In the first part of the paper, a mixing layer of tanh y form is considered, and twodimensional solutions of the non-linear inviscid equations are found representing periodic perturbations from the neutral wave of linearized stability theory. To second order in amplitude the solutions are equivalent to the equilibrium state calculated by Schade (1964), who studied the development of perturbations in time and found an evolution towards that equilibrium state. The present calculation, however, is taken to fourth-order in amplitude. It is noted that the solutions presented in this paper are regular, even though viscosity is ignored; and the relationships to the singular (if inviscid) time-dependent solutions of Schade are explained. Such regular, inviscid solutions have been found only for odd velocity profiles, such as tanh y.

Although the details of the velocity distributions are not of the form observed experimentally, it is shown that the amplitude ratios of fundamental and first harmonic, for a given absolute amplitude, are comparable with those observed.

In part 2 some exact non-linear solutions are presented of the inviscid, incompressible equations of fluid flow in two or three spatial dimensions. They illustrate the flows of part 1, since they are periodic in one co-ordinate (x), have a shear in another (y) and are independent of the third. Included, as two-dimensional cases, are (i) the tanh y velocity distribution for a flow wholly in the x-direction, (ii) the well-known solution for the flow due to a set of point vortices equi-spaced on the axis, and (iii) an example of linearized hydrodynamic (Orr-Sommerfeld) stability theory. The flows may involve concentrations of vorticity. In three-dimensional cases the z component of velocity is even in y, whereas the x component is odd. Consequently, the class of flows represents, in general, small or large periodic perturbations from a skewed shear layer. Time-dependent solutions, representing waves travelling in the x direction may be obtained by translation of axes.

The linearized problem of two-dimensional gravity waves in a viscous incompressible stratified fluid occupying the upper half-space z > 0 is investigated. It is assumed that the dynamic viscosity coefficient μ is constant and that the density distribution ρ(z) is exponential. This leads to a fourth-order differential equation in the z co-ordinate, the coefficients of which depend on ρ(z) and on a dimensionless parameter ε which is proportional to μ/σ, σ being the frequency of the oscillation. The problem is solved for small ε. It is found that there is a region in which the solutions behave like certain solutions of the inviscid problem (with ε = 0). However, when the solutions of the inviscid problem are wave-like in z, they do not satisfy the radiation condition. This is because the viscosity, in addition to damping the motion for large z, reflects waves. The appropriate solution of the inviscid problem consists, therefore, of an incident and a reflected wave. As μ → 0, the ratio of the amplitudes of the reflected and the incident wave approaches exp (− 2π2H/Λ), where Λ is the vertical wavelength, and H the density scale height. The solution, however, does not have a limit since the reflecting layer shifts, altering the phase of the reflected wave. The results of the analysis are supplemented by a number of numerically computed solutions, which are then used to discuss the validity of the linearization.

A linear theory for steady motions in a rotating stratified fluid is presented, valid under the assumption that ε < E, where ε and E are respectively the Rossby and Ekman numbers. The fact that the stable stratification inhibits vertical motions has important consequences and many features of the dynamics of homogeneous rotating fluids are no longer present. For instance, in addition to the absence of the Taylor-Proudman constraint, it is found that Ekman layer suction no longer controls the interior dynamics. In fact, the Ekman layers themselves are frequently absent. Furthermore, the vertical Stewartson boundary layers are replaced by a new kind of boundary layer whose structure is characteristic of rotating stratified fluids. The interior dynamics are found to be controlled by dissipative processes.

Measurements in three boundary layers, one with constant free-stream velocity and two with power-law variations of free-stream velocity giving ‘moderate’ and ‘strong’ adverse pressure gradients, are presented and discussed. Several unifying features of the turbulent motion, expected to appear in all boundary layers not too far from equilibrium, are identified. The intensity spectra at higher wavenumbers follow the Kolmogorov inertial-subrange law, although the Reynolds number is not particularly high even by laboratory standards: in addition the smaller-scale motion in the outer layer is determined entirely by the local shear stress and the boundary-layer thickness. The large eddy motion increases in strength relative to the general turbulence level as the general turbulence level increases, and the limited evidence available suggests that the large eddies are similar to those in the free mixing layer. In all cases the large eddies contribute a significant proportion of the shear stress in the outer layer.

As a contribution to the breakdown phenomenon of vortices in a two-dimensional free boundary layer, this paper deals with the question whether a single cylindrical (i.e. two-dimensional) vortex can become unstable. For this reason a single vortex, as it occurs in a free boundary layer, is approximated by an axisymmetrical vortex model. The inviscid stability theory of rotating fluids is then applied to this vortex model. By general stability criteria it was found that a vortex consisting of vorticity of one sign only is stable according to the Rayleigh criterion, but, if the vorticity has an extremum value outside the axis, may become unstable with respect to cylindrical disturbances. Furthermore, stability calculations for three special types of vortex were performed. It was found that they were more unstable with respect to cylindrical disturbances than to three-dimensional ones.

It is shown that for a dynamical system admitting wave propagation modes (i.e. a wave-guide) the cross-power spectral density for stationary random fluctuations in the system will be dominated by the waves if they are lightly damped, the reason being that these can correlate over large distances of the order the inverse of the damping ratio. For a turbulent shear flow the wave propagation constant is obtained approximately from the solution of the Orr-Sommerfeld problem for the mean flow. Numerical calculations for a flat-plate boundary layer produce results for the streamwise dependence of the cross-power spectral density for the surface pressure fluctuations in good qualitative and quantitative agreement with measurements. An exception is the convection velocity for which the theory predicts a value that is somewhat too low.

For the flow of a stably-stratified fluid in the inlet region of a rectangular duct, it is shown experimentally that the upper and lower critical Reynolds numbers are functions of the interfacial Froude number F, and that if F is large they are lower than for a homogeneous flow. In stratified flows the disturbances leading to turbulent flow sometimes arise at the interface and lead to interfacial waves, whose wavelength at breaking is equal to the conduit depth.

A theoretical discussion is given of the motion of a fluid contained in a tube forming a closed loop that is heated from below and cooled from above. The fluid is assumed to have uniform temperature over each cross-section, and the heat transfer is assumed proportional to the difference between the local temperatures of the fluid and the tube. The latter temperature is prescribed. The system has one steady solution with warm fluid rising in one branch and cold fluid sinking in the other. This solution may, however, become unstable in an oscillatory manner. A weak instability takes the form of pulsations, the motion being always of one sign, while a strong instability takes the form of oscillations with zero mean motion. These oscillations are irregular and do not repeat themselves even over very long times.

These unstable motions are associated with thermal anomalies in the fluid that are advected materially around the loop. The anomalies amplify through the correlated variations in flow rate. A warm pocket of fluid creates maximum flow rate going through the upper part and minimum flow rate going through the lower part of the loop. Accordingly it passes quicker through the heat sink than through the heat source, and the latter becomes more effective. Similarly, the heat sink acts more effectively on a cold pocket of fluid.

The curve of neutral stability is worked out as a function of the two parameters of the problem, a non-dimensional gravity and a non-dimensional friction coefficient. The instability has also been studied by direct numerical time integration of the model equations.

It is suggested that the mechanism of instability found for this model operates also in more complicated systems, and can explain the pulsative type of motions observed recently in certain convection experiments.

We consider, in this paper, boundary-layer flows which are induced when an isothermal rigid-body rotation is disturbed by heating the fluid. The basic rigid-body rotation is sustained by one or more rotating planes at which the temperature differences are initiated. Conditions of both uniform and unsteady heating are discussed.

The problem of the stability of a fluid with time-dependent heating has been investigated by Morton (1957), Lick (1965) and Foster (1965). Morton and Lick assumed that the rate of change of the temperature profile is small compared with the growth rate of the disturbances (quasi-static assumption). This assumption is invalid near the onset of instability (as defined by ∂/∂t = 0), and Foster has therefore used an initial-value approach.

In this paper the range of validity of the quasi-static assumption is discussed, and results of a time-scaled analysis and calculations based on this are compared with the work of Foster; the agreement is found to be good. We restrict our attention to a semi-infinite fluid initially at a constant temperature; at time t = 0 a temperature difference ΔT is applied at the (lower) horizontal boundary (case (A) of Foster).

Stuart (1960) has developed a theory of the stability of plane Poiseuille flow to periodic disturbances of finite amplitude which, in the neighbourhood of the neutral curve, leads to an equation of the Landau (1944) type for the amplitude A of the disturbance: \[ d|A|^2/dt = k_1|A|^2 - k_2|A|^4. \] If k2 is positive in the supercritical region (R > RC) where k1 is positive, then, according to Stuart, there is a possibility of the existence of periodic solutions of finite amplitude which asymptotically approach a constant value of (k1/k2)½. We have evaluated the coefficient k2 and found that there indeed exists a zone in the (α, R)-plane where it is positive. This is the zone inside the dashed curve shown in figure 1, with the region of instability predicted by the linear theory included inside the ‘neutral curve’. Stuart's theory and Eckhaus's generalization thereof could apply in the overlapping zone just above the lower branch of the neutral curve.

This paper considers the effect of a pressure point moving over the surface of a fluid which is composed of two layers of different densities, the lower layer being of infinite depth. The method is the same as that used by Crapper (1964) for the same problem with a uniform fluid. The resulting waves are in two sets, the first being the normal surface wave for the uniform ocean, with a slight change in amplitude, and the second an internal wave. The crests of the internal wave have a pattern similar to that of the surface wave for a shallow uniform fluid.

The wake behind a two-dimensional bluff body has been photographed in water in order to investigate the modifying effect of base bleed. With this technique it has been possible to measure velocities at chosen points and from these make simultaneous estimates of the base pressure and of the strength of individual vortices in the wake. The results suggest that the observed decay in the vortex structure of the wake is related to variations in the conditions of mixing between the base fluid and the external stream.

When a solute is injected into a straight circular tube through which a solvent is in steady laminar flow it spreads out longitudinally under the combined effect of molecular diffusion and advection with the flow. Taylor (1953) showed that, provided there is no density difference between the solute and the solvent, the distribution of mean concentration satisfies a diffusion equation with a certain longitudinal diffusivity for large times and with respect to axes moving with the discharge velocity. In the present paper it is shown by a general argument that this statement remains true if the pipe is uniformly curved. An expression is given for the diffusivity, valid when the radius of curvature is sufficiently large. For all common liquids and most gases the diffusivity is reduced by the curvature. The rest of the paper deals with the effects of buoyancy forces caused by a density difference between the solute and the solvent, when the tube is horizontal. It is shown that in general a longitudinal diffusivity does not exist and an equation is derived that replaces the diffusion equation if the effects of buoyancy are small. A prediction from this equation is that buoyancy should not have a noticeable effect on the longitudinal dispersion for Péclet numbers near a certain value at which the two opposing influences of horizontal spreading due to gravity and lateral mixing due to secondary flow are in balance. This prediction is consistent with some observations made by Reejhsinghani, Gill & Barduhn (1966).

The equilibrium state of a rapidly rotating fluid, heated uniformly from above and cooled uniformly from below while contained in a cylinder with insulated side-walls is studied.

The circulations which are produced by the resulting stratification are studied over a wide range of parameters and it is shown that many of the features of the linear theory of rotating stratified fluid flows found in earlier studies reappear in this non-linear problem.

These include the gradual disappearance of Ekman layer suction and O(1) Ekman layers as the stratification increases, and the determination of the interior flow by the side-wall boundary layers in conjunction with the Ekman layers.

It is suggested therefore, that studies of rotating stratified flows which are unbounded laterally may frequently be defective and lead to solutions which are not the limit of any physically realizable experiment.

The occurrence of detached shear layers should, according to straightforward theoretical arguments, often characterize hydrodynamical motions in a rapidly rotating fluid. Such layers have been produced and studied in a very simple system, namely a homogeneous liquid of kinematical viscosity v filling an upright, rigid, cylindrical container mounted coaxially on a turn-table rotating at Ω0 rad/s about a vertical axis, and stirred by rotating about the same axis at Ω1 rad/s a disk of radius a cm and thickness b’ cm immersed in the liquid with its plane faces parallel to the top and bottom end walls of the container. By varying Ω0, Ω1 and a, ranges of Rossby number, the modulus of ε ≡ (Ω1 + Ω0)/½ (Ω1 + Ω0), from 0·01 to 0·3, and Ekman number, E ≡ 2v/a2(Ω1 + Ω0), from 10−5 to 5 × 10−4 were attained. Although the apparatus was axisymmetric, only when |ε| did not exceed a certain critical value, |εT|, was the flow characterized by the same property of symmetry about the axis of rotation. Otherwise, when |ε| > |εT|, non-axisymmetric flow occurred, having the form in planes perpendicular to the axis of rotation of a regular pattern of waves, M in number, when ε was positive, and of a blunt ellipse when ε was negative.

The axial flow in the axisymmetric detached shear layer, and the uniform rate of drift of the wave pattern characterizing the non-axisymmetric flow when ε is positive, depend in relatively simple ways on ε and E. The dependence of|εT| on E can be expressed by the empirical relationship |εT| = AEn, where A = 16·8 ± 2·2 and n = 0·568 ± 0·013 (= (4/7) × (1·000 − (0·005 ± 0·023))!), standard errors, 25 determinations. M does not depend strongly on E but generally decreases with increasing ε.

The wake of a disk at an angle to a stream contains marked periodic motions which arise from the regular shedding of vortices from the trailing edge. The vortices are in the form of a chain of irregular rings, each one linked to the succeeding one, and they move downstream at about 0·6 of free-stream velocity. The prominence of the vortex shedding increases as the angle of incidence (measured from the normal) increases up to at least 50°. The shedding frequency increases with the angle of incidence, but by a suitable choice of reference velocity and length scale, may be described by a wake Strouhal number which has the constant value 0·21 for all angles of incidence above zero, up to at least 40°.

Axially-symmetric bodies at zero incidence shed vortices in a similar manner, except that the orientation of the plane of vortex shedding is not fixed and varies from time to time.

The gasdynamic problem of collapsing shocks and detonation waves having spherical or cylindrical symmetry is considered near the point or axis of symmetry. The solution basic to this work is the self-similar flow of a collopsing symmetrical shock wave with counterpressure neglected. The focusing effect as the flow progresses causes the front to accelerate and its velocity is singular at the instant of collapse. In the present work the perturbations, due to counter-pressure and also to a uniform heat release, which give rise to essentially identical mathematical solutions, are evaluated. The basic self-similar solution is investigated in detail over a range of values of the specific heat ratio.

The principal characteristics of jet penetration are the appearance of free stream-lines at the sides of the jet and of a dividing streamline, which separates the jet and penetrated fluid. Kinematic analysis of such flow via free-streamline theory and the notched hodograph is developed with one unspecified parameter, the ratio of jet to counterstream velocity in the steady flow case. The kinetics of the problem, appearing when the jet and penetrated (or counterstream) fluid differ in density, is simply related to the kinematic solution via the square root of the density ratio. Experiments, both steady state and transient with several liquids, are presented which generally verify the theory. The experiments also yield information on the magnitude of the parameter and indicate its variation with the density ratio.

Using the grey gas approximation, the effect of radiative heat loss on axially symmetric flows is studied. Using an expansion procedure about the axis of symmetry, a numerical solution for the stagnation region is found taking the shock to be spherical. The results of this calculation are compared with the results of Lighthill's non-radiative constant density solution.