An investigation is made of the scattering effect of a random ocean bottom of constant average depth upon the propagation of shallow water waves. Of particular concern is the case of long-distance propagation, in which the conventional perturbation schemes fail to apply. The approximation scheme employed is basically one of selective summation of the type used in other areas of physics such as the theory of many-body interactions. Results are obtained for the average wave and the two-point correlation function of the wave field for the case when the ocean statistics are homogeneous and isotropic. The application of the results to the case of a tsunami is discussed.

The propagation of plane hydromagnetic waves in a fluid rotating with angular velocity ω and permeated by a magnetic field B = {Bx(z), By(z), 0} varying in both magnitude and direction with z is studied by techniques recently applied to the propagation of internal gravity waves in a shear flow (Bretherton 1966; Booker & Bretherton 1967). Particular attention is paid to a class of 'slow’ hydromagnetic waves of interest in connexion with the dynamics of the earth's liquid core. While, in general, rotation permits propagation across the lines of force, there is associated with each wave a ‘critical level’ z = zc, which acts as a valve by effectively permitting the wave to penetrate it from one side only. A slow hydromagnetic wave with frequency ω and wavenumber components k,l normal to the magnetic field gradient can only effectively penetrate its critical level if its propagation speed across field lines W is such that Wωz,(ωx,k+ ωy,l)ω < 0. The phenomenon of ‘critical-layer absorption’ evidently does not in general require the presence of a mean shear flow; a non-uniform magnetic field gives rise to similar effects provided that some other restoring mechanism (in this case the Coriolis force) is available to permit hydromagnetic waves to propagate across field lines.

It has been observed that standing surface waves in water may be excited by acoustic fields of very much higher frequency. No special relationship between the two frequencies appears to be required, but there is such a relationship between the spatial variations of the acoustic and surface wave modes. Another requirement is that the lower frequency should be comparable with the resonant and width of the acoustic response of the system. An explanation of such phenomena is proposed and is tested on a somewhat idealized model by the use of techniques which could be extended to deal with more realistic situations. The model serves to explain qualitatively the available experimental observations. It is suggested that the phenomenon of spatial resonance is not confined to the interaction between water waves and acoustic fields, but may occur generally in systems having modes with related spatial patterns but greatly different frequencies.

Experiments on the two-dimensional turbulent wake generated by pairs of cylinders of unequal diameter have revealed some interesting flow characteristics. The wake width grew asymmetrically in the downstream direction, spread rate and entrainment coefficients proving larger on the small diameter cylinder side. Mean velocity profiles were also skewed to this side while maximum values of Reynolds stresses were larger on the other. Close to the cylinder, a region or turbulent ‘energy reversal’ was measured. The level of turbulence and the diffusion mechanism were high at this point and some comments are made concerning the structure of the flow under these conditions.

The phenomenon was observed during experiments in which a beaker containing water was vibrated in one of its bell modes (the inextensional flexural vibrations of the wall). For certain combinations of driving force and frequency, a standing water wave of large amplitude was generated whose peripheral wavenumber might be either zero (i.e. the wave was radially symmetric) or twice that of the bell mode. This relationship between the wavenumbers of the bell mode and water wave, and the fact that the driving frequency was many times that of the water wave, indicated that this was an instance of a general mechanism that has been studied theoretically by Mahony & Smith (1972). For a model situation, allowing for dissipative effects and nonlinear coupling between nearly resonant oscillations at greatly differing frequencies, they derived a relation-ship between the driving force and frequency representing conditions of neutrals tability (i.e. such that the rate of energy transfer from the high frequency to the low frequency oscillations is zero). The aim of the experimental observations reported here was to check this relationship and other predictions of their theory.

A set of partial differential equations of the Navier-Stokes type is derived for external rarefied gas flows at all Knudsen numbers. Only the expressions of the stress tensor and heat flux vector are different from the customary Navier-Stokes relations and Fourier law. The new expressions are calculated from an approximate distribution function constructed through analysis of the BGK model of the Boltzmann equation so that it retains the qualitative property of the collision term and reproduces accurately the two extreme continuum and free-molecule regimes. Explicit forms of the stress tensor and heat flux components are given for low-speed two-dimensional flows. Solutions for vanishing Mach number and compressibility effects are then discussed. For a nearly isothermal cylinder, the present formulation leads to only one governing equation of the Navier-Stokes type for all flow regimes.

The stability of progressive internal waves of modes 1 and 3, propagating down a long tank filled with a linearly stratified salt water solution, is studied theoretically and experimentally. Examination of the spectra of the waves shows when a1 > 10−2, where a is the wave amplitude and l is the vertical wavenumber, that single internal waves excite waves of several resonant triads, where the excited waves belong to that set of triads with the largest theoretical growth rates. For example, a wave of mode 3 with a non-dimensional frequency around 0.66 excites waves of the following triads: (5,8,3), (6,9,3), (8,11,3), (9,12,3) and (10,13,3), where the integers are mode numbers. The spontaneous appearance of these naturally excited triads greatly complicates attempts to isolate and study preselected wave interactions. In one case, when waves of mode 1 and 3 with al > 10−2 were generated simultaneously while tuned to the (1,3,4,7) multiple resonance, the fastest growing wave was neither a wave of mode 4 located at the difference frequency nor a wave of mode 7 at the sum frequency, but rather a wave of mode 9 located at a frequency slightly above that of the 4-wave.

Extensive hot-wire auto- and cross-correlation measurements obtained in a fully developed compressible turbulent boundary layer are presented. A tentative mechanism of turbulence production and growth in hypersonic flow suggested by these measurements is developed. This flow model is consistent with previous observations in incompressible flows. Detailed measurements of the mean properties of the hypersonic turbulent boundary layer are also presented and compared with results from various transformation and finite-difference prediction methods. It is shown that none of the theories predict all the properties of the hypersonic turbulent boundary layer and that additional measurements are needed to provide more adequate physics of turbulent processes for use in the various theories.

The calculation of the pattern of waves produced by a point disturbance in a steady field which may be non-uniform can be performed straightforwardly by a single process of integration along the characteristic rays. For illustration, the method is applied to gravity waves produced by a source moving either in a straight line or in a circular path, to the symmetrical waves produced by a source in an expanding sheet and to the waves resulting from an instantaneous disturbance in :I stratified fluid.

An analysis of existing data on low Reynolds number flows strongly suggests that the conclusion of Simpson (1970) concerning the variation of von Kármás constant κ with Reynolds number is not correct. This implies that Coles’ (1962) assumption of the validity of the logarithmic velocity profile at low Reynolds numbers is correct and, moreover, that the inference drawn by Coles and later authors regarding the presence of viscous effects in the outer layer is valid. The analysis shows that these viscous effects are not present in duct flows, so that they are presumably associated with the presence of a turbulent-irrotational interface; it is argued that the ‘viscous superlayer’ can affect a large part of the outer layer at low Reynolds numbers. The data analysis incidentally shows that the viscous sublayer is more strongly affected by shear-stress gradients or transverse wall curvature than is the rest of the inner layer.

When similarity solutions are used to describe convective plumes or thermals, there is always found to be a discrepancy between the work done by buoyancy forces and the kinetic energy of mean motion. It is the main purpose of this paper to set down the ratio of these quantities for a wide variety of forms of buoyant elements and environmental stabilities. For consistency, the remaining fraction of the energy must appear as turbulent kinetic energy and eventually be dissipated, but these processes are not investigated in detail. The results are shown to have some relevance to the problem of convectively driven mixing across a density interface, where the largest scales of motion are dominant, and to the understanding of the transition zone between two self-preserving states of turbulent convection.

The Navier-Stokes like equation derived by Atassi & Shen for low-speed rarefied flows is analysed for a circular geometry. Asymptotic solutions for both Kn [Lt ] 1 and Kn B [Gt ] 1 are deduced using the method of matched asymptotic expansions. For Kn [Lt ] 1, fust- and second-order solutions are obtained and compared with the corresponding BGK treatment. For Kn [Gt ] 1, this equation is used only to describe the far field, the inner solution being of the free-molecule type. The matching procedure leads to a blockage correction for the drag and heat-transfer rate. Results are compared with other theories and known experimental data.

The flow induced by the differential rotation of a cylindrical depression of radius a in one of two parallel rigid planes rapidly rotating about their common normal at speed Q is studied. A Taylor column bounded by the usual Stewartson layers arises, but the shear-layer structure is rather different from any previously studied. The Ei-layers (E = v/ωa2) smooth the discontinuity in the geostrophic flow, but the way in which this is accomplished is related to the possible singu-larities of the E1/3-layer solutions. The fact that the 1/4-layer is partially free and partially attached to a vertical boundary accounts for the new joining conditions for the 1/4-layer. The drag on a right circular cylindrical bump in uniform flow is given in addition to some general comments on the applicability of these joining conditions to the motion of an axisymmetric object of quite general shape.

Measurements of the thickness and the stability of thin films of liquid (1–150 μmthick) formed on a rotating horizontal disk are presented and correlated in terms of an asymptotic-expansion solution of the thin-film equations. Water, various alcohols and water with wetting activities were used to cover a range of viscosity (1-2.5cP) and surface tension (20-72 dynes/cm). Smooth flow was found to occur in a region defined by the flow rate, rotational speed and physical properties of the liquid. Outside this region various wave patterns were observed (concentric, spiral and irregular waves). A linear theory of the stability of the film based on an extension of classical stability theories for plane films on inclined planes is given and contrasted with the experimental results. Surface phenomena associated with the use of wetting agents were found to have a strong effect on the stability of the film.

A fully developed turbulent boundary layer was subjected to a strongly favourable pressure gradient in order to investigate the role of the large eddy structure during ‘relaminarization’. Measurements of%he mean velocity profiles indicated that the ‘law of the wall’ disappeared in the region of the maximum pressure gradient. The three fluctuating velocity components and the tangential Reynolds stress were obtained to determine more precisely the nature of the decay of the turbulent structure. These measurements indicated that the absolute level of the velocities and stress were approximately constant along a mean streamline except near the wall. However, the relative levels were decreasing, as reported previously by several authors. The intermittency factor γ decreased along the mean streamlines until most of the boundary layer had only a negligible turbulence level. Space-time auto- and cross-correlations of u, v and I (the intermittency function) of the large-scale structure were obtained in the region of maximum pressure gradient and are compared with those measured in a zero pressure gradient flow.

This paper describes an experimental study of the distortion of grid-generated turbulence as it approaches the stagnation region of a two-dimensional bluff body. When Lx/D [Gt ] 1, where Lx, is the scale of turbulence and D is a typical body dimension, along the mean stagnation streamline $(\overline{u^2})^{\frac{1}{2}$ attenuates like the mean flow, whereas if Lx/D [Lt ] 1 the turbulence is distorted by the mean flow field and $(\overline{u^2})^{\frac{1}{2}}$ will amplify because of vortex stretching. When Lx/D = O(1) there is found to be a combination of these effects with attenuation of energy at low wavenumbers and amplification at high wavenumbers. Measurements of the pressure fluctuations at the stagnation point show that at low wavenumbers the level of the pressure fluctuations can be predicted by a direct application of Bernoulli's equation.

Several solutions for the solitary wave have been attempted since the work of Boussinesq in 1871. Of the approximate solutions, most have obtained series expansions in terms of wave amplitude, these being taken as far as the third order by Grimshaw (1971). Exact integral equations for the surface profile have been obtained by Milne-Thomson (1964,1968) and Byatt-Smith (1970), and these have been solved numerically. In the present work an exact operator equation is developed for the surface profile of steady water waves. For the case of a solitary wave, a form of solution is assumed and coefficients are obtained numerically by computer to give a ninth-order solution. This gives results which agree closely with exact numerical results for the surface profile, where these are available. The ninth-order solution, together with convergence improvement techniques, is used to obtain an amplitude of 0.85for the solitary wave of greatest height and to obtain refined approximations to physical quantities associated with the solitary wave, including the surface profile, speed of the wave and the drift of fluid particles.

New eigenvalue bounds are derived for the linear stability of inviscid parallel flows, both for homogeneous and for stratified fluids. The usefulness of these bounds, as compared with that of previous results, is assessed for several examples. For homogeneous fluids the new upper bounds for the imaginary part ci of the complex phase velocity are sometimes better than previous criteria. For both homogeneous and stratified flows, the new upper bounds for the wave-number α of neutrally stable disturbances improve on previous results, giving values within 10% of the known exact solution in several cases.

The nonlinear Boltzmann equation has been solved for shock waves in a gas of elastic spheres. The solutions were made possible by the use of Nordsieck's Monte Carlo method of evaluation of the collision integral in the equation. Accurate solutions were obtained by the same numerical procedure for eight values of the upstream Mach numbers M1 ranging from 1.1 to 10, even though the corresponding degree of departure from equilibrium varies by a factor greater than 100. Many more characteristics of the internal structure of the shock waves have been calculated from the solutions than have hitherto been available. Each solution of the Boltzmann equation requires about 108 multiplications to obtain statistical errors of 3% in values of the velocity distribution function and collision integral and much smaller errors in the moments of these functions. The reciprocal shock thickness is in agreement with that of the Mott-Smith shock (u2 moment) from M1 = 2.5-8. The density profile is asymmetric with an upstream relaxation rate (measured as density change per mean free path) approximately twice as large as the downstream value for weak shocks and equal to the downstream value for strong shocks. The temperature density relation is in agreement with that of the Navier-Stokes shocks for Mach numbers in the range 1·1-1-56. The Boltzmann reciprocal shock thickness is smaller than the Navier-Stokes value in this range of Mach number because the viscosity-temperature relation computed is not constant as predicted by the linearized theory.

The velocity moments of the distribution function are, like the Mott-Smith shock, approximately linear with respect to the number density; however, the deviations from linearity are statistically significant. Four functionals of the distribution function that are discussed show maxima within the shock. The entropy is a good approximation to the Boltzmann function for all M1. The solutions obtained satisfy the Boltzmann theorem for all Mach numbers. The ratio of total heat flux q to qx (that associated with the longitudinal degree of freedom) correlates well with local Mach number for all Ml in accordance with a relation derived by Baganoff & Nathenson (1970). The Chapman-Enskog linearized theory predicts that this ratio is constant. The (effective) transport coefficients are larger than the Chapman-Enskog equivalents by as much as a factor of three at the mid-shock position.

At M1 = 4, and for 40% of the velocity bins, the distribution function is different from the corresponding Mott-Smith value by more than three times the 90% confidence limit. The r.m.8. value of the percentage difference in distribution functions is 15% for this Mach number. At MI = 1-59, the half width and several other characteristics of the function \[ \int f\,dv_y dv_z \] differ from that of the Chapman-Enskog first iterate, and many of the deviations are in agreement with an experiment by Muntz & Harnett (1970).

It is shown that some variational methods of quantum mechanics can be re-formulated for application to the problem of the diffusion of marked particles in a fluid in random motion. As an example, a variational derivation of the Wiener-Hermite approximation procedure is given. The simplest such approxi- mation is examined and it is made plausible that it gives the exact form for both the one- and two-particle distribution functions in a certain limiting situation.