The steady two-dimensional mixed-convection flow past a heated or cooled horizontal plate of finite length is analysed for large Péclet numbers and weak buoyancy effects. The plate is assumed to be aligned with the free stream. It is shown that the hydrostatic pressure jump across the wake, combined with the Kutta condition, gives rise to a potential flow that can be determined by distributing vortices in the wake as well as in the plate. Solutions in closed form are obtained for two cases, i.e. laminar flow of a fluid with very small Prandtl number and turbulent flow. As a result, a lift force opposite to the buoyancy force and a tangential force opposite to the viscous drag force are found. For the case of laminar flow of a fluid with very small Prandtl number, closed-form solutions are also obtained for the temperature distribution and the heat transfer rate, respectively.

Experiments conducted on mixing across a stable density interface in a turbulent Taylor–Couette flow show, for the first time, experimental evidence of an increase in mixing efficiency at large Richardson numbers. With increasing buoyancy gradient the buoyancy flux first passes a maximum, then decreases and at large values of the buoyancy gradient the flux increases again. Thus, the curve of buoyancy flux versus buoyancy gradient tends to be N-shaped (rather than simply bell shaped), a behaviour suggested by the model of Balmforth et al. (J. Fluid Mech. vol. 428, 1998, p. 349). The increase in mixing efficiency at large Richardson numbers is attributed to a scale separation of the eddies active in mixing at the interface; when the buoyancy gradient is large mean kinetic energy is injected at scales much smaller than the eddy size fixed by the gap width, thus decreasing the eddy turnover time. Observations show that there is no noticeable change in interface thickness when the mixing efficiency increases; it is the mixing mechanism that changes. The curves of buoyancy flux versus buoyancy gradient also show a large variability for identical experimental conditions. These variations occur at time scales one to two orders of magnitude larger than the eddy turnover time scale.

Theoretical and experimental investigations are described of the linearized wave equation of supersonic flow with a single relaxation process. Firing-range experiments using a known model gas mixture with a single relaxation process are discussed. Qualitatively, it is shown that weak conical and spherical waves decay as predicted by theory. In addition, this decay can be determined quantitatively for conical disturbances; and, satisfactory agreement with theory is found. For application to unknown relaxation processes, it is suggested that observation of the damping of weak waves provides an additional tool for the study of fast rate processes.

The transition from laminar to complex spatio-temporal dynamics of plumes due to a localized buoyancy source is studied numerically. Several experiments have reported that this transition is sensitive to external perturbations. Therefore, a well-controlled set-up has been chosen for our numerical study, consisting of a localized heat source at the bottom of an enclosed cylinder whose sidewall is maintained at a fixed temperature which varies linearly up the wall. Restricting the dynamics to the axisymmetric subspace, the first instability is to a puffing state. However, for smaller Grashof numbers, the plume becomes unstable to three-dimensional perturbations and a swirling plume spontaneously appears. The next bifurcation, viewed in the rotating frame where the plume is stationary, also exhibits puffing and suggests a connection between the unstable axisymmetric puffing solution and the swirling plume. Further bifurcations result in quasi-periodic states with a very low-frequency modulation, and these eventually become spatio-temporally complex.

Vibrationally relaxing flow through a nozzle is examined in the case when the amount of energy in the lagging mode is small. It is shown that there exists a ‘boundary-layer’ region in which relatively large departures from equilibrium occur. The position of this region is given by the type of criterion that has previously been used to predict the onset of ‘freezing’. An analytical solution for the distribution of the vibrational energy in the nozzle is obtained for a particular nozzle geometry, and an expression for the final asymptotic ‘frozen’ value of the vibrational energy far downstream is found. This asymptotic solution can be obtained from conditions at the ‘freezing’ point provided a suitable boundary condition is applied there.

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship between pseudoenergy and pseudomomentum , where c is the horizontal phase speed in the direction of symmetry associated with , has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.

The impulsive spin-up of a two-component superfluid and its container is solved analytically for the first time in arbitrary geometry, generalizing the extensively studied case of single-fluid spin-up. The superfluid is modelled by the Hall–Vinen–Bekarevich–Khalatnikov (HVBK) two-fluid equations, neglecting non-hydrodynamic processes like vortex tension and pinning, which are weak in certain applications (e.g. neutron stars) and confined to boundary layers in the inviscid HVBK component in other applications (e.g. helium II spin-up in smooth-walled containers). Both components of the flow are found to be columnar. The spin-up time depends on the geometry, mutual friction coefficients, $B$, ${B}^{\prime } $, the Ekman number $E$, and the superfluid density fraction ${\rho }_{n} $. For $B\sim O(1)$, the inviscid component undergoes Ekman pumping due to strong coupling to the viscous component, and the azimuthal velocities are ‘locked together’ during the spin-up. For $B\lesssim {E}^{1/ 2} $, there is no Ekman pumping in the inviscid component and the inviscid azimuthal velocity spins up via mutual friction on a combination of the mutual friction and Ekman time scales. The spin-up process is studied in spheres, cylinders (with co- and counter-rotating lids), and cones, and occurs faster in spheres and cones. Ekman pumping is (anti-) clockwise adjacent to the (lower-) upper-right boundary, and mirrored about the rotation axis. The solution obtained by Reisenegger (J. Low Temp. Phys., vol. 92, 1993, p. 77) between slowly accelerating parallel plates is recovered in the associated limit. The hydrodynamic torque on the container decays dual-exponentially with time in a cylinder but not in a sphere or cone, where it is a superposition of exponentials whose time scales vary with latitude. The torque is a good diagnostic of the flow; e.g. it is steepest initially in a sphere with strong mutual friction.

Detailed observations have been performed on the evolution of a viscous catenary, a rope of high-viscosity fluid suspended from two points falling under gravity. Stroboscopic imaging techniques are used to obtain the position and shape of the strand as a function of time. Depending on their initial thickness and profile, the filaments are observed to evolve into either a quasi-catenary, or other, more complex shapes. A conceptually simple, energy-based theory is developed and compared with observations. It is shown to describe reasonably, except for a scaling in the time scale, the catenary-like regime.

A uniformly valid asymptotic solution has been constructed for three-dimensional jet-flapped wings by the method of matched asymptotic expansions for high aspect ratios. The analysis assumes that the flow is inviscid and incompressible and is formulated on the thin airfoil theory in accordance with the well-established Spence (1961) theory in two dimensions.

A simple method emerges in treating the bound vortices along the jet sheet which forms behind the wing with the aid of the following physical picture. Three distinct flow regions—namely inner, outer and Trefitz—exist in the problem. Close to the wing the flow approximates to that in two dimensions. Therefore, Spence's solution in two dimensions applies. In the outer region a wing shrinks to a line of singularities from which the main disturbances of flow in this region arise. In particular, we find that the shape of the jet sheet, hence the strength of vortices, is now predetermined by the strength of the singularities there. Hence a complete flow field in the outer region can now be determined first by evaluating the flow due to various degrees of singularities along this line and then adding the effect of the jet bound vortices which is now known. Far removed from the wing, the well-known Trefftz region exists in which calculations of aerodynamic forces can be most easily done.

The result has been applied to various wing planforms such as cusped, elliptic and rectangular wings. The present result breaks down for rectangular wings. However, we can apply Stewartson's (1960) solution for lifting-line theory for semi-infinite rectangular wings, because, to this second-order approximation it is established that the jet sheet in the outer region makes no contribution to lift, with the direct contribution of the deflected jet at the exit being cancelled by the reduced circulation in the jet vortices. This result for the rectangular wing gives excellent agreement with the experiments made on a rectangular wing, while the result for elliptic wings underestimates them considerably.

Numerical finite-difference results from the full axisymmetric incompressible Navier–Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a long cylindrical container. The governing parameters are the Ekman number, E = ν*/(Ω*a*2), Rossby number, Ro = W*/(Ω*a*), and the dimensionless height of the container, 2H (the scaling length is the radius of the particle, a*; Ω* is the container angular velocity, W* is the particle axial velocity and ν* the kinematic viscosity). The study concerns the flow field for small values of E and Ro while HE is of order unity, and hence the appearance of a free Taylor column (slug) of fluid ‘trapped’ at the particle is expected. The numerical results are compared with predictions of previous analytical approximate studies. First, developed (quasi-steady-state) cases are considered. Excellent agreement with the exact linear (Ro = 0) solution of Ungarish & Vedensky (1995) is obtained when the computational Ro = 10−4. Next, the time-development for both an impulsive start and a start under a constant axial force is considered. A novel unexpected behaviour has been detected: the flow field first attains and maintains for a while the steady-state values of the unbounded configuration, and only afterwards adjusts to the bounded container steady state. Finally, the effects of the nonlinear momentum advection terms are investigated. It is shown that when Ro increases then the dimensionless drag (scaled by μ*a*W*) decreases, and the Taylor column becomes shorter, this effect being more pronounced in the rear region (μ* is the dynamic viscosity). The present results strengthen and extend the validity of the classical drag force predictions and therefore the issue of the large discrepancy between theory and experiments (Maxworthy 1970) concerning this force becomes more acute.

The surge studied in this paper is due to the sudden closure of the sink of a bathtub vortex. Experimental evidence established that the surge is not initially a wave but is in fact a slug of fluid which has, for a very short period of time, a distinctive motion of its own. The purpose of this paper is to present the salient features of an experimental investigation of the effect of circulation, geometry, and viscosity on the speed of the surge and to report some rather unexpected observations.

The influence of time-periodic vibrations on long Marangoni waves in two-layer films is investigated. The problem is governed by a system of nonlinear equations obtained in the framework of the lubrication approximation. Periodic boundary conditions are applied on the boundaries of the computational region. The development of instabilities is investigated by means of nonlinear simulations. Excitation of two-dimensional and three-dimensional subharmonic wavy regimes is studied. A new phenomenon, the excitation of nonlinear waves with a temporal period that is four times larger than that of the gravity modulation, is revealed.

Numerical simulations of the surface quasigeostrophic patch indicate the development of a scale-invariant singularity of the boundary curvature in finite time, with some evidence of universality across a variety of initial conditions. At the time of singularity, boundary segments are shown to possess an exact and simple analytic form, described by branches of a logarithmic spiral centred on the point of singularity. The angles between the branches depend non-trivially on the shape of the smooth connecting boundary as the singularity is approached, but are independent of the global boundary.

This paper describes an experimental investigation of the conditions for which the asymptotic description of longitudinal dispersion given by Taylor (1954) would apply. At non-dimensional times following the release of a dye pulse that are significantly larger than those previously investigated, the integrated concentration curves were observed to be skewed. At relatively short times from release the concentration curves appear to be well described by the models presented by Sullivan (1971) and by Chatwin (1973). Some features of the asymptotic behaviour, namely the translation of the modal value of the integrated concentration curve at the discharge velocity and the constant temporal growth rate of the variance, are observed at the longest times following release. On the basis of these observations it is estimated that a non-dimensional time interval of tu*/d = O(105/R*), where R* = u*d/v, u* is the friction velocity, v the kinematic viscosity and d the tube diameter, is required for the Taylor result to become applicable. Thus application of Taylor's theory is significantly restricted in turbulent flows, especially those with irregular boundaries and those that are not stationary. There the variations in the flow must be small with respect to an equivalent ‘development time’ if a value of the ‘local’ longitudinal diffusion coefficient is to have meaning.

Split-film electrochemical transducer theory is briefly outlined so as to show its ability to separate statistical properties of each component of the surface velocity gradient in regions as different as a boundary layer, a separated zone and a three-dimensional separation line. A lot of measurements of the flow field near a cylinder which are not yet available elsewhere are carried out everywhere on the cross-section of a yawed cylinder. A direct verification of the independence principle and Wild's thermal analogy in separated zones is executed. The periodic and random parts of each component of the surface velocity gradient are separated. Cross-spectral analysis between any pair of points shows that the periodic surface flow is coherent upon any cross-section and in a large region in the spanwise direction. The stationary wave system is investigated in space and time. The reconstitution, instant by instant, of the averaged integral wall streamlines leads to an understanding of the synchronization of the natural oscillations. The four main stagnation or separation lines quickly move from one extreme position to the other, and at certain times contrary vortices stretching in the direction of the generators can appear. In the rearward stagnation region random small-scale fluctuations are probably turbulent, but in the intermediate region high-level random fluctuations in the spanwise direction are much more coherent and probably contribute to keeping the flow in phase at large distances.

An investigation has been made of the structure of the motion above a heated plate inclined at a small angle (about 10°) to the horizontal. The turbulence is considered in terms of the similarities to and differences from the motion above an exactly horizontal surface. One effect of inclination is, of course, that there is also a mean motion.

Accurate data on the mean temperature field and the intensity of the temperature fluctuations have been obtained with platinum resistance thermometers, the signals being processed electronically. More approximate information on the velocity field has been obtained with quartz fibre anemometers. These results have been supplemented qualitatively by simultaneous observations of the temperature and velocity fluctuations and also by smoke experiments.

The principal features of the flow inferred from these observations are as follows. The heat transfer and the mean temperature field are not much altered by the inclination, though small, not very systematic, variations may result from the complexities of the velocity field. This supports the view that the mean temperature field is largely governed by the large-scale motions. The temperature fluctuations show a systematic variation with distance from the lower edge and resemble those above a horizontal plate when this distance is large. The largescale motions of the turbulence start close to the lower edge, but the smaller eddies do not attain full intensity until the air has moved some distance up the plate. The mean velocity receives a sizable contribution from a ‘through-flow’ between the side-walls. Superimposed on this are developments that show that the momentum transfer processes are complex and certainly not capable of representation by any simple theory such as an eddy viscosity. On the lower part of the plate there is surprisingly large acceleration, but further up the mixing action of the small eddies has a decelerating effect.

Numerical studies of two-dimensional, transonic flows of dense gases of retrograde type, known as BZT gases, around thin airfoils are presented. The computations are guided by a recent asymptotic theory of Rusak & Wang (1997). It provides a uniformly valid solution of the flow around the entire airfoil surface which is composed of outer and inner solutions. A new transonic small-disturbance (TSD) equation solver is developed to compute the nonlinear BZT gas flow in the outer region around most of the airfoil. The flow in the inner region near the nose of the airfoil is computed by solving the problem of a sonic flow around a parabola. Numerical results of the composite solutions calculated from the asymptotic formula are compared with the solutions of the Euler equations. The comparison demonstrates that, in the leading order, the TSD solutions of BZT gas flows represent the essence of the flow character around the airfoil as computed from the Euler equations. Furthermore, guided by the asymptotic formula, the computational results demonstrate the similarity rules for transonic flows of BZT gases. There are differences between the self-similar cases that may be related to the error associated with the accuracy of the asymptotic solution. A discussion on the flow patterns around an airfoil at transonic speeds and at various upstream thermodynamic conditions is also presented. The paper provides important guidelines for future studies on this subject.

Steady detonation in multi-dimensional flow is controlled by the chemical energy release that occurs in a subsonic elliptic flow region known as the detonation driving zone (DDZ). It is the region encompassing the detonation shock and sonic flow locus (in the frame of the detonation shock). A detonation that is strongly confined by material surrounding the explosive has the shock and sonic locus separated at the material interface. Information about the material boundary is traditionally believed to influence the DDZ structure via the subsonic flow on the boundary ahead of the sonic locus. A detonation that is weakly confined has the detonation shock and sonic locus intersecting at the material boundary. The sonic nature of the flow at the intersection point on the boundary is believed to isolate the DDZ structure from the material properties of the confinement. In this study, we examine the paths of characteristics propagating information about the confinement through the supersonic hyperbolic flow region that exists beyond the sonic locus, and determine whether these paths may impinge on the sonic locus and consequently influence the DDZ structure. Our configuration consists of a solid wall boundary deflected through a specified angle on detonation shock arrival, so that the streamline turning angle of the wall at the explosive edge is unambiguously defined. By varying the wall deflection angle from small through large values, we can systematically capture the evolution of the DDZ structure and the characteristic flow regions that influence its structure for strongly to weakly confined detonations. In all strong and weak confinement cases examined, we find that a subset of characteristics from the supersonic flow regions always impinge on the sonic locus. Limiting characteristics are identified that define the boundary between characteristics that impinge on the sonic surface and those that propagate information downstream of the sonic surface. In combination with an oblique-shock polar analysis, we show that the effects on the DDZ of characteristic impingement can be significant.

Measurements of the intermittency factor γ and in particular the crossing frequency fγ of the turbulent/non-turbulent interface in the outer regions of various turbulent shear flows depend strongly on the settings of the intermittency meter used. Two methods of calibrating an intermittency meter of conventional design are described. In the first, turbulent and non-turbulent signals are simulated and switched at random times using an analog computer. Particular attention is given to the spectra of the switching and turbulent signals but the non-turbulent signal is assumed to have the same spectrum as the turbulent signal. In the second method, the same switching process is applied to two real signals, obtained in the fully turbulent and irretational flow regions associated with a turbulent jet with a co-flowing external air stream. A rather simple calibration procedure derived using the results of both methods is applied to the measurements of γ and fγ in the same jet. It is suggested that the simulation process adopted here could be useful in inferring properties of intermittent turbulent flows.

The velocity fluctuations of particles in a low-Reynolds-number fluidized bed have important similarities and differences with the velocity fluctuations in a low-Reynolds-number sedimenting suspension. We show that, like sedimentation, the velocity fluctuations in a fluidized bed are described well by the balance between density fluctuations due to Poisson statistics and Stokes drag. However, unlike sedimentation, the correlation length of the fluctuations in a fluidized bed increases with volume fraction. We argue that this difference arises because the relaxation time of density fluctuations is completely different in the two systems.