The velocity and pressure fields produced in a homogeneous rapidly rotating fluid driven by an isolated buoyant parcel are investigated. Gravity and rotation are allowed to have arbitrary orientations and the parcel shape is assumed Gaussian. Inertial forces and time-dependent effects are ignored. The linear problem is easily solved by three-dimensional Fourier transform, and the inversion is facilitated by assuming the Ekman number, E, to be very small. In this limit the fields form a Taylor column extended in the direction of the rotation axis. In the absence of rigid boundaries no boundary layers occur. The velocity and pressure in the vicinity of the parcel are found in closed form while elsewhere (within the Taylor column) they are expressed in terms of relatively simple scalar integrals which are easily evaluated.

Within the buoyant parcel, the momentum balance is baroclinic, involving Coriolis, pressure and buoyancy forces. Outside the parcel, the balance is geostrophic at unit order. The viscous force is important at order E and determines the axial structure of the Taylor column. In contrast to the case of flow driven by a rigid body, no ‘Taylor slug’ of recirculating flow occurs. The velocity and pressure decay algebraically with distance from the parcel, with the scale of variation being a/E in the axial direction and a in the radial direction, where a is the parcel radius. In the vicinity of the parcel, the return flow occurs in a broad region surrounding the parcel. The structure of flow in the vicinity of the parcel is independent of the Ekman number. This return flow sweeps the fringes of the parcel backward, making the net rise speed significantly slower than that of a rigid sphere of identical buoyancy. The return flow also acts to deform the parcel; this deformation is quantified.

This study investigates the sound produced when a jet, issued from a circular nozzle or hole in a plate, goes through a similar hole in a second plate. The sound, known as a hole tone, is encountered in many practical engineering situations. Direct computations of a hole tone feedback system were conducted. The mean velocity of the air jet was 10 m s−1. The nozzle and the end plate hole both had a diameter of 51 mm, and the impingement length between the nozzle and the end plate was 50 mm. The computational results agreed well with past experimental data in terms of qualitative vortical structures, the relationship between the most dominant hole tone peak frequency and the jet speed, and downstream growth of the mean jet profiles. Based on the computational results, the shear-layer impingement on the hole edge, the resulting propagation of pressure waves and the associated vortical structures are discussed. To extract dominant unsteady behaviours of the hole tone phenomena, a snapshot proper orthogonal decomposition (POD) analysis of pressure fluctuation fields was conducted. It was found that the pressure fluctuation fields and the time variation of mass flows through the end plate hole were dominantly expressed by the first and second POD modes, respectively. Integrating the computational results, an axisymmetric throttling mechanism linking mass flow rates through the hole, vortex impingement and global pressure propagation, is proposed.

We analyze the global linear stability of the axisymmetric flow around a spinning bullet-shaped body of length-to-diameter ratio $L/D=2$, as a function of the Reynolds number, $Re=\rho w_{\infty } D /\mu $, and of the rotation parameter $\varOmega =\omega D/(2 w_{\infty })$, in the ranges $Re<450$ and $0\leq \varOmega \leq 1$. Here, $w_{\infty }$ and $\omega $ are the free-stream and the body rotation velocities respectively, and $\rho $ and $\mu $ are the fluid density and viscosity. The two-dimensional eigenvalue problem (EVP) is solved numerically to find the spectrum of complex eigenvalues and their associated eigenfunctions, allowing us to explain the different bifurcations from the axisymmetric state observed in previous numerical studies. Our results reveal that, for the parameter ranges investigated herein, three global eigenmodes, denoted low-frequency (LF), medium-frequency (MF) and high-frequency (HF) modes, become unstable in different regions of the $(Re,\varOmega )$-parameter plane. We provide precise computations of the corresponding neutral curves, that divide the $(Re,\varOmega )$-plane into four different regions: the stable axisymmetric flow prevails for small enough values of $Re$ and $\varOmega $, while three different frozen states, where the wake structures co-rotate with the body at different angular velocities, take place as a consequence of the destabilization of the LF, MF and HF modes. Several direct numerical simulations (DNS) of the nonlinear state associated with the MF mode, identified here for the first time, are also reported to complement the linear stability results. Finally, we point out the important fact that, since the axisymmetric base flow is $SO(2)$-symmetric, the theory of equivariant bifurcations implies that the weakly nonlinear regimes that emerge close to criticality must necessarily take the form of rotating-wave states. These states, previously referred to as frozen wakes in the literature, are thus shown to result from the base-flow symmetry.

The motion induced in a layer of Boussinesq fluid by moving periodic thermal waves is obtained by numerically solving the complete nonlinear two-dimensional momentum and temperature equations. Three sets of boundary conditions are treated: rigid upper and lower boundaries with symmetrical heating; free upper boundary and rigid lower boundary with heating only at the top; free upper and lower boundaries with symmetrical heating. The nonlinear streamline patterns show that, when the velocity fluctuations are larger than the phase speed of the thermal wave and the mean flow, the convection cells have shapes governed by fluctuating nonlinear interactions. Significant mean velocities can be created even without the characteristic tilt in the convection cells expected on the basis of linear theory. Nonlinear interactions can lead to a mean shear even in the absence of motion of the thermal source. When the viscous diffusion time across the fluid layer is less than or of the same order as the period of the thermal wave, the order of magnitude of the induced mean velocity does not exceed that of the phase speed of the wave, even for intense thermal forcing.

We study the fluid dynamics of rolling wheels at Reynolds number (where is the Reynolds number based on the wheel diameter), with the objective of characterizing the various regimes of steady and unsteady motion. Regardless of the Reynolds number, the flow is found to separate approximately upstream of the apex of the wheel, where a saddle point in the pseudo-streamtrace pattern is observed. Under the flow conditions here essayed, the drag coefficient steadily decreases with , and the lift coefficient remains strictly positive. The positive lift provided by the rolling wheel is associated with the presence of a strong (positive) peak of the static pressure in the upstream proximity of the contact point with the ground, which we interpret as the result of the impingement of flow particles entrained in the boundary layer that develops on the front part of the wheel. Steady laminar flow is observed up to , which is characterized by a three-dimensional wake whose length increases with the Reynolds number. Unsteadiness is first observed at , under which conditions the flow retains planar symmetry, and is characterized by the quasi-periodic shedding of hairpin vortices. Transition to three-dimensional flow happens at , in which case a sinuous mode of instability in the wheel wake is established, which modulates the shedding of the hairpins, and which causes the onset of a non-zero side force. At the highest Reynolds number considered here () the wake exhibits some characters of turbulence, with wide-band frequency spectra, and its topology entirely changes, becoming split into two parts, and being much shortened compared to the lower- cases. Despite the limitation of the study to low Reynolds numbers we find that, once significant three-dimensionality and scale separation are established in the wheel wake, the nature of the flow becomes qualitatively similar to fully developed turbulent flow. In perspective, this observation opens interesting avenues for the prediction of unsteady flow around rotating tyres at moderate computational cost.

We consider the dynamical processes by which a dense, porous object floating in a body of liquid becomes waterlogged and sinks. We first generalize the classic model of capillary rise in a porous medium to present an analytically tractable model of the process, which is valid for objects that are very shallow compared to their horizontal extent. We find an analytical expression for the time taken for the object to sink under this approximation. We use a series of boundary integral simulations to show that decreasing the horizontal extent of the object decreases the time taken to sink. We find that the results of these numerical simulations are in good quantitative agreement with a series of laboratory experiments. Finally, we discuss the implications of our work for pumice fragments, often found floating in open water after a volcanic eruption, occasionally even supporting plant, animal and human remains.

We consider the effect of high rotation rates on two liquid layers that initially form concentric cylinders, centred on the axis of rotation. The configuration may be thought of as a fluid–fluid centrifuge. There are two types of perturbation to the interface that may be considered, an azimuthal perturbation around the circumference of the interface and a varicose perturbation in the axial direction along the length of the interface. It is the first of these types of perturbation that we consider here, and so the flow may be considered essentially two-dimensional, taking place in a circular domain. A linear stability analysis is carried out on a perturbation to the hydrostatic background state and a fourth-order Orr–Sommerfeld-like equation that governs the system is derived. We consider the dynamics of systems of stable and unstable configurations, inviscid and viscous fluids, immiscible fluid layers with surface tension and miscible fluid layers that may have some initial diffusion of density. In the most simple case of two layers of inviscid fluid separated by a sharp interface with no surface tension acting, we show that the effects of the curvature of the interface and the confinement of the system may be characterised by a modified Atwood number. The classical Atwood number is recovered in the limit of high azimuthal wavenumber, or the outer fluid layer being unconfined. Theoretical predictions are compared with numerical experiments and the agreement is shown to be good. We do not restrict our analysis to equal volume fluid layers and so our results also have applications in coating and lubrication problems in rapidly rotating systems and machinery.

Corrected relationships are presented between integral properties of periodic gravity surface waves in the case of a non-zero mean Eulerian velocity. Also considered is the special case of a reference frame in which the mean wave momentum vanishes.

In this paper a detailed analysis of undular bore dynamics in channels of variable cross-section is presented. Two undular bore regimes, low Froude number (LFN) and high Froude number (HFN), are simulated with a Serre–Green–Naghdi model, and the results are compared with the experiments by Treske (1994). We show that contrary to Favre waves and HFN bores, which are controlled by dispersive non-hydrostatic mechanisms, LFN bores correspond to a hydrostatic phenomenon. The dispersive-like properties of the LFN bores is related to wave refraction on the banks in a way similar to that of edge waves in the near shore. A fully hydrostatic asymptotic model for these dispersive-like bores is derived and compared to the observations, confirming our claim.

We numerically solve the nonlinear two-fluid Hall–Vinen–Bekharevich–Khalatnikov (HVBK) equations for superfluid helium confined inside a short Couette annulus. The outer cylinder and the ends of the annulus are held fixed whilst the inner cylinder is rotated. This simple flow configuration allows us to study how the vortex lines respond to a shear in the presence of boundaries. It also allows us to investigate further the boundary conditions associated with the HVBK model. The main result of our investigation is the anomalous motion of helium II when compared to a classical fluid. The superfluid Ekman cells always rotate in the opposite sense to a classical Navier–Stokes fluid due to the mutual friction between the two fluids, whilst the sense of rotation of the normal fluid Ekman cells depends on the parameter range considered. We also find that the tension of the vortex lines forces the superfluid to rotate about the inner cylinder almost like a rigid column.

The flow field around pairs of small particles moving and rotating in a shear flow close to a wall at low but finite Reynolds number (Re) is computed as a function of time by means of the lattice-Boltzmann technique. The total force and torque acting on each particle is computed at each time step and the position of the particles is updated. By considering the lift force and the disturbances induced by the particles, the trajectories of the pair of particles are explained as a function of the distances from the wall and the Reynolds number. It is shown that when particles are positioned in a particular form, they collide forming strings. In particular, we are interested in particle-bridge formation in shear flows, and two collided particles (a string) can be considered as a nucleus of a particle bridge.

We consider the two-dimensional inviscid flow that occurs when a fluid, initially at rest around a slender wedge-shaped void, is allowed to recoil under the action of surface tension. As noted by Keller & Miksis (1983), a similarity scaling is available, with lengths scaling like $t^{2/3}$. We find that an asymptotic balance is possible when the wedge semi-angle, $\alpha$, is small, in an inner region of $O(\alpha^{4/9})$, a distance of $O(\alpha^{-2/9})$ from the origin, which leads to a simpler boundary value problem at leading order. Although we are able to reformulate the inner problem in terms of a complex potential and reduce it to a single nonlinear integral equation, we are unable to find a solution numerically. This is because, as noted by Vanden-Broeck, Keller & Milewski (2000), and reproduced here numerically using a boundary integral method, the free surface is self-intersecting for $\alpha\,{<}\, \alpha_0\,{\approx}\,2.87^\circ$. Since disconnected solutions are only possible when there is a void inside the initial wedge, we consider the effect of an inviscid low-density fluid inside the wedge. In this case, a solution is available for a slightly smaller range of wedge semi-angles, since the flow of the interior fluid sucks the free surfaces together, with the exterior flow seeing pinch-off at a finite angle.

We conclude that for $\alpha\,{<}\,\alpha_0$, we must introduce the effect of viscosity at small times in order to regularize the initial value problem. Since the solution for $t\,{\ll}\,1$ in the presence of viscosity is simply connected (Billingham 2005), the free surface must first pinch off at some finite time and then continue to do so at a sequence of later times. We investigate this using boundary integral solutions of the full inviscid initial value problem, with smooth initial conditions close to those of the original problem. In addition, we show that the inner asymptotic scalings that we developed for the steady problem can also be used in this time-dependent problem. The unsteady inner equations reduce to those for steady unidirectional flow outside a region of constant pressure, and can be solved numerically.

We also show how the slender wedge solution can be related to the small-time behaviour of two coalescing drops, and describe the relationship between our solutions and those found by Duchemin, Eggers & Josserand (2003), for which a similar unsteady inner region exists. In each case, the free surface pinches off repeatedly, and no similarity solution exists.

We explore the evolution of the gravest internal Kelvin wave in a two-layer rotating cylindrical basin, using direct numerical simulations (DNS) with a hyper-viscosity/diffusion approach to illustrate different dynamic and energetic regimes. The initial condition is derived from Csanady’s (J. Geophys. Res., vol. 72, 1967, pp. 4151–4162) conceptual model, which is adapted by allowing molecular diffusion to smooth the discontinuous idealized solution over a transition scale, ${\it\delta}_{i}$, taken to be small compared to both layer thicknesses $h_{\ell },\ell =1,2$. The different regimes are obtained by varying the initial wave amplitude, ${\it\eta}_{0}$, for the same stratification and rotation. Increasing ${\it\eta}_{0}$ increases both the tendency for wave steepening and the shear in the vicinity of the density interface. We present results across several regimes: from the damped, linear–laminar regime (DLR), for which ${\it\eta}_{0}\sim {\it\delta}_{i}$ and the Kelvin wave retains its linear character, to the nonlinear–turbulent transition regime (TR), for which the amplitude ${\it\eta}_{0}$ approaches the thickness of the (thinner) upper layer $h_{1}$, and nonlinearity and dispersion become significant, leading to hydrodynamic instabilities at the interface. In the TR, localized turbulent patches are produced by Kelvin wave breaking, i.e. shear and convective instabilities that occur at the front and tail of energetic waves within an internal Rossby radius of deformation from the boundary. The mixing and dissipation associated with the patches are characterized in terms of dimensionless turbulence intensity parameters that quantify the locally elevated dissipation rates of kinetic energy and buoyancy variance.

Oceanic internal waves forced by a latitude-independent wind field travelling eastward at speed U is investigated, extending the hydrostatic f-plane model of Kundu & Thomson (1985). The ocean has a well-mixed surface layer overlying a stratified interior with a depth-dependent buoyancy frequency N(z), and f can vary with latitude. Solutions are found by decomposition into vertical normal modes. Problems discussed are (i) the response to a slowly moving line front, and (ii) the response in a variable f ocean.

For the slowly moving line front assuming a depth-independent N, the trailing waves are found to have large frequencies, and the vertical acceleration ∂w/∂t is important (that is the dynamics are non-hydrostatic) if the frequency ω is larger than a few times (Nf)½. The wake contains waves associated with all vertical modes, in contrast to hydrostatic dynamics in which slowly moving line fronts do not generate trailing waves of low-order modes. It is argued that slowly moving wind fields can provide an explanation for the frequently observed broad peak in the spectrum of vertical motion at a frequency somewhat smaller than N, and of the vertical coherence of the associated waves in the upper ocean.

To study lower-frequency internal waves, the hydrostatic constant-f model of Kundu & Thomson is extended to variable f. Various sections through such a flow clearly illustrate the development of a meridional wavelength λy = 2π/βt as predicted by D'Asaro (1989), in addition to the zonal wavelength λx due to translation of the wind. The two effects combine to cause a greater horizontal inhomogeneity, so that energy from the surface layer descends quickly, travelling equatorward and downward. Since waves at any point arrive from different latitudes, spectra no longer consist of discrete peaks but are more continuous and broader than those in the constant-f model. The waves are more intermittent because of the larger spectral width, and vertically less correlated in the thermocline because of a larger bandwidth of vertical modes. The vertical correlation in the deep ocean, however, is still high because the response is dominated by one or two low-order modes after 30 days of integration. As U decreases, the larger bandwidth of frequency increases the intermittency, and the larger bandwidth of vertical wavenumber decreases the vertical correlation. A superposition of travelling wind events intensifies the high-frequency end of the spectrum; a month-long travelling series of realistic strength can generate waves with amplitudes of order 4 cm/s in the deep ocean.

It is suggested that propagating winds and linear dynamics are responsible for the generation of a large fraction of internal waves in the ocean at all depths. The main effect of nonlinearity and mean flow may be to shape the internal wave spectra to a ω-2 form.

Branching tube flows are examined, for one mother to two, three or more daughter tubes. The case of many daughters (abrupt multi-branching) models blood flow through an arteriovenous malformation in the brain, while that of very few daughters (gradual branching) applies elsewhere in physiology and surgical grafting, as well as other applications including industrial ones. Theory and computation are presented for two- and three-dimensional motions, under the viscous and inviscid effects of small changes in mass flux between the daughter tubes, area expansion and turning of the flow. Specific configurations for which flow solutions are obtained are (a) with two large daughters, (b) with one small daughter/side branch, and (c) with multiple small daughters.

The numerous physical mechanisms acting concern overall upstream influence and through-flow, and flow separation and criteria for its avoidance, as well as criteria for the amount of turning and area expansion possible without energy loss and other factors associated with separation, and the role of the branching geometry versus that of the mass-flux distribution in the daughters. In particular, configuration (a) allows substantial separation-free turning and expansion only with certain shaping of the outer wall and an area expansion ratio typically less than 1.2, whereas more daughters involve a balance between geometry and mass flux. In (b), an abrupt pressure jump is induced at the mouth of the small daughter, near which mass-flux effects tend to dominate over geometrical shaping effects. In (c), as the number of daughters increases, the amount of separation-free turning and expansion is found to increase substantially, and the distributed mass-flux influence readily overrides the geometrical influence throughout the branching; there is also an integrated upstream effect of the multi-branching on the incident mother flow even though each daughter flow acts as if independent. Tentative designs based on wall shaping, flux distributions and divider placement are considered for flow improvement/surgery.

A laboratory investigation has been undertaken to quantify water wave attenuation rates as a function of rainfall rate. Vertical artificial rainfall is shown to generate weak near-surface velocity fluctuations that decline systematically away from the free surface and are independent of rainfall rate across the range of rainfall rates investigated (40–$170~\mathrm{mm} ~{\mathrm{h} }^{- 1} $). In the absence of rain, the observed attenuation of gravity waves is at levels consistent with classical viscous theory, but with a systematic finite-amplitude effect observed above a mean steepness of 0.10. Wave attenuation rates were found to be independent of the mean wave steepness and identical when artificial rainfall rates of 108 and $141~\mathrm{mm} ~{\mathrm{h} }^{- 1} $ were applied. Reassessment of complementary theoretical and experimental studies of individual droplets impacting on undisturbed water surfaces indicates that above a weak threshold rainfall rate of $30~\mathrm{mm} ~{\mathrm{h} }^{- 1} $, the surface irradiation becomes so frequent that droplet-generated violent surface motions directly interact with the incoming droplets. Present evidence is that a matching of time scales develops between the incoming surface irradiation and surface water motions generated by antecedent droplets as the rainfall rate increases. Consequently, at high rainfall rates, a highly dissipative surface regime is created that transmits little of the incident rainfall kinetic energy to the aqueous layers below. Rainfall-induced wave attenuation rates are compared with measurements of other wave attenuation processes to obtain a hierarchy of strength in both the laboratory and the field. Comparison is also made with wave attenuation theories that incorporate momentum and energy flux considerations. Rain-induced wave attenuation rates are weak or very strong depending on whether they are expressed in terms of energy scaling obtained from above or below the surface respectively, due to the high dissipation rate that occurs in the vicinity of the interface.

The flow inside a square cavity, generated by a uniform oscillatory motion of the upper cavity wall, is investigated. Numerical solutions of the full unsteady Navier–Stokes equations are computed for a selection of Reynolds’ numbers and frequency parameters in the range 0–600. These results show that the steady streaming component of the flow, in general, comprises two main, equal, contrarotating eddies, situated in the upper half of the cavity (one on either side of the cavity centreline) and two weaker, equal, contrarotating eddies, situated in the lower section of the cavity (again one on either side of the cavity centreline).

The boundary layer that forms on the walls of a shock tube, after the diaphragm which initially separates two gases at different pressures is burst, is investigated. Both the driver and driven gases are assumed to have the same thermal properties and the shock tube wall is maintained at constant temperature. Crocco variables are used and a method is presented for solving the compressible boundary-layer equations within the tube in similarity variables. Three cases, corresponding to different initial pressure ratios of the driver and driven gases, are calculated which are representative of weak and medium-strength shock and expansion waves.