This paper develops a one-dimensional extension to classical layered hydraulics that incorporates non-hydrostatic effects. General results for a homogeneous layer in a multi-layer steady flow are applied to single- and two-layer flow over a two-dimensional sill. The equation obtained for single-layer flows is the same as that obtained by Naghdi & Vongsarnpigoon (1986) using the direct theory of constrained fluid sheets, and compares very well with the laboratory measurements of Sivakumaran et al. (1983). The new equation derived for two-layer flows provides excellent agreement with the laboratory measurements of Lawrence (1993). Accurate solutions are obtained for a regime of two-layer flow whose behaviour cannot be explained, even qualitatively, using classical hydraulic theory.

A two-roll apparatus is used to explore experimentally the detailed fluid mechanics of meniscus roll coating in which inlets are starved and flow rates are small. Both forward and reverse modes of operation (with contra- and co-rotating rolls) are investigated using optical sectioning combined with dye injection and particle imaging techniques. That part of parameter space where meniscus coating occurs is identified by varying the roll separation and roll speeds and hence flow rate and capillary number.

Key features of the flow structures identified in the forward mode include two large eddies (each with saddle point, separatrix and sub-eddies), a primary fluid transfer jet and the existence of two critical flow rates associated with the switching-on of a second fluid transfer jet and the switching-off of the primary transfer jet followed by a change in the flow structure. In the reverse mode, the key features are a single large eddy consisting of two sub-eddies, a saddle point and separatrix, a primary fluid transfer jet and once again two critical flow rates. These correspond to (i) the switching-on of a secondary transfer jet and (ii) the disappearance of a saddle point at the nip resulting in the merger of the primary and secondary transfer jets.

Measurements of film thickness and meniscus location made over a range of speed ratios and capillary numbers are compared with theoretical predictions. A plate–roll apparatus is used to confirm the presence, for very small flow rates, of a sub-ambient, almost linear, pressure profile across the bead. Investigated also is the transition from inlet-starved to fully flooded roll coating as flow rate is increased and the changes in flow structure and pressure profile are observed.

We consider the stability with respect to three-dimensional perturbations of columnar vortices on the f-plane and as a function of the strength of a stabilizing density stratification parallel to the axis of the vortex. We seek to understand the dynamics of the processes through which such a vertically oriented barotropic vortex may be destabilized. As models of the basic vorticity distribution we consider both Kelvin–Helmholtz vortices in shear and ‘Kida-like’ vortices in strain. In the case of rotating unstratified flow, an isolated anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations by small values of the background rotation, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as expected on the basis of the Taylor–Proudman theorem. The dominant instability mechanism which drives the destruction of anticyclonic vortices in the presence of slow background rotation may be understood to constitute a three-dimensional inertial (centrifugal) instability. Through explicit analysis we show that sufficiently strong density stratification stabilizes the two-dimensional columnar structures to disruption by this and additional modes of instability that exist even in the absence of rotation. We furthermore demonstrate that there exists a second fundamental mode of instability in the presense of background rotation which affects only anticyclonic vortex columns whose cross-sections are elliptical. Only when the ellipticity of the vortex is sufficiently high does this mode dominate the centrifugal mode. The process whereby anticyclonic vortices may be selectively destroyed appears to provide a possible explanation of an asymmetry that is sometimes observed to be characteristic of the atmospheric von Kármán vortex streets that are observed in the lee of oceanic islands. The anticyclonic branch of the street often seems to be absent. More generally, the centrifugal mechanism for the selective destruction of anticyclones discussed herein very clearly explains a number of recent results obtained from both laboratory experiments and numerical simulations.

Aviscoelastic model (a mixing-length model with relaxation) is developed to investigate the effect of turbulent advection on the mean flow perturbation and the drag force induced by turbulent shear flow over an undulating surface. The relaxation term is proportional to the ratio of eddy turnover time to travelling time; accordingly, near the surface, the relaxation model reduces to an eddy-viscosity or mixing-length model, while far from the surface it reduces to a rapid-distortion model.

The linear governing equations are transformed into streamline coordinates and solved through matched asymptotic expansions. According to order-of-magnitude estimates in Belcher, Newley & Hunt (1993), the drag force contributed by nonlinear shear stress is of the same order as that contributed by asymmetric pressure arising from the leeward thickening of the perturbed boundary layer. The nonlinear analysis in the present model confirms this estimate in most cases. Our analytical results show a dip in shear stress at the interface between the inner and outer layers and provide evidence that this dynamical feature is related to eddy advection. Numerical calculation using a shooting method gives results that compare well with the analysis.

When deep-water surface gravity waves traverse an area with a curved or otherwise variable current, the current can act analogously to an optical lens, to focus wave action into a caustic region. In this region, waves of surprisingly large size, alternatively called freak, rogue, or giant waves are produced. We show how this mechanism produces freak waves at random locations when ocean swell traverses an area of random current. When the current has a constant (possibly zero) mean with small random fluctuations, we show that the probability distribution for the formation of a freak wave is universal, that is, it does not depend on the statistics of the current, but only on a single distance scale parameter, provided that this parameter is finite and non-zero. Our numerical simulations show excellent agreement with the theory, even for current standard deviation as large as 1.0 m s−1. Since many of these results are derived for arbitrary dispersion relations with certain general properties, they include as a special case previously published work on caustics in geometrical optics.

The stability of continuously stratified vortices with large displacement of isopycnal surfaces on the f-plane is examined both analytically and numerically. Using an appropriate asymptotic set of equations, we demonstrated that sufficiently large vortices (i.e. those with small values of the Rossby number) are unstable. Remarkably, the growth rate of the unstable disturbance is a function of the spatial coordinates. At the same time, the corresponding boundary-value problem for normal modes has no smooth square-integrable solutions, which would normally be regarded as stability.

We conclude that (potentially) stable vortices can be found only among ageostrophic vortices. Since this assumption cannot be verified analytically due to complexity of the primitive equations, we verify it numerically for the particular case of two-layer stratification.

As a sequel to earlier work on viscous flow through random beds of fixed spheres, the flow through beds of fixed cylindrical fibres is studied by the same method. Several distributions of orientation are considered. The aim is to find the shielding radius and drag per unit length as a function of volume fraction occupied by the fibres, in the semi-dilute situation. The first approximation is obtained from the drag on a very long cylinder resulting from the uniform flow at infinity of a viscous fluid in the presence of Darcy resistance. Estimates are made of the effects of finite length, and of curvature of the fibres. Finally the effect of a neighbouring cylinder is considered, to obtain the second-stage approximation for straight fibres. Comparison is made with some experimental and numerical results for unidirectional fibres and for plane pads.

The stability of an incompressible swept attachment-line boundary layer flow is studied numerically, within the Görtler–Hämmerlin framework, in both the linear and nonlinear two-dimensional regimes in a self-consistent manner. The initial-boundary-value problem resulting from substitution of small-amplitude excitation into the incompressible Navier–Stokes equations and linearization about the generalized Hiemenz profile is solved. A comprehensive comparison of all linear approaches utilized to date is presented and it is demonstrated that the linear initial-boundary-value problem formulation delivers results in excellent agreement with those obtained by solution of either the temporal or the spatial linear stability theory eigenvalue problem for both zero suction and a layer in which blowing is applied. In the latter boundary layer recent experiments have documented the growth of instability waves with frequencies in a range encompassed by that of the unstable Görtler–Hämmerlin linear modes found in our simulations. In order to enable further comparisons with experiment and, thus, assess the validity of the Görtler–Hämmerlin theoretical model, we make available the spatial structure of the eigenfunctions at maximum growth conditions.

The condition on smallness of the imposed excitation is subsequently relaxed and the resulting nonlinear initial-boundary-value problem is solved. Extensive numerical experimentation has been performed which has verified theoretical predictions on the way in which the solution is expected to bifurcate from the linear neutral loop. However, it is demonstrated that the two-dimensional model equations considered do not deliver subcritical instability of this flow; this strengthens the conjecture that three-dimensionality is, at least partly, responsible for the observed discrepancy between the linear theory critical Reynolds number and the subcritical turbulence observed either experimentally or in three-dimensional numerical simulations. Further, the present nonlinear computations demonstrate that the unstable flow has its line of maximum amplification in the neighbourhood of the experimentally observed instability waves, in a manner analogous to the Blasius boundary layer. In line with previous eigenvalue problem and direct simulation work, suction is observed to be a powerful stabilization mechanism for naturally occurring instabilities of small amplitude.

The absolute/convective instability properties of the Batchelor vortex are determined by direct numerical simulation of the linear impulse response. A novel decomposition procedure is applied to the computed wavepacket in order to retrieve the complex wavenumber and frequency prevailing along each spatio-temporal ray. In particular, the absolute wavenumber and frequency observed in the laboratory frame are determined as a function of swirl parameter and external flow. The introduction of a moderate amount of swirl is found to strongly promote absolute instability. In the case of wakes, the transitional helical mode that first undergoes a switch-over to absolute instability is found to be m=−1 without requiring any external counterflow. In the case of jets, the transitional helical mode is very sensitive to swirl and varies in the range −5[les ]m[les ]−1. Only a slight amount of external counterflow (1.5% of centreline velocity) is then necessary to trigger absolute instability. The results of this numerical procedure are in good qualitative and quantitative agreement with those obtained by direct application of the Briggs–Bers criterion to the inviscid dispersion relation (Olendraru et al. 1996). Implications for the dynamics of swirling jets and wakes are discussed.

This paper describes an investigation of the dynamics and acoustics of cloud cavitation, the structures which are often formed by the periodic breakup and collapse of a sheet or vortex cavity. This form of cavitation frequently causes severe noise and damage, though the precise mechanism responsible for the enhancement of these adverse effects is not fully understood. In this paper, we investigate the large impulsive surface pressures generated by this type of cavitation and correlate these with the images from high-speed motion pictures. This reveals that several types of propagating structures (shock waves) are formed in a collapsing cloud and dictate the dynamics and acoustics of collapse. One type of shock wave structure is associated with the coherent collapse of a well-defined and separate cloud when it is convected into a region of higher pressure. This type of global structure causes the largest impulsive pressures and radiated noise. But two other types of structure, termed ‘crescent-shaped regions’ and ‘leading-edge structures’ occur during the less-coherent collapse of clouds. These local events are smaller and therefore produce less radiated noise but the interior pressure pulse magnitudes are almost as large as those produced by the global events.

The ubiquity and severity of these propagating shock wave structures provides a new perspective on the mechanisms reponsible for noise and damage in cavitating flows involving clouds of bubbles. It would appear that shock wave dynamics rather than the collapse dynamics of single bubbles determine the damage and noise in many cavitating flows.

This work involves the study of the development of Type 1 stationary and travelling cross-flow modes in the three-dimensional boundary layer over a rotating disk. In order to control the characteristics of the stationary modes, we utilized organized patterns of roughness which were applied to the disk surface. These consisted of ink dots which were equally spaced in the azimuthal direction at a fixed radius in order to enhance particular azimuthal wavenumbers. Logarithmic spiral patterns of dots were also used to enhance azimuthal wave angles. Velocity fluctuation time series were decomposed into the components corresponding to the stationary and travelling modes using the instantaneous disk position as a reference. Their development was documented through the linear and nonlinear stages leading to turbulence. The linear stage agreed well with linear stability predictions for both modes. In the nonlinear stage we documented a triad coupling between pairs of travelling modes and a stationary mode. The strongest of these was a difference interaction which led to the growth of a low-azimuthal-number, stationary mode. This mode had the largest amplitude and appeared to dominate transition. In retrospect, we can observe the signs of this mechanism in past flow visualization (Kobayashi, Kohama & Takamadate 1980), and it can account for the ‘jagged’ front normally associated with cross-flow-dominated transition on swept wings.

Two branches of gravity–capillary solitary water waves are known to bifurcate from a train of infinitesimal periodic waves at the minimum value of the phase speed. In general, these solitary waves feature oscillatory tails with exponentially decaying amplitude and, in the small-amplitude limit, they may be interpreted as envelope-soliton solutions of the nonlinear Schrödinger (NLS) equation such that the envelope travels at the same speed as the carrier oscillations. On water of infinite depth, however, based on the fourth-order envelope equation derived by Hogan (1985), it is shown that the profile of these gravity–capillary solitary waves actually decays algebraically (like 1/x2) at infinity owing to the induced mean flow that is not accounted for in the NLS equation. The algebraic decay of the solitary-wave tails in deep water is confirmed by numerical computations based on the full water-wave equations. Moreover, the same behaviour is found at the tails of solitary-wave solutions of the model equation proposed by Benjamin (1992) for interfacial waves in a two-fluid system.

This paper formulates a model of mixing in a stratified and turbulent fluid. The model uses the horizontally averaged vertical buoyancy gradient and the density of turbulent kinetic energy as variables. Heuristic ‘mixing-length’ arguments lead to a coupled set of parabolic differential equations. A particular form of mechanical forcing is proposed; for certain parameter values the relationship between the buoyancy flux and the buoyancy gradient is non-monotonic and this leads to an instability of equilibria with linear stratification. The instability results in the formation of steps and interfaces in the buoyancy profile. In contrast to previous ones, the model is mathematically well posed and the interfaces have an equilibrium thickness that is much larger than that expected from molecular diffusion.

The turbulent mixing process can take one of three forms depending on the strength of the initial stratification. When the stratification is weak, instability is not present and mixing smoothly homogenizes the buoyancy. At intermediate strengths of stratification, layers and interfaces form rapidly over a substantial interior region bounded by edge layers associated with the fluxless condition of the boundaries. The interior pattern subsequently develops more slowly as interfaces drift together and merge; simultaneously, the edge layers advance inexorably into the interior. Eventually the edge layers meet in the middle and the interior pattern of layers is erased. Any remaining structure subsequently decays smoothly to the homogeneous state. Both the weak and intermediate stratified cases are in agreement with experimental phenomenology. The model predicts a third case, with strong stratification, not yet found experimentally, where the central region is linearly stable and no steps form there. However, the edge layers are unstable; mixing fronts form and then erode into the interior.

The long-time limit of the response of incompressible three-dimensional boundary layer flows on infinite swept wedges and infinite swept wings to impulsive forcing is examined using causal linear stability theory. Following the discovery by Lingwood (1995) of the presence of absolute instabilities caused by pinch points occurring in the radial direction in the boundary layer flow of a rotating disk, we search for pinch points in the cross flow direction for both the model Falkner–Skan–Cooke profile of a swept wedge and for a genuine swept-wing configuration. It is shown in both cases that, within a particular range of the parameter space, the boundary layer does indeed support pinch points in the wavenumber plane corresponding to the crossflow direction. These crossflow-induced pinch points do not constitute an absolute instability, as there is no simultaneous pinch occurring in the streamwise wavenumber plane, but nevertheless we show here how they can be used to find the maximum local growth rate contained in a wavepacket travelling in any given direction. Lingwood (1997) also found pinch points in the chordwise wavenumber plane in the boundary layer of the leading-edge region of a swept wing (i.e. at very high flow angles). The results presented in this paper, however, demonstrate the presence of pinch points for a much larger range of flow angles and pressure gradients than was found by Lingwood, and indeed describe the flow over a much greater, and practically significant, portion of the wing.