Numerical and experimental techniques were used to study the physics of flow separation for steady internal flow in a 45° junction geometry, such as that observed between two pipes or between the downstream end of a bypass graft and an artery. The three-dimensional Navier–Stokes equations were solved using a validated finite element code, and complementary experiments were performed using the photochromic dye tracer technique. Inlet Reynolds numbers in the range 250 to 1650 were considered. An adaptive mesh refinement approach was adopted to ensure grid-independent solutions. Good agreement was observed between the numerical results and the experimentally measured velocity fields; however, the wall shear stress agreement was less satisfactory. Just distal to the ‘toe’ of the junction, axial flow separation was observed for all Reynolds numbers greater than 250. Further downstream (approximately 1.3 diameters from the toe), the axial flow again separated for Re [ges ] 450. The location and structure of axial flow separation in this geometry is controlled by secondary flows, which at sufficiently high Re create free stagnation points on the model symmetry plane. In fact, separation in this flow is best explained by a secondary flow boundary layer collision model, analogous to that proposed for flow in the entry region of a curved tube. Novel features of this flow include axial flow separation at modest Re (as compared to flow in a curved tube, where separation occurs only at much higher Re), and the existence and interaction of two distinct three-dimensional separation zones.

We investigate the effect of the finite horizontal boundary properties on the critical Rayleigh and wave numbers for controlled Rayleigh–Bénard convection in an infinite horizontal domain. Specifically, we examine boundary thickness, thermal diffusivity and thermal conductivity. Our control method is through perturbation of the lower-boundary heat flux. A linear proportional-differential control method uses the local amplitude of a shadowgraph to actively redistribute the lower-boundary heat flux. Realistic boundary conditions for laboratory experiments are selected. Through linear stability analysis we examine, in turn, the important boundary properties and make predictions of the properties necessary for successful control experiments. A surprising finding of this work is that for certain realistic parameter ranges, one may find an isola to time-dependent convection as the primary bifurcation.

The evaporating meniscus of a perfectly wetting fluid exhibits an apparent contact angle Θ that is a function of superheat. Existing theory predicts Θ and the heat flow from the contact region as part of the solution of a free-boundary problem. That theory admits the possibility that much of the heat flow occurs at the nanometre scale [lscr ]Θ at which Θ is determined. Here, the heat flow at that scale is proved negligible in typical applications. A phenomenological model of the contact region then holds since the part of the wetting film thinner than [lscr ]Θ can be replaced by an apparent contact line. Self-consistency arguments are used to derive conditions under which (i) the phase interface can be taken as linear with assumed contact angle Θ; (ii) the heat flux to the liquid side of the phase interface is given by Newton's law of cooling with predicted heat transfer coefficient h; and (iii) the temperature satisfies Laplace's equation within the phases. When these conditions are met, prediction of the heat flow is decoupled from the physically non-trivial problem of predicting Θ. Next, this conduction theory is used to find the heat flow from the contact region of a meniscus on a conductive slab. The solution depends on Θ, the liquid–solid conductivity ratio k = K[lscr ]/Ks and a Biot number [Bscr ] = hd/K[lscr ] based on slab thickness d. Asymptotic and numerical analysis is used to find the temperature in the double limit [Bscr ]−1 → 0 and k → 0. The solution has an inner-and-outer structure, and properties of the inner region prove universal. Formulae given here for the heat flow and contact line temperature on a slab thus apply to more complex geometries. Further, the solution explains the main features seen in published simulations of evaporation from conductive solids. Near the contact line, the solid temperature varies rapidly on the scale d of the slab thickness, but varies slowly with respect to the liquid temperature. The solid temperature thus proves uniform at the scale on which Θ is determined. Lastly, the quantitative predictions of the simplified model are verified against both new and published numerical solutions of the existing theory. In typical applications, the new formulae give the heat flow and contact line temperature with an error of about 10%. This error is due to the approximations made to derive the simplified model, rather than to those made to solve it.

The method of matched asymptotic expansions is used to describe the sound generated by the interaction between a short-wavelength gust (reduced frequency k, with k [Gt ] 1) and an airfoil with small but non-zero thickness, camber and angle of attack (which are all assumed to be of typical size O(δ), with δ [Lt ] 1) in transonic flow. The mean-flow Mach number is taken to differ from unity by O(δ2/3), so that the steady flow past the airfoil is determined using the transonic small-disturbance equation. The unsteady analysis is based on a linearization of the Euler equations about the mean flow. High-frequency incident vortical and entropic disturbances are considered, and analogous to the subsonic counterpart of this problem, asymptotic regions around the airfoil highlight the mechanisms that produce sound. Notably, the inner region round the leading edge is of size O(k−1), and describes the interaction between the mean-flow gradients and the incident gust and the resulting acoustic waves. We consider the preferred limit in which kδ2/3 = O(1), and calculate the first two terms in the phase of the far-field radiation, while for the directivity we determine the first term (δ = 0), together with all higher-order terms which are at most O(δ2/3) smaller – in fact, this involves no fewer than ten terms, due to the slowly-decaying form of the power series expansion of the steady flow about the leading edge. Particular to transonic flow is the locally subsonic or supersonic region that accounts for the transition between the acoustic field downstream of a source and the possible acoustic field upstream of the source. In the outer region the sound propagation has a geometric-acoustics form and the primary influence of the mean-flow distortion appears in our preferred limit as an O(1) phase term, which is particularly significant in view of the complicated interference between leading- and trailing-edge fields. It is argued that weak mean- flow shocks have an influence on the sound generation that is small relative to the effects of the leading-edge singularity.

The two-dimensional scattering of water waves over a finite region of arbitrarily varying topography linking two semi-infinite regions of constant depth is considered. Unlike many approaches to this problem, the formulation employed is exact in the context of linear theory, utilizing simple combinations of Green's functions appropriate to water of constant depth and the Cauchy–Riemann equations to derive a system of coupled integral equations for components of the fluid velocity at certain locations. Two cases arise, depending on whether the deepest point of the topography does or does not lie below the lower of the semi-infinite horizontal bed sections. In each, the reflected and transmitted wave amplitudes are related to the incoming wave amplitudes by a scattering matrix which is defined in terms of inner products involving the solution of the corresponding integral equation system.

This solution is approximated by using the variational method in conjunction with a judicious choice of trial function which correctly models the fluid behaviour at the free surface and near the joins of the varying topography with the constant-depth sections, which may not be smooth. The numerical results are remarkably accurate, with just a two-term trial function giving three decimal places of accuracy in the reflection and transmission coefficents in most cases, whilst increasing the number of terms in the trial function results in rapid convergence. The method is applied to a range of examples.

In strongly stratified geophysical fluids such as the stratosphere and the ocean, the vertical mixing of tracers is largely due to patches of turbulence that are intermittent in time and space. Heuristic models for this type of mixing are studied which extend that of Dewan (1981a). The recognition that, in these models, fluid particles undergo continuous-time random walks allows the derivation of closed-form results for the particle-position statistics. The particle dispersion is shown generally to be diffusive in the long-time limit. However, the early-time, non-diffusive regime is also analysed, since a time-scale estimate indicates its practical importance, in particular for stratospheric mixing.

Because the restratification of fluid patches previously homogenized by turbulence takes a finite time, the probability for a fluid region to become turbulent may depend on the time elapsed since it has last been turbulent. This introduces a ‘memory effect’ whose consequences for the tracer mixing are analysed in detail using a simple non-Markovian model.

The heuristic models studied allow the large-scale dispersive effects of the turbulent patches to be inferred from the properties of individual patches. This highlights those properties that might most usefully be determined from investigations of the dynamics of the turbulent patches themselves.

The information exchange dynamics between different spatial scales of a turbulent field is experimentally investigated, with particular reference to the energy cascade process. This is done by use of the collective light scattering (CLS) diagnostic, a new optical setting tuned for the observation of atomic number density fluctuations at macroscopic scales. A two-channel scattering device is described that provides simultaneously the fluctuations at two different spatial scales. The turbulent system is an axisymmetric air jet. With this system, time correlations between signal amplitudes for two different scales show a characteristic time width. This time does not depend on the observed scales: it is close to the turbulence production time. Nevertheless, signal amplitude autocorrelation shows a shorter characteristic time. The autocorrelation time scaling law behaves almost like the eddy turnover time from Kolmogorov theory.

The subject of this study is a steady two-dimensional incompressible flow past a rapidly rotating cylinder with suction. The rotation velocity is assumed to be large enough compared with the cross-flow velocity at infinity to ensure that there is no separation. High-Reynolds-number asymptotic analysis of incompressible Navier–Stokes equations is performed. Prandtl's classical approach of subdividing the flow field into two regions, the outer inviscid region and the boundary layer, was used earlier by Glauert (1957) for analysis of a similar flow without suction. Glauert found that the periodicity of the boundary layer allows the velocity circulation around the cylinder to be found uniquely. In the present study it is shown that the periodicity condition does not give a unique solution for suction velocity much greater than 1/Re. It is found that these non-unique solutions correspond to different exponentially small upstream vorticity levels, which cannot be distinguished from zero when considering terms of only a few powers in a large Reynolds number asymptotic expansion. Unique solutions are constructed for suction of order unity, 1/Re, and 1/√Re. In the last case an explicit analysis of the distribution of exponentially small vorticity outside the boundary layer was carried out.

A fuel cell is an electrochemical device that converts chemical energy into electrical energy. The essential difference between a fuel cell and a battery is that the fuel and oxidant are continually supplied to a fuel cell whereas a battery has a finite amount of fuel stored within it which eventually becomes exhausted, after which the battery has to be recharged or replaced. We consider here a tubular cell which is run on methane. A mathematical model for the reaction and transport processes in the cell is presented. This takes into account diffusion, advection and reaction as well as the electrochemical flow of oxygen. When the device is running steadily, we perform an analysis, using the method of matched asymptotic expansions to find the flow field, the mass fractions of each chemical species and the electrical power output. These results are confirmed by a numerical technique and compared with experimental results.

The flow around two circular cylinders of equal diameter, arranged in a staggered configuration, was investigated using flow visualization and particle image velocimetry for centre-to-centre pitch ratio P/D = 1[ratio ]0 to 5.0 and angle of incidence. α = 0° to 90°. Experiments were conducted within the low subcritical Reynolds number regime, from Re = 850 to 1900. Nine flow patterns were identified, and processes of shear layer reattachment, induced separation, vortex pairing and synchronization, and vortex impingement, were observed. New insight was gained into previously published Strouhal number data, by considering the flow patterns involved. The study revealed that vortex shedding frequencies are more properly associated with individual shear layers than with individual cylinders; more specifically, the two shear layers from the downstream cylinder often shed vortices at different frequencies.

A body immersed in an ocean of large depth is assumed to vibrate and to radiate a time-harmonic acoustic field of small amplitude in the presence of gravity waves of small amplitude. Assuming both waves to have lengths of the same order (which in practice corresponds to very low acoustic frequencies) it is shown that the diffraction of acoustic waves by the corrugated free surface generates a second-order acoustic pressure field p2. The computation of p2 involves a difficulty: a non-homogeneous Dirichlet condition to be satisfied on the mean free surface up to infinity which implies the absence of any clear indication about the condition that should be imposed at infinity to have a well-posed problem. In order to get an insight into this difficult problem the simple case of a point source is studied. We first judiciously choose one solution and then show it is the physical solution using a limiting-amplitude procedure. Coming back to the general case of a vibrating body the calculation of p2 is split into two successive steps: the first one consists in computing an explicit convolution product via numerical methods of integration, the second one is a standard radiation problem that is solved using a method coupling a Green integral representation and finite elements. A peak of the second-order pressure appears just above the vibrating body.

The same concepts also apply to other second-order scattering problems, such as the sea-keeping of weakly immersed submarines.

The cross-stream migration of a deformable drop in two-dimensional Hagen–Poiseuille flow at finite Reynolds numbers is studied numerically. In the limit of a small Reynolds number (< 1), the motion of the drop depends strongly on the ratio of the viscosity of the drop fluid to the viscosity of the suspending fluid. For viscosity ratio 0.125 a drop moves toward the centre of the channel, while for ratio 1.0 it moves away from the centre until halted by wall repulsion. The rate of migration increases with the deformability of the drop. At higher Reynolds numbers (5–50), the drop either moves to an equilibrium lateral position about halfway between the centreline and the wall – according to the so-called Segre–Silberberg effect or it undergoes oscillatory motion. The steady-state position depends only weakly on the various physical parameters of the flow, but the length of the transient oscillations increases as the Reynolds number is raised, or the density of the drop is increased, or the viscosity of the drop is decreased. Once the Reynolds number is high enough, the oscillations appear to persist forever and no steady state is observed. The numerical results are in good agreement with experimental observations, especially for drops that reach a steady-state lateral position. Most of the simulations assume that the flow is two-dimensional. A few simulations of three-dimensional flows for a modest Reynolds number (Re = 10), and a small computational domain, confirm the behaviour seen in two dimensions. The equilibrium position of the three-dimensional drop is close to that predicted in the simulations of two-dimensional flow.

The methods of formal matched asymptotics are used to investigate the motion of a vortex in shallow inviscid fluid of varying depth and zero Froude number in the limit as the vortex core radius tends to zero. To leading order the vortex is driven by local gradients in the logarithm of the depth along an isobath (or depth contour). A further term in the vortex velocity is calculated in which effects arising from the global bottom topography, other vortices and the vortex core structure appear. The evolution of the vortex core structure is then calculated. A point-vortex model is formulated which describes the motion of a number of small vortices in terms of the bottom topography, the inviscid flows around the vortices and their evolving core structure. A numerical method for solving this model is discussed and finally some numerical simulations of the motion of vortex pairs over a varying bottom topography are presented.