We have applied the methods of classical statistical mechanics to derive the inviscid equilibrium states for one- and two-layer nonlinear quasi-geostrophic flows, with and without bottom topography and variable rotation rate. In the one-layer case without topography we recover the equilibrium energy spectrum given by Kraichnan (1967). In the two-layer case, we find that the internal radius of deformation constitutes an important dividing scale: at scales of motion larger than the radius of deformation the equilibrium flow is nearly barotropic, while at smaller scales the stream functions in the two layers are statistically uncorrelated. The equilibrium lower-layer flow is positively correlated with bottom topography (anticyclonic flow over seamounts) and the correlation extends to the upper layer at scales larger than the radius of deformation. We suggest that some of the statistical trends observed in non-equilibrium flows may be looked on as manifestations of the tendency for turbulent interactions to maximize the entropy of the system.

A numerical model is developed to simulate three-dimensional Bénard convection. This model is used to investigate thermal convection in air for Rayleigh numbers between 4000 and 25000. According to experiments, this range of Rayleigh numbers in air covers three regimes of thermal convection: (i) steady two-dimensional convection, (ii) time-periodic convection and (iii) aperiodic convection. Numerical solutions are obtained for each of these regimes and the results are compared with the available experimental data and theoretical predictions.

At the Rayleigh number Ra = 4000 the present model is able to produce experimentally realistic wavelengths for the two-dimensional convection. The small amplitude wave disturbances at Ra = 6500 have period τ = 0·24. When they become finite amplitude travelling waves, the period is τ = 0·27. These values are in good agreement with theoretical and experimental results. A detailed study of the form of these waves and of their energetics is given in appendix A. As the Rayleigh number is increased to Ra = 9000 and 25 000, the convection manifests progressively more complex spatial and temporal variations.

The vertical heat transport and other mean properties of the convection are calculated for the range of Ra considered and compared with experimental and theoretical data. A detailed comparison is also made between the mean properties of two- and three-dimensional convection at the larger values of Ra. It is found that the heat flux Nu is nearly independent of the dimensionality of the convection.

The hydrodynamic stability of a pure liquid undergoing steady rapid evaporation at reduced pressure is examined using linear stability analysis. Results show that the rapidly evaporating liquid is unstable to local variations in evaporation rate, local surface depressions being produced by the force exerted on the surface by the rapidly departing vapour and sustained liquid flows being driven by the resultant shear exerted on the liquid surface by the vapour. The coupling of this ‘differential vapour recoil’ mechanism to the Marangoni effect is investigated and the importance of inertial heat transfer, fluid inertia and viscous dissipation at the interface to system stability is resolved.

A vertical plate oscillating vertically in a statically stably-stratified fluid induces an internal wave damped by viscous forces. A two-dimensional linear stability analysis of this time-dependent state shows that the wave is highly unstable when the buoyancy and forcing frequencies are comparable. This gravitational (buoyancy) instability is due to the presence of the background stratification. The neutral curve is calculated and the system energetics are explored. Excellent agreement is obtained with the recent experimental observations of Robinson & McEwan.

We discuss non-axisymmetric instability of a thin cylindrical sheet of a gas which rotates concentrically with a rigidly rotating environment. For the sake of simplicity, we restrict ourselves to a linear single-mode analysis of a two-dimensional disturbance for which the axial component of the wavenumber vector vanishes. We further restrict ourselves to two limiting cases. In case 1 the gas can be treated as incompressible, while in case 2 the effect of radial stratification caused by the centrifugal force is extremely strong. In case 1 there are two unstable modes: a travelling and a stationary disturbance with respect to a system of co-ordinates which rotates with the environment. For each disturbance, we show the domains of instability on ρ vs. ω diagrams, where ρ is the density and ω is the angular velocity of the sheet non-dimensionalized with respect to those of the environment. A negative-viscosity phenomenon is also described. In case 2 both the travelling and the stationary disturbances are stabilized by the strong radial stratification. An outline of a WKB method of approximation is given.

The stability of the decaying laminar flow in a suddenly blocked channel is investigated. The partial differential system governing the stability of the flow is solved using a WKB type of approach. It is shown that the first term of the WKB expansion of the disturbance velocity field is just that obtained by a quasi-steady approach. The flow is found to be unstable at quite small Reynolds numbers. This instability is associated with the inflexional nature of the velocity profiles of the decaying flow.

Slow flow of a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions. The main result is an integral equation for the force per unit length exerted on the body by the fluid. The novelty is that the body is permitted to twist and dilate in addition to undergoing the translating, bending and stretching, which have been considered by others. The method of derivation is relatively simple, and the resulting integral equation does not involve the limiting processes which occur in the previous work.

The overall pattern of normal modes of parallel flow of inviscid stratified fluid is examined. For a given flow and wavenumber the modes are divided into five classes, some of which may be empty: (i) a finite class of non-singular unstable modes; (ii) a conjugate finite class of non-singular damped stable modes; (iii) a finite class of singular stable modes, each of these having a branch point and being the limit of unstable modes; (iv) a discrete class of modified internal gravity waves, these being non-singular stable modes (if the density decreases with height everywhere); (v) a continuous class of singular stable modes. The modified internal gravity waves are described asymptotically for large values of the Richardson number. These asymptotic results are related to and extended by numerical calculations for a sinusoidal basic velocity profile and a Bickley jet. The wave speeds for small values of the Richardson number are found to depend only upon the local behaviour of the mean flow near an overall simple maximum or minimum of the velocity profile. Finally some difficulties in the use of the Howard formula for perturbation at a curve of marginal stability are elucidated.

When fluid in an annulus between two cylinders is set in motion by whirling movements of one or both of the cylinders, dynamic forces are imposed by the fluid on the cylinders. Knowledge of these forces is frequently important, indeed often critical, to the engineer designing rotor systems or journal bearings. Quite general solutions of the Navier-Stokes equations are presented for this problem and are limited only by restrictions on the amplitude of the whirl motion. From these solutions, the forces are derived under a wide variety of circumstances, including large and small annular widths, high and low Reynolds numbers and with and without a mean flow created by additional net rotation of one or both of the cylinders.

This study was motivated by the focal tendency of atherosclerotic plaque to appear near arterial junctions. A two-dimensional bifurcation was selected to provide preliminary information on branch flow dynamics. Superseding previous steady flow estimates in similar branches, digital solutions for two flux waves have been obtained to reveal temporal and spatial distributions of shear and separation in pulsatile flow. Marked differences from steady flow are illustrated as well as variation due to flux-wave shape.

A numerical investigation of vortex breakdown has been undertaken in an attempt to understand its properties, and the mechanisms responsible for it. Solutions of the full steady axisymmetric Navier—Stokes equations for breakdown in an unconfined viscous vortex have been obtained for core Reynolds numbers up to 200, for a two-parameter family of assumed upstream velocity distributions. Diffusion and convection of vorticity away from the vortex core, and the strong coupling between the circumferential and axial velocity fields in highly-swirling flows, are shown to lead to stagnation and reversal of the axial flow near the axis. The various theories of vortex breakdown are considered in light of the present numerical solutions.

When Rayleigh-Bénard convection is generated under random conditions, the finite amplitude rolls and bimodal flow are observed to possess randomly placed dislocations where the rolls fit together poorly. The dislocations move into the small wavelength convection, and hence provide a size-adjustment mechanism. It is observed that the dimensionless speed of the movement is smaller for larger Prandtl number fluid.

A more rigorous proof is given of the validity of a generalized Prandtl—Glauert technique for caculating the solution to time-dependent small perturbation flows. The method, first used by Miles (1950a) for the airfoil problem and later applied by Amiet & Sears (1970) to more general problems, allows calculation of the term of first order in frequency. Anomalous behaviour for the two-dimensional problem is examined in detail and found to be limited to those two-dimensional cases which include shed vorticity downstream of the body. This anomaly, which precludes using the method for these cases, results from the need to satisfy a velocity boundary condition on the body. For this purpose the velocity must be calculated from the basic variable, the pressure, through an integrated form of the momentum equation. It is in thus calculating the velocity that the anomaly occurs. The method can be applied to both the two-dimensional case without shed vorticity and the general three-dimensional case.

The dispersion relation for a random gravity wave field is derived using the complete system of nonlinear equations. It is found that the generally accepted dispersion relation is only a first-order approximation to the mean value. The correction to this approximation is expressed in terms of the energy spectral function of the wave field. The non-zero mean deviation is proportional to the ratio of the mean Eulerian velocity at the surface and the local phase velocity. In addition to the mean deviation, there is a random scatter. The root-mean-square value of this scatter is proportional to the ratio of the root-mean-square surface velocity and the local phase velocity. As for the phase velocity, the nonzero mean deviation is equal to the mean Eulerian velocity while the root-mean-square scatter is equal to the root-mean-square surface velocity. Special cases are considered and a comparison with experimental data is also discussed.

A flow-visualization study has shown that strong Kármán vortices develop behind the blunt trailing edge of a plate when the free-stream velocities over both surfaces are equal and that the vortices tend to disappear when the surface velocities are unequal. This observation provides an explanation for the occurrence and disappearance of certain discrete tones often found to be present in the noise spectra of coaxial jets. Both the vortex formation and the tones occur at a Strouhal number based on the lip thickness and the average of the external steady-state velocities of about 0.2.

Results from theoretical calculations of the vortex formation, based on an inviscid incompressible analysis of the motion of point vortices, were in good agreement with the experimental observations.

Experiments reported previously determined the detention time of airborne smoke particles momentarily trapped in the wake bubble behind a flat disk normal to smooth air flow. The dimensionless group H, the product of the detention time td and mainstream air velocity U divided by the disk diameter D, was found to be 7.44 for all combinations of U, D and the Reynolds number, a result that was consistent with a suggested physical model for particle transport across the bubble boundary. The work is now extended into the regime of turbulent free-stream flow, where H is seen to decrease with an increasing level of turbulence while the base pressure coefficient becomes more negative. At the same time, the length of the bubble decreases, as does the bubble shape factor (the ratio of bubble volume to surface area, non-dimensionalized with respect to D). A simple theoretical relationship between H and the base pressure coefficient is argued, and is found to be in good agreement with experiment.

An important conclusion from this work is that the free-stream turbulence parameter $\Lambda \equiv l_fk^{\frac{1}{2}}_f/DU $ (where lf and kf are the length scale and the kinetic energy of the free-stream turbulence respectively) controls the properties of the flow about the disk.

This work has potential applications in several areas of topical technological interest.

This paper describes a prototype model experiment designed to test the principle that the ‘excess’ noise of a jet issuing from a conical nozzle can be significantly diminished by reducing the maximum pressure gradient in the flow. The experiment uses a water jet containing flow inhomogeneities in the form of air or helium bubbles exhausting through a conventional conical nozzle or a specially contoured ‘bellmouth’ nozzle. It is argued that the level of the internally generated noise is controlled by the mean-flow pressure gradient, and substantial reductions in the sound level are recorded with the bellmouth nozzle. Certain features of the acoustic pressure signatures of the two-phase flow are examined in detail, in particular a rather surprising absolute difference in the sound pressure levels produced when helium rather than air bubbles are used under otherwise identical mean-flow conditions. Theoretical arguments are advanced which appear to explain the principal features of the observations.

The exact equations of steady inviscid flow past a ship hull are formulated in a reference domain, onto which the flow domain is mapped. A thin-ship perturbation analysis is performed in the reference domain, and the first- and second-order solutions are derived. The classical thin-ship theory is obtained as the consistent, mapping-independent, perturbation solution in the physical space. Guilloton's method is interpreted as an inconsistent, mapping-dependent, second-order approximation. A new inconsistent approximation is obtained by exploiting the freedom in the mapping of the flow domain onto the reference domain. Further improvements are suggested.

Methods of linear hydrodynamic stability are applied to analyse the stability of the isothermal Newtonian tubular film flow model of the industrial film-blowing process. The infinitesimal disturbances are assumed to possess axisymmetry. The relevant eigenvalue problem is formulated. A straightforward numerical scheme is developed to deal with the problem. Results are presented in the form of neutral-stability curves in the relevant parameter space. The significance and relevance of these to the industrial process are briefly discussed.

The flow field produced by a shock wave reflecting from a helium gas-liquid interface was investigated using a cryogenic shock tube. Incident and reflected shock waves were observed in the gas; transmitted first- and second-sound shocks were observed in the liquid. Wave diagrams are constructed to compare the data with theoretical wave trajectories. Qualitative agreement between data and theory is shown. Quantitative differences between data and theory indicate a need for further analysis of both the gas-liquid interface and the propagation of nonlinear waves in liquid helium.

This work was a first step in the experimental investigation of a complex non-equilibrium state. The results demonstrate clearly the usefulness of the cryogenic shock tube as a research tool. The well-controlled jump in temperature and pressure across the incident shock wave provides unique initial conditions for the study of dynamic phenomena in superfluid helium.