The interaction between a conventional rectangular (primary) air jet and a co-flowing synthetic jet is investigated experimentally. The nozzles of both jets have the same long dimension but the aspect ratio of the synthetic jet orifice is 25 times larger. Detailed particle image velocimetry (PIV) measurements of the flow in the midspan plane show that primary jet fluid is directed into the synthetic jet orifice and the interaction between the jets leads to the formation of a closed recirculating flow domain. The concomitant formation of a low-pressure region between the jets results in deflection of the primary jet toward the actuator jet despite the absence of an extended control surface (e.g. a diffuser or collar) and is balanced by a force on the primary jet conduit. For a given synthetic jet strength and primary jet speed, the vectoring force depends mainly on the volume flow rate of primary jet fluid that is diverted into the synthetic jet actuator. This flow rate is regulated by restricting the flow of entrained ambient fluid using a small streamwise extension of the synthetic jet orifice that scales with the orifice width. The response of the primary jet to the imposed vectoring is investigated using stepped modulation of the driving signal. The characteristic vectoring time and vectoring angle decrease monotonically with primary jet speed.

The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

This paper discusses a potential-vorticity-conserving approach to modelling nonlinear internal gravity waves in a rotating Boussinesq fluid. The focus of the work is on the pseudo-plane motion (motion in the x, z-plane), for which we present a broad range of numerical results. In this case there are two material coordinates, the density and the y-component of the velocity in the inertial frame of reference, which are related to the x and z displacements of fluid particles relative to a reference configuration. The amount of potential vorticity within a fluid region bounded by isosurfaces of these material coordinates is proportional to the area within this region, and is therefore conserved as well. Two new potentials, defined in terms of the displacements and combining the vorticity and density fields, are introduced as new dependent variables. These potentials entirely govern the dynamics of internal gravity waves for the linearized system when the basic state has uniform potential vorticity. The final system of equations consists of three prognostic equations (for the potential vorticity and the Laplacians of the two potentials) and one diagnostic equation, of Monge–Ampère type, for a third potential. This diagnostic equation arises from the nonlinear definition of potential vorticity. The ellipticity of the Monge–Ampère equation implies both inertial and static stability. In three dimensions, the three potentials form a vector, whose (three-dimensional) Laplacian is equal to the vorticity plus the gradient of the perturbation density.

Numerical simulations are carried out using a novel algorithm which directly evolves the potential vorticity, in a Lagrangian manner (following fluid particles), without diffusion. We present results which emphasize the way in which potential vorticity anomalies modify the characteristics of internal gravity waves, e.g. the propagation of internal wave packets, including reflection, refraction, and amplification. We also show how potential vorticity anomalies may generate internal gravity waves, along with the subsequent ‘geostrophic adjustment’ of the flow to a ‘balanced’ wave-less state. These examples, and the straightforward extension of the theoretical and numerical approach to three dimensions, point to a direct and accurate means to elucidate the role of potential vorticity in internal gravity wave interactions. As such, this approach may help a better understanding of the observed characteristics of internal gravity waves in the oceans.

Spin-up in a rectangular container with a free surface is investigated numerically and experimentally. In the formulation of two-dimensional numerical computation, we use a potential-like function in addition to the stream function to deal with the first-order Ekman pumping model. It is shown that our numerical results are in good agreement with those obtained by the experiment when either the leading-order or first-order pumping model is used. On the other hand, when no pumping effect is considered the numerical results show, except in the initial development, a considerable discrepancy from those of the experiment. Our attention in this study is focused on clarifying the physical mechanism of cyclonic vortex merging. At low Reynolds numbers and/or liquid depths the Ekman pumping damps the vortical flows fast, resulting in non-merging. At moderate Reynolds numbers, it enhances merging because the cyclonic vortices expand due to the Ekman pumping. We discuss the influence of various parameters, including Reynolds number, Rossby number, and dimensionless liquid depth, on the evolution of the vortical flows.

This study details experiments investigating a previously unrecognized surge instability on a cavitating propeller in a water tunnel. The surge instability is explored through visual observation of the cavitation on the propeller blades and in the tip vortices. Similarities between the instability and previously documented cavitation phenomena are noted. Measurements of the radiated pressure are obtained, and the acoustic signature of the instability is identified. The magnitudes of the fluctuating pressures are very large, presumably capable of producing severe hull vibration on a ship.

The origins of this instability are explored through separate investigation of the cavitation dynamics and the response of the water tunnel to volumetric displacement in the working section. Experiments are conducted to quantify the dynamics of the propeller cavitation. Finally, a model is developed for the complete system, incorporating both the cavitation and facility dynamics. The model predicts active system dynamics (linked to the mass flow gain factor familiar in the context of pump dynamics) and therefore potentially unstable behaviour for two distinct frequency ranges, one of which appears to be responsible for the instability.

The flow in an infinite slab of rectangular cross-section is investigated numerically by a finite volume method. Two facing walls which move parallel to each other with the same velocity, but in opposite directions, drive a plane flow in the cross-section of the slab. A linear stability analysis shows that the two-dimensional flow becomes unstable to different modes, depending on the cross-sectional aspect ratio, when the Reynolds number is increased. The critical mode is found to be stationary for all aspect ratios. When the separation of the moving walls is larger than approximately twice the height of the cavity, the basic flow forms two vortices, each close to one of the moving walls. The instability of this flow is of centrifugal type and similar to that in the classical lid-driven cavity problem with a single moving wall. When the moving walls are sufficiently close to each other (aspect ratio less than 2) the two vortices merge and form an elliptically strained vortex. Owing to the dipolar strain this flow becomes unstable through the elliptic instability. When both moving walls are very close, the finite-length plane-Couette flow becomes unstable by a similar elliptic mechanism near both turning zones. The critical mode produces wide streaks reaching far into the cavity. For a small range of aspect ratios near unity the flow consists of a single vortex. Here, the strain field is dominated by a four-fold symmetry. As a result the instability process is analogous to the instability of a Rankine vortex in an quadripolar strain field, resulting from vortex stretching into the four corners of the cavity.

We investigate the stability and dynamics of three-dimensional limit-cycle states in flow past a circular cylinder using low-dimensional modelling. High-resolution direct numerical simulations are employed to obtain flow snapshots from which the most energetic modes are extracted using proper orthogonal decomposition. We show that the limit cycle is reproduced very accurately with only twenty three-dimensional modes. The addition of two-dimensional modes to the Karhunen–Loeve expansion basis improves the ability of the model to capture the three-dimensional bifurcation, including the discontinuity in the Strouhal number discovered experimentally.

We present experimental measurements of velocity and temperature fields in horizontal planes crossing a cylindrical Rayleigh–Bénard convection cell in steady rotation about its vertical axis. The range of dimensionless rotation rates Ω is from zero to 5×104 for a Rayleigh number R = 3.2×108. The corresponding range of convective Rossby numbers is ∞ > Ro > 0.06. The patterns of velocity and temperature and the flow statistics characterize three basic flow regimes. For Ro [Gt ] 1, the flow is dominated by vortex sheets (plumes) typical of turbulent convection without rotation. The flow patterns for Ro ∼ 1 are cyclone-dominated, with anticyclonic vortices rare. As the Rossby number continues to decrease, the number of anticyclonic vortex structures begins to grow but the vorticity PDF in the vicinity of the top boundary layer still shows skewness favouring cyclonic vorticity. Velocity-averaging near the top of the cell suggests the existence of a global circulation pattern for Ro [Gt ] 1.

Activation-energy asymptotics is employed to explore effects of the Lewis number, the ratio of thermal to fuel diffusivity, in a one-dimensional model of steady motion of edges of reaction sheets. The propagation velocity of the edge is obtained as a function of the relevant Damköhler number, the ratio of the diffusion time to the chemical time. The results show how Lewis numbers different from unity can increase or decrease propagation velocities. Increasing the Lewis number increases the propagation velocity at large Damköhler numbers and decreases it at small Damköhler numbers. Advancing-edge and retreating-edge solutions are shown to exist simultaneously, at the same Damköhler number, if the Lewis number is sufficiently large. This multiplicity of solutions has a bearing on potential edge-flame configurations in non-uniform flows.

A rapid-distortion model is developed to investigate the interaction of weak turbulence with a monochromatic irrotational surface water wave. The model is applicable when the orbital velocity of the wave is larger than the turbulence intensity, and when the slope of the wave is sufficiently high that the straining of the turbulence by the wave dominates over the straining of the turbulence by itself. The turbulence suffers two distortions. Firstly, vorticity in the turbulence is modulated by the wave orbital motions, which leads to the streamwise Reynolds stress attaining maxima at the wave crests and minima at the wave troughs; the Reynolds stress normal to the free surface develops minima at the wave crests and maxima at the troughs. Secondly, over several wave cycles the Stokes drift associated with the wave tilts vertical vorticity into the horizontal direction, subsequently stretching it into elongated streamwise vortices, which come to dominate the flow. These results are shown to be strikingly different from turbulence distorted by a mean shear flow, when ‘streaky structures’ of high and low streamwise velocity fluctuations develop. It is shown that, in the case of distortion by a mean shear flow, the tendency for the mean shear to produce streamwise vortices by distortion of the turbulent vorticity is largely cancelled by a distortion of the mean vorticity by the turbulent fluctuations. This latter process is absent in distortion by Stokes drift, since there is then no mean vorticity.

The components of the Reynolds stress and the integral length scales computed from turbulence distorted by Stokes drift show the same behaviour as in the simulations of Langmuir turbulence reported by McWilliams, Sullivan & Moeng (1997). Hence we suggest that turbulent vorticity in the upper ocean, such as produced by breaking waves, may help to provide the initial seeds for Langmuir circulations, thereby complementing the shear-flow instability mechanism developed by Craik & Leibovich (1976).

The tilting of the vertical vorticity into the horizontal by the Stokes drift tends also to produce a shear stress that does work against the mean straining associated with the wave orbital motions. The turbulent kinetic energy then increases at the expense of energy in the wave. Hence the wave decays. An expression for the wave attenuation rate is obtained by scaling the equation for the wave energy, and is found to be broadly consistent with available laboratory data.

Shoreline boundary conditions for nearshore hydrodynamic models are discussed on the basis of the swash zone equations of Brocchini & Peregrine (1996). Swash zone flows are investigated further using the shallow water equations. Results from numerical computations are used to guide approximation to provide more practical boundary conditions for wave-averaged flows. Approximate boundary conditions, valid for small values of the rate of change of the mean water volume in the swash zone, are found which allow explicit computation of a non-zero mean water depth at the mean shoreline. This is computed in terms of the local height of the short waves. Implementation issues are also discussed.

The behaviour of an inviscid gravity current which is released from behind a lock and then propagates over a horizontal boundary at the base of a stratified ambient fluid is considered. An extension of the shallow-water formulation for a homogeneous ambient to the stratified case is developed, without using any additional adjustable parameters. Attention is focused on the initial ‘slumping’ stage of a rectangular current which is typified by a constant speed of propagation. The analytical results are in good agreement with, and give a firm theoretical interpretation of, the corresponding experiments and numerical simulations of Maxworthy et al. (2002). Finite-difference solutions of the Navier–Stokes equations, using a different technique from that used by Maxworthy et al. (2002), are also presented and provide both good agreement with their results and further validation of the present shallow-water approach. The differences between currents in a homogeneous and stratified ambient, and possible implementation of the results to other configurations, are discussed.

This work is an experimental study of the turbulent vortex structures, and heat and momentum transport in the wake of two side-by-side circular cylinders. The spacing T between the cylinder axes was varied from 1.5d to 3d (d is the cylinder diameter). Both cylinders were slightly heated. A movable three-wire probe measured the velocity and temperature fluctuations, and an X-wire provided a phase reference. Measurements were conducted at x/d = 10, 20 and 40 at a Reynolds number of 5800 (based on d and the free-stream velocity U∞). At T/d = 1.5, the phase-averaged velocity and temperature fields display a single vortex street. The two rows of vortices exhibit a significant difference in the maximum vorticity, size and lateral distance from the flow centreline. As T/d is increased to 3.0, the flow is totally different. Two antiphase streets occur initially. They are less stable, with vortices weakening faster, than the street at T/d = 1.5. By x/d = 40, one street only is identifiable. Effective vorticity flux density indicates that, while the outer vortex nearer to the free stream interacts largely with the adjacent oppositely signed inner vortices located near the flow centreline, the inner vortex interacts with the cross-stream inner vortices as well as with adjacent outer vortices. As a result, vorticity associated with the inner vortex is annihilated quicker than that associated with the outer vortex, leading to the early disappearance of inner vortices and formation of a single street. The contribution of the coherent motion of various Reynolds-averaged quantities such as the momentum and heat fluxes has also been quantified and discussed in conjunction with the vortex structures of the flow and temperature fields.

Many practical applications, such as in blade cascades and turbomachinery, involve inhomogeneous turbulent shear flows subjected simultaneously to multiple strains. In principle, the applied strain can be combined to yield an effective strain. However, no simple stress–strain relation is capable of establishing turbulent stress or energy balance in the mean or on an instantaneous basis. In the current investigation, a turbulent boundary layer is examined in the presence of convex curvatures of different strengths combined with streamwise (favourable and adverse) pressure gradients, with various values of pressure gradient ratio, (∂P/∂s)/(∂P/∂n). Measurements of the mean and turbulent parameters and flux Richardson number show appreciable changes, mainly in the outer portion of the boundary layer (y+ > 100). The turbulent burst frequency, particularly at the location of application of the additional strain rate, also changes relative to its value with wall curvature alone.

Three primary observations from these experiments are as follows: (i) in all cases, the mean velocity profile and all of the measured Reynolds stresses collapse in the near-wall region using standard inner scaling; (ii) the applied strains combine nonlinearly, with one of the strains dominating the local flow during its development; (iii) the ratio of the radial to axial pressure gradient magnitude influences both classical turbulence correlations and mean flow, as well as the physical production cycle of turbulence; and (iv) application rate of newly introduced strain rates is at least as important as their magnitudes.

We present a mathematical model for the drainage of a surfactant-stabilized foam lamella, including capillary, Marangoni and viscous effects and allowing for diffusion, advection and adsorption of the surfactant molecules. We use the slender geometry of a lamella to formulate the model in the thin-film limit and perform an asymptotic decomposition of the liquid domain into a capillary-static Plateau border, a time-dependent thin film and a transition region between the two. By solving a quasi-steady boundary-value problem in the transition region, we obtain the flux of liquid from the lamella into the Plateau border and thus are able to determine the rate at which the lamella drains.

Our method is illustrated initially in the surfactant-free case. Numerical results are presented for three particular parameter regimes of interest when surfactant is present. Both monotonic profiles and those exhibiting a dimple near the Plateau border are found, the latter having been previously observed in experiments. The velocity field may be uniform across the lamella or of parabolic Poiseuille type, with fluid either driven out along the centreline and back along the free surfaces or vice versa. We find that diffusion may be negligible for a typical real surfactant, although this does not lead to a reduction in order because of the inherently diffusive nature of the fluid–surfactant interaction. Finally, we obtain the surprising result that the flux of liquid from the lamella into the Plateau border increases as the lamella thins, approaching infinity at a finite lamella thickness.

In this paper it is shown that the two-dimensional time-periodic vortex shedding régime observed in the cylinder wake at moderate Reynolds numbers may be interpreted as a nonlinear global structure and its naturally selected frequency obtained in the framework of hydrodynamic stability theory. The frequency selection criterion is based on the local absolute frequency curve derived from the unperturbed basic flow fields under the assumption of slow streamwise variations. Although the latter assumption is only approximately fulfilled in the vicinity of the obstacle, the theoretically predicted frequency is in good agreement with direct numerical simulations for Reynolds numbers Re > 100.

Direct numerical simulations of the velocity and temperature fields for turbulent flow in a channel are used to examine the influence of Prandtl number Pr on turbulent transport. The Reynolds number, based on the half-height of the channel and the friction velocity, is Reτ = 150. Prandtl numbers of 1.0, 0.3, 0.1, 0.05, 0.025 were studied. The bottom and the top walls were kept at constant temperatures of +Tw and −Tw. The influence of Pr on Reynolds transport, on the turbulent diffusivity, ατ, and on the spectral density function of the temperature fluctuations was studied. The observation that spatial variations of the ratio of the turbulent diffusivity to the value observed at Pr = 1.0 are not large is used to propose a method for calculating average temperature fields. The decrease in ατ with decreasing Pr is related to observations of the increased damping of high-wavenumber temperature fluctuations. Molecular conductivity, at smaller Pr, is pictured to act as a filter that renders high-frequency velocity fluctuations ineffective in transporting heat.