Faraday waves are standing waves which arise through a parametric instability on the surface of a vertically oscillated fluid layer. They can emerge with various symmetries, simply square to $N$-fold rotationally symmetric, which for $N > 3$ are quasi-crystalline. In an experiment with a very large aspect ratio we determine the boundaries of the stability regions of waves with different rotational symmetries in the driving frequency–amplitude parameter plane. We find a remarkable agreement with a recent theory by Chen & Viñals (1999) who predict the stability boundaries at the onset amplitude. We argue why such agreement can only be observed in a very large experiment. The main nonlinear damping mechanism of the waves is a three-wave resonance. We devise a simple model that captures this mechanism and that can explain quantitatively the change of the symmetry of the waves with fluid depth. Detailed information about the surface is obtained by scanning the wave field and measuring the phase of subharmonic and harmonic components. Also the results of these measurements compare very favourably to the theoretical predictions.

Understanding the rates of cooling and solidification in laminar flows down sloping channels is central to predicting the advance of lava flows. The mechanisms involved include thermal convection and a competition between shear strain rate and the rate of formation of solid at the chilled surface of the flow. We report experiments in which polyethylene glycol wax flows in a laminar fashion down an inclined, open channel of rectangular cross-section under cold water. Two distinctly different flow regimes are recognized: ‘tube’ flow in which solidification of the flow surface creates a stationary roof while melt continues to flow through a relatively well-insulated ‘tube’ beneath, and a ‘mobile crust’ regime in which a solid surface crust develops only in the centre of the channel. In the latter regime the crust is carried down the channel, separated from the walls by crust-free shear regions in which cooling produces only dispersed fragments of solid owing to the effects of shearing. This flow structure is quasi-invariant over a large distance downstream. We show that thermal convection takes place in organized rolls that have axes aligned with the shear flow, and conclude that transition between the two flow regimes occurs at a critical value of the combined parameter $\vartheta \,{=}\, \psi (\hbox{\it Ra}{/}R_0)^{1/3}$, where $\psi \,{=}\, U_0 t_s{/}H_0$ is the ratio of a surface solidification timescale $t_{s}$ to a shearing timescale $H_0 {/} U_0 $, $H_{0}$ and $U_{0}$ are the flow depth and centreline surface velocity in the absence of solidification, {\it Ra} is a Rayleigh number and $R_{0}$ is a constant.

The transition to turbulence in the incompressible flow around a NACA0012 wing at high incidence is studied by DNS in the Reynolds number range 800–10000. Two main routes are identified for the two-dimensional transition mechanisms: that to aperiodicity beyond the von Kármán mode via a period-doubling scenario and the development of a shear-layer instability, forced by the fundamental oscillation of the separation point downstream of the leading edge. The evolution of the global parameters as well as the variation law of the shear-layer instability wavelength are quantified. The history of the three-dimensional transition mechanisms from a nominally two-dimensional flow structure is identified beyond the first bifurcation, as well as the preferred spanwise wavelengths.

Direct numerical simulations are performed to study the dynamics of an inhomogeneous stratified shear flow that models an atmospheric jet centred at the tropopause across which the density stratification is non-uniform. Small to moderate background stratifications are selected, and simulations are conducted for a range of Reynolds and Froude numbers. A spectral domain decomposition method that is particularly suitable for simulations of non-uniformly stratified shear flows is developed to simulate the desired turbulent jet, and quasi-equilibrium flow fields are obtained by long-time integration of governing equations. The structures of the mean flow and turbulence fields are calculated, which are interpreted using relevant length scales (Ozmidov, buoyancy, shear, Ellison) and Richardson number profiles. The ratios of the Ellison to buoyancy scales are much smaller than unity at the jet core and approach unity at the edges, confirming that mechanical turbulence prevails in the jet core, while nonlinear waves and stratification effects are dominating at the jet edges. The jet core is found to support sustained mechanical (active) turbulence, outside which lay a region of patchy turbulence and nonlinear gravity wave activity characterized by spatially decaying velocity fluctuations and strong temperature fluctuations. Detailed energy budgets show how energy is partitioned within the flow, including the transport of energy from the jet to its immediate vicinity by nonlinear gravity waves.

The effect of an insoluble surfactant on the gravity-driven flow of a liquid film down an inclined wall with periodic undulations or indentations is investigated in the limit of vanishing Reynolds number. A perturbation analysis for walls with small-amplitude sinusoidal corrugations reveals that the surfactant amplifies the deformation of the film surface, though it also renders the film thickness more uniform over the inclined surface. The effect of the surfactant is most significant when the film thickness is less than half the wall period. To explain the deforming influence of the surfactant, a linear stability analysis of film flow down an inclined plane is undertaken for two-dimensional perturbations. The results reveal the occurrence of a Marangoni normal mode whose rate of decay is lower than that of the single mode occurring in the absence of surfactants. Numerical methods based on a combined boundary-element/finite-volume method are implemented to compute flow down a periodic wall with large-amplitude corrugations or semi-circular depressions. In the case of a wavy wall, it is found that the shape of the film surface is described well by the linear perturbation expansion for small and moderate wave amplitudes. Streamline patterns reveal that, although the effect of the surfactant on the shape of the film surface is generally small, Marangoni tractions may have a profound influence on the kinematics by causing the onset of regions of recirculating flow.

Recent experimental observations using tracer visualization techniques have suggested that even for low Reynolds number flows, the steady vortex breakdown bubble in a torsionally driven cylinder is not axisymmetric and has an inflow/outflow asymmetry at its tail. These results seem to suggest that this type of vortex breakdown is closer in structure to that observed in the open vortex tube than previously thought. In this paper we examine numerically the effect of imperfections in the rig geometry, in particular quantifying the effect of a very slight misalignment between the axis of rotation of the end plate and the cylinder axis. It is shown that even slight misalignments, which are unlikely to be measurable experimentally, can produce the main details of non-axisymmetric open breakdown bubbles that have been observed in many experiments. This provides a resolution to the mismatch between the axisymmetric and closed bubble predictions of three-dimensional numerical simulations and stability analysis, and the observations of open bubbles often apparent even in carefully controlled experiments.

Both lighter- and hydrophobic heavier-than-liquid particles will float on liquid–air surfaces. Capillary forces cause the particles to cluster in typical situations identified here. This kind of clustering causes particles to segregate into islands and bands of high concentrations in thin liquid films rimming the inside of a slowly rotating cylinder partially filled with liquid. A second regime of particle segregation, driven by secondary motions induced by off-centre gas bubbles in a more rapidly rotating cylinder at higher filling levels, is identified. A third regime of segregation of bidisperse suspensions is found in which two layers of heavier-than-liquid particles that stratify when there is no rotation, segregate into alternate bands of particles when there is rotation.

A series of scale-model experiments investigated the scouring mechanisms associated with a tsunami impinging on a coastal cylindrical structure. Since scaling effects are significant in sediment transport, a large-scale sediment tank was used. Video images from inside the cylinder elucidated the vortex structures and the time development of scour around the cylinder. The scour development and mechanisms differed according to the sediment substrate – sand or gravel. For gravel, the most rapid scour coincided with the greatest flow velocities. On the other hand, for the sand substrate, the most rapid scour occurred at the end of drawdown – after flow velocities had subsided and shear stresses were presumed to have decreased. This behaviour can be explained in terms of pore pressure gradients. As the water level and velocity subside, the pressure on the sediment bed decreases, creating a vertical pressure gradient within the sand and decreasing the effective stress within the sand. Gravel is too porous to sustain this pressure gradient. During drawdown, the surface pressure decreases approximately linearly from a sustained peak at $\uDelta P$ to zero over time $\uDelta T$. The critical fraction $\Lambda $ of the buoyant weight of sediment supported by the pore pressure gradient can be estimated as \[ \Lambda = \frac{2}{\sqrt \pi} \frac{\uDelta P}{\gamma_b \sqrt {c_v \uDelta T}}, \] in which $\gamma_{b}$ is the buoyant specific weight of the saturated sediment and $c_{v}$ is the coefficient of consolidation. Much deeper scour was observed where $\Lambda $ exceeded one-half.

Active feedback control was applied to suppress oscillations in thermocapillary convection in a half-zone liquid bridge. The experiment is on a unit-aspect-ratio liquid bridge where the most unstable azimuthal mode has wavenumber 2 when control is absent. Active control was realized by locally modifying the surface temperature using the local temperature measured at different locations fed back through a simple control law. The performance of the control process was quantified by analysing local temperature signals, and the flow structure was simultaneously identified by flow visualization. With optimal placement of sensors and heaters, proportional control can raise the critical Marangoni number by more than 40%. The amplitude of the oscillation can be suppressed to less than 30% of the initial value for a wide range of Marangoni number, up to 90% of the critical value. The proportional control was tested for a period-doubling state and it stabilized the oscillation to a periodic state. Weakly nonlinear control was applied by adding a cubic term to the control law to improve the performance of the control and alter the bifurcation characteristics.

We investigate the three-dimensional instability of finite-amplitude standing surface waves under the influence of gravity. The analysis employs the transition matrix (TM) approach and uses a new high-order spectral element (HOSE) method for computation of the nonlinear wave dynamics. HOSE is an extension of the original high-order spectral method (HOS) wherein nonlinear wave–wave and wave–body interactions are retained up to high order in wave steepness. Instead of global basis functions in HOS, however, HOSE employs spectral elements to allow for complex free-surface geometries and surface-piercing bodies. Exponential convergence of HOS with respect to the total number of spectral modes (for a fixed number of elements) and interaction order is retained in HOSE. In this study, we use TM-HOSE to obtain the stability of general three-dimensional perturbations (on a two-dimensional surface) on two classes of standing waves: plane standing waves in a rectangular tank; and radial/azimuthal standing waves in a circular basin. For plane standing waves, we confirm the known result of two-dimensional side-bandlike instability. In addition, we find a novel three-dimensional instability for base flow of any amplitude. The dominant component of the unstable disturbance is an oblique (standing) wave oriented at an arbitrary angle whose frequency is close to the (nonlinear) frequency of the original standing wave. This finding is confirmed by direct long-time simulations using HOSE which show that the nonlinear evolution leads to classical Fermi–Pasta–Ulam recurrence. For the circular basin, we find that, beyond a threshold wave steepness, a standing wave (of nonlinear frequency $\Omega$) is unstable to three-dimensional perturbations. The unstable perturbation contains two dominant (standing-wave) components, the sum of whose frequencies is close to $2\Omega$. From the cases we consider, the critical wave steepness is found to generally decrease/increase with increasing radial/azimuthal mode number of the base standing wave. Finally, we show that the instability we find for both two- and three-dimensional standing waves is a result of third-order (quartet) resonance.

Internal gravity waves propagate away from a localized monochromatic disturbance inside beams, which develop around a St Andrew's Cross in two dimensions and around a double cone in three dimensions. The structure of the beams depends on three mechanisms, which couple together the different directions of propagation of the waves within the fluid. These mechanisms are associated with the size of the disturbance, the start-up of the motion and the viscosity of the fluid, respectively. The present paper considers each mechanism in isolation, for three-dimensional generation. The analysis is asymptotic and relies on far-field and large-time approximations. For each mechanism, three expressions of the waves are found: one, exact for an extended disturbance, that involves all the wavenumber vectors satisfying the dispersion relation; and two others, respectively uniform and non-uniform asymptotic expansions, that involve only the wavenumber vectors associated with group velocity vectors pointing toward the observer. For each mechanism, profiles of pressure and velocity are presented. A new time-independent characterization of the waves is introduced, in terms of the intensity or average energy flux; it is applied to the definition of the beam width. For an extended disturbance this width is a constant, the diameter of the disturbance. For an impulsive start-up the width increases linearly with the distance from the disturbance, and decreases in inverse proportion to the time elapsed since the start-up. For a viscous fluid the width increases as the one-third power of the distance. In all three cases, for a disturbance of multipolar order $2^n$, at constant strength, the power output varies in inverse proportion to the $(2n+1)$th power of the beam width.

A linear stability analysis of a shear flow in the presence of a continuous but steep variation of viscosity between two layers of nearly uniform viscosity is presented. This instability is investigated in relation to the known interfacial instability for the parallel flow of two superposed fluids of different viscosity. With respect to this configuration, the stability of our problem depends on two new parameters: the interface thickness $\delta$ and the Péclet number $\hbox{\it Pe}$, which accounts for diffusion effects when viscosity perturbations, coupled to the velocity perturbations, are allowed. We show that instability still exists for the continuous viscosity profile, provided the thickness of the interface is small enough and $\hbox{\it Pe}$ sufficiently large. Small and large wavenumbers are found to be stable, at variance with the discontinuous configuration. Of particular interest is also the possibility of obtaining higher growth rates than in the discontinuous case for suitable $\hbox{\it Pe}$ and $\delta$ ranges.

Random motions of irrotational gravity water surface waves on deep water are formulated using the so-called Wiener–Hermite functional series expansion, based on the ‘ideal random process’, i.e. the white noise. Such a procedure is known to differ fundamentally from moment expansions such as Gram–Charlier or Edgeworth series. The applications concern ‘free waves’ which are homogeneous in the horizontal plane and stationary in time. Starting from the basic hydrodynamic equation and boundary conditions, the general procedure for obtaining the equations for the deterministic kernels is described. First, the expansion is carried out with no approximation of the hydrodynamic equations but the expansion is limited to the first order. This defines the Gaussian part of the wave field. As expected, the nonlinearity of the hydrodynamic equations has effects on the dispersion relation through explicit frequency and acceleration terms whose physical interpretations are discussed. No attempt is made to solve the highly complicated coupled nonlinear integral kernels equations. Instead, Dirac kernel functions are chosen à priori as an approach to a narrowband random wave field. In this case, the nonlinearity is found to be characterized by a ‘statistical wave steepness’ having an upper limit value of order 0.42. As a second example, a non-Gaussian field is determined on the basis of the hydrodynamic equations truncated at second order in the wave amplitude. In the case of Dirac first-order kernels, the second-order nonlinear effect results in the generation of the second harmonic of the fundamental wave component. The ratio between the energy levels of these two components is found to compare well with standard results from laboratory experiments.

A flow-Mach-number-induced hysteresis phenomenon, in the shock-on-shock interaction of conical shock waves, is investigated numerically, by solving the Euler equations, using a W-modification of the non-stationary Godunov method with second-order accuracy both in space and time. The investigation reveals a multi-path hysteresis loop. It is shown that there are flow Mach number ranges in which three different wave configurations can be obtained for identical flow conditions. This study complements an earlier study by Ben-Dor et al. (2001) in which an angle-of-incidence-induced hysteresis was investigated both numerically and experimentally over a similar geometry. Based on the experimental findings of Ben-Dor et al.'s (2001) study, it is hypothesized that, in fact, four different wave configurations, three inviscid and one viscous, can be obtained for identical flow conditions. Since the geometry under investigation resembles supersonic intakes, this finding is relevant to their performance in supersonic/hypersonic flight.

The issue of the parameterization of small-scale (‘subgrid’) turbulence is addressed in the context of passive scalar transport. We focus on the Kraichnan advection model which lends itself to the analytical investigation of the closure problem. We derive systematically the dynamical equations which rule the evolution of the coarse-grained scalar field. At the lowest-order approximation in $l/r$, $l$ being the characteristic scale of the filter defining the coarse-grained scalar field and $r$ the inertial-range separation, we recover the classical eddy-diffusivity parameterization of small scales. At the next-leading order a dynamical closure is obtained. This outperforms the classical model and is therefore a natural candidate for subgrid modelling of scalar transport in generic turbulent flows.

At high Reynolds number, the flow of an incompressible viscous fluid over a lifting surface is a rich blend of fluid dynamic phenomena. Here, boundary layers formed at the leading edge develop over both the suction and pressure sides of the lifting surface, transition to turbulence, separate near the foil's trailing edge, combine in the near wake, and eventually form a turbulent far-field wake. The individual elements of this process have been the subject of much prior work. However, controlled experimental investigations of these flow phenomena and their interaction on a lifting surface at Reynolds numbers typical of heavy-lift aircraft wings or full-size ship propellers (chord-based Reynolds numbers, $Re_C {\sim} 10^7{-}10^8$) are largely unavilable. This paper presents results from an experimental effort to identify and measure the dominant features of the flow over a two-dimensional hydrofoil at nominal $Re_C$ values from near one million to more than 50 million. The experiments were conducted in the US Navy's William B. Morgan Large Cavitation Channel with a solid-bronze hydrofoil (2.1 m chord, 3.0 m span, 17 cm maximum thickness) at flow speeds from 0.25 to 18.3 m s$^{-1}$. The foil section, a modified NACA 16 with a pressure side that is nearly flat and a suction side that terminates in a blunt trailing-edge bevel, approximates the cross-section of a generic naval propeller blade. Time-averaged flow-field measurements drawn from laser-Doppler velocimetry, particle-imaging velocimetry, and static pressure taps were made for two trailing-edge bevel angles (44$ ^\circ$ and 56$ ^\circ$). These velocity and pressure measurements were concentrated in the trailing-edge and near-wake regions, but also include flow conditions upstream and far downstream of the foil, as well as static pressure distributions on the foil surface and test section walls. Observed Reynolds-number variations in the time-averaged flow over the foil are traced to changes in suction-side boundary-layer transition and separation. Observed Reynolds-number variations in the time-averaged near wake suggest significant changes occur in the dynamic flow in the range of $Re_C$ investigated.

In this study we consider the separated flow of an inviscid fluid around a moving flat plate. The motion of the plate, which is initially started from rest, is prescribed and unconstrained and we set ourselves the task of fully characterizing the resulting motion in the surrounding fluid. To do this we use a boundary integral representation for the complex-conjugate velocity field $\Phi(z,t)$ and require that the force and torque on the plate be determined as part of the solution. The flow solution is assumed to consist of a bound vortex sheet coincident with the plate and two free vortex sheets that emanate from each of the plate's two sharp edges. The time evolution of these vortex sheets is then considered in general. For physical reasons, the flow solution is required to satisfy the unsteady Kutta condition, which states that $\Phi(z,t)$ must be bounded everywhere, and the rigorous imposition of this condition then yields two types of additional constraint. The first governs the rate at which circulation is shed from the plate's edges and the second ensures that the free vortex sheets are shed tangentially. In fact, all the familiar flow characteristics associated with the imposition of the steady Kutta condition are rigorously shown to have exact parallels in the unsteady case. In addition, explicit expressions for the normal force and torque on the plate are derived. An asymptotic solution to the full system of evolution equations is developed for small times $t > 0$ and a simplified version of this solution is used as an initial condition for a desingularized numerical treatment of the full problem. A fast numerical algorithm is proposed and implemented and the results of several example calculations are presented. The featured examples are limited to high effective angles of attack due to the occurrence of a specific type of event that prevents further time-integration of the evolution equations using the current numerical method. The event corresponds physically to a situation in which a Lagrangian point placed at one of the plate's edges moves onto instead of away from the edge.