The thermally driven motion of water contained in a rotating annulus of square cross-section and having a free surface is investigated by numerical integration of the three-dimensional non-linear Navier–Stokes equations. The nature of steady wave flow is examined in detail and a comparison made with the corresponding axisymmetric solution in parameter space.

The steady wave solution proves to be consistent kinematically, dynamically and energetically with Lorenz's hypothesis that the wave can be attributed to the baroclinic instability mechanism. The deviaboricThe deviation from the zonal mean. wave possesses some of the characteristics of the theoretical Eady wave and it is possible to define the complete deviaboric wave structure by means of two-dimensional quasi-phase, amplitude diagrams. These diagrams may also typify the nature of certain solutions to the non-separable baroclinic instability problem.

The wave motion is almost completely independent of the side boundary layers which make little contribution to the characteristics and energetics of the deviakoric flow. These side layers are approximately axisymmetric and appear qualitatively indistinguishable from their counterparts in the axisymmetric solution. However, significant Ekman layer features appear in the deviatoric wave structure.

Away from the boundaries the dynamical balance of terms is hydrostatic and quasi-geostrophic with changes of vertical vorticity influenced by stretching and viscous diffusion. Heat conduction is completely unimportant except in the side boundary layers.

The angular momentum transport by the deviatoric motion is largest at the free surface and is mainly against the angular momentum gradient. A strong outward deviatoric flux of momentum is found in the Ekman layer.

The dissipation of deviatoric kinetic energy occurs in the Ekman layer and jet whilst most of the dissipation of the mean kinetic energy occurs in the boundary layer of the inner wall.

The large differences between the axisymmetric and zonal mean states indicate that linear baroclinic instability analysis of the axisymmetric state is not strictly relevant to an understanding of the wave formation. The character of the wave suggests that the mean environment with which the deviatoric wave interacts is the wave-present zonal mean state. Only a non-linear finite amplitude baroclinic instability analysis (as yet undeveloped) could possibly explain the wave formation.

The propagation at high Reynolds number of a heavy, two-dimensional gravity current of given initial volume at the base of a uniform flow is considered. An experimental setup is described for which a known volume of fluid is rapidly introduced halfway down a 9 m channel in which there is a uniform flow of water. The density excess of the released fluid is produced by either dissolving salt or suspending particles in water. The upstream and downstream propagation of the current was measured for different initial salt concentrations, particle sizes and concentrations. A simple box model for the motion of and deposit from the gravity current is constructed. The analytical results obtained compare well with our numerical solutions of one-layer and two-layer models incorporating the appropriate shallow-water equations. Both sets of results are in very good agreement with the experimental data.

The present study is concerned with radiation-induced natural convection in a water-filled triangular enclosure with a sloping bottom, which is directly relevant to buoyancy-driven flows in littoral regions. An improved scaling analysis is carried out to reveal more detailed features of the flow than a previously reported analysis. Two critical functions of the Rayleigh number with respect to the horizontal position are derived from the scaling for identifying the distinctness and stability of the thermal boundary layer. Four flow scenarios are possible, depending on the bottom slope and the maximum water depth. For each flow scenario, the flow domain may be composed of multiple subregions with distinct thermal and flow features, depending on the Rayleigh number. The dividing points between neighbouring subregions are determined by comparisons of the critical functions of the Rayleigh number with the global Rayleigh number. Position-dependent scales have been established to quantify the flow properties in different subregions. The different flow regimes for the case with relatively large bottom slopes and shallow waters are examined in detail. The present scaling results are verified by numerical simulations.

A linear stability analysis of incipient channellization on hillslopes is performed using the shallow-water equations and a description of the erosion of a cohesive bed. The base state consists of a laterally uniform Froude-subcritical sheet flow down a smooth, downward-concave hillslope profile. The downstream boundary condition consists of the imposition of a Froude number of unity. The process of channellization is thus driven from the downstream end. The flow and bed profiles describe a base state that migrates at constant, slow speed in the upstream direction due to bed erosion. Transverse perturbations corresponding to a succession of parallel incipient channels are introduced. It is found that these perturbations grow in time, so describing incipient channellization, only when the characteristic spacing between incipient channels is on the order of 6–100 times the Froude-critical depth divided by the resistance coefficient. The characteristic wavelength associated with maximum perturbation growth rate is found to scale as 10 times the Froude-critical depth divided by the resistance coefficient. Evaluating the friction coefficient as on the order of 0.01, an estimate of incipient channel spacing on the order of 1000 times the Froude-critical depth is obtained. The analysis reveals that downstream-driven channellization becomes more difficult as (a) the critical shear stress required to erode the bed becomes so large that it approaches the Froude-critical shear stress reached at the downstream boundary and (b) the Froude number of the subcritical equilibrium flow attained far upstream approaches unity. Alternative mechanisms must be invoked to explain channellization on slopes high enough to maintain Froude-supercritical sheet flow.

A free falling, absorbing liquid drop hit by a nanosecond laser pulse experiences a strong recoil pressure kick. As a consequence, the drop propels forward and deforms into a thin sheet which eventually fragments. We study how the drop deformation depends on the pulse shape and drop properties. We first derive the velocity field inside the drop on the time scale of the pressure pulse, when the drop is still spherical. This yields the kinetic energy partition inside the drop, which precisely measures the deformation rate with respect to the propulsion rate, before surface tension comes into play. On the time scale where surface tension is important, the drop has evolved into a thin sheet. Its expansion dynamics is described with a slender-slope model, which uses the impulsive energy partition as an initial condition. Completed with boundary integral simulations, this two-stage model explains the entire drop dynamics and its dependence on the pulse shape: for a given propulsion, a tightly focused pulse results in a thin curved sheet which maximizes the lateral expansion, while a uniform illumination yields a smaller expansion but a flat symmetric sheet, in good agreement with experimental observations.

The effect of harmonic oscillations in the magnitude of the surface temperature on the free-convection laminar velocity and temperature boundary layers on a flat plate is analysed. Low-and high-frequency solutions are developed separately. The results obtained are in striking contrast to the corresponding results for forced-convection flows.

The investigation to be described here is a wide-ranging experimental study aimed at determining both the details of the flow field and the pressure drop and friction factor characteristics for turbulent flow in eccentric annular ducts. The experiments were performed utilizing three annular ducts of different diameter ratios; in each case the eccentricity was varied from zero (concentric annulus) to unity (walls in contact). To provide the broadest possible perspective, the measurements of the velocity field are presented in three different ways. First, contour maps showing lines of constant velocity are constructed. From these are deduced circumferential distributions of the local shear stress on the bounding walls. Velocity profiles along lines normal to the walls are represented in terms of both law-of-the-wall variables and defect-law variables. Neither of these representations affords complete agreement with the universal circular-tube distributions. In general, the defect law provides a somewhat closer correlation of the results for the eccentric annulus with those for the tube. The experimental findings do not substantiate a prior analytical model which assumes that the universal law of the wall applies on all lines normal to the bounding walls of the annulus. Friction factors, based on static pressure measurements, are shown to decrease with increasing eccentricity. The measured friction factors are in fair agreement with those of analysis. Hydrodynamic development lengths, deduced from entrance-region pressure data, are found to increase with increasing eccentricity. Circumferential pressure variations also increase with eccentricity.

This paper presents a new experimental time-domain technique for the evaluation of the large-scale structure in a turbulent flow. The technique is demonstrated by hot-wire anemometry for a circular jet flow at a moderate Reynolds number of 104, and the large-scale structure identified is compared successfully with smoke flow-visualization observations. The temporal and spatial relationships of the separate large-scale flow events have been derived, and this information enabled the evaluation of the nonlinear spatial development of the large-scale flow structure.

Results are presented from a computational study of the flow over a forward-facing step in a plane channel. The aim of the study is to gain better insight into the three-dimensionality that is typically observed in the separation region of flows over steps and ribs, and in similar configurations. We consider laminar flow at a Reynolds number of 330, based on step height and bulk velocity of the oncoming flow, and the step is assumed to be infinitely extended in the spanwise direction. High-resolution simulations are undertaken using a mixed spectral/spectral-element code. Moreover, a linear stability study of the flow at the step is performed. The results show that, in the case considered, the three-dimensionality is not related to some absolute instability of the separation bubble in front of the step; rather, it is a sensitive reaction of the flow to three-dimensional perturbations present in the oncoming stream. It is demonstrated that disturbance amplitudes of less than 1% of the mean flow (at, say, 10 step heights ahead of the step) already suffice to produce a visibly three-dimensional structure of the separation zone. If the disturbance level is systematically decreased, the three-dimensional state evolves to an almost two-dimensional recirculation. Here, the key finding is that the intensity of the flow response is proportionate to the amplitude of the inflow disturbance, meaning that the breakup of the flow in the step region is a linear (i.e. small) perturbation of the two-dimensional base flow. A comparison of the present simulation results with experimental data shows close agreement concerning, for example, the flow topology in the step region, and the spanwise spacing of the characteristic streaks that form further downstream.

As two fluid particles separate in time, the entire spectrum of eddy motions is being sampled from the smallest to the largest scales. In large-scale geophysical systems for which the Earth rotation is important, it has been conjectured that the relative diffusivity should vary respectively as $D^{2}$ and $D^{4/3}$ for distances respectively smaller and larger than a well-defined forcing scale of the order of the internal Rossby radius (with $D$ the r.m.s. separation distance). Particle paths data from a mid-latitude float experiment in the central part of the North Atlantic appear to support these statements partly: two particles initially separated by a few km within two distinct clusters west and east of the mid-Atlantic ridge, statistically dispersed following a Richardson regime ($D^{2}\,{\sim}\,t^{3}$ asymptotically) for r.m.s. separation distances between 40 and 300 km, in agreement with a $D^{4/3}$ law. At early times, and for smaller separation distances, an exponential growth, in agreement with a $D^{2}$ law, was briefly observed but only for the eastern cluster (with an e-folding time around 6 days). After a few months or separation distances greater than 300 km, the relative dispersion slowed down naturally to the Taylor absolute dispersion regime.

A novel, digital, hot-wire anemometer technique for the simultaneous measurement of the instantaneous streamwise and lateral velocity fields in high-intensity turbulent flows is discussed. It involves the use of a three-wire probe comprising two 45° slanted hot wires and a normal hot wire. A comprehensive and systematic examination of several factors that can affect the fidelity of the streamwise and lateral velocity waveforms is developed to assess the performance of the new technique as well as hot-wire systems generally. These factors are: (i) rectification, which stems from the inherent insensitivity of hot wires to the direction of the instantaneous (total) velocity vector in a turbulent flow; (ii) spanwise velocity fluctuations; (iii) axial cooling of hot wires; (iv) unpredictable variations in one of four hot-wire calibration parameters; (v) random hot-wire calibration errors; (vi) spanwise separation of the hot wires. Relevant hot-wire anemometer-response equations relating instantaneous anemometer output voltages to instantaneous flow velocities were established on the basis of extensive voltage-velocity calibration data pertaining to hot wires orientated with respect to the calibration flow velocity at various yaw and pitch angles ranging from 0° to 90°. Simulated Gaussian (streamwise, lateral and spanwise) velocity fields appropriate to flows with turbulence intensity levels varying between 5 and 80% and Reynolds shear-stress coefficients varying between 0.1 and 0.5 were generated by means of a digital computer, and the associated anemometer-voltage signals computed in accordance with the response equations subject to different combinations of the first four of the aforementioned factors. In order to take into account the effects of the last two factors, viz calibration errors and spanwise wire separation, uncorrelated Gaussian ‘noise’ fluctuations were superimposed on the above voltage signals. Estimates of the known (simulated) streamwise and lateral velocity signals were then determined by simultaneous solution of (a) the actual instantaneous response equations, (b) approximate versions of them, and (c) linearized versions of them. The results indicate that reasonably accurate estimates of velocity signals from a turbulent flow can be obtained by means of conventional hot-wire anemometer techniques – which assume that anemometer voltage fluctuations are linear functions of corresponding velocity fluctuations – only if the turbulent intensity level of the flow does not exceed about 20%. In marked contrast, the 3-wire anemometer technique introduced here can be used to measure streamwise and lateral velocity signals simultaneously with a high degree of accuracy for turbulence-intensity levels of up to 40%. In addition, this technique is capable of yielding high-fidelity streamwise velocity waveforms for levels in excess of 70%.

An investigation of the local receptivity of a Blasius boundary layer to a harmonic vortical disturbance is presented as a step towards understanding boundary-layer receptivity to free-stream turbulence. Although there has been solid experimental verification of the linear theory describing acoustic receptivity of boundary layers, this was the first experimental verification of the mechanism behind local receptivity to a convected disturbance. The harmonic wake from a vibrating ribbon positioned upstream of a flat plate provided the free-stream disturbance. Two-dimensional roughness elements on the surface of the plate acted as a local receptivity site. Hot-wire measurements in the boundary layer downstream of the roughness confirmed the generation of Tollmien–Schlichting (TS) instability waves by an outer-layer interaction between the long-wavelength convected disturbance and the short-scale mean-flow distortion due to the roughness. The characteristics of the instability waves were carefully measured to ensure that their behaviour was correctly modelled by linear stability theory. This theory was then used to determine the immeasurably small initial wave amplitudes resulting from the receptivity process, from wave amplitudes measured downstream. Tests were performed to determine the range of validity of the linear assumptions made in current receptivity theories. Experimental data obtained in the linear regime were then compared to theoretical results of other authors by expressing the experimental data in the form of an efficiency function which is independent of the free-stream amplitude, roughness height and roughness geometry. Reasonable agreement between the experimental and theoretical efficiency functions was obtained over a range of frequencies and Reynolds numbers.

In this investigation we explore the effect of unsteady vortical structures on the adiabatic wall temperature distribution in an impinging jet. Treating first the simpler case of a free jet, we introduce a conceptual model for the separation of the total temperature, appealing to the dynamics of particle pathlines and vortex rings in the jet. The presence of a region of higher total temperature on the inside of the jet and a region of lower total temperature toward the jet periphery, predicted by the model, exhibits good agreement with the experimental data taken at high subsonic Mach number. The results from a numerical simulation further confirm the theoretical expectations.

Through a similar argument, we show that when a thermally insulated flat plate is inserted into the jet, the wall temperature distribution is modified by the presence of secondary vortical structures, which are induced near, and swept over, the plate surface. When the plate is near the jet nozzle, a region of lower wall temperature, attributable to these additional vortices, is observed in the experimental data. When the plate is further from the nozzle, no secondary vortices are formed and no region of lowered wall temperature is measured. Self-sustaining acoustic resonance, when it occurs, is found to alter significantly this picture of the wall temperature distribution.

Although the scope of this work is limited to free and impinging jets, this present topic, along with the previously reported mechanism of the Eckert–Weise effect, exemplifies the wider family of problems in which unsteady vortical structure strongly affects the wall temperature and heat transfer.

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 trajectories of 20 and 40 $\umu$m radius water droplets colliding with a gas–water interface are observed to determine the conditions under which drops will bounce or coalesce on impact and the apparent coefficient of restitution of the drops that bounce. The experiments were performed in a pressure chamber so that the pressure and composition of the gas could be varied to explore the effects of the viscosity and mean-free path of the gas. The impact velocity is varied by producing drops with a velocity larger than their terminal velocity using a piezoelectric drop generator and adjusting the distance between the generator and the liquid interface. The geometry of the collisions is axisymmetric and the Weber number, $\hbox{\it We}\,{=}\,\rho_{l}U^{2}a/\sigma$, is O (1) or smaller, so as to facilitate comparison with a theory for weakly deformable gas–liquid interfaces. Here, $\rho_{l}$ is the liquid density, $U$ the impact velocity, $a$ the drop radius and $\sigma$ the surface tension. After a low-Weber-number drop coalesces, a smaller, daughter drop is emitted from the interface with a velocity higher than the incident velocity of the mother drop. The daughter drop radius is about 0.55$a$ and the daughter velocity is 0.38($\sigma /(\rho_{l}a))^{1/2}$ for We$\,{<}\,0.01$.

The experimental results are compared with a theory in which the small deformations of the drop and surface are expanded in Legendre polynomials and Fourier modes, respectively, the non-continuum lubrication stresses are computed in the thin gas film between the drop and interface, and the liquid flow is approximated as an inviscid potential flow. The coefficient of restitution decreases with increasing Weber number and becomes insensitive to the viscosity of the gas at Weber numbers larger than about 1. At smaller Weber numbers, drops in a less viscous gas lose less energy during the collision. Drops are observed to undergo a transition from coalescence to bouncing as the drop velocity (Weber number) is increased. However, the marginal condition for drop bouncing is much more sensitive to gas mean-free path (Knudsen number) and gas viscosity (Ohnesorge number) than to Weber number. The Knudsen and Ohnesorge numbers are defined as $\hbox{\it Kn}\,{=}\,{\lambda/a}$ and $\hbox{\it Oh}\,{=}\,\mu_{g}/(\rho_{l}a\sigma )^{1/2}$ where $\lambda $ is the mean-free path and $\mu_{g}$ is the gas viscosity. Theory and experiment show similar trends of increasing critical Weber number with decreasing Ohnesorge number and increasing Knudsen number. Theoretical results are also derived for the coalescence–bounce transition and coefficient of restitution for head-on collisions of equal sized drops.

To understand the fluid dynamics of a biologically inspired unsteady low-aspect-ratio propulsor, unsteady pressure distributions were measured and compared with time-averaged thrust performance and wake visualizations. The experiments were performed on rigid rectangular panels with different aspect ratios, pitching in a uniform flow. Panel aspect ratio and pitching amplitude were shown to affect the magnitude and time dependence of the pressure distribution on the panel surface, the vorticity generation on the panel, and thrust production. A new scaling is proposed that includes these parameters and collapses the oscillating pressure magnitude and the thrust coefficient.

This paper exploits the turbulent flow control method using streamwise travelling waves (Quadrio et al. J. Fluid Mech., vol. 627, 2009, pp. 161–178) to study the effect of Reynolds number on turbulent skin-friction drag reduction. Direct numerical simulations (DNS) of a turbulent channel flow subjected to the streamwise travelling waves of spanwise wall velocity have been performed at Reynolds numbers ranging from $\mathit{Re}_{{\it\tau}}=200$ to 1600. To the best of the authors’ knowledge, this is the highest Reynolds number attempted with DNS for this type of flow control. The present DNS results confirm that the effectiveness of drag reduction deteriorates, and the maximum drag reduction achieved by travelling waves decreases significantly as the Reynolds number increases. The intensity of both the drag reduction and drag increase is reduced with the Reynolds number. Another important finding is that the value of the optimal control parameters changes, even in wall units, when the Reynolds number is increased. This trend is observed for the wall oscillation, stationary wave, and streamwise travelling wave cases. This implies that, when the control parameters used are close to optimal values found at a lower Reynolds number, the drag reduction deteriorates rapidly with increased Reynolds number. In this study, the effect of Reynolds number for the travelling wave is quantified using a scaling in the form $\mathit{Re}_{{\it\tau}}^{-{\it\alpha}}$. No universal constant is found for the scaling parameter ${\it\alpha}$. Instead, the scaling parameter ${\it\alpha}$ has a wide range of values depending on the flow control conditions. Further Reynolds number scaling issues are discussed. Turbulent statistics are analysed to explain a weaker drag reduction observed at high Reynolds numbers. The changes in the Stokes layer and also the mean and root-mean-squared (r.m.s.) velocity with the Reynolds number are also reported. The Reynolds shear stress analysis suggests an interesting possibility of a finite drag reduction at very high Reynolds numbers.

Model equations for three-dimensional, inviscid flow between two arbitrary, time-varying material surfaces are derived using a ‘direct’ or variational approach due to Kantorovich. This approach results in a hierarchy of approximate theories, each of a higher level of spatial approximation and complexity. It can be shown that the equations are equivalent in substance to ‘the theory of directed fluid sheets’ of Green & Naghdi (1974, 1976).

The theory can be used to study the propagation of long waves in water of finite depth and, as such, competes with theories derived using the classical Rayleigh–Boussinesq perturbation methods. In order to demonstrate that there is an advantage to the present approach, we compare predictions for steady, two-dimensional waves over a horizontal bottom. Numerical solutions indicate that the direct theory converges more rapidly than the perturbation theories. Also, the equations of the higher-order direct theories contain singularities related to waves of limiting height, and indeed such waves can be predicted with relative accuracy. Finally, the range of applicability of the direct theory is far greater: waves as short as three times the water depth can be modelled. This is essentially a deep-water condition, well beyond the range of convergence of the Rayleigh–Boussinesq approach.

We consider the slow deformation of a relatively inviscid conducting drop surrounded by a viscous insulating fluid subject to a uniform electric field. The general behaviour is to deform and elongate in the direction of the field. Detailed numerical computations, based on a boundary integral formulation, are presented. For fields below a critical value, we obtain the evolution of the drop to an equilibrium shape; above the critical value, we calculate the drop evolution up to breakup. At breakup it appears that smaller droplets are emitted from the ends of the drop with a charge greater than the Rayleigh limit. As the electric field strength is increased the ejected droplet size decreases. A further increase in field strength results in the mode of breakup changing to a thin jet-like structure being ejected from the end. The shape of all drops is very close to spheroidal up to aspect ratios of about 5. Also, for fields just above the critical value there is a period of slow deformation which increases in duration as the critical field strength is approached from above. Slender-body theory is also used to model the drop behaviour. A similarity solution for the slender drop is obtained and a finite-time singularity is observed. In addition, the general solution for the slender-body equations is presented and the solution behaviour is examined. The slender-body results agree only qualitatively with the full numerical computations. Finally, a spheroidal model is briefly presented and compared with the other models.

Flow control induced by a single wire that is attached on the outer surface and parallel to the span of a stationary circular cylinder is investigated experimentally. The Reynolds number has a value of 10 000 and the wire diameter is nearly two orders of magnitude smaller than the cylinder diameter, while being larger than the thickness of the unperturbed boundary layer forming around the cylinder. A technique of high-image-density particle image velocimetry is used to characterize mean and unsteady structures of the separating shear layer and the near wake. Only one of the shear layers is directly perturbed by the surface wire. This disturbance, however, has important global consequences over the entire near wake, provided that the wire is located within a certain range of angular positions with respect to the approach flow. Over this range, there are two angles that can be defined as critical on the basis of the streamwise extent of the near-wake structure. In a simplified sense, these critical angles are associated with significant extension and contraction of the near wake, relative to the wake in the absence of the effect of a surface disturbance. The critical angle of the wire that yields the most significant extension of the near wake is also found to lead to bistable oscillations of the separating shear layer at irregular time intervals, much longer than the time scale associated with the classical Kármán vortex shedding. The foregoing two critical states of extension and contraction of the near wake are, respectively, linked to attenuation or enhancement of the Kármán instability. Moreover, the onset of the shear-layer instability, Reynolds stress, Strouhal number and the transverse extent of shear-layer flapping are all shown to depend on the angular position of the wire within the defined range of angles.