An arbitrarily-shaped, closed container completely filled with fluid is considered. It is assumed that the fluid is originally in a stably-stratified state of rest, and that at an initial instant the temperature of the container walls is impulsively changed. The ensuing unsteady laminar motion is found by solving the linearized Boussinesq equations governing buoyancy-driven flows. A ‘boundarylayer/inviscid-interior’ decomposition leads to a modified asymptotic expansion scheme of analysis. The boundary-layer concept is valid only for large values of the Rayleigh number, and, in addition, we limit the Prandtl number to order unity. It is found that the inviscid interior region heats up by means of a convection process that is driven by suction induced by the boundary layer. The inviscid, adiabatic interior responds to a special horizontal ‘average’ value of the container temperature perturbation. The boundary layer smears out, or averages, any circumferential variation in this perturbation, so that the interior, in effect, responds to an isothermal boundary in each horizontal plane. The interior temperature and vertical velocity component are expressed simply in terms of this horizontal ‘average’ container temperature. The horizontal velocity potential is governed by a Poisson equation, whose solution is developed for several specific geometries to illustrate the nature of the flow.

The paper presents a comprehensive analysis of gas motion created by a strong explosion in an atmosphere whose density is an exponential function of altitude. For the near field (i.e. short times after initiation), an exact analytical solution of the equations of motion is obtained by means of a perturbation technique. For the far field (i.e. long times after initiation), a similarity solution associated with a logarithmic front trajectory is derived. The two are shown to be well matched with each other. Finally, a fully-algebraic approximate solution is given that qualitatively reproduces all the salient features of the exact and asymptotic solutions, while quantitatively it is in fair agreement with their results.

The flow near the bow of a steadily turning ship is analysed using a modified slender-body theory. The rate of change of flow quantities in the longitudinal (x) direction is assumed to be greater than that implied by ‘conventional’ slender-body theory. As a consequence some features of high Froude number flow are apparent which cannot be predicted by the ‘conventional’ theory. The modified slender-body theory proposed requires the solution of a two-dimensional Laplace equation (in y and z) but its free-surface condition still involves an x derivative. A Fourier-transform method is used to solve this problem. A simple bow configuration of constant draft is analysed and numerical results for the free-surface elevation are presented.

The interaction between internal gravity waves in a rotating frame and the mean flow is discussed for the case when the properties of the mean flow vary slowly on a scale determined by the local wave structure. The principle of conservation of wave action is established. It is shown that the main effect of the waves on the Lagrangian mean velocity is due to an appropriate ‘radiation stress’ tensor. A circulation theorem and a potential-vorticity equation are derived for the mean velocity.

The exit condition at the trailing edge of a nozzle for slightly unsteady flow has been investigated experimentally. This problem plays a crucial role in sound transmission through nozzles with flow. The measuring technique used is new and is based on the synchronization of a laser beam to the wave motion of a small smoke filament in the boundary layer leaving the nozzle. The resolution of the jet flow deflexion measurements is of the order of 1–3μm. The authors found the jet deflexion envelope to have a nearly parabolic shape near the nozzle edge. The size of this ‘parabolic’ region decreases with decreasing Strouhal number. This statement applies to the motion of the exterior border of the boundary layer at the dividing streamline between flow originating from the interior of the nozzle and flow coming from outside. It was found that the unsteady flow problem near the edge remains linear for fluctuating velocities of small magnitude.

The paper examines how a passive contaminant disperses along the axis of a tube in which the flow is driven by a longitudinal pressure gradient varying harmonically with time. This problem is of intrinsic interest and is relevant to some important practical problems. Two examples are dispersion in estuaries and in the blood stream. By means both of statistical arguments and an analysis like that used by Taylor (1953) in the case of a steady pressure gradient it is shown that eventually the mean distribution of concentration satisfies a diffusion equation (and is therefore a Gaussian function of distance along the axis) with an effective longitudinal diffusion coefficient K(t) which is a harmonic function of time with a period equal to one half of that of the imposed pressure gradient. Contrary to the supposition made in most previous work on this problem it is shown by examining some special cases that the harmonic terms in K(t) may have a noticeable effect on the dispersion of the contaminant and in particular on the rate at which it is spreading axially. The size of the effect depends on both the frequency and the Schmidt number and is particularly large at low frequencies. The paper concludes with an analysis of a model of dispersion in estuaries which has been used frequently and it is concluded that here too oscillatory effects may often be noticeable.

In order to prevent stagnation and improve water quality, the intakes of many water-supply reservoirs have the form of momentum jets directed approximately radially into the storage. An analysis which idealizes this flow as consisting of a turbulent jet issuing horizontally from a circular orifice into a large rotating basin of deep water shows that the jet path is a spiral whose length scale L depends upon the rate of rotation and the kinematic momentum of the jet. Good agreement is found with flow-visualization experiments when the basin depth h is ‘large’ (h/L [gsim ] 0·21). For small depths (h/L [lsim ] 0·024) the flow tends to be twodimensional and the jet path is found to be straight. Full-scale reservoirs are usually shallow enough that the effect of the earth's rotation on the jet path is likely to be small. However these reservoirs are not inordinately shallow and tests with distorted hydraulic models are likely to show significant effects of rotation and can be misleading to the unsuspecting.

A Mach-Zehnder interferometer was used to study the interaction between convection currents in the sea water and the ice-water interface during freezing. The temperature at the ice-water interface during the early stages of freezing was approximately 0 °C, not the equilibrium freezing temperature. Salt plumes were observed to appear first at specific sites on the ice-water interface. This is contrary to earlier results reported by Farhadieh & Tankin (1972). The salinity profiles in the salt plumes were measured using interferograms in conjunction with Abel inversion. The maximum salinity, along the centre-line of the salt plume, is an indication of the salinity of the brine trapped within the ice.

The linear stability of a shear layer of an inviscid compressible fluid is considered. It is shown that there is instability of two-dimensional disturbances at all values of the Mach number, contrary to previous results for a vortex sheet. The difference arises from the discovery of a second unstable mode. This mode is supersonic, decays weakly with distance from the shear layer, and is not governed by the principle of exchange of stabilities. Detailed numerical and asymptotic results are given for the hyperbolic-tangent shear layer.

A shock-dynamic model based on the symmetrical expansion of the critical shock is used to analyse the progressive decay of an originally planar shock wave through a large and abrupt area change. This is tested against measurements of shock strength made along the axes of area changes where the shock waves are free to expand in two or three dimensions.

The critical shock is defined as the configuration when the decaying shock wave at the axis first becomes curved. The axial shock begins to decay less than one diameter from the entrance of the area change. Differences between the experimental onset of decay and the theoretical position of the critical shock are accounted for by the non-ideal behaviour of a practical pressure transducer.

The model predicts that, when the shock wave is decaying symmetrically, there is a linear relationship between a derived function ε of the decaying shock strength and the distance from the area change. This is confirmed experimentally for all the shocks studied. The quantitative application of the results in three dimensions up to 400 mm enables accurate predictions of experimental results at 1 m for M < 2·0. Also, the model may be applied to three-dimensional results to predict accurately equivalent results in two dimensions.

The numerical values of ε are based on the equivalence of the ratio of the shock areas and the ratio of their Chisnell (1957) functions. Hence correlations between experimental results and predictions of the model are evidence that Chisnell's theory can be extended to include large and abrupt area changes.

The paper discusses a theoretical model of statistically steady flow in a strongly stratified estuary. A halocline is assumed to be present and the lower layer is taken to be deep and non-turbulent. The outflowing upper fluid mixes with the salty lower fluid and the flux of the brackish water increases with distance from the head of the estuary. The mixing is assumed to be similar to that in laboratory models of mixing across density interfaces.

Two equations of mass conservation are used, one for the steady-state mass flux across a vertical section from top to bottom of the channel and one for the mass flux into a section of the upper fluid. A buoyancy conservation equation is used for the buoyancy flux across a vertical section. A final equation is obtained by integrating the horizontal equation of motion across a section of the upper fluid. The flow in this layer is assumed to be opposed by a frictional force proportional to the square of the velocity averaged over the layer. The pressure-gradient force arising from the slope of the free surface is solved for in terms of the thickness of the upper layer, the buoyancy difference across the interface, the slope of the interface and the horizontal density gradient in the upper layer. The derivation shows that the horizontal pressure-gradient force vanishes in the lower layer.

The mathematical problem reduces to two ordinary differential equations for the flux in the upper layer and its thickness. Attention is confined to the solution for subcritical flow, in which the interface falls with distance from the head, reaching a maximum depth at a certain section of the estuary. Beyond this the interface rises. At the mouth, where, by definition, the width of the estuary increases rapidly, it is shown that there must be a transition from subcritical to supercritical flow. This condition, applied to the solution for uniform width, determines a remaining unknown related to the depth of the halocline at the head of the estuary and the complete solution is obtained as a function of the freshwater influx per unit width, the r.m.s. turbulent velocity, the estuary length and the buoyancy of sea water.

The solution is complicated but has reasonable behaviour for variations of the given parameters of the problem. A basic feature for values of the constants appropriate to fjord-type estuaries is the dominance of friction, omitted in an earlier, incomplete investigation by Stommel. This is also revealed by the large drop in the free surface over the length of the estuary.

A comparison with two estuaries, Oslofjord and Knight Inlet, British Columbia, indicates that the former is very different from the model of this paper but that the latter may have a similar nature.

The work reported in part 1 has been extended to cover flows past circular cylinders with wake splitter plates having 2 [les ] L/D [les ] 7 and to include flows past normal flat plates with splitter plates having L/D [les ] 3·7. Pressure distributions and wake Strouhal numbers were measured and visualization studies carried out. The results obtained indicate that no further changes would be produced by lengthening the splitter plates beyond the limits tested.

The combined results of parts 1 and 2 provide coherent descriptions of the effects of wake splitter plates for all values of L/D of significance for the two profiles, which are representative of two distinct classes of bluff bodies, viz., those with cross-sections of curvilinear shape for which the flow separation points are not determined uniquely by the geometry and those for which the separation points are fixed.

The effect of the simultaneous action of a uniform magnetic field and a uniform angular velocity on the linear stability of the Bénard layer to time-dependent convective motions is examined in the Boussinesq approximation. Four models, characterized by the relative directions of the magnetic field, angular velocity and gravitational force, are discussed under a variety of boundary conditions. Apart from a few cases, the treatment applies when the Taylor number T and the Chandrasekhar number Q (the square of the Hartmann number) are large. (These parameters are dimensionless measures of angular velocity and magnetic field, respectively.)

It is shown that the motions at the onset of instability can be of three types. If the Coriolis forces dominate the Lorentz forces, the results for the rotating non-magnetic case are retained to leading order. If the Coriolis and Lorentz forces are comparable, the minimum temperature gradient required for instability is greatly reduced. Also, in this case, the motions that ensue at marginal stability are necessarily three-dimensional and the Taylor-Proudman theorem and its analogue in hydromagnetics are no longer valid. When the Lorentz forces dominate the Coriolis forces, the results obtained are similar to those for the magnetic non-rotating case at leading order.

The most unstable mode is identified for all relations T = KQα, where K and α are positive constants, taking into account both time-dependent and time-independent motions

Various types of boundary layers developing on different boundaries are also examined.

This paper reports measurements of turbulent quantities in an axisymmetric wall jet subjected to an adverse pressure gradient in a conical diffuser, in such a way that a suitably defined pressure-gradient parameter is everywhere small. Self-similarity is observed in the mean velocity profile, as well as the profiles of many turbulent quantities at sufficiently large distances from the injection slot. Autocorrelation measurements indicate that, in the region of turbulent production, the time scale of ν fluctuations is very much smaller than the time scale of u fluctuations. Based on the data on these time scales, a possible model is proposed for the Reynolds stress. One-dimensional energy spectra are obtained for the u, v and w components at several points in the wall jet. It is found that self-similarity is exhibited by the one-dimensional wavenumber spectrum of $\overline{q^2}(=\overline{u^2}+\overline{v^2}+\overline{w^2})$, if the half-width of the wall jet and the local mean velocity are used for forming the non-dimensional wavenumber. Both the autocorrelation curves and the spectra indicate the existence of periodicity in the flow. The rate of dissipation of turbulent energy is estimated from the $\overline{q^2}$ spectra, using a slightly modified version of a previously suggested method.

This paper reports experimental investigations on the characteristic decay and the radial-type decay regions of a three-dimensional isothermal turbulent wall jet in quiescent surroundings. The velocity and the length scale behaviour for both the longitudinal and the transverse directions are studied, and compared with the results of other workers. The estimated skin friction is discussed in relation to the available data from earlier investigations. Wall jet expansion rates and the behaviour of skin friction are also discussed. The rate of approach of turbulence components to a self-similar form is found to be influenced by the fact that the expansion rate of the wall jet in the longitudinal direction is different from that in the transverse.

An analysis of the response of an X-probe which takes into account the axial sensitivity k, the effect of the w component of velocity and the effective rectification by the hot wire is presented. Owing to rectification points in the measured u, v ‘phase plane’ are confined within a certain sector, thus leading to distortions in the measured joint probability density functions. Numerical computations show that high turbulence intensities lead to large errors in second-order moments measured by cross-wire probes; for instance, the error in the measured correlation (over and above that due to k) can be as high as 28% when the turbulence intensity is 35%.

A stably stratified free shear layer is created in a continuously circulating water channel in the laboratory. Two streams of salt water of different concentrations are brought together at the entrance to the open channel and a layered uniform flow field with a distinct sharp interface is produced in the test section. The maximum density difference between the two layers is Δρx = 0·0065ρw, where ρw is the density of water. The velocity of each layer can be adjusted at will to create free shear across the interface. At the end of the open channel, a mechanical device to separate the layers for recirculation is provided. The resulting flow field has a viscous region approximately 15 times larger than the scale of the salinity diffusion layer. Visual observations are made with hydrogen bubbles and dye traces. Interfacial waves are initiated by artificial excitation. The perturbation frequencies range from 0·476 to 10·40Hz. The measured wavelengths range from 0·46 to 3·02 cm. Damped waves as well as growing waves are observed at various exciting frequencies. Velocity profiles and instantaneous velocities are measured by a hot-film anemometer designed for use in salt water. Experimental values of the Richardson number, the dominant parameter characterizing the instability process, range from 1·23 to 14·45. The stability boundary is determined experimentally. Comparisons with Hazel's numerical results and the earlier results of Scotti & Corcos for low values of the Richardson number are also made.

The paper concerns one- and two-dimensional models of steady seepage flow in unconfined aquifers and the relationship between them. The first part gives a new proof of Charnyi's result that one- and two-dimensional theory yield the same value for the flow rate in a horizontal aquifer or porous bed between vertical ends and shows the extent to which it can be generalized to non-uniform or anisotropic media. The second part solves the highly two-dimensional problem of flow from a line source (line of springs) in an otherwise impermeable, sloping stratum and compares the result with the predictions of a one-dimensional Dupuit–Pavlovsky approach. Confirmatory experiments using the Hele Shaw analogue of seepage flow are also reported.

Supercavitating hydrofoils of large aspect ratio operating near a free surface are investigated, assuming an inviscid and irrotational flow with the effects of gravity and surface tension neglected. The flow near the foil, treated as two-dimensional, is solved by a nonlinear free-streamline theory, then a three-dimensional ‘downwash’ correction is made using Prandtl's lifting-line theory. The strength of the lifting-line vortex is determined by information from the two-dimensional solution through a matching procedure, in which the inverse of aspect ratio is used as a small parameter for asymptotic expansions. The analysis incorporates a free-surface reference level to determine the submergence depth of the foil. The present method can be applied to any type of foil having an arbitrary planform or profile shape, including a rounded leading edge, a twist and even a small dihedral angle, within the assumption of large aspect ratio. Numerical computations made on rectangular flat-plate hydrofoils show excellent agreement of results with existing experimental data, even for large angles of attack and relatively low aspect ratios. The pressure distributions, shapes of the cavity and free surface are also calculated as a function of spanwise position.

Pressure and pressure-velocity space correlations are calculated, using rapid-distortion theory, for turbulence in a uniform shear flow. It is found that pressure fluctuations remain correlated over significantly greater distances than do velocity fluctuations. When these predictions are used as a model for turbulence in free turbulent shear flows, it is found that the predicted scale of the pressure fluctuations is larger than the flow width. It is proposed that pressure fluctuations remain highly correlated right across free shear flows. Predictions from the theory are then compared with various experimental situations in which reasonable qualitative agreement is to be expected, and this is found.