The interaction of tidal currents with sea-floor topography results in the radiation of internal gravity waves into the ocean interior. These waves are called internal tides and their dissipation due to nonlinear wave breaking and concomitant three-dimensional turbulence could play an important role in the mixing of the abyssal ocean, and hence in controlling the large-scale ocean circulation.

As part of on-going work aimed at providing a theory for the vertical distribution of wave breaking over sea-floor topography, in this paper we investigate the instability of internal tides in a very simple linear model that helps us to relate the formation of unstable regions to simple features in the sea-floor topography. For two-dimensional tides over one-dimensional topography we find that the formation of overturning instabilities is closely linked to the singularities in the topography shape and that it is possible to have stable waves at the sea floor and unstable waves in the ocean interior above.

For three-dimensional tides over two-dimensional topography there is in addition an effect of geometric focusing of wave energy into localized regions of high wave amplitude, and we investigate this focusing effect in simple examples. Overall, we find that the distribution of unstable wave breaking regions can be highly non-uniform even for very simple idealized topography shapes.

We develop a model for the dynamics of a reactive gravity-driven flow in a porous layer of finite depth, accounting for the change in permeability and density across the dissolution front. We identify that the two controlling parameters are the mobility ratio across the reaction front and the ratio of the buoyancy-driven flow to the fluid injection rate. We present some numerical solutions for the evolution of a two-dimensional dissolution front, and develop an approximate analytic solution for the limit of large injection rate compared to the buoyancy-driven flow. The model predictions are compared with some new analogue laboratory experiments in which fresh water displaces a saturated aqueous solution initially confined within a two-dimensional reactive permeable matrix composed of salt powder and glass ballotini. We also present self-similar solutions for an axisymmetric gravity-driven reactive current moving through a porous layer of finite depth. The solutions illustrate how the reaction front becomes progressively wider as the ratio of the buoyancy-driven flow to the injection rate increases, and also as the mobility contrast across the front increases.

In the present study, flow simulation has been carried out in a backward-facing step channel defined by an expansion ratio of 2.02 and a spanwise aspect ratio of 8 to provide the physical insight into the longitudinal and spanwise flow motions and to identify the presence of Taylor–Görtler-like vortices. The Reynolds numbers have been taken as 1000 and 2000, which fall in the category of transitional flow. The present simulated results were validated against the experimental and numerical data and the comparison was found to be satisfactory. The simulated results show that the flow becomes unsteady and exhibits a three-dimensional nature with the Kelvin–Helmholtz instability oscillations and Taylor–Görtler-Like longitudinal vortices. The simulated data were analysed to give an in-depth knowledge of the complex interactions among the floor and roof eddies, and the spiralling spanwise flow motion. Destabilization of the present incompressible flow system, with the amplified Reynolds number due to the Kelvin–Helmholtz and Taylor–Görtler instabilities, is also highlighted. A movie is available with the online version of the paper.

We consider the linear temporal stability of a Couette flow of a Maxwell gas within the gap between a rotating inner cylinder and a concentric stationary outer cylinder both maintained at the same temperature. The neutral curve is obtained for arbitrary Mach (Ma) and arbitrarily small Knudsen (Kn) numbers by use of a ‘slip-flow’ continuum model and is verified via comparison to direct simulation Monte Carlo results. At subsonic rotation speeds we find, for the radial ratios considered here, that the neutral curve nearly coincides with the constant-Reynolds-number curve pertaining to the critical value for the onset of instability in the corresponding incompressible-flow problem. With increasing Mach number, transition is deferred to larger Reynolds numbers. It is remarkable that for a fixed Reynolds number, instability is always eventually suppressed beyond some supersonic rotation speed. To clarify this we examine the variation with increasing (Ma) of the reference Couette flow and analyse the narrow-gap limit of the compressible TC problem. The results of these suggest that, as in the incompressible problem, the onset of instability at supersonic speeds is still essentially determined through the balance of inertial and viscous-dissipative effects. Suppression of instability is brought about by increased rates of dissipation associated with the elevated bulk-fluid temperatures occurring at supersonic speeds. A useful approximation is obtained for the neutral curve throughout the entire range of Mach numbers by an adaptation of the familiar incompressible stability criteria with the critical Reynolds (or Taylor) numbers now based on average fluid properties. The narrow-gap analysis further indicates that the resulting approximate neutral curve obtained in the (Ma, Kn) plane consists of two branches: (i) the subsonic part corresponding to a constant ratio (Ma/Kn) (i.e. a constant critical Reynolds number) and (ii) a supersonic branch which at large Ma values corresponds to a constant product Ma Kn. Finally, our analysis helps to resolve some conflicting views in the literature regarding apparently destabilizing compressibility effects.

Breakup of electrified jets is important in applications as diverse as electrospraying, electroseparations and electrospray mass spectrometry. Breakup of a perfectly conducting, incompressible Newtonian liquid jet surrounded by a passive insulating gas that is stressed by a radial electric field is studied by a temporal analysis. An initially quiescent jet is subjected to axially periodic shape perturbations and the ensuing dynamics are followed numerically until pinch-off by both a three-dimensional but axisymmetric (two-dimensional) and a one-dimensional slender-jet algorithm. Results computed with these algorithms are verified against predictions from linear theory for short times. Breakup times, ratios of the sizes of the primary to satellite drops formed at pinch-off, and the Coulombic stability of these drops are reported over a wide range of electrical Bond numbers, NE (ratio of electric to surface tension force), Ohnesorge numbers, NOh (ratio of viscous to surface tension force), and disturbance wavenumbers, k. Effect of surface charge on interface overturning is investigated. Furthermore, the influence of electrostatic stresses on the dynamics of pinch-off and the mechanisms of satellite drop formation is also addressed.

Drop impact on a water surface can be followed by underwater sounds originating not at the drop impact but when the entrained bubbles oscillate. Although the sound mechanism in the regular bubble entrainment region is well-known, there is less knowledge on the impact phenomena in the irregular bubble entrainment region where various situations can exist, such as many types of bubble formation or even no bubble generation under some conditions. In the present study, the aim is to clarify the dynamics of the water surface after the impact of a primary drop, mainly with diameter 5.2, 5.7 and 6.2mm, each of which is accompanied by a single satellite drop. Special attention was paid to the breakup behaviour of the water surface for Froude number Fr < 300. It was found that three underwater sounds were generated for a single drop impact, besides the sound due to impact itself. The first two were audible to the human ear, but the third one was almost inaudible. The first underwater sound resulted from the oscillation of a single air bubble formed as a result of the satellite drop impact on the bottom of the contracting cavity, and the second sound was due to the oscillation of air bubbles generated during the collapse of the water column. The formation of these air bubbles strongly depends on the Froude number, Weber number (or Bond number) and the aspect ratio of the drop at impact, although involving probability characteristics. Furthermore it is suggested that an air bubble entrapped in a water column plays an important role in increasing the probability of contact between the column surface and the curved free surface. A Japanese Suikinkutsu was introduced as an application of drop-impact-induced sounds. Using an open-type Suikinkutsu an additional experiment was carried out with larger drops with average diameters of 6.2, 7.2 and 7.8, mm.

A direct numerical simulation of turbulent flow in a straight square duct was performed in order to determine the minimal requirements for self-sustaining turbulence. It was found that turbulence can be maintained for values of the bulk Reynolds number above approximately 1100, corresponding to a friction-velocity-based Reynolds number of 80. The minimum value for the streamwise period of the computational domain is around 190 wall units, roughly independently of the Reynolds number. We present a characterization of the flow state at marginal Reynolds numbers which substantially differs from the fully turbulent one: the marginal state exhibits a four-vortex secondary flow structure alternating in time whereas the fully turbulent one presents the usual eight-vortex pattern. It is shown that in the regime of marginal Reynolds numbers buffer-layer coherent structures play a crucial role in the appearance of secondary flow of Prandtl's second kind.

This paper presents the application of feedback control to spatially developing boundary layers. It is the natural follow-up of Högberg & Henningson (J. Fluid Mech. vol. 470, 2002, p. 151), where exact knowledge of the entire flow state was assumed for the control. We apply recent developments in stochastic models for the external sources of disturbances that allow the efficient use of several wall measurements for estimation of the flow evolution: the two components of the skin friction and the pressure fluctuation at the wall. Perturbations to base flow profiles of the family of Falkner–Skan–Cooke boundary layers are estimated by use of wall measurements. The estimated state is in turn fed back for control in order to reduce the kinetic energy of the perturbations. The control actuation is achieved by means of unsteady blowing and suction at the wall. Flow perturbations are generated in the upstream region in the computational box and propagate in the boundary layer. Measurements are extracted downstream over a thin strip, followed by a second thin strip where the actuation is performed. It is shown that flow disturbances can be efficiently estimated and controlled in spatially evolving boundary layers for a wide range of base flows and disturbances.

The stability of pipe flow when mildly deviating from the developed Poiseuille profile by a non-axisymmetric azimuthally periodic distortion is examined. The motivation for this is to consider deviations, the origin of which may be attributed to small-amplitude disturbances having sinusoidal periodicity along the azimuthal coordinate, which are known to be the ones most amplified by the transient growth linear mechanism. A mathematical technique for finding the minimum energy density of azimuthally periodic deviations triggering exponential instability is presented. The results show that owing to bifurcations multiple solutions of optimal deviations exist. As the Reynolds number is increased additional bifurcations appear and create more distinct solutions. The different solutions correspond to different radial distributions of the deviations, and at Reynolds numbers of about 2000 they are distributed over less than a half of the pipe radius. It is found that the dependence of the optimal deviation velocity leading to instability on the Reynolds number Re is approximately 20/Re. A comparison to axisymmetric base-flow deviations shows that the minimum energy required for an azimuthally periodic deviation to trigger instability is almost twice that for the axisymmetric flow. However, azimuthally periodic deviations, which are shown to have a streaky pattern, may have a role in the self-sustaining process. They may be formed as a result of a transient growth amplification of initial streamwise rolls and can produce, via self-interactions between the resulting growing waves, patterns of streamwise rolls as well.

We consider free convection driven by a heated vertical plate immersed in a nonlinearly stratified medium. The plate supplies a uniform horizontal heat flux to a fluid, the bulk of which has a stable stratification, characterized by a non-uniform vertical temperature gradient. This gradient is assumed to have a typical length scale of variation, denoted Z0, while Ψ0 is the order of magnitude of the related heat flux that crosses the medium vertically.

We derive an analytic solution to the Boussinesq equations that extends the classical solution of Prandtl to the case of nonlinearly stratified media. This novel solution is asymptotically valid in the regime RaS ≫ 1, where RaS denotes the Rayleigh number of nonlinear stratification, based on Z0, Ψ0, and the physical properties of the medium.

We then apply the new theory to the natural convection affecting the vapour phase in a liquefied pure gas tank (e.g. the cryogenic storage of hydrogen). It is assumed that the cylindrical storage tank is subject to a constant uniform heat flux on its lateral and top walls. We are interested in the vapour motion above a residual layer of liquid in equilibrium with the vapour. High-precision axisymmetric numerical computations show that the flow remains steady for a large range of parameters, and that a bulk stratification characterized by a quadratic temperature profile is undoubtedly present. The application of the theory permits a comparison of the numerical and analytic results, showing that the theory satisfactorily predicts the primary dynamical and thermal properties of the storage tank.

There is an upper bound to the amount of power that can be generated by turbines in tidal channels as too many turbines merely block the flow. One condition for achievement of the upper bound is that the turbines are deployed uniformly across the channel, with all the flow through them, but this may interfere with other uses of the channel. An isolated turbine is more effective in a channel than in an unbounded flow, but the current downstream is non-uniform between the wake of the turbines and the free stream. Hence some energy is lost when these streams merge, as may occur in a long channel. We show here, for ideal turbine models, that the fractional power loss increases from 1/3 to 2/3 as the fraction of the channel cross-section spanned by the turbines increases from 0 to close to 1. In another scenario, possibly appropriate for a short channel, the speed of the free stream outside the turbine wake is controlled by separation at the channel exit. In this case, the maximum power obtainable is slightly less than proportional to the fraction of the channel cross-section occupied by turbines.

The geometric properties are quantified for concentration iso-surfaces of a high-Schmidt-number passive scalar field produced by an iso-kinetic source with an initial finite characteristic length scale released into the inertial layer of fully developed open-channel-flow turbulent boundary layers. The coverage dimension and other measures of two-dimensional transects of the passive scalar iso-surfaces are found to be scale dependent. The coverage dimension is around 1.0 at the order of the Batchelor length scale and based on our data increases in a universal manner to reach a local maximum at a length scale around the Kolmogorov scale. We introduce a new parameter called the coverage length underestimate, which demonstrates universal behaviour in the viscous–convective regime for these data and hence is a potentially useful practical tool for many mixing applications. At larger scales (in the inertial–convective regime), the fractal geometry measures are dependent on the Reynolds number, injection length scale, and concentration threshold of the iso-surfaces. Finally, the lacunarity of the iso-surface structure shows that the instantaneous scalar field is most inhomogenous around the length scale corresponding to the Kolmogorov scale.

Nonlinear mechanisms leading to the ejection of dipolar vortices from a laminar wall jet are being investigated using highly accurate Navier–Stokes simulations. With a set of well-defined numerical experiments for a forced Glauert wall jet, the nonlinear resonant interaction between the large-amplitude harmonic disturbance and a small-amplitude wave packet is systematically explored using two-dimensional simulations. Generated by a small-amplitude pulse, the wave packet experiences rapid resonant growth in the subharmonic part of its spectrum resulting in vortex mergings and, ultimately, the ejection of a pair of counter-rotating vortices from the wall jet. This two-dimensional subharmonic instability, if not mitigated by competing three-dimensional instabilities, can lead to the detachment of the entire wall jet from the surface. As shown using three-dimensional direct numerical simulations, vortex ejection still occurs in a forced transitional wall jet if the two-dimensional wave packet can reach a large amplitude level upstream of the region of three-dimensional turbulent breakdown. Movies are available with the online version of the paper.

The flow near the trailing edge of a horizontal plate in a uniform parallel stream under a small angle of attack in the limit of large Reynolds number and large Grashof number is considered. Applying the concept of interacting boundary layers, a triple-deck problem taking the hydrostatic pressure perturbation into account can be formulated. However, it turns out that the interaction pressure is discontinuous at the trailing edge and thus new sublayers to resolve the discontinuity are introduced.

Linear internal waves in inviscid bounded fluids generally give a mathematically ill-posed problem since hyperbolic equations are combined with elliptic boundary conditions. Such problems are difficult to solve. Two new approaches are added to the existing methods: the first solves the two-dimensional spatial wave equation by iteratively adjusting Cauchy data such that boundary conditions are satisfied along a predefined boundary. After specifying the data in this way, solutions can be computed using the d'Alembert formula.

The second new approach can numerically solve a wider class of two dimensional linear hyperbolic boundary value problems by using a ‘boundary collocation’ technique. This method gives solutions in the form of a partial sum of analytic functions that are, from a practical point of view, more easy to handle than solutions obtained from characteristics. Collocation points have to be prescribed along certain segments of the boundary but also in the so-called fundamental intervals, regions along the boundary where Cauchy data can be given arbitrarily without over-or under-determining the problem. Three prototypical hyperbolic boundary value problems are solved with this method: the Poincaré, the Telegraph, and the Tricomi boundary value problem. All solutions show boundary-detached internal shear layers, typical for hyperbolic boundary value problems. For the Tricomi problem it is found that the matrix that has to be inverted to find solutions from the collocation approach is ill-conditioned; thus solutions depend on the distribution of the collocation points and need to be regularized.

Particle image velocimetry (PIV) measurements were performed in a wave tank under water waves propagating and breaking on a 1/15 sloping beach. The wave transformation occurred in the surf zone over a large domain covering several wavelengths from incipient breaking to swash zone beyond the shoreline. PIV spatial interrogation windows must be small enough to obtain accurate velocities, and one window covers only a small part of the domain. To overcome this problem and to measure the instantaneous velocity field over the whole surf zone area, we have split the full field into 14 overlapping smaller windows of the same size. Local measurements were synchronized with each other using pulsed TTL triggers and wave gauge data. The full velocity field was then reconstructed at every time step by gathering the 14 PIV fields. We then measured the complete space–time evolution of the velocity field over the whole surf zone. We determined also the ensemble-period-average and phase-average components of the flow with their associated fluctuating parts. We used the PIV images and velocity measurements to estimate the void fraction in each point of the surf zone. Special attention was given to the calculation of the spatial derivatives in order to obtain relevant information on vorticity and on the physical terms that appear in the fluctuating kinetic energy transport equation.

The rheological properties of a cell suspension may play an important role in the flow field generated by populations of swimming micro-organisms (e.g. in bioconvection). In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). Effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The three-dimensional movement of 64 identical squirmers in a simple shear flow field, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The computation utilizes a database of pairwise interactions that has been constructed by the boundary element method. The restriction to pairwise additivity of forces is expected to be justified if the suspension is semi-dilute. The results for non-bottom-heavy squirmers show that the squirming does not have a direct influence on the apparent viscosity. However, it does change the probability density in configuration space, and thereby causes a slight decrease in the apparent viscosity at O(c2), where c is the volume fraction of spheres. In the case of bottom-heavy squirmers, on the other hand, the stresslet generated by the squirming motion directly contributes to the bulk stress at O(c), and the suspension shows strong non-Newtonian properties. When the background simple shear flow is directed vertically, the apparent viscosity of the semi-dilute suspension of bottom-heavy squirmers becomes smaller than that of inert spheres. When the shear flow is horizontal and varies with the vertical coordinate, on the other hand, the apparent viscosity becomes larger than that of inert spheres. In addition, significant normal stress differences appear for all relative orientations of gravity and the shear flow, in the case of bottom-heavy squirmers.

The diffusive behaviour of swimming micro-organisms should be clarified in order to obtain a better continuum model for cell suspensions. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). Effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The three-dimensional movement of 64 or 27 identical squirmers in a fluid otherwise at rest, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The computation utilizes a database of pairwise interactions that has been constructed by the boundary element method. In the case of (non-bottom-heavy) squirmers, both the translational and the orientational spreading of squirmers is correctly described as a diffusive process over a sufficiently long time scale, even though all the movements of the squirmers were deterministically calculated. Scaling of the results on the assumption that the squirmer trajectories are unbiased random walks is shown to capture some but not all of the main features of the results. In the case of (bottom-heavy) squirmers, the diffusive behaviour in squirmers' orientations can be described by a biased random walk model, but only when the effect of hydrodynamic interaction dominates that of the bottom-heaviness. The spreading of bottom-heavy squirmers in the horizontal directions show diffusive behaviour, and that in the vertical direction also does when the average upward velocity is subtracted. The rotational diffusivity in this case, at a volume fraction c=0.1, is shown to be at least as large as that previously measured in very dilute populations of swimming algal cells (Chlamydomonas nivalis).

The normal modes of a horizontally uniform, vertically sheared flow over a sloping bottom are considered in two active layers underneath a deep motionless third layer. The variations of the layer thickness are assumed to be small to analyse the sixth-order eigenvalue problem for finite-Froude-number typical for oceanic currents. The dispersion curves for the Rossby waves and the Poincaré modes of inertia–gravity waves (IGW) are investigated to identify the different types of instabilities that occur if there is a pair of wave components which have almost the same Doppler-shifted frequency related to crossover of the branches when the Froude number increases. Simple criteria for ageostrophic instabilities due to a resonance between the IGW and the Rossby wave because of the thickness gradient in either the lower or middle layer, are derived. They exactly correspond to violation of sufficient Ripa's conditions for the flow stability. In both cases the growth rate and the interval of unstable wavenumbers are shown to be proportional to the square root of the corresponding gradient of the layer thickness. These types of ageostrophic instability can coexist (and with Kelvin–Helmholtz instability). However, their role in generating unbalanced motions and mixing processes in geophysical fluids appears limited due to small growth rates and narrow intervals of the unstable wavenumbers in comparison to Kelvin–Helmholtz instability.