Consider a two-dimensional bubble moving with speed U through an unbounded, inviscid fluid. Let all lengths be normalized by T/ρU2 where T is the surface tension. Then the shape of the bubble depends on a single parameter Γ = 2Δp/ρU2 − 1, where Δp = pb − p∞ is the difference between the bubble pressure and the ambient pressure. We obtain solutions for the bubble shape over the whole range of Γ-values that are physically relevant. The formulation involves a mapping from an auxiliary circle plane ζ where the flow field is known. The problem then reduces to solving an infinite set of nonlinear algebraic equations for the coefficients in the mapping function.

To a first approximation, when Γ → ∞, the bubble takes an elliptical shape of aspect ratio $(1 + {\textstyle\frac{2}{3}}\Gamma^{-1})/(1 - {\textstyle\frac{2}{3}}\Gamma^{-1})$ flattened in the flow direction. The solution correct to order Γ−5 is then obtained which is fairly accurate for Γ as low as 2. When Γ = 0 the exact, nonlinear solution for the bubble shape is given by $x = \frac{1}{3}(\frac{1}{3}\cos\phi - \frac{1}{27}\cos 3\phi), y=\frac{1}{3}(\frac{5}{3}\sin\phi + \frac{1}{27}\sin 3\phi)$. We can then obtain a perturbation solution for Γ → 0 correct to order Γ6. This solution, useful in the range 0.75 > Γ > − 0.4537, even gives reasonable descriptions of non-convex bubble shapes for Γ < 0 down to the pinch-off limit Γ* when the bubble ceases to be simply connected. It is remarkable that a simple analytical representation correct to order Γ2 analytically yields a value for Γ* of − 0.4548, i.e. within 0.3% of the correct value; naturally, the higher-order approximations are even more accurate. While the present results eliminate the need for direct numerical computations over most of the range of Γ, such results, too, are presented. Finally, the dependence of the bubble geometrical parameters, Weber number and added mass on Γ is determined.

When a well-mixed suspension of small particles is emplaced below a clear fluid whose density is greater than that of the interstitial fluid, but less than that of the bulk suspension, the subsequent settling of the dense particles releases buoyant interstitial fluid and drives convection in the overlying layer. Mixing of interstitial fluid and some entrained particles into the overlying fluid causes the density of the overlying fluid to evolve with time and changes the rate of descent of the interface between the sedimenting and convecting regions. These effects are investigated experimentally in a simple rectangular geometry using suspensions of spherical glass particles. It is found that the convecting region is well mixed in both composition and particle concentration and that the interfacial velocity may be predicted from the instantaneous (uniform) bulk density of the upper layer and the distribution of the particle settling velocities. In the case of an overlying density gradient, the convection does not extend through the depth of the overlying fluid but erodes the base of the gradient to form a well-mixed layer between the gradient and the sedimenting fluid. On completion of the first cycle of sedimentation-driven convection, sedimentation from this well-mixed layer produces further cycles of sedimentation-driven convection, which are of successively decreasing intensity and increasing duration. Whether the overlying fluid is uniform or stratified, both theory and experiment show that the particles that are lifted into the convection are smaller on average than those which settle at the base of the lower layer. Thus, when the lifted particles are eventually allowed to settle there is a discontinuity generated in the variation of the size distribution of particles with height in the final sedimented pile. This phenomenon may be an important mechanism for secondary layering in the deposits from turbidity currents and pyroclastic flows.

A study is made of the asymptotic equations governing gravity waves on water which cause a transition from one surface level to a slightly higher one and approach a steady wave-form as time increases, at least at the head of the wave. A two-parameter family of limit processes is surveyed in each of which the time scale and the horizontal length scale tend to infinity in a definite relation to the amplitude, as that tends to zero. Small-amplitude linearization is shown to be possible at most during a transitory stage of the wave development. Arbitrarily close approach to steadiness at the head of the wave is found to imply that a substantial part of the transition wave must be ultimately governed by a nearsteady variant of the non-linear equation of Korteweg & de Vries (1895) and must take the form of a train of cnoidal waves characterized by a parameter which changes slightly from crest to crest.

This paper considers the slow motion of a sphere, permanently and uniformly magnetized in one direction, in a viscous electrically conducting medium. The line of the magnetic poles is assumed to be parallel to the direction of the motion of the sphere. The velocity and pressure fields are calculated by two iterations. The distortion of the magnetic field is also calculated. An expression is obtained for total drag due to the viscous, pressure and magnetic forces.

This paper presents an experimental study on the propagation speed of gravity currents at moderate values of a gravity Reynolds number. Two cases are considered: gravity currents propagating along a rigid boundary and intrusive gravity currents. For the first case, a semi-empirical formula for the front propagation speed derived from simple energy arguments is shown to capture well the effect of flow deceleration because of viscous dissipation. In the second case, the propagation speed is shown to agree with the one predicted for energy-conserving virtually inviscid flows (Shin, Dalziel & Linden, J. Fluid Mech. vol. 521, 2004, p. 1), which implies that the losses due to vorticity generation and mixing at the liquid–liquid interface play only a minor role in the total balance of energy.

The thermo-fluid-dynamic field that arises when an infinite thick plate is impulsively accelerated to a constant speed in a laminar regime is studied, taking into account the coupling of the convection and conduction in the fluid with the conduction in the solid. Two significant cases are discussed depending on the boundary condition imposed on the unwetted side of the plate: constant temperature or adiabatic wall. The work is particularly focused on analysing the singularities arising in the field at the initial time. For this purpose an exact analytical solution of the problem governed by the Navier–Stokes equations with constant properties and by the energy equations in the fluid and in the solid is proposed and discussed. The non-dimensional parameter governing the conjugated effects is shown to be the ratio between the thermal effusivities in the fluid and in the solid. The results have also been extended to the analysis of compressible flows by the Stewartson–Dorodnitsin transformation.

Energy & ability theory is employed to study the finite-amplitude stability of a viscous incompressible fluid occupying the space between a pair of concentric cylinders when the inner-cylinder angular velocity varies linearly with time. For the case with a fixed outer cylinder and increasing inner-cylinder speed, we find an enhancement of stability, consistent with a linear-theory result due to Eagles. When the inner-cylinder speed decreases, we find an initially decreased stability bound, indicating the possibility of hysteresis, while, if the inner cylinder is allowed to reverse direction and linearly increase in speed, we find significant stability enhancement.

The properties of a dilute bidisperse particle–gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit τv [Lt ] τc, where τc is the time between successive collisions of a particle, and τv is the viscous relaxation time. In this limit, the particles relax close to their terminal velocity between successive collisions, and we use a perturbation analysis in the small parameter ε, which is proportional to τv/τc, about a base state in which all the particles settle at their terminal velocities. The mean velocities of the two species are O(ε) different from their terminal velocities, and the mean-square velocities are O(ε) smaller than the square of the terminal velocity. The distribution functions for the two species, which incorporate the first effects of collisions between particles settling at their terminal velocities, are derived. The velocity distribution is highly anisotropic in this limit, and the mean-square velocity in the vertical direction is twice that in the horizontal plane. The distribution function for each species is singular at its terminal velocity, and the distributions are non-zero in a finite region in velocity space between the two terminal velocities.

Fields of statistically steady wind-generated waves produced in the NASA-Wallops wind wave facility, were perturbed by the injection of groups of longer waves with various slopes, mechanically generated at the upwind end of the tank. The time histories of the surface displacements were measured at four fetches in ensembles consisting of 100 realizations of each set of experimental conditions, the data being stored and analysed digitally. The overall interaction was found to have four distinct phases. (i) When the longer waves overtake the pre-existing wind-generated waves, during the first half of the group where successive crests are increasing in amplitude, vigorous wave breaking near the crests reduces the energy density and $\overline{\zeta^2}$ in the wind waves while straining by the orbital velocities of the group reduces their wavelengths near the crests; the ‘significant slope’ $2\pi(\overline{\zeta^2})^{\frac{1}{2}}/\lambda $ at the crests is found to be very nearly constant and equal to the initial, undisturbed value. After the maximum wave of the group has passed, breaking appears to virtually cease but the earlier energy loss results in suppression of the short waves. The overall suppression by a group of waves is significantly less than that measured by Mitsuyasu (1966) and Phillips & Banner (1974) in a continuous train of waves whose slope is equal to the maximum in the group. A simple description of this phase of the interaction, involving constant significant slope of the breaking waves over the leading half of the group and conservation of action thereafter, gives suppression ratios close to those measured. (ii) Once the group has passed, the surface is much smoother and the waves begin to regenerate under the continued influence of the wind but at rates considerably slower than those suggested by Plant's formula, using the averaged value of u*. This is qualitatively consistent with a locally reduced surface stress as the wind blows from rougher water well behind the group to the smoother surface immediately behind it. (iii) The regeneration is interrupted by the arrival of a wave energy front moving down the tank, across which the energy density rises abruptly to values up to six times greater than in the undisturbed field. At the same time, the dominant frequencies just behind the wave energy front are lower than in the initial field, and the significant slope $(\overline{\zeta^2})^{\frac{1}{2}} \sigma^2/g $ is, within experimental uncertainty, again identical to that in the initial field. The front was found to propagate notably faster than the appropriate group velocity (g/2σ) and it is suggested that this is the combined result of dispersion, nonlinearity and wind amplification, together with wind-induced drift in the tank. Finally, (iv) the energy density gradually subsides and the dominant wave frequency increases as the wind waves relax towards their undisturbed state, the relaxation seeming to be essentially complete when energy packets arriving at a point have originated at the upwind end of the tank, rather than at the wave energy front.

In this paper a model is presented that incorporates characteristic features of the turbulent structures as revealed by recent experimental observations. The principal characteristic features are the occurrence of periodicity and intermittency, not only at the edge of the boundary layer, but also close to the wall.

By an averaging procedure, equations are derived for the large-scale part of the turbulent motion. The unknown terms, representing the small-scale turbulent stress, are assumed to be zero except in the so-called burst regions which occupy only a small fraction of the total flow field and arise from local instability of the large-scale flow field. In the model distinction is made between a thin layer near the wall where viscous forces play an important role and the remaining part of the boundary layer where inviscid equations are valid. The momentum transport takes place in three different stages with different mechanisms. First of all the fluid in the wall region is retarded by viscous forces and collected in long narrow regions (streaks). After this a rapid exchange takes place in the burst regions where the low-momentum fluid is ejected into the outer region. Finally the large-scale structures in the outer region take over the transport.

It turns out that the transport properties of a turbulent boundary layer can be calculated reasonably well with this deterministic model. It can be concluded that the coherent part of the turbulent motion is very important in the transport process.

This paper presents numerical calculations of the self-excited oscillations of an incompressible planar jet impinging upon a wedge for a Reynolds-number range of 250–650. For this Reynolds-number range these flows are experimentally observed to be two-dimensional and laminar. A finite-difference vorticity/stream-function formulation of the Navier-Stokes equations is employed. The self-sustained flow oscillations result in not just one but several well-defined flow frequency components due to nonlinear interaction of two primary components: the most unstable frequency (β) of the jet shear layer and a low-frequency modulating component ($\frac{1}{3}\beta $). The modulating component results from vortex-vortex interaction at the impingement edge of both like and counter-rotating vortices. Although the interaction pattern varies through the Reynolds-number range studied, the pattern adjusts itself to maintain the modulating component $\frac{1}{3}\beta $ which has a strong upstream influence. The numerical results, in agreement with experimental results, strongly suggest the occurrence of such phenomena as frequency jumps and hysteresis. Pressure at the wedge surface has been calculated and compared with experimental results. Numerical results for wedge torque and lift, which have not been experimentally measured, have also been obtained.

We study the streaming flow past a rapidly rotating circular cylinder. The starting point is the full continuity and momentum equations without any approximations. We assume that the solution is a boundary-layer flow near the cylinder surface with the potential flow outside the boundary layer. The order of magnitude of the terms in the continuity and momentum equations can be estimated inside the boundary layer. When terms of the order of $\delta/a$ and higher are dropped, where $\delta$ is the boundary-layer thickness and $a$ is the radius of the cylinder, the equations used by M. B. Glauert (Proc. R. Soc. Lond. A, vol. 242, 1957, p. 108) are recovered. Glauert's solution ignores the irrotational rotary component of the flow inside the boundary layer, which is consistent with dropping $\delta/a$ terms in the governing equations.

We propose a new solution to this problem, in which the velocity field is decomposed into two parts. Outside the boundary layer, the flow is irrotational and can be decomposed into a purely rotary flow and a potential flow past a fixed cylinder. Inside the boundary layer, the velocity is decomposed into an irrotational purely rotary flow and a boundary-layer flow. Inserting this decomposition of the velocity field inside the boundary layer into the governing equations, we obtain a new set of equations for the boundary-layer flow, in which we do not drop the terms of the order of $\delta/a$ or higher. The pressure can no longer be assumed to be a constant across the boundary layer, and the continuity of shear stress at the outer edge of the boundary layer is enforced. We solve this new set of equations using Glauert's method, i.e. to expand the solutions as a power series of $\alpha \,{=}\, 2 U_0 /Q$, where $U_0$ is the uniform stream velocity and $Q$ is the circulatory velocity at the outer edge of the boundary layer. The pressure from this boundary-layer solution has two parts, an inertial part and a viscous part. The inertial part comes from the inertia terms in the momentum equations and is in agreement with the irrotational pressure; the viscous part comes from the viscous stress terms in the momentum equations and may be viewed as a viscous pressure correction, which contributes to both drag and lift. Our boundary-layer solution is in reasonable to excellent agreement with the numerical simulation in the companion paper by Padrino & Joseph (2006).

Accurate real-time simulations and forecasting of phase-revolved ocean surface waves require nonlinear effects, both geometrical and kinematic, to be accurately represented. For this purpose, wave models based on a Lagrangian steepness expansion have proved particularly efficient, as compared to those based on Eulerian expansions, as they feature higher-order nonlinearities at a reduced numerical cost. However, while they can accurately model the instantaneous nonlinear wave shape, Lagrangian models developed to date cannot accurately predict the time evolution of even simple periodic waves. Here, we propose a novel and simple method to perform a Lagrangian expansion of surface waves to second order in wave steepness, based on the dynamical system relating particle locations and the Eulerian velocity field. We show that a simple redefinition of reference particles allows us to correct the time evolution of surface waves, through a modified nonlinear dispersion relationship. The resulting expressions of free surface particle locations can then be made numerically efficient by only retaining the most significant contributions to second-order terms, i.e. Stokes drift and mean vertical level. This results in a hybrid model, referred to as the ‘improved choppy wave model’ (ICWM) (with respect to Nouguier et al.’s J. Geophys. Res., vol. 114, 2009, p. C09012), whose performance is numerically assessed for long-crested waves, both periodic and irregular. To do so, ICWM results are compared to those of models based on a high-order spectral method and classical second-order Lagrangian expansions. For irregular waves, two generic types of narrow- and broad-banded wave spectra are considered, for which ICWM is shown to significantly improve wave forecast accuracy as compared to other Lagrangian models; hence, ICWM is well suited to providing accurate and efficient short-term ocean wave forecast (e.g. over a few peak periods). This aspect will be the object of future work.

In view of the importance of concentration fluctuations in practical and theoretical problems of turbulent diffusion, there are presented here some invariant properties of the distribution of fluctuations associated with a cloud of contaminant containing a finite quantity Q of material. These properties are invariant provided only that Q is conserved, no assumption whatsoever being made about the random turbulent velocity field. Consequences of the results for (i) steady plumes, (ii) the representation of the distribution of concentration by series of (generalized) Hermite polynomials, and (iii) the relationship of the ensemble mean concentration with the distance–neighbour function, are discussed using experimental evidence.

Practical prediction of structural vibrations due to a turbulent boundary layer currently depends on empirical representations of the unsteady wall pressures. Improvements in these representations would be greatly facilitated if a simple, physically based model were available to test ad hoc assumptions and provide rigorous interpolation of experimental data. A possible candidate is the attached-eddy model, developed from Townsend's initial ideas by Perry and co-workers in the context of turbulence velocity spectra. This approach employs the superposition of contributions from individual ‘eddies’, of varying size, to yield its predictions. It is shown here that the same methodology can be applied for wall pressures, once the field due to an eddy has been obtained via solution of the governing Poisson equation. Comparisons with large-eddy simulation and experimental data, spanning a two-decade Reynolds number range, show remarkably good agreement, given the simplicity of the model. It is concluded that this approach has the potential to provide useful physical insight and, subject to its extension to a time-resolved form, improvements to existing empirical formulations.

The problem solved is that of the interaction between a laminar boundary layer on a semi-infinite flat plate and an oncoming shear flow of finite lateral dimensions bounded by uniform irrotational flow extending to infinity. The pressures along the plate and upstream of the same are deduced (to a linearized approximation) in the form of a Fourier integral based on the solution of a simpler periodic flow problem. It is found that while the assumption of an infinite, uniform shear flow gives asymptotically correct interaction pressure gradients on the plate near the leading edge, the pressure level even there (compared to upstream infinity) is strongly influenced by the boundedness of the external shear. At distances from the leading edge which are large compared to the lateral extent of the shear flow, the pressure gradients along the plate are shown to be vanishingly smaller than in the infinite shear case.

We investigate the generation conditions of a high-Reynolds-number swirling jet experiment, based on a rotating honeycomb device and using a final contraction. Using hot-wire measurements, we first show that for high swirl levels, the flow at the jet exhaust may exhibit fully developed turbulence in the whole plane. By analysing the fluctuation levels obtained for several values of the contraction ratio, ranging from 4 to 18.4, we prove that this turbulence does not result from upstream-propagating disturbances initiated in the jet, but originates in the pipe flow upstream of the exit plane. Using stereo particle image velocimetry, we then measure the flow in the constant-cross-section pipe located between the rotating honeycomb outlet and the contraction. This investigation is supplemented with simplified numerical simulations of the mean flow. The pipe flow dynamics is found to result from the interplay of a rich variety of complex phenomena, which are independent of the contraction ratio in the range considered here. In the near-wall region, centrifugal instability occurs in the form of intermittent azimuthal vortices, starting from moderate swirl levels and persisting for all higher levels. As the flow exiting from the honeycomb has a swirl level high enough to reach the subcritical regime, a complex mean flow organization is observed, dominated by the presence of large-amplitude axisymmetric Kelvin wave trains. Gradients in the resulting flow lead to the appearance of generalized centrifugal instabilities in an annular region in the rotational core, starting in the early subcritical regime. As the swirl level is further increased, large-scale, high-amplitude axisymmetric and simple spiral perturbations add to the global dynamics, leading to an overall very high fluctuation level. Consideration of the turbulent spectra in the jet exit planes suggests that the simple spiral coherent structure could be the resonant response of the flow to the periodic excitation by the rotating honeycomb. Overall, the study illustrates why a swirling jet experiment should exclude the use of a final contraction in order to guarantee smooth flow conditions in the exit at high swirl.

Motivated by the recent paper of Hills & Moffatt (2000), we investigate the Stokes flow in a trihedral corner formed by three mutually orthogonal planes, induced by a non-zero velocity distribution over one of the walls of the corner. It is shown that the local behaviour of the velocity field near the edges of the corner, where a discontinuity of the boundary velocity is assumed, coincides with the Goodier–Taylor solution for a two-dimensional wedge. Analysis of the streamline patterns confirms the existence of eddies near the stationary edge in the flow, induced either by uniform translation of one of the walls of the corner in the direction perpendicular to its bisectrix or by uniform rotation of a side about the vertex of the corner. These flows are shown to be quasi-two-dimensional. If the wall rotates about a centre displaced from the vertex, the induced flow is essentially three-dimensional. In the antisymmetric velocity field, a stagnation line appears composed of stagnation points of different types. Otherwise the three-dimensionality manifests itself in a non-closed spiral shape of the streamlines.