Experimental visualization of natural convective flow was carried out by using several kinds of fluid contained in a narrow vertical rectangular cavity with one vertical wall heated, the opposing vertical wall cooled and the upper and lower walls insulated. The effects of the Prandtl number Pr of the working fluid and the width of the cavity W on the flow pattern are discussed qualitatively in the present paper. The occurrence of flow patterns consisting of unicellular flow, steady secondary flow, tertiary flow and transition (from laminar to turbulent) flow is categorically demonstrated by the photographs taken. Moreover, experimental measurements of the net heat transfer through the vertical fluid layer are given for aspect ratios of 6-30 and Prandtl numbers of 4-12 500.

A generalization of the isotropic theory of Batchelor & Proudman (1954) is developed to estimate the effect of sudden but arbitrary three-dimensional distortion on homogeneous, initially axisymmetric turbulence. The energy changes due to distortion are expressed in terms of the Fourier coefficients of an expansion in zonal harmonics of the two independent scalar functions that describe the axisymmetric spectral tensor. However, for two special but non-trivial forms of this tensor, which represent possibly the simplest kinds of non-isotropic turbulence and specify the angular distribution but not the wavenumber dependence, the energy ratios have been determined in closed form. The deviation of the ratio from its isotropic value is the product of a factor containing R, the initial value of the ratio of the longitudinal to the transverse energy component, and another factor depending only on the geometry of the distortion. It is found that, in axisymmetric and large two-dimensional contractions, the isotropic theory gives nearly the correct longitudinal energy, but (when R > 1) over-estimates the increase in the transverse energy; the product of the two intensities varies little unless the distortion is very large, thus accounting for the stress-freezing observed in rapidly accelerated shear flows.

Comparisons with available experimental data for the spectra and for the energy ratios show reasonable agreement. The different ansatzes predict results in broad qualitative agreement with a simple strategem suggested by Reynolds & Tucker (1975), but the quantitative differences are not always negligible.

A theory is developed for turbulence in a stably stratified fluid, for example in the experiments of Rouse & Dodu and of Turner where there is no shear and the turbulence is induced by a source of energy near the lower boundary of the fluid. A growing mixed layer of thickness D appears in the lower portion of the fluid and is separated from the non-turbulent fluid above, in which the buoyancy gradient is given, by an interfacial layer of thickness h. The lower mixed layer has a very weak buoyancy gradient and the large buoyancy difference across the interfacial layer is Δb.

As indicated by the experiments of Thompson & Turner and Hopfinger & Toly, and derived by the author in a recent paper, if u is the root-mean-square horizontal velocity and l is the integral length scale, the eddy viscosity ul is a constant in a homogeneous fluid agitated by a grid. When there is stratification, the theory indicates that the fluid motion is unaffected by buoyancy forces in the mixed layer, so that ul should again be constant in the lower portions of the mixed layer. Since l is proportional to distance, we may conveniently suppose that the source of the disturbances is at a level z = 0 where u is infinite in accordance with uz = K. Thus we may take K to be a fundamental parameter characterizing the turbulent energy source. Then z is distance above the plane of the virtual energy source. If the non-turbulent fluid has uniform buoyancy, DΔb = U2 may be shown to be constant. In general, whether constant or not, U may be taken to be a fundamental parameter expressing the stability. The quantity $\hat{R}i = U^{2}D^{2}/K^2$ is the most fundamental of the several Richardson numbers that have been introduced in this problem because, with its use, ‘constants’ of proportionality do not depend on the molecular coefficients of viscosity or diffusion (for high Reynolds number turbulence) or on the geometry of the grid.

The theory contains a number of results:

1. $u_e D/K \sim \hat{R}i^{-\frac{7}{4}},$ where $u_e = dD/dt$ is the entrainment velocity. Integration yields $D \infty t^{\frac{2}{11}}$ for a homogeneous upper fluid and $D \infty t^{\frac{2}{9}}$ for a linear upper density field. This $-\frac{7}{4}$ entrainment law compares with a $-\frac{3}{2}$ law suggested by several experimenters.

2. Turbulence in the interfacial layer is intermittent with intermittency factor $I_3 \sim \hat{R}i^{-\frac{3}{4}}$. The turbulent patches have dimension $\sigma_3 \sim D\hat{R}i^{-\frac{3}{4}}$.

3. If the (equal) root-mean-square velocities in an infinite homogeneous fluid at a distance D from the grid are denoted by u1 ∼ v1 ∼ w1, we find that the r.m.s. velocities near the interface are u2 ∼ u1, v2 ∼ u1 and $w_2 \sim u_1 \hat{R}i^{-\frac{1}{4}}$.

4. The buoyancy flux q2 near the interface may be expressed as $q_2 \sim w^3_2/D$.

5. h ∼ D, as observed in several experiments.

In this paper we obtain an explicit representation for the unsteady motion on a transversely sheared mean flow that corresponds to the gustline motion on a uniform mean flow. The important features of this motion are discussed. It is shown that its velocity, pressure and vorticity are all induced by a certain disturbance field that is a linear combination of the vorticity and particle-displacement fields and is everywhere frozen in the mean flow. The general ideas are illustrated by considering the scattering of a gust by a half-plane embedded in a shear flow.

Spark photography with a sensitive schlieren system has been used to show the interaction between an incident acoustic wave and the flow around an airfoil trailing edge. Impact of the wave did not show any significant observable effect on the wake or trailing-edge boundary layer. The intensity of the wave diffracted from the edge varied considerably with the prevailing flow conditions. In the event of unsteadiness in the flow or boundary-layer separation the diffracted wave was strongly visible. In smooth flows with attached boundary layers the diffracted wave was very weak. These observations tend to support the recent conclusion of Howe (1976) that trailing-edge flows are quieter if they do not show singular behaviour. This is in contrast to the predictions of earlier theoretical models of the edge diffraction problem.

Experiments have shown that the two-dimensional flow near a forward stagnation line may be unstable to three-dimensional disturbances. The growing disturbance takes the form of secondary vortices, i.e. vortices more or less parallel to the original streamlines. The instability is usually confined to the boundary layer and the spacing of the secondary vortices is of the order of the boundary-layer thickness. This situation is analysed theoretically for the case of infinitesimal disturbances of the type first studied by Görtler and Hämmerlin. These are disturbances periodic in the direction perpendicular to the plane of the flow, in the limit of infinite Reynolds number. It is shown that the flow is always stable to these disturbances.

A model for the movement of a small viscous droplet on a surface is constructed that is based on the lubrication equations and uses the dynamic contact angle to describe the forces acting on the fluid at the contact line. The problems analysed are: the spreading or retraction of a circular droplet; the advance of a thin two-dimensional layer; the creeping of a droplet or cell on a coated surface to a region of greater adhesion; the distortion of droplet shape owing to surface contamination. Relevant biological problems concerning cell movement and adhesion are described.

The two-dimensional wake of a thin flat plate parallel to the stream was maintained laminar and steady at Reynolds numbers up to 3000 in a low-turbulence wind tunnel. Velocity distributions in the wake were measured in detail for Reynolds numbers from 20 to 3000. One of the interesting results is the appearance of a velocity overshoot, namely that the velocity in the outer part of the shear layer exceeds that of the uniform flow in the vicinity of the trailing edge. Comparisons between the experimental results and Goldstein's theoretical predictions show good agreement in the far wake irrespective of the Reynolds number, but not in the near wake even at higher Reynolds numbers, in particular immediately behind the trailing edge.

The mechanisms involved in the spreading of one liquid on the surface of another are considered for the case of a positive spreading coefficient. Taking into account the effects of gravity, capillary pressure and the spreading coefficient, the general equations determining the spreading are derived. These are then solved for the situation where the spreading liquid is introduced onto the surface of a uniformly flowing substratum at a position upstream of a barrier on the substratum surface. In this manner a steady-state situation is achieved for which the variation of spreading-liquid thickness with position is obtained as a function of the deposited volume of spreading liquid and the velocity of the substratum.

The results of a companion paper are extended to encompass the flow about smooth, but otherwise general body shapes. The wave behaviour depends on three important parameters, namely the body thickness ratio ε, the quantity δ, which is proportional to the difference between the frozen and equilibrium sound speeds, and the ratio λ of a relaxation time to a characteristic flow time. Both analytical and numerical solutions have been obtained; account is taken of nonlinearity for complete spectra of the three parameters, enabling an assessment to be made of the evolution of the wave forms for a host of situations. In particular, it is possible to predict the structures of the shock waves in various regions, and it transpires that under certain conditions vibrational relaxation can overwhelm other dissipative effects.

Flow in a partly liquid-filled, rotating, horizontal cylinder is analysed by means of finite-element numerical simulation. Of alternative methods for locating the free surface, a boundary collocation scheme with Newton-Raphson iteration converges. This method forces the residual in the normal-stress boundary condition to zero at a finite set of points on the liquid meniscus. Solutions of the steady, two-dimensional, incompressible flow problem show circumferential variation of the liquid-film thickness and corresponding pressure and velocity fields, including recirculation zones. The complications of an unknown meniscus location and a nonlinear normal-stress condition when surface tension is significant are illustrated. The finite-element method proves an effective and convenient tool for such flows, in which inertial, gravitational, pressure, viscous and capillary forces are all important.

Water-wave experiments are presented showing the evolution of finite amplitude waves in relatively shallow water when no solitons are present. In each case, the initial wave is rectangular in shape and wholly below the still water level; the amplitude of the wave is varied. The asymptotic solution of the Korteweg-de Vries (KdV) equation in the absence of solitons (Ablowitz & Segur 1976) is compared with observed evolution. In addition, the asymptotic solution of the linearized KdV equation (a linear dispersive model) is compared with both the KdV solution and experiments. This comparison is a natural consequence of the fact that, in the absence of solitons, the asymptotic solutions of the KdV equation and its linearized version are qualitatively similar. Both the experiments and the model equations suggest that the asymptotic wave structure consists of a negative triangular wave, travelling with a speed (gh)½, followed by a train of modulated oscillatory waves which travel more slowly. Quantitative comparisons are made for the amplitude, shape and decay rate of the leading wave and the amplitude, dominant wavenumbers and velocities of the trailing wave groups. Over the parameter range of the experiments, asymptotic KdV theory predicts more closely the observed behaviour. The leading wave is observed to decay more rapidly than the trailing wave groups; hence the leading wave becomes less prominent with time. This is in agreement with the KdV solution, whereas just the opposite is predicted by linear theory. Linear predictions for the trailing wave groups are accurate only when they agree with the KdV predictions. Both models predict the evolution of short waves in the trailing wave region. When the short waves are unstable (k gt; 1·36), either group disintegration or focusing into envelope solitons is possible. Both of these phenomena are observed in the experiments; neither is predicted by long-wave models. The nonlinear Schrödinger equation is reviewed and tested as a model of these unstable wave groups. There is some evidence that the KdV equation and the nonlinear Schrödinger equation can be patched together to provide an asymptotic description of these unstable groups.

Model equations which describe the evolution of long-wave initial data in water of uniform depth are tested to determine explicit criteria for their applicability. We consider linear and nonlinear, dispersive and non-dispersive equations. Separate criteria emerge for the leading wave and trailing oscillations of the evolving wave train. The evolution of the leading wave depends on two parameters: the volume (non-dimensional) of the initial data and an Ursell number based on the amplitude and length of the initial data. The magnitudes of these two parameters determine the appropriate model equation and its time of validity. For the trailing oscillatory waves, a local Ursell number based on the amplitude of the initial data and the local wavelength determines the appropriate model equation. Finally, these modelling criteria are applied to the problem of tsunami propagation. Asymptotic (t → ∞) linear dispersive theory does not appear to be applicable for describing the leading wave of tsunamis. If the length of the initial wave is approximately 100 miles, the leading wave is described by a linear non-dispersive model from the source region until shoaling occurs near the coastline. For smaller lengths (∼ 40 miles) a linear dispersive (but not asymptotic) model is applicable. The longer-period oscillatory waves following the leading wave, which can induce harbour resonance, apparently require a nonlinear dispersive model.

The instability of steady natural convection of a stably stratified fluid between vertical surfaces maintained at different temperatures is analysed. The linear stability theory is employed to obtain the critical Grashof and Rayleigh numbers, for widely varying levels of the stable background stratification, for Prandtl numbers ranging from 0·73 to 1000 and for the limiting case of infinite Prandtl number. The energetics of the critical disturbance modes also are investigated. The numerical results show that, if the value of the Prandtl number is in the low to moderate range, there is a transition from stationary to travelling-wave instability if the stratification exceeds a certain magnitude. However, if the Prandtl number is large, the transition, with increasing stratification, is from travelling-wave to stationary instability. The theoretical predictions are in excellent agreement with the experimental observations of Elder (1965) and of Vest & Arpaci (1969), for stationary instability, and in fair to good agreement with the experimental results of Hart (1971), for travelling-wave instability.

The rate at which a particle in suspension can capture diffusing solute molecules can be increased by stirring the fluid. It is shown that the increase is closely related to the total power expended, per unit volume of fluid, by the stirring device. The increase in diffusion current to a particle is related to the local rate of deformation of the fluid Ω (which, in turn, determines the dissipation) by a function F(Ωa2/D), where a is the particle radius and D the diffusion constant for the molecule in question. The function F has been determined experimentally by investigating the corresponding problem in heat transfer. In a fluid of viscosity η, stirring which is vigorous enough to double the mean diffusion current to a particle must entail a power dissipation, per unit volume, not less than 500ηD2/a4. For particles a few microns in size, or smaller, in water, effective stirring is not feasible. The results can be used also to predict the effect of stirring on the coagulation of similar particles. To double, by stirring, the rate at which particles of radius b form dimers requires a stirring power proportional to b−6, and is not feasible in water if b is less than 10−5 cm.

A transformation technique is used to solve the problem of steady nonlinear surface waves where the restoring force is either gravity or surface tension. An exact nonlinear integro-differential equation is found which yields known approximate solutions. Extensions to the method to account for more complicated geometries are also illustrated. The equation is solved numerically and results in agreement with previous solutions are obtained. In the case of capillary waves, the existence of two types of wave of greatest height is clearly indicated.

The resonant interactions between topographic planetary waves in a continuously stratified fluid are investigated theoretically. The interacting waves form a resonant triad and travel along a channel with a uniformly sloping bottom. The basic state stratification in the channel is characterized by a constant buoyancy frequency. The existence of solutions to the quadratic resonance conditions is established graphically. Each wave by itself is a bottom-intensified oscillation of the type discovered by Rhines (1970) except for the addition of a small positive frequency correction. This correction must be included to satisfy higher-order terms in the bottom boundary condition. For strong stratification (r2 [Gt ] L2, where r = internal deformation radius and L = channel width), the waves are strongly bottom-trapped and this frequency correction is negligible. For weak stratification (r2 [Lt ] L2) the waves are barotropic and the frequency correction is O(δ), where δ = fractional change in depth across the channel. In many oceanic contexts, δ lies in the range 0·1-0·4 and therefore this correction can produce a significant change in the phase speed. The amplitudes of the waves in the triad obey the classical gyroscopic equations usually encountered in quadratic resonance problems. In particular, the amplitudes evolve on the slow time scale \[ t=O(1/f_0\delta^2), \] which for our scaling assumptions is also O(1/f0Ro), where Ro is the Rossby number.

The results are applied to the Norwegian continental slope region. It is shown that, in this vicinity, there may exist resonant triads consisting of two short, high-frequency waves (periods around 3-4 days) and one long, low-frequency wave (period around 9 days).

Structure functions of turbulent temperature and velocity fluctuations are measured both for the atmosphere, in the surface layer over land, and for the laboratory, in the inner region of a thermal boundary layer and on the axis of a heated jet. Even-order temperature structure functions, up to order eight, generally compare favourably with the analysis of Antonia & Van Atta over the inertial subrange. The Reynolds number dependence of these structure functions, as predicted by the analysis, is in qualitative agreement with the measured data. Odd-order temperature structure functions depart significantly from the isotropic value of zero, particularly at large time delays. This departure is reasonably well predicted, over the inertial subrange, by postulating a simple ramp model for the temperature fluctuations. Assumptions involved in this model are directly tested by measurements in the heated jet. The ramp structure does not seriously affect either the even-order temperature structure functions or the mixed velocity-temperature functions, which include even-order moments of the temperature difference.

The problem of a liquid jet moving at hypersonic speed into a gas is considered in a frame of reference in which the tip of the jet is at rest. The liquid jet flow is assumed to be inviscid, irrotational, incompressible and two-dimensional although an approximate extension to the axially symmetric case is developed. The air flow in the hypersonic shock layer is analysed using the modified Newtonian theory. The condition of continuity of pressure at the gas-liquid interface then allows a solution to the potential problem in the liquid to be found by transforming to the hodograph plane. The resulting jet shape is presented graphically in terms of the relevant parameters.

The application of the method to penetration problems is also discussed and comparisons made with experimental results and ‘exact’ solutions.

An axisymmetric gravity wave, for which each of nonlinearity, dispersion and radial spreading is weak but significant, is determined as a similarity solution with slowly varying amplitude $\frac{3}{4}{\cal S}a $ and length scale l, where $a/d \propto (r/d)^{\frac{2}{3}}, l/d \propto (r/d)^{\frac{1}{3}}, r$ is the radius, d is the depth, and [Sscr ] is the family parameter of the solutions. It is shown that the free-surface displacement η(r,t) is either a wave of elevation (η ≥ 0) or a wave of depression (η ≤ 0) and that (|η|/)½ satisfies a Painlevé equation that is a nonlinear generalization of Airy's equation. Representative numerical solutions and asymptotic approximations for small and large [Sscr ] are presented. It is shown that the similarity solution conserves energy but not mass, in consequence of which (in order to obtain a complete solution to a well-posed initial-value problem) it must either be accompanied by some other component or components or be driven by a source (or sink) in some interior domain in which the implicit restriction r [Gt ] d is violated. A linear model is developed that is valid for r [lsim ] d and compensates for the mass defect of, and matches, the nonlinear similarity solution for |[Sscr ]| [Lt ] 1.