We use a high-resolution spectral numerical scheme to solve the two-dimensional equations of motion for the flow of a uniformly stratified Boussinesq fluid over isolated bottom topography in a channel of finite depth. The focus is on topography of small to moderate amplitude and slope and for conditions such that the flow is near linear resonance of either of the first two internal wave modes. The results are compared with existing inviscid theories: the steady hydrostatic analysis of Long (1955), time-dependent linear long-wave theory, and the fully nonlinear, weakly dispersive resonant theory of Grimshaw & Yi (1991). For the latter, we use a spectral numerical technique, with improved accuracy over previously used methods, to solve the approximate evolution equation for the amplitude of the resonant mode. Also, we present some new results on the modal similarity of the solutions of Long and of Grimshaw & Yi. For flow conditions close to linear resonance, solutions of Grimshaw & Yi's evolution equation compare very well with our fully nonlinear numerical solutions, except for very steep topography. For flow conditions between the first two resonances, Long's steady solution is approached asymptotically in time when the slope of the topography is sufficiently small. For steeper topography, the flow remains unsteady. This unsteadiness is manifested very clearly as periodic oscillations in the drag, which have been observed in previous numerical simulations and tow-tank experiments. We explain these oscillations as mainly due to the internal waves that according to linear theory persist longest in the neighbourhood of the topography.

The interaction of sound waves with an aqueous suspension of solid particles was analysed experimentally and theoretically. A heretofore unreported effect of particle concentration in the vicinity of a wave-tube centreline was observed. The phenomenon is related to the combined effect of Rayleigh-type acoustic streaming, jet-like streaming (quartz wind) and drift forces occurring in the presence of a sonic wave in the suspension-filled tube.

An analysis for predicting the interaction of a steady oblique shock wave and a planar mixing region is presented. Specifically, an equation for the shock curvature was obtained from the shock wave and isentropic wave difference equations which govern the shock transmission within a region of varying Mach number. The effects of nonuniform gas composition within the mixing region were assessed using a similar treatment; however, the wave equations were expanded in terms of a varying ratio of specific heats instead of a varying Mach number. An expression for the shock-induced vorticity due to velocity and density gradients within the mixing region was also obtained. This expression provides a means of estimating the possible mixing augmentation induced in various shock wave-mixing region interactions. When the velocity and density gradients within the mixing region oppose each other, it is demonstrated that the pre-shock vorticity may be attenuated by the shock. Applications of the analysis are discussed with reference to specific examples involving mixing augmentation and shock oscillation.

We have studied numerically the stability of a two-dimensional Couette flow in a polytropic fluid subjected to a localized shear, using a pseudo-spectral method (Fourier-Chebyshev). The polytropic index has been chosen equal to 2 and a radial force (pseudo-gravity) is introduced in order to perform comparisons with the shallow water experimental results. When the Reynolds number is not too low, the initial flow which is purely azimuthal becomes unstable and a stable rotating pattern is formed, with a number of azimuthal modes which decreases when the Mach number increases. A qualitative agreement is found with the experimental results, although the spatial resolution constraint strongly limits the numerical Reynolds and Mach numbers. From the variation of the linear growth rate of the unstable modes with the Mach number, we are able to show the transition between a flow subjected to Kelvin-Helmholtz instability towards one essentially driven by a centrifugal instability, which is efficient for rotating supersonic flows if the angular momentum decreases outwards. The latter situation may occur for some flows in astrophysical disks.

Numerical calculations are performed for the problem of penetration into a vortex core of a blade travelling normal to the vortex axis, where the plane formed by the blade span and the direction of blade motion coincides with the normal plane of the vortex axis at the point of penetration. The calculations are based on a computational method, applicable for unsteady three-dimensional flow past immersed bodies, in which a collocation solution of the vorticity transport equation is obtained on a set of Lagrangian control points. Differences between this method and other Lagrangian vorticity-based methods in the literature are discussed. Lagrangian methods of this type are particularly attractive for problems of unsteady vortex-body interaction, since control points need only be placed on the surface of the body and in regions of the flow with non-negligible vorticity magnitude. The computations for normal blade-vortex interaction (BVI) are performed for an inviscid fluid and focus on the relationship between the vortex core deformation due to penetration of the blade into the vortex ambient position and the resulting unsteady pressure field and unsteady force acting on the blade. Computations for cases with different vortex circulations are performed, and the accuracy of an approximate formulation using rapid distortion theory is assessed by comparison with the full computational results for unsteady blade force. The force generated from blade penetration into the vortex ambient position is found to be of a comparable magnitude to various other types of unsteady BVI forces, such as that due to cutting of the vortex axial flow.

The structure of round jets in cross-flow was studied using flow visualization techniques and flying-hot-wire measurements. The study was restricted to jet to freestream velocity ratios ranging from 2.0 to 6.0 and Reynolds numbers based on the jet diameter and free-stream velocity in the range of 440 to 6200.

Flow visualization studies, together with time-averaged flying-hot-wire measurements in both vertical and horizontal sectional planes, have allowed the mean topological features of the jet in cross-flow to be identified using critical point theory. These features include the horseshoe (or necklace) vortex system originating just upstream of the jet, a separation region inside the pipe upstream of the pipe exit, the roll-up of the jet shear layer which initiates the counter-rotating vortex pair and the separation of the flat-wall boundary layer leading to the formation of the wake vortex system beneath the downstream side of the jet.

The topology of the vortex ring roll-up of the jet shear layer was studied in detail using phase-averaged flying-hot-wire measurements of the velocity field when the roll-up was forced. From these data it is possible to examine the evolution of the shear layer topology. These results are supported by the flow visualization studies which also aid in their interpretation.

The study also shows that, for velocity ratios ranging from 4.0 to 6.0, the unsteady upright vortices in the wake may form by different mechanisms, depending on the Reynolds number. It is found that at high Reynolds numbers, the upright vortex orientation in the wake may change intermittently from one configuration of vortex street to another. Three mechanisms are proposed to explain these observations.

A drop that falls into a deep liquid can either coalesce with the receiving liquid and form a vortex ring or splash. Which phenomenon actually occurs depends on the impact conditions. When the impact conditions are gradually changed the transition between coalescence and splashing proceeds via a number of intermediate steps. These are studied by means of high-speed photography of the normal impact of water drops on a plane water surface. The characteristics of different flows that appear in the transitional regime and possible mechanisms causing these flows are discussed in detail. The phenomena considered include the rise of thick jets and the ejection of high-rising thin jets out of the impact crater, the entrainment of gas bubbles, crater dynamics, crown formation and the generation of splash droplets. Finally, a classification of the phenomena characteristic of the transitional regime is given.

Inertial-range scaling laws for two- and three-dimensional turbulence are re-examined within a unified framework. A new correction to Kolmogorov's k−5/3 scaling is derived for the energy inertial range. A related modification is found to Kraichnan's logarithmically corrected two-dimensional enstrophy-range law that removes its unexpected divergence at the injection wavenumber. The significance of these corrections is illustrated with steady-state energy spectra from recent high-resolution closure computations. Implications for conventional numerical simulations are discussed. These results underscore the asymptotic nature of inertial-range scaling laws.

Making use of numerical continuation techniques as well as bifurcation theory, both one- and two-dimensional travelling wave solutions of the ensemble-averaged equations of motion for gas and particles in fluidized beds have been computed. One-dimensional travelling wave solutions having only vertical structure emerge through a Hopf bifurcation of the uniform state and two-dimensional travelling wave solutions are born out of these one-dimensional waves. Fully developed two-dimensional solutions of high amplitude are reminiscent of bubbles. It is found that the qualitative features of the bifurcation diagram are not affected by changes in model parameters or the closures. An examination of the stability of one-dimensional travelling wave solutions to two-dimensional perturbations suggests that two-dimensional solutions emerge through a mechanism which is similar to the overturning instability analysed by Batchelor & Nitsche (1991).

In the kinematic dynamo problem a fluid motion is specified arbitrarily and the induction equation is solved for non-decaying magnetic fields; it forms part of the larger magnetohydrodynamic (MHD) dynamo problem in which the fluid flow is buoyancy-driven. Although somewhat restrictive, the kinematic problem is important for two reasons: first, it suffers from numerical difficulties that are holding up progress on the MHD problem; secondly, for the geodynamo, it is capable of reproducing details of the observable magnetic field. It is more efficient to study these two aspects for the kinematic dynamo than for the full MHD dynamo. We explore solutions for a family of fluid flows in a sphere, first studied by Kumar & Roberts (1975), that is heuristically representative of convection in a rotating sphere such as the Earth's core. We guard against numerical difficulties by comparing our results with well-understood solutions from the axisymmetric (αω) limit of Braginskii (1964a) and with solutions of the adjoint problem, which must yield identical eigenvalues in an adequate numerical treatment. Previous work has found a range of steady dipolar solutions; here we extend these results and find solutions of other symmetries, notably oscillatory and quadrupolar fields. The surface magnetic fields, important for comparison with observations, have magnetic flux concentrated by downwelling flow. Roberts (1972) found that meridional circulation promoted stationary solutions of the αω-equations, preferred solutions being oscillatory when no such circulation was present. We find analogous results for the full three-dimensional problem, but note that in the latter case the ‘effective’ meridional circulation arising from the non-axisymmetric convection (a concept made precise in the asymptotic limit of Braginskii 1964a) must be considered. Thus stationary solutions are obtained even in the absence of ‘true’ meridional circulation, and the time-dependence can be controlled by the strength of the convection as well as by the meridional circulation. The preference for fields of dipole or quadrupole parity is largely controlled by the sign of the velocity: a reversal of velocity from a case favouring a dipole will favour quadrupole parity, and vice versa. For the comparison problem of Proctor (1977b) this symmetry is exact; for the physical problem the boundary conditions make a difference. The boundary effect is first removed by surrounding the dynamo region with a thick layer of quiescent conducting fluid, and then studied numerically by progressively reducing the thickness of this layer to zero. The insulating boundary contributes to the difficulty of obtaining dynamo action, and to the numerical difficulties encountered. The effect of an inner boundary on dynamo action is also considered, but found to be slight.

Wind-tunnel experiments were conducted to examine the effect of grid-generated turbulence on lift forces at sections of a circular cylinder. Turbulence of longitudinal intensity between 0.6% and 18% was employed, with cylinder Reynolds numbers in the range 1 × 105 to 5 × 105. Addition of low-intensity turbulence had the primary effect of inducing the critical transition at Reynolds numbers below that for smooth flow; above transition there was little difference between the forces experienced by the cylinder in smooth or turbulent flow, with no sign of organized vortex shedding.

At higher turbulence intensities effects consistent with a return to organized vortex shedding were observed, particularly for the highest intensity and at the upper end of the Reynolds number range; lift coefficients were greater than in smooth supercritical flow, with a broad spectral peak centred near a Strouhal number of 0.23 accompanied by an increase in spanwise correlation lengths of lift force.

High-resolution numerical simulations (with up to 2563 modes) are performed for three-dimensional flow driven by the large-scale constant force fy = F cos(x) in a periodic box of size L = 2π (Kolmogorov flow). High Reynolds number is attained by solving the Navier-Stokes equations with hyperviscosity (-1)h+1Δh (h = 8). It is shown that the mean velocity profile of Kolmogorov flow is nearly independent of Reynolds number and has the ‘laminar’ form vy = V cos(x) with a nearly constant eddy viscosity. Nevertheless, the flow is highly turbulent and intermittent even at large scales. The turbulent intensities, energy dissipation rate and various terms in the energy balance equation have the simple coordinate dependence a + b cos(2x) (with a, b constants). This makes Kolmogorov flow a good model to explore the applicability of turbulence transport approximations in open time-dependent flows. It turns out that the standard expression for effective (eddy) viscosity used in K-[Escr ] transport models overpredicts the effective viscosity in regions of high shear rate and should be modified to account for the non-equilibrium character of the flow. Also at large scales the flow is anisotropic but for large Reynolds number the flow is isotropic at small scales. The important problem of local isotropy is systematically studied by measuring longitudinal and transverse components of the energy spectra and crosscorrelation spectra of velocities and velocity-pressure-gradient spectra. Cross-spectra which should vanish in the case of isotropic turbulence decay only algebraically but somewhat faster than corresponding isotropic correlations. It is verified that the pressure plays a crucial role in making the flow locally isotropic. It is demonstrated that anisotropic large-scale flow may be considered locally isotropic at scales which are approximately ten times smaller than the scale of the flow.

We use three-dimensional simulations to study compressible convection in a rotating frame with magnetic fields and overshoot into surrounding stable layers. The, initially weak, magnetic field is amplified and maintained by dynamo action and becomes organized into flux tubes that are wrapped around vortex tubes. We also observe vortex buoyancy which causes upward flows in the cores of extended downdraughts. An analysis of the angles between various vector fields shows that there is a tendency for the magnetic field to be parallel or antiparallel to the vorticity vector, especially when the magnetic field is strong. The magnetic energy spectrum has a short inertial range with a slope compatible with k+1/3 during the early growth phase of the dynamo. During the saturated state the slope is compatible with k−1. A simple analysis based on various characteristic timescales and energy transfer rates highlights important qualitative ideas regarding the energy budget of hydromagnetic dynamos.

The purpose of this analytical work is twofold: first, to clarify the physical mechanisms triggering the one-dimensional instabilities of plane detonations in gases; secondly to provide a nonlinear description of the longitudinal dynamics valid even far from the bifurcation. The fluctuations of the rate of heat release result from the temperature fluctuations of the shocked gas with a time delay introduced by the propagation of entropy waves. The motion of the shock is governed by a mass conservation resulting from the gas expansion across the reaction zone whose position fluctuates relative to the inert shock. The effects of longitudinal acoustic waves are quite negligible in pistonsupported detonations at high overdrives with a small difference of specific heats. This limit leads to a useful quasi-isobaric approximation for enlightening the basic mechanism of galloping detonations. Strong nonlinear effects, free from the spurious singularities of the square-wave model, are picked up by considering two different temperature sensitivities of the overall reaction rate: one governing the induction length, another one the thickness of the exothermic zone. A nonlinear integral equation for the longitudinal dynamics of overdriven detonations is obtained as an asymptotic solution of the reactive Euler equations. The analysis uses a distinguished limit based on an infinitely large temperature sensitivity of the induction kinetics and a small difference of specific heats. Comparisons with numerical calculations show a satisfactory agreement even outside the limits of validity of the asymptotic solution.