We present a detailed study of the turbulent wake behind a quarter-ring curved cylinder at Reynolds number $Re=3900$ (based on cylinder diameter and incoming flow velocity), by means of direct numerical simulation. The configuration is referred to as a concave curved cylinder with incoming flow aligned with the plane of curvature and towards the inner face of the cylinder. Wake flows behind this configuration are known to be complex, but have so far only been studied at low $Re$. This is the first direct numerical simulation investigation of the turbulent wake behind the concave configuration, from which we reveal new and interesting wake dynamics, and present in-depth physical interpretations. Similar to the low-$Re$ cases, the turbulent wake behind a concave curved cylinder is a multi-regime and multi-frequency flow. However, in addition to the coexisting flow regimes reported at lower $Re$, we observe a new transitional flow regime at $Re=3900$. The flow field in this transitional regime is dominated not by von Kármán-type vortex shedding, but by periodic asymmetric helical vortices. Such vortex pairs exist also in some other wake flows, but are then non-periodic. Inspections reveal that the periodic motion of the asymmetric helical vortices is induced by vortex shedding in its neighbouring oblique shedding regime. The oblique shedding regime is in turn influenced by the transitional regime, resulting in a unified and remarkably low dominating frequency in both flow regimes. Owing to this synchronized frequency, the new wake dynamics in the transitional regime might easily be overlooked. In the near wake, two distinct peaks are observed in the time-averaged axial velocity distribution along the curved cylinder span, while only one peak was observed at lower $Re$. The presence of the additional peak is ascribed to a strong favourable base pressure gradient along the cylinder span. It is noteworthy that the axially directed base flow exceeded the incoming velocity behind a substantial part of the quarter-ring and even persisted upwards along the straight vertical extension. As a by-product of our study, we find that a straight vertical extension of 16 cylinder diameters is required in order to avoid any adverse effects from the upper boundary of the flow domain.

A hydrophilic or a hydrophobic long rigid solid plate of finite width, forming a meniscus with a liquid in a uniform gravitational field is considered. The one-dimensional meniscus with prescribed heights of the triple point from the far-field liquid surface is investigated analytically using the Young–Laplace equation. It is found that for a hydrophilic plate, the vertical force necessary to break the meniscus during removal of the plate from the liquid is larger than the force necessary to break the meniscus during submersion of the plate into the liquid. Furthermore, the capillary force on the plate reaches a maximum before the meniscus collapses during removal, but no maximum exists before collapse during submersion. The reverse is true when the plate is hydrophobic. The study is then extended to investigate the interaction force between two plates, each forming a meniscus with the liquid. The elevations of the plates from the far-field liquid surface are prescribed, in contrast to earlier studies where interaction between long cylinders floating under self weight was considered. Here, the menisci are determined exactly using the Young–Laplace equation. It is shown that for prescribed plate elevations, there can be at most two possible pairs of menisci between them. Each pair bifurcates from a meniscus that is determined by the elevations of the plates and the gap between them. Furthermore, as known for solids floating under self-weight, the horizontal component of the interaction force is attractive for similar menisci (e.g. when the two plates are equally displaced in or out of the liquid), and repulsive when they form opposite menisci. It is shown that if the two menisci are of the same type, but not similar (e.g. one plate is pushed more into the liquid than the other), then the force is attractive at long distances, and may be repulsive at shorter distances with a stable equilibrium at a finite distance between the plates, depending on the elevations of the plates. Such interaction can be between two hydrophilic or two hydrophobic plates or between a hydrophilic and a hydrophobic plate.

Contour dynamics is used to compute the two-dimensional (f-plane) motion of an initially circularly symmetric barotropic eddy with piecewise-uniform vorticity as it is advected around a circular obstacle by a uniform upstream current. For grazing incidence of this ‘shielded’ eddy (compensating positive and negative vorticity) the main effect of the vortex images (inside the obstacle) is to change the speed of those particles in the outer portion of the eddy that are closest to the obstacle; a lesser velocity is induced on the oppositely signed vortices near the eddy centre. The result is a systematic separation of the centroids of the ± vortices in the eddy, and the eddy emerges far downstream with an invariant dipole moment (m = 1 azimuthal mode). This causes the eddy to move with a constant velocity V normal to the uniform basic flow. The ratio of the numerically computed V to the accompanying far-field dipole moment agrees with a previous analytical theory for a completely isolated eddy subjected to a small-amplitude m = 1 initial disturbance. The scattering effect might be realizable in a rotating homogeneous fluid by translating a cylinder relative to an otherwise stationary eddy. Application to a density-stratified model is suggested.

A new method for treating boundary-value problems in gas-kinetic theory has been developed. The new method has the advantage of reproducing the bulk or asymptotic flow properties accurately whilst giving a realistic description of the behaviour of the molecular distribution function in the neighbourhood of a wall. As an example, the Kramers, or slip-flow, problem is solved for a general specular-diffuse boundary condition and some new expressions for the slip coefficient, flow speed and molecular distribution function at the surface are derived.

A brief discussion of the eigenvalue spectrum of the associated Boltzmann equation is given and its physical significance pointed out.

Certain analogies between this problem and the Milne problem in neutron transport theory are demonstrated.

If the viscosity and specific weight of a fluid are variable, the equations governing its flow in a porous medium are non-linear and in general very difficult to solve. It has been found, however, that steady flows of a fluid of variable viscosity but constant specific weight can be reduced to those of a homogeneous fluid by a remarkably simple transformation, which indicates that the flow patterns of the fluid are the same as those of a homogeneous fluid with the same boundary conditions, and that only the speed need be modified. The speed of the actual flow is obtained by dividing the speed of the homogeneous-fluid flow by a factor proportional to the actual viscosity. The transformation is also used to derive the equations governing steady two-dimensional flows and steady axisymmetric flows of a fluid of variable viscosity and specific weight. In a good many cases of practical importance these equations are exactly linear, in spite of the fact that the governing equations obtained without the use of the above-mentioned transformation are non-linear. An exact solution for a steady two-dimensional flow with prescribed boundary conditions is given. Two inverse methods for generating exact solutions for two-dimensional flows are presented, together with two illustrative examples. The theory also applies to Hele-Shaw flows, so that it can be easily verified in the laboratory.

This work aims at understanding the viscous effects of the outer potential flow on Prandtl's boundary layer. For a body moving with a constant velocity in an otherwise quiescent liquid, the non-zero viscous dissipation of the outer potential flow gives rise to an additional drag, increasing the drag calculated from the boundary layer alone. The drag is considered in three cases here, on a two-dimensional circular gas bubble in a streaming flow, at the edge of the boundary layer around a rapidly rotating cylinder in a uniform flow, and on an airfoil in a streaming flow. The drag may be computed using the dissipation method or the viscous pressure correction of the irrotational pressure. Such a pressure correction can be induced by the discrepancy between the irrotatinal shear stress and the zero shear stress at a fluid–gas interface, or by the discrepancy between the shear stress evaluated from the boundary-layer solution and that evaluated from the outer potential flow solution at the edge of the boundary layer.

An analytical model, based on the Fokker–Planck equation, is constructed of the dye visualization expected near a three-dimensional stagnation point in a swirling fluid flow. The model is found to predict dye traces that oscillate in density and position in the meridional plane in which swirling flows are typically visualized. Predictions based on the model are made for the steady vortex breakdown bubble in a torsionally driven cylinder and compared with computational fluid dynamics predictions and experimental observations. Previous experimental observations using tracer visualization techniques have suggested that even for low-Reynolds-number flows, the steady vortex breakdown bubble in a torsionally driven cylinder is not axisymmetric and has an inflow/outflow asymmetry at its tail. Recent numerical and theoretical studies show that the asymmetry of the vortex breakdown bubble, and consequently its open nature, can be explained by the very small imperfections that are present in any experimental rig. Distinct from this, here it is shown that even for a perfectly axisymmetric flow and breakdown bubble, the combined effect of dye diffusion and the inevitable small errors in the dye injection position lead to the false perception of an open bubble structure with folds near the lower stagnation point. Furthermore, the asymmetries in the predicted flow structures can be remarkably similar to those observed in flow observations and computational predictions with geometric asymmetries of the rig. Thus, when interpreting dye-visualization patterns in steady flow, even if axisymmetric flow can be achieved, it is important to take into account the relative diffusivity of the dye and the accuracy of its injection.

Theoretical formulation describing the wave generation by a slender solid landslide is presented. The slide motion is coupled with the surface wave dynamics and both the fluid viscous (lubricative) resistance and the solid friction are considered.

Numerical results are obtained for a cosine function shape slide. Numerical results show clearly the importance of the dynamic nonlinear coupling. The effects of lubricative resistance, solid friction, initial landslide speed and the slide height are discussed.

In this paper we investigate some effects of a boundary forcing on 2-dimensional cellular patterns in instabilities above threshold. Boundary forcing is modelled as an inhomogeneous boundary condition on the slowly varying amplitude A, i.e. $A = \lambda {\rm e}^{{\rm i}\phi_0}$ on boundaries. The relevant range is λ = O(ε½), where ε is the relative distance to the linear-instability threshold. A wavenumber-selection mechanism then occurs, resulting in a band of selected wavenumbers of width proportional to λ. For large values of λε−½ it is shown that no stationary solution exists outside the band of Eckhaus-stable wavenumbers (Eckhaus 1965). For finite geometries of size L, a nonlinear analogue of ‘quantization’ of modes is investigated. The amplitude equation (equivalent to a space-dependent Ginzburg–Landau model) is analysed by an expansion in powers of exp(-L/ζ), where ζ is the coherence length. The range λ = O(ε) is also investigated. A correction to previous theories of wavenumber selection through boundaries (Cross, Daniels, Hohenberg & Siggia 1983a; Pomeau & Zaleski 1981) is calculated. The latter results are general and assume only the existence of a higher-order stationary amplitude equation, which is recast in a form consistent with its boundary conditions.

The effect of an applied transverse magnetic field on the development of a two-dimensional jet of incompressible fluid is examined. The jet is prescribed in terms of its mass flux ρQ0 and its lateral scale d at an initial section x = 0. The three dimensionless numbers characterizing the problem are a Reynolds number R = Q0/ν, a magnetic Reynolds number Rm = μσQ0, and a magnetic interaction parameter N = σB20d2/ρQ0, where ρ represents density, σ conductivity, μ permeability and B0 applied field strength, and it is assumed that \[ R_m \ll 1,\quad R\gg 1,\quad N\ll 1. \] It is shown that when M2 = RN [Gt ] 1, an inviscid treatment is appropriate, and that the effect of the magnetic field is then to destroy the jet momentum within a distance of order N−1 in the downstream direction. A general solution for inviscid development is obtained, and it is shown that a large class of velocity profiles (though not all of them) are self-preserving.

When M2 [Lt ] 1, it is shown that the viscous similarity solution obtained by Moreau (1963a, b) is relevant. This solution is re-derived and re-interpreted; it implies that the jet momentum is destroyed within a distance of order $R^{\frac{1}{4}}N^{-\frac{3}{4}}$ in the downstream direction.

Some further aspects of the jet annihilation problem are qualitatively discussed in § 4, viz. the nature of the overall flow field, the effect of the presence of distant boundaries, the effect of increasing Rm to order unity and greater, and the effect of oblique injection. Finally the development of a jet of conducting fluid into a nonconducting environment is considered; in this case the jet is not stopped by the magnetic field unless a return path outside the fluid for the induced current is available.

The extended channel model derived and analysed in two previous articles is further developed by investigating free oscillations and forced motion in natural enclosed basins. Firstly, a zeroth-order model is analysed. In this model, field variables are expressed as a product of a single known cross-sectional shape function and an unknown function of time and the co-ordinate along the lake axis. Conditions are discussed under which this zeroth-order model is meaningful, and it is shown that under normal circumstances Coriolis effects must be ignored. Subsequently the general Nth-order channel model is applied to the Lake of Lugano. It is shown that eigedrequencies and amphidromic systems are well predicted in such channel-like lakes. The paper ends with a discussion on the selection of shape functions and with further applications and limitations of the channel model.

We report on new experiments designed to investigate bed destabilization processes in a two-dimensional wave flume physical model of a beach. The mobile bed consists of non-cohesive granular material of low density. The wave conditions are provided by repeating a cycle of waves made of two bichromatic groups of different period. The horizontal and vertical velocities are acoustically profiled vertically from free-stream elevation down to the still bed level in the surf zone. Additional measurements of the fluid pressure at positions closely aligned horizontally and vertically in and slightly above the sediment bed are undertaken. Mobile bed interfaces, still bed and top interface, are detected via acoustic and optical methods. Both methods are cross-compared and give similar results. Flow turbulence over the bed is analysed, the Reynolds turbulent shear stress is found negligible compared to the orbital flow induced momentum diffusion. The shear stress and the horizontal pressure gradient are computed at near-bed elevation and used in the bed incipient plug flow model of Sleath (Cont. Shelf Res., vol. 19 (13), 1999, pp. 1643–1664). Both the model and the measurements confirm that destabilization occurs when the non-dimensional pressure gradient (or Sleath number) exceeds the threshold value of 0.3 which is simultaneous with strong flow acceleration. The near-bottom fluid shear stress detected during these flow accelerations at steep wave fronts is found experimentally to be negative, which is retrieved with an unsteady plug flow model. This is suggesting that the fluid above the bed resists the sediment layer motion at these particular phases.

The asymptotic solution for the incompressible viscous flow past a semi-infinite flat plate constructed by Goldstein (1956, 1960) can be valid only if the solutions of certain ordinary differential equations obey certain constraints (given in Goldstein 1956, 1960). In this paper, we construct the solutions of these equations, show that the necessary constraints are met, and hence establish the validity of the asymptotic solution up to the order considered. The manner in which undetermined constants appear in the solution are discussed.

The thermocapillary and shear-induced instabilities of a thin heated layer of liquid bounded from the top by a deformable free surface and at the bottom by a horizontally oscillating plate are studied for both Earth-bound and microgravity conditions. Finite-wavelength thermocapillary convection can be stabilized very significantly by the oscillatory shear, whereas shear-induced instabilities are greatly stabilized if the Marangoni number is negative. For long-wavelength thermocapillary convection, oscillatory shear can stabilize or destabilize the basic state, depending primarily on the imposed forcing frequency. With microgravity, significant stabilization of the dominant long-wavelength convection can be achieved by carefully selecting the imposed frequency.

The present paper treats the large Reynolds number scaling of coherent structures in stably stratified flows sheared between two horizontally placed walls. Three-dimensional steady solutions are used to confirm the theoretical scaling. For small values of the Richardson number, the previously known scaling based on the vortex–wave interaction/self-sustaining process is found to give excellent predictions of the numerical results. When the Richardson number is increased, the maximum size of the vortices is limited by the Ozmidov scale. The largest possible Richardson number to sustain the vortices is predicted to be of order unity when the typical length scale of the vortices reaches the Kolmogorov scale. The minimum-scale vortices are governed by unit Reynolds number Navier–Stokes equations.

A homogeneous turbulent shear flow in its asymptotic stage of development was subjected to an additional (longitudinal) strain by passing the flow through gradual contraction in the direction perpendicular to that of the mean shear. Two contractions, of area ratio 1.4 and 2.6, were used. Mean velocity and turbulent stress (both normal and shear) distributions were measured at several streamwise locations in the contraction region. The mean velocity distributions agree quite well with calculations based on the (inviscid) Bernoulli equation. Until at least half-way down the contraction with the larger area ratio, the rapid-distortion calculations considering only the streamwise acceleration were found to be reasonably successful in predicting the turbulent intensities. For the smaller-area-ratio contraction, corrections for the ‘natural development’ of the shear flow become important nearly everywhere. Similar calculations considering the shear as the only straining mechanism are generally less successful, although the shear strain rate is at least as rapid as, or even more so than, the longitudinal one. The pressure-rate-of-strain covariance terms estimated from the approximate component energy balance were used to test the adequacy of three models with varying degrees of complexity. Although none of these models appears general enough, their performance is generally adequate for the lower-area-ratio contraction; perhaps not surprisingly, the more complex the model the better its performance.