We perform direct numerical simulations of a rigid sphere translating parallel to a flat wall in an otherwise quiescent ambient fluid. A spectral element method is employed to perform the simulations with high accuracy. For $Re\,{<}\,100$, we observe the lift coefficient to decrease with both Reynolds number and distance from the wall. In this regime the present results are in good agreement with the low-Reynolds-number theory of Vasseur & Cox (1977), with the recent experiments of Takemura & Magnaudet (2003) and with the simulations of Kim et al. (1993). The most surprising result from the present simulations is that the wall-induced lift coefficient increases dramatically with increasing $Re$ above about 100. Detailed analysis of the flow field around the sphere suggests that this increase is due to an imperfect bifurcation resulting in the formation of a double-threaded wake vortical structure. In addition to a non-rotating sphere, we also simulate a freely rotating sphere in order to assess the importance of free rotation on the translational motion of the sphere. We observe the effect of sphere rotation on lift and drag forces to be small. We also explore the effect of the wall on the onset of unsteadiness.

In spite of its practical relevance, little is known about the turbulence characteristics in sharp open-channel bends, which may largely be attributed to a lack of accurate experimental data. This paper reports an experimental investigation of the turbulence structure in one cross-section of an open-channel bend. The flow pattern in this section is characterized by a bicellular pattern of cross-stream circulation (secondary flow) and, in the outer part, a strongly reduced turbulence activity as compared with straight uniform shear flow. The turbulence structure differs fundamentally from that in straight uniform shear flow. The velocity fluctuations are atypically coherent over the channel width, whence the measured signal is decomposed into slow width-coherent fluctuations and a fast background signal. The width-coherent fluctuations reflect a bulk spatio-temporal oscillation of the pattern of circulation cells whereas the background signal represents developed turbulence. A spectral analysis shows that the width-coherent fluctuations have the characteristics of a wavelike motion, i.e. they contribute significantly to the turbulent normal stresses but only weakly to the shear stress, whereas the background turbulence is characterized by efficient shear stress generation. The reduced turbulence activity and the tendency of the secondary-flow pattern to oscillate are both effects of the streamline curvature. Similar observations on reduced turbulence activity and the tendency to wavelike motion have been reported in literature for flows in curved wind tunnels and density-stratified flows. Our experimental results indicate that these phenomena are potentially important in curved open-channel flows, where they affect the mixing and transport capacity of the flow.

Self-similar plane solutions for the inertial stage of gravity currents are related to the initial parameters and a coefficient that is determined by the boundary condition at the front. Different relations have been proposed for the boundary condition in terms of a Froude number at the front, none of which have a sound theoretical or experimental basis. This paper focuses on considerations of the appropriate Froude number based on results of lock-exchange experiments in which extended inertial gravity currents are generated in a rectangular cross-section channel. We use ‘top-hat’ vertical density profiles of the currents to obtain an ‘equivalent’ depth, defined by profiles having the same buoyancy at every position as the real profiles. As in previous work, our experimental results show that in the initial constant-velocity phase the Froude number can be defined in terms of the lock depth. However, as the current enters the similarity phase when the initial release conditions are no longer relevant, we find that the Froude number is more appropriately defined in terms of the maximum height of the head. Strictly speaking, the self-similar solution to the shallow-water equations requires a front condition that uses the height at the rear of the head. We find that this rear Froude number is not constant and is a function of the head Reynolds number over the range 400–4500.

The propagation of a liquid-filled crack from an over-pressured source into a semi-infinite uniform elastic solid is studied. The fluid is lighter than the solid and propagates due to its buoyancy and to the source over-pressure. The role of this over-pressure at early and late times is considered and it is found that the combination of buoyancy and over-pressure leads to significantly different behaviour from buoyancy or over-pressure alone. Lubrication theory is used to describe the flow, where the pressure in the fluid is determined by the elastic deformation of the solid due to the presence of the crack. Numerical results for the evolution of the crack shape and speed are obtained. The crack grows exponentially at early times, but at later times, when buoyancy becomes important, the crack growth accelerates towards a finite-time blow-up. These results are explained by asymptotic similarity solutions for early and late times. The predictions of these solutions are in close agreement with the full numerical results. A different case of crack geometry is also considered in order to highlight connections with previous work. The geological application to magma-filled cracks in the Earth's crust, or dykes, is discussed.

Steady three-dimensional fully nonlinear gravity–capillary solitary waves are calculated numerically in infinite depth. These waves have decaying oscillations in the direction of propagation and monotone decay perpendicular to the direction of propagation. They travel at a velocity $U$ smaller than the minimum velocity $c_{min}$ of linear gravity–capillary waves. It is shown that the structure of the solutions in three dimensions is similar to that found by Vanden-Broeck & Dias (J. Fluid Mech. vol. 240, 1992, pp. 549–557) for the corresponding two-dimensional problem.

To study the waves and runup/rundown generated by a sliding mass, a numerical simulation model, based on the large-eddy-simulation (LES) approach, was developed. The Smagorinsky subgrid scale model was employed to provide turbulence dissipation and the volume of fluid (VOF) method was used to track the free surface and shoreline movements. A numerical algorithm for describing the motion of the sliding mass was also implemented.

To validate the numerical model, we conducted a set of large-scale experiments in a wave tank of 104m long, 3.7m wide and 4.6m deep with a plane slope (1:2) located at one end of the tank. A freely sliding wedge with two orientations and a hemisphere were used to represent landslides. Their initial positions ranged from totally aerial to fully submerged, and the slide mass was also varied over a wide range. The slides were instrumented to provide position and velocity time histories. The time-histories of water surface and the runup at a number of locations were measured.

Comparisons between the numerical results and experimental data are presented only for wedge shape slides. Very good agreement is shown for the time histories of runup and generated waves. The detailed three-dimensional complex flow patterns, free surface and shoreline deformations are further illustrated by the numerical results. The maximum runup heights are presented as a function of the initial elevation and the specific weight of the slide. The effects of the wave tank width on the maximum runup are also discussed.

We present high-precision measurements of the Nusselt number $\cal{N}$ as a function of the Rayleigh number $R$ for cylindrical samples of water (Prandtl number $\sigma \,{=}\, 4.38$) with diameters $D \,{=}\, 49.7, 24.8,$ and 9.2cm, all with aspect ratio $\Gamma \,{\equiv}\, D/L \,{\simeq}\,1$ ($L$ is the sample height). In addition, we present data for $D \,{=}\, 49.7$ and $\Gamma \,{=}\, 1.5, 2, 3,$ and 6. For each sample the data cover a range of a little over a decade of $R$. For $\Gamma \,{\simeq}\,1$ they jointly span the range $10^7 \lesssim R \lesssim 10^{11}$. Where needed, the data were corrected for the influence of the finite conductivity of the top and bottom plates and of the sidewalls on the heat transport in the fluid to obtain estimates of $\cal{N}_{\infty}$ for plates with infinite conductivity and sidewalls of zero conductivity. For $\Gamma \,{\simeq}\,1$ the effective exponent $\gamma_{\hbox{\scriptsize\it eff}}$ of ${\cal N}_{\infty} \,{=}\, N_0 R^{\gamma_{\hbox{\scriptsize\it eff}}}$ ranges from 0.28 near $R \,{=}\, 10^8$ to 0.333 near $R \,{\simeq}\,7\,{\times}\,10^{10}$. For $R \lesssim 10^{10}$ the results are consistent with the Grossmann–Lohse model. For larger $R$, where the data indicate that ${\cal N}_\infty(R) \,{\sim}\, R^{1/3}$, the theory has a smaller $\gamma_{\hbox{\scriptsize\it eff}}$ than $1/3$ and falls below the data. The data for $\Gamma \,{>}\, 1$ are only a few percent smaller than the $\Gamma \,{=}\, 1$ results.

We consider perhaps the simplest non-trivial problem in advection–diffusion – a finite absorber of arbitrary cross-section in a steady two-dimensional potential flow of concentrated fluid. This problem has been studied extensively in the theory of solidification from a flowing melt, and it also arises in advection–diffusion-limited aggregation. In both cases, the fundamental object is the flux to a circular disk, obtained by conformal mapping from more complicated shapes. Here, we construct an accurate numerical solution by an efficient method that involves mapping to the interior of the disk and using a spectral method in polar coordinates. The method combines exact asymptotics and an adaptive mesh to handle boundary layers. Starting from a well-known integral equation in streamline coordinates, we also derive high-order asymptotic expansions for high and low Péclet numbers ($Pe$). Remarkably, the ‘high’$Pe$ expansion remains accurate even for such low $Pe$ as $10^{-3}$. The two expansions overlap well near $Pe\,{=}\,0.1$, allowing the construction of an analytical connection formula that is uniformly accurate for all $Pe$ and angles on the disk with a maximum relative error of 1.75%. We also obtain an analytical formula for the Nusselt number ($Nu$) as a function of $Pe$ with a maximum relative error of 0.53% for all possible geometries after conformal mapping. Considering the concentration disturbance around a disk, we find that the crossover from a diffusive cloud (at low $Pe$) to an advective wake (at high $Pe$) occurs at $Pe\,{\approx}\,60$.

The dynamics of flexible parachute canopies and vortex shedding in their near wake are studied experimentally in a water tunnel. The velocity field was measured by particle image velocimetry for two different canopy diameters. The periodic oscillation of the canopy diameter about a mean value which is referred to as ‘breathing’ has a non-dimensional frequency, based on the free-stream velocity and the mean canopy projected diameter, of approximately 0.55 for the range of Reynolds numbers examined. The dimensionless breathing frequency observed in the experiments is consistent with the values for larger canopies. The shear layer emanating from the canopy rolls up and sheds symmetric vortex rings. The frequency of vortex shedding was measured to be the same as the canopy breathing frequency. This Strouhal number is unique in the sense that it is much higher than those associated with rigid axisymmetric bluff bodies such as disks and spheres. The canopy breathing is shown to stem from the cyclical variation of suction pressure, resulting from the passage of vortex rings, on the exterior surface of the canopy. The added mass associated with the breathing of the canopy is found to be accountable for up to 40% of the canopy drag fluctuations in the range of parameters investigated.

Temporal linear stability modes depending on two space directions are computed for a two-dimensional boundary-layer flow along a flat plate. The spatial structure of each individual temporally stable mode is shown to be reminiscent of the spatial exponential growth of perturbations along the flat plate, as predicted by local analyses. It is shown using an optimal temporal growth analysis, that an appropriate superposition of a moderate number of temporal modes gives rise to a spatially localized wave packet, starting at inflow and exhibiting transient temporal growth when evolving downstream along the plate. This wave packet is in qualitative agreement with the convectively unstable disturbance observed when solving the Navier–Stokes equations for an equivalent initial condition.

It has been recognized that particle inertia throws dense particles out of regions of high vorticity and leads to an accumulation of particles in the straining-flow regions of a turbulent flow field. However, recent direct numerical simulations (DNS) indicate that the tendency to cluster is evident even at particle separations smaller than the size of the smallest eddy. Indeed, the particle radial distribution function (RDF), an important measure of clustering, increases as an inverse power of the interparticle separation for separations much smaller than the Kolmogorov length scale. Motivated by this observation, we have developed an analytical theory to predict the RDF in a turbulent flow for particles with a small, but non-zero Stokes number. Here, the Stokes number ($\hbox{\it St}$) is the ratio of the particle's viscous relaxation time to the Kolmogorov time. The theory approximates the turbulent flow in a reference frame following an aerosol particle as a local linear flow field with a velocity gradient tensor and acceleration that vary stochastically in time. In monodisperse suspensions, the power-law dependence of the pair probability is seen to arise from a balance of an inward drift caused by the particles' inertia that scales linearly with the particle separation distance and a pairwise diffusion owing to the random nature of the flow with a diffusivity that scales quadratically with the particle separation distance. The combined effect leads to a power law behaviour for the RDF with an exponent, $c_1$, that is proportional to $\hbox{\it St}^2$. Predictions of the analytical theory are compared with two types of numerical simulation: (i) particle pairs interacting in a local linear flow whose velocity varies according to a stochastic velocity gradient model; (ii) particles interacting in a flow field obtained from DNS of isotropic turbulence. The agreement with both types of simulation is very good. The theory also predicts the RDF for unlike particle pairs (particle pairs with different Stokes numbers). In this case, a second diffusion process occurs owing to the difference in the response of the pair to local fluid accelerations. The acceleration diffusivity is independent of the pair separation distance; thus, the RDF of particles with even slightly different viscous relaxation times undergoes a transition from the power law behaviour at large separations to a constant value at sufficiently small separations. The radial separation corresponding to the transition between these two behaviours is predicted to be proportional to the difference between the Stokes numbers of the two particles. Once again, the agreement between the theory and simulations is found to be very good. Clustering of particles enhances their rate of coagulation or coalescence. The theory and linear flow simulations are used to obtain predictions for the rate of coagulation of particles in the absence of hydrodynamic and colloidal particle interactions.

The generation of a gravity current by the release of a semi-infinite region of buoyant fluid of depth $H$ overlying a deeper, denser and quiescent lower layer in a rotating channel of width $w$ is considered. Previous studies have focused on the characteristics of the gravity current head region and produced relations for the gravity current speed $c_{b}$ and width $w_b$ as a functions of the local current depth along the wall $h_b$, reduced gravity $g^\prime$, and Coriolis frequency $f$. Here, the dam-break problem is solved analytically by the method of characteristics assuming reduced-gravity flow, uniform potential vorticity and a semigeostrophic balance. The solution makes use of a local gravity current speed relation $c_{b} \,{=}\, c_b(h_b,\ldots)$ and a continuity constraint at the head to close the problem. The initial value solution links the local gravity current properties to the initiating dam-break conditions. The flow downstream of the dam consists of a rarefaction joined to a uniform gravity current with width $w_b$ (${\le}\, w$) and depth on the right-hand wall of $h_b$, terminated at the head moving at speed $c_b$. The solution gives $h_b$, $c_b$, $w_b$ and the transport of the boundary current as functions of $w/L_R$, where $L_R \,{=}\, \sqrt{g^\prime H}/f$ is the deformation radius. The semigeostrophic solution compares favourably with numerical solutions of a single-layer shallow-water model that internally develops a leading bore. Existing laboratory experiments are re-analysed and some new experiments are undertaken. Comparisons are also made with a three-dimensional shallow-water model. These show that lateral boundary friction is the primary reason for differences between the experiments and the semigeostrophic theory. The wall no-slip condition is identified as the primary cause of the experimentally observed decrease in gravity current speed with time. A model for the viscous decay is developed and shown to agree with both experimental and numerical model data.

The effect of initial conditions on the growth rate of turbulent Rayleigh–Taylor (RT) mixing has been studied using carefully formulated numerical simulations. A monotone integrated large-eddy simulation (MILES) using a finite-volume technique was employed to solve the three-dimensional incompressible Euler equations with numerical dissipation. The initial conditions were chosen to test the dependence of the RT growth coefficient ($\alpha_{b})$ and the self-similar parameter ($\beta_{b}\,{=}\,\lambda_{b}/h_{b})$ on (i) the amplitude, (ii) the spectral shape, (iii) the longest wavelength imposed, and (iv) mode-coupling effects. With long wavelengths present in the initial conditions, $\alpha _{b}$ was found to increase logarithmically with the initial amplitudes, while $\beta_{b}$ is less sensitive to amplitude variations. The simulations are in reasonable agreement with the predictions for $\alpha_{b}$ from a recently proposed model, but not for $\beta_{b}$. In the opposite limit where mode-coupling dominates, no such dependence on initial amplitudes is observed, and $\alpha_{b}$ takes a universal lower-bound value of ${\sim}\,0.03\,{\pm}\,0.003$. This may explain the low values of $\alpha _{b}$ reported by most numerical simulations that are initialized with annular spectra of short-wavelength modes and hence evolve purely through mode-coupling. Small-scale effects such as molecular mixing and kinetic energy dissipation showed a weak dependence on the structure of initial conditions. Initial density spectra with amplitudes distributed as $k^{0}$, $k^{-1}$ and $k^{-2}$ were used to investigate the role of the spectral slopes on the development of turbulent RT mixing. Furthermore, in a separate study, the longest wavelength imposed in the initial wavepacket was also varied to determine its effect on $\alpha_{b}$. It was found that the slopes of the initial spectra, and the longest wavelength imposed had little effect on the RT growth parameters.

Particle image velocimetry (PIV) has been applied to the study of acoustic flow of liquid in a standing wave tube. Even though liquid compressibility is very small, the liquid must be treated as compressible in this case. With the finite compressibility of liquid in mind, a series of different standing wave modes can be formed by pressure waves emanated at specific driving frequencies from a bimorph piezo disk at the end of the tube. In this paper, the first three natural standing wave modes were visualized using 1 μm diameter fluorescent microspheres seeded in the liquid. The variation of the flow field in the acoustic boundary layer near the wall was measured using PIV. Water was first used as a working fluid. Experiments were then carried out with a glycerol–water mixture (50%–50% by volume) to examine the effect of viscosity change on the wave propagation and flow structure inside the tube. The experimental results are compared with theoretical model predictions.

This paper presents an investigation of the nonlinear steepening of a gasdynamic disturbance propagating in a steady non-uniform base flow. The base flow is the steady compressible flow of a gas in a variable-area duct. The quasi-one-dimensional continuity, momentum and energy equations for the unsteady disturbance in homentropic flow are solved using the method of characteristics (wave front expansion technique). A closed-form solution for the slope of the disturbance at the wave front is obtained. The solution admits singularity for a compressive disturbance, which is responsible for the formation of shock in the flow. The solution is general and is applicable in any range of Mach number of the base flow. A special case of the steady gas flow in a convergent–divergent duct (C-D nozzle), where the flow makes a transition from subsonic to supersonic and vice versa, is investigated.

I calculate the optimal upper bound, subject to the assumption of streamwise invariance, on the long-time-averaged buoyancy flux ${\cal B}^*$ within the flow of an incompressible stratified viscous fluid of constant kinematic viscosity $\nu$ and depth $h$ driven by a constant surface stress $\tau\,{=}\,\rho u^2_\star$, where $u_\star$ is the friction velocity with a constant statically stable density difference $\Delta \rho$ maintained across the layer. By using the variational ‘background method’ (due to Constantin, Doering and Hopf) and numerical continuation, I generate a rigorous upper bound on the buoyancy flux for arbitrary Grashof numbers $G$, where $G\,{=}\,\tau h^2/(\rho \nu^2)$. As $G \,{\rightarrow}\, \infty$, for flows where horizontal mean momentum balance, horizontally averaged heat balance, total power balance and total entropy flux balance are imposed as constraints, I show numerically that the best possible upper bound for the buoyancy flux is given by ${\cal B}^* \,{\leq}\, {\cal B}^*_{\hbox{\scriptsize max}}\,{=}\,u_{\star}^4/(4\nu)+ O(u_{\star}^3/h)$. This bound is independent of both the overall strength of the stratification and the layer depth to leading order. This bound is associated with a velocity profile that has the scaling characteristics of a somewhat decelerated laminar, linear velocity profile.

The effect of angle of attack on the acoustic receptivity of the boundary layer over two-dimensional parabolic bodies is investigated using a spatial solution of the Navier–Stokes equations. The free stream is decomposed into a uniform flow with a superposed periodic velocity fluctuation of small amplitude. The method follows that of Haddad & Corke (1998) and Erturk & Corke (2001) in which the solution for the basic flow and linearized perturbation flow are solved separately. Different angles of incidence of the body are investigated for three leading-edge radii Reynolds numbers. For each, the angle of attack ranges from $0^{\circ}$ to past the angle where the mean flow separates. The results then document the effect of the angle of incidence on the leading-edge receptivity coefficient ($K_{{\hbox{\scriptsize{\it LE}}}}$), and in the case of the mean flow separation, on the amplitude of Tollmien–Schlichting (T-S) waves at the linear stability Branch II location ($K_{II}$). For angles of attack before separation, we found that the leading-edge receptivity coefficient, $K_{{\hbox{\scriptsize{\it LE}}}}$, increased with angle of incidence which correlated with an increase in the pressure gradient at the physical leading edge. When a separation zone formed at larger angles of incidence, it became a second site of receptivity with a receptivity coefficient that exceeded that of the leading edge. This resulted in dramatic growth of the T-S waves with Branch II amplitudes more than 100 times larger than those at angles just before separation, and 1000 times more than those at $0^{\circ}$ angle of attack.

A vertically standing freely-rotating ellipsoidal vortex of uniform anomalous potential vorticity in a rotating stratified fluid under quasi-geostrophic conditions of small Rossby and Froude numbers steadily rotates without change of form. The vortex can have arbitrary axis lengths, but must have one axis parallel to the vertical $z$-axis along the direction of gravity. The rotation rate is proportional to the potential vorticity anomaly but otherwise depends on only two independent aspect ratios characterizing the shape of the vortex. The linear stability of this class of vortex equilibria was first determined semi-analytically more than a decade ago. It was found that vortices are unstable over a wide range of the parameter space and are stable only when strongly oblate and of nearly circular cross-section.

New results, presented here, using a complementary approach and backed by nonlinear simulations of the full quasi-geostrophic equations indicate that these ellipsoidal vortices are in fact stable over a much wider range of parameter space. In particular, a mode previously thought to be unstable over much of the parameter space is evidently stable. Moreover, we have determined that this mode is just the difference between two neighbouring equilibrium states having slightly different horizontal aspect ratios; hence, this mode must be neutrally stable. Agreement is found for all other modes. However, by an independent analysis considering only ellipsoidal (though time-varying) disturbances, we have identified one unstable mode as purely ellipsoidal, i.e. it does not change the form of the ellipsoid, only its shape. Under this instability, the vortex quasi-periodically tilts over while undergoing mild changes in shape.

The range of parameters leading to non-ellipsoidal instabilities turns out to be narrow, with instability principally occurring for highly eccentric (horizontally squashed, prolate) vortices. The long-term fate of these instabilities is examined by nonlinear contour-dynamical simulations. These reveal a wealth of complex phenomena such as the production of a sea of small-scale vortices, yet, remarkably, the dominant vortex often tends to relax to a stable rotating ellipsoid.

This work is concerned with the linearized theory of water waves applied to the motion of a floating structure that restricts in some way the motion of a portion of the free surface (an example of such a structure is a floating torus). When a structure of this type is held fixed in incident monochromatic waves, or forced to move time harmonically with a prescribed velocity, the amplitude of the fluid motion will have local maxima at certain frequencies of the forcing. These resonances correspond to poles of the scattering and radiation potentials when extended to the complex frequency domain. It is shown in this work that, in general, the positions of these poles in the scattering and radiation potentials will not coincide with the positions of the poles that appear in the velocity potential for the coupled problem obtained when the structure is free to move. The poles of the potential for the coupled problem are associated with the solution for the structural velocities of the equation of motion. When physical quantities such as the amplitude of the fluid motion are examined as a function of (real) frequency, there will in general be a shift in the resonant frequencies in going from the radiation and scattering problems to the coupled problem. The magnitude of this shift depends on the geometry of the structure and how it is moored.