Experiments were undertaken to investigate dam-break flows where a finite volume of highly viscous fluid (glucose with viscosity μ ≈ 350 Pa s) maintained behind a lock gate was released into a horizontal or inclined flume. The resulting sequence of flow-depth profiles was tracked using a three-dimensional visualization system. In the low-Reynolds-number and high-capillary-number limits, analytical solutions can be obtained from the Navier–Stokes equations using lubrication theory and matched asymptotic expansions. At shallow slopes, similarity solutions can also be worked out. While the variation in the front position scaled with time as predicted by theory for both horizontal and sloping flumes, there was a systematic delay in the front position observed. Moreover, taking a closer look at the experimental flow-depth profiles shows that they were similar, but they noticeably deviated from the theoretical similarity form for horizontal planes. For sloping beds, the flow-depth profile is correctly predicted provided that different scalings are used at shallow and large slopes.

Horizontal cylinders floating on liquid surfaces are mundanely observed, whose examples include the legs of aquatic arthropods and floating larvae, twigs and hairs. We study the force and energy required to lift the cylinder clear from the water surface, to evaluate the role of wettability, especially superhydrophobicity, in the adhesion of floating cylinders. We find that a drastic degree of energy saving is achieved when lifting a superhydrophobic cylinder as compared with a cylinder with moderate wettability. This can serve as a starting point to understand how the superhydrophobicity of the legs of water-walking insects help to propel them efficiently.

A linear stability analysis shows that the jet in crossflow is characterized by self-sustained global oscillations for a jet-to-crossflow velocity ratio of 3. A fully three-dimensional unstable steady-state solution and its associated global eigenmodes are computed by direct numerical simulations and iterative eigenvalue routines. The steady flow, obtained by means of selective frequency damping, consists mainly of a (steady) counter-rotating vortex pair (CVP) in the far field and horseshoe-shaped vortices close to the wall. High-frequency unstable global eigenmodes associated with shear-layer instabilities on the CVP and low-frequency modes associated with shedding vortices in the wake of the jet are identified. Furthermore, different spanwise symmetries of the global modes are discussed. This work constitutes the first simulation-based global stability analysis of a fully three-dimensional base flow.

Review of three studies devoted to the available potential energy (APE) leads to the proposal of a diagnosis for APE, well-suited for rotating stratified flows within the primitive equations (PE) framework in which anharmonic effects (due to large vertical displacements of isopycnals) are permitted. The chosen diagnosis is based on the APE definition of Holliday & McIntyre (J. Fluid Mech., vol. 107, 1981, pp. 221–225) and uses the background stratification of Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128). Subsequent evaluation of the APE in a PE direct simulation (1/100°, 200 levels) of oceanic mesoscale turbulence indicates that anharmonic effects are significant. These effects are due to large vertical displacements of the isopycnals and the curvature of the background density profile.

A three-dimensional direct numerical simulation (DNS) is applied to flows inside and outside an evaporating spherical water droplet in air, and the effect of relative humidity on the heat transfer between the droplet and ambient air is investigated. The initial air temperature is set to be 15 K higher than the initial droplet temperature. The results show that the local evaporation heat loss indicates the maximum on the front of the droplet and decreases on going from the front to the rear, and the reduction on the rear becomes marked for high-droplet Reynolds numbers. This is because the evaporation rate is suppressed on the rear surface by the presence of flow separations behind the droplet. The droplet temperature decreases in the low-humidity condition, whereas it increases in the high-humidity condition. This difference is caused by the heat balance of evaporation heat loss and convective heat gain from ambient air at the droplet surface.

Numerous biomedical and industrial applications require separation or sorting of particles in systems in which it is undesirable to allow particle adhesion to a surface, such as a centrifuge wall and a filter fibre. Such systems typically involve either adhesive particles which could easily foul such surfaces or very delicate particles as is the case with suspensions of biological cells. The current study explores an approach for particle separation based on exposure to an oscillating straining flow, which would be typical for peristaltic and other types of contractive wall motions in a channel or tube. We find that particles immersed in an oscillating straining flow are attracted to the nodal points of the straining field, a phenomenon which we refer to as ‘oscillatory clustering’. A simplified theory of this process is developed for cases with isolated particles immersed in an unbounded uniform straining flow, in which the particle motion is found to be governed by a damped Mathieu equation. Moreover, the drift velocity imposed on particles through oscillatory clustering is sufficient to suspend them against a downward gravitational force in a limit-cycle oscillatory path. Theoretical approximations for the average suspension height and oscillation amplitude are obtained. A discrete-element method (DEM) for colliding and adhesive particles is then employed to examine oscillatory clustering for more realistic systems in which particles collide with each other and with container walls. The DEM is used to examine oscillatory clustering of a particle suspension in an oscillating box and for standing peristaltic waves in a channel, both with and without particle adhesion forces and gravitational forces.

We consider a small droplet of water sitting on top of a heated superhydrophobic surface. A toroidal convection pattern develops in which fluid is observed to rise along the surface of the spherical droplet and to accelerate downwards in the interior towards the liquid/solid contact point. The internal dynamics arise due to the presence of a vertical temperature gradient; this leads to a gradient in surface tension which in turn drives fluid away from the contact point along the interface. We develop a solution to this thermocapillary-driven Marangoni flow analytically in terms of streamfunctions. Quantitative comparisons between analytical and experimental results, as well as effective heat transfer coefficients, are presented.

Multidimensional instability of planar detonations, leading to cellular structures, is studied analytically near Chapman–Jouguet conditions, in the limit of small heat release, with small (Newtonian) differences between heat capacities, by using an expansion in a small parameter representing the ratio of the heat release to the thermal enthalpy of the fresh mixture. In this limit, the dynamics of detonations is governed by the interaction between the acoustic waves and the heat-release rate inside the inner detonation structure, the entropy–vorticity wave playing a negligible role at leading order. This situation is just opposite from that considered in our 1997 study of strongly overdriven detonations. The present analysis offers a step towards improving our understanding of the cellular structures of ordinary detonations, for which both the entropy–vorticity waves and the acoustic waves are involved in the instability mechanism. The relevant bifurcation parameter is identified, involving the degree of overdrive and the sensitivity of the rate of heat release to temperature at the Neumann state, and the onset of the instability is studied analytically for a realistic model of the inner structure of gaseous detonations.

Mixing of a passive scalar in the peripheral region close to a wall is investigated by means of accurate direct numerical simulations of both a three-dimensional Couette channel flow at low Reynolds numbers and a two-dimensional synthetic flow. In both cases, the resulting phenomenology can be understood in terms of the theory recently developed by Lebedev & Turitsyn (Phys. Rev. E, vol. 69, 2004, 036301). Our results prove the robustness of the identified mechanisms responsible for the persistency of scalar concentration close to the wall with important consequences in completely different fields ranging from microfluidic applications to environmental dispersion modelling.

The equilibrium position of a spherical air bubble in a solid body rotating flow around a horizontal axis is investigated experimentally. The flow without bubbles is checked to be solid body rotating. The area of influence of the bubble is characterized to determine for each bubble whether the incoming flow is perturbed or not. The demineralized water used is shown to Tbe contaminated, and spinning of the bubble's interface is observed and measured. From the measurement of the bubble's equilibrium position, drag and lift coefficients are determined. They appear to be dependent on two dimensionless numbers. Eo the Eötvös number and Rω the rotational Reynolds number (or Taylor number Ta) can be varied independently by changing the control parameters, and for that reason are the convenient choice for experiments. (Re, Ro) with Ro the Rossby number is an equivalent choice generally adopted in the literature for numerical simulations, and Re denotes the Reynolds number. When using this second representation, the Ro number appears to be an indicator of the influence on the force coefficients of the shear, of the curvature of the streamlines of the flow and of the bubble's spinning. The bubble's spinning effect on the lift force is far from trivial. Its contribution explains the important gap between lift values for a bubble (not spinning) in a clean fluid and for a bubble (spinning) in a contaminated fluid as present.

Binder & Vanden-Broeck (2005) showed there are no subcritical or critical solutions satisfying the radiation condition for steady flows past a flat plate. By using a weakly nonlinear analysis, it is shown that such flows exist for a curved plate. Fully nonlinear solutions are computed by a boundary integral equation method, and new nonlinear solutions for supercritical and generalized critical flows past a curved plate are presented.

We investigate, via perturbation analyses, the mechanisms of nonlinear resonant interaction of surface-interfacial waves with a rippled bottom in a two-layer density-stratified fluid. As in a one-layer fluid, three classes of Bragg resonances are found to exist if nonlinear interactions up to the third order in the wave/ripple steepness are considered. As expected, the wave system associated with the resonances is more complicated than that in a one-layer fluid. Depending on the specifics of the resonance condition, the resonance-generated wave may be a surface or internal mode and may be transmitted or reflected. At the second order, class I Bragg resonance occurs involving two surface and/or internal waves and one bottom-ripple component. The interaction of an incident surface/internal wave with the bottom ripple generates a new surface or internal wave that may propagate in the same or the opposite direction as the incident wave. At the third order, class II and III Bragg resonances occur involving resonant interactions of four wave/ripple components: two surface and/or internal waves and two bottom-ripple components for class II resonance; three surface and/or internal waves and one bottom-ripple components for class III resonance. As in class I resonance, the resonance-generated wave in class II resonance has the same frequency as that of the incident wave. For class III resonance, the frequency of the resonant wave is equal to the sum or difference of the two incident wave frequencies. We enumerate and represent, using Feynman-like diagrams, the possible cases and combinations for Bragg resonance up to the third order (in two dimensions). Analytical regular perturbation results are obtained and discussed for all three classes of Bragg resonances. These are valid for limited bottom patch lengths and initial/finite growth of the resonant waves. For long bottom patches, a uniformly valid solution using multiple scales is derived for class I resonance. A number of applications underscoring the importance and implication of these nonlinear resonances on the evolution of ocean waves are presented and discussed. For example, it is shown that three internal/surface waves co-propagating over bottom topography are resonant under a broad range of Bragg conditions. The present study provides the theoretical basis and understanding for the companion paper (Alam, Liu & Yue 2008), where a direct numerical solution for the general nonlinear problem is pursued.

We develop a direct numerical method to study the general problem of nonlinear interactions of surface/interfacial waves with variable bottom topography in a two-layer density stratified fluid. We extend a powerful high-order spectral (HOS) method for nonlinear gravity wave dynamics in a homogeneous fluid to the case of a two-layer fluid over non-uniform bottom. The method is capable of capturing the nonlinear interactions among large number of surface/interfacial wave mode and bottom ripple components up to an arbitrary high order. The method preserves exponential convergence with respect to the number of modes of the original HOS and the (approximately) linear effort with respect to mode number and interaction order. The method is validated through systematic convergence tests and comparison to a semi-analytic solution we obtain for an exact nonlinear Stokes waves on a two-layer fluid (in uniform depth). We apply the numerical method to the three classes of generalized Bragg resonances studied in Alam, Liu & Yue (J. Fluid Mech., vol. 624, 2009, p. 225), and compare the perturbation predictions obtained there with the direct simulation results. An important finding is possibly the important effect of even higher-order nonlinear interactions not accounted for in the leading-order perturbation analyses. To illustrate the efficacy of the numerical method to the general problem, we consider a somewhat more complicated case involving two incident waves and three bottom ripple components with wavenumbers that lead to the possibility of multiple Bragg resonances. It is shown that the ensuing multiple (near) resonant interactions result in the generation of multiple new transmitted/reflected waves that fill a broad wavenumber band eventually leading to the loss of order and chaotic motion.

The growth, oscillation and collapse of vortex cavitation bubbles are examined using both two- and three-dimensional numerical models. As the bubble changes volume within the core of the vortex, the vorticity distribution of the surrounding flow is modified, which then changes the pressures at the bubble interface. This interaction can be complex. In the case of cylindrical cavitation bubbles, the bubble radius will oscillate as the bubble grows or collapses. The period of this oscillation is of the order of the vortex time scale, τV = 2πrc/uθ, max, where rc is the vortex core radius and uθ, max is its maximum tangential velocity. However, the period, oscillation amplitude and final bubble radius are sensitive to variations in the vortex properties and the rate and magnitude of the pressure reduction or increase. The growth and collapse of three-dimensional bubbles are reminiscent of the two-dimensional bubble dynamics. But, the axial and radial growth of the vortex bubbles are often strongly coupled, especially near the axial extents of the bubble. As an initially spherical nucleus grows into an elongated bubble, it may take on complex shapes and have volume oscillations that also scale with τV. Axial flow produced at the ends of the bubble can produce local pinching and fission of the elongated bubble. Again, small changes in flow parameters can result in substantial changes to the detailed volume history of the bubbles.

The effects of free-surface proximity on the flow field around tidal stream turbines are modelled using actuator disc theory. Theoretical results are presented for a blocked configuration of tidal stream turbines such as a linear array that account for the proximity of the free surface and the seabed. The theoretical results are compared to open channel flow experimental results in which the flow field has been simulated using a porous disc and strip. These results are complemented by more detailed measurements of the performance of a model horizontal-axis turbine carried out in a water flume and a wind tunnel. The two sets of experiments represent highly blocked and effectively unblocked cases, respectively. The theoretical model of the effects of free-surface proximity provides a blockage correction for axial induction that can be incorporated in blade element momentum codes. The performance predictions obtained with such a code are in good agreement with the experimental results for CP and CT at low tip-speed ratios. The agreement weakens with increasing tip-speed ratio, as the wake of turbine enters a reversed flow state. A correction following the philosophy of Maskell is applied to CT in this region, which provides a better agreement.

A small-deformation perturbation analysis is developed to study the effect of surfactant on drop dynamics in viscous flows. The surfactant is assumed to be insoluble in the bulk-phase fluids; the viscosity ratio and surfactant elasticity parameters are arbitrary. Under small-deformation conditions, the drop dynamics are described by a system of ordinary differential equations; the governing equations are given explicitly for the case of axisymmetric and two-dimensional imposed flows. Analytical results accurate to third order in the flow-strength parameter (capillary number) are derived (i) for the stationary drop shape and surfactant distribution in simple shear and axisymmetric straining flows, and (ii) for the rheology of a dilute emulsion in shear flow which include a shear-thinning viscosity and non-zero normal stresses. For drops with clean interfaces, the small-deformation theory presented here improves the results of Barthès-Biesel & Acrivos (J. Fluid Mech., vol. 61, 1973, p. 1). Boundary integral simulations are used to test our theory and explore large-deformation conditions.

In this paper we take up the question of the spectral stability of travelling water waves from a new point of view, namely that the spectral data of the water-wave operator linearized about fully nonlinear Stokes waves is analytic as a function of a height parameter. This observation was recently made rigorous by the author using a boundary perturbation approach which is amenable to approximation by a stable high-order numerical method. Using this algorithm, we investigate, for both super- and sub-harmonic disturbances, the evolution of the spectrum, in particular the ‘first collision’ of eigenvalues and the ‘smallest singularity’ in the perturbation expansion. The former is studied in response to MacKay & Saffman's (1986) work on the water-wave problem which demonstrated that instability can only arise after the collision of two eigenvalues of opposite Krein signature. However, we present results which show, quite explicitly, that eigenvalue collision (even of opposite Krein signature) is insufficient to conclude instability. With this in mind, we have identified a new criterion for the loss of spectral stability, namely the appearance of a singularity in the expansion of the spectral data (as a function of the height parameter mentioned above). We give some heuristic reasons why this should be so, and then provide complete numerical spectral stability results for four representative depths, two above (h = ∞, 2) and two below (h = 1, 1/2) Benjamin's (1967) critical value, hc ≈ 1.363, above which the Benjamin–Feir instability emerges. We find that the strongest (two-dimensional) instability appears to be among the long waves, but we notice that there is a sharp difference between ‘shallow-water’ and ‘deep-water’ waves in that first eigenvalue collision and smallest expansion singularity are synonymous for shallow water, while this is not so in deep water where ‘windows of stability’ beyond the first eigenvalue collision exist.

We examine the collective dynamics of polarizable, Brownian, sedimenting rods of high aspect ratio. Previous work of Koch and Shaqfeh (J. Fluids Mech., vol. 209, 1989 pp. 521–542) has shown that in the absence of Brownian motion, sedimenting suspensions of rods are unstable to concentration fluctuations and form dense streamers via interparticle hydrodynamic interactions. Recently, Saintillan, Shaqfeh & Darve (Phys. Fluids, vol. 18 (121701), 2006b p. 1) demonstrated that electric fields can act to stabilize these non-Brownian suspensions of polarizable rods through induced-charge electrokinetic rotation, which forces particle alignment. In this paper, we employ a mean-field linear stability analysis as well as Brownian dynamics simulations to study the effect of thermal motion on the onset of instability. We find that in the absence of electric fields, Brownian motion consistently suppresses instability formation through randomization of particle orientation. However, when electric fields are applied, thermal motion can act to induce instability by counteracting the stabilizing effect of induced-charge orientation.

This work explores fractal geometrical properties of scalar turbulent interfaces derived from experimental two-dimensional spatial images of the scalar field in separated shear layers at large Reynolds numbers. The resolution of the data captures the upper three decades of scales enabling examination of multiscale geometrical properties ranging from the largest energy-containing scales to inertial scales. The data show a −5/3 spectral exponent over a wide range of scales corresponding to the inertial range in fully developed turbulent flows. For the fractal aspects, we utilize two methods as it is known that different methods may lead to different fractal aspects. We use the recently developed method for fractal analysis known as the Multiscale-Minima Meshless (M3) method because it does not require the use of grids. We also use the conventional box-counting approach as it has been frequently employed in various past studies. The outer scalar interfaces are identified on the basis of the probability density function (p.d.f.) of the scalar field. For the outer interfaces, the M3 method shows strong scale dependence of the generalized fractal dimension with approximately linear variation of the dimension as a function of logarithmic scale, for interface-fitting reference areas, but there is evidence of a plateau near a dimension D ~ 1.3 for larger reference areas. The conventional box-counting approach shows evidence of a plateau with a constant dimension also of D ~ 1.3, for the same reference areas. In both methods, the observed plateau dimension value agrees with other studies in different flow geometries. Scalar threshold effects are also examined and show that the internal scalar interfaces exhibit qualitatively similar behaviour to the outer interfaces. The overall range of box-counting fractal dimension values exhibited by outer and internal interfaces is D ~ 1.2–1.4. The present findings show that the fractal aspects of scalar interfaces in separated shear layers at large Reynolds number with −5/3 spectral behaviour can depend on the method used for evaluating the dimension and on the reference area. These findings as well as the utilities and distinctions of these two different definitions of the dimension are discussed in the context of multiscale modelling of mixing and the interfacial geometry.