Nonlinear wave-kinetic interactions are analysed by examining the propagation of finite-amplitude waves in a gaseous medium undergoing non-equilibrium exothermic reaction. An exact nonlinear wave equation is developed, and the various coupling mechanisms are identified.

An approximate equation, which takes into account the chemical and transport effects, is derived for high-frequency weak nonlinear waves. The equation is numerically integrated to predict the amplification rates of weak shock pulses and changes in their wave forms under different reaction conditions.

Dramatic nonlinear amplification is predicted for mixtures of high activation energies. Furthermore, the amplification rates are enhanced with increased shock strength and pulse duration. In the latter instance, a threshold value is identified, above which wave amplification is a maximum.

Steady, three-dimensional convection in rectangular boxes of fluid-saturated porous material with square horizontal cross-section heated from below is found to be non-unique. The properties of a special class of solutions exhibiting a high degree of symmetry are determined as a function of box size and Rayleigh number. The stability of these solutions to general three-dimensional perturbations is also determined. In some cases, when these solutions are found to be unstable, the alternative forms of three-dimensional convection are presented. Multiple three-dimensional steady states are given for a few particular values of box size and Rayleigh number.

Experiments are described which show that large-scale coherent structures exist in the wakes behind three-dimensional blunt bodies. Using a ‘flying hot-wire’ apparatus, the vortex-shedding cycle has been described by phase-averaged vector fields allowing an animation of large-scale motions to be produced. It is found that the large-scale structures retain their identity for long streamwise distances and contribute significantly to the Reynolds stress.

Using critical-point theory as described recently by Perry, Lim & Chong (1980), the effect of phase ‘jitter’ on ‘washout’ has been analysed. Furthermore, it is found from critical-point theory that the large-scale motions possess the same geometrical features as the low-Reynolds number (unsteady laminar flow) wake results of Perry & Lim (1978) and Perry et al. (1980).

The shear-free turbulent boundary layer is calculated by the large-eddy simulation technique. The filtered Navier-Stokes equations are used; the method of integration employs Fourier expansions in the homogeneous directions and finite differences in the cross-stream direction. Results indicate that the simulation is capable of predicting the primary Reynolds-number effects.

The structure of fully developed turbulence in a smooth circular tube has been studied for a Reynolds number of 69000 (based on centre-line velocity and radius) and at a distance from the wall of y+ = 70. The data were taken as correlations of the longitudinal component of turbulence in narrow frequency bands, the longitudinal and transverse separations being varied simultaneously. Fourier transformation of these correlations defines power spectral density functions with frequency ω and longitudinal and transverse wavenumbers kx and kz as the independent variables. In this form the data show the distribution of convection velocity among waves of different size and inclination as well as defining the coherence lengths associated with such wave packets.

Essential features of a geometrically similar wave description of the turbulence are discussed, such a model allowing considerable simplification in the description of the turbulence both for two-point and three-point space-time correlations of the velocity field. Morrison & Kronauer (1969) predicted that the wave convection velocity should depend only on total wavenumber k in a specific manner related to the mean velocity profile. The experimentally determined convection velocities contradict this prediction. An alternative formulation for convection velocity involving an additional empirical function of frequency S(ω), fits the data for the range of experimentation. Unfortunately the results provide no information on the functional dependence (if any) of convection velocity on distance from the wall.

The problem of nonlinear wave-wave interactions is reformulated, in a Eulerian framework, for two classical geophysical systems: barotropic Rossby waves and internal gravity waves on a vertical plane. The departure of the dynamical fields from the equilibrium state is expanded in the linear-problem eigenfunctions, using their properties of orthogonality and completeness. The system is then completely described by the expansion amplitudes, whose evolution is controlled by a system of equations (with quadratic nonlinearity) which is an exact representation of the original model equations. There is no a priori need for the usual multiple-time-scale analysis, or any other perturbation expansion, to develop the theory. These or other approximations (like truncation of the expansion basis or the Boltzmann equation for a stochastic description) can, if desired, be performed afterwards.

The evolution of the system is constrained mainly by the conservation of energy E and pseudo-momentum P, properties related to time and space homogeneity of the model equations. Conservation of E and P has, in turn, some interesting consequences: (a) a generalization of Fjortoft's theorem, (b) a class of exact nonlinear solutions (which includes the system of one single wave), and (c) conservation of E and P in an arbitrarily truncated system (which is useful in the development of approximations of the problem).

The properties of all possible resonant triads are shown and used to estimate the order of magnitude of off-resonant coupling coefficients.

The results are used in two different problems: the stability of a single wave (maximum growth rates are evaluated in both the strong- and weak-interactions limits) and the three-wave system. The general solution (for any initial condition and for both the resonant and off-resonant cases) of the latter is presented.

Experimental results concerning the onset of instability in a spherical gap are presented, for the case in which the gap width is not too large with respect to the radius. The two spheres may be rotated in the same direction or in opposite directions. The resulting flow field and some special observations are described. The first onset of instability is recorded and the results are summarized in stability diagrams. It is shown that Taylor's calculations for rotating cylinders are also valid for rotating spheres. Finally, further experimental results by several authors concerning the onset of instabilitty between rotating cylinders are compared with those for rotating spheres.

In an attempt to explain the mechanics of turbulence suppression previously observed by us in jets under controlled excitation, the near fields of four circular jets, a plane jet and a plane mixing layer, all with laminar efflux boundary layers, have been explored through hot-wire measurements. It is shown that controlled excitation, induced acoustically as well as by vibrating ribbons, can reduce turbulence intensities in all these flows. Reduction by as much as 80% is observed, the maximum decrease occurring at about 400θe downstream from the exit; θe is the initial shear-layer momentum thickness. The suppression effect is a maximum for excitation at the Strouhal number Stθ(≡ fθe/Ue) of about 0.017. In the jets, the turbulence suppression is observed over the range 0·75 ≤ x/D ≤ 8, while for the plane mixing layer it could be detected as far downstream as x ≅ 6000θe.

The flow-fields with and without excitation for a typical case of turbulence suppression have been studied in detail. Spectra of the u signal and time-averaged field data indicate that excitation at Stθ ≅ 0·017 suppresses the formation of naturally occurring energetic vortices - an observation confirmed by flow-visualization experiments and by study of the large-scale coherent structures of the shear layer, educed through conditional-sampling measurements. Excitation at Stθ ≅ 0·017 produces a rapid growth of the shear layer instability mode, and consequently, its saturation, roll-up and breakdown occurs much earlier in x than is found to occur naturally (at Stθ ≅; 0·012). The suppression effect is apparently a consequence of earlier transition of the shear layer vortices, which otherwise naturally grow to larger sizes and survive for larger x, as well as being due to the prevention of successive pairing of these structures.

The measurements in a highly curved mixing layer reported by Castro & Bradshaw (1976) are used to evaluate the performance of a calculation method based on the solution of modelled transport equations for the Reynolds stresses and the dissipation rate of turbulent energy. The model reproduces the suppression of turbulence by stabilizing curvature and, downstream of the curved region, where the flow returns asymptotically to being a plane mixing layer, calculated values of turbulent intensity and shear stress overshoot the plane-layer values in accordance with the experimental observations. The results are compared with those obtained by Townsend (1980) from a rapid-distortion model which correctly predicts the streamwise variation of the shear stress to intensity ratio. By contrast, calculations based on a conventional two-equation eddy-viscosity model fail badly to account for curvature effects on this flow.

The axially symmetric Stokes-flow problems occurring when a point source, rotlet or stokeslet is situated along the axis through the centre of a circular hole in a solid plane wall are examined. Exact solutions of the governing equations are obtained in terms of toroidal coordinates and their use in modelling the flows caused by a small particle translating and rotating near to a filter pore is considered. First-order expressions are derived for the effects of the wall and hole upon the hydrodynamic force and torque on the particle for situations in which the particle dimensions are small in comparison with its distance from the solid portion of the plane wall. The resulting expressions apply to any centrally symmetric particle, not necessarily axisymmetric. Finally, expressions are derived for the motion of a neutrally buoyant sphere suspended in a flow through a hole. It is demonstrated that such a particle will generally migrate across the streamlines of the undisturbed flow - away from or towards the symmetry axis of the flow, according as the particle is approaching or receding from the hole. Such migratory motion may be of importance in the flow of suspensions through orifices and stenoses.

Studies of the morphology of fragmentation of liquid drops in a liquid medium due to shock-induced flows are reported. For such high-density-ratio and high-interfacialtension systems, the fragmentation is found to be primarily due to the penetration of the drop by unstable waves. An envelope of conditions which characterize drop breakup in these systems has been deduced. Analytical models, which involve the time constants of both the drop-piercing Taylor instability and the wake-forming shear effects, reflect our experimental results well. In addition, it is found that the mercury/water system studied here should simulate the uranium dioxide/sodium pair, which is of interest in safety studies of fast reactors.

Laminar flows in annular ducts of rectangular cross-section subjected to a constant axial magnetic field B0 are considered. The equations of flow are treated by a perturbation method involving infinite series expansions in ascending powers of the ratio Λ = a/R0 (where a and R0 are respectively the height and the mean radius of the annular duct).

The leading terms of the series are calculated in the range of high values of the Hartmann number M by means of a boundary-layer technique. When M is large, the secondary flow pattern exhibits two profoundly distinct features. Firstly, in the Hartmann and interior regions, secondary flows have a one-dimensional structure and involve no inertial effects if the curvature of the duct is small enough; secondly, in thin layers near the walls parallel to the magnetic field, the secondary flow pattern is dramatically influenced by the conductivities of the walls: varying these conductivities gives rise to either one or several counter-rotating eddies. When the Reynolds number of the flow increases, inertial effects emanating from these layers penetrate the core of the duct by convective transport. Order-of-magnitude arguments show that the mean velocity is affected by secondary flow effects when $KM^{-\frac{5}{4}}$ becomes large, where K is the Dean number of the flow.

We examine the Galerkin (including single-mode and Lorenz-type) equations for convection in a sphere to determine which physical processes are neglected when the equations of motion are truncated too severely. We test our conclusions by calculating solutions to the equations of motion for different values of the Rayleigh number and for different values of the limit of the horizontal spatial resolution. We show how the gross features of the flow such as the mean temperature gradient, central temperature, boundary-layer thickness, kinetic energy and temperature variance spectra, and energy production rates are affected by truncation in the horizontal direction. We find that the transitions from steady-state to periodic, and then to aperiodic convection depend not only on Rayleigh number but also very strongly on the horizontal resolution of the calculation. All of our models are well resolved in the vertical direction, so the transitions do not appear to be due to poorly resolved boundary layers. One of the effects of truncation is to enhance the high-wavenumber end of the kinetic energy and thermal variance spectra. Our numerical examples indicate that, as long as the kinetic energy spectrum decreases with wavenumber, a truncation gives a qualitatively correct solution.

We study a steady hypersonic rotating flow of a perfect gas past a system of thin stationary scoops in a gas centrifuge of annulus type. The gas is assumed inviscid; its ratio of specific heats is assumed to be approximately 1.

The scoops are set at zero angle of attack and are periodic with respect to the azimuthal variable. The flow is assumed to be a three-dimensional small perturbation on a basic state of rigid-body rotation. New scaling laws are proposed as appropriate to realistic operating conditions of gas centrifuges. Basic equations, boundary conditions and shock conditions are linearized for a weakly hypersonic flow by an analytical procedure similar to that used in the thin-wing approximation in high speed aerodynamics. The solution of the basic equations is obtained by the eigenfunction expansion method. The solution provides us a simple addition theorem for the scoop drag which makes the resultant drag of a system of several scoops equal to the product of the number of scoops and the drag of a standard system with a single scoop. The solution makes it clear that despite the above addition theorem, the scoops interact in their effects on the flow.

A study is made of intrusions of fixed volumes of fluid of constant density into a uniformly stratified and uniformly rotating environment. In particular, the eventual steady-state configuration is sought for the ideal fluid case. Exact two-dimensional solutions can be found using a co-ordinate transformation which converts the equations satisfied outside the intrusion into the Cauchy-Riemann equations. The same technique does not, however, yield linear equations in the axisymmetric case.

Effects of friction are also considered for cases where the quasi-geostrophic approximation can be made. Vertical advection by the viscosity-induced motion is shown to have the same effect on the temperature field as a greatly enhanced lateral diffusion, and therefore tends to spread the intrusion out laterally.

An obstacle in westerly flow on a periodic β-plane can generate resonant Rossby waves and cause large perturbations to the flow field. The pattern and strength of the flow can vary markedly in response to relatively small changes in the forces driving the system. The aim of this paper is to develop a simple theory valid in these circumstances. Such a theory has application to the dynamics of planetary scale quasi-steady perturbations in the atmosphere.

A series of models for barotropic quasi-geostrophic flow in an annulus is presented. An implicit quasi-linear model, with zonally averaged flow parametrized in terms of topography shape and net zonal mass transport, gives good agreement with nonlinear calculations. In one case stable multiple equilibria are predicted and confirmed, but the multiplicity regime is small.

Laboratory experiments have been made to investigate the development of internal gravity waves as they approach a critical layer where their phase speed is equal to that of the mean flow. The waves are produced in the accelerating flow of a stratified fluid in a long tilted tube in which the lower boundary has sinusoidal corrugations. As found in earlier experiments, the waves are not observed to propagate beyond the critical layer. Near the layer their amplitude increases, with the development of regions in which the fluid is gravitationally unstable. Kelvin-Helmholtz instability is not observed, perhaps because of viscous effects.

A model is devised which describes the weakly nonlinear development of the waves. This is solved numerically. The results compare favourably with the experiments until gravitational instability is imminent. The numerical model is used to estimate both the second order Eulerian ‘jet’, which develops below the critical layer, and the Stokes drift. In the cases examined, the maximum drift is stronger than the jet and opposite in direction. The numerical model predicts the regions of wave breaking quite well.

Internal gravity waves in the ocean may be modified by transient critical layers, for example those caused by vertically-propagating near-inertial oscillations.

This paper is concerned with the problem of the transition to planing of a boat. A steady-state nonlinear solution of the problem is obtained using a theory of fluid sheets for two-dimensional motion of an incompressible inviscid fluid contained in the paper of Green & Naghdi (1977), together with appropriate jump conditions demanded by the theory. The motion of the fluid is coupled with the motion of a free-floating body and detailed analysis is undertaken pertaining to such features as the sinkage, the bow-up trim angle and the determination of the propulsion force. In particular, the results show that the governing mechanism of the hump speed phenomenon is the change in the bow-up trim angle of the boat. The differential equations and the relevant boundary conditions of the problem are reduced to a system of essentially algebraic equations whose solutions are obtained numerically.

A nonlinear steady state solution is obtained for an incompressible, inviscid fluid in a waterfall under the action of gravity, using a special case of the system of differential equations of a recent theory of fluid sheets derived by a direct approach. This analytical solution has a simple form and offers some advantages over previous solutions of the problem either by asymptotic techniques or by numerical procedures. A comparison with experimental results is also indicated.

The stability of a rotating doubly diffusive fluid is considered. It is shown that (1) a non-rotating layer can be destabilized by rotation, (2) a rotating layer can be destabilized by the addition of a bottom-heavy solute gradient, and (3) under some conditions, three thermal Rayleigh numbers are required to specify linear stability criteria. Numerical results are presented on the basis of which the explanation by Acheson (1979) of the second of these three anomalies can be assessed, and Acheson's explanation is adapted to the two other anomalies.