This work was motivated by recent experimental results on the spectra of fluctuating temperature gradients in a heated turbulent boundary layer obtained by Sreenivasan, Danh & Antonia. Standard techniques of turbulence theory are used herein to derive expressions relating the individual one-dimensional spectra of each of the three components of the spatial gradient ∂θ/∂xi in a locally isotropic turbulent scalar field. The results of the isotropic theory explain all of the new observed features of the temperature-gradient spectra. The spectra of ∂θ/∂y and ∂θ/∂z decrease monotonically with increasing wavenumber, in contrast to the well-known behaviour of the spectrum of ∂θ/∂x, which reaches a maximum value at roughly one-tenth the Kolmogorov wavenumber. The spectra of ∂θ/∂y and ∂θ/∂z are relatively rich in low frequency energy and relatively poor in high frequency energy compared with the spectrum of ∂θ/∂x. The absolute magnitudes of the spectra of ∂θ/∂y and ∂θ/∂z calculated from the spectrum of ∂θ/∂x using the isotropic relations are in generally good agreement with the corresponding measured spectra for a large range of wavenumbers, indicating second-order spectral local isotropy of the fine-scale scalar structure for sufficiently large wavenumbers. The form of the spectra of ∂θ/∂y and ∂θ/∂z in the inertial subrange is derived analytically.

This paper describes measurements of noise from two-phase flow over hydrofoils. The experiments were performed in a variable-pressure water tunnel which was acoustically calibrated so that sound power levels could be deduced from the sound measurements. It is partially reverberant in the frequency range of interest.

Cavitation was generated on a hydrofoil in the presence of either a separated laminar boundary layer or a fully turbulent attached boundary layer. The turbulent boundary layer was formed downstream of a trip which was positioned near the leading edge. High-speed photographs show the patterns of cavitation which were obtained in each case. The noise is shown to depend on the type of cavitation produced; and for each type, the dependence on speed and cavitation index has been determined. Dimensionless spectral densities of the sound are shown for each type of flow.

Flow of a homogeneous inviscid fluid down a rotating channel of slowly varying crosssection is considered, with particular reference to conditions under which the flow is ‘hydraulically controlled’. This problem is a member of a general class of problems of which gas flow through a nozzle and flow over a broad-crested weir are examples (Binnie 1949). A general discussion of such problems gives the means for determining the position of the control section (which is generally flow dependent) and shows that at this position there always exist long-wave disturbances with zero phase speed (i.e. disturbances are always ‘critical’ at the control section). The general theory is applied to the rotating-channel problem for the case of uniform potential vorticity. For this problem, three parameters are needed to specify the upstream flow, and the control theory gives a relationship between these parameters which depends on the geometry of the channel.

The development of the flow field of a jet emanating from a point source of momentum in an infinite incompressible fluid of density σ is considered. The flow field is assumed to be due to the application of a constant force F0 at the origin. The problem is formulated in terms of the dimensionless variable λ = (vt)½/r, where v is the kinematic viscosity of the fluid, t the time from the application of the force and r the distance from the origin. At a station r the flow field is dipolar, with the dipole axis in the direction of F0, for all t satisfying the inequalities vt Lt; r2 and F0t2 [Lt ] 4πρr4. Also, at a given time t the streamlines of the developing flow field in a section through the axis of symmetry of the problem form closed loops about a stagnation point. If this occurs at λ = λm, the stagnation point propagates to infinity, along a straight line emanating from the origin, with speed v½/2λmt½, where λm = λm(F0) decreases as F0 increases. The larger F0 is the faster the steady state is established.

Laboratory experiments on swirling flows through tubes often exhibit a phenomenon called vortex breakdown, in which a bubble of reversed flow forms on the axis of swirl. Mager has identified breakdown with a discontinuity in solutions of the quasicylindrical flow equations. In this study we define a tornado-like vortex as one for which the axial velocity falls to zero for sufficiently large radius, and seek to clarify the conditions under which the solution of the quasi-cylindrical flow equations can be continued indefinitely or breaks down at a finite height. Vortex breakdown occurs as a dynamical process. Hence latent-heat effects, though doubtless important to the overall structure and maintenance of the tornado, are neglected here on the scale of the breakdown process. The results show that breakdown occurs when the effective axial momentum flux (flow force) is less than a critical value; for higher values of the flow force, the solution continues indefinitely, with Long's (1962) similarity solution as the terminal state. When applied to the conditions of the 1957 Dallas tornado, the computed breakdown location is in agreement with Hoecker's analysis of the observations.

The breakdown of Rossby waves in a bounded system is studied for the case in which the wave amplitude is small. In a very long, laterally bounded, channel all waves are unstable via second-order resonant interactions except those of wavenumber π/L in the cross-channel direction (where L is the channel width), which are stable if their longitudinal wavenumber is greater than 0·681π/L. These waves are, however, unstable to weaker side-band interactions, so that all waves with non-zero longitudinal wavenumber are unstable. The transition from sideband to triad instability occurs where the group velocity of the basic wave is equal to the velocity of long waves.

This paper investigates the flow near the summit of steep, progressive gravity wave when the crest is still rounded but the flow is approaching Stokes's corner flow. The natural length scale in the neighbourhood of the summit is seen to be l =q2/2g, where g denotes gravity and q is the particle speed at the crest in a reference frame moving with the wave speed. We show that a class of self-similar smooth local flows exists which satisfy the free-surface condition and which tend to Stokes's corner flow when the radial distance r becomes large compared withl. The behaviour of the solution at large values of r/l is shown to depend on the roots of the transcendental equation \[ K \tan h K = \pi/2\surd{3}. \] The two real roots correspond to a damped oscillation of the free surface decaying like (l/r)½. The positive imaginary roots correspond to perturbations vanishing like higher negative powers of r.

The complete flow is calculated by transforming the domain onto the interior of a circle in the complex plane and expanding the potential at the surface in a Fourier series. The computation is checked by an independent method, based on approximating the flow by a sequence of dipoles. The profile of the surface is found to intersect its asymptote at large values of r/l. This implies that the maximum slope slightly exceeds 30°. The computed value 30·37° is in close agreement with that obtained by extrapolating the maximum slopes of steep gravity waves, as calculated by previous authors. The vertical acceleration of a particle at the crest is 0·388g. In the far field, however, the acceleration tends to the value ½g corresponding to the Stokes corner flow.

The flow of viscous lubricant in narrow gaps is considered for those geometries in which cavitation arises. A detailed review is presented of those boundary conditions which have been proposed for terminating the lubrication regime (i.e. those valid where the cavity forms). Finally it is shown that a uniform cavity-fluid interface remains stable to small disturbances provided that \[ \frac{d}{dx}\left(P+\frac{T}{r}\right) < 0, \] in which T and r represent the surface tension of the fluid and the radius of curvature of the interface respectively whilst dP/dx is the gradient of fluid pressure immediately upstream of the interface.

The steady and unifrom flow of a viscous fluid past a unifrom cavity in a gemoetry with small, yet arbitrary, film thickness is considered. A mathematical model for describing steady perturbations to such a flow is presented in which the perturbation to the cavity-fluid interface is represented by a small amplitude harmonic wave of wavenumber n. A linearized perturbation analysis then permits the formulation of a boundary-value problem involving the homogeneous Reynolds equation, the solution to which determines both n and the perturbed pressure field.

Numerical and approximate analytic solutions are determined for the cylinderplane geometry in which fluid flows between a rotating cylinder and a Perspex block. Whilst these compare well with experimental data over the whole range \[ 0.1 < \eta U/T < 3, \] closest agreement between theory and experiment is attained for small values of both ηU/T and n.

An extension is made of the α-effect model of the earth's dynamo into the nonlinear regime following the prescription of Malkus & Proctor (1975). In this model, the effects of small-scale dynamics on the α-effect are suppressed, and the global effects of induced velocity fields examined in isolation. The equations are solved numerically using finite-difference methods, and it is shown that viscous and inertial forces are unimportant in the final equilibration, as suggested in the above paper.

A study is made of the extent to which local boundary geometry can influence separation in a two-dimensional or an axisymmetric Stokes flow. It is shown that a Stokes flow can separate from a point on a smooth body at an arbitrary angle, which can be determined only by reference to the global solution for the flow past the body, and the dominant mode in the stream function near a point of separation is O(r3) in the distance r from the separation point. When the body has a protruding cusped edge it is shown that separation can occur at an arbitrary inclination to the edge which must again be determined from the global solution. In this case the stream function is O(r3/2) near the edge. When the flow is locally within a wedge-shaped region of angle β, where β ≠ π or 2π, and β > 146·3°, it is shown that the dominant modes near the vertex of the wedge are non-separating modes. It follows that, in general, a Stokes flow around such a wedge cannot separate from the vertex. This conclusion is illustrated by reference to the global solution for uniform axisymmetric flow past a spherical lens, in which the structure of the flow near the rim is examined in detail. In the case of a body having a sharp edge of small but non-zero angle protruding into the flow, so that β is very close to 2π, it is shown that separation occurs exceedingly near to the edge. This happens, for example, in the flow past a thin concave-convex lens, for which separation occurs near the rim on the concave side. The analysis also suggests that a similar separation occurs very near the rim on the flatter side of a thin asymmetric biconvex lens. However, for the symmetric biconvex lens, and, as a special case, the circular disk, no separation occurs on either side near the rim. For β < 146·3·, streaming flow into the vertex of a wedge does not occur because of the presence of an infinite set of vortices, and the possibility of separation at the vertex in the sense discussed here does not arise.

The nonlinear growth of turbulent jets documented by Kotsovinos (1976) is tentatively attributed to draughts in the laboratory, caused principally by the jet itself.