional space–time POD formulation applies even for systems with probability distributions that are not heavy-tailed, i.e. for systems in which events overlap and occur in rapid succession.

A pressure impulse model is presented for wave impact on vertical circular cylinders. Pressure impulse is the time integral of the pressure during an impact of short time scale. The model is derived for a simplistic geometry and has relative impact height, crest length and cylinder radius as effective variables. The last parameter, the maximum angle of impact, is free and can be calibrated to yield the right force impulse. A progression of simpler pressure impulse models are derived in terms of a three-dimensional box generalization of the two-dimensional wall model and an axisymmetric model for vertical cylinders. The dependence on the model parameters is investigated in the simpler models and linked to the behaviour of the three-dimensional cylinder model. The model is next validated against numerical results for a wave impact for a phase- and direction-focused wave group. The maximum impact angle is determined by calibration against the force impulse. A good match of the pressure impulse fields is found. Further comparison to the force impulse of two common models in marine engineering reveals improved consistency for the present model. The model is found to provide a promising representation of the pressure impulse field, based on a limited number of input parameters. Its further validation and potential as a robust tool in force and response prediction is discussed.

We present a scaling theory that links the frequency of long frontal waves to the kinetic energy decay rate and inverse transfer of potential energy in freely evolving equivalent barotropic turbulence. The flow energy is predominantly potential, and the streamfunction makes the dominant contribution to potential vorticity (PV) over most of the domain, except near PV fronts of width $O(L_{D})$, where $L_{D}$ is the Rossby deformation length. These fronts bound large vortices within which PV is well-mixed and arranged into a staircase structure. The jets collocated with the fronts support long-wave undulations, which facilitate collisions and mergers between the mixed regions, implicating the frontal dynamics in the growth of potential-energy-containing flow features. Assuming the mixed regions grow self-similarly in time and using the dispersion relation for long frontal waves (Nycander et al., Phys. Fluids A, vol. 5, 1993, pp. 1089–1091) we predict that the total frontal length and kinetic energy decay like $t^{-1/3}$, while the length scale of the staircase vortices grows like $t^{1/3}$. High-resolution simulations confirm our predictions.

Results are reported from linear and nonlinear stability analyses of pulsatile flow in a straight channel with a smooth constriction with 50 % occlusion. Physical mechanisms based on the energy variation of the two-dimensional modes are proposed to elucidate the equilibrium states and changes in the flow-field behaviour. When investigating how the energy of the two-dimensional modes varied with Reynolds number, an energy minimum was observed before the primary instability, so that it marked the Reynolds number value at which the flow field started to exhibit one pulse front per cycle downstream of the constriction. In addition, this flow is characterized by a bistable system in two dimensions, whose equilibrium states are related to the energy level of the two-dimensional mode. The energy difference between the equilibrium states surprisingly decreases from the lower to the higher Reynolds number, which indicates that the system should converge to a unique solution at high Reynolds number. Regarding the flow’s nature, the equilibrium state with higher energy presented a vortex pair soon after the constriction, with vorticity opposite to that of the base flow separating shear layers.

The partial coalescence dynamics of a compound drop in a liquid pool is numerically investigated. We study the effect of the ratio of the inner to outer radii $(R_{r})$ of the compound drop while maintaining a constant liquid volume in the outer shell of the compound droplet. It is observed that for small values of the radius ratio, the coalescence dynamics is similar to that of a ‘simple’ drop, but the partial coalescence is suppressed for large values of $R_{r}$. Increasing the value of $R_{r}$ decreases the distance migrated by the inner bubble in the downward direction inside the pool. The location of the bubble after coalescence is found to play an important role in the pinch-off process of the satellite drop. The influence of the governing dimensionless parameters on the coalescence dynamics has also been investigated.