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 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.

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.

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.

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).

A hypersonic gun tunnel has been used to investigate turbulent shock-boundary-layer interaction at a wedge-compression corner. The results extent the Mach number range of existing data, in particular of incipient separation.

The nature of a three-dimensional viscous flow along a corner near its junction has been clarified in this paper by constructing a Stokes slow-flow solution. We have further demonstrated that this Stokes solution can be matched onto an inertial-flow solution in principle by establishing an overlap domain along one sector of an inertial-flow region, namely along the flow symmetry line. This Stokes solution reveals a remarkably complex structure of the flow as characterized by a separating streamwise velocity profile in addition to a sequence of Moffatt's viscous eddies in a cross-flow plane.

Theoretical results based on the methods of Freeman and Garr & Marrone show that the stand-off distance and flow pattern of non-equilibrium dissociating flow of nitrogen over the front part of a blunt body can be correlated in terms of a single reaction rate parameter ω taking account of parameters describing the speed, density, dissociation and temperature of the free stream. The density pattern, which is sensitive to the reaction rate, consists of two distinct regions dominated by the effects of reaction and pressure respectively. The shape and size of these regions depend on Q. Experimental results obtained by optical interferometry in a free-piston shock tunnel confirm the theoretical results. A scale effect consistent with the induction time phenomenon suggested by Shui, Appleton & Keck modifies the theoretical results considerably in the case of small models.

If a region of reverse flow remains confined within a boundary layer the conventional boundary-layer equations should continue to apply downstream of the point of detachment of the surface streamline (ω = 0). Nevertheless, standard numerical techniques fail in the presence of backflow since these methods become highly unstable and, in addition, neglect the upstream flow of information. A procedure for numerically integrating the boundary-layer equations through a region of reverse flow which takes downstream influence into account is therefore presented. This method is then applied to the problem of uniform flow past a parallel flat plate of finite length whose surface has a constant velocity directed opposite to that of the main stream. Although singularities occur at both the point of detachment (xs) and reattachment (xr) of the ω = 0 streamline, this integration technique provides a solution which ceases to apply only in the close proximity of these singular points. From this solution it is evident that, throughout a large portion of the separated region, the flow is strongly affected by conditions near xr, thereby demonstrating the importance of allowing information to be transmitted upstream in a region of backflow. Near (xs), however, it is found that, in spite of the presence of reverse flow, the solution has a self-similar form in this particular example.