It is proposed that a spilling breaker can be regarded as a turbulent gravity current riding down the forward slope of a wave, and can be treated using methods which have been successful in other contexts. The whitecap retains its identity because it is lighter than the water below, owing to the trapping of air bubbles, and information from laboratory experiments is used to estimate the density of the air-water mixture in different circumstances. Entrainment of water from below, at a rate E(Ri0,) which is a function of the overall Richardson number Ri0, has two opposing effects. It provides increasing mass and buoyancy fluxes which can produce an accelerating flow and it also gives rise to a drag, because the entrained fluid has upslope momentum (inco-ordinates moving with the wave crest).

A similarity solution is obtained under the assumptions that the flow is steady in time, and that the slope and the density difference remain constant. In this solution, the thickness of the whitecap is proportional to the distances measured from the crest of the wave. The tangential velocity is proportional to s½. Since the velocity in a Stokes 120° angle is also proportional to s½ this implies that such a flow can start from a small disturbance with zero flux, and propagate with constant acceleration. An important consequence of the analysis is that solutions of this kind are possible only when the slope and the density difference between the whitecap and the water below are sufficiently large; otherwise the upward drag dominates, and a self-sustaining flow cannot form. For a slope of 30°, near the crest of the breaking wave, the theory predicts that a density difference greater than 8% is required to sustain a steady motion, at which point the downslope velocity is 12% of the opposing velocity at the wave surface. A 'starting plume’ model of the advancing front of the breaker is also discussed, which suggests that this too will be accelerating uniformly, but will have a velocity somewhat less than that in the flow behind.

A comparison with the laboratory observations of Kjeldsen & Olsen verifies several features of the model, including the order of magnitude of the relative velocities in the whitecaps and the wave beneath. It also reveals the intermittent nature of the flow, which is here explained as due to the intermittent rounding of the wave crest due to damping of the wave by the whitecap on the forward face.

The high-frequency component of the random solution of a model problem is shown to be statistically orthogonal to the Gaussian component. This is shown to be a consequence of the existence of an equilibrium range. It is concluded that random flow fields can be viewed as being approximately Gaussian only in a very special sense and, in particular, that Wiener–Hermite expansions can provide a useful description only of large-scale hydrodynamical phenomena.

In two previous papers (Proudman 1970; Dowden 1972) it has been shown that some of the phenomena of turbulence a t high Reynolds numbers can be modelled by a suitable chosen member of the class of ν-fluids. These are non-Newtonian fluids all of whose properties depend only on a single dimensional constant whose dimensions are those of viscosity. The purpose of this paper is to construct an equation to model homogeneous turbulence in the presence of a spatially constant rate of deformation in the limit of infinite Reynolds number.

The equation employed is that of a doubly degenerate third-order v-fluid (in Proudman's classification) in the limit ν → 0. In such a fluid the stress tensor S is governed by an equation of the form \[ A\dot{S}\ddot{S}+B\dot{S}^2+Cu^{\prime}S\dot{S}+D\dot{u}^{\prime}S^2 + Eu^{\prime 2}S^2 = 0, \] where A, B,…, E are isotropic tensor constants of the fluid, u′ is the total rate of deformation tensor and dots denote time derivatives. A list of properties required of the equation and its solution is proposed, and the most general form of A, B,…, E is given consistent with these requirements. Computed solutions of this equation are compared with the results of experiments on homogeneous turbulence, and are found to agree well with them.

Recent field observations in the polar oceans show that the hollow tubes of ice called ice stalactites form around streamers of cold brine rejected by the growing sea ice. In a laboratory study of this process, we inject cold, dense brine a t a constant salinity, temperature and volume flux into an insulated tank of sea water held at its freezing point, then photograph the resultant stalactite growth. Because the inner wall temperature of the stalactite remains on the salinity-determined freezing curve, as the stalactite grows and the temperature deficit of the brine goes into the growth of ice, the inner wall melts to dilute and cool the adjacent brine back to its freezing point. This melting means that both the inner and outer stalactite radii increase with time. The radius of the stalactite tip, which is constant for each experiment, is shown to be controlled by the onset of a convective instability. If the tip becomes too large, overturning occurs and the sea-water intrusion freezes, reducing the radius of the tip so that the flow leaving the tip is marginally stable. Inside the stalactite, since the inner radius increases with time, both theory and experiment show the interior flow to be convectively unstable. The present study also derives a solution from the constant-heat-flux Graetz solution for the growth in both length and side-wall area of the stalactite. The experiments show that away from the stalactite base and the very beginning of the experiment this solution, with convection accounted for by an adjustable coefficient, describes the experimental growth. Finally, analysis of the experiments shows that as much as 50% of the ice represented by the cold brine does not go into the stalactite, rather the ice goes directly into the ocean as loose crystals.

The analytical characteristic method is an effective method for computing non-linear effects in inviscid supersonic flow problems. Although only linear equations have to be solved, the results are essentially nonlinear, in the sense that the functional relations between physical state variables and space co-ordinates are nonlinear in the small perturbation parameter introduced, like the thickness ratio or incidence of a wing. This holds even for the first-order approximation of the method.

In the case of two-dimensional (plane or axisymmetric) flow the independent variables are characteristic co-ordinates, i.e. they are chosen so as to be constant along corresponding characteristic lines. The space co-ordinates are considered as dependent variables. In three dimensions there is no unique definition of a characteristic co-ordinate system, because the manifold of characteristic surfaces or bi-characteristics is larger than is necessary for defining a co-ordinate system. The success of a three-dimensional analytical characteristic method, however, depends on the proper choice of the co-ordinate system.

The present analytical Characteristic method for three-dimensional flow is based on the fact that three-dimensional flow behaves locally like axisymmetric flow if it is considered in the osculating plane. The corresponding ‘distance from the axis’ is a function of space depending on the flow field. No change of pressure occurs normal to the osculating plane and in isentropic flow no change of speed either. Therefore no co-ordinate perturbation is performed in this normal direction. In the osculating plane the analytical characteristic methodis applied locally as in axisymmetric flow. In the large the space co-ordinates are obtained by integration along the main bi-characteristics.

As an example the flow field on the suction side of a flat delta wing with sub-sonic leading edges is computed. As a main result one obtains shock waves in the neighbourhood of the leading edges following the expansion.

Forced nonlinear acoustic oscillations near the resonant frequency of closed and open tubes are studied experimentally. In particular, the motion in tubes terminated with different orifice plates is studied, and comparison is made with second- and third-order theories of the motion which contain an adjustable end-wall reflexion coefficient.

It is found that oscillations at resonance in an open tube exhibit remarkably large amplitudes despite the fact that in some cases shock waves are emitted from the open end. For oscillations at resonance in a closed tube, the effect of substituting an orifice plate for the solid end wall is to reduce the amplitude and thicken the compressive portion of the shock waves which occur under these conditions. In both the open-tube and closed-tube experiments the reflexion coefficients which are evaluated by fitting theory to experiment are found to increase with increasing amplitude, in agreement with the observations of previous investigators (Ingard & Ising 1967). In fact, for the open end the same linear dependence upon amplitude is observed, but the constant of proportionality is different. Qualitative differences are observed between the reflexion coefficients of a given orifice at the open-end and the closed-end resonant frequencies; at the open-end frequency the reflexion from the given orifice is less ideal than at the closed-end frequency. The implications of reflexion coefficients dependent on the wave forms are discussed.

Measurements of the mean flow, intermittent structure and turbulent fluctuations were made in a cold-wall boundary layer at a stream Mach number of 9·4 and Reynolds number based on momentum thickness of 36 800. For these conditions, the r.m.s. sublayer thickness was 32 times smaller than that of the boundary layer proper, and the interfacial standard deviation of the latter was about three times proportionately smaller than has been found at low speeds. The mean flow data, which extended well into the sublayer, revealed a large increase in static pressure from the layer edge to the wall and a quadratic law relation between the total temperature and velocity. While the transformed velocity profile was in good agreement with the incompressible law of the wake, no indication of a linear variation of velocity in the sublayer was detected.

Hot-wire fluctuation data, interpreted with the use of appropriate assumptions concerning the nature of the sound field, indicated that the turbulence is dominated by high-frequency pressure fluctuations whose magnitude at the wall and beyond the layer edge agree with extrapolation of data acquired at supersonic speeds. The static temperature fluctuations agreed with expectations from adiabatic, supersonicdata apparently because they were suppressed by the cooled-wall condition. The fluctuations in the longitudinal velocity component were generally small and differed little from lower Mach number results. The high turbulence Reynolds numbers found generated an inertial-subrange spectral decay, while the longitudinal integral scales were found independent of turbulence mode and about one-fifth the boundary-layer thickness.

A theoretical analysis has been made of turbulent viscous interaction on iso-thermal surfaces at hypersonic speeds. The important parameters governing the effects of incidence and displacement have been obtained under both strong and weak interaction conditions for flat-plate flows. A more general expression relating boundary-layer growth to the external pressure field and effective body shape has been obtained. The method is applied to the wedge compression corner problem and the results compared with some experimental data.

Laplace transforms and regular double asymptotic expansions are used to solve the problem of ordinary chemical mass transport in a permeable tube, where there is small radial convection through the membrane wall and where the length-to-diameter ratio is large. The system is taken to be dilute and Newtonian and the solution is found to higher order in two small parameters. Results indicate that the exit concentration decreases markedly as the diameter, membrane permeability and tube length increase, and that changes in mass transport owing to variations in radial convection are much more significant than those due to the same order of magnitude changes in the resistance of the chemical solute to passage through the membrane (transmittance). In addition, the maximum effects of changes in the radial convection and transmittance are not at the membrane itself (r = 1), but rather roughly at radial values of 0·6 and 0, respectively.

Davis has shown by means of a lengthy calculation that, for two-dimensional oscillations in a canal of width 2a, the mth eigenvalue has the form \[ N_m = {\textstyle\frac{1}{2}}m\pi - \frac{\lambda_1+\lambda_2}{4m\pi}+o\bigg(\frac{1}{m}\bigg), \] where λ1/a and λ2/a are the curvatures of the bounding cross-sectional curve C at its vertical intersections with the free surface. Here the same result is obtained more simply.

The linear stability of the developing flow of an incompressible fluid in the entrance region of a circular tube is investigated. The case of non-axisymmetric small disturbances is considered in the analysis. The main-flow velocity distribution used in the stability calculations is that from the solution of the linearized momentum equation. The eigenvalue problem consisting of the disturbance equations and the boundary conditions is solved by a direct numerical integration scheme along with an iteration procedure. An orthonormalization method is employed to remove the ‘parasitic errors’ inherent in the numerical integration of the coupled disturbance equations. The flow is found to be unstable to non-axisymmetric disturbances with an azimuthal wavenumber of one. Neutral-stability curves and critical Reynolds numbers at various axial locations are presented. A comparison of these results is made with those for axisymmetric disturbances reported by Huang & Chen. It is found that the first instability of the flow is due to non-axisymmetric disturbances and occurs in the entrance region of the pipe with a minimum critical Reynolds number of 19 780.

An extensive experimental study of the near wake of a 9° half-angle sharp cone at incidence at M∞, = 7 has been conducted. The model was suspended by six thin wires. The flow field in the plane of symmetry was surveyed from 0 to 2·5 diameters downstream of the base in the axial direction, and to the bow shock in the transverse direction. For three values of incidence (0° 5° and 15°))measurements were made of the static pressure, Pitot pressure and stagnation temperature, from which the streamline pattern was determined. In addition the u. = 0 line was found independently. The results of the measurements were also used to trace lines of constant Pitot pressure showing the shock waves, and lines of constant stagnation temperature in the viscous wake.