The effect of surface topography on an otherwise two-dimensional boundary-layer flow is investigated. The flow is assumed to be steady, laminar and incompressible, and is described by triple-deck theory. The basic problem reduces to the solution of a form of the nonlinear three-dimensional boundary-layer equations, together with an interaction condition. The solutions are obtained by a spectral method, with the computations carried out iteratively in Fourier-transform space. Numerical results are presented for several cases including three-dimensional separation. Comparison is made with the predictions of linearized theory. The decay corridor observed by Smith is confirmed for one localized configuration, but not for another having a broader height distribution.

In this paper we investigate the effect of vertical rotation on the linear stability of an unbounded region of vertically stratified fluid, which has compensating horizontal temperature and salinity gradients, so there is no overall horizontal density gradient. We find the most unstable perturbations for given linear vertical and horizontal gradients and show how the addition of rotation affects the results found for the non-rotating case by Holyer (1983), using molecular diffusivities. By using a transformation that, for the non-oscillatory instability, links the rotating case to the non-rotating case we show that the growth rate, the across-front slope and the wavenumber of the intrusion with the maximum growth rate is unchanged. The basic difference between the non-rotating and the rotating case, for the non-oscillatory instability, is that in the rotating case the interleaving layers slope both along and perpendicular to the direction of the horizontal temperature and salinity gradients and not just along them. The oscillatory instability has no simple transformation between the rotating and the non-rotating cases, and the addition of rotation changes the growth rate and the wavenumber of the instabilities.

A method of constructing plane potential flows past doubly periodic arrays of cylindrical obstacles is described. It is based on Rankine's well-known technique of determining by simple superposition the shape of an oval-shaped obstacle simultaneously with the flow round the obstacle. This paper deals with the use of such potentials in calculating effective medium conductivities for a class of doubly periodic two-phase materials which contain either non-conducting or perfectly conducting cylindrical inclusions embedded within a conducting matrix phase.

It is known from recent experiments that the disturbance due to a slender ship advancing in a shallow channel is essentially one-dimensional in the horizontal plane. In particular solitons can be radiated upstream in a transient manner. In this note we develop a theory for soliton radiation by slender bodies. It is shown that, when the ship speed is in the transcritical range, one-dimensional upstream influence can occur even when the channel width is nearly of the order of the ship length but much greater than the ship beam. The theory is also extended to one or more ships travelling in the same channel at near-critical speeds.

Structural characteristics of transitionally rough and fully rough turbulent boundary layers are presented. These were measured in flows at different roughness Reynolds numbers developing over uniform spheres roughness. Inner regions of the longitudinal component of normal Reynolds stress profiles and log regions of mean profiles continuously change in the transitionally rough regime, as the roughness Reynolds number, Rek, varies. These properties asymptotically approach fully rough behaviour as Rek increases, and smooth behaviour at low Rek Profiles of other Reynolds-stress tensor components, turbulence kinetic energy, turbulence-kinetic-energy production, and the turbulence-kinetic-energy dissipation are also given, along with appropriate scaling variables. Fully rough, one-dimensional spectra of longitudinal velocity fluctuations from boundary-layer inner regions are similar to smooth-wall results for k1y > 0.2 when non-dimensionalized using distance from the wall y as the lengthscale, and (τ/ρ)½ as the velocity scale, where τ is local shear stress, ρ is static density, and k1 is one-dimensional wavenumber in the flow direction.

This paper is concerned with the evolution of small-amplitude, long-wavelength, resonantly forced oscillations of a liquid in a tank of finite length. It is shown that the surface motion is governed by a forced Korteweg—de Vries equation. Numerical integration indicates that the motion does not evolve to a periodic steady state unless there is dissipation in the system. When there is no dissipation there are cycles of growth and decay reminiscent of Fermi–Pasta–Ulam recurrence. The experiments of Chester & Bones (1968) show that for certain frequencies more than one periodic solution is possible. We illustrate the evolution of two such solutions for the fundamental resonance frequency.

A rotating tank is filled from the outside with a mixture of particles and fluid. Under certain attainable conditions on the times for filling, separation, and spin-up, theory implies that the filling process acts like a centripetal separator in which heavy particles are actually concentrated at the inward-moving front. Centrifugal settling in the interior is counteracted by mass transport in the rotating boundary layers to produce this unusual volume-fraction distribution.

The electrophoretic velocity of a gas bubble is difficult to measure, since we must also contend with velocities due to buoyancy. One way to avoid this problem is to spin the electrophoretic cell about a horizontal axis. Centrifugal forces keep the bubble on the centreline of the cell. The price to be paid is the creation of Taylor columns which alter the hydrodynamic drag on the bubble.

Here we modify two analyses by Moore & Saifman (1968, 1969) to include electrical effects. Motion of the body is assumed to be slow and steady, and the Ekman number small. The electrical double-layer thickness is small compared with the thickness of the Ekman layer. It is assumed that the presence of surfactants makes the gas-water interface rigid, and a no-slip boundary condition is applied.

We predict that the electrophoretic velocity U should be proportional to εEψ0(aν)−l (ν/Ω)½, where E is the applied electric field, ε the permittivity of the suspending fluid, ψ0 the ζ potential at the surface of the bubble, a the bubble radius, ν the fluid viscosity, ν the kinematic viscosity and Ω the rate of rotation. There is reasonable agreement with some, but not all, published experimental results.

According to the theory of Ikeda, Parker & Sawai (1981), meander migration rate at a point depends on a convolution integral of channel curvature from that point upstream. The problem can be quantified in terms of the bend equation. The time development of periodic bend trains of finite amplitude is analysed using the method of two-timing. The results apply near the critical wavenumber for the growth of bends of infinitesimal amplitude.

A finite-amplitude equilibrium state bifurcating from the null state at the critical wavenumber was delineated by Parker, Diplas & Akiyama (1983): they called the resulting solution the Kinoshita curve. It is found herein that this equilibrium state is unstable. Bends of longer Cartesian wavelength grow to cutoff. Shorter bends are obliterated. Nevertheless, in either case, the bend train tends towards the shape of the Kinoshita curve.

The theory suggests that some growing bends may be stabilized by local obstructions to downstream migration. The obstructions would cause an effective reduction in Cartesian wavelength, moving the bend from the unstable regime to the stable regime. A rather crude check of bend shape and rates of deformation generally lends support to the analysis.

Experimental results are reported on hydrodynamic interactions between a solid plate with a spherical particle attached to it and a rigid sphere moving parallel to the plate. Trajectories and velocities of the moving sphere were determined by taking single-frame multiple-image photographs using stroboscopic light.

Sphere—sphere hydrodynamic interactions were detectable on the background of plate—sphere interactions for initial dimensionless sphere—wall separations Z0 < 4.9. The sphere trajectories were found to be symmetrical for, Z0 ≥ 2.3 and asymmetrical otherwise. For asymmetrical trajectories the sphere velocity was larger after the encounter than prior to it. It was concluded that surface roughness of the spheres was responsible for the observed deviations from symmetry.

Numerical calculations were performed to obtain sphere trajectories and velocities. The calculations agree with the experimental data for dimensionless distances between sphere centres r > 2.5. For r < 2.5 the numerical results were in fair agreement with the data when Z0 [gsim ] 2.9. For smaller Z0, theoretical predictions were inaccurate.

A time-dependent extension of the reduced wave equation of Berkhoff is developed for the case of waves propagating over a bed consisting of ripples superimposed on an otherwise slowly varying mean depth which satisfies the mild-slope assumption. The ripples are assumed to have wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter along with the bottom slope. The theory is verified by showing that it reduces to the case of plane waves propagating over a patch of sinusoidal ripples, which vary in one direction and extend to ± ∞ in the transverse direction, studied recently by Davies & Heathershaw and Mei. We then formulate and use coupled parabolic equations to study propagation over patches of arbitrary form in order to study wave reflection.

Coupled equations governing the forward- and back-scattered components of a linear wave propagating in a region with varying depth may be derived from a second-order wave equation for linear wave motion. In this paper previous studies are extended to the case of weakly nonlinear Stokes waves coupled at third order in wave amplitude, using a Lagrangian formulation for irrotationaj motions. Comparison with previous computational and experimental results are made.

A second-order modelling technique is used to investigate the behaviour of homogeneous scalar turbulence. Special attention is paid to the influence of timescale ratio on scalar flux relaxation. We develop a model for the scalar flux equation in a homogeneous turbulence and consider both a scalar field without mean-scalar gradients and one with constant mean-scalar gradients based on Sirivat & Warhaft (1981) experiments. Good agreement with experiment in all the cases is obtained.

This paper is concerned with theoretical predictions, given the upstream conditions from a rigid obstacle of arbitrary shape, of the downstream flow beyond the obstacle for an incompressible inviscid fluid sheet under the action of gravity. The fluid sheet flows upstream over a level bottom, continues to flow over (or under) an obstacle leading to a downstream region over a level bottom. In the absence of surface tension, a nonlinear st., ady-state solution of the problem is used to predict the downstream values of the free-surface wave height for the full range of the far-upstream Froude number. The general results obtained are then applied to a special case of fluid flowing over a stationary hump leading to a supercritical flow far downstream and detailed numerical comparison is made with available experimental results, with very good agreement.

A laboratory investigation of the influence of wind on the evolution of mechanically generated regular (m.g.r.) waves is reported. Surface elevation measurements were made at four fetches for steep (0.1 < $\overline{ak}$ < 0.2, 2 Hz) m.g.r. waves, moderate (15 < u* < 25 cm s−1, 3–6 Hz) wind waves, and combinations of the m.g.r. and wind waves. The m.g.r. wave spectra exhibit Benjamin—Feir sidebands that grow exponentially with fetch and whose growth rate increases as the initial wave steepness increases. As fetch increases for the wind cases, total energy increases and the frequency of the spectral maximum downshifts, but no spectral lines representing Benjamin—Feir sidebands were detected even though the wave steepness and fetch were similar to the m.g.r. waves whose spectra displayed sidebands. As wind speed increased over the m.g.r. waves, sideband magnitude, sideband growth rate and low-frequency perturbation components associated with the instability mechanism were reduced.

The energetics of an unswept wing of finite span oscillating harmonically in combined pitch and heave in in viscid incompressible flow are determined in closed form. The calculations are based on a recently developed low-frequency unsteady lifting-line theory. The energetic calculations for the wing consist of sectional and total values of thrust, leading-edge suction force, power required to maintain the wing oscillations, and energy-loss rate due to vortex shedding in the wake, where the latter quantity is only defined for the entire wing. These results are used to analyse the optimum motion of a wing oscillating harmonically: optimum motion minimizes the power input for fixed average total thrust. The optimum solution is found to be unique (at least for low reduced frequencies), in contrast to the two-dimensional optimum, which is non-unique. Numerical results are presented for the energetics and optimum motion of an elliptic wing.

To understand better the structure of the known solution for the optimum motion of an oscillating two-dimensional airfoil, the solution is recast in terms of the normal modes of the energy-loss-rate matrix. It is found that one of the modes, termed here the ‘invisible mode’, plays a central role in the optimum solution and is responsible for its non-uniqueness. The three-dimensional optimum, which is unique, does not have an invisible mode.

This paper is devoted to the study of the mean and fluctuating pressure field that develop on the surface of a square cylinder (side D) placed in homogeneous flows with different intensities ($u^{\prime}/\overline{U}_{\infty}$) and ratios of turbulence scales to cylinder scale L11/D. In these experiments $u^{\prime}/\overline{U}_{\infty}$ varies from 3 to 17.5% and L11/D varies from 0.1 to 2. This range of experimental conditions is more extensive than that of Lee (1975). Lee's results are largely confirmed, in particular the influence of varying turbulence intensity on the mean pressure distribution in the recirculation region is much stronger than that of varying the integral scale. The theory proposed by Durbin & Hunt (1980) for the prediction of the pressure spectrum at the stagnation point has been shown to be adequate for low frequencies but only qualitative at high frequencies. Finally, measurements of cross-spectra, taken for different points on the same cross-section, have shown how the fluctuating surface pressure field downwind of separation has a narrowband component at the Strouhal frequency that is sensitive to the turbulence, and a broadband component generated by the recirculating flow that is insensitive to the upwind turbulence.

The spatial decay and structural evolution of grid-generated turbulence under the effect of buoyancy was studied in a ten-layer, salt-stratified water channel. The various density gradients were chosen such that the initial overturning turbulent scale was slightly smaller than any of the respective buoyancy scales. The observed general evolution of the flow from homogeneous turbulence to a composite of fossil turbulence or quasi-two-dimensional turbulence and internal wavefield is in good agreement with the predictions of Gibson (1980) and the lengthscale model of Stillinger, Helland & Van Atta (1983). The effect of the initial size of the turbulent lengthscale compared with the buoyancy scale on the decay and evolution of the turbulence is investigated and the observed influence on the rate of decay of both longitudinal and vertical velocity fluctuations pointed out by Van Atta, Helland & Itsweire (1984) is shown to be related to the magnitude of the initial internal wavefield at the grid. An attempt is made to remove the wave-component kinetic energy from the vertical-velocity-fluctuation data of Stillinger, Helland & Van Atta (1983) in order to obtain the true decay of the turbulent fluctuations. The evolution of the resulting fluctuations is similar to that of the present large-grid data and several towed-grid experiments. The rate of destruction of the density fluctuations (active-scalar dissipation rate) is estimated from the evolution equation for the potential energy, and the deduced Cox numbers are compared with those obtained from oceanic microstructure measurements. The classical Kolmogorov and Batchelor scalings appear to collapse the velocity and density spectra better than the buoyancy scaling proposed by Gargett, Osborn & Nasmyth (1984). The rise of the velocity spectra at low wavenumbers found by Stillinger, Helland & Van Atta (1983) is shown to be related to internal waves.

Several conditional-sampling techniques are applied to data bases generated by large-eddy and direct numerical simulations. It is shown that the bursting process is associated with well-organized vortical structures inclined at about 45° to the wall. These vortical structures are identified by examining the vortex lines of three-dimensional instantaneous and ensemble-averaged vorticity fields. Two distinct horseshoe-shaped vortical structures corresponding to the sweep and ejection events are detected. These vortical structures are associated with high Reynolds shear stress and hence make a significant contribution to turbulent-energy production.

Using the averaging method and a perturbation technique originally due to Melnikov (1963), we show that an N-degree-of-freedom model of weakly nonlinear surface waves due to Miles (1976) has transverse homoclinic orbits. This implies that Smale horseshoes, and hence sets of chaotic orbits, exist in the phase space. In this particular example, an irregular ‘sloshing’ of energy between two modes of oscillation results. We briefly discuss the relevance of our results to recent experimental work on parametrically excited surface waves.