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 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.

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.

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.

A submerged sphere is considered to be absorbing power from an incident wave through an integrated mooring and power take-off system. It is shown that the power absorbed (as characterized by the absorption length) depends on the hydrodynamic properties of the sphere; in particular on the added-mass and damping coefficients. These coefficients are determined and the results used to study the power absorption properties of the sphere. Curves are given showing the variation of the absorption length with wavenumber, for differing depths of submergence.

The flow of closely fitting neutrally buoyant elastic spheres through a circular cylindrical tube is considered under the assumptions that the Reynolds’ equation is valid in the fluid and equations of linear elasticity hold in the solid. Computations are carried out for several values of Poisson's ratio. The results are compared with the results of previous models on elastic compressible particles.

The response of a heated circular cylinder to impulsive and sinusoidal variations in the velocity of flow past it has been simulated by numerical integration of the governing equations. The fluid has been treated as viscous and incompressible and as having constant properties. The range of Reynolds number investigated was 1 [Lt ] R [Lt ] 40. Since vortex shedding normally does not occur in this range, the flows were treated as symmetrical. The thermal and flow transients are presented for the following cases.: (i)impulsive starts from rest to final steady state Reynolds numbers 1, 5, 10, 26·67; (ii)impulsive increases in velocities of 50% magnitude from steady state Reynolds numbers 1, 10 and 26·67; (iii)sinusoidal variation in velocity with amplitude of 10% impressed on a mean flow at Reynolds number 10.

Results are also given for the thermal transients associated with instantaneous changes in cylinder temperature at Reynolds numbers 1, 5 and 40. The results obtained for transient and steady state flow parameters are in agreement with those obtained numerically and experimentally by other workers and the results for steady state heat flux from the cylinder are in agreement with experimental values. The new results obtained for heat transfer in unsteady flows provides information which is relevant to the operation of hot-wire anemometers.

The well-known class of self-similar solutions for an ideal polytropic gas sphere of radius R(t) expanding into a vacuum with velocity $u(r, t) = r\dot{R}/R$ is shown to be convectively unstable. The physical mechanism results from the buoyancy force experienced by anisentropic distributions in the inertial (effective gravitational) field. An equation for the perturbed displacement ξ(r, t), derived from the linearized fluid equations in Lagrangian co-ordinates, is solved by separation of variables. Because the basic state is non-steady, the perturbations do not grow exponentially, but can be expressed in terms of hypergeometric functions. For initial density profiles \[ \rho_0(r)\sim(1-r^2/r^2_0)^{\kappa}, \] modes with angular dependence Ylm(θ, ϕ) are unstable provided l > 0 and κ < 1/(γ − 1), where γ is the ratio of specific heats. For large l, the characteristic growth time of the perturbations varies as l−½ and the amplification increases exponentially as a function of l. The radial eigenfunctions are proportional to rl, and the compressibility and vorticity are both non-zero.