We consider the problem of acoustic radiation generated by a spatial instability wave on a weakly developing shear flow. Assuming a local WKBJ approximation for the instability wave near its maximum, we compute the acoustic pressure field by using a Fourier transform along the streamwise direction. When the instability wave is close to transonic near its maximum amplitude, approximations for this pressure field are obtained by a steepest descent method. A branch cut and several saddle points are shown possibly to contribute to the approximation. A detailed analysis of these contributions is provided. The modifications of the acoustic field when we pass from subsonic to supersonic are examined. In particular, the superdirective character of the acoustic field of subsonic instability waves and the directivity pattern of supersonic waves are shown to be both compatible with our mathematical description and associated with a single saddle-point contribution.

The acoustic near field is also shown to possess a caustic around which a specific approximation is derived. In a large region of the physical space, the near field is composed of two saddle-point contributions. Close to the shear flow, one of these contributions degenerates into a branch-point contribution which always becomes dominant over the instability wave downstream of a location that is computed. An interesting phenomenon is observed in certain regions downstream of the maximum: the transverse behaviour of the instability wave has to be exponentially growing far from the shear layer to match the acoustic field. We demonstrate that this phenomenon neither requires a branch-point contribution nor a supersonic instability wave.

The problem of stability of a water-coated ice layer is investigated for a free-surface flow of a thin water film down an inclined plane. An asymptotic (double-deck) theory is developed for a flow with large Reynolds and Froude numbers which is then used to investigate linear two-dimensional, three-dimensional and nonlinear two-dimensional stability characteristics. A new mode of upstream-propagating instability arising from the interaction of the ice surface with the flow is discovered and its properties are investigated. In the linear limit, closed-form expressions for the dispersion relation and neutral curves are obtained for the case of Pr = 1. For the general case, the linear stability problem is solved numerically and the applicability of the solution with Pr = 1 is analysed. Nonlinear double-deck equations are solved with a novel global-marching-type scheme and the effects of nonlinearity are investigated. An explanation of the physical mechanism leading to the upstream propagation of instability waves is provided.

Two-dimensional variable-density Navier–Stokes simulations have been conducted in order to investigate the effects of a slope on the classical lock-exchange flow. Simulations of full lock releases show that the flow goes through an initial quasi-steady phase that is characterized by a constant front velocity. This quasi-steady front velocity has a maximum for slope angles around 40°, and it persists up to a dimensionless time of the order of 10. The flow subsequently undergoes a transition to a second phase with a larger, unsteady, front velocity. These computational findings were confirmed by experimental observations of lock-exchange flows in a tube of circular cross-section.

The reason for the observed transition from a quasi-steady front velocity to a larger, unsteady, value is found in the continuous acceleration of the stratified fluid layers connecting the two fronts by the streamwise component of the gravity vector. This acceleration leads to a situation where the fluid layers behind the current front move faster than the front itself. Initially the resulting addition of fluid to the current front from behind affects only the size of the front, while its velocity remains unchanged. Eventually, the current front is unable to absorb more fluid from behind and its velocity has to increase, thereby triggering the transition to the second, unsteady, phase. The transition time is determined as a function of the slope and the density ratio of the two fluids. For increasing density contrast, the transition is seen to occur earlier for the denser current.

Conceptually simple models based on the analysis by Thorpe (1968) are compared with simulation results for the flow in the region connecting the fronts. For the early stages of the flow a two-layer stratification model is found to be appropriate, while the later stages require a three-layer stratification model, owing to the intense mixing in the central part of the channel cross-section. These models are employed to estimate the time after which the accelerating stratified fluid layers will affect the velocities of the current fronts. They provide upper and lower estimates for the transition time which are in good agreement with the simulation results.

A Floquet stability analysis of the transition to three-dimensionality in the wake of a cylinder forced to oscillate transversely to the free stream has been undertaken. The effect of varying the oscillation amplitude is determined for a frequency of oscillation close to the natural shedding frequency. The three-dimensional modes that arise are identified, and the effect of the oscillation amplitude on their structure and growth rate quantified.

It is shown that when the two-dimensional wake is in the 2S configuration (which is similar to the Kármán vortex street), the three-dimensional modes that arise are similar in nature and symmetry structure to the modes in the wake of a fixed cylinder. These modes are known as modes A, B and QP and occur in this order with increasing Re. However, increasing the amplitude of oscillation causes the critical Reynolds number for mode A to increase significantly, to the point where mode B becomes critical before mode A. The critical wavelength for mode A is also affected by the oscillation, becoming smaller with increasing amplitude. Elliptic instability theory is shown also to predict this trend, providing further support that mode A primarily arises as a result of an elliptic instability.

At higher oscillation amplitudes, the spatio-temporal symmetry of the two-dimensional wake changes and it takes on the P + S configuration, with a pair of vortices on one side of the wake and a single vortex on the other side, for each oscillation cycle. With the onset of this configuration, modes A, B and QP cease to exist. It is shown that two new three-dimensional modes arise from this base flow, which we call modes SL and SS. Both of these modes are subharmonic, repeating over two base-flow periods. Also, either mode can be the first to become critical, depending on the amplitude of oscillation of the cylinder.

The emergence of these two new modes, as well as the reversal of the order of inception of the three-dimensional modes A and B, leads to the observation that for an oscillating cylinder wake there are four different modes that can lead the transition to three-dimensionality, depending on the amplitude of oscillation. Therefore this type of flow provides a good example for studying the effect of mode-order inception on the path taken to turbulence in bluff-body wakes.

For the range of amplitudes studied, the maximum Re value for which the flow remains two-dimensional is 280.

Large-eddy simulations of a cavity configuration yielding a mean flow that exhibits spanwise asymmetry are carried out. Results from the computations reveal that the asymmetry is due to a bifurcation of the whole flow field inside the cavity. It is demonstrated that the bifurcation originates in an inviscid confinement effect induced by the lateral walls. The branch of the bifurcation can be selected by slightly altering the incoming mean flow. Further investigations show that underlying steady spanwise modulations of velocity are amplified under the influence of the lateral walls. The modulation of the streamwise velocity component has the largest energy content and its dominant wavelength contaminates both vertical velocity and pressure. Complementary to these linear interactions, nonlinear energy transfers from streamwise velocity to pressure are also found. A transient analysis highlights the stiff transition from a symmetrical two-structure non-bifurcated flow to a stable unsymmetrical one-and-a-half-structure bifurcated flow. The switch to the bifurcated flow induces an alteration of the Rossiter aero–acoustic loop yielding a change in the dominant Rossiter mode and the appearance of a nonlinear harmonic of the first mode.

Experiments conducted on mixing across a stable density interface in a turbulent Taylor–Couette flow show, for the first time, experimental evidence of an increase in mixing efficiency at large Richardson numbers. With increasing buoyancy gradient the buoyancy flux first passes a maximum, then decreases and at large values of the buoyancy gradient the flux increases again. Thus, the curve of buoyancy flux versus buoyancy gradient tends to be N-shaped (rather than simply bell shaped), a behaviour suggested by the model of Balmforth et al. (J. Fluid Mech. vol. 428, 1998, p. 349). The increase in mixing efficiency at large Richardson numbers is attributed to a scale separation of the eddies active in mixing at the interface; when the buoyancy gradient is large mean kinetic energy is injected at scales much smaller than the eddy size fixed by the gap width, thus decreasing the eddy turnover time. Observations show that there is no noticeable change in interface thickness when the mixing efficiency increases; it is the mixing mechanism that changes. The curves of buoyancy flux versus buoyancy gradient also show a large variability for identical experimental conditions. These variations occur at time scales one to two orders of magnitude larger than the eddy turnover time scale.

High-resolution two- and three-dimensional numerical simulations are performed of first-mode internal gravity waves interacting with a shelf break in a linearly stratified fluid. The interaction of nonlinear incident waves with the shelf break results in the formation of upslope-surging vortex cores of dense fluid (referred to here as internal boluses) that propagate onto the shelf. This paper primarily focuses on understanding the dynamics of the interaction process with particular emphasis on the formation, structure and propagation of internal boluses onshelf. A possible mechanism is identified for the excitation of vortex cores that are lifted over the shelf break, from where (from the simplest viewpoint) they essentially propagate as gravity currents into a linearly stratified ambient fluid.

We study water-immersed granular avalanches in a long rectangular cell of small thickness. By video means, both the angle of the granular pile and the velocity profiles of the grains across the depth are recorded as a function of time. These measurements give access to the instantaneous granular flux. By inclining the pile at initial angles larger than the maximum angle of stability, avalanches are triggered and last for a long time, up to several hours for small grains, during which both the slope angle and the granular flux relax slowly. We show that the relaxation is quasi-steady so that there is no inertia: the relaxation at a given time is controlled only by the slope angle at that time. This allows us to adapt a frictional model developed recently for dry or water-immersed grains flowing in stationary conditions. This model succeeds well in reproducing our unsteady avalanche flows, namely the flowing layer thickness, the granular flux and the temporal relaxation of the slope. When a water counter-flow is applied along the pile, the granular avalanches are slowed down and behave as if granular friction were increased by an amount proportional to the water flow. All these findings are also reproduced well with the same friction model by taking into account the additional fluid force.

A rotating suspension of non-neutrally buoyant particles, confined by a horizontal cylinder, can be unstable to axial perturbations in concentration. A highly regular pattern of particle density and fluid flow then coexists in a non-equilibrium stationary state. The density profile along the cylinder axis is roughly sinusoidal, with a well-defined wavelength close to the cylinder diameter, and has a magnitude of approximately 30% of the average number density. We have used numerical simulations within the Stokes-flow approximation to investigate the mechanism underlying axial-band formation. Our results show that bands develop from an inhomogeneous particle distribution in the radial plane, which is itself driven by the competition between gravity and the viscous drag of the rotating fluid. We have discovered that the mean angular velocity of the particles is an order parameter which distinguishes between a low-frequency segregated phase and a high-frequency dispersed phase, where the particles fill the whole volume uniformly. The order parameter is a function of a single dimensionless frequency, with a characteristic length that is the mean interparticle separation. As the rotational frequency increases, the particle distribution becomes more homogeneous and the band structure disappears. Hydrodynamic diffusion stabilizes the suspension against centrifugal forces, allowing for a uniformly dispersed phase that can be used to grow three-dimensional cell cultures in an artificial microgravity environment.

The receptivity problem of plane Bingham–Poiseuille flow with respect to weak perturbations is addressed. The relevance of this study is highlighted by the linear stability analysis results (spectra and pseudospectra). The first part of the present paper thus deals with the classical normal-mode approach in which the resulting eigenvalue problem is solved using the Chebychev collocation method. Within the range of parameters considered, the Poiseuille flow of Bingham fluid is found to be linearly stable. The second part investigates the most amplified perturbations using the non-modal approach. At a very low Bingham number (B ≪ 1), the optimal disturbance consists of almost streamwise vortices, whereas at moderate or large B the optimal disturbance becomes oblique. The evolution of the obliqueness as function of B is determined. The linear analysis presented also indicates, as a first stage of a theoretical investigation, the principal challenges of a more complete nonlinear study.

A tube filled with a perfectly wetting liquid falls axially under its own weight. In its gravity-free reference frame, the liquid interface is deformed by surface tension into a hemispherical shape. On impact of the tube on a rigid floor, the interface curvature reverses violently, forming a concentrated jet. If the contact angle at the tube wall is such that the interface is flat, the liquid rebounds as a whole with the tube, with no deformation. We analyse this phenomenon using an impulse pressure description, providing an exact description of the initial liquid velocity field at the impact, supported by high-speed image velocimetry measurements. This initial dynamics is insensitive to liquid surface tension and viscosity.

This study attempts to explain the evolutionary pattern of a series of well-mixed layers separated by thin high-gradient interfaces frequently observed in stratified fluids. Such layered structures form as a result of the instability of the equilibrium with uniform stratification, and their subsequent evolution is characterized by a sequence of merging events which systematically increase the average layer thickness. The coarsening of layers can take one of two forms, depending on the realized vertical buoyancy flux law. Layers merge either when the high-gradient interfaces drift and collide, or when some interfaces gradually erode without moving vertically. The selection of a preferred pattern of coarsening is rationalized by the analytical theory – the merging theorem – which is based on linear stability analysis for a series of identical layers and strongly stratified interfaces. The merging theorem suggests that the merger by erosion of weak interfaces occurs when the vertical buoyancy flux decreases with the buoyancy variation across the step. If the buoyancy flux increases with step height, then coarsening of a staircase may result from drift and collision of the adjacent interfaces. Our model also quantifies the time scale of merging events and makes it possible to predict whether the layer merging continues indefinitely or whether the coarsening is ultimately arrested. The merging theorem is applied to extant one-dimensional models of turbulent mixing and successfully tested against the corresponding fully nonlinear numerical simulations. It is hypothesized that the upscale cascade of buoyancy variance associated with merging events may be one of the significant sources of the fine-scale (∼ 10m) variability in the ocean.

It is shown by direct numerical simulation that the preferential concentration of small heavy particles in homogeneous isotropic developed turbulence has a self-similar multi-scale nature when the particle relaxation time is within the inertial time scales of the turbulence. This is shown by the pair correlation function of the particle distribution extending over the entire inertial range, and the probability density function of the volumes of particle voids taking a power-law form. This self-similar multi-scale nature of particle clustering cannot be explained only by the centrifugal effect of the smallest-scale (i.e. the Kolmogorov scale) eddies, but also by the effect of co-existing self-similar multi-scale coherent eddies in the turbulence at high Reynolds numbers. This explanation implies that the preferential concentration of particles takes place even when the relaxation time of particles is much larger than the Kolmogorov time, provided it is smaller than the longest time scale of the turbulence, since even the largest-scale eddies bring about particle clustering.

The governing equations for the two-point correlations of the turbulent fluctuating velocity in the temporally evolving wake were analysed to determine whether they could have equilibrium similarity solutions. It was found that these equations could have such solutions for a finite-Reynolds-number wake, where the two-point velocity correlations could be written as a product of a time-dependent scale and a function dependent only on similarity variables. It is therefore possible to collapse the two-point measures of all the scales of motions in the temporally evolving wake using a single set of similarity variables. As in an earlier single-point analysis, it was found that the governing equations for the equilibrium similarity solutions could not be reduced to a form that was independent of a growth-rate dependent parameter. Thus, there is not a single ‘universal’ solution that describes the state of the large-scale structures, so that the large-scale structures in the far field may depend on how the flow is generated.

The predictions of the similarity analysis were compared to the data from two direct numerical simulations of the temporally evolving wakes examined previously. It was found that the two-point velocity spectra of these temporally evolving wakes collapsed reasonably well over the entire range of scales when they were scaled in the manner deduced from the equilibrium similarity analysis. Thus, actual flows do seem to evolve in a manner consistent with the equilibrium similarity solutions.

The governing equations for the two-point velocity correlations in the far field of the axisymmetric jet are examined and it is shown that these equations can have equilibrium similarity solutions for jets with finite Reynolds number that retain a dependence on the growth rate of the jet. The two-point velocity correlation can be written as the product of a scale that depends on the downstream position of the two points and a function that only depends on the similarity variables. Physically, this result implies that the turbulent processes producing and dissipating energy at the different scales of motion, as well as transferring energy between the different scales of motion, are in equilibrium as the flow evolves downstream. A particularly interesting prediction from the analysis is that the two-point similarity solutions depend only on the separation distance between the points in the streamwise similarity coordinate (i.e. υ = ξ′ − ξ), that is, the logarithm of the streamwise coordinate itself (i.e. ξ = ln x1, where x1 is measured from a virtual origin). Thus, the measures of the turbulence are homogeneous in the streamwise similarity coordinate.

The predictions from the similarity analysis for the streamwise two-point velocity correlation were compared with combined hot-wire and LDA measurements on the centreline of a round jet at a Reynolds number of 33000, and with two-point velocity correlations computed from PIV measurements in a round jet at a Reynolds number of 2000 performed by Fukushima et al. In both cases, the measured two-point velocity correlations in the streamwise direction collapsed when they were scaled in the manner predicted by the similarity analysis. The results provide further evidence that the equilibrium similarity hypothesis does describe the development of the flow in fully developed turbulent round jets and that the two-point correlations are statistically homogeneous in the streamwise similarity coordinate.

Slender-body asymptotic theory is applied to determine the electro-magneto-phoretic motion of a freely suspended elongated particle which is arbitrarily oriented relative to uniformly applied electric and magnetic fields.

The elliptic instability of a Batchelor vortex subject to a stationary strain field is considered by theoretical and numerical means in the regime of large Reynolds number and small axial flow. In the theory, the elliptic instability is described as a resonant coupling of two quasi-neutral normal modes (Kelvin modes) of the Batchelor vortex of azimuthal wavenumbers m and m + 2 with the underlying strain field. The growth rate associated with these resonances is computed for different values of the azimuthal wavenumbers as the axial flow parameter is varied. We demonstrate that the resonant Kelvin modes m = 1 and m = −1 which are the most unstable in the absence of axial flow become damped as the axial flow is increased. This is shown to be due to the appearance of a critical layer which damps one of the resonant Kelvin modes. However, the elliptic instability does not disappear. Other combinations of Kelvin modes m = −2 and m = 0, then m = −3 and m = −1 are shown to become progressively unstable for increasing axial flow. A complete instability diagram is obtained as a function of the axial flow parameter for several values of the strain rate and Reynolds number.

The numerical study considers a system of two counter-rotating Batchelor vortices in which the strain field felt by each vortex is due to the other vortex. The characteristics of the most unstable linear modes developing on the frozen base flow are computed by direct numerical simulations for two axial flow parameters and compared to the theory. In both cases, a very good agreement is obtained for the most unstable modes. Less unstable modes are also identified in the numerics and shown to correspond to peculiar resonances involving Kelvin modes from branches of different labels.

We investigate the evolution of a finite release of fluid into an infinite, two-dimensional, horizontal, porous slab saturated with a fluid of different density and viscosity. The vertical boundaries of the slab are impermeable and the released fluid spreads as a gravity current along a horizontal boundary. At early times the released fluid fills the entire height of the layer, and the governing equation admits a self-similar solution that is a function of the viscosity ratio between the two fluids. This early similarity solution describes a tilting interface with tips propagating as x ∝ t1/2. At late times the released fluid has spread along the boundary and the height of the current is much smaller than the thickness of the layer. The governing equation simplifies and admits a different similarity solution that is independent of the viscosity ratio. This late similarity solution describes a point release of fluid in a semi-infinite porous half-space, where the tip of the interface propagates as x ∝ t1/3. The same simplification of the governing equation occurs if the viscosity of the released fluid is much higher than the viscosity of the ambient fluid. We have obtained an expression for the time when the solution transitions from the early to the late self-similar regime. The transition time increases monotonically with increasing viscosity ratio. The transition period during which the solution is not self-similar also increases monotonically with increasing viscosity ratio, for mobility ratios larger than unity. Numerical computations describing the full evolution of the governing equation show good agreement with the theoretical results. Estimates of the spreading of injected fluids over long times are important for geological storage of CO2, and for the migration of pollutants in aquifers. In all cases it is important to be able to anticipate when the spreading regime transitions from x ∝ t1/2 to x ∝ t1/3.

A theoretical formulation is proposed for forced mass transport by pressure gradients in a liquid binary mixture around a spherical bubble undergoing volume oscillations in a sound field. Assuming the impermeability of the bubble wall to both species, diffusion driven by pressure gradients and classical Fick-diffusion must cancel at the bubble wall, so that an oscillatory concentration gradient arises in the vicinity of the bubble. The Péclet number pe is generally high in typical situations and Fick diffusion cannot restore equilibrium immediately, so that an asymptotic average concentration profile may progressively build up in the liquid over large times. Such a behaviour is reminiscent of the so-called rectified diffusion problem, leading to slow growth of a gas bubble oscillating in a sound field. A rigorous method formerly proposed by Fyrillas & Szeri (J. Fluid Mech. vol. 277, 1994, p. 381) to solve the latter problem is used to solve the present one. It is based on splitting the problem into a smooth part and an oscillatory part. The smooth part is solved by a multiple scales method and yields the slowly varying average concentration field everywhere in the liquid. The oscillatory part is obtained by matched asymptotic expansions in terms of the small parameter pe−1/2: the inner solution is required to satisfy the oscillatory balance between pressure diffusion and Fick diffusion at the bubble wall, while the outer solution is required to be zero. Matching both solutions yields a unique splitting of the problem. The final analytical solution, truncated to leading order, compares successfully to direct numerical simulation of the full convection–diffusion equation. The analytical expressions for both smooth and oscillatory parts are calculated for various sets of bubble parameters: driving pressure, frequency and ambient radius. The smooth problem always yields an average depletion of the heaviest species at the bubble wall, only noticeable for large molecules or nano-particles. For driving pressures sufficiently high to yield inertial oscillations of the bubble, the oscillatory problem predicts a periodic peak excess concentration of the heaviest species at the bubble wall at each collapse, lingering on several tens of the time of the characteristic duration of the bubble rebound. The two effects may compete for large molecules and practical implications of this segregation phenomenon are proposed for various processes involving acoustic cavitation.

A comprehensive study of the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. The effect of viscosity stratification, heat diffusivity and of buoyancy are estimated separately, with some unexpected results. From linear stability results, it has been accepted that heat diffusivity does not affect stability. However, we show that realistic Prandtl numbers cause a transient growth of disturbances that is an order of magnitude higher than at zero Prandtl number. Buoyancy, even at fairly low levels, gives rise to high levels of subcritical energy growth. Unusually for transient growth, both of these are spanwise-independent and not in the form of streamwise vortices. At moderate Grashof numbers, exponential growth dominates, with distinct Poiseuille–Rayleigh–Bénard and Tollmien–Schlichting modes for Grashof numbers up to ∼ 25 000, which merge thereafter. Wall heating has a converse effect on the secondary instability compared to the primary instability, destabilizing significantly when viscosity decreases towards the wall. It is hoped that the work will motivate experimental and numerical efforts to understand the role of wall heating in the control of channel and pipe flows.