The conditional sampling technique is used to measure ensemble averages of the longitudinal and normal velocity fluctuations u and v respectively and of the Reynolds shear stress fluctuations uv both within the turbulent and irrotational regions near the outer edge of a turbulent boundary layer. The measurements are made in both a smooth- and a rough-wall boundary layer under zero-pressure-gradient conditions. The smooth- and rough-wall results are qualitatively similar but the magnitude of the rough-wall averages is higher than that of the smooth- wall averages, corresponding with the higher value of wall shear stress on the rough surface. The maximum shear stress value encountered within a burst represents a significant proportion of the wall shear stress.

The statistical properties of the turbulence within the burst are close but not quite identical to the nearly Gaussian properties of the inner region of the boundary layer. During an attempt to distinguish between bursts of different ages or strengths at the time of measurement, it was found that bursts of relatively short duration travel at much the same longitudinal velocity as the local mean U and contribute little to the local shear stress. The longer and less frequent bursts have a mean velocity smaller than U and a maximum shear stress comparable to the shear stress at the wall.

A blob of Newtonian fluid is sandwiched in the narrow gap between two plane parallel surfaces so that, a t some initial instant, its plan-view occupies a simply connected domain D0. Further fluid, with the same material properties, is injected into the gap at some fixed point within D0, so that the blob begins to grow in size. The domain D occupied by the fluid at some subsequent time is to be determined.

It is shown that the growth is controlled by the existence of an infinite number of invariants of the motion, which are of a purely geometric character. For sufficiently simple initial domains D0 these allow the problem to be reduced to the solution of a finite system of algebraic equations. For more complex initial domains an approximation scheme leads to a similar system of equations to be solved.

The singularity of the Ekman layer at the equator of a rotating gravitating sphere makes it difficult to satisfy a prescribed stress boundary condition at the surface of a layer of liquid on the sphere. The equations of motion are investigated for a homogeneous ocean with vertical and lateral eddy viscosities. The horizontal Coriolis terms are not neglected. A linear equation for the boundary layer is obtained and a solution of the equation for the boundary-layer part of the velocity field is found in closed form. This is valid in a parameter range which includes the previous solutions of Stewartson and Gill as limiting cases.

An exact formula is obtained for the term of order c2 in the expression for the bulk stress in a suspension of force-free spherical particles in Newtonian ambient fluid, where c is the volume fraction of the spheres and c [Lt ] 1. The particles may be of different sizes, and composed of either solid or fluid of arbitrary viscosity. The method of derivation circumvents the familiar obstacle, of non-absolutely convergent integrals representing the effect of all pair interactions in which one specified particle takes part, by the judicious use of a certain quantity which is affected by the presence of distant particles in a similar way and whose mean value is known exactly. The bulk stress is in general of non-Newtonian form and depends on the statistical properties of the suspension which in turn are dependent on the type of bulk flow.

The formula contains two functions which are parameters of the flow field due to two spherical particles immersed in fluid in which the velocity gradient is uniform at infinity. One of them, p(r, t), represents the probability density for the vector r separating the centres of the two particles. The variation of p(r, t) for a moving material point in r-space due to hydrodynamic action is found in terms of a function q(r), and this gives p(r, t) explicitly over the whole of the region of r-space occupied by trajectories of one particle centre relative to another which come from infinity. In a region of closed trajectories, steady-state hydrodynamic action alone does not determine the relation between the values of p (r, t) for different material points. The function q(r) is singular when the spheres touch, and the contribution of nearly-touching spheres to the bulk stress is evidently important. Approximate numerical values of all the relevant functions are presented for the case of rigid spherical particles of uniform size.

In the case of steady pure straining motion of the suspension, all trajectories in r-space come from infinity, the suspension has isotropic structure and the stress behaviour can be represented (to order c2) in terms of an effective viscosity ${\mathop\mu\limits^{*}}$. It is estimated from the available numerical data that for a suspension of identical rigid spherical particles \[ {\mathop\mu\limits^{*}}/\mu = 1 + 2.5c + 7.6c^2, \] the error bounds on the coefficient of c2 being about ∓ 0.8. In the important case of steady simple shearing motion, there is a region of closed trajectories of one sphere centre relative to another, of infinite volume. The stress system is here not of Newtonian form, and numerical results are not obtainable until the probability, density p(r, t) can be made determinate in the region of closed trajectories by the introduction of some additional physical process, such as three-sphere encounters or Brownian motion, or by the assumption of some particular initial state.

In the analogous problem for an incompressible solid suspension it may be appropriate to assume that for many methods of manufacture p(r, t) is uniform over the accessible part of r-space, in which event the solid suspension has ‘Newtonian’ elastic behaviour and the ratio of the effective shear modulus to that of the matrix is estimated to be 1 + 2·5c + 5·2c2 for a suspension of identical rigid spheres.

A free-streamline theory is presented for the separated flow past two-dimensional bluff bodies attached to a long plane wall on which a turbulent boundary layer has developed. The non-uniform velocity profile in the turbulent boundary layer which would be measured if the bluff bodies were absent has been replaced by a hypothetical inviscid parallel shear flow which has a constant vorticity. This model admits analytical solutions and automatically yields closed streamlines in front of the bluff bodies such as the normal plate and the semicircular projection, which are geometrically very similar to observed front separation bubbles. The present theory involves three or four parameters which must be determined on the basis of experimental information, the number of parameters depending upon the shape of bluff bodies. Two typical examples of bluff bodies, i.e. the normal plate and the semicircular projection, are worked out. Pressure distributions around these bodies predicted by the present theory are found to give a good agreement with experimental measurements.

Following the previous velocity-field study (Ishigaki 1970), this paper studies how the temperature field in the laminar boundary layer near a two-dimensional stagnation point responds to the main-stream oscillation. The time-mean temperature field is of particular interest and is studied in detail. The velocity field is treated as known and is taken from the previous paper. In § 3 the solutions over the whole frequency range are obtained under the assumption of small amplitude oscillation and the results are compared with the existing approximate solutions for low and high frequency in terms of heat transfer. Time-mean heat transfer decreases at low frequency, but slightly increases at high frequency. Two factors that cause time-mean modification of the temperature field are examined quantitatively. In § 4 the finite amplitude case is treated under the assumption of high-frequency oscillation and a few examples of the time-mean temperature profile are shown.

The linear theory of rotating stratified fluids is applied to describe the steady axisymmetric motion of a stratified fluid in a rotating annulus for values of the stratification parameter $\overline{S} = \sigma N^2/(2\Omega)^2$ that are order one or larger: $\overline{S} \geqslant O(1)$. The motion is mechanically driven by either an applied velocity or by an applied stress at the top surface. The side walls are thermally insulated. Primary attention is given to a study of the meridional, or upwelling circulation. Simple analytical solutions are obtained with the aid of the narrow-gap approximation and a justifiable assumption of an infinite depth. In that case the meridional circulation is confined to a surface Ekman layer and to a Lineykin layer of thick-ness of $O(\overline{S}^{-\frac{1}{2}})$. It is shown how the solutions for the stress-driven upwelling flow in the annulus apply to a two-dimensional linear model of coastal upwelling in a stratified ocean.

The boundary layer that forms on the walls of a shock tube, after the diaphragm which initially separates two gases at different pressures is burst, is investigated. Both the driver and driven gases are assumed to have the same thermal properties and the shock tube wall is maintained at constant temperature. Crocco variables are used and a method is presented for solving the compressible boundary-layer equations within the tube in similarity variables. Three cases, corresponding to different initial pressure ratios of the driver and driven gases, are calculated which are representative of weak and medium-strength shock and expansion waves.

The problem considered is that of the steady motion of a series of neutrally buoyant, flat-faced, rigid, cylindrical capsules along the axis of a pipeline under the influence of a hydraulic pressure gradient. The Navier-Stokes equations are non-dimensionalized and expressed in central-difference form. Numerical solutions are found by the method of relaxation for Reynolds numbers up to 20 000 and a close agreement is obtained with readings from a laboratory apparatus for Reynolds numbers up to 2200.

The flow is examined in detail and the existence of toroidal vortices between successive capsules is demonstrated. Their shape is shown to be increasingly influenced by inertial forces as the Reynolds number increases, but the overall pressure gradient is not greatly dependent on the Reynolds number.

An attempt has been made to identify the geometric character of the random fine-structure regions dispersed in fully turbulent fluid. The technique was measure-ment of two-position coincidence functions for the presence of velocity fine-structure. In this primitive approach, we tried to distinguish among only three possible categories for the random shapes: (a) ‘blobs’, (b) ‘rods’ and (c) 'slabs’, depending on whether three mean orthogonal dimensions of the domains were such that (a) all were of the same order, (b) one was an order larger than the other two, or (c) one was an order smaller than the other two.

Highly idealized paradigms for these three categories were studied analytically: the two-position coincidence functions were computed for the cases of (a) spheres, (b) circular cylinders and (c) plane slabs, each field containing randomly sized elements distributed randomly in space with a homogeneous and isotropic distribution. Comparison of the measured coincidence functions with these three paradigms suggests that the fine-structure regions are more nearly of ‘rod-like’ geometry than like either of the other two. No attempt was made to distinguish shapes which might be called 'strips’.

A circular cylinder is positioned horizontally in an incompressible stably stratified fluid which has a constant Brunt-Väisälä frequency. A vertical two-dimensional internal wave is produced when the cylinder is oscillated at this natural frequency. A small amplitude viscous similarity solution which explains the main features of the internal wave is presented.

Slow motion of a thin plate at a finite angle of attack in a rotating container filled with a viscous incompressible fluid is analysed. The Rossby and Ekman numbers are assumed to be small. The solution method is developed by studying horizontal translation of an elliptical plate. The plate is shown to carry a stagnant Taylor column with it as it moves. Detailed analysis of the structure of the vertical shear column bounding the Taylor column is circumvented by integrating the equations of motion across the shear column. A jump condition based upon mass conservation in the shear column which relates the geostrophic regions inside and outside the Taylor column results. This jump condition and its method of derivation can be used to analyse arbitrary (slow) motion of any thin plate at any angle of attack.

The fluid motion resulting when a disk moves using all six degrees of freedom at an infinitesimal angle of attack is discussed. The forces and moments on the disk are calculated and the streamlines of the geostrophic flow are displayed.

It was suggested by Proudman (1970) that many of the phenomena of turbulence at high Reynolds number could be modelled by a suitably chosen member of a class of non-Newtonian fluids, ν-fluids, all of whose properties depend only on a single dimensional constant with the dimensions of viscosity. This paper investigates the relaxation of homogeneous stress in a doubly degenerate third-order ν-fluid (which is the simplest member of the class that can possibly be used to model turbulence) in the limit ν → 0.

The equation which governs the stress tensor S is of the form \[ AS\ddot{S} + B\dot{S}^2 = 0, \] where A and B are isotropic tensor constants of the fluid; its differential structure can be of four distinct types. A list of properties required of the solutions of the equation is set out, and it is shown that only one of the four types has all the properties demanded of it. The behaviour of solutions of this equation is found to be consistent with the theoretical and experimental results on the decay of homogeneous turbulence.

This paper describes the behaviour of large amplitude, long gravity waves as they move over a horizontal bed into a region where the flow is steady but sheared in a vertical direction. A new class of exact solutions to the nonlinear hydraulic flow equations is derived. These solutions describe progressing waves and are sufficiently general to allow both the shape of the free surface at any instant and the shear profile of the undisturbed flow to be specified. The waves are examples of neutrally stable disturbances in the sense that they neither grow nor decay in amplitude, although, like simple waves on an unsheared flow, they can break.

Part 3 of this study treats a prototype with insulating side walls at z = ± 1 for all x and insulating top and bottom walls a ty = ± a for x < 0 and at y = ± (a + bx) for x > 0, where the applied magnetic field is in the y direction and the flow is in the x direction. In the diverging portion (x > 0) of this duct, the entire mass flux is carried by high-velocity jets adjacent to the side walls, while the fluid elsewhere is stagnant. In the constant-area portion (x < 0), the fully developed flow is severely disturbed as it approaches the join at x = 0, and high-velocity jets occur even before the top and bottom walls begin to diverge. The analysis presented in Part 3 is not valid in the limit b → 0, and the object of this paper is to reconcile the stagnant core flow for b ≠ 0 with the fully developed flow for b = 0. Conditions are such that inertia forces are negligible.

The fist transitional stage occurs when 1 [Gt ] b [Gt ] M−½, where M is the (large) Hartmann number. The upstream disturbance disappears, and downstream each of the O(M−½) side-wall boundary layers splits into an O(b−lM−½) outer layer and an O(M−½) inner layer. The fluid outside these layers is still stagnant and an O(bM−½) velocity in the outer sublayers accounts for the mass flux. The viscous inner sublayers reduce the velocity in the outer sublayers to zero at the side walls.

The second transitional stage occurs when b = O(M−½). The outer sublayers spread across the entire duct so that none of the fluid is stagnant, and an O(1) core velocity accounts for the mass flux. This analysis is valid no matter how small b becomes, and as b → 0 the fully developed solution is recovered every-where.

The flow of an incompressible viscous fluid past a deforming sphere is studied for small values of the Reynolds number. The deformation is assumed to be radial but is otherwise quite general. The case of S = O(l), where S is the Strouhal number, is investigated in detail. In particular, the drag is obtained up to O(R2 In R), where R is the Reynolds number.

A class of similarity solutions is derived for the equations of simple waves on inviscid shear flows discovered by Blythe, Kazakia & Varley (1972). Such solutions all have propagation speeds proportional to the square root of the depth and contain the uniform flow solution as a special case.

The development of magnetohydrodynamic flow due to the passage of a uniform electric current past a sphere immersed in an incompressible viscous conducting fluid extending to infinity is considered. The flow field is the response of the fluid to the Lorentz force set up by the electric current and the associated magnetic field. The solution is based on the assumption that the flow field is weak and has a negligible effect on the electromagnetic variables. We also assume that the convection terms in the momentum equation are negligible. The solution, obtained by means of Laplace transforms, is analytic except for the evaluation of an integral which is done numerically. It is shown that the flow field spreads radially from the sphere into the fluid. The rate of development of the flow field increases with the ratio of the conductivity of the sphere to that of the fluid.

Asymptotic expansion techniques are used, in the limit of large Reynolds number, to study the structure of fully turbulent shear layers. The relevant Reynolds number characterizes the ratio of the local turbulent stress to the local laminar stress, so that a relatively thick outer defect layer, in which, to lowest order, there is a balance between turbulent stress and convection of momentum, may be distinguished from a relatively thin wall layer, in which, to lowest order, there is a balance between turbulent and laminar stresses. The two cases examined are channel (or pipe) flow and two-dimensional boundary-layer flow with an applied pressure gradient, upstream of any separation. Attention, for these two cases, is confined to the flow of incompressible constant property fluids. Closure is effected through the introduction of an eddy-viscosity model formulated with sufficient generality for most existing models to be special cases. Results are carried to higher orders of approximation to indicate what properties for the friction velocity, integral thicknesses, and velocity profiles, and what conditions for similarity are implied by current eddy-viscosity closures.

Plane Poiseuille flow in which the pressure gradient has a small amplitude time-periodic component in addition to a constant component is considered. The velocity field close to the boundaries, arising from a small amplitude high frequency disturbance to the flow, is calculated to second order in the modulation amplitude. The energy-transfer integral for the disturbance is then calculated to the same order. It is found that, if the thickness of the disturbance shear wave relative to that of the modulation shear wave is greater than ½, the modulation inhibits energy transfer into the disturbance and so stabilizes the flow.