Measurements have been made of the wavelength of Taylor vortices between rotating cylinders. It is shown that the relaxation time of such a vortex system is approximately L2/6v, where L is the length of the vortex column and v is the kinematic viscosity. Previous measurements reported in the literature have not been steady-state measurements because of the long relaxation time. The present data are accurate to 1% and extend to 40 times the critical Taylor number. The variation of wavelength with Taylor number is linear and the slope is exceedingly small and negative. The non-uniqueness of wave-number observed by Coles (1965) in doubly periodic flows is here shown to occur in the rotationally symmetric case. It is argued that variational methods are inapplicable in determining the wave-number of finite-amplitude secondary flows. The experimental results show that the wave-number is determined uniquely by the initial conditions of the system. It is suggested that any method which neglects the time-dependent behaviour of the system cannot select the final state from the manifold of solutions which occur in non-linear problems.

The stability of almost fully developed viscous flow in a rotating pipe is considered. In cylindrical polar co-ordinates (r, ø, z) this flow has the velocity components \[ \{W_0o(1),\quad\Omega r[1+o(\epsilon)],\quad W_0[1-r^2/r^2_0+o(1)]\},_{+}^{+} \] where ε = Wo/2Ωr0 and is bounded externally by the rigid cylinder r = r0, which rotates about its axis with angular velocity Ω. In the limit of small ε, the disturbance equations can be solved in terms of Bessel functions and it is shown that, in that limit, the flow is unstable for Reynolds numbers R = Wor0/v greater than Rc [asymp ] 82[sdot ]9. The unstable disturbances take the form of growing spiral waves, which are stationary relative to the rotating cylinder and the critical disturbance at R = Rc has azimuthal wave-number 1 and axial wavelength 2πr0/ε. Furthermore, it is shown that the most rapidly growing disturbance for R > Rc has an azimuthal wave-number which increases with R. Some of the problems involved in testing the results by experiment are discussed and a possible application to the theory of vortex breakdown is mentioned. In an appendix this instability is shown to be an example of inertial instability.

The closure mechanism of the human aortic valve is investigated experimentally with a rigid-walled model placed in a pulsatile water-tunnel. It is shown that the valve is controlled by a fluid feed-back system incorporating a stagnation point at the downstream end of each sinus and a trapped vortex within it, and that threequarters of the valve's closure is accomplished during forward flow, requiring only very little reversed flow to seal it. The experiments are complemented by solutions of the inviscid-flow equations, based on a Hill spherical vortex model.

It has been found experimentally that under a lid of excellent heat conductivity the wavelength of convective motions on a uniformly heated plane increases with increasing supercritical Rayleigh numbers. Under an insulating lid, the wavelength at the critical Rayleigh number is larger than under a well conducting lid. When heating from below is applied rapidly the wavelength is less than in the corresponding stationary case.

The analysis of the sound field generated by the passage of isotropic turbulence through a shock of finite strength (Ribner 1953, 1954) has been extended to provide the flux of acoustic energy emanating from unit area on the downstream side of the shock. This is motivated by the problem of estimating the sound power emerging from a supersonic jet containing shock waves. The energy flux is found to vary almost linearly with shock density ratio, reaching a maximum at infinite Mach number of 0[sdot ]062 of the flux of turbulence kinetic energy convected into unit area of the shock.

Direct comparison with a result obtained by Lighthill (1953) is misleading. His energy relations, reckoned relative to a frame moving with the fluid, must be converted to the shock-fixed frame used herein. The converted results of his theory (weak shocks) and the results of our theory (arbitrary shocks) appear to show a similar asymptotic behaviour for vanishing shock strength; they diverge with increasing shock strength.

An experiment has been conducted modelling a two-dimensional mixed region collapsing in a continuously density-stratified medium. The process of collapse can be divided into three stages. Empirical formulae have been derived to describe the process of the first two stages, during which gravitational effects determine the modelling criterion. The collapse in the final stage is complicated by viscous effects; a ratio between the Froude number and the Reynolds number seems to provide a probable modification of the time scale.

The pattern of internal waves generated by the initial impulsive collapse of this mixed region can be represented by moving rays connecting either wave crests or troughs. These rays move away from the collapse centre and at the same time decrease their slopes from the horizontal. A simpler steady-state wave pattern generated by an oscillating plunger has also been studied. Taken together, these experimental results are interpreted to show that the energy density of the collapse is skewed toward high frequencies and is peaked at 8/10 of the Brunt-Väisälä frequency of the medium.

This paper describes a floating-element skin friction meter which has been designed for use in adverse pressure gradients. The effects of secondary forces on the element, which arise from the pressure gradient, are examined in some detail. The limitations of various methods of measuring wall shear stress are discussed and the results from the floating element device are compared with measurements taken in a two-dimensional boundary layer using Preston tubes and velocity profiles. As it is planned to use the instrument later for direct measurements of the shear stress in three-dimensional boundary layers, the relevance of the instrument to this situation is also discussed.

Let u(x) be the velocity field in a fluid of infinite extent due to a vorticity distribution w(x) which is zero except in two closed vortex filaments of strengths K1, K2. It is first shown that the integral \[ I=\int{\bf u}.{\boldmath \omega}\,dV \] is equal to αK1K2 where α is an integer representing the degree of linkage of the two filaments; α = 0 if they are unlinked, ± 1 if they are singly linked. The invariance of I for a continuous localized vorticity distribution is then established for barotropic inviscid flow under conservative body forces. The result is interpreted in terms of the conservation of linkages of vortex lines which move with the fluid.

Some examples of steady flows for which I ± 0 are briefly described; in particular, attention is drawn to a family of spherical vortices with swirl (which is closely analogous to a known family of solutions of the equations of magnetostatics); the vortex lines of these flows are both knotted and linked.

Two related magnetohydrodynamic invariants discovered by Woltjer (1958a, b) are discussed in ±5.

The effect of a streamwise pressure gradient on the velocity profile in the viscous sublayer of a turbulent flow along a smooth wall in two-dimensional flow is estimated. In the analysis, a similarity argument is used and the necessary empirical information obtained from a constant pressure flow. An allowance is made for the departure from the wall value of the gradient of total shear stress normal to the wall. The results of analysis were used to generate new additive constants for use with Townsend's modified law of the wall velocity profile and subsequently Townsend's profile is found to be in good agreement with the measured velocity profiles in an adverse pressure gradient.

The refraction of progressive internal waves on sloping bottoms is treated for the case of constant Brunt—Väisälä frequency. In two dimensions simple, explicit expressions for the changing wavelengths and amplitudes are found. For small slopes, the solutions reduce to simple propagating waves at infinity.

The singularity along a characteristic is shown to be removable, though the solutions are now inhomogeneous waves. The viscous boundary layers of the wedge geometry are briefly considered with the inviscid solutions remaining as interior solutions.

A theory valid for small slopes is obtained for three-dimensional waves. The waves are refracted in the usual manner, turning parallel to the beach in shallow water.

A theoretical study is made of the free periods of oscillation of an incompressible inviscid fluid, bounded by two rigid concentric spheres of radii a, b (a > b), and rotating with angular velocity Ω about a common diameter. An attempt is made to use the Longuet-Higgins solution of the Laplace tidal equation as the first term of an expansion in powers of the parameter ε = (a − b)/(a + b), of the solution to the full equations governing oscillations in a spherical shell. This leads to a singularity in the second-order terms at the two critical circles where the characteristic cones of the governing equation touch the shell boundaries.

A boundary-layer type of argument is used to examine the apparent non-uniformity in the neighbourhood of these critical circles, and it is found that, in order to remove the singularity in the pressure, an integrable singularity in the velocity components must be introduced on the characteristic cone which touches the inner spherical boundary. Further integrable singularities are introduced by repeated reflexion at the shell boundaries, and so, even outside the critical region the velocity terms contain what may reasonably be described as a pathological term, generally of order ε½ compared to that found by Longuet-Higgins, periodic with wavelength O(εa) in the radial and latitudinal directions.

Some consequences of this result are discussed.

Among the parameters which determine the erosion damage sustained by the walls of a nozzle in which a mixture of gas and particles is flowing, is the angle between the direction of the particle flow and the wall surface at the moment of impact. In this work an approximate analytical solution is made for a number of gas particle flows to determine broadly the features on which particle trajectory depends and some experimental results are given which confirm the theoretical computations. It is shown that the divergent region of a conical nozzle is unlikely to suffer a severe particle attack but that for parallel flow convergent-divergent nozzles the convex region near the exit may be affected. The choke, on the other hand, is most susceptible to particle attack even by fairly small particles. It may be said, in general, that any particle which enters the choke section with a velocity which, in the absence of effects from the gas would allow the particle to strike the choke wall, will in fact hit the wall at some point along the length of the choke.

Treated in the present paper are the second-order effects, i.e. the effect of displacement thickness, longitudinal curvature, external shear and slip, on the two-dimensional laminar boundary-layer flow of incompressible fluid. The analysis is developed in terms of the stream-function co-ordinates proposed for the analysis of the rotational flow by the senior author previously. An inverse problem is set in the co-ordinates plane, and, within the framework of the second-order approximation, the problem is reduced to solving a parabolic partial differential equation for the total head. An implicit finite-difference scheme is then devised to solve the equation, and the practical computations are carried out for the flow along a flat plate or the one around a parabolic body, by making use of an electronic computer.

When rotating Couette flow becomes unstable a periodic vortex structure is formed. For the wide-gap case, this flow is steady for a rather large range of the Taylor number above onset. In the region of finite amplitude instability the wave-numbers of the periodic structure are not unique. It is shown empirically that the non-uniqueness is not an end effect but a bonafide property of the flow and that the wave-form is a unique function of the wavelength. Data is presented to demonstrate the interval over which the wave-numbers can be varied when the parameters of the system are fixed. The large effect on the wave-form of small changes in the wavelength is also illustrated. These conclusions are based on extensive measurements of the azimuthal drift velocity for a particular mode of secondary flow.

A buoyancy-driven system can be unstable due to two different mechanisms—one mechanical and the other involving buoyancy forces. The mechanical instability is of the type normally studied in connexion with the Orr-Sommerfeld equation. The buoyancy-driven instability is rather different and is related to the ‘Coriolis’-driven instability of rotating fluids. In this paper, the stability of a buoyancy-driven system, recently called a ‘buoyancy layer’, is examined for the whole range of Prandtl numbers, s. The buoyancy-driven instability becomes increasingly important as the Prandtl number is increased and so particular interest is attached to the limit in which the Prandtl number tends to infinity. In this limit, the system is neutrally stable to first order, but second-order effects render the flow unstable at a Reynolds number of order σ-½. Consequences of the results for the stability of convection in a vertical slot are examined.

A reacting flow free of molecular transport exhibits noteworthy behaviour in the neighbourhood of a blunt, symmetrical stagnation point. A local analytical study using the Lighthill-Freeman gas model reveals the basic structure of such a flow. Chemical activity is found to affect some, but not all, of the local characteristics of the flow. Unaffected are the pressure and velocity fields near the stagnation point, where the pressure varies quadratically and the velocity varies linearly as in an inert flow. In addition, the stagnation point is found to be in chemical equilibrium for all non-zero reaction rates. On the other hand the density, temperature, and concentration fields are affected by the non-equilibrium reactions. The extent of this effect can be predicted on the basis of a reaction parameter that measures the rate of reaction in terms of the velocity gradient at the stagnation point. A rapidly reacting flow (with reaction parameter greater than unity) approaches the stagnation point with vanishing gradients of density and temperature, whereas a slowly reacting flow approaches with infinite gradients. The flow field is represented mathematically by functions that are regular along the body but non-analytic in the normal direction. Numerical computations support the validity of the local closed-form solution, and provide information on the local effects of the chemical history of the flow.

The vibrational relaxation frequency of carbon dioxide has been determined by measuring the rate of change of thermal emission in shock waves near 4±3μ. This method of measuring the relaxation frequency depends mainly on the degree of excitation of the asymmetric stretching mode of the molecule, and the results are compared with those of earlier density measurements made in the same shock tube. The gas samples used are not optically thin, and it is shown that self-absorption can be taken into account. The results imply that the relaxation frequency of the asymmetric stretching mode is about 70% of that of the bending mode.

An exact solution is obtained for the two-dimensional flow of an elastico-viscous (Walters fluid B’) incompressible fluid past an infinite porous wall under the following conditions: (i) the suction velocity normal to the plate oscillates in magnitude but not in direction about a non-zero mean; (ii) the free-stream velocity oscillates in time about a constant mean.

The response of the skin-friction to the fluctuating stream and suction velocity is studied for variations in the suction parameter A, the elasticity parameter k and the frequency parameter μ. It is found that the back-flow at the wall is enhanced by k. For the same value of A, the amplitude of the skin-friction decreases with increasing k. Also an increase in k and μ leads to a decrease in the phase of the skin-friction. For moderately large A and k, the phase of the skin-friction may be completely negative.

Electrochemical techniques have been used to measure the velocity gradients at the surface of a cylinder for Reynolds numbers from 5 × 103 to 105. This is a companion study to that already reported by Dimopoulos & Hanratty (1968) for a Reynolds number range of 60–360. The use of a specially designed sandwich electrode enabled the direction of the velocity gradient as well as its magnitude to be measured. Of particular interest is the region of definite length after separation where the velocity gradient is negative, followed by an ill-defined region where the flow moves in the positive direction. Still farther downstream the direction of flow changes with time in an irregular fashion. The measured velocity gradients prior to separation are described satisfactorily by boundary-layer theory. The presence of a splitter plate in the rear of the cylinder eliminates periodic fluctuations in the wake and has a significant effect on the boundary layer prior to separation.