Measurements of the mean flow, intermittent structure and turbulent fluctuations were made in a cold-wall boundary layer at a stream Mach number of 9·4 and Reynolds number based on momentum thickness of 36 800. For these conditions, the r.m.s. sublayer thickness was 32 times smaller than that of the boundary layer proper, and the interfacial standard deviation of the latter was about three times proportionately smaller than has been found at low speeds. The mean flow data, which extended well into the sublayer, revealed a large increase in static pressure from the layer edge to the wall and a quadratic law relation between the total temperature and velocity. While the transformed velocity profile was in good agreement with the incompressible law of the wake, no indication of a linear variation of velocity in the sublayer was detected.

Hot-wire fluctuation data, interpreted with the use of appropriate assumptions concerning the nature of the sound field, indicated that the turbulence is dominated by high-frequency pressure fluctuations whose magnitude at the wall and beyond the layer edge agree with extrapolation of data acquired at supersonic speeds. The static temperature fluctuations agreed with expectations from adiabatic, supersonicdata apparently because they were suppressed by the cooled-wall condition. The fluctuations in the longitudinal velocity component were generally small and differed little from lower Mach number results. The high turbulence Reynolds numbers found generated an inertial-subrange spectral decay, while the longitudinal integral scales were found independent of turbulence mode and about one-fifth the boundary-layer thickness.

To date, analytical models of internal tide generation by two-dimensional ridges have considered only idealized shapes. Here, we advance the Green function approach to address the generation of internal tides by two-dimensional topography of arbitrary shape, employing the Wentzel-Kramers-Brillouin (WKB) approximation to consider the impact of non-uniform stratifications. This allows for a more accurate analytical estimation of tidal conversion rates. Studies of single and double ridges reveal that the conversion rate and the nature of the radiated internal tide can be sensitive to the topographic shape, particularly around criticality and when there is interference between wave fields generated by neighbouring ridges. The method is then applied to the study of two important internal tide generation sites, the Hawaiian and Luzon Ridges, where it captures key features of the generation process.

We consider laminar displacement flows in narrow eccentric annuli, oriented horizontally, between two fluids of Herschel–Bulkley type, (i.e. including Newtonian, power-law and Bingham models). This situation is modelled via a Hele-Shaw approach. Whereas slumping and stratification would be expected in the absence of any imposed flow rate, for a displacement flow we show that there are often steady-state travelling wave solutions in this displacement. These may exist even at large eccentricities and for large density differences between the fluids. When heavy fluids displace light fluids, annular eccentricity opposes buoyancy and steady states are more prevalent than when light fluids displace heavy fluids. For large ratios of buoyancy forces to viscous forces we derive a lubrication-style displacement model. This simplification allows us to find necessary and sufficient conditions under which a displacement can be steady, which can be expressed conveniently in terms of a consistency ratio. It is interesting that buoyancy does not appear in the critical conditions for a horizontal well. Instead a competition between fluid rheologies and eccentricity is the determining factor. Buoyancy acts only to determine the axial length of the steady-state profile.

Inviscid computational results are presented on a self-propelled swimmer modelled as a virtual body combined with a two-dimensional hydrofoil pitching intermittently about its leading edge. Lighthill (Proc. R. Soc. Lond. B, vol. 179, 1971, pp. 125–138) originally proposed that this burst-and-coast behaviour can save fish energy during swimming by taking advantage of the viscous Bone–Lighthill boundary layer thinning mechanism. Here, an additional inviscid Garrick mechanism is discovered that allows swimmers to control the ratio of their added-mass thrust-producing forces to their circulatory drag-inducing forces by decreasing their duty cycle, $DC$, of locomotion. This mechanism can save intermittent swimmers as much as 60 % of the energy it takes to swim continuously at the same speed. The inviscid energy savings are shown to increase with increasing amplitude of motion, increase with decreasing Lighthill number, $Li$, and switch to an energetic cost above continuous swimming for sufficiently low $DC$. Intermittent swimmers are observed to shed four vortices per cycle that give rise to an asymmetric time-averaged jet structure with both momentum surplus and deficit branches. In addition, previous thrust and power scaling laws of continuous self-propelled swimming are further generalized to include intermittent swimming. The key is that by averaging the thrust and power coefficients over only the bursting period then the intermittent problem can be transformed into a continuous one. Furthermore, the intermittent thrust and power scaling relations are extended to predict the mean speed and cost of transport of swimmers. By tuning a few coefficients with a handful of simulations these self-propelled relations can become predictive. In the current study, the mean speed and cost of transport are predicted to within 3 % and 18 % of their full-scale values by using these relations.

We consider the onset of thermosolutal (double-diffusive) convection of a binary fluid in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the bounding surfaces, in the case when the solubility of the dissolved component depends on temperature. We use a linear stability analysis to investigate how the dissolution or precipitation of this component affects the onset of convection and the selection of an unstable wavenumber; we extend this analysis using a Galerkin method to predict the structure of the initial bifurcation and compare our analytical results with numerical integration of the full nonlinear equations. We find that the reactive term may be stabilizing or destabilizing, with subtle effects particularly when the thermal gradient is destabilizing but the solutal gradient is stabilizing. The preferred spatial wavelength of convective cells at onset may also be substantially increased or reduced, and strongly reactive systems tend to prefer direct to subcritical bifurcation. These results have implications for geothermal-reservoir management and ore prospecting.

Under some conditions of temperature and flow an ice-water interface in the presence of a turbulent stream has been observed to be unstable. In this paper the source and the conditions for the instability were investigated for a well-defined turbulent boundary-layer flow. It was found that the instability resulted from the interaction that occurs between a wavy surface and a turbulent flow over it. Such an interaction results in a heat transfer variation which is 90 to 180 degrees out of phase with the surface wave shape – a result which is consistent with the calculations of Thorsness & Hanratty (1979a, b).

The main factor controlling damping of the instability at an ice-water interface was found to be the rate at which heat is conducted away from the interface into the ice.

In the past it has been found that when an ice layer is melting, that is when the heat conduction in the ice is small, the ice surface is highly unstable. In the present study it was found that for a sufficiently large temperature ratio (Tf−Tw)/(T∞−Tf), a steady-state ice layer is also unstable. Furthermore it is predicted, from the present observations, that a growing ice layer with a ratio of ice-side to water-side heat fluxes of up to 2.3 could be unstable.

Under sufficiently unstable conditions waves on the ice surface grow to an amplitude at which flow separations occur near the wave crests. This results in a ‘rippled’ ice surface pattern very similar to the patterns observed on mobile bed surfaces (Kennedy 1969) or surfaces which are being dissolved into a flowing stream (Allen 1971). The development of a ‘rippled’ ice surface results in a very substantial increase in the mean heat-transfer rate which would have an important influence on predictions of ice formation in the presence of a turbulent stream.

The topology of the organized motion has been obtained in the slightly heated self-preserving far wake of a circular cylinder at a Reynolds number, based on the cylinder diameter, of about 1200. In a frame of reference moving with the organized motion, the toplogy in the plane of main shear reduces to a succession of centres and saddles, located at about the wake half-width. Centres are identifiable by large values of spanwise vorticity associated with the coherent large-scale motion. Saddles occur at the intersection of converging and diverging separatrices, the latter being identifiable with the high strain rate due to the large-scale motion. Large values of the longitudinal turbulence intensity associated with the smaller-scale motion occur at the centres. High values of the normal and shear stresses, the temperature variance and heat fluxes associated with the large-scale motion occur on either side of each saddle point along the direction of the diverging separatrix. Contours for the production of energy and temperature variance associated with the small-scale motion are aligned along the diverging separatrices, and have maxima near the saddle point. Contours for one component of the dissipation of small-scale temperature variance also have a high concentration along the diverging separatrix. Flow visualizations in the far wake suggest the existence of groups of three-dimensional bulges which are made up of clusters of vortex loops.

The wake characteristic of groups of normal flat plates, consisting of two, three, or four plates placed side by side with slits in between, have been investigated experimentally. When the ratio of the slit width to the plate width (slit ratio) was small, the gap flows were observed to be biased either upward or downward in a stable way, leading to multiple, stable flow patterns for a single slit-ratio value. Some regularities were recognized in the gap-flow directions and the appearance of the flow patterns. The plates on the biased side showed high drag and regular vortex shedding, while those on the unbiased side showed the opposite. The origin of the biased flow has also been investigated with water-tank experiments, numerical calculations and wind-tunnel experiments. The-results showed that the origin of biasing is strongly related to the vortex shedding of each plate of a row.

Using the slender-body theory for Stokes flow, the equations of motion are developed for a small flexible inextensible thread. The nearly straight thread is examined analytically, and is shown to straighten rapidly. In a simple shear flow the distortions decay less rapidly, but rapidly enough not to rotate the thread through the plane of flow. Numerical studies of simple shear with more substantially distorted threads show the same qualitative behaviour. Additionally some differences are revealed in the nonlinear regime between the buckling and stretching processes which occur in the compressive and tensile quadrants of the flow.

This paper presents a generalization of the classical results of T. B. Benjamin (J. Fluid. Mech. vol. 31, 1968, p. 209) concerning the propagation of a steady gravity current into a homogeneous ambient, to the case of a stratified ambient. The current of thickness $h$ and density $\roh_c$ propagates, with speed $U$, at the bottom of a long horizontal channel of height $H$, into the unperturbed ambient whose density increases linearly from $\roh_o$ (at the top) to $\roh_b$ (at the bottom). The reduced gravity is $g^{\prime} \,{=}\, (\roh_c/\roh_o -1)g$ and the governing parameters are $a \,{=}\, h/H$ and $S \,{=}\, (\roh_b-\roh_o)/(\roh_c-\roh_o)$, with $0<a<1, 0 < S < 1 $; here $g$ is the acceleration due to gravity. For a Boussinesq high-Reynolds two-dimensional configuration, a flow-field solution of Long's model, combined with flow-force balance over the width of the channel, are used for obtaining the desired results, in particular: $\hbox{\it Fr} \,{=}\, U/(g^{\prime}h)^{1/2}$, head loss (dissipation), and criticality of $U$ with respect to the fastest internal wave mode. The classical results of Benjamin are fully recovered for $S \rightarrow 0$. For small $S$ and fixed $a$, the values of $\hbox{\it Fr}$ and head loss are shown to decrease with $S$ like $(1-2S/3)^{1/2}$ and $(1-2S/3)$, respectively, and the propagation is supercritical. For larger $S$ several solutions are possible (for a given geometry $a$), mostly in the subcritical regime. Considerations for the physical acceptability of the multiple results are presented, and the connection with observations from lock-release experiments are discussed. The conclusion is that the present results provide a reliable and versatile generalization of the classical unstratified problem.

We consider the motion of a fluid-immersed negatively buoyant particle in the vicinity of a thin compressible elastic wall, a situation that arises in a variety of technological and natural settings. We use scaling arguments to establish different regimes of sliding, and complement these estimates using thin-film lubrication dynamics to determine an asymptotic theory for the sedimentation, sliding and spinning motions of a cylinder. The resulting theory takes the form of three coupled nonlinear singular-differential equations. Numerical integration of the resulting equations confirms our scaling relations and further yields a range of unexpected behaviours. Despite the low-Reynolds-number feature of the flow, we demonstrate that the particle can spontaneously oscillate when sliding, can generate lift via a Magnus-like effect, can undergo a spin-induced reversal effect and also shows an unusual sedimentation singularity. Our description also allows us to address a sedimentation–sliding transition that can lead to the particle coasting over very long distances, similar to certain geophysical phenomena. Finally, we show that a small modification of our theory allows us to generalize the results to account for additional effects such as wall poroelasticity.

Surface streamline patterns on a spheroid have been examined at several angles of attack. Most of the tests were performed at low Reynolds numbers in a hydraulic flume using coloured dye to make the surface flow visible. A limited number of experiments was also carried out in a wind tunnel, using wool tufts, to study the influence of Reynolds number and turbulent separation. The study has verified some of the important qualitative features of three-dimensional separation criteria proposed earlier by Maskell, Wang and others. The observed locations of laminar separation lines on a spheroid at various incidences have been compared with the numerical solutions of Wang and show qualitative agreement. The quantitative differences are attributed largely to the significant viscous-inviscid flow interaction which is present, especially at large incidences.

An experimental study of transition from steady laminar to chaotic flow in a horizontal annulus between concentric cylinders is conducted for 0.90×105[les ]RaL[les ]3.37×105. Qualitative information on the averaged thermal field is obtained by an interferometric method, whereas the kinematics are visualized by using smoke lines. The transition from steady to unsteady mono- and multi-periodic regimes is accurately studied by a CTA and CCA combined hot-wire technique. Different data analyses are performed in order to differentiate the forms of transition.

The behaviour of a pair of corotating vortices in the wake of a flapped airfoil is experimentally studied in a water towing tank. Reynolds numbers based on total circulation of the vortices range from 1.0×104 to 6.4×104. Planar velocity vector fields and their gradients are derived from PIV images using an adaptive Lagrangian parcel tracking algorithm. Isovorticity surfaces are extracted from time series of planar vorticity data. The behaviour of the vortices is tracked by using various moments of both the probability density distribution and the spatial distribution of their streamwise vorticity. All vortices show a Lamb–Oseen circulation distribution when they are clearly identifiable. Further, vortices from the apless wing exhibit Lamb–Oseen velocity and vorticity distributions with slow growth. All corotating vortex pairs are observed to merge at about 0.8 orbit periods. First-order statistics of the flow field remain invariant during the merger. The higher-order moments of the vorticity distribution show strong time dependence, which implies three-dimensionality of the flow resulting from vortex stretching. The strengths of the individual vortices before merger are constant, and the total circulation before and after merger remains constant within the range of observations. The trajectory of the centre of vorticity remains unaffected by the merger process. The merger is preceded by a splitting of the weaker vortex into filaments which, depending on the relative strengths of the vortices, can occur in the radial direction, the axial direction, or a combination of the two. Mechanisms contributing to the merger dynamics are discussed.

A perturbation expansion of the Boltzmann equation is used to derive constitutive relations for the granular flow of rough spheres in the limit where the energy dissipation in a collision is small compared to the energy of a particle. In the collision model, the post-collisional relative normal velocity at the point of contact is $-e_n$ times the pre-collisional normal velocity, and the post-collisional relative tangential velocity at the point of contact is $-e_t$ times the pre-collisional relative tangential velocity. A perturbation expansion is employed in the limit $(1 - e_n)\,{=}\,\varepsilon^2 \ll 1$, and $(1 - e_t^2) \propto \varepsilon^2 \ll 1$, so that $e_t$ is close to $\pm 1$. In the ‘rough’ particle model, the normal coefficient of restitution $e_n$ is close to 1, and the tangential coefficient of restitution $e_t$ is close to 1. In the ‘partially rough’ particle model, the normal coefficient of restitution $e_n$ is close to 1; and the tangential coefficient of restitution $e_t$ is close to $-1$ if the angle between the relative velocity vector and the line joining the centres of the particles is greater than the ‘roughness angle’ (chosen to be $(\upi/4)$ in the present calculation), and is close to 1 if the angle between the relative velocity vector and the line joining the centres is less than the roughness angle. The conserved variables in this case are mass and momentum; energy is not a conserved variable in the ‘adiabatic limit’ considered here, when the length scale is large compared to the ‘conduction length’. The results for the constitutive relations show that in the Navier–Stokes approximation, the form of the constitutive relation is identical to that for smooth particles, but the coefficient of shear viscosity for rough particles is 10%–50% higher than that for smooth particles. The coefficient of bulk viscosity, which is zero in the dilute limit for smooth particles, is found to be non-zero for rough and partially rough particles, owing to the transport of energy between the translational and rotational modes. In the Burnett approximation, there is an antisymmetric component in the stress tensor for rough and partially rough particles, which is not present for smooth particles.

The constitutive relations are used to analyse the ‘core region’ of a steady granular flow down an inclined plane, where there is a local balance between the production of energy due to the mean shear and the dissipation due to inelastic collisions. It is found that realistic results, such as the decrease in density upon increase in the angle of inclination near close packing, are obtained for the rough and partially rough particle models when the Burnett coefficients are included in the stress tensor, but realistic results are not obtained using the constitutive relations for smooth particles. This shows that the flow dynamics is sensitive to the numerical values of the viscometric coefficients, and provides an indication of the minimal model required to capture the flow dynamics.

We present a continuum model for melt water drainage through a spatially distributed system of connected subglacial cavities, and consider in this context the complications introduced when effective pressure or water pressure drops to zero. Instead of unphysically allowing water pressure to become negative, we model the formation of a partially vapour- or air-filled space between ice and bed. Likewise, instead of allowing sustained negative effective pressures, we allow ice to separate from the bed at zero effective pressure. The resulting model is a free boundary problem in which an elliptic obstacle problem determines hydraulic potential, and therefore also determines regions of zero effective pressure and zero water pressure. This is coupled with a transport problem for stored water, and the coupled system bears some similarities with Hele-Shaw and squeeze-film models. We present a numerical method for computing time-dependent solutions, and find close agreement with semi-analytical travelling wave and steady-state solutions. As may be expected, we find that ice–bed separation is favoured by high fluxes and low ice surface slopes and low bed slopes, while partially filled cavities are favoured by low fluxes and high slopes. At the boundaries of regions with zero water or effective pressure, discontinuities in water level are frequently present, either in the form of propagating shocks or as stationary hydraulic jumps accompanied by discontinuities in potential gradient.

The impact of a nearly cylindrical water mass on a water surface is studied both experimentally and theoretically. The experiments consist of the rapid release of water from the bottom of a cylindrical container suspended above a large water tank and of the recording of the free-surface shape of the resulting crater with a high-speed camera. A bubble with a diameter of about twice that of the initial cylinder remains entrapped at the bottom of the crater when the aspect ratio and the energy of the falling water mass are sufficiently large. Many of the salient features of the phenomenon are explained on the basis of simple physical arguments. Boundary-integral potential-flow simulations of the process are also described. These numerical results are in fair to good agreement with the observations.

Very-large-scale motions (VLSMs) and large-scale motions (LSMs) coexist at moderate Reynolds numbers in a very long open channel flow. Direct numerical simulations two-way coupled with inertial particles are analysed using spectral information to investigate the modulation of VLSMs. In the wall-normal direction, particle distributions (mean/preferential concentration) exhibit two distinct behaviours in the inner flow and outer flow, corresponding to two highly anisotropic turbulent structures, LSMs and VLSMs. This results in particle inertia’s non-monotonic effects on the VLSMs: low inertia (based on the inner scale) and high inertia (based on the outer scale) both strengthen the VLSMs, whereas moderate and very high inertia have little influence. Through conditional tests, low- and high-inertia particles enhance VLSMs following two distinct routes. Low-inertia particles promote VLSMs indirectly through the enhancement of the regeneration cycle (the self-sustaining mechanism of LSMs) in the inner region, whereas high-inertia particles enhance the VLSM directly through contribution to the Reynolds shear stress at similar temporal scales in the outer region. This understanding also provides more general insight into inner–outer interaction in high-Reynolds-number, wall-bounded flows.

We examine the dynamics of a binary mixture in a cubic cell subjected to a temperature differential and oscillatory forcing. The Soret effect, which is negative in the present study, provides a coupling mechanism by which a temperature gradient establishes a concentration gradient in a mixture. We present the results of experiments that were performed on the International Space Station (ISS) and compare the observations with the results of direct numerical simulations. The evolution of temperature and concentration fields is investigated by optical digital interferometry. One advantage of the experimental technique is the observation of the fields along two perpendicular directions of the cell, allowing us to restore the three-dimensional field. Experimental evidence disproves speculations that the ISS microgravity environment always affects diffusion-controlled processes. Furthermore, we demonstrate that imposed vibrations with constant frequency and amplitude create slow mean flows and that they do influence the diffusion kinetics. The perturbation of the diffusive fields scales as the square of the vibrational velocity. In addition to calculations of the full three-dimensional Navier–Stokes equations, a two-time-scale computational methodology is used for situations in which the forcing period is very small compared to the natural time scales of the problem. The simulations show excellent agreement with experimental observations.

An inconsistency in a previous stability analysis of a cylindrical vortex sheet is resolved. It is found that disturbances with azimuthal wavenumbers m = 1 and 2 are stable, whereas previously the stability of these modes was uncertain. It has been hypothesized that the multiple-vortex phenomenon is initiated by such an instability (Ward 1972). Since double vortices (m = 2) arise, we are led to consider a central downdraft surrounded by a uniform updraft; for a non-zero vertical wavenumber γ, the modes m = 1 and 2 are destabilized. Our theory is supported by the observation that double vortices tend to form as intertwining spirals (i.e. they have vertical structure).