A new economical finite-difference method is described for the calculation of three-dimensional heated surface jets discharging into stagnant water. The equations solved are for continuity, lateral and longitudinal momentum, and thermal energy. The turbulent shear stresses and heat fluxes in these equations are determined with a turbulence model involving simplified forms of the transport equations for these stresses and fluxes and the solution of differential transport equations for the turbulent kinetic energy κ and the rate of its dissipation ε. The experimentally observed entrainment reduction due to buoyancy is reproduced by this model. The predictions are compared in detail with the recent measurements of Pande & Rajaratnam, which are judged to be superior to those of other investigators. The agreement is generally satisfactory.

The streamlines due to a stokeslet on the axis in a finite, semi-infinite and infinite cylinder are obtained together with the case of a Stokes-doublet and source-doublet in an infinite cylinder. In the infinite and semi-infinite cylinder examples an infinite set of toroidal eddies are obtained. The eddies alternate in sign and the magnitude of the stream function decays exponentially with distance from the driving singularity. In the finite cylinder a primary interior eddy adjacent to the singularity is always obtained and, depending on location of the singularity within the cylinder and the ratio of cylinder length to radius, a finite number of secondary interior eddies. In the case of long cylinders, the eddies are generated along the axis, whereas, for squat cylinders, secondary eddies occur in the radial direction. The interior eddies emerge from the corner as the length of the cylinder is increased. Moffatt corner eddies exist but they are very much smaller than the interior eddies.

Flows in thin-walled, collapsible tubes are of fundamental importance to various physiologic phenomena and clinical devices.

A one-dimensional, unsteady theory is developed for flows generated either by externally applied pressures or by body forces. Part 1 deals with small-amplitude, linearized flows, part 2 with large amplitude, nonlinear flows. Experimental results for a tube collapsing under external pressure are given in part 3, together with theoretical interpretations and comparative results of numerical simulations.

Several new and unanticipated phenomena are revealed. These are in part associated with the highly nonlinear ‘equation of state’ (transmural pressure versus area) for a partially collapsed tube, and in part with whether the flow speed is sub- or supercritical relative to the speed of area waves. For instance, a flow produced by a spatially uniform external pressure applied to a limited region becomes choked at a flow-limiting throat at which point the fluid speed reaches the local wave speed. This throat forms at the edge of the pressurized region. The critical velocity can be exceeded with the application of certain types of spatially graded external pressures.

The adiabatic approximation for a solitary wave in a channel of gradually varying breadth b and uniform depth is tested by experiment and by numerical solution of the generalized Korteweg-de Vries (KdV) equation. The results for a linearly diverging channel show good agreement with the prediction α (dimensionless wave amplitude)$\propto b^{-\frac{2}{3}}$. The experiments and numerical solutions for the linearly converging channel show that the wave growth is well approximated by α ∞ b−½. The discrepancy between the diverging and converging channels is shown to be due to nonlinear effects associated with the choice of the spatial variable as the slow variable in the generalized KdV equation. The measured and computed profiles display the predicted ‘shelves’ of elevation and depression in the converging and diverging channels, respectively.

The far field of a lifting three-dimensional wing in transonic flow is analysed. The boundary-value problem governing the flow far from the wing is derived by the method of matched asymptotic expansions. The main result is to show that corrections which are second order in the near field make a first-order contribution to the far field. The present study corrects and simplifies the work of Cheng & Hafez (1975) and Barnwell (1975).

Previous experimental work on the diffraction of a detonation wave at a large and abrupt area change in a tube, has shown that every system is characterized by a critical tube diameter at which quenching of the detonation occurs. Zeldovich, Kogarko & Simonov (1956) established that the critical tube diameter, for the oxy-acetylene system with varying dilution of nitrogen, lies between 500 and 700 times the one-dimensional induction zone length. Later, Mitrovanov & Soloukhin (1964) discovered that, for the same system, the critical diameter is 10 or 13 times the transverse wave spacing for a flat channel and cylindrical tube respectively. The two results are shown to be equivalent and are confirmed by further experiments in a 75 × 6 mm channel in which the flow is two-dimensional.

Smoked foil and schlieren records show that, for supercritical waves, re-ignition occurs at sites along the wedge formed by the head of the expansion from the diffracting aperture and criticality is attained when the site is located at the apex of the wedge. A universal feature of re-initiation, which is also observed in liquid and solid explosives, is the sudden appearance of a transverse detonation which sweeps through the compressed, but unreacted, gas of the dissociated shock-reaction zone regime; this is signalled by the appearance of fine triple-point writing on smoked-foil records.

A criterion for re-initiation is formulated by equating the critical velocity gradient which characterizes the decay of the wavefront in a cell, to that obtaining in the diffracted shock front at the head of the expansion fan; an expression for the latter is derived from Whitham's (1957) theory for non-reactive shocks. The prediction of the criterion is in good agreement with observation.

For a particle with two or three perpendicular planes of symmetry rotating at low Reynolds number in a Couette flow field, there are three orbits in which the motion is simply periodic. The three scalars Bi involved in Bretherton's shape tensor are found experimentally from periods of rotation in these orbits. Experiments on right parallelepipeds, ranging from thin platelets through cubes to rectangular section rods, are described. Each of the Bi is found to depend on the aspect ratio of one of the particle cross-sections, with only slight influence from its third dimension. Results are expressed in terms of a relationship between this aspect ratio and that of an equivalent ellipse, incorporating a weak function of the third particle dimension. The equations of motion governing a general doubly periodic motion, and incorporating experimental Bi values, are integrated numerically and compared satisfactorily with experimental observations.

The high-Reynolds-number (K) flow through a symmetric channel, with walls whose shape is time dependent, is studied. The distortion of the walls is of non-dimensional height $O(K^{-\frac{1}{3}}$) and length O(1), this particular size of perturbation being chosen such that (for the first regime of unsteadiness studied) the effects of the unsteadiness, viscous diffusion and advection all interact nonlinearly in the region of the fluid near the walls.

For this first regime of unsteadiness the problem is solved numerically. This leads on to analytic descriptions for progressively faster time variations of wall shape, and in fact the entire range of unsteadiness is covered for this particular size of distortion.

When heat is applied at the bottom of a stable salinity gradient a series of layers with uniform temperature and salinity is formed. The evolution of this system is investigated in the laboratory and a numerical model of the process is developed. New layers are formed sequentially at the top of a growing convection region while lower down adjacent layers merge. For given fluid properties the convection depends upon one parameter Q, which is proportional to the (suitably non-dimensionalized) ratio of the salinity gradient to the heat flux. We find that the depth of the top of the convecting region and the number of layers present increase like the square root of time over the range of Q examined. This permits the definition of an effective conductivity, KT, for the total series of layers which is directly proportional to κT, the molecular thermal diffusivity, and inversely proportional to Q. The vertical growth of the layers is thus retarded by increasing Q. The average thickness of the layers decreases with increasing salinity gradient and appears to be independent of the applied heat flux.

Solutions to Long's equation for a stably stratified incompressible fluid traversing a mountain range are obtained for various terrain shapes and amplitudes when the horizontal scale is large compared to the vertical wavelength. Nonlinear lower and upper (radiative) boundary conditions are utilized and found to have a strong influence on the wave structure at large amplitudes. The results for symmetric and asymmetric mountain profiles reveal that the wave amplitude and wave drag are significantly enhanced for mountains with gentle windward and steep leeward slopes. These results confirm and explain those obtained by Raymond (1972) using a different solution method. Several results obtained by Smith (1977) from perturbation analysis are also confirmed and extended to large amplitudes. The methods are also applied to investigate the nonlinear nature of the singularity predicted by linear theory for flow over a step.

The effects of thermally insulating boundaries on rapidly and almost rigidly rotating gas flows are examined. It is shown that, on a thermally insulating boundary, all boundary layers disappear to zeroth order and that the geostrophic flow alone satisfies the kinematical boundary condition on such a boundary. The temperature gradient of the geostrophic flow is on a horizontal thermally insulating boundary corrected by a weak Ekman layer of strength E½ where E is the Ekman number. On a vertical thermally insulating boundary, the temperature gradient of the geostrophic flow is in the general case corrected by E¼ and $E^{\frac{1}{3}}$ Stewartson layers of strengths E¼ and $E^{\frac{1}{3}}$ respectively.

A model for the kinematics of a turbulent flow close to a solid boundary is explored. The field is assumed to be homogeneous in the direction of mean flow. The equations of motion are solved numerically for a flow which is periodic in time and in a direction transverse to the direction of mean flow. The period is taken to be the time interval between ‘bursts’ and the wavelength, the spacing of the streaky structure close to the wall observed by a number of investigators. Good agreement is obtained between the calculated flow field and experimental results, especially for y+ < 15. This agreement suggests that the flow oriented eddies in the viscous wall region can be represented by a model which views the flow in this region to be coherent and to be associated with spanwise flow deviations in a well-mixed outer region. The model allows for the periodic movement of low momentum fluid from the wall, which, because of the assumption of a well mixed outer region, gives rise to a shear layer. This seems to correspond to the observed ‘bursting’ phenomenon. The calculations confirm the suggestion by Fortuna and Hanratty (Fortuna 1970; Hanratty, Chorn & Hatziavramidis 1977) that the secondary flow in the viscous wall region generated by these spanwise flow deviations gives rise to the development of large velocity fluctuations in the direction of mean flow and accounts for the experimentally observed maximum in the velocity fluctuations close to the wall. Also, the comparison of calculations with measurements of the average velocity and with an experimental quadrant analysis of the Reynolds stress suggests that the secondary flow is making a major contribution to the Reynolds stress.

In order to elucidate the general Stokes flow characteristics present for slender bodies of finite centre-line curvature the singularity method for Stokes flow has been employed to construct solutions to the flow past a slender torus. The symmetry of the geometry and absence of ends has made a highly accurate analysis possible. The no-slip boundary condition on the body surface is satisfied up to an error term of O(ε2 In ε), where ε is the slenderness parameter (ratio of cross-sectional radius to centre-line radius). This degree of accuracy makes it possible to determine the force per unit length experienced by the torus up to a term of O(ε2). A comparison is made between the force coefficients of the slender torus to those of a straight slender body to illustrate the large differences that may occur as a result of the finite centre-line curvature.

A model equation is derived to study trapped nonlinear waves with a turning effect, occurring in disturbances induced on a two-dimensional steady flow. Only unimodal disturbances under the short wave assumption are considered, when the wave front of the induced disturbance is plane. In the neighbourhood of certain special points of sonic-type singularity, the disturbances are governed by a single first-order partial differential equation in two independent variables. The equation depends on the steady flow through three parameters, which are determined by the variations of velocity and depth, for example (in the case of long surface water waves), along and perpendicular to the wave front. These parameters help us to examine various relative effects. The presence of shocks in a continuously accelerating or decelerating flow has been studied in detail.

A numerical method is presented for the computation of two-dimensional periodic progressive surface waves propagating under the combined influence of gravity and surface tension. The dynamic boundary equation is used in its exact nonlinear form. The procedure involves a boundary-integral formulation coupled with a Newtonian iteration. Solutions of high accuracy can be achieved over much of the range of wavelengths and heights including limiting waves. A number of different continuous families of solutions have been produced, all of which ultimately exhibit closed bubbles at their troughs. The so-called critical wavelengths are less important than have been previously assumed; the number of possible wave forms does increase with increasing wavelength, however.

The linear perturbations of the flow of a non-diffusive fluid are considered. The classification of the normal modes of parallel flow of an inviscid stratified fluid presented by Banks, Drazin & Zaturska (1976) is here extended to encompass modes which propagate at infinity. When the basic flow is unbounded and the buoyancy frequency is non-zero at infinity the five classes presented earlier are augmented by three further classes: for a given flow and wavenumber they are (a) a continuous class of non-singular stable modes which are modifications of internal gravity waves by shear; (b) a continuous class of stable modes which are singular at each critical layer but otherwise similar to those of class (a); and (c) a finite number of marginally stable singular modes with over-reflexion. This classification is illustrated by many new results. Some asymptotic properties of the stable and unstable modes are found for large values of the Richardson number and for long waves. Two prototype problems, in which the basic flows are a piecewise-linear shear layer and a triangular jet, are solved analytically. The modified internal gravity waves for a Bickley jet with uniform buoyancy frequency are treated to illustrate the complementary nature of the propagating and evanescent modes. This treatment is both analytical and numerical. The general ideas are further illustrated by a numerical study of the stability characteristics of a hyperbolic-tangent shear layer. Finally the modes for a basic flow of boundary-layer type are found in exact terms of a hypergeometric function.

Numerical solutions of the inviscid equations that describe standing waves of finite amplitude on deep water are reported. The calculations suggest that standing waves exist of steepness, height and energy greater than the limiting wave of Penney & Price (1952). The computed profiles are found to be consistent with Taylor's (1953) experimental observations.

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

Natural convection flows induced by localized heating of the base of a thermally stratified enclosure are examined. The enclosure is a vertical circular cylinder with height equal to radius. The wall temperature increases linearly with height, and a small heat source is centrally located on the floor. Parameters of the problem are the ambient stratification rate (Γ−1), the Prandtl number (Pr), and a Grashof number (Gr) based on the temperature and the diameter of the heated spot. Over wide ranges of Γ and Gr, vertically layered convection cells are observed in the upper part of the enclosure in both laboratory experiments and numerical calculations. For the case of strong buoyancy and weak stratification, plume-like flows exist immediately above the heat source. Streak photographs are in qualitative accord with the numerical calculations, except for a range of Gr when an azimuthal rotation of the laboratory plume is observed. All flows are otherwise laminar. The theoretical results reveal a √Gr scaling at large Gr for the velocities and heat transfer rates, and a correlation of strongly stratified, viscous flows with the group Gr Pr Γ−1.

The existence of a liquefaction shock wave, a compression shock which converts vapour into liquid, has recently been predicted on physical grounds. The liquefaction shock was experimentally produced as the reflected shock at the closed end of a shock tube. Measurements of pressure, temperature, index of refraction and shock velocity confirm the existence of the shock and its general conformity to classical Rankine-Hugoniot conditions, with a discrepancy ∼ 10°C between measured and predicted liquid temperatures. Photographic observations confirmed the existence of a clear liquid phase and revealed the (unanticipated) presence of small two-phase torus-form rings. These rings are interpreted as vortices and are formed in or near the shockfront (∼ 50 rings/mm2 are visible near the shockfront at any given time). Separate experiments with the incident shock under conditions of partial liquefaction produced a fog behind the shock: measurements of laser-beam attenuation yielded the thickness of the condensation zone and estimates of the droplet size (∼ 10−7 m).