The Bragg scattering of random, non-stationary surface gravity waves by random topography on a gently sloping bottom is investigated. A correction is given of previously published expressions for the triad wave–wave–bottom interaction source term in the spectral energy balance equation, and the result is reconciled with deterministic theories for the reflection of waves from sinusoidal seabed undulations. For both normal and oblique incidence, the stochastic and deterministic theories are equivalent in the limit of long propagation distances. Even for relatively short distances (for example two bottom undulations), the reflected energy predicted by the stochastic source term formulation is generally within 15% of values predicted by deterministic theories. The detuning of Bragg resonance by refraction and shoaling is discussed, suggesting practical validity conditions for the stochastic theory. The effect of bottom scattering on swell propagation is illustrated with numerical model computations for the North Carolina continental shelf using high-resolution bathymetry and an efficient semi-implicit scheme to evaluate the bottom scattering source term and integrate the energy balance equation. Model results demonstrate the importance of forward scattering of waves that propagate at large oblique angles over bottom features with typical scales of one to several surface wavelengths. This process contributes significantly to the directional spread of swell on the continental shelf by diffusing energy, in the spectrum, around the mean wave direction. Back-scattering, caused by bottom features with crests parallel to those of the surface waves and wavelengths close to half the surface wavelength, is weak, owing to the sharp roll-off of the bottom elevation spectrum at high wavenumbers. Model predictions are consistent with field measurements.

Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as Rγ say, with γ < 0), so that small but finite perturbations lead to transition. Trefethen et al. (1993) conjectured that in fact γ <−1. Subsequent numerical experiments by Lundbladh, Henningson & Reddy (1994) indicated that for streamwise initial perturbations γ =−1 and −7/4 for plane Couette and plane Poiseuille flow respectively (using subcritical Reynolds numbers for plane Poiseuille flow), while for oblique initial perturbations γ =−5/4 and −7/4 Here, through a formal asymptotic analysis of the Navier–Stokes equations, it is found that for streamwise initial perturbations γ =−1 and −3/2 for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations γ =−1 and −5/4. Furthermore it is shown why the numerically determined threshold exponents are not the true asymptotic values.

The dynamic equation for the second-order moment of a passive scalar increment is investigated in the context of DNS data for decaying isotropic turbulence at several values of the Schmidt number Sc, between 0.07 and 7. When the terms of the equation are normalized using Kolmogorov and Batchelor scales, approximate independence from Sc is achieved at sufficiently small r/ηB (r is the separation across which the increment is estimated and ηB is the Batchelor length scale). The results imply approximate independence of the mixed velocity-scalar derivative skewness from Sc and underline the importance of the non-stationarity. At small r/ηB, the contribution from the non-stationarity increases as Sc increases.

The formation of solitary internal waves with trapped cores via shoaling is investigated numerically. For density fields for which the buoyancy frequency increases monotonically towards the surface, sufficiently large solitary waves break as they shoal and form solitary-like waves with trapped fluid cores. Properties of large-amplitude waves are shown to be sensitive to the near-surface stratification. For the monotonic stratifications considered, waves with open streamlines are limited in amplitude by the breaking limit (maximum horizontal velocity equals wave propagation speed). When an exponential density stratification is modified to include a thin surface mixed layer, wave amplitudes are limited by the conjugate flow limit, in which case waves become long and horizontally uniform in the centre. The maximum horizontal velocity in the limiting wave is much less than the wave's propagation speed and as a consequence, waves with trapped cores are not formed in the presence of the surface mixed layer.

Numerical simulations are used to study homogeneous, forced turbulence in three-dimensional rotating, stably stratified flow in the Boussinesq approximation, where the rotation axis and gravity are both in the zˆ-direction. Energy is injected through a three-dimensional isotropic white-noise forcing localized at small scales. The parameter range studied corresponds to Froude numbers smaller than an O(1) critical value, below which energy is transferred to scales larger than the forcing scales. The values of the ratio N/f range from ≈1/2 to ∞, where N is the Brunt–Väisälä frequency and f is twice the rotation rate. For strongly stratified flows (N/f[Gt ]1), the slow large scales generated by the fast small-scale forcing consist of vertically sheared horizontal flow. Quasi-geostrophic dynamics dominate, at large scales, only when 1/2 [les ] N/f [les ] 2, which is the range where resonant triad interactions cannot occur.

In this paper we study the fluidization of 1204 spheres at Reynolds numbers in the thousands using the method of distributed Lagrange multipliers. The results of the simulation are compared with an experiment. This is the first direct numerical simulation of a fluidized bed at the finite Reynolds numbers encountered in applications. The simulations are processed to give straight lines in log–log plots leading to power laws as in the celebrated experimental correlations of Richardson & Zaki (1954). The numerical method allows the first direct calculation of the slip velocity and other averaged values used in two-fluid continuum models. The computation and the experiment show that a single particle may be in balance with respect to weight and drag for an interval of fluidizing velocities; the expectation that the fluidizing velocity is unique is not realized. The numerical method reveals that the dynamic pressure decreases slowly with the fluidizing velocity. Tentative interpretations of these new results are discussed.

The two-dimensional steady potential free surface flow due to a pressure distribution moving at a constant velocity at the surface of a fluid of infinite depth is considered. The effects of gravity and surface tension are included in the dynamic boundary condition. The fully nonlinear problem is solved numerically by a boundary integral equation method and the results are compared with those of the linear theory of Rayleigh (1883). It is found that for some values of the capillary number, the nonlinear solutions do not approach the linear solution of Rayleigh as the magnitude of the pressure distribution approaches zero. Appropriate linear and nonlinear solutions are constructed.

Careful analysis of new field observations on the geometry of meandering tidal channels reveals that the values of meander wavelengths as well as curvatures, conveniently scaled by local channel width, fall within a fairly restricted range, suggesting the existence of some mechanistic process controlling meander formation. A three-dimensional model is then proposed to predict flow and bed topography in weakly meandering tidal channels. The analysis is developed for meander wavelengths much smaller than the tidal wavelength and assuming that sediment is transported both as bedload and as suspended load. Both conditions are typically met in nature. Due to the symmetry of the meander pattern investigated, the theory indicates that oscillations associated with the basic flow give rise to symmetric oscillations of the point bar–pool pattern around the locations of maximum curvature. However, no net migration in a tidal cycle is present, at least for periodic tides with zero mean. Suspended load leads both to an enhanced bottom deformation and to a downstream shifting of the position of the point bar. The model then provides the basis of a planimetric instability theory of the type developed for river meanders (Blondeaux & Seminara 1985). Though the available data do not yet allow a detailed quantitative comparison, it is shown that the wavelengths selected by the ‘bend mechanism’ are somewhat larger than those typically encountered in nature. The geomorphology of the process of meander formation in tidal environments is then discussed and, upon comparison with observational evidence, points out the need for various developments of the present model.

Density-driven instabilities between miscible fluids in a vertical Hele-Shaw cell are investigated by means of experimental measurements, as well as two- and three-dimensional numerical simulations. The experiments focus on the early stages of the instability growth, and they provide detailed information regarding the growth rates and most amplified wavenumbers as a function of the governing Rayleigh number Ra. They identify two clearly distinct parameter regimes: a low-Ra, ‘Hele-Shaw’ regime in which the dominant wavelength scales as Ra−1, and a high-Ra ‘gap’ regime in which the length scale of the instability is 5±1 times the gap width. The experiments are compared to a recent linear stability analysis based on the Brinkman equation. The analytical dispersion relationship for a step-like density profile reproduces the experimentally observed trend across the entire Ra range. Nonlinear simulations based on the two- and three-dimensional Stokes equations indicate that the high-Ra regime is characterized by an instability across the gap, wheras in the low-Ra regime a spanwise Hele-Shaw mode dominates.

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).

In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

We present an experimental investigation of turbulent fountains produced when dense fluid is injected upwards into either a homogeneous or a stratified environment at an angle to the vertical. We determine the initial height above the source at which the flow first comes to rest, the final height of the fountain, and the height at which the mixed fluid finally intrudes when the environment is stratified. The initial fountain height is found to decrease monotonically as the angle of inclination is increased. In contrast, the final fountain height is found to increase and then to decrease, with the maximum height occurring at an angle of inclination of about 10°. This behaviour is due a combination of the decreased interaction between the upflow and the downflow, and the decreased vertical momentum of the injected fluid, as the angle of inclination is increased.

An analytical study is made of the transient adjustment process of an initially stationary, stably stratified fluid in a square container. The boundary walls are highly conducting. The overall Rayleigh number Ra is large. Flow is initiated by the simultaneous switch-on of a temperature increase (δT) at the vertical wall and a forced vertical throughflow (Ra−1/4δw) at the horizontal walls. The principal characteristics are found by employing the matched asymptotic expansion method. The flow field is divided into the inviscid interior, vertical boundary layers and horizontal boundary layers and analyses are conducted for each region. The horizontal boundary layers are shown to be of double-layered structure, and explicit solutions are secured for these layers. Evolutionary patterns of velocity and temperature in the whole flow domain are illustrated. Both opposing (δw/δT > 0) and cooperating (δw/δT < 0) configurations are considered. The flow character in the opposing configuration can be classified into (a) a forced-convection dominant mode (δw/δT > 1/ √2), (b) a buoyancy-convection-dominant mode (0 < δw/δT < 1/√2), and (c) a static mode (δw/δT ≈ 1/√2). Global evolutionary processes are depicted, and physical rationalizations are provided.

We consider the instability of two vortex pairs in a counter-rotating configuration. Such configurations model vortex wakes observed behind aircraft with inboard vortices produced at the inboard flap edges and at the tips of the horizontal tail. The instability potential is characterized by an optimal perturbation analysis. This extends the analysis of Fabre & Jacquin (2000) which was restricted to particular stationary configurations, and that of Crouch (1997) which considered co-rotating configurations. A complete mapping of the optimal perturbations is presented. The optimal perturbations grow faster than the Crow (1970) instability. However, they correspond to short-wavelength perturbations mainly affecting the weaker inboard vortices. A possible strategy which consists of forcing a long wavelength is then investigated. Application of both the optimal and the long-wave optimal perturbations to reduction of vortex wake danger is discussed.

Direct simulations of the turbulent shear layer are performed for subsonic to supersonic Mach numbers. Fully developed turbulence is achieved with profiles of mean velocity and turbulence intensities that agree well with laboratory experiments. The thickness growth rate of the shear layer exhibits a large reduction with increasing values of the convective Mach number, Mc. In agreement with previous investigations, it is found that the normalized pressure–strain term decreases with increasing Mc, which leads to inhibited energy transfer from the streamwise to cross-stream fluctuations, to the reduced turbulence production observed in DNS, and, finally, to reduced turbulence levels as well as reduced growth rate of the shear layer. An analysis, based on the wave equation for pressure, with supporting DNS is performed with the result that the pressure–strain term decreases monotonically with increasing Mach number. The gradient Mach number, which is the ratio of the acoustic time scale to the flow distortion time scale, is shown explicitly by the analysis to be the key quantity that determines the reduction of the pressure–strain term in compressible shear flows. The physical explanation is that the finite speed of sound in compressible flow introduces a finite time delay in the transmission of pressure signals from one point to an adjacent point and the resultant increase in decorrelation leads to a reduction in the pressure–strain correlation.

The dependence of turbulence intensities on the convective Mach number is investigated. It is found that all components decrease with increasing Mc and so does the shear stress.

DNS is also used to study the effect of different free-stream densities parameterized by the density ratio, s = ρ2/ρ1, in the high-speed case. It is found that changes in the temporal growth rate of the vorticity thickness are smaller than the changes observed in momentum thickness growth rate. The momentum thickness growth rate decreases substantially with increasing departure from the reference case, s = 1. The peak value of the shear stress, uv, shows only small changes as a function of s. The dividing streamline of the shear layer is observed to move into the low-density stream. An analysis is performed to explain this shift and the consequent reduction in momentum thickness growth rate.

When a drop impacts on a liquid layer it ejects a thin horizontal sheet of liquid, emanating from the neck region connecting the two liquid masses. Dual-frame imaging and pulsed lasers are used to study the origin, speed and evolution of this ejecta sheet for a range of viscosities. The initial ejecta speed can be more than 10 times the impact velocity of the drop. Visualizations using fluorescent dye show the sheet originating from the underlying liquid layer, not the drop liquid. The sheet undergoes a characteristic instability, bending out of its plane and hitting the bottom layer. For some impact conditions the sheet folds in on itself.

The velocity fields of a turbulent wake behind a flat plate obtained from the direct numerical simulations of Moser et al. (1998) are used to study the structure of the flow in the intermittent zone where there are, alternately, regions of fully turbulent flow and non-turbulent velocity fluctuations on either side of a thin randomly moving interface. Comparisons are made with a wake that is ‘forced’ by amplifying initial velocity fluctuations. A temperature field T, with constant values of 1.0 and 0 above and below the wake, is transported across the wake as a passive scalar. The value of the Reynolds number based on the centreplane mean velocity defect and half-width b of the wake is Re ≈ 2000.

The thickness of the continuous interface is about 0.07b, whereas the amplitude of fluctuations of the instantaneous interface displacement yI(t) is an order of magnitude larger, being about 0.5b. This explains why the mean statistics of vorticity in the intermittent zone can be calculated in terms of the probability distribution of yI and the instantaneous discontinuity in vorticity across the interface. When plotted as functions of y−yI the conditional mean velocity 〈U〉 and temperature 〈T〉 profiles show sharp jumps at the interface adjacent to a thick zone where 〈U〉 and 〈T〉 vary much more slowly.

Statistics for the conditional vorticity and velocity variances, available in such detail only from DNS data, show how streamwise and spanwise components of vorticity are generated by vortex stretching in the bulges of the interface. While mean Reynolds stresses (in the fixed reference frame) decrease gradually in the intermittent zone, conditional stresses are roughly constant and then decrease sharply towards zero at the interface. Flow fields around the interface, analysed in terms of the local streamline pattern, confirm and explain previous results that the advancement of the vortical interface into the irrotational flow is driven by large-scale eddy motion.

Terms used in one-point turbulence models are evaluated both conventionally and conditionally in the interface region, and the current practice in statistical models of approximating entrainment by a diffusion process is assessed.

We consider the statics of compound droplets made of two immiscible fluids on a rigid substrate, in the limit when gravity is dominated by capillarity. In particular, we show that the merging of four phases along a single contact line is a persistent and robust phenomenon from a mechanical and thermodynamic perspective; it can and does occur for a range of interfacial energies and droplet volumes. We give an interpretation for this in the context of the macroscopic Young–Laplace law and its microscopic counterpart due to van der Waals, and show that the topological transitions that result can be of either a continuous or discontinuous type depending on the interfacial energies in question.

This paper presents the force coefficients (added mass and damping) for a circular cylinder oscillating horizontally in a uniformly stratified fluid of limited depth and in a continuously stratified fluid with a smooth pycnocline. The frequency-dependent added mass and damping are evaluated from Fourier transforms of the experimental records of impulse response functions. The stratification is shown to have a strong effect on the fluid–body interaction. It is found that, when the characteristic vertical extent of stratification (depth of uniformly stratified fluid or pycnocline thickness) decreases, the power radiated with internal waves is reduced and the maximum of the frequency spectrum of wave power shifts toward lower frequency. The results of experiments are compared with available theoretical predictions.