The flow of a rotating, two-layer fluid system over long ridges of constant cross-section is considered. Homogeneous incompressible fluids of constant, but different, density are confined between two ‘infinite’ horizontal plane surfaces which rotate at a constant angular velocity about a vertical axis. The ridge is located on the lower surface while upstream of the ridge each fluid is in uniform motion in a direction normal to the ridge. Solutions are obtained for both an f-plane and a β-plane under the following restrictions: E [Lt ] 1, Ro ∼ E½, S ∼ O(1), H/L ∼ O(1), d/L ∼ O(1) and h/L ∼ E½ where E is the Ekman number, Ro is the Rossby number, S is a stratification parameter, H/L is the two-fluid depth to ridge width ratio, d/L is the lower fluid depth to ridge width ratio and h/L is the aspect ratio of the ridge. This set of restrictions assures that viscosity is important in considering the dynamics of the system. Furthermore the restrictions are ones that make laboratory experimentation feasible. Solutions are also presented for the non-viscous case (i.e. E = O), and are compared with their viscous counterparts. The importance of viscosity in this physical system is thus demonstrated.

The displacement height appears in the logarithmic velocity profile for rough-wall boundary layers as a reference height for the vertical co-ordinate. It is shown that this height should be regarded as the level at which the mean drag on the surface appears to act. The equations of motion then show that this also coincides with the average displacement thickness for the shear stress.

A simple analytical model, experimental results and dimensional analysis are all used to indicate how the displacement height depends upon the detailed geometry of the roughness elements.

A method of treating turbulent pair dispersion and scalar diffusion is presented. Use is made of Kraichnan's form of Richardson's diffusion equation by relating the turbulent pair diffusivity to single-time Eulerian velocity statistics (which are presumed known) by means of a statistical independence hypothesis. In this procedure the diffusivity itself is coupled to solutions of the diffusivity equation in a self-consistent way.

The method is applied to both two-and three-dimensional flow. In three-dimensional inertial-range and dissipative-range turbulence the turbulent pair diffusivity is determined and used to find the values of the coefficients of the scalar spectrum in the $k^{-\frac{5}{3}}$ and k−1 ranges with good agreement with experiment. The Obukhov–Corrsin constant is found to be 0·49 and the Batchelor constant is √5. In two-dimensional turbulence the results are compared with constant-pressure balloon dispersion experiments. Results are also found for the rate of decay of scalar intensity in the special case where the initial scalar spectrum peaks in the inertial range.

In this paper, an experimental determination of the three-dimensional velocity distribution due to the physiological pulsatile flow of a Newtonian, incompressible fluid at various locations in a curved tube of circular cross-section is presented. Our results show four interesting features of the pulsatile flow development in the curved tube. First is the presence of a reversed flow along the inner wall of the tube during the diastolic (deceleration) phase of the pulsatile flow cycle. Second is that the flow does not appear to be fully developed in the curved tube through the cross-section whose L/a ratio is equal to 16·9, the final location at which measurements were made in this study, where L is the axial length and a is the radius of the curved tube. A third feature observed is the vacillation of the peak axial velocity across the horizontal diameter of the tube from the upstream to the downstream region in the curved tube. In the upstream region (L/a = 3·4), the maximum axial velocity measured occurred nearest to the outer wall. The maximum axial velocity shifted towards the inner wall in the middle of the tube (L/a = 10·2), while in the downstream region (L/a = 16·9), the maximum axial velocity measured was again near the outer wall. Finally, trapped vortical motions are observed to occur at the inner wall of the tube in the downstream region.

The details of a spark-induced ‘spot’ in an axisymmetric mixing layer of a 12·7 cm diameter (D) free air jet have been educed in different azimuthal planes and at three streamwise stations corresponding to x/D = 1·5, 3·0, 4·5. The measurement technique and the spot properties in the plane of the spark at the three stations were discussed in parts 1 and 2 (Sokolov et al. 1980 and Hussain, Kleis & Sokolov 1980, hereinafter referred to as I and II, respectively). The present part describes the azimuthal structure of the spot and its initiation mechanism.

It is shown that the distributions of phase-average longitudinal and lateral velocities, the intermittency and the coherent Reynolds stress within the spot are essentially the same in various azimuthal planes at each streamwise location. The spark induces a local boundary-layer spot on the nozzle wall and simultaneously triggers the instability of the free shear layer downstream from the lip. The boundarylayer spot persists initially in the free shear layer but decays downstream due to the lack of a sustaining mechanism. The mixing-layer spot – the result of a roll-up of a natural instability mode triggered in the free shear layer by the acoustic disturbance radiated from the spark – grows downstream and undergoes intense interactions, remaining essentially axisymmetric and travelling at about 60% of the core fluid velocity. Velocity signals in different azimuthal planes of the free shear layer show that the natural instability of the jet occurs axisymmetrically on an instantaneous basis even though the jet diameter is considerably larger than the instability wavelength. The natural instability is amplitude-modulated in a random manner; this modulation is also essentially axisymmetric.

The large-scale coherent structures produced by the intense localized spark are not only axisymmetric on the phase-average basis, but are also individually axisymmetric in the laminar instability region.

The dispersion of a spot of contaminant in a high-Péclet-number laminar flow is studied by means of the ray method developed by Cohen & Lewis (1967). This method is free from the usual severe restrictions on the time range. Thus, it is possible to investigate strong shear-distortions of the concentration distribution. Also, the effects of rigid boundaries can be allowed for simply by including reflected rays. Three examples are studied in detail: stagnation flow, a point vortex and plane Poiseuille flow.

An approximate integral-equation approach is used to model the growth and collapse of a vapour cavity in close proximity to an initially plane free surface. By comparison with experiment, it is shown to predict all the salient features of the bubble and freesurface interaction, provided that the complete nonlinear Bernoulli pressure condition is applied on both surfaces. Features observed and predicted include the formation of an accelerating liquid jet in the bubble and a pronounced spike in the free surface during the collapse phase of the bubble's life. If the bubble is initially sufficiently close to the free surface, it will become ‘entrained’ in the raised free surface with a veneer of liquid separating the two free surfaces.

Previous work by the present authors on the onset of convection in a layered porous medium heated from below is extended to an investigation of the heat transported by convection at slightly supercritical Rayleigh numbers.

The two-dimensional convection patterns and associated values of the critical Rayleigh number, cell width and slope of the Nusselt-number graph are calculated for two- and three-layer configurations over a wide range of layer depth and permeability ratios. The results show that the commonly studied problem of a homogeneous layer bounded above and below by impermeable boundaries is a special case, in that the slope of the Nusselt-number graph at the critical point is nearly independent of cell width. For a homogeneous layer with a permeable upper boundary, and for multi-layered systems, the slope of this graph depends strongly on cell width.

Three-dimensional numerical integrations of the Navier-Stokes equations have been made for parameters corresponding to some previous laboratory studies of transverse flow past obstacles in a rotating fluid. In the laboratory experiments the character of the flow was found to depend upon the parameter [Sscr ]L = L/DR, where R is the Rossby number U0/ΩL, L is the horizontal scale of the obstacle, D the depth of the fluid, U0 the flow speed and ω the angular rate of rotation. For [Sscr ]L [Gt ] 1 the flows appeared twodimensional and our results confirm the applicability of this assumption in previous asymptotic theories. For [Sscr ]L ∼ 1 a leaning disturbance is produced which can look columnar in character (‘leaning Taylor column’) and our results enable a detailed examination of this structure. To clarify the importance of nonlinear effects in the leaning Taylor column we compare them with the predictions of a linear inertial wave theory. This theory is valid only for small obstacle slopes but provided it includes the effects of viscosity it gives good predictions of the amplitude of the disturbances. The main difference between the viscous linear theory and the Navier-Stokes solution is a flow asymmetry of nonlinear origin. The role of viscosity is important but passive in the sense that it does not alter the flow structure near the obstacle but progressively dissipates the disturbance with increasing distance from the obstacle. This viscous confinement of the disturbance makes the lee wave flow structure look columnar and is important in allowing some laboratory flows to seem unbounded. The results also confirm the conjecture of Mason (1975, 1977) that the large drag forces occurring in these flows are due to inertial wave radiation.

Expressions are derived for the n-dimensional Stokes velocity and pressure fields (and stream function) corresponding, respectively, to the translation and rotation of a hypersphere in a viscous fluid at rest at infinity. These are utilized to calculate the force and first antisymmetric stress moment (‘torque’) on the n-dimensional hypersphere. They are also utilized to derive the generalizations of Faxén's laws for the force and stress moment corresponding to arbitrarily prescribed velocity fields on the hypersphere surface and at infinity.

The paper considers the problem of calculating the statistical characteristics of a passive scalar dispersed by a homogeneous turbulence field. In many turbulent shear flows the time-scale for the evolution of the scalar field is intrinsically related to that of the turbulent velocity field. This is by no means always the case, however, and it is at this more general situation that the present work is aimed. An approximate transport equation for the rate of dissipation of scalar variance is proposed which, it is argued, must contain (at least) two sink terms one of which responds to the time scale of the velocity field while the other reflects that of the scalar field itself. The model has been applied to the limited number of homogeneous scalar flows for which data are available and achieves satisfactory agreement as judged by the evolution of the mean-square scalar variance.

Measurements, including the six components of the Reynolds-stress tensor, have been made along three generators of a centre-mounted circular cylinder with an elliptical nose cone. The pressure distribution was axisymmetric upstream and asymmetric downstream. The streamwise adverse pressure gradient led to almost zero skin friction in the direction of the limiting streamline, and the circumferential pressure gradient led to skew angles up to 30° in the vicinity of the wall. Special emphasis was laid on measurements in the wall region (y+ > 4), on the repeatability of these measurements and on an estimate of error bounds. The turbulence level encountered (up to 60% close to the wall) was much higher than in other threedimensional boundary-layer measurements. It has been shown that available measuring techniques have to be improved considerably or even abandoned if used under these conditions. Previous measurements of collateral velocity profiles in three-dimensional boundary layers will probably now have to be corrected for severe aerodynamic interference effects.

It has been shown that the normal stresses $\overline{u^{\prime 2}_i}$ and the shear stress component $\overline{u^{\prime}v^{\prime}}$ behave qualitatively much as those in a two-dimensional adverse pressuregradient boundary layer. The other components, $\overline{v^{\prime}w^{\prime}}$ and $\overline{u^{\prime}w^{\prime}}$, both characteristic of three-dimensional flow and caused by the circumferential pressure gradient, are influenced in different ways by the streamwise and circumferential pressure gradients. Spectra of u′-fluctuations are again similar to those obtained from two-dimensional boundary layers.

Mean velocity profiles obey the linear and logarithmic law of the wall known from two-dimensional boundary layers both along a line of symmetry and in the threedimensional boundary layer. This may be because the streamwise pressure gradient dominates over the circumferential pressure gradient in this experiment.

Finally it has been found that the skew angle γ of the Reynolds shear stress vector leads the skew angle ζ of the resultant velocity gradient or ‘mean shear’, both having the opposite sign of the skew angle β of the mean velocity vector except close to the wall. The ratio of Reynolds shear stress and turbulent kinetic energy is no longer ‘approximately’ constant as is assumed for two-dimensional boundary layers.

The effect of rotation on the flow around an obstacle in a Hele Shaw cell is considered. For the flow outside the strip-shaped boundary layer near the obstacle, it is shown that the rotating and non-rotating cases are related by a simple transformation. The flow in the boundary layer near the obstacle is computed numerically and its relation to the Stewartson E¼ and $E^{\frac{1}{3}}$ layers is briefly discussed.

This paper complements an earlier study of the mean velocities in turbulent flows in a flat channel, one of whose walls can move relative to the other, so that the role of the stress gradient within the wall layers can be varied widely and in a controlled manner.

Measurements of longitudinal, normal and lateral velocity fluctuation intensities (u′,v′,w′) and of shear stresses have been made in essentially fully developed flows established by various combinations of pressure gradient and wall velocity The channel aspect ratio (breadth/height) has been varied between 12 and 28 and the development ratio (development length/height) between 20 and 45. The introduction of a turbulence-generating grid at the entrance to the duct increases the effective development length.

The study has considered twenty-six flows that are two-dimensional in the mean, which have been established by blowing and relative motion either in the same direction or directly opposed. Empirical descriptions, based on similarity laws incorporating either the wall stress or the local stress, are developed for the turbulence near the walls and in the core. The profiles of u′, v′ and w′ coalesce, to a reasonable approximation, when normalized with appropriate length and velocity scales. Extensive ‘plateau’ regions are identified, in which the scaled intensities are sensibly constant.

A number of quantities characteristic of the structure of the turbulence are considered, in order to elucidate the effect of the stress gradient on the wall layer, and stages in the erosion of the constant-stress layer are identified.

A nonlinear theory is presented for the resonance of long gravity waves trapped on an uneven bottom when a long packet of short swells is incident. By allowing the trapped wave to be comparable in amplitude to the incident swells, the transient evolution of trapped waves is studied from initial growth through maturity to final decay, for swell packets of finite duration. For a totally submerged ridge, it is found that, while the trapped waves are resonated by second-order periodic modulations of the swell envelope, energy transfer to short swells and nonlinear emission of long waves act as damping mechanisms to limit the amplification. Bottom friction is not qualitatively crucial and affects the results only quantitatively. For a closed beach, breaking of short swells is dealt with empirically but resonant modulation is still the primary factor in exciting surf beats.

Calculations of the primary electroviscous effect have previously been restricted to spherical particles. Here we examine a suspension of randomly orientated rod-shaped particles of length l. Two competing mechanisms are present in a linear velocity field. The flow relative to each rod is largest at the rod ends. When l is large the consequent large distortion of the charge cloud increases the electroviscous effect. On the other hand, the given total charge on a uniformly charged rod is spread more thinly as l increases, and this tends to reduce the electroviscous effect. The balance of these two mechanisms is examined.

An analysis is presented which describes the slow-time evolution of an internal gravity wave in an arbitrarily specified stratification. The weakly nonlinear description of a single-wave mode, governed by the nonlinear Schrödinger equation, breaks down when certain resonant conditions are satisfied. One such condition occurs when the group velocity of the wavetrain is equal to the phase velocity of a higher-mode long wave of the system. The resonant interaction occurs on a faster time scale and is described by a coupled pair of nonlinear partial differential equations governing the evolution of both the short-wave and the long-wave modes. This long-wave/short-wave interaction is pursued further in an experimental investigation by measuring the modal interchange of energy between two internal waves of disparate length and time scales. The resulting data are compared with numerical solutions of the long-wave/short-wave resonant interaction equations. In general, the agreement between the theory and the experiment is reasonably good in the range of operating conditions for which the theory is valid.

A simple derivation of recent shallow-flow equations with bed curvature is presented. These equations are applied to steady flow over a high overflow spillway crest, to obtain the head–discharge relationship and the crest-pressure distribution in good agreement with experiment. It is inferred that the equations are valid for quite large negative curvature. On the other hand, their application to steady flow over a spillway toe indicates their validity for positive curvature is more limited.

An experimental investigation is described of the velocity field in a steady, spillingtype breaker, generated on a steady current by a submerged hydrofoil. Velocities have been measured with a laser–doppler system, and analysed with respect to mean and r.m.s. values as well as Reynolds stresses. The results indicate that the turbulent flow field downstream of the initiation of the separation at the surface resembles that in a self-similar turbulent wake.

In an experimental and theoretical study we model the convection generated in the polar oceans when a fresh-water ice wall melts in salt water of uniform far-field temperature T∞, and salinity S∞. Our laboratory results show that there are three different flow regimes which depend on T∞ and S∞. First, when T∞ and S∞ lie between the maximum density curve and the freezing curve, the flow is only upward. Secondly, for the oceanic case 30 [les ] S∞ [les ] 35‰ and T∞ < 20°C, the flow consists of a laminar bidirectional flow at the bottom of the ice and a turbulent upward flow along the remainder of the ice wall. The laminar flow consists of an upward flowing layer approximately 2 mm thick inside of a downward flowing outer layer approximately 10 mm thick. Thirdly, for the same range of S∞ but for T∞ > 20°C, the flow reverses: at the top of the ice there is a laminar bidirectional flow above a downward turbulent flow. To model the turbulent upward flow theoretically, we numerically solve the governing equations in similarity form with a spatially varying eddy diffusivity that depends on the density difference between the ice-water interface and the far-field. The laboratory data then allows us to evaluate the dependence of eddy diffusivity on T∞ and S∞. The results show that the magnitude of the eddy diffusivity is of the same order as the molecular viscosity and that both mass injection at the interface and opposed buoyancy forces must be included in a realistic flow model. Finally, we use an integral approach to predict the far-field conditions that yield the high-temperature flow reversal and obtain a result consistent with our observations.