We investigate the changes to a fully developed turbulent boundary layer caused by the presence of a two-dimensional moving wave of wavelength L = 2π/k and amplitude a. Attention is focused on small slopes, ak, and small wave speeds, c, so that the linear perturbations are calculated as asymptotic sequences in the limit (u* + c)/UB(L) → 0 (u* is the unperturbed friction velocity and UB(L) is the approach-flow mean velocity at height L). The perturbations can then be described by an extension of the four-layer asymptotic structure developed by Hunt, Leibovich & Richards (1988) to calculate the changes to a boundary layer passing over a low hill.

When (u* + c)/UB(L) is small, the matched height, zm (the height where UB equals c), lies within an inner surface layer, where the perturbation Reynolds shear stress varies only slowly. Solutions across the matched height are then constructed by considering an equation for the shear stress. The importance of the shear-stress perturbation at the matched height implies that the inviscid theory of Miles (1957) is inappropriate in this parameter range. The perturbations above the inner surface layer are not directly influenced by the matched height and the region of reversed flow below zm: they are similar to the perturbations due to a static undulation, but the ‘effective roughness length’ that determines the shape of the unperturbed velocity profile is modified to zm = z0 exp (kc/u*).

The solutions for the perturbations to the boundary layer are used to calculate the growth rate of waves, which is determined at leading order by the asymmetric pressure perturbation induced by the thickening of the perturbed boundary layer on the leeside of the wave crest. At first order in (u* + c)/UB(L), however, there are three new effects which, numerically, contribute significantly to the growth rate, namely: the asymmetries in both the normal and shear Reynolds stresses associated with the leeside thickening of the boundary layer, and asymmetric perturbations induced by the varying surface velocity associated with the fluid motion in the wave; further asymmetries induced by the variation in the surface roughness along the wave may also be important.

This paper gives analytical and numerical solutions for both westward and eastward flows past obstacles on a β-plane. The flows are considered in the quasi-geostrophic limit where nonlinearity and viscosity allow deviations from purely geostrophic flow. Asymptotic solutions for the layer structure in almost-inviscid flow are given for westward flow past both circular and more elongated cylindrical obstacles. Structures are given for all strengths of nonlinearity from purely linear flow through to strongly nonlinear flows where viscosity is negligible and potential vorticity conserved. These structures are supported by accurate numerical computations. Results on detraining nonlinear western boundary layers and corner regions in Page & Johnson (1991) are used to present the full structure for eastward flow past an obstacle with a Huff rear face, completing previous analysis in Page & Johnson (1990) of eastward flow past obstacles without rear stagnation points. Viscous separation is discussed and analytical structures proposed for separated flows. These lead to predictions for the size of separated regions that reproduce the behaviour observed in experiments and numerical computations on β-plane flows.

In flows of granular material, collisions between individual particles result in the movement of particles in directions transverse to the bulk motion. If the particles were distinguishable, a macroscopic overview of the transverse motions of the particles would resemble a self-diffusion of molecules as occurs in a gas. The present granular- flow study includes measurements of the self-diffusion process, and of the corresponding profiles of the average velocity and of the streamwise component of the fluctuating velocity. The experimental facility consists of a vertical channel fed by an entrance hopper that is divided by a splitter plate. Using differently-coloured but otherwise identical glass spheres to visualize the diffusion process, the flow resembles a classic mixing-layer experiment. Unlike molecular motions, the local particle movements result from shearing of the flow; hence, the diffusion experiments were performed for different shear rates by changing the sidewall conditions of the test section, and by varying the flow rate and the channel width. In addition, experiments were also conducted using different sizes of glass beads to examine the scaling of the diffusion process. A simple analysis based on the diffusion equation shows that the thickness of the mixing layer increases with the square-root of downstream distance and depends on the magnitude of the velocity fluctuations relative to the mean velocity. The results are also consistent with other studies that suggest that the diffusion coefficient is proportional to the particle diameter and the square-root of the granular temperature.

Upward flow of air can support a thin layer of liquid on a plane wall against gravity. Such apparently stationary layers are for example sometimes seen on the windscreen of a car travelling at high speed in rain. We solve here the two-dimensional case of a layer whose length is finite, but significantly greater than the meniscus length. The flow is steady, with a fixed layer boundary, inside which there is a steadily circulating viscous liquid, and outside which the air exerts a traction which is assumed to have a known (small) constant drag coefficient CD. The air also exerts a non-uniform pressure on the liquid layer, of a magnitude determined by the shape of the layer, and the relationship between these two quantities can be obtained by thin-airfoil theory. In the lubrication approximation, the problem can be reduced to a nonlinear singular integro-differential equation to determine the unknown shape of the layer boundary. This equation is solved numerically for various (small) wall angles, for cases where the effect of surface tension is confined to a small meniscus region near the layer's leading edge. The numerical results indicate that solutions exist only for walls whose inclination is less than 0.70 C½D, and, for a range of inclinations below that maximum value, that two distinct steady solutions can exist at each inclination.

A new simulation method is presented for low-Reynolds-number flow problems involving elongated particles in an unbounded fluid. The technique extends the principles of Stokesian dynamics, a multipole moment expansion method, to ellipsoidal particle shapes. The methodology is applied to prolate spheroids in particular, and shown to be efficient and accurate by comparison with other numerical methods for Stokes flow. The importance of hydrodynamic interactions is illustrated by examples on sedimenting spheroids and particles in a simple shear flow.

The short-time limit of the hydrodynamic transport properties is calculated for crystalline dispersions of parallel prolate spheroids using a moment expansion technique similar in concept to the simulation method known as Stokesian dynamics. The concentration dependence of the sedimentation rate, the hindered diffusivity and the Theological behaviour of face-centred lattices are examined for concentrations up to regular close packing (74% by volume). The influence of the detailed microstructure of the dispersion is also investigated by considering different arrangements of parallel ellipsoids. Useful reference configurations are proposed as standard geometries for regular arrays of prolate spheroids.

We present an experimental study of the scaling laws for the front (or cap) of an isolated, laminar starting plume. The scaling relations are formulated and measured experimentally over a range of power, fluids, and heaters. The results are that the cap rises at constant velocity, grows diffusively in width, and its temperature depends inversely on height. This extends analytic results by Batchelor (1954) for the column (stem) below the front. The source size determines initial conditions for the cap, but does not affect it in the far field. The shape of the front is fitted by a model of potential flow. The interaction between plume caps is complex, but with simple underlying dynamics. We conjecture that some of our conclusions can be applied to a distribution of plumes, as in soft turbulent convection.

A spectacular feature of transient cavity collapse in the neighbourhood of a rigid boundary is the formation of a high-speed liquid jet that threads the bubble and ultimately impacts upon the side of the bubble nearest to the boundary. The bubble then evolves into some toroidal form, the flow domain being doubly connected. In this work, the motion of the toroidal bubble is computed by connecting the jet tip to the side of the bubble upon which it impacts. This connection is via a cut introduced into the flow domain and across which the potential is discontinuous, the value of this discontinuity being equal to the circulation in the flow. A boundary integral algorithm is developed to account for this geometry and some example computations are presented. Consideration of the pressure field in the fluid has implications for possible damage mechanisms to structures due to nearby cavity collapse.

The two-dimensional acoustics problem of the scattering of an obliquely incident plane wave by a row of equally-spaced circular cylinders is solved using multipole expansions. The method is superior to existing techniques available for this problem as it allows the far-field behaviour of the solution to be evaluated in a straightforward manner, and extensive results for the reflection and transmission coefficients are given. The problem described above has a direct analogue in the theory of water waves and this is also discussed.

A theory is developed for time-dependent coherent structures in a marginally stable atmospheric zonal flow. The coherent structures have the form of solitary waves travelling in the zonal direction. Analytical solutions are found for stationary solitary waves but these are shown to be always unstable. The instability manifests itself either as a fission of the structure subsequently emitting two oppositely directed travelling solitary waves or as an implosion in which the structure becomes increasingly more narrow and intense. Which of the two occurs depends sensitively on initial conditions. These solitary waves are stable in head-on collisions only if their joint zonally integrated amplitude is less than a critical value; otherwise, the implosion instability occurs. General initial conditions can give rise to solitary waves which either split, implode, or break down to form a train of nonlinear wave packets. A scenario for the birth and decay of isolated disturbances is given, utilizing the slow parametric transit of the marginal stability curve of the background zonal flow.