A linear stability analysis is applied to a stratified, thermally radiating, un-bounded shear layer. Temperature disturbances are assumed to be optically thin. Both viscous and inviscid neutral stability boundaries are determined numerically for hyperbolic-tangent mean velocity and potential-temperature profiles. For these profiles, long-wavelength disturbances are completely destabilized (the critical Richardson number Ric → ∞ as the wavenumber k → 0) in the inviscid limit. A similar situation is found for the case of discontinuous step-function profiles. However, in contrast to the non-radiating problem, the functional form of the neutral stability boundary is not the same for both the continuous and discontinuous profiles in the limit k → 0. Application of the viscous results to the atmospheres of the earth and Venus yield critical Richardson numbers in excess of ¼.

We analyse the effects of small, circularly symmetric topography on the slow flow of an inviscid, incompressible, diffusionless, horizontally uniform, baroclinic current and show that the vertical influence depends primarily on three parameters: a stratification measure S (the square of the ratio of buoyancy frequency times height scale to Coriolis parameter times length scale), a topographic parameter β (ratio of scaled topographic height multiplied by scaled bottom current to Rossby number ε) and the scaled upstream shear u′0(z) (the dimensional upstream shear divided by the ratio of the r.m.s. upstream flow speed to height scale).

Investigating a linear stratification model we find that the topographic effect is depth independent if S [lsim ] ε and a Taylor column, as indicated by the appearance of closed streamlines above the bump, exists when β > 2. Moderate stratification (S ∼ 1) causes the flow to be fully three-dimensional and the Taylor column to be a conical vortex whose height depends on β S and u′0). The results are compared with Davies's (1971, 1972) experiments.

Our results tend to support the Taylor column theory of Jupiter's Great Red Spot but effects due to variations in the Coriolis parameter with latitude have been (unjustifiably) ne glected. Using typical values for the earths oceans we find that Taylor columns of significant height could be found there. Some pertinent observations from the ocean are discussed.

In this paper, the first of two parts on solutions to the non-relativistic de Hoffmann-Teller relations, a unified notation is developed and used to produce new analytical solutions for one-dimensional unsteady and two-dimensional crossed-fields representations. A simple numerical solution method for the non-aligned two-dimensional representation is also included. These solutions are explicit in the upstream parameters and a single strength parameter. Unification of reality, entropy and evolutionary conditions is also carried out. The resulting single criterion is used to derive algorithms by which the correct solution is obtained, in every case without previous computation and elimination of non-physical solutions. Classifications of the solutions are discussed in relation to previous schemes.

The force acting on a fish-like body with combined thickness and lifting effects is analysed on the assumption of inviscid flow. A general expression is developed for the pressure force on the body, which is analogous to the momentum-flux analysis for non-lifting bodies in classical hydrodynamics. For bodies with constant volume, the mean drag (or propulsive) force is expressed in terms of a contour integral around the vortex sheet behind the body. Attention is focused on the case of steady-state motion with constant angle of attack, and the induced drag is analysed for finned axisymmetric bodies using the slender-body approximation developed by Newman & Wu (1973). Unlike earlier results of Lighthill (1970), the lift–drag ratio in this case depends on the body thickness.

It is shown in this note that the velocity of unstable or neutral waves in plane Poiseuille flow or plane Couette-Poiseuille flow, or of axisymmetric waves in Poiseuille flow, stable or unstable, must lie within the range of the velocity of the flow.

A survey of the density field in the immediate vicinity of Jarvis Island was made in April 1971. The island is isolated and is situated in the Pacific Equatorial Undercurrent, thus presenting the opportunity to study a high Reynolds number (∼ 109) stratified shear flow. Large deflexions of the isopycnals were observed near the island. Solutions derived from the perfect-fluid equations of Drazin (1961) axe in reasonable agreement with the observed mass field in the upstream region. Downstream, a wake region was apparent, but the dynamics of the wake are obscure. A coefficient of pressure deficit in the wake is in the same range as values computed from homogeneous laboratory flows, but baroclinic effects are observed.

We apply the perturbation theory which was recently developed and justified by Joseph & Sattinger (1972) to determine the form of the time-periodic solutions which bifurcate from plane Poiseuille flow. The results a t lowest significant order seem to be in good agreement with those following from the formal perturbation method of Stuart (1960) as extended by Reynolds & Potter (1967). Given the numerical results of the present calculation, the rigorous theory guarantees that the only time-periodic solution which bifurcates from laminar Poiseuille flow is a two-dimensional wave. The wave which bifurcates at the lowest Reynolds number exists, but it is unstable when its amplitude is small. Solutions which escape the small domain of attraction of laminar Poiseuille flow snap through this unstable time-periodic solution with a small amplitude to solutions of larger amplitudes.

Surface-active impurities may collect as a stationary film on the lowest part of a bubble rising in liquid while the remainder of the surface moves freely. Numerical approximations for the motion are available if the Reynolds number is low, but they fail for small films. We give the steady-state asymptotic solution for that case, and obtain the perturbation of the drag coefficient from its value for a completely free surface. It depends on the amount by which the surface tension is reduced at the rear stagnation point. This reduction has usually been taken to be the maximum possible for the particular impurity; we consider also the case where dilution is so great that that maximum cannot be reached because the impurity would then be diffusing off the surface at the rear faster than onto it elsewhere.

It is proved that a critical level, at which a wave packet is neither reflected nor transmitted, can exist only if the wave normal curve, which is formed by taking the cross-section through the wave normal surface in the plane of propagation, possesses an asymptote which is parallel to the direction of variation of the properties of the medium through which the wave packet moves. This condition, when applied to various types of hydromagnetic waves (such as hydromagnetic waves of the inertial or gravity type, or slow magnetoacoustic waves), shows that critical levels for such waves can exist only if the direction of spatial variations of the medium is perpendicular to the ambient magnetic field. Provided that the angle between the gravitational acceleration, or the rotation axis, and the magnetic field is not zero, hydromagnetic critical levels, characteristic of the gravity or inertial type, act like ‘valves’ in the sense that the wave packet can pierce the critical level from one side and is captured from the other side. It is also pointed out that critical-level behaviour is to some extent a consequence of the WKBJ approximation since the other limit, namely when the waves feel an almost discontinuous behaviour in the properties of the medium, gives markedly different results, particularly in the presence of streaming, which can give rise to the phenomenon of wave amplification.

Experiments studying steady supercritical Taylor vortex flow have been made using pairs of long cylinders with two different radius ratios, three fluids of different viscosities and three different end boundaries for the fluid column. The emphasis in these experiments is on the determination of the wavelength of the Taylor vortices and the size of the end rings. The wavelength which one measures in a finite cylinder differs from the wavelengths found theoretically for infinitely long cylinders. Provided that the end effects were properly taken into account, the wavelength of singly periodic Taylor vortices in aninfinitely long cylinder was found to remain constant between T/Tc = 1 and T/Tc, ≈ 80 in experiments with radius ratios η = 0·505 and η = 0·727. Further studies of Taylor vortex flow at very high Taylor numbers, where the vortices are either doubly periodic or truly turbulent, showed that the wavelength increases under these conditions. However, the observed wavelengths were no longer unique but distributed statistically around a wavelength larger than the critical wavelength.

A gas in a tube is excited by a reciprocating piston operating at or near a resonant frequency. Damping is introduced into the system by two means: radiation of energy from one end of the tube and rate dependence of the gas. These define a lumped damping coefficient. It is shown that in the small rate limit the signal in the periodic state suffers negligible distortion in one travel time, and hence its propagation according to acoustic theory is valid. The shape of the signal is determined by a nonlinear ordinary differential equation. The small rate condition provides a test of the applicability of the theory to given experimental conditions. When there is no damping, shocks are a feature of the flow for frequencies in the resonant band. For a given amount of damping an upper bound on the piston acceleration which ensures shockless motion is given. The resonant band is analysed for both damped and undamped cases. The predictions of the theory are compared with experiment.

The present paper considers the behaviour of a two-dimensional inviscid bubble when placed in viscous fluid whose velocity at large distances varies parabolically, and whose motion is governed by the equations of Stokes flow. For arbitrary values of the surface tension a t the bubble interface, this free boundary problem can be reduced to a coupled pair of transcendental equations.

When surface tension effects are large, the cross-section of the bubble is nearly circular. In the symmetric situation, with the bubble at the centre of a parabolic velocity profile, a reduction of the surface tension first produces a fattening at the rear of the bubble which then distorts further to form a re-entrant cavity. The solution also shows that the bubble moves faster than the undisturbed fluid velocity at its centre. When in an asymmetric position, the bubble has a drift velocity taking it towards the symmetric position.

The asymptotic solution for large time of the initial-value problem for weakly nonlinear wave systems is obtained by the method of matched asymptotic expansions in the case in which the linearized problem is slightly unstable. The linearized solution is valid until its small exponential growth overcomes the algebraic decay due to the dispersion of the initial energy. For larger times the nonlineax terms become important, but there are no additional dispersive or diffusive effects. For the non-diffusive case an exact solution which enables the explicit verification of the asymptotic results is found.

The horizontal, linearly stratified, non-diffusive, high Reynolds number flow of an unbounded fluid over a two-dimensional vertical barrier is studied for a range of internal Froude numbers (Fi) under the Oseen and Boussinesq approximations. For F1 > 0·47 the most prominent feature of the flow is the system of large amplitude lee waves located downstream of the barrier with crests tilted in the upstream direction. For 0·47 < F1 < 0·6 the crests actually extend upstream of the barrier and appear as flows of alternating direction over the barrier. For 0·47 < Fi < 0·5 reversed flows due to these waves actually extend far upstream. For Fi < 0·47 a blocking column upstream of the obstacle, as well as large amplitude lee waves, is present. For even smaller 3. the amplitude of the lee-wave system diminishes but the blocking column remains. It is also shown that the steady-state solution obtained by Trustrum (1964, 1971) for the density and pressure field is drastically altered if a small viscosity is retained in the transient analysis.

A slightly stratified shear flow is considered when the effects of nonlinearity, viscosity and thermal diffusivity are in balance in the critical layer. Finite amplitude essentially non-diffusive neutral waves exist only if the mean temperature, velocity and vorticity profiles are distorted such that small jumps in these quantities occur across the critical layer.

A numerical method for integrating the unsteady twodimensional boundarylayer equations using a second-order-accurate implicit method, which allows for arbitrary mesh spacing in the space and time variables, is developed. A unique feature of the method is the use of an asymptotic solution valid at the downstream end of the integration mesh which permits backflow to be taken into account. Newton's iterative technique is used to solve the nonlinear finite-difference equations a t each computation step, using a rapid algorithm for solving the resulting linearized equations. The method is applied to a flow which is periodic in time and contains regions of backflow. The numerical computations are compared with known numerical and asymptotic solutions and the agreement is excellent.

The cumulant-discard approach is used to predict the third- and fourth-order moments and the probability density of turbulent Reynolds shear stress fluctuations uv, the streamwise and normal velocity fluctuations being represented by u and v respectively. Measurements of these quantities in a turbulent boundary layer are presented, with the required statistics of uv obtained by the use of a high-speed digital data-acquisition system. Including correlations between u and u up to the fourth order, the cumulant-discard predictions are in close agreement with the measurements in the inner region of the layer but only qualitatively follow the experimental results in the outer intermittent region. In this latter region, predictions for the third- and fourth-order moments of uv are also obtained by assuming that the properties of both turbulent and irrotational fluctuations are Gaussian and by using some of the available conditional averages of u, v and uv.

An experimental study was made of the interaction produced by the collision, in still air, of two diametrically opposed plane turbulent wall jets. Distributions of time-average velocity, longitudinal turbulence and static pressure on the wall were measured. Simple universal relations giving the position of interaction and the maximum wall static pressure are deduced from the measurements and are explained in terms of the ‘frozen-flow’ concept. There is little or no loss of momentum due to mixing in the interaction process. The free jet that results from the interaction has the same velocity distribution as a conventionally generated two-dimensional free jet, but its rate of spread and turbulence level are about three times larger.

The structure of a supersonic laminar boundary layer near a flat plate is examined when fluid is injected into it with velocity of O(ε3U*∞) over a distance of O(L). Here U*∞ is the undisturbed fluid velocity, L the length of the plate and ε−8 is a representative Reynolds number. An essential requirement of the theory is that separation must have occurred upstream of the blow through a free interaction. It is assumed that between separation and the blow the reversed flow region has a wedge-like shape, of semi-angle in which O(ε2), the fluid velocity has decayed to insignificant values at points just upstream of the blowing region. The blown fluid fills this wedge and the favourable pressure gradient necessary to drive this fluid downstream causes the boundary of the wedge to curve until at the end of the blow it is parallel to the plate. Explicit expressions for the pressure variation and boundary-layer thickness are worked out using a (crucially) modified form of the Cole-Aroesty theory. The relation. between the strong injection studied here and massive injection, when the blowing velocity is of O(U*∞), is also discussed.

Linearized wave-stress tensors derived from Hamilton's variational principle may be asymmetric. If interpreted as momentum fluxes, they would lead to lack of conservation of orbital angular momentum. It is shown that changes in internal angular momentum or spin of waves and torque coupling to external fields can adequately provide conservation of total angular momentum in such cases, Examples are given for acoustic, internal gravity, Rossby and plasma waves.