The buoyancy flux ratio, r = αFT/βFS, across a salt-sugar fingering interface is determined from accurate measurements of the density variations in a two-layer run-down experiment. Since r is close to 1, the net buoyancy flux, βFS − αFT, which is proportional to 1 − r, gives r with a resolution that is much improved over that of previous measurements. The value of r is found to decrease from r ≈ 0·94 ± 0·01 at Rρ = 1·02 to r = 0·88 ± 0·01 at Rρ = 2, where Rρ is the density anomaly ratio. The latter is smaller than the previous (constant) value of 0·92. The buoyancy flux due to sugar is found to decrease rapidly with increasing Rρ, varying approximately as R−6ρ.

The motion of the head of a gravity current travelling down a slope of angle θ to the horizontal is investigated in the laboratory. The head is produced by suddenly initiating a buoyancy flux from a line source at the top of the slope. It is found that for very small slopes (θ [les ] 0.5°) the head decelerates with distance from the source, but at greater slopes the buoyancy force is large enough to overcome frictional effects and a steady head velocity results. Over a wide range of slope angles the front velocity Uf, non-dimensionalized by the cube root of the buoyancy flux (g′0Q)1/3, is almost independent of the slope angle and Uf/(g’0Q)1/3 = 1.5 ± 0.2 for 5° [les ] θ [les ] 90°. This result is shown to follow from some simple analysis which relates the velocity of the front to the following flow. For a Boussinesq plume the front velocity is found to be approximately 60% of the mean velocity of the following flow. This means that the head increases in size as it travels down the slope, both by direct entrainment into the head itself and by addition of fluid from the following flow. We find that direct entrainment increases with increasing slope and accounts for one-tenth of the growth of the head at 10° and about two-thirds at 90°.

The waves propagating from an oscillating plane piston into a vibrationally relaxing gas are calculated by an exact numerical method ignoring viscosity and heat conduction. Secondary effects due to the starting of the piston from rest and to acoustic streaming can be eliminated from the calculated flows, leaving a truly periodic progressive wave which can be analysed and compared with approximate solutions. It is found that for moderate amplitude waves nonlinearity is only important as a convective effect which produces higher harmonics, whereas dissipation is adequately described by linear theory.

This paper presents the first ‘exact’ solutions to the creeping-flow equations for the transverse motion of a sphere of arbitrary size and position between two plane parallel walls. Previous solutions to this classical Stokes flow problem (Ho & Leal 1974) were limited to a sphere whose diameter is small compared with the distance of the closest approach to either boundary. The accuracy and convergence of the present method of solution are tested by detailed comparison with the exact bipolar co-ordinate solutions of Brenner (1961) for the drag on a sphere translating perpendicular to a single plane wall. The converged series collocation solutions obtained in the presence of two walls show that for the best case where the sphere is equidistant from each boundary the drag on the sphere predicted by Ho & Leal (1974), using a first-order reflexion theory, is 40 per cent below the true value when the walls are spaced two sphere diameters apart and is one order-of-magnitude lower at a spacing of 1.1 diameters.

Exact solutions are presented for the three-dimensional creeping motion of a sphere of arbitrary size and position between two plane parallel walls for the following conditions: (a) pure translation parallel to two stationary walls, (b) pure rotation about an axis parallel to the walls, (c) Couette flow past a rigidly held sphere induced by the motion of one of the boundaries and (d) two-dimensional Poiseuille flow past a rigidly held sphere in a channel. The combined analytic and numerical solution procedure is the first application for bounded flow of the three-dimensional boundary collocation theory developed in Ganatos, Pfeffer & Weinbaum (1978). The accuracy of the solution technique is tested by detailed comparison with the exact bipolar co-ordinate solutions of Goldman, Cox & Brenner (1967a, b) for the drag and torque on a sphere translating parallel to a single plane wall, rotating adjacent to the wall or in the presence of a shear field. In all cases, the converged collocation solutions are in perfect agreement with the exact solutions for all spacings tested. The new collocation solutions have also been used to test the accuracy of existing solutions for the motion of a sphere parallel to two walls using the method of reflexions technique. The first-order reflexion theory of Ho & Leal (1974) provides reasonable agreement with the present results for the drag when the sphere is five or more radii from both walls. At closer spacings first-order reflexion theory is highly inaccurate and predicts an erroneous direction for the torque on the sphere for a wide range of sphere positions. Comparison with the classical higher-order method of reflexions solutions of Faxen (1923) reveals that the convergence of the multiple reflexion series solution is poor when the sphere centre is less than two radii from either boundary.

Solutions have also been obtained for the fluid velocity field. These solutions show that, for certain wall spacings and particle positions, a separated region of closed streamlines forms adjacent to the sphere which reverses the direction of the torque acting on a translating sphere.

The energetics of a linear array of hollow or stagnant-cored vortices of finite cross-section in an ideal fluid is studied in this paper. The results are useful in a discussion of the amalgamation of vortex structures in a turbulent mixing layer.

The effect of homogeneous strain on passive scalar fluctuations, and the resultant evolution of the scalar field when the strain is removed, is experimentally studied by passing thermal fluctuations in decaying grid turbulence through a four-to-one axisymmetric contraction. Using a mandoline (Warhaft & Lumley 1978a) to vary the scale size of the initial thermal fluctuations and hence the pre-contraction mechanical/thermal time-scale ratio, r, it is shown, for values of r greater than unity, that as r is increased so is the post-contraction thermal decay rate, i.e. the contraction does not cause the thermal-fluctuation decay rate to equilibrate to a constant value. In these experiments the post-contraction thermal decay rate is always greater than the pre-contraction decay rate, i.e. the contraction accelerates the thermal-fluctuation decay. Moreover, the mechanical/thermal time-scale ratio in the post-contraction region is driven further from unity. In terms of scale size the uniform strain has the effect of increasing the thermal length scale by an amount equal in value to the contraction ratio if the pre-contraction thermal length scale is comparable to that of the pre-contraction velocity scale. However, if the pre-contraction thermal length scale is smaller than the pre-contraction velocity scale then the effect of the contraction on the thermal scale is less marked. The contraction induces significant negative cross-correlation ρuθ between the longitudinal velocity u and thermal fluctuations θ even if the pre-contraction cross-correlation is close to zero. The magnitude of ρuθ and hence the post-contraction heat flux is varied and the coherence structure is studied. It is shown that the thermal-fluctuation decay rate is insensitive to the magnitude of the heat flux, the latter of which decays rapidly compared to the relatively slow decay of turbulence energy in the post-contraction region. It is also shown that ρuθ tends towards zero in this axisymmetric homogeneous flow at a faster rate than in isotropic turbulence. In accord with previous investigations, the return toward isotropy of the velocity field is very slow.

The only solutions to date which describe nonlinear resonant acoustic oscillations are those for which the distortion of the travelling waves is negligible. Many experiments do not comply with this small-rate restriction. A finite-rate theory of resonance for an inviscid gas, in which the intrinsic nonlinearity of the waves is taken into account, necessitates the construction of periodic solutions of a nonlinear functional equation. This is achieved by introducing the notion of a critical point of the functional equation, which corresponds physically to a resonating wavelet. In a finite-rate theory a wave may break in a single cycle in the tube, and thus there may be more than one shock present even at fundamental resonance. Discontinuous solutions of the functional equation are constructed which satisfy the weak shock conditions.

A simplified model problem has recently been suggested by Schaeffer (1980) in order to explain the results obtained by Benjamin (1978) in his investigation of Taylor vortices in short cylinders. In particular Schaeffer reproduces the results obtained by Benjamin for cylinders so short that only two-cell or four-cell flows are possible. The model given by Schaeffer has artificial conditions imposed on the fluid velocity field at the end walls. These conditions depend on a parameter α and reduce to the no-slip condition when α = 1. If α = 0 the conditions require that the normal component of the velocity and the normal derivative of the tangential velocity vanish at the ends. In this case the onset of Taylor vortex-like motion occurs as a bifurcation from purely circumferential flow. If α is now taken to be small and positive, there is no bifurcation and the circulatory flow develops smoothly. We shall use perturbations method for the case of small α. The imperfect bifurcation problem which we obtain predicts some results consistent with those of Benjamin.

Experimental results for the release of a fixed volume of one homogeneous fluid into another of slightly different density are presented. From these results and those obtained by previous experiments, it is argued that the resulting gravity current can pass through three states. There is first a slumping phase, during which the current is retarded by the counterflow in the fluid into which it is issuing. The current remains in this slumping phase until the depth ratio of current to intruded fluid is reduced to less than about 0.075. This may be followed by a (previously investigated) purely inertial phase, wherein the buoyancy force of the intruding fluid is balanced by the inertial force. Motion in the surrounding fluid plays a negligible role in this phase. There then follows a viscous phase, wherein the buoyancy force is balanced by viscous forces. It is argued and confirmed by experiment that the inertial phase is absent if viscous effects become important before the slumping phase has been completed. Relationships between spreading distance and time for each phase are obtained for all three phases for both two-dimensional and axisymmetric geometries. Some consequences of the retardation of the gravity current during the slumping phase are discussed.

The low-frequency character of two model problems is exploited in order to illustrate the acoustic consequences of the interactions between chemically reacting (or relaxing) inhomogeneities and flames or constrictions in ducts. The monopole of the former is associated with heat transfer in a fluid which exhibits variations in its specific heats, while in the latter there is an extension of the classical phenomenon associated with the pulsations of an inhomogeneity of the fluid compressibility. This second mechanism is found to be insignificant, but the heat-conduction source is considered to be very powerful at sufficiently low Mach numbers; in fact, to first order it is independent of the flow Mach number for laminar, as well as a certain class of turbulent, flows.

Solutions to the steady advection–diffusion equation in a branching channel are obtained for both uniform and spatially varying flow fields and for two channel geometries. An interesting feature of the solutions is that anisotropy of the dispersion coefficients in the direction of the streamlines may be accounted for. The analysis reveals that mixing is confined to a distance, b2U/π2KN, downstream of the junction in the advection-dominated case and a distance, KS/U, upstream in the diffusion-dominated situation, KS and KN being the diffusivities along and across the flow respectively, U the characteristic velocity of the flow, and b the breadth of channel downstream of the junction.

Flow visualization and laser-doppler anemometry have been used to determine the flow structure and measure the velocity distribution inside a rotating cylindrical cavity with an outer to inner radius ratio of 10, and an axial spacing to inner radius ratio of 2·67. A flow structure comprising an inner layer, Ekman layers, an outer layer and an interior potential core has been confirmed for the cases where the inlet air enters the cavity either axially, through a central hole, or radially, through a central gauze tube, and leaves radially through a series of holes in the peripheral shroud. Velocity measurements in the laminar Ekman layers agree well with the ‘modified linear theory’, and long-and short-wavelength disturbances (which have been reported by other experimenters) have been observed on the Ekman layers when the radial Reynolds number exceeds a critical value. The phenomenon of reverse flow in the Ekman layers and the possibility of ingress of external fluid through the holes in the shroud have also been observed.

The well-known Whitham theory may be applied to shocks diffracting over concave corners, provided that the diffraction results in Mach reflexion. This paper compares the theory with data obtained during experiments with diffracting shocks. If an incident shock is classified as weak or strong in the strict sense defined by von Neumann, then it is found that the Whitham theory accurately determines the Mach number of the Mach stem at the wall for both the weak and the strong cases. The theory also has some further value for strong shocks but not for weak; it is not applicable when the diffraction is regular.

A two-dimensional stability analysis is presented of flow of low Froude number over an erodible bed. Particular regard is given to the modelling of the turbulent flow close to the bed. In contrast to previous theories that use a constant eddy-viscosity approach the present theory predicts the occurrence of two separate modes of instability, with wavelengths related to the roughness of the bed and the depth of the flow. It is postulated that these two modes correspond to the formation of ripples and dunes respectively. The results are strongly dependent on the two parameters z0, the roughness length of the bed, and β, the effect of the local bed slope on the bed-load transport. Using physically plausible estimates for these parameters the results of the analysis are in good agreement with observations for both ripples and dunes.

Boundary-layer structure of prograde and retrograde rotating flows past a cylinder on a β-plane is investigated. It is found that β inhibits boundary-layer separation for prograde flows but it exerts no influence on the boundary-layer structure for retrograde flows. The results agree with the few available experimental observations.

An efficient relaxation method is developed for computing the properties of a family of vortex pairs with distributed vorticity, propagating without change of shape through a homogeneous, inviscid fluid. The numerical results indicate that a steady state exists even when the gap between vortices is arbitrarily small, and that as the gap closes the steady state approaches a limiting vortex pair with a cusp on the axis of symmetry. Comparison is made with an approximate theory due to Saffman, and agreement is found to be good until the vortices are almost touching. The energy of members of the family is computed, and possible means of experimental production are discussed.

Under some conditions of temperature and flow an ice-water interface in the presence of a turbulent stream has been observed to be unstable. In this paper the source and the conditions for the instability were investigated for a well-defined turbulent boundary-layer flow. It was found that the instability resulted from the interaction that occurs between a wavy surface and a turbulent flow over it. Such an interaction results in a heat transfer variation which is 90 to 180 degrees out of phase with the surface wave shape – a result which is consistent with the calculations of Thorsness & Hanratty (1979a, b).

The main factor controlling damping of the instability at an ice-water interface was found to be the rate at which heat is conducted away from the interface into the ice.

In the past it has been found that when an ice layer is melting, that is when the heat conduction in the ice is small, the ice surface is highly unstable. In the present study it was found that for a sufficiently large temperature ratio (Tf−Tw)/(T∞−Tf), a steady-state ice layer is also unstable. Furthermore it is predicted, from the present observations, that a growing ice layer with a ratio of ice-side to water-side heat fluxes of up to 2.3 could be unstable.

Under sufficiently unstable conditions waves on the ice surface grow to an amplitude at which flow separations occur near the wave crests. This results in a ‘rippled’ ice surface pattern very similar to the patterns observed on mobile bed surfaces (Kennedy 1969) or surfaces which are being dissolved into a flowing stream (Allen 1971). The development of a ‘rippled’ ice surface results in a very substantial increase in the mean heat-transfer rate which would have an important influence on predictions of ice formation in the presence of a turbulent stream.

A backwash vortex has been observed under waves running up a sloping bed. The vortex is formed at the edge of the swash zone when backwash turns back into an oncoming wave bore. The vortex eroded a bed consisting of glass beads, and formed a step which induced the formation of sand ripples.

The present study examines the flow past slender bodies possessing finite centre-line curvature in a viscous, incompressible fluid without any appreciable inertia effects. We consider slender bodies having arbitrary centre-line configurations, circular transverse cross-sections, and longitudinal cross-sections which are approximately elliptic close to the body ends (i.e. prolate-spheroidal body ends). The no-slip boundary condition on the body surface is satisfied, using a convenient stepwise procedure, to higher orders in the slenderness parameter (ε) than has previously been possible. In fact, the boundary condition is satisfied up to an error term of O(ε2) by distributing appropriate stokeslets, potential doublets, rotlets, sources, stresslets and quadrupoles on the body centre-line. The methods used here produce an integral equation valid along the entire body length, including the ends, whose solution determines the stokeslet strength or equivalently the force per unit length up to a term of O(ε2). The O(ε2) correction to the stokeslet strength is also found. The theory is used to examine the motion of a partial torus and a helix of finite length. For helical bodies comparisons are made between the present theory and the resistive-force theory using the force coefficients of Gray & Hancock and Lighthill. For the motion considered the Gray & Hancock force coefficients generally underestimate the force per unit length, whereas Lighthill's coefficients provide good agreement except in the vicinity of the body ends.