The dynamic contact angle for a contact line moving over a solid surface with random sparse spots of roughness is determined theoretically in the limit of zero capillary number. The model exhibits many of the observed characteristics of moving contact lines on real rough surfaces, including contact-angle hysteresis and stick-slip. Several types of rough surface are considered, and a comparison is made between periodic and random rough surfaces.

Experimental measurements are presented for the hydrodynamic torque exerted on a stationary sphere situated at the axis of a slowly rotating viscous liquid at small rotary sphere Reynolds numbers (Re < 0.1) as a function of depth of submersion of the sphere below the free surface. Effects of free-surface proximity on the torque furnished the impetus for the study. Experiments were performed for different depths of sphere immersion beneath the free surface, varying from full to partial submersion. Rotation rates were maintained sufficiently low to approximate a planar interface. Torque measurements agreed well with existing theoretical predictions for both the interface-straddling and fully submerged sphere cases. In particular, the predicted continuity of the torque and its derivative at the interface-penetration point (where the sphere first starts to protrude through the free surface) was observed. Free-surface curvature as well as meniscus-curvature effects upon the torque were found to be negligible in the experiments, including even the extreme case where the sphere was in the almost fully withdrawn configuration.

An experimental study has been made of the development of a turbulent boundary layer in an adverse pressure gradient after an interaction with a normal shock wave that was strong enough to separate the boundary layer locally. The pressure gradient applied to the layer was additional to the pressure gradients induced by the shock wave. Measurements were taken for several hundreds of layer thicknesses downstream of the interaction. To separate the effects of shock wave and pressure gradient a second set of observations were made in a reference layer that developed in the same adverse pressure gradient without first interacting with a normal shock wave. It is shown that the adverse pressure gradient impressed on the flow downstream of the shock has a major effect on the structure of the interaction region and the growth of the layer through it. Consequently, existing results for interactions without a postshock pressure gradient should not be used as a model for predicting practical flows, which typically have strong pressure gradients applied downstream of the shock wave. It is also shown that the shock wave produces a pronounced stabilizing effect on the downstream flow, which can be attributed to the streamwise vortices shed into the flow from the separated region formed by the shock wave. The implications of this result for nominally two-dimensional flow situations and to flows involving weak interactions without local separations are discussed.

Attached cavitation on a wall with continuous curvature is investigated on the basis of experiments carried out on various bodies (circular and elliptic cylinders, NACA 16 012 foil). Visualization of the boundary layer by dye injection at the leading edge shows that a strong interaction exists between attached cavitation and the boundary layer. In particular, it is shown that the cavity does not detach from the body at the minimum pressure point, but behind a laminar separation, even in largely developed cavitating flow. A detachment criterion which takes into account this link between attached cavitation and boundary layer is proposed. It consists of connecting a cavitating potential-flow calculation and a boundary-layer calculation. Among all the theoretically possible detachment points, the actual detachment point is chosen to be the one for which the complete calculation predicts a laminar separation just upstream. This criterion, applied to the NACA foil, leads to a prediction which is in good agreement with experimental results.

Slender laminar and turbulent, plane and axisymmetric jets emerging from orifices in plane or conical walls are studied at large distances from the orifices. The entrainment of momentum coupled with the entrainment of volume into a jet is determined, and its effect on the flow field is analysed by combining inner and outer expansions with a multiple scaling approach.

In turbulent (plane or axisymmetric) jets, the axial velocity decreases more rapidly than predicted by classical boundary-layer solutions, and the momentum flux vanishes as the distance from the orifice tends to infinity. The analysis unveils a source of discrepancies in previous experimental data on turbulent jets.

In a laminar plane jet, the momentum flux changes but little. In a laminar axisymmetric jet, the momentum flux changes slowly, yet considerably. When a critical distance from the orifice is approached, the momentum flux in the jet becomes very small, the jet diameter very large, and a toroidal viscous eddy is predicted. The structure of the flow field is briefly discussed.

The flow induced by a laminar round glycerol jet emerging from a plane wall is investigated. The flow field (jet region as well as outer flow field) is visualized by means of very small bubbles suspended in the fluid. For slender jets the observed flow pattern disagrees with both potential-flow theory and Squire's ‘exact’ solution, but it is in excellent agreement with results of an asymptotic analysis by Schneider (1981). At a downstream distance that depends exponentially on the square of the Reynolds number, the jet develops into a toroidal eddy as predicted by Schneider (1985). In an apparatus of reasonable size the eddy can be observed only in a very limited Reynolds-number regime.

The steady compressible flow of an ideal gas in a rotating annulus with thermally conducting walls is considered for small Rossby number ε and Ekman number E and moderate rotational Mach numbers M. Attention is focused on nonlinear effects which show up when σ and εM2 are not small (σ = ε/HE½, H is the dimensionless height of the container). These effects are not properly predicted by the classical linear perturbation analysis, and are treated here by quasi-linear extensions.

The extra work required by these extensions is only the numerical solution of one ordinary differential equation for the pressure.

Numerical solutions of the full Navier–Stokes equations in the nonlinear range are presented, and the validity of the present approach is confirmed.

An asymptotic scheme is derived for calculating values of the ‘reflected’ Stokeslet-field velocity dyadic V and its gradient ${\boldmath \nabla V}$ back at the Stokeslet location for situations in which this singular point lies in close proximity to the wall of an infinitely long circular cylinder. The asymptotic formulas furnished by this scheme permit calculations of first- and second-order wall effects in the non-dimensional parameter κ = a/R0 (a ≡ characteristic particle radius, R0 ≡ cylinder radius) upon the Stokes resistance of a particle of arbitrary shape, location and orientation when translating and/or rotating near the wall of an otherwise quiescent fluid-filled or fluid-surrounded circular cylinder. This reflection-type calculation is applicable for circumstances in which the inequalities κ [Lt ] 1 and 1 − β [Lt ] 1 are each separately satisfied, while simultaneously K/(1 − β) [Lt ] 1. (Here β = b/R0 is the fractionally eccentric Stokeslet location, or equivalently the centre of reaction of the particle, with b its distance from the tube axis.) The main result of this paper is the development of the pair of asymptotic wall-correction formulas \[ W_{jk}\sim {}_0C_{jk}(1-\beta)^{-1}+{}_1C_{jk}+{}_2C_{jk}(1-\beta)+O(1-\beta)^2 \] and \[ W_{jk,\,l}\sim {}_0D_{jkl}(1-\beta)^{-2}+{}_1D_{jkl}(1-\beta)^{-1}+{}_2D_{jkl}+O(1-\beta) \] to particle resistance, with W ≡ Wjk and ${\boldmath \nabla W}\equiv W_{kl,j}$ respectively the non-dimensional normalized wall-effect dyadic and its gradient at the Stokeslet location β. The numerical, rational fraction, β-independent, nCjk and nDjkl coefficients (n = 0, 1, 2) appearing above are evaluated by solving a recursive sequence of Stokes-flow boundary-value problems in the semi-infinite fluid domain bounded by a plane wall. These simple asymptotic formulas are shown to agree excellently in the range near β = 1 with existing values derived from the exact solution of the original circular-cylinder boundary-value problem, involving tedious infinite-series summations of complicated Bessel-function integrands extended over infinite integration domains. Generalizations of the scheme to particle motion in the space external to a circular cylinder is briefly sketched, as too is the case of cylinders of non-circular cross-section.

The Reynolds-averaged Navier–Stokes equations are solved numerically for supersonic flow over a blunt fin mounted on a flat plate. The fin shock causes the boundary layer to separate, which results in a complicated, three-dimensional shock-wave and boundary-layer interaction. The computed results are in good agreement with the mean static pressure measured on the fin and the flat plate. The main features, such as peak pressure on the fin leading edge and a double peak on the plate, are predicted well. The role of the horseshoe vortex is discussed. This vortex leads to the development of high-speed flow and, hence, low-pressure regions on the fin and the plate. Different thicknesses of the incoming boundary layer have been studied. Varying the thicknesses by an order of magnitude shows that the size of the horseshoe vortex and, therefore, the spatial extent of the interaction are dominated by inviscid flow and only weakly dependent on the Reynolds number. Coloured graphics are used to show details of the interaction flow field.

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.

The effects of internal circulation in bubbles and droplets have been analysed by means of a semi-analytical series-truncation method. The equations of motion are transformed into a series of coupled, ordinary, nonlinear differential equations by use of orthogonal sets. These infinite-series equations are then truncated adequately and solved numerically. Using this series-truncation method, we have evaluated the effects of different ratios (between the continuous and dispersed phases) of both density and viscosity for the flows of low Reynolds numbers. For all the density ratios investigated, the density difference has almost no effect on the drag coefficient at low Reynolds numbers. The shear stress and the drag coefficient increase with increasing viscosity ratio of droplet to ambience and decrease with increasing Reynolds number.

An extensive set of numerical simulations is performed to synthesize the behaviour of a barotropic flow over isolated topography on an f-plane and on a β-plane. The model is based on the quasi-geostrophic vorticity equation, where the dissipation terms have been retained. The use of open boundary conditions. following the method described by Orlanski (1976), allows detailed simulation of time-dependent flows over long periods.

On the f-plane, the ultimate solution is always characterized by a typical vorticity field with an anticyclonic vortex trapped over the topography, but different transient regimes occur, related to the importance of advection versus topography effect: direct advection of the positive vortex for strong flows; eddy interactions and double-vortex-structure appearance for weaker flows; oscillatory regimes with topographic trapped-waves generation for very strong vorticity-interaction cases.

On the β-plane, and for prograde flows, the situation is complicated by a Rossby wave pattern extending mainly downstream but also having an upstream component corresponding to zonal waves. For retrograde flows the obstacle does not excite Rossby waves but a transient response with zonal waves whose lifetime depends on the nonlinearity.

The hydrodynamic interaction between two spheres in a Brinkman medium has been calculated using both the method of reflections and the boundary collocation technique. In particular, calculation of the forces and dipoles for two spheres in a uniform stream and linear field show that the method of reflections converges more rapidly than in the Stokes case, owing to screening of interactions, and that the boundary collocation technique produces accurate solutions at almost all separations (except touching) with relatively few collocation points.

The permeability of a random array of fixed spheres has been calculated over the range of volume fractions from dilute to almost closest packing, by assuming pairwise-additive (low-Reynolds-number) hydrodynamic interactions within an effective medium. Non-convergent pair interactions arising from the long-range decay of the Stokeslet were removed by renormalizing the Stokes equation to determine the permeability of the effective medium, i.e. to include the mean screening effect of the other spheres. Pair interactions in this Brinkman medium were calculated by the method of reflections in the far field and boundary collocation in the near field.

The permeability predicted by the theory asymptotes correctly to established results for dilute arrays, and compares favourably (within 15%) with the Carman correlation for volume fractions between 0.3 and 0.5. The magnitude also falls within the range of exact results for periodic arrays at the higher concentrations, but our model does not reproduce the dependence on structure.

Use of the Brinkman equation with an effective viscosity leads to an apparent slip velocity at the boundary of a porous medium. Our calculation of the bulk stress via volume averaging determines the effective viscosity and hence the slip coefficient unambiguously for dilute porous media. However, at concentrations corresponding to the available experimental results the lengthscale characterizing pressure or velocity gradients becomes comparable to the interparticle spacing, and the averaging technique fails. Indeed the Brinkman equation itself is no longer valid.

Interactions between a few fingers in sharply stratified Hele-Shaw flow are investigated by numerical integration of the initial-value problem. It is shown that fingers evolving from an initial perturbation of an unstable interface consisting of a single wave are rather insensitive to variations of the control parameters governing the flow. Initial perturbations with at least two waves, on the other hand, lead to important finger-interaction and selection mechanisms at finite amplitude. On the basis of the results reported here many features of an earlier numerical study of the ‘statistical-fingering’ regime can be rationalized.

In an earlier paper (Walton 1982) we showed that, when a fluid layer is heated non-uniformly from below in such a way that the vertical temperature difference maintained across the layer is a slowly varying monotonic function of a horizontal coordinate x, then convection occurs for x > xc, where xc is the point where the local Rayleigh number is equal to the critical value for a uniformly heated layer. Furthermore, the amplitude of the convection increases smoothly from exponentially small values for x [Lt ] xc and asymptotes to a value given by Stuart–Watson theory for x [Gt ] xc.

At the present time no solutions of this kind are available for a class of problems in which the onset of instability is heavily influenced by a shear flow (e.g. Görtler vortices in a boundary layer on a curved wall, convection in a heated Blasius boundary layer). In a first step to bridge the gap between these problems and in order to elucidate the difficulties associated with the presence of a shear flow, we investigate the effect of a (weak) shear flow on our earlier convection problem. We show that the onset of convection is delayed and that it appears more suddenly, but still smoothly. The role of horizontal diffusion is shown to be of paramount importance in enabling a solution of this kind to be found, and the implications of these results for instabilities in higher-speed shear flows are discussed.

This paper uses numerical simulation to analyse the effects of uniform rotation on homogeneous turbulence. Both large-eddy and full simulations were made. The results indicate that the predominant effect of rotation is to decrease the rate of dissipation of the turbulence and increase the lengthscales, especially those along the axis of rotation. These effects are a consequence of the reduction, due to the generation of inertial waves, of the net energy transfer from large eddies to small ones. Experiments are also influenced by a more complicated interaction between the rotation and the wakes of the turbulence-generating grid which modifies the nominal initial conditions in the experiment. The latter effect is accounted for in simulations by modifying the initial conditions. Finally, a two-equation model is proposed that accounts for the effects of rotation and is able to reproduce the experimental decay of the turbulent kinetic energy.

This paper presents a comparison between theory and experiment for the in-line forces on cylinders of general cross-section in planar oscillatory flows of small amplitude. The theoretical analysis evaluates corrections to the standard inviscid inertial force at low Keulegan-Carpenter numbers which arise from the presence of viscous laminar boundary layers and from the development of vortex shedding. The boundary-layer contribution due to both skin friction and displacement effects is calculated to first order in the Stokes parameter β−½. The contribution to the in-line force from separation and vortex shedding, for which the results presented only apply to sharp-edged bodies, is taken from previous work on vortex shedding from isolated edges using the discrete vortex modelling technique. The resulting force has components both in phase with the fluid acceleration (inertia) and in phase with the velocity (drag).

The theoretical results are compared to measurements taken in a [xcup ]-tube water channel on a number of cylinders of different cross-section including circular cylinders and sharp-edged sections. The comparisons suggest that the theory is valid for Keulegan–Carpenter numbers below about 3 and for moderately high values of the β parameter.

The paper reports numerical solutions to a semi-elliptic truncation of the Navier–Stokes equations for the case of developing laminar flow in circular-sectioned bends over a range of Dean numbers. The ratios of bend radius to pipe radius are 7:1 and 20:1, corresponding with the configurations examined experimentally by Talbot and his co-workers in recent years. The semi-elliptic treatment facilitates a much finer grid than has been possible in earlier studies. Numerical accuracy has been further improved by assuming radial equilibrium over a thin sublayer immediately adjacent to the wall and by re-formulating the boundary conditions at the pipe centre.

Streamwise velocity profiles at Dean numbers of 183 and 565 are in excellent agreement with laser-Doppler measurements by Agrawal, Talbot & Gong (1978). Good, albeit less complete, accord is found with the secondary velocities, though the differences that exist may be mainly due to the difficulty of making these measurements. The paper provides new information on the behaviour of the streamwise shear stress around the inner line of symmetry. Upstream of the point of minimum shear stress, our numerical predictions display a progressive shift towards the result of Stewartson, Cebici & Chang (1980) as the Dean number is successively raised. Downstream of the minimum, however, in contrast with the monotonic approach to an asymptotic level reported by Stewartson, the numerical solutions display a damped oscillatory behaviour reminiscent of those from Hawthorne's (1951) inviscid-flow calculations. The amplitude of the oscillation grows as the Dean number is raised.

The flow-visualization methods of dye injection, hydrogen-bubble generation and paraffin mist are employed to investigate radial flow between parallel circular disks with a steady influx. Three distinct flow patterns are observed in the range of Re between 1.5 and 50. (1) Steady flow without boundary-layer separation and re-attachment, for Re < Rec. (2) A self-controlled flow oscillation which decays further downstream, in the range of Rec [les ] Re < Ret. (3) A self-sustained flow fluctuation which develops into a laminar-turbulent transition with a reverse transition further downstream, when Re [ges ] Ret. Rec and Ret are the critical and transition Reynolds number, respectively.

The oscillating flows are caused by a vortex street consisting of vortices (i.e. separating annular bubbles) that separate periodically and alternately from both disks. Finite-difference solutions of the unsteady vorticity transport equation broadly agree with certain experimental observations. The study concludes that the separation and reattachment of shear layers in the radial flow through parallel disks are unsteady phenomena and the sequence of nucleation, growth, migration and decay of the vortices is self-sustained.