The linear Jeans stability problem in a turbulent medium is treated using a description of the large-scale motions, with the response of turbulence on the small scales being treated using a renormalization approach. This treatment shows how turbulence at scales smaller than the potentially collapsing scale builds up a turbulent pressure force which effectively resists compression, if the kinetic energy is sufficient to balance the gravitational attraction.

Progressive radial cross-waves in a deep, periphractic wavetank are investigated on the assumption that the vertical component of the capillary force vanishes at the wavemaker. For a cylindrical wavemaker, the envelope of the radial cross-wave is shown to obey an evolution equation that differs from the cubic Schrödinger equation only in the presence of a factor 1/R in the cubic term, where R is a slow radial variable. Weak, linear damping is incorporated, and the transition conditions at which the directly forced concentric wave loses stability to a parametrically forced cross-wave are obtained. The cylindrical problem is used to develop an asymptotic approximation to the corresponding problem for a spherical wavemaker. The theory is compared with the experiments of Tatsuno, Inoue & Okabe (1969). The theoretical predictions of resonant wavenumbers are consistent with their data, but the corresponding predictions of wavemaker amplitudes, on the assumption of linear damping that is confined to an inextensible (fully contaminated) free-surface boundary layer, are an order of magnitude smaller than those observed by Tatsuno et al. (1969). This underprediction of the transition amplitudes may be due to nonlinear phenomena — in particular, nonlinear effects at the contact line and ‘undersurface flows’ (Taneda 1991) – that are not comprehended by the theoretical model.

The evolution of small counter-rotating circumferential vortices in Taylor–Couette flow was examined using the laser induced fluorescence and alumina particle flow visualization techniques. The objective of the study was to critically evaluate the hypothesis of Barcilon et al. (1979) and Barcilon & Brindley (1984) that Görtler vortices form close to the cylinder walls at moderately high Taylor numbers. Three radius ratios spanning an order of magnitude, 0.084 [les ] Rn/Rt [les ] 0.877, were examined over a Taylor number range of 3 × 104 [les ] Ta [les ] 3 × 108. Still-photograph sequences taken from video records of the LIF experiments are presented showing vortex pairs close to the inner cylinder wall at Taylor numbers an order of magnitude smaller than those reported by Barcilon and co-workers. Measurements of the core-to-core separation between counter-rotating vortices were made in order to estimate the wavenumber of the instability. These measurements agree remarkably well with the theoretical analysis of Barcilon and co-workers particularly for the small- and medium-gap experiments. The present measurements indicate that there is a $-\frac{1}{3}$ power law relationship between the Görtler wavelength and Taylor number. This is consistent with the work of Barcilon & Brindley (1984). However, the present study indicates that the Görtler vortices first form at the inner cylinder wall, and that a full theoretical treatment must include inner-wall effects.

Experiments have been carried out on a vertical jet of helium issuing into a co-flow of air at a fixed exit velocity ratio of 2.0. At all the experimental conditions studied, the flow exhibits a strong self-excited periodicity. The natural frequency behaviour of the jet, the underlying flow structure, and the transition to turbulence have been studied over a wide range of flow conditions. The experiments were conducted in a variable-pressure facility which made it possible to vary the Reynolds number and Richardson number independently. A stroboscopic schlieren system was used for flow visualization and single-component laser-Doppler anemometry was used to measure the axial component of velocity. The flow exhibits several interesting features. The presence of co-flow eliminates the random meandering typical of buoyant plumes in a quiescent environment. The periodicity of the helium jet under high-Richardson-number conditions is striking. Under these conditions transition to turbulence consists of a rapid but highly structured and repeatable breakdown and intermingling of jet and free-stream fluid. At Ri = 1.6 the three-dimensional structure of the flow is seen to repeat from cycle to cycle. The point of transition moves closer to the jet exit as either the Reynolds number or the Richardson number increases. The wavelength of the longitudinal instability increases with Richardson number. At low Richardson numbers, the natural frequency scales on an inertial timescale, τ1 = D/Uj where D is the jet diameter and Uj is the mean jet exit velocity. At high Richardson number, the natural frequency scales on a buoyancy timescale, τ2 = [ρjD/g(ρ∞ -ρj)]½ where g is the gravitational acceleration and ρj and ρ∞ are the jet and free-stream densities respectively. The transition from one flow regime to another occurs over a narrow range of Richardson numbers from 0.7 to 1. A buoyancy Strouhal number is used to correlate the high-Richardson-number frequency behaviour.

A spectral collocation and matrix eigenvalue method is used to study the linear stability of the trailing line vortex model of Batchelor. For both the inviscid and viscous stability problem, the entire unstable region in the swirl/axial wavenumber parameter space is mapped out for various azimuthal wavenumbers m. In the inviscid case, the non-axisymmetric perturbation with azimuthal wavenumber m = 1 has an unstable region of larger extent than any other, with an unusual two-lobed structure; also, the location and numerical value of the maximum disturbance growth rate previously reported for this case are shown to be incorrect. Exploiting the increasingly localized structure of perturbation eigenfunctions allows accurate results to be obtained up to values of m more than 3 orders of magnitude larger than previously, and the results for the most unstable mode are in excellent agreement with the asymptotic theory of Leibovich & Stewartson. A viscous analysis of these fundamentally inviscid modes reveals that the critical Reynolds number at which instability first occurs increases as O(m2) for m [Gt ] 1, and finds the critical values of swirl and wavenumber, which approach limiting values as m → ∞.

In the viscous case, the instabilities for m = 0 and 1 recently reported by Khorrami are found via a simplified numerical approach and the entire unstable region for each of these modes is mapped out over a wide range of Reynolds numbers. The critical Reynolds numbers for these modes are found to be 322.42 and 17.527, respectively, the latter having been unreported previously. The instabilities persist in the limit of large Reynolds number, with corresponding disturbance growth rates decreasing roughly as 1/Re. In addition to the primary mode, a new family of long-wave viscous instabilities is found for the m = 1 case.

A high-order spectral method originally developed to simulate nonlinear gravity wave–wave interactions (Dommermuth & Yue 1987a), is here extended to study nonlinear interactions between surface waves and a body. The present method accounts for the nonlinear interactions among NF wave modes on the free surface and NB source modes on the body surface up to an arbitrary order M in wave steepness. By using fast-transform techniques, the operational count per time step is only linearly proportional to M and NF (typically NF [Gt ] NB). Significantly, for a (closed) submerged body, the exponential convergence with respect to M, NF (for moderately steep waves), and NB is obtained. To illustrate the usefulness of this method, we apply it to study the diffraction of Stokes waves by a submerged circular cylinder. Computations up to M = 4 are performed to obtain the nonlinear steady and harmonic forces on the cylinder and the transmission and reflection coefficients. Comparisons to available measurements as well as existing theoretical/computational predictions are in good agreement. Our most important result is the quantification of the negative horizontal drift force on the cylinder which is fourth order in the incident wave steepness. It is found that the dominant contribution of this force is due to the quadratic interaction of the first- and third-order first-harmonic waves rather than the self-interaction of the second-order second-harmonic waves, which in fact reduces the negative drift force.

The motion of buoyant transient cavities with non-condensible contents is investigated numerically using a boundary-integral method. The bubble contents are described by an adiabatic gas law. Motion is considered in the neighbourhood of a rigid boundary, in an axisymmetric geometry. We investigate whether the non-condensible contents will resist the formation of jets. It is found that jets form upon collapse and, in general, completely penetrate the bubble before it rebounds, but circumstances are identified under which the non-spherical bubble will rebound prior to this occurrence. In these cases the bulk of the jet growth occurs upon rebound. Furthermore, the interaction between the buoyancy force causing jet formation upwards, and the Bjerknes attraction of the rigid boundary causing jet formation towards it, is investigated and general principles discussed which allow the behaviour to be interpreted. The concept of the Kelvin impulse is utilized.

The problem of convection driven by radial buoyancy in a rotating cylindrical annulus with conical end surfaces represents one of the basic models of rotating fluid dynamics with applications to convection in planets and stars. Although only two-dimensional equations govern the flow in the limit of high rotation rates, a surprising variety of different states of motion can be found. In this paper earlier numerical work is extended by the consideration of rigid boundary conditions at the cylindrical walls and by a study of spatially modulated convection. Of particular interest is the case of curved conical end surfaces which appears to promote the formation of separate cylindrical convection layers.

The linear stability of the Batchelor (1964) vortex is investigated. Particular emphasis is placed on modes found recently in a numerical study by Khorrami (1991). These modes have a number of features very distinct from those found previously for this vortex, including (i) exhibiting small growth rates at large Reynolds numbers and (ii) susceptibility to destabilization by viscosity. In this paper these modes are described using asymptotic techniques, producing results which compare very favourably with fully numerical results at large Reynolds numbers.

Fields of statistically steady wind-generated waves produced in the NASA-Wallops wind wave facility, were perturbed by the injection of groups of longer waves with various slopes, mechanically generated at the upwind end of the tank. The time histories of the surface displacements were measured at four fetches in ensembles consisting of 100 realizations of each set of experimental conditions, the data being stored and analysed digitally. The overall interaction was found to have four distinct phases. (i) When the longer waves overtake the pre-existing wind-generated waves, during the first half of the group where successive crests are increasing in amplitude, vigorous wave breaking near the crests reduces the energy density and $\overline{\zeta^2}$ in the wind waves while straining by the orbital velocities of the group reduces their wavelengths near the crests; the ‘significant slope’ $2\pi(\overline{\zeta^2})^{\frac{1}{2}}/\lambda $ at the crests is found to be very nearly constant and equal to the initial, undisturbed value. After the maximum wave of the group has passed, breaking appears to virtually cease but the earlier energy loss results in suppression of the short waves. The overall suppression by a group of waves is significantly less than that measured by Mitsuyasu (1966) and Phillips & Banner (1974) in a continuous train of waves whose slope is equal to the maximum in the group. A simple description of this phase of the interaction, involving constant significant slope of the breaking waves over the leading half of the group and conservation of action thereafter, gives suppression ratios close to those measured. (ii) Once the group has passed, the surface is much smoother and the waves begin to regenerate under the continued influence of the wind but at rates considerably slower than those suggested by Plant's formula, using the averaged value of u*. This is qualitatively consistent with a locally reduced surface stress as the wind blows from rougher water well behind the group to the smoother surface immediately behind it. (iii) The regeneration is interrupted by the arrival of a wave energy front moving down the tank, across which the energy density rises abruptly to values up to six times greater than in the undisturbed field. At the same time, the dominant frequencies just behind the wave energy front are lower than in the initial field, and the significant slope $(\overline{\zeta^2})^{\frac{1}{2}} \sigma^2/g $ is, within experimental uncertainty, again identical to that in the initial field. The front was found to propagate notably faster than the appropriate group velocity (g/2σ) and it is suggested that this is the combined result of dispersion, nonlinearity and wind amplification, together with wind-induced drift in the tank. Finally, (iv) the energy density gradually subsides and the dominant wave frequency increases as the wind waves relax towards their undisturbed state, the relaxation seeming to be essentially complete when energy packets arriving at a point have originated at the upwind end of the tank, rather than at the wave energy front.

The dynamic interaction between a rigid porous structure (porosity ϕ) and its saturating fluid is studied. From the microscopic conservation laws and constitutive relations, macroscopic equations are derived. An averaging technique proposed and discussed by for example Lévy, Auriault and Burridge & Keller is used, from which we reformulate the theory by Johnson, Koplik & Dashen. The macroscopic equations then serve to describe the high-frequency behaviour of an oscillating fluid within a porous sample. This behaviour may be characterized by the length parameter Λ and by the tortuosity parameter α∞. It is shown that Λ and α∞, which describe an oscillatory flow behaviour, may be evaluated on the basis of steady potential flow theory. Numerical results are then presented for several pore geometries, and for these geometries, the steady-state permeability k0 is computed numerically also. The parameter 8α∞k0/ϕΛ2, first introduced by Johnson et al., is then evaluated and appears to be weakly dependent on pore geometry. This implies that for many porous media the dynamic interaction is described by an approximate scaling function. New experimental data, concerning oscillating flows through several porous media, are presented. Within limits of accuracy, these data are in agreement with the approximate scaling function.

The disturbance wave pattern produced from a harmonic point source in a compressible flat-plate boundary layer is computed using linear stability theory and the direct numerical integration approach. Receptivity coefficients are computed for a wide band of least-stable spanwise modes generated at the source, which are followed in the streamwise direction in order to study the wave-interference pattern. The effect of boundary-layer growth on the development of linear waves is determined by using the method of multiple scales. Results are presented for Mach numbers of 0, 2 and 7. It is found that disturbances spread in wedge-shaped regions behind the source and the wedge angle decreases with Mach number. The lateral spreading angle for the instability waves turns out to be quite close to the angle found experimentally for lateral contamination of turbulence. It is found that, owing to wave cancellation, the computed maximum disturbance amplitude is significantly lower than that obtained by following the most-amplified single normal mode.

An amplitude evolution equation is derived for roll waves that occur in a uniform open-channel flow down an incline. A periodic series of roll waves, which is a manifestation of the instability of high-velocity turbulent flow, is shown to be itself unstable to subharmonic disturbances. These disturbances develop into longer and higher roll waves. In this way roll waves tend to increase in size as time goes on, in agreement with experimental evidence.

The hydrodynamic equations describing the fluid motion in a narrow cell and in a porous medium become identical in the limit of infinite height-to-width ratio (Hele-Shaw limit) in the first and zero permeability in the second case. The properties of the convection onset are, however, indistinguishable from an experimental point of view away from these limits. For realistic (rigid, impermeable) boundary conditions the critical Rayleigh number, the critical wavenumber and in the case of a Hopf bifurcation additionally the critical frequency are derived in both cases for a binary fluid mixture. For the porous medium the assumption of zero permeability is usually a very good approximation. The critical values for the porous medium are compared to those of the narrow cell for different height-to-width ratios and the connection with experiments is discussed.

A simpler and more rigorous derivation is presented for the LET (Local Energy Transfer) theory, which generalizes the theory to the non-stationary case and which corrects some minor errors in the original formulation (McComb 1978), Previously, ad hoc generalizations of the LET theory (McComb & Shanmugasundaram 1984) gave good numerical results for the free decay of isotropic turbulence. The quantitative aspects of these previous computations are not significantly affected by the present corrections, although there are some important qualitative improvements.

The revised LET theory is also extended to the problem of passive scalar convection, and numerical results have been obtained for freely decaying isotropic turbulence, with Taylor–Reynolds numbers in the range 5 [les ] Rλ [les ] 1060, and for Prandtl numbers of 0.1, 0.5 and 1.0. At sufficiently high values of the Reynolds number, both velocity and scalar spectra are found to exhibit Kolmogorov-type power laws, with the Kolmogorov spectral constant taking the value α = 2.5 and the Corrsin–Oboukhov constant taking a value of β = 1.1.

Experiments are described which demonstrate higher-order Bragg resonant interactions between linear gravity waves and doubly sinusoidal beds. These higher-order effects, which include harmonic and subharmonic Bragg reflections, have been observed by making very precise measurements in a wave tank. Subharmonic reflection was found to be very large, even for small bottom undulation amplitudes. The experimental data are compared with the predictions of a numerical model based on the full potential theory of linear waves.

The late stages of transition, from the Λ-vortex stage up to turbulence, are investigated by postprocessing data from a direct numerical simulation of the complete K-type transition process in plane channel flow at a Reynolds number of 5000 (based on channel half-width and laminar centreline velocity). The deterministic roll-up of the high-shear layer that forms around the Λ-vortices is examined in detail. The new vortices arising from this process are visualized by plotting three-dimensional surfaces of constant pressure. Five vortices are observed, with one of these developing into a strong hairpin-shaped vortex. Interactions between the different vortices, and between the two channel halves, are found to be important. In the very last stage of transition second-generation shear layers are observed to form and roll up into new vortices. It is postulated that at this stage a sustainable mechanism of wall-bounded turbulence exists in an elementary form. The features which are locally present include high wall shear, sublayer streaks, ejections and sweeps. Large-scale energetic vortices are found to be an important part of the mechanism by which the turbulence spreads to other spanwise positions. The generality of the findings are discussed with reference to data from simulations of H-type and mixed-type transition.

In this paper we consider the two-dimensional boundary-value problem that arises when the Helmholtz equation is solved in a parallel-plate waveguide on the centreline of which is placed an obstacle that is symmetric about the centreline but which has otherwise arbitrary shape. The normal derivative of the unknown potential ϕ is specified on the surface of the obstacle. Two problems are considered in detail. First the problem of determining any trapped-mode wavenumbers is considered and secondly the problem of the scattering of an incident wave by the obstacle is examined. The solutions to these problems are sought using integral equations. Both problems have relevance in acoustics and in water-wave theory.

The stability of a boundary layer on a heated flat plate is investigated in the linear regime. The flow is shown to be unstable to longitudinal vortex structures which develop in a non-parallel manner in the streamwise direction. Solutions of the non-parallel equations are obtained numerically at O(1) values of the appropriate stability parameter, i.e. the Grashof number. We investigate the particular cases in which instability is induced by localized or distributed wall roughness or non-uniform wall heating. The case when the vortices are induced by free-stream disturbances is also considered. We then investigate the high-Grashof-number limit and the fastest growing mode. The fastest growing mode is found to be governed by a quasi-parallel theory and occurs at high wavenumbers. The wavenumber and growth rate of the fastest growing mode are found in closed form. At low wavenumbers the vortex instability is shown to be closely related to Tollmein–Schlichting waves; the effect of wall heating or cooling on the latter type of instability is discussed.

The investigation focuses on solitary-wave solutions of an approximate pseudo-differential equation governing the unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. The validity of this model equation is shown to depend on the assumption that T/g(ρ2-ρ1)h2 [Gt ] 1, where T is the interfacial surface tension, ρ2 − ρ1 the difference between the densities of the fluids and h the undisturbed thickness of the upper layer.

Various properties of solitary waves are demonstrated. For example, they have oscillatory outskirts and their velocities of translation are less than the minimum velocity of infinitesimal waves. Also, they realise respective minima of an invariant functional for fixed values of another such functional, being in consequence orbitally stable. Explicit non-trivial solutions of the equation in question are unavailable, but an existence theory is presented covering both periodic and solitary waves of permanent form.