A well-tested integral method has been used to calculate turbulent boundary-layer development for the distribution of external velocity given by U ∝ x−0.255. The results suggest that different values of the initial momentum thickness, so long as this is below some critical value, produce a range of equilibrium layers having widely different values of the form parameter G. For values of the initial momentum thickness greater than the critical value, layers are produced which proceed more or less rapidly to separation. These results provide a plausible explanation for conflicting experimental observations made in the past.

Additional calculations for the flows U ∝ x−0.15 and U ∝ x−0.35 suggest that, in the first case, a unique equilibrium condition is approached whatever the initial momentum thickness unless this exceeds some critical value; in the second case no equilibrium condition appears possible.

Calculated flow properties are compared with measurements obtained in two-dimensional isothermal wakes with and without recirculation. The equations of continuity and momentum were solved numerically together with equations which formed a turbulence model. Calculations were made using three turbulence models: the first comprised transport equations for turbulence kinetic energy and the rate of turbulence dissipation; the second and third comprised equations for the rate of turbulence dissipation and two forms of Reynolds-stress equations characterized by different redistribution terms. The results show that, for wakes without recirculation, the particular turbulence model is less important than the boundary condition assumed in the plane of the trailing edge of the body; though the Reynolds-stress models do, of course, provide a better representation of the individual normal stresses. In the case of wakes with recirculation, both the length of the recirculation region and the rate of spread of the downstream wake are underestimated. The second discrepancy is particularly evident and appears to stem from the form of the dissipation equation. A suggestion for improving the modelling of this equation is provided together with necessary justification.

A multi-layer model is used to describe a ‘two-dimensional’ continuously stratified fluid. We use a momentum theorem in each layer to derive an ordinary differential equation describing the vertical structure behind a jump. This equation is compared with the corresponding equation for continuous flow. As one would expect from the classical one-layer theory, they are identical up to second order for weak disturbances. The energy change across a jump is also derived. By requiring that energy be lost through a jump, we calculate when a weak jump is possible in general. Algorithms for computing jumps of arbitrary strength are given. To ensure that the flow after the jump is stable and also for a numerical reason to be stated in § 8, we require that the Richardson number after the jump be equal to or greater than S¼S. Numerical examples are given to show the range of parameters within which jumps are possible; the velocity profiles related to different kinds of jumps also appear. Since hydraulic jumps in a continuously stratified fluid have not yet been observed in any laboratory, it should be of interest to verify these calculations experimentally.

An estimate is obtained of the heat transfer from a constant-temperature hot wire in a two-dimensional low Reynolds number flow (R [Lt ] 1). The flow has a small sinusoidally fluctuating velocity superimposed on the mean velocity. For the fluctuating components it is shown that at low frequencies the heat transfer is in phase with the velocity, the magnitude of the heat transfer being given by the result for steady heat transfer, but at high frequencies the heat transfer lags behind the velocity, the relative magnitude of the heat transfer decreasing as (frequency)−1.

A numerical exploration of multiple-cell solutions for the flow between a rotating and a stationary disk is carried out systematically by the continuation method. The paper confirms and extends the work of Mellor, Chapple & Stokes. The results include the discovery of one-, two-, three- and five-cell solution regions which have not been reported before.

The stability of natural convection in fluid between two parallel vertical plates is investigated theoretically. The two plates are maintained at different temperatures and a uniform stable temperature gradient β is present in the vertical direction. The Prandtl number of the fluid is fixed at 7.5. An orthonormalization method is used in numerical integrations of the disturbance equations. It is shown how the critical Grashof number varies with β for both stationary and travelling disturbances. It is found that for β ≤ 7.1 × 10−3 the convection is unstable to stationary disturbances and for β ≥ 7.1 × 10−3 it is unstable to travelling disturbances. The critical Grashof number is given by \[ G_c = \left\{\begin{array}{@{}l@{\quad}c@{\quad}l@{}} 500 & {\rm for} & \beta < 1.0\times 10^{-3},\\ 1.3\times 10^6\beta^3 & {\rm for} &\beta > 4.1\times 10^{-2}, \end{array}\right. \] and even for intermediate values of β the variation of Gc is rather simple but not monotonic.

Some insight into the mechanism of solid-particle transport by peristalsis is sought experimentally through a two-dimensional model study (§ 2). The peristaltic wave is characterized by a single bolus sweeping by the particle, resulting in oscillatory motion of the particle. Because of fluid-particle interaction and the significant curvature in the wall wave, the peristaltic flow is highly nonlinear and time dependent.

For a neutrally buoyant particle propelled along the axis of the channel by a single bolus, the net particle displacement can be either positive or negative. The instantaneous force acting upon the particle and the resultant particle trajectory are sensitive to the Reynolds number of the flow (§ 3 and 4). The net forward movement of the particle increases slightly with the particle size but decreases rapidly as the gap width of the bolus increases. The combined dynamic effects of the gap width and Reynolds number on the particle displacement are studied (§ 5). Changes in both the amplitude and the form of the wave have significant effects on particle motion. A decrease in wave amplitude along with an increase in wave speed may lead to a net retrograde particle motion (§ 6). For a non-neutrally buoyant particle, the gravitational effects on particle transport are modelled according to the ratio of the Froude number to the Reynolds number. The interaction of the particle with the wall for this case is also explored (§ 7).

When the centre of the particle is off the longitudinal axis, the particle will undergo rotation as well as translation. Lateral migration of the particle is found to occur in the curvilinear flow region of the bolus, leading to a reduction in the net longitudinal transport (§ 8). The interaction of the curvilinear flow field with the particle is further discussed through comparison of flow patterns around a particle with the corresponding cases without a particle (§ 9).

The paper describes an experimental study of the structure of air-water mixtures flowing vertically. Resistivity probe techniques were applied to measurements of local void properties, including void fraction, gas-phase convection velocity, bubble size distributions and space-time correlation functions. The axial development of flows for six different air-water mixing conditions were examined. Measurements up to 108 diameters from inlet indicated that, while flow patterns for the different mixers may be significantly different initially, the flows tend to develop towards a common structure determined only by the flux rates of the two phases. This was evidenced by the convergence of the void and velocity profiles, and particularly the bubble size distributions, as the flow developed. Estimates of bubble sizes for these more developed conditions, based on a balance of energy between the interfacial structure and the turbulent structure, gave values of diameters which were on average 13% above experimental values. The void distributions obtained for bubbly flow conditions, after an adequate settling length, appear to be characterized by a local minimum at the centre of the pipe.

Internal Alfvén-acoustic-gravity waves propagating in an isothermal, perfectly electrically conducting, plane stratified, inviscid, compressible atmosphere permeated by a horizontal stratified magnetic field in which the mean horizontal velocity U(z) depends on the height z only exhibit singular properties at the Doppler-shifted frequencies \[ \Omega_{d} = 0,\quad\pm\Omega_A,\quad\pm\Omega_A/(1+M^2)^{\frac{1}{2}},\quad\pm (\Omega_c/2^{\frac{1}{2}})[1+M^2\pm \{(1+M^2)^2 - 4\Omega^2_A/\Omega^2_c\}^{\frac{1}{2}}]^{\frac{1}{2}} \] where ΩA is the Alfvén frequency, Ωc the sonic frequency and M the magnetic Mach number. The phenomenon of critical-layer absorption is studied using the momentum-transport approach of Booker & Bretherton (1967), the wave-packet approach (which is a consequence of the WKBJ approximation) of Bretherton (1966) and the technique involving wave normal curves of McKenzie (1973). The absorption effects are also illustrated, following Acheson (1972), by drawing ray trajectories. We find that the waves are absorbed at the critical levels Ωd = ± ΩA and ± ΩA/(1 + M2)½, and in particular we observe that these levels do not act like valves as observed by Acheson (1972). We also conclude that the combined effect of velocity shear and density and magnetic-field stratification is to increase the number of absorption levels.

The linearized theory of axisymmetric hypersonic source flow past a slender pointed body is treated by using linearized potential theory, assuming that the hypersonic parameter is less than unity. The governing equation in the transformed co-ordinates with the ratio of specific heats γ = 1.5 shows a modified form relative to the equation for parallel flow, whereas that with γ = 2 shows essentially the same form. Some numerical calculations for a cone and a cone-cylinder are presented, and the surface pressure distributions obtained for both γ = 1.5 and 2 show that there is a considerable pressure-distribution difference between source flow and parallel flow. The pressure on a cone in hypersonic source flow approaches zero downstream, while, if the cone has a cylindrical afterbody, the pressure recovers to the free-stream value quickly. This suggests that the static pressure probe for supersonic parallel flow is useful also in source flow.

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.

Three turbulent shear flows with quadratic mean-velocity profiles are generated by using an appropriately designed honeycomb and parallel-rod grids with adjustable rod spacing. The details of two of the flow fields, with quadratic mean-velocity profiles with constant positive mean-shear gradients ($\partial^2\overline{U}_1/\partial X^2_2 >0$), are obtained, and include, in the mean flow direction, the development and distribution of mean velocities, fluctuating velocities, Reynolds stresses, microscales, integral scales, energy spectra, shear correlation coefficients and two-point spatial velocity correlation coefficients. A third flow field is generated with a quadratic mean velocity profile with constant negative mean-shear gradient ($\partial^2\overline{U}_1/\partial X^2_2 < 0$), to investigate in the mean flow direction the effect of the change in sign on the resulting field. An open-return wind tunnel with a 2 × 2 × 20 ft test-section is used.