The flows caused by a uniformly expanding circular cylinder or sphere in a perfect gas at rest at various (from small to very large) expansion velocities are analysed by a method of successive approximations in which the first approximation represents the incompressible flow. The shock waves which form around the bodies are also treated. The results for a sphere, even up to the third approximation, agree closely with Taylor's (1946) calculations for all the expansion rates.

The manner in which fluid driven through a channel of width a responds in anticipation of a severe asymmetric distortion (e.g. to the wall or interior conditions downstream) is discussed when the oncoming flow is fully developed, the characteristic Reynolds number K is large and the whole motion remains laminar. Far ahead of the disturbance, at distances $O(aK^{\frac{1}{7}})$, there occurs a free interaction which triggers a small displacement in the core, generating viscous layers near both walls; the relatively large induced pressure gradient acting across the channel is then found to sustain the growth of this displacement. Numerical solutions of the fundamental nonlinear problem show that one layer separates in a regular fashion, but that beyond separation the other layer, under compression, produces a singularity in the interaction. Analysis of the singularity based on a self-similar structure in the partly reversed flow then leads to a description of the nonlinear flow features nearer the distortion which seems to have strong physical significance. The major implication is that the flow will have already separated at one wall, and developed a definite nonlinear character quite distinct from that of the original Poiseuille flow, long before it reaches finite distances from the distortion. The separation point is predicted to be a distance $0.49 aK^{\frac{1}{7}} (+ O(a))$ ahead of the particular finite distortion. Comparisons of this and other predictions with computed solutions of the full Navier–Stokes equations show reasonable agreement.

We consider the flow of rapidly rotating fluid over topography in a circular basin. The equations of motion (here the inviscid quasi-geostrophic vorticity equations) can be integrated exactly for certain zonally averaged currents. The assumption of the existence of a specified zonal current is equivalent to the assumption of no upstream influence in the unbounded case. It is unlikely that such solutions can be realized in experiments with real fluids for the presence of viscosity, however small, causes ‘zonal influence’ independent of the magnitude of the viscosity at times larger than the spin-up time. For times smaller than the spin-up time decaying transients can cause zonal influence which increases in magnitude with decreasing viscosity.

The growth of the unsteady boundary layer on an infinite rotating disk in a counter-rotating fluid is examined numerically and analytically. The numerical computations indicate that the boundary layer breaks down when ωt* ≈ 2·36 in a novel way: the displacement thickness, as well as all the velocity components, becomes infinite. This numerical solution is fitted to an asymptotic expansion which contains the singularities found in the numerical integrations, and it is concluded that the solution of the unsteady similarity equations does break down at a finite time as the numerical results indicate. This problem is placed in a physically more realistic context by considering numerically the unsteady boundary layer which develops on a finite rotating disk in a counter-rotating fluid. It is found that the breakdown of the solution occurs at the axis at the same time, and thus the concept of a thin boundary layer in this more realistic problem is also destroyed in a finite time.

Numerical calculations of the Landau constant are presented for the case of a shear layer of finite Reynolds number, having the velocity profile $\overline{u} = \tan h\,y$. It is found that this parameter has a strong dependence on the Reynolds number for Re [les ] 100. In particular, the Landau constant is reduced by 43% from its inviscid value when Re = 40, the latter value being typical of many experiments. This percentage, however, is based upon a calculation in which the mean-flow distortion has been neglected. A rough estimate of the latter effect indicates that it could possibly increase the value of the Landau constant sufficiently that the net influence of a finite Reynolds number would be of a smaller magnitude.

The motion of a two-dimensional packet of capillary–gravity waves on water of finite depth is studied. The evolution of a packet is described by two partial differential equations: the nonlinear Schrödinger equation with a forcing term and a linear equation, which is of either elliptic or hyperbolic type depending on whether the group velocity of the capillary–gravity wave is less than or greater than the velocity of long gravity waves. These equations are used to examine the stability of the Stokes capillary–gravity wave train. The analysis reveals the existence of a resonant interaction between a capillary–gravity wave and a long gravity wave. The interaction requires that the liquid depth be small in comparison with the wavelength of the (long) gravity waves and the evolution equations describing the dynamics of this interaction are derived.

A theory for the evolution of the wind drift current and of the Langmuir circulations in infinitely deep water of constant density is presented. The model improves and extends a recent quasi-steady theory of Craik & Leibovich which asserts that the Langmuir circulations arise from a nonlinear interaction between surface waves and the frictional wind drift current. In turn, the development of the wind drift should be strongly influenced by Langmuir circulations, when they are present, and the two current systems are therefore treated here as a single inseparable system driven by a prescribed wind stress and surface wave field. Mixing by the vertical motions in the Langmuir circulations is shown to yield solutions for the wind drift, obtained both analytically and numerically, which are consistent with experiments and with field observations. The model yields a streaky flow pattern with a mean motion much like a turbulent wall layer, although the model is deterministic. In particular, it is found that a ‘viscous sublayer’ joins surface water to a logarithmic ‘inertial sublayer’ below. The scaling rules that emerge from the theory allow the surface speed of the wind drift to reach nearly full development in a matter of minutes.

Uniformly valid asymptotic solutions for linearized disturbances valid in a finite region containing the critical point u = c are developed using the method of multiple scales for a given isoenergetic basic flow. While uniformity of approximation will certainly improve the quantitative accuracy of a typical computation of stability characteristics of the flow, a major feature of such solutions is that they afford a more accurate description of the behaviour of the linearized disturbances in the neighbourhood of the critical point u = c.

Turbulent entrainment at the density interface of a stable two-layer stratified fluid is studied in the laboratory, a constant surface stress being applied at the free surface. Conservation of mass requires that the overall Richardson number Ri = Dgδρ/ρu*2 is constant in each experiment, where D is the depth of the mixed layer, gδρ/ρ the buoyancy difference and u* the friction velocity. If the entrainment rate E = ue/u* is a function only of Ri, it is therefore constant in each experiment and can be measured with a greater accuracy than has previously been attained. The functional dependence of ue/u* on Ri is established over the range 30 < Ri < 1000; it is found not to follow any simple power law. The entrainment rates are considerably higher than those measured by Kato & Phillips (1969), for which the fluid below the mixed layer was linearly stratified. Such a condition allows internal gravity waves to be radiated downwards and the reduction in entrainment rate is consistent with that found by Linden (1975).

A two-dimensional configuration has been investigated in which air flows through a convergent nozzle and expands abruptly into a rectangular duct of larger cross-section which terminates in a plenum chamber. Three different types of oscillation have been observed in the downstream duct. At low plenumchamber pressures an oscillation occurs towards the exit of the duct as the boundary layer of the flow becomes alternately separated and attached. At increasing plenum pressure a shock-pattern oscillation takes place in which a change from a normal shock to oblique shocks occurs during a cycle. At still greater plenum pressures a base-pressure oscillation occurs which influences the entire duct flow downstream of the abrupt change in cross-section. The amplitudes of the oscillation can be as high as 10% of the rest state, and the frequency of the base-pressure oscillations can be predicted approximately from one-dimensional gasdynamic theory.

The unsteady duct phenomena have been studied by synchronizing instantaneous pressures measured by quartz pressure transducers with interferograms obtained with a Mach–Zehnder interferometer.

Work recently reported by the authors (Perry & Abell 1975) on smooth-walled pipe flow showed support for the Townsend (1976) structural similarity principle as regards viscosity not being directly relevant in controlling the mean relative motions and the energy-containing turbulent motions. The work also supported a universal spectral behaviour in the wall region of the flow. In many hypotheses for rough-walled pipe flow, surface roughness, like viscosity, enters the problem only via external boundary conditions. Data obtained in a rough pipe are reported here and on first appearance the results seem to contradict the Townsend hypothesis and to threaten the very foundation upon which many similarity laws for rough-walled flows are based. However, on closer examination of the spectrum scaling of smooth-walled pipe flow the low and high wavenumber energy not necessarily associated with the universal similarity range can be accounted for. The broad-band longitudinal turbulence results for a rough-walled pipe can then be predicted from the smooth-wall scaling. The conclusion is that, despite the apparent anomalies, the turbulence structure in a rough pipe appears to follow the same scaling laws as for a smooth pipe, given a sufficient length of flow development in both cases. The deduced functional forms are consistent with Townsend's (1976) attached-eddy hypothesis.

A Lavrentiev model of the flow about a blunt two-dimensional body with a separation bubble is considered. Physical bases of the model are discussed in relation to other wake models. The Lavrentiev wake bubble contains a pair of closed free streamlines enclosing the regions of vorticity. It is shown, by means of conformal mapping, that the complex potential can be expressed in terms of elliptic functions, and a one-parameter family of exact solutions has been constructed for a normal flat plate and truncated wedges, for both an unbounded and a bounded stream. A procedure for relating the value of the parameter to the Reynolds number of the real fluid flow is indicated.