An air–water coupled model is developed to investigate wind-wave generation processes at low wind speed where the surface wind stress is about 0.089 dyn cm−2 and the associated surface friction velocities of the air and the water are u*a~8.6 cms−1 and u*w~0.3 cms−1, respectively. The air–water coupled model satisfies continuity of velocity and stress at the interface simultaneously, and hence can capture the interaction between air and water motions. Our simulations show that the wavelength of the fastest growing waves agrees with laboratory measurements (λ~8–12 cm) and the wave growth consists of linear and exponential growth stages as suggested by theoretical and experimental studies. Constrained by the linearization of the interfacial boundary conditions, we perform simulations only for a short time period, about 70s; the maximum wave slope of our simulated waves is ak~0.01 and the associated wave age is c/u*a~5, which is a slow-moving wave. The effects of waves on turbulence statistics above and below the interface are examined. Sensitivity tests are carried out to investigate the effects of turbulence in the water, surface tension, and the numerical depth of the air domain. The growth rates of the simulated waves are compared to a previous theory for linear growth and to experimental data and previous simulations that used a prescribed wavy surface for exponential growth. In the exponential growth stage, some of the simulated wave growth rates are comparable to previous studies, but some are about 2–3 times larger than previous studies. In the linear growth stage, the simulated wave growth rates for these four simulation runs are about 1–2 times larger than previously predicted. In qualitative agreement with previous theories for slow-moving waves, the mechanisms for the energy transfer from wind to waves in our simulations are mainly from turbulence-induced pressure fluctuations in the linear growth stage and due to the in-phase relationship between wave slope and wave-induced pressure fluctuations in the exponential growth stage.

We present a series of laboratory experiments in which a steady stream of air is supplied through a small hole in the wall of a vertical pipe of rectangular cross-section down which there is a steady flux of water. For a range of liquid flow rates, the air forms a steady standing bubble whose nose is attached to the point of air supply. The steady bubble sheds a flux of much smaller air bubbles at its base, located downstream of the air injection point. The minimum liquid speed for which steady standing bubbles develop occurs at a particular Froude number of the liquid flow, Frd = U/ = 0.38, where U is the upstream speed, g the acceleration due to gravity and d the width of the cell. These trapped bubbles are distinct from the freely rising Taylor bubble, in that the Froude number at the nose is variable. Also, on a length scale greater than that influenced by surface tension, we find that the bubble nose asymptotes to a cusp-like shape, with an angle that decreases with Frd. We show that numerical solutions of the potential flow equations replicate the bubble shape and angle of the cusp, which appear independent of the gas flux. We also find that there is a minimum gas flux for which these standing bubbles develop. As the gas flux decreases below this threshold, the standing bubbles become unstable and, instead, a much shorter oscillating bubble develops. This produces a wake which has similarities with that formed downstream of a cylinder in a confined channel, but which also carries bubbles downstream. We also find that with sufficiently small gas flux, no bubble develops. For liquid flow rates smaller than the critical value, Frd < 0.38, we find that the bubbles become unstable and detach from the injection point and rise up the tube.

In this paper we consider the dynamics of bubbles near a kidney stone subjected to a lithotripter shock wave. We address the effect of kidney stone geometry and composition on the cavitation potential near the stone in a shock wave lithotripter. The analysis is based on the previously developed work metric in which the work done on a bubble by the lithotripter shock wave (LSW) is used to determine the maximum radius the bubble achieves. Results of the reflection of the LSW from cylindrical kidney stones with proximal surfaces of varying geometry show that the presence of the stone enhances bubble growth near the stone and decreases growth further away, owing to constructive and destructive interference, respectively. These effects hold true regardless of the shape and curvature of the face, and are strongest for stones with concave faces and higher reflection coefficients. A consequence of the analysis is an elucidation of the mechanism for enhanced cavitation activity and creation of deep craters on the proximal side of a kidney stone, as have been observed in recent experiments.

In this paper we use the boundary integral method to model the non-spherical collapse of bubbles excited by lithotripter shock waves near a rigid boundary. The waves we consider are representative of those developed by shock wave lithotripsy or shock wave therapy devices, and the rigid boundaries we consider are representative of kidney stones and reflective bony tissue. This study differs from previous studies in that we account for the reflection of the incident wave and also the asymmetry of the collapse caused by the presence of the rigid surface. The presence of the boundary causes interference between reflected and incident waves. Quantities such as kinetic energy, Kelvin impulse and centroid translation are calculated in order to illuminate the physics of the collapse process. The main finding is that the dynamics of the bubble collapse depend strongly on the distance of the bubble relative to the wall when reflection is taken into account, but much less so when reflection is omitted from the model. The reflection enhances the expansion and subsequent collapse of bubbles located near the boundary owing to constructive interference between incident and reflected waves; however, further from the boundary, the dynamics of collapse are suppressed owing to destructive interference of these two waves. This result holds regardless of the initial radius of the bubble or its initial state at the time of impact with the lithotripter shock wave. Also, the work done by the lithotripter shock wave on the bubble is shown to predict strongly the maximum bubble volume regardless of the standoff distance and the presence or absence of reflection; furthermore, allowing for non-sphericity, these predictions match almost exactly those of a previously developed spherical model.

Under fair weather conditions, local flow patterns in areas of complex topography are driven by diurnal heating and cooling. If the topography is basin-shaped, downslope flows occurring at night accumulate (pool) in the basin valley to form a stable layer of cold air. During the morning transition, this cold pool is destroyed by the onset of turbulent convection and upslope flow. A series of laboratory experiments was conducted to identify mechanisms responsible for the breakup of cold pools. An idealized V-shaped tank filled with thermally stratified water and heated with an approximately uniform bottom heat flux was used. Temperature measurements and dye visualization were used for flow diagnostics. Mechanisms of cold pool destruction were identified and placed in the context of previously proposed mechanisms. A new mechanism was identified, wherein a dominant intrusion emanating from the upslope flow plays a dynamically important role in cold pool destruction. The results are expected to help develop subgrid parameterizations for meso-scale weather forecasting models, which are notorious for giving poor predictions during the morning transition period in complex terrain.

Steady inlet flow through a circular tube with an axisymmetric blockage of varying size is studied both numerically and experimentally. The geometry consists of a long, straight tube and a blockage, semicircular in cross-section, serving as a simplified model of an arterial stenosis. The stenosis is characterized by a single parameter, the aim being to highlight fundamental behaviours of constricted flows, in terms of the total blockage. The Reynolds number is varied between 50 and 2500 and the stenosis degree by area between 0.20 and 0.95. Numerically, a spectral-element code is used to obtain the axisymmetric base flow fields, while experimentally, results are obtained for a similar set of geometries, using water as the working fluid. At low Reynolds numbers, the flow is steady and characterized by a jet flow emanating from the contraction, surrounded by an axisymmetric recirculation zone. The effect of a variation in blockage size on the onset and mode of instability is investigated. Linear stability analysis is performed on the simulated axisymmetric base flows, in addition to an analysis of the instability, seemingly convective in nature, observed in the experimental flows. This transition at higher Reynolds numbers to a time-dependent state, characterized by unsteadiness downstream of the blockage, is studied in conjunction with an investigation of the response of steady lower Reynolds number flows to periodic forcing.

Independent poloidal and azimuthal body forces are induced in a liquid metal cylinder by travelling and rotating magnetic fields of different frequencies, respectively. The bulk axial and azimuthal velocities are measured by the ultrasound Doppler method. Particle image velocimetry is used to observe the upper free surface velocity distribution. The transition from the poloidal to the azimuthal body force governed regime occurs at a fixed ratio of the respective force magnitude of around 100. This transition is marked by the formation of a concentrated vortex revealing several similarities to intense atmospheric vortices. The vortex structure is controlled by a relatively weak azimuthal force while the maximum speed of the swirl is mainly governed by the poloidal one. Under a certain force ratio the average axial velocity changes its direction in the vortex core, resembling the subsidence in an eye of a tropical cyclone or a large tornado. Multiple moving vortices encircle the vortex core in this regime.

A computational fluid dynamics cavitation model based on the Eulerian–Lagrangian approach and suitable for hole-type diesel injector nozzles is presented and discussed. The model accounts for a number of primary physical processes pertinent to cavitation bubbles, which are integrated into the stochastic framework of the model. Its predictive capability has been assessed through comparison of the calculated onset and development of cavitation inside diesel nozzle holes against experimental data obtained in real-size and enlarged models of single- and multi-hole nozzles. For the real-size nozzle geometry, high-speed cavitation images obtained under realistic injection pressures are compared against model predictions, whereas for the large-scale nozzle, validation data include images from a charge-coupled device (CCD) camera, computed tomography (CT) measurements of the liquid volume fraction and laser Doppler velocimetry (LDV) measurements of the liquid mean and root mean square (r.m.s.) velocities at different cavitation numbers (CN) and two needle lifts, corresponding to different cavitation regimes inside the injection hole. Overall, and on the basis of this validation exercise, it can be argued that cavitation modelling has reached a stage of maturity, where it can usefully identify many of the cavitation structures present in internal nozzle flows and their dependence on nozzle design and flow conditions.

We investigate similarity solutions for the outer part of a zero-pressure-gradient turbulent boundary layer in the limit of infinite Reynolds number. Previous work by George (Phil. Trans. R. Soc. vol. 365, 2007 p. 789) has suggested that the only appropriate velocity scale for the outer region is U1, the free-stream velocity. This is based on the fact that scaling with U1 leads to a mathematically valid similarity solution of the momentum equation for the outer region in the asymptotic limit of infinite Reynolds number. Here we show that the classical scaling using the friction velocity also leads to a valid similarity solution for the outer flow in this limit. Therefore on this basis it is not possible to dismiss the friction velocity as a possible scaling as has been suggested by George (2007) and others. We show that both the free-stream velocity and the friction velocity are potentially valid scalings according to this theoretical criterion.

This paper presents numerical simulations of two-dimensional incompressible flow around two circular cylinders in relative motion, which may result in impact. Viscous flow computations are carried out using a streamfunction–vorticity method for two equal-diameter cylinders undergoing a two-dimensional impact in otherwise stationary fluid and for cases of similar impact of two cylinders in a steady incident flow. These results are supported by potential flow calculations carried out using a Möbius conformal transformation and infinite arrays of image singularities. The inviscid flow results are compared with other published work and show that the inviscid forces induced on the cylinders have an inverse square root singularity with respect to the time to impact. All impacts considered in this paper result from steady motion of the cylinders along the line joining their centres.

Thinning rates of liquid lamellae in surfactant-free non-Newtonian gas–liquid foams, appropriate for ceramic or polymer melts and also in metals near the melting point, are derived in two dimensions by matched asymptotic analysis valid at small capillary number. The liquid viscosity is modelled (i) as a power-law function of the shear rate and (ii) by the Ellis law. Equations governing gas–liquid interface dynamics and variations in liquid viscosity are derived within the lamellar, transition and plateau border regions of a corner of the liquid surrounding a gas bubble. The results show that the viscosity varies primarily in the very short transition region lying between the lamellar and the Plateau border regions where the shear rates can become very large. In contrast to a foam with Newtonian liquid, the matching condition which determines the rate of lamellar thinning is non-local. In all cases considered, calculated lamellar thinning rates exhibit an initial transient thinning regime, followed by a t−2 power-law thinning regime, similar to the behaviour seen in foams with Newtonian liquid phase. In semi-arid foam, in which the liquid fraction is O(1) in the small capillary number, results explicitly show that for both the power-law and Ellis-law model of viscosity, the thinning of lamella in non-Newtonian and Newtonian foams is governed by the same equation, from which scaling laws can be deduced. This result is consistent with recently published experimental results on forced foam drainage. However, in an arid foam, which has much smaller volume fraction of liquid resulting in an increase in the Plateau border radius of curvature as lamellar thinning progresses, the scaling law depends on the material and the thinning rate is not independent of the liquid viscosity model parameters. Calculations of thinning rates, viscosities, pressures, interface shapes and shear rates in the transition region are presented using data for real liquids from the literature. Although for shear-thinning fluids the power-law viscosity becomes infinite at the boundaries of the internal transition region where the shear rate is zero, the interface shape, the pressure and the internal shear rates calculated by both rheological models are indistinguishable.

The results of an experimental investigation to determine the influence of two physiologically relevant inlet conditions on the flow physics downstream of an idealized stenosis are presented. The two inlet conditions are an asymmetric mean inlet velocity profile and an asymmetric mean inlet velocity profile plus secondary flow, as found downstream of a bend. The stenosis is modelled as an axisymmetric 75% area reduction occlusion with a length-to-diameter ratio of 2. The flow was forced by a 10-harmonic carotid artery-inspired waveform with mean, maximum and minimum Reynolds numbers of 364, 1424 and 24, respectively, and a Womersley number of 4.6. Laser Doppler velocimetry and particle image velocimetry were used to characterize the spatial and temporal evolution of a baseline case (no disturbances) as well as the two physiologically relevant inlet conditions. The asymmetric inlet velocity profile was found to reduce the region of influence of the stenosis by forcing the stenotic jet towards the tube wall via an induced non-uniform radial pressure gradient, similar to the Coanda effect. Curvature-induced secondary flow was found to play a minor role in the near-stenosis region. Vortex ring formation was relatively unaffected by the mean velocity gradient and secondary flow. Evidence of remnants of the starting vortex ring was observed far downstream in all cases.

We analyse the exchange of energy for an axisymmetric gravity current, released instantaneously from a lock, propagating over a horizontal boundary at high Reynolds number. The study is relevant to flow in either a wedge or a full circular geometry. Attention is focused on effects due to a linear stratification in the ambient. The investigation uses both a one-layer shallow-water model and Navier–Stokes finite-difference simulations. There is fair agreement between these two approaches for the energy changes of the dense fluid (the current). The stratification enhances the accumulation of potential energy in the ambient and reduces the energy decay (dissipation) of the two-fluid system. The total energy of the axisymmetric current decays considerably faster with distance of propagation than for the two-dimensional counterpart.

A steady theory is presented for gravity currents propagating with constant speed into a stratified fluid with a general density profile. Solution curves for front speed versus height have an energy-conserving upper bound (the conjugate state) and a lower bound marked by the onset of upstream influence. The conjugate state is the largest-amplitude nonlinear internal wave supported by the ambient stratification, and in the limit of weak stratification approaches Benjamin's energy-conserving gravity current solution. When the front speed becomes critical with respect to linear long waves generated above the current, steady solutions cannot be calculated, implying upstream influence. For non-uniform stratification, the critical long-wave speed exceeds the ambient long-wave speed, and the critical-Froude-number condition appropriate for uniform stratification must be generalized. The theoretical results demonstrate a clear connection between internal waves and gravity currents. The steady theory is also compared with non-hydrostatic numerical solutions of the full lock release initial-value problem. Some solutions resemble classic gravity currents with no upstream disturbance, but others show long internal waves propagating ahead of the gravity current. Wave generation generally occurs when the stratification and current speed are such that the steady gravity current theory fails. Thus the steady theory is consistent with the occurrence of either wave-generating or steady gravity solutions to the dam-break problem. When the available potential energy of the dam is large enough, the numerical simulations approach the energy-conserving conjugate state. Existing laboratory experiments for intrusions and gravity currents produced by full-depth lock exchange flows over a range of stratification profiles show excellent agreement with the conjugate state solutions.

The physical picture of a fluid particle pair as a couple of material points rotating around their centre of mass is proposed to model turbulent relative dispersion in the inertial range. This scheme is used to constrain the non-uniqueness problem associated to the Lagrangian models in the well-mixed class and the properties of the stochastic process derived are analysed with respect to some turbulent velocity characteristics. A simple illustrative Markov model is developed in stationary homogeneous isotropic turbulence and the particle separation statistics are compared with direct numerical simulation data. In spite of the simplicity of the model, a consistent comparison is observed in the inertial range, supporting the formulation proposed.

Effects of random bathymetric irregularities on wave generation by transcritical ship motion in a shallow channel are investigated. Invoking Boussinesq approximation in shallow waters, it is shown that the wave evolution is governed by an integro-differential equation combining features of Korteweg–deVries and Burgers equations. For an isolated ship, the bottom roughness weakens the transient waves radiated both fore and aft. When many ships advance in tandem, a steady mount of high water can be formed in front and a depression behind. Wave forces on both an isolated ship and a ship in a caravan are obtained as functions of the mean-square roughness, ship speed and the blockage coefficient.

The effect of confining a gas bubble between two parallel walls was investigated for the inertia-dominated regime characterized by high Reynolds and low Weber numbers. Single bubble experiments were performed with non-polar liquids such that the bubble surface could be considered clean; hence, shear free. The drag coefficient was found to be the result of two main effects: the Reynolds number and the confinement. The total drag could be written as the product of the corresponding unconfined drag, which depended mainly on the Reynolds number, and a function F(s)=1 + κs3. The confinement parameter s was defined as the ratio of the bubble radius to the gap width. The value of the constant κ depended on the way in which the bubbles moved within the gap, which was found to be either in a rectilinear (κ≈8) or oscillatory trajectory (κ≈80). For Re < 70, and a range of values of the confinement parameter, the bubbles followed a rectilinear path. For this regime, numerical simulations were performed to obtain the drag force on the bubble directly; a reasonable agreement was found with experiments. Moreover, a comparison of these results with a potential-flow-based model indicated that the vorticity produced at the walls induced a significant part of the drag. For Re > 70, oscillations were observed in the bubble trajectory. In all cases, the oscillation occurred in a zigzag manner. Near the transition the bubbles oscillated but did not reach the walls; for larger Reynolds numbers, the bubbles collided repeatedly with the walls as they ascended. The instability, which is different from the well-known unconfined path instability, resulted from the reversal of sign of the wall-induced lift force: for low Reynolds number, the walls have a stabilizing effect because of the repulsive nature of the lift force between the walls and the bubble, while for high Reynolds number the lift is attractive and trajectories become unstable. Considering a model for the lift force of a bubble moving near a wall, the conditions for the transition were identified. A reasonable agreement between the model and experiments was found.

We report on the results of a numerical study of nearly immiscible contrasted density currents aimed at shedding light on the influence of wall effects on current dynamics in the lock-exchange configuration. The numerical approach is an interface-capturing method which does not involve any explicit reconstruction of the interface. Navier–Stokes equations are solved on a fixed grid and a hyperbolic equation is used for the transport of the local volume fraction of one of the fluids. This allows us to describe the density currents for the complete range of density contrast 10−3≤ρL/ρH≤0.99 (ρL and ρH being the density of the light and heavy fluids) and a wide range of Reynolds number 70≤Re≤5×104 (based on the channel height and the viscosity of the heavy fluid). The use of free-slip vs. no-slip boundary conditions enables us to separate the dissipation at the interface from the dissipation at the boundaries. Present results reveal that wall effects play a significant role on the propagation of contrasted density currents, unlike dissipation at the interface. It is first shown that when wall friction can be neglected, theoretical models based on the inviscid shallow-water approximations and Benjamin's steady-state result describe fairly well the light and heavy front velocities of density currents for the complete range of density ratio. However, when wall friction cannot be neglected, the results depart significantly from the prediction of inviscid theories. It is observed that most of the dissipation in highly contrasted currents takes place at the bottom wall and is a maximum at the head of the heavy current. This dissipation is shown to be responsible for the decrease of the front velocity. We propose a simple model based on Benjamin's analysis that includes wall friction. Keeping in mind the simplicity and limitations of the present model, the prediction of the front velocity of both the heavy and light currents is observed to be in good agreement with the numerical results for the complete range of density contrast. This gives further support to the idea that wall effects are the crucial ingredient for accurately predicting the front velocity of highly contrasted density currents.