By making simple assumptions, an analytical theory is deduced for the mean velocity behind a two-dimensional obstacle (of height h) placed on a rigid plane over which flows a turbulent boundary layer (of thickness δ). It is assumed that h [Gt ] δ, and that the wake can be divided into three regions. The velocity deficit − u is greatest in the two regions in which the change in shear stress is important, a wall region (W) close to the wall and a mixing region (M) spreading from the top of the obstacle. Above these is the external region (E) in which the velocity field is an inviscid perturbation on the incident boundary-layer velocity, which is taken to have a power-law profile U(y) = U∞(y − y1)n/δn, where n [Gt ] 1. In (M), assuming that an eddy viscosity (= KhU(h)) can be defined for the perturbed flow in terms of the incident boundary-layer flow and that the velocity is self-preserving, it is found that u(x,y) has the form $\frac{u}{U(h)} = \frac{ C }{Kh^2U^2(h)} \frac{f(n)}{x/h},\;\;\;\; {\rm where}\;\;\;\; \eta = (y/h)/[Kx/h]^{1/(n+2)}$, and the constant which defines the strength of the wake is $C = \int^\infty_0 y^U(y)(u-u_E)dy$, where u = uE(x, y) as y → 0 in region (E).

In region (W), u(y) is proportional to In y. By considering a large control surface enclosing the obstacle it is shown that the constant of the wake flow is not simply related to the drag of the obstacle, but is equal to the sum of the couple on the obstacle and an integral of the pressure field on the surface near the body.

New wind-tunnel measurements of mean and turbulent velocities and Reynolds stresses in the wake behind a two-dimensional rectangular block on a roughened surface are presented. The turbulent boundary layer is artificially developed by well-established methods (Counihan 1969) in such a way that δ = 8h. These measurements are compared with the theory, with other wind-tunnel measurements and also with full-scale measurements of the wind behind windbreaks.

It is found that the theory describes the distribution of mean velocity reasonably well, in particular the (x/h)−1 decay law is well confirmed. The theory gives the correct self-preserving form for the distribution of Reynolds stress and the maximum increase of the mean-square turbulent velocity is found to decay downstream approximately as $ (\frac{x}{h})^{- \frac{3}{2}} $ in accordance with the theory. The theory also suggests that the velocity deficit is affected by the roughness of the terrain (as measured by the roughness length y0) in proportion to In (h/y0), and there seems to be some experimental support for this hypothesis.

An investigation of the hydrodynamic stability of swirling flows having arbitrary Rossby numbers is described. A necessary condition for instability is derived for rigidly rotating flows and this condition is further refined in the specific case of a parabolic axial flow. Numerical results are presented for two azimuthal wave-numbers corresponding to the maximum growth rates of unstable perturbations as a function of Rossby number. It is found that the largest growth rates occur when the Rossby number is O(1) and that instability persists for surprisingly large values of this parameter. Previous explanations of the instability mechanism are discussed and it is concluded that these are only adequate in the limit of small Rossby number.

The stretching of line elements, surface elements and wave vectors by a random, isotropic, solenoidal velocity field in D dimensions is studied. The rates of growth of line elements and (D – 1)-dimensional surface elements are found to be equal if the statistics are invariant to velocity reversal. The analysis is applied to convection of a sparse distribution of sheets of passive scalar in a random straining field whose correlation scale is large compared with the sheet size. This is Batchelor's (1959) κ−1 spectral regime. Some exact analytical solutions are found when the velocity field varies rapidly in time. These include the dissipation spectrum and a joint probability distribution that describes the simultaneous effect of Stretching and molecular diffusivity κ on the amplitude profile of a sheet. The latter leads to probability distributions of the scalar field and its space derivatives. For a growing κ−1 range at zero κ, these derivatives have essentially lognormal statistics. In the steady-state κ−1 regime at κ > 0, intermittencies measured by moment ratios are much smaller than for lognormal statistics, and they increase less rapidly with the order of the derivative than in the κ = 0 case. The κ > 0 distributions have singularities a t zero amplitude, due to a background of highly diffused sheets. The results do not depend strongly on D. But as D → ∞, temporal fluctuations in the stretching rates become negligible and Batchelor's (1959) constant-strain dissipation spectrum is recovered.

An integral analysis of the type used to predict the flow of co-flowing jets has been applied to the problem of a sudden enlargement in a pipe (Borda–Carnot expansion). This technique successfully predicts all the overall flow parameters of interest (e.g. reattachment lengths, pressure profile, etc.). The analysis indicates that the downstream conditions (up to reattachment) are insensitive to wall shear and the point of minimum pressure does not coincide with the location of the maximum return-flow velocity.

An experimental study of the occurrence of temperature oscillations in molten gallium contained in a rectangular boat and heated from the side is reported. The dependence of the critical temperature difference across the boat for which oscillations set in upon the boat dimensions and upon the strength of a transverse magnetic field is described. The dependence of the frequency of oscillation on these parameters is also reported together with measurements of the variation of the phase of the oscillations over the top surface of the melt. The results are discussed in relation to the theory in the companion paper by Gill (1974).

The early and intermediate development of a highly accelerated (or decelerated) turbulent boundary layer is analysed. For sufficiently large accelerations (or pressure gradients) and for total normal strains which are not excessive, the equation for the Reynolds shear stress simplifies to give a stress that remains approximately constant as it is convected along streamlines. The theoretical results for the evolution of the mean velocity in favourable and adverse pressure gradients agree well with experiment for the cases considered. A calculation which includes mass injection at the wall is also given.

In this paper, the change in energy dissipation due to a small hump on a body in a uniform steady flow is calculated. The result is used in conjunction with the variational methods of optimal control to obtain the optimality conditions for four minimum-drag problems of fluid mechanics. These conditions imply that the unit-area profile of smallest drag has a front end shaped like a wedge of angle 90°.

We expand the equations describing plane Poiseuille flow in Fourier series in the co-ordinates in the plane parallel to the bounding walls. There results an infinite system of equations for the amplitudes, which are functions of time and of the cross-stream co-ordinate. This system is drastically truncated and the resulting set of equations is solved accurately by a finite difference method. Three truncations are considered: (I) a single mode with dependence only on the downstream co-ordinate and time, (II) the mode of (I) plus its first harmonic, (III) a single three-dimensional mode. For all three cases, for a variety of initial conditions, the solutions evolve to a steady state as seen in a particular moving frame of reference. No runaways are encountered.

Thermal oscillations have been found to occur during crystal growing, thereby introducing undesirable striations in the solid crystal produced. A reason for the existence of such oscillations is given in this paper, and relevant experiments discussed.

Flow visualization and laser-anemometry measurements are reported in the flow downstream of a plane 3: 1 symmetric expansion in a duct with an aspect ratio of 9·2: 1 downstream of the expansion. The flow was found to be markedly dependent on Reynolds number, and strongly three-dimensional even well away from the channel corners except at the lowest measurable velocities. The measurements at a Reynolds number of 56 indicated that the separation regions behind each step were of equal length. Symmetric velocity profiles existed from the expansion to a fully developed, parabolic profile far downstream, although there were substantial three-dimensional effects in the vicinity of the separation regions. The velocity profiles were in good agreement with those obtained by solving the two-dimensional momentum equation. At a Reynolds number of 114, the two separation regions were of different lengths, leading to asymmetric velocity profiles; three dimensional effects were much more pronounced. At a Reynolds number of 252, a third separation zone was found on one wall, downstream of the smaller of the two separation zones adjacent to the steps. As at the lower Reynolds numbers, the flow was very stable. At higher Reynolds numbers the flow became less stable and periodicity became increasingly important in the main stream; this was accompanied by a highly disturbed fluid motion in the separation zones, as the flow tended towards turbulence.

Measurements of the heat flux and temperature gradient in a layer of saline solution stably stratified by Soret diffusion and heated from below exhibit the onset of convection by the growth of very small oscillations which eventually trigger finite amplitude convection. The critical Rayleigh numbers and periods of oscillation are consistent with published theoretical calculations. Several non-linear modes are observed, even for the same heat flux, near the point of onset.

Plane turbulent mixing between two streams of different gases (especially nitrogen and helium) was studied in a novel apparatus. Spark shadow pictures showed that, for all ratios of densities in the two streams, the mixing layer is dominated by large coherent structures. High-speed movies showed that these convect at nearly constant speed, and increase their size and spacing discontinuously by amalgamation with neighbouring ones. The pictures and measurements of density fluctuations suggest that turbulent mixing and entrainment is a process of entanglement on the scale of the large structures; some statistical properties of the latter are used to obtain an estimate of entrainment rates. Large changes of the density ratio across the mixing layer were found to have a relatively small effect on the spreading angle; it is concluded that the strong effects, which are observed when one stream is supersonic, are due to compressibility effects, not density effects, as has been generally supposed.

The approximations implicit in Bénard convection have been modified to include viscous dissipation. It is shown that both the influence of an adiabatic temperature gradient and of viscous dissipation are governed by the same dimensionless parameter Di = αgh/cp. Numerical calculations of finite amplitude convection are given for finite values of Di. It is found that increasing Di decreases flow velocities and finally stabilizes the flow.

Stokes flow is analysed for a combination body, consisting of a sphere attached to a slender body, translating along its axis in an infinite and otherwise un-disturbed fluid. The cross-section of the after-body, or tail, is circular; the radius, while not necessarily constant, is small compared with the radius of the spherical head. The tail is represented by a distribution of Stokeslets of strength per unit length F(z), located and directed along its axis. The interactive effect of head-tail attachment is manifested by the presence of image singularities located within the sphere. The image system for a single tail Stokeslet must be such that the no-slip condition is satisfied on the surface of the sphere. It is shown that this system consists of a Stokeslet, a Stokes doublet (stresslet only) and a source doublet located a t the image point. The strength F(z) is obtained by applying the no-slip condition to the combination body. The solution follows the lines of traditional slender-body theory, an expansion being performed in ascending powers of the reciprocal of the logarithm of the aspect ratio. The integral force parameters and F(z) are obtained to second order. The interactive effect is assessed, and the results are discussed in the context of a sedimenting micro-organism, such as a spermatozoon. The drag on the combination body is shown to be less by around 10% than the sum of the drags on an isolated sphere and tail. This drag, for a sperm-shaped body, is divided approximately equally between head and tail.

A turbulence representation, consisting of a generalized set of transport equations for the Reynolds stress tensor and the turbulence energy decay rate, is applied to the study of convective heat transport between parallel plates at moderate Rayleigh numbers, 5 × 103 ≤ Ra ≤ 6·4 × 105. A series of heat flux transitions, in good agreement with those observed experimentally, is detect, ed in this study and found to correlate with changes in the turbulence structure. In the order of increasing Rayleigh number these structural changes correspond to: the transition from laminar to turbulent flow, the transition from low to locally high intensity turbulence, the transition to uniformly high intensity turbulence, and the transition from a buoyancy dominated turbulence to a shear dominated turbulence. An analysis is made of the effect of each of these transitions on the mechanism for heat transfer between the plates.

The steady streaming generated in the boundary layer on a cylinder performing simple harmonic motion in a viscous incompressible fluid which is otherwise at rest is investigated in the case where the Reynolds number Rs associated with this streaming is large. Comparison is made between experimental results obtained here and the theories of Riley (1965) and Stuart (1966). This comparison shows good agreement between the theories and the experiment close to the cylinder, but away from the cylinder significant discrepancies are observed. Possible reasons for these discrepancies are discussed.

This paper investigates the non-uniform motion of a thin plate of finite aspect ratio, with a rounded leading edge and sharp trailing edge, executing heaving and pitching oscillations at zero mean lift. Such vertical motions characterize the horizontal lunate tails with which cetacean mammals propel themselves, and the same motions, turned through 90° to become horizontal motions of sideslip and yaw, characterize the vertical lunate tails of certain fast-swimming fishes. An oscillating vortex sheet consisting of streamwise and spanwise components is shed to trail behind the body and it is this additional feature of the streamwise component resulting from the finiteness of the plate that makes this study a generalization of the two-dimensional treatment of lunate-tail propulsion by Lighthill (1970). The forward thrust, the power required, the energy imparted to the wake and the hydromechanical propulsive efficiency are determined for this general motion as functions of the physical parameters defining the problem: namely the aspect ratio, the reduced frequency, the feathering parameter and the position of the pitching axis. The dependence of the thrust coefficient and propulsive efficiency on these physical parameters, for the complete range of variation consistent with the assumptions of the problem, has been depicted graphically.

Measurements have been made of the phase difference between the signals of two hot wires separated in the stream direction and placed in and close to the turbulent wake of a two-dimensional bluff body. Results are presented in the form of the wavelength and the phase speed of perturbations at the vortex shedding frequency for various positions in the flow. It is shown that values of the phase speed may be obtained that are in agreement with published values of the vortex transport velocity. It is necessary to ensure that the hot wires are close to the paths of the centres of the vortices in the wake, otherwise spurious results caused by the influence of the bound vorticity at the model will be obtained.

A study has been made of the flow and disintegration of thin liquid sheets in combustion gases up to temperatures of 950°C. It is found that below 300°C sheet breakdown occurs through the growth of antisymmetric Kelvin–Helmholtz waves. Above this temperature high frequency symmetric waves and localized disturbances are superimposed on the sheet and disintegration then occurs by the combined action of aerodynamic waves and perforations, the contribution of the latter predominating with increasing temperature. It is demonstrated that the new wave system is electrohydrodynamic in origin, the electric field being generated by the charged species present in the gas. The drop size is found to be critically dependent upon the nature of the disintegration process.