The wave-action conservation equation for water waves is always derived from a Lagrangian for irrotational flow. This is quite satisfactory if the whole flow-field (i.e. waves and background current) is irrotational, but is inadequate for a background current with a large-scale (vertical) vorticity, even if the flow has negligible vorticity on the local scale of a few wavelengths. A wave-action conservation equation is derived for this case and equations governing the flow and the waves are given in a simple form closely parallel to the irrotational flow equations.

A nonlinear solution is constructed representing the steady flow field generated in viscous incompressible fluid in a spherical envelope by a constant point force F0 acting at the centre O of the envelope. Our analysis shows that when F0 = O(3ν2ρ), where v is the coefficient of kinematic viscosity and ρ the density of the fluid, the linear solution, which is symmetric about a plane through O perpendicular to the force, represents a reasonable approximation to the velocity field. As F0 increases the velocity field develops an asymmetry and the centre of the eddy, that exists in a meridian section, is displaced towards the direction of the force and is closer to the boundary. Also as F0 increases, on the axis of symmetry, the fluid speed per unit force decreases behind the force and increases ahead of it and percentage-wise the increase is larger further from O.

Steady finite amplitude two-dimensional solutions are obtained for the problem of convection in a horizontal fluid layer heated from below and rotating about its vertical axis. Rigid boundaries with prescribed constant temperatures are assumed and the solutions are obtained numerically by the Galerkin method. The existence of steady subcritical finite amplitude solutions is demonstrated for Prandtl numbers P < 1. A stability analysis of the finite amplitude solutions is performed by superimposing arbitrary three-dimensional disturbances. A strong reduction in the domain of stable rolls occurs as the rotation rate is increased. The reduction is most pronounced at low Prandtl numbers. The numerical analysis confirms the small amplitude results of Küppers & Lortz (1969) that all two-dimensional solutions become unstable when the dimensionless rotation rate Ω exceeds a value of about 27 at P ≃ ∞. A brief discussion is given of the three-dimensional time-dependent forms of convection which are realized at rotation rates exceeding the critical value.

Measurements, including one-point double, triple or quadruple mean products of velocity fluctuations, have been made in low-speed turbulent, boundary layers on flat surfaces downstream of concave or convex bends with turning angles of 20 or 30 degrees, the length of the curved region being at most 6 times the boundary-layer thickness at entry. These short bends approximate to ‘impulses’ of curvature, and the object of the work was to investigate the impulse response of the boundary layer, essentially the decay of structural changes downstream of the bends. The work can be regarded as a sequel, with much more detailed measurements, to the study by So & Mellor (1972, 1973, 1975) who investigated the response to step increases of curvature: turbulent boundary layers being nonlinear systems, responses to several kinds of curvature history are needed to assemble an adequate description of the flow. The most striking feature of the ‘impulse’ response is that the decay of the high turbulent intensity found at exit from the concave bends is not monotonic; the Reynolds stresses in the outer layer collapse to well below the level at entry, and are still falling slowly at the end of the test rig although in principle they must recover eventually. On the convex (stabilized) side the flow recovers, monotonically in the main, from a low level of turbulent intensity at the exit. The pronounced second-order response on the concave side can be explained qualitatively by interaction between the shear stress and the mean shear and is not peculiar to curved flows, but in the present cases the response is complicated by large changes in the dimensionless structure parameters related to double or triple mean products of velocity fluctuations. Strong spanwise variations, due presumably to longitudinal vortices, further complicate the flow in the concave bends, and decay only very slowly downstream.

This paper is concerned with laboratory measurements of turbulence in a mixed layer and through a density interface, in the absence of a mean flow. The present results confirm the conclusion of Hopfinger & Toly (1976) that the turbulence in a mixed layer is not significantly affected by the slow entrainment of fluid across the bounding density interface. In a homogeneous fluid the turbulence intensity is found to be surprisingly non-uniform, even two grid mesh distances away from the grid. Velocity measurements have also been taken through a density interface (at the entraining boundary of a mixed layer) and the turbulence here varies in a manner that bears some resemblance to the theory of Hunt & Graham (1978). These velocity measurements were taken with a laser-Doppler anemometer and they were made possible by a novel experimental technique which eliminates the refractive index variations which normally occur in a turbulent, density-stratified liquid flow.

Measurements have been made in the flow over an axisymmetric cylinder-flare body, in which the boundary layer developed in axial flow over a circular cylinder before diverging over a conical flare. The lateral divergence, and the concave curvature in the transition section between the cylinder and the flare, both tend to destabilize the turbulence. Well downstream of the transition section, the changes in turbulence structure are still significant and can be attributed to lateral divergence alone. The results confirm that lateral divergence alters the structural parameters in much the same way as longitudinal curvature, and can be allowed for by similar empirical formulae. The interaction between curvature and divergence effects in the transition section leads to qualitative differences between the behaviour of the present flow, in which the turbulence intensity is increased everywhere, and the results of Smits, Young & Bradshaw (1979) for a two-dimensional flow with the same curvature but no divergence, in which an unexpected collapse of the turbulence occurred downstream of the curved region.

The problem of the strong velocity disturbances remaining after the laminarization of an electroconductive fluid flow in a transverse magnetic field was investigated experimentally. The turbulence intensities, two-point correlations, and energy spectra of the velocity pulsations were measured using two-electrode probes and hot-film sensors. The two hypotheses with regard to a physical explanation of the persistence of the disturbances were examined, and it was concluded that these disturbances are generated mainly at the entry of the flow into the magnetic-field region. An almost undisturbed laminarized flow was obtained by eliminating the entry effects.

Definitions of energy density, energy flux and momentum flux for capillary—gravity waves are derived by integration of the equations of motion and also by Whitham's averaged Lagrangian method. We then confirm recent results due to Hogan (1979) both in the general case and in the case of pure capillary waves. Comparison with the Lagrangian results also allows us to give general definitions of ‘wave action density’ and ‘wave action flux’.

The crest of a steep, symmetric gravity wave is shown to be closely approximated by the expression \[ x+iy = \frac{\alpha +\gamma i\chi}{(\beta + i\chi)^{\frac{1}{3}}}, \] where x, y are co-ordinates in the vertical plane, χ is the complex velocity potential and α, β, γ are certain constants. This expression is asymptotically correct both for small and for large values of |χ|; and the free surface agrees with the exact profile calculated by Longuet-Higgins & Fox (1977) everywhere to within 1·5 per cent. The pressure at the surface is constant to within 5 per cent.

Experiments have been performed to determine the basic mechanism of heating in resonance tubes of square section with constant area excited by underexpanded jet flows. The jet flow between the nozzle exit and the tube inlet plays a key role in the performance of a resonance tube. A detailed and systematic investigation of the unsteady complex shock structure in this part of the flow region has led to a better understanding of the fundamental mechanisms associated with the gas heating in such tubes. A study of the effects of tube location in relation to free-jet shock location (without the presence of the resonance tube) has shed further light on the underlying mechanism of sustained oscillations of the flow in a resonance tube.

In an effort to determine the characteristics of the various types of convection that can occur in a fluid-saturated porous medium heated from below, a Galerkin approach is used to investigate three-dimensional convection in a cube and two-dimensional convection in a square cross-section. Strictly two-dimensional, single-cell flow in a square cross-section is steady for Rayleigh numbers R between 4π2 and a critical value which lies between 300 and 320; it is unsteady at higher values of R. Double-cell, two-dimensional flow in a square cross-section becomes unsteady when R exceeds a value between 650 and 700, and triple-cell motion is unsteady for R larger than a value between 800 and 1000. Considerable caution must be exercised in attributing physical reality to these flows. Strictly two-dimensional, steady, multicellular convection may not be realizable in a three-dimensional geometry because of instability to perturbations in the orthogonal dimension. For example, even though single-cell, two-dimensional convection in a square cross-section is steady at R = 200, it cannot exist in either an infinitely long square cylinder or in a cube. It could exist, however, in a cylinder whose length is smaller than 0.38 times the dimension of its square cross-section. Three-dimensional convection in a cube becomes unsteady when R exceeds a value between 300 and 320, similar to the unicellular two-dimensional flow in a square cross-section. Nusselt numbers Nu, generally accurate to 1%, are given for the strictly two-dimensional flows up to R = 1000 and for three-dimensional convection in cubes up to R = 500. Single-cell, two-dimensional, steady convection in a square cross-section transports the most heat for R < 97; this mode of convection is also stable in square cylinders of arbitrary length including the cube for R < 97. Steady three-dimensional convection in cubes transports more heat for 97 [lsim ] R [lsim ] 300 than do any of the realizable two-dimensional modes. At R [gsim ] 300 the unsteady modes of convection in both square cylinders and cubes involve wide variations in Nu.

A simple analysis shows that with a disturbance present the potential jump in a steady flow in a canal is expressed in terms of (1) the effective volume (displaced volume and added mass/density of fluid) and (2) the depth Froude number for either a submerged body or a body with thin waterplane area. For a ship moving in a canal, the expression for potential jump contains a contribution from the line integral term along the intersection between the ship hull and the free surface. When a pressure distribution is given on the free surface, the potential jump can be expressed explicitly in terms of the depth Froude number and the total pressure force, regardless of the shape of the pressure distribution. From the present relations, the added mass of a ship in steady motion in a canal is computed from the potential jump computed previously by the author for various Froude numbers. This added mass plays an essential role in the equation of motion initially when a sudden external force is applied to a steady moving ship. The present analysis is complimentary to that of Newman (1976) and the extension of that to the three-dimensional case. As practical applications of the potential jump, which has had a limited interest, we proposed approximate formulas for speed correction and sinkage of a ship in a towing tank experiment. Also proposed is an approximate formula for the speed correction in a wind tunnel experiment. The present approximate formula is compared with ‘exact’ numerical results obtained by the localized finite element method for both towing tank and wind tunnel experiments. The present speed correction formula is also compared with existing approximate formulas for a wind tunnel experiment. The present formulas compare favourably with the exact numerical results.

This paper makes use of the ease of modelling topographic Rossby waves in a laboratory context to investigate the ability of these waves to generate strong zonal mean flows when the geostrophic (f/H) contours are closed. A zonally travelling wave is forced in a narrow latitude band of a ‘polar beta plane’. Stronger signals occur when the motion of the driving is retrograde and at the phase speed of the gravest free modes. An important zonal westward mean flow occurs in the free interior while a compensating eastward jet is found at forced latitudes. The dependence of the mean flow strength upon the wave steepness indicates that genuine rectification processes are indeed taking place when the fluid is stirred by purely oscillating devices.

This general tendency for topographic Rossby waves to transfer energy to zonal components is first analysed theoretically by investigating a side-band instability mechanism within an unforced fluid. Among the products of the interactions between a primary wave of wavenumber k and its side bands of wavenumber k ± δk, the zonal flow is prominent. Wave steepnesses of order (|δk|/|k|)½ only are required for zonal energy to grow whereas non-zonal components of scale longer or shorter than the primary wave need huge steepnesses [of order (|δk|/|k|−3/2] for amplification. This supplements the earlier notion that ‘nearly zonal’ waves may be generated by weak resonant interaction.

For gentle driving certain classical aspects of Rossby wave propagation can be checked against the experiments. The linear theory provides also a convenient framework to discuss the meridional structure of the wave-induced Reynolds stress. For more energetic driving, a test of the potential vorticity mixing theory can be carried out and sheds further light upon the rectification mechanisms.

The far turbulent wake of a slightly heated cylinder was studied using our specially developed four-wire probe and digital processing of signals obtained. The processing method included a simultaneous solution of full nonlinear response equations for the four sensors yielding values of instantaneous velocity vector and temperature. The accuracy of results was improved substantially by the subtraction of free-stream velocity and temperature variations, first-order corrections for the d.c. drift of the signals, the elimination of 60 Hz related noise, and instantaneous correction for the streamwise displacement between the sensor wires. A low temperature threshold was used to discriminate between turbulent and potential flow.

The frequency of crossing of the turbulent-potential interface and statistics of turbulent and potential zone intervals are presented. Conditionally sampled fluctuating velocity and temperature fields are given for turbulent and potential zones as well as for a fluid at the downstream and upstream interface of the turbulent zones. Averaging was done with respect to conventional (Reynolds) as well as local zone or point means. The results on turbulence structure up to second-order correlations are presented in this paper.

It is hoped that the data will be useful for those interested in methods of prediction of turbulence, especially those based on the consideration of large structures. Researchers more interested in the physical phenomena involved might also find the data suitable for further analysis and interpretation.

We report on a set of turbulent flow experiments of the Taylor type in which the fluid is contained between a rotating inner circular cylinder and a fixed concentric outer cylinder, focusing our attention on very large Taylor number values, i.e. \[ 10^3 \leqslant T/T_c \leqslant 10^5, \] where Tc is the critical value of the Taylor number T for onset of Taylor vortices. At such large values of T, the turbulent vortex flow structure is similar to the one observed when T – Tc is small and this structure is apparently insensitive to further increases in T. These flows are characterized by two widely separated length scales: the scale of the gap width which characterizes the Taylor vortex flow and a much smaller scale which is made visible by streaks in the form of a ‘herring-bone’-like pattern visible at the walls. These are conjectured to be Görtler vortices which arise as a result of centrifugal instability in the wall boundary layers. Ideas of marginal instability by which we postulate that both the Taylor and Görtler vortex structures are marginally unstable on their own scale seem to provide good quantitative agreement between predicted and observed Görtler vortex spacings.

A plane turbulent buoyant jet discharging vertically into a two-dimensional channel of confined depth is considered. The channel opens at both ends into a large outside reservoir, thus defining a steady symmetrical flow field within the channel. The analysis is aimed at two aspects, the stability and the bulk mixing characteristics of the discharge. A stable discharge configuration is defined as one in which a buoyant surface layer is formed which spreads horizontally and does not communicate with the initial buoyant jet region. On the other hand, the discharge configuration is unstable when a recirculating cell exists on both sides of the jet efflux.

It is shown that discharge stability is only dependent on the dynamic interaction of three near-field regions, a buoyant jet region, a surface impingement region and an internal hydraulic jump region. The buoyant jet region is analysed with the assumption of a variable entrainment coefficient in a form corresponding to an approximately constant jet-spreading angle as confirmed by different experimental sources. The properties of surface impingement and internal jump regions are determined on the basis of control volume analyses. Under the Boussinesq approximation, only two dimensionless parameters govern the near-field interaction; these are a discharge densimetric Froude number and a relative depth. For certain parameter combinations, namely those implying low buoyancy and shallow depth, there is no solution to the conjugate downstream condition in the hydraulic jump which would satisfy both momentum and energy conservation principles. Arguments are given which interpret this condition as one which leads to the establishment of a near-field recirculation cell and, thus, discharge instability.

The far-field boundary conditions, while having no influence on discharge stability, determine the bulk mixing characteristics of the jet discharge. The governing equations for the two-layered counterflow system in the far field are solved. The strength of the convective transport, and hence the related dilution ratio, is governed by another non-dimensional parameter, the product of the relative channel length and the boundary friction coefficient.

Experiments in a laboratory flume, covering a range of the governing parameters, are in excellent agreement with the theoretical predictions, both the stability criterion and the bulk mixing characteristics.

The method of characteristics is used to calculate the unsteady one-dimensional flow produced by the convection of a temperature disturbance, contained between two contact surfaces, in high subsonic flow through a nozzle. The results show the development of the pressure disturbances which are associated with the force perturbation needed to accelerate the temperature disturbance at the same rate as the surrounding gas and propagate in both the upstream and downstream directions. Using a simple flow model, expressions for the mean pressure disturbances are derived and shown to be in good agreement with numerical solutions. Other numerical solutions which show the effect of changing the various flow parameters and the nozzle shape are presented. In all of the cases considered, the pressure disturbances caused by a 10 % change in the inlet temperature are significantly large and may be an important part of ‘excess’ noise.

The flow of a homogeneous viscous liquid towards a sink in the interior of a rotating basin with a free surface, a horizontal bottom and a vertical side wall is considered. The conditions assumed are such that an Ekman layer occurs at the bottom beyond a small distance from the sink. A first-order correction to the Ekman model accounting for the influence of the inertial terms in the equations of motion is given for a special case. It is shown theoretically and experimentally that eccentric withdrawal from a circular basin causes a vortex at the sink and a counter-rotating gyre attached to the far wall.

The use of flagella by sessile organisms to generate feeding currents is analysed. The organism consists of a spherical cell body (radius A) to which a smooth flagellum (radius a, length L) is attached radially. The cell body is a height H above the plane substrate to which it is rigidly attached via a stalk. The organism propagates plane sinusoidal waves (amplitude α, wavenumber k) from base to tip. The flagellum is represented by distributions of stokeslets and dipoles along its centre-line. The cell body and substrate are modelled by employing an approximate form of the Green's function for the sphere in the half space. The error terms in the model are O(a/L) and O(A2/H2). The analysis and method of solution are adapted from Higdon (1979).

The mean flow rate and power consumption are calculated for a wide range of parameters. Optimal motions are determined with the criterion of minimizing the power required to achieve a given flow rate. The optimum wave has maximum slope in the range 2 < αk < 2·5 (compared to the optimum value αk = 1 for swimming). The optimum number of waves Nλ increases linearly with flagellar length for L/A > 10 and is approximately constant, Nλ = 1, for shorter flagella. The optimum flagellar length is in the range 5 < L/A < 10. There is no optimum flagellar radius a/A. For optimal efficiency, the height H should be greater than or equal to the length of the flagellum.

The optimum values of the parameters are compared to the values for the choano-flagellates described by Lapage (1925) and Sleigh (1964). Excellent agreement is found between the predicted optima and the observed values. The calculated velocity field closely resembles the flow described by Sleigh and Lapage.

Even very small Coriolis forces are shown to alter significantly the nature of the upstream wake of an object in slow (small Froude number) translation through a non-diffusive stratified fluid. If the Ekman number is of order one, the far upstream extent of the wake is reduced. If the fluid rotation is sufficient to make the Ekman number small, the contraction of the wake is much greater. We study a particular case in detail; the Ekman number is small enough to make horizontal boundary layers Ekman layers. In this case, the wake is confined to the vicinity of the object, the upstream flow arising from a combination of Ekman pumping and baroclinic vorticity generation. The upstream flow is described by an eigenfunction whose amplitude is dependent on object geometry. If the object is a semi-infinite rectangular parallelepiped, that amplitude is determined by detailed examination of the shear layer at the face of the parallelepiped and its interaction with the Ekman layer on the top surface of the object