In steady, two-dimensional, inviscid flows it is well-known that, in the absence of rotational forcing, the vorticity is constant along streamlines. In a bounded domain the streamlines are necessarily closed. In some circumstances, investigated in this paper, this behaviour is exhbited also by forced viscous flows, when the variation of vorticity across the streamlines is determined by a balance between viscous diffusion and the forcing. Similar results hold in axisymmetry. For such flows, an iterative process for finding the vorticity as a function of the stream function is described. The method applies whenever the viscous boundary condition can be expressed in terms of the vorticity or tangential stress rather then the tangential velocity. When it is applicable, the iterative method is faster than direct solution of the Navier-Stokes equations at high Reynolds numbers. As an example, the method is used to calculate the flow in a model of the electromagnetic stirring process. In this model, a conducting fluid in an elliptical region is driven by a rotating magnetic field and resisted by a surface stress. The functional dependence of the vorticity on the stream function is found for various values of the magnetic skin depth, surface stress and eccentricity of the ellipse. The form of the flow is discussed with particular reference to whether it consists of a single circulatory region or separates into two or more such regions.

In steady, two-dimensional, inertia-dominated flows it is well known that the vorticity is constant along the streamlines, which, in a bounded domain, are necessarily closed. For inviscid flows, the variation of vorticity across the streamlines is arbitrary, while for forced, weakly dissipitative flows, it is determined by the balance between viscous diffusion and the forcing. This paper discusses the linear stability of flows of this type to two-dimensional disturbances. Arnol'd's stability theorems are discussed. An alternative functional to Arnol'd's is found, which gives the same stability criteria and which permits a representation of the problem in terms of a Schrödinger equation. Conditions for stability are derived from this functional. In particular it is shown that total flow reversals are potentially unstable. The results are illustrated with respect to the geometrically simple case when the streamlines are circular and the forcing is due to a rotating magnetic field, for which case the stability regions are calculated as a function of two parameters. It is shown that the entire theory, including Arnol'd's theorems, applies also to poloidal axisymmetric flows.

A general method for computing the hydrodynamic interactions among an infinite suspension of particles immersed between two infinite plane boundaries, under the condition of vanishing particle Reynolds number, is presented. The method accounts for both near-field particle-particle and particle-boundary lubrication effects as well as dominant many-body effects, which include reflections with both particles and boundaries. Through relative motion of the boundaries, a bulk shear flow can be generated, and the resulting particle motions, as well as the forces exerted by the boundaries on the fluid, computed. Knowledge of the boundary forces allows for the calculation of the suspension viscosity. The simulation method is applied to several example problems; in one, the resuspension of a sediment layer of particles is illustrated. The general method can also be extended to dynamically simulate suspensions immersed in a pressure driven flow between two walls or through a tube.

The problem is considered of topographic waves propagating on depth gradients in rotating domains. A variational principle is derived for the eigenfunctions and eigenfrequencies of normal modes on a domain, and applied to subdomains of the whole domain. Considering suitable boundary conditions on the open boundary between the subdomain and the remainder of the whole domain gives upper and lower bounds and estimates for the frequencies of normal modes localized in the subdomain without the complication of solving over the whole domain. It is shown that applying a zero-mass-flux condition at the open boundary leads to a lower bound on the frequency whereas requiring a particular form of the energy flux to vanish identically at each point of the boundary provides an upper bound.

The topographic wave equation is solved in a domain consisting of a channel with a terminating bay zone. For exponential depth profiles the problem reduces to an algebraic eigenvalue problem. In a flat channel adjacent to a shelf–like bay zone the solutions form a countably infinite set of orthogonal bay modes: the spectrum of eigenfrequencies is purely discrete. A channel with transverse topography allows wave propagation towards and away from the bay: the spectrum has a continuous part below the cutoff frequency of free channel waves. Above this cutoff frequency a finite number (possibly zero) of bay-trapped solutions occur. Bounds for this number are given. At particular frequencies below the cutoff the system is in resonance with the incident wave. These resonances are shown to be associated with bay modes.

An inviscid three-dimensional vortex-sheet model for the near field of a strong jet issuing from a pipe into a crossflow is derived. The solution for this model shows that the essential mechanisms governing this idealized flow are the distortion of the main transverse vorticity by the generation of additional axial and transverse vorticity within the pipe owing to the pressure gradients induced by the external flow, and the convection of both components of vorticity from the upwind side of the jet to its downwind side.

The deformation of the cross-section of the jet which is predicted by this model is compared with the deformation predicted by the commonly used time-dependent two-dimensional vortex-sheet model. Differences arise because the latter model does not take into account the effects of the transport of the transverse component of vorticity. The complete three-dimensional vortex-sheet model leads to a symmetrical deformation of the jet cross-section and no overall deflection of the jet in the direction of the stream.

To account for viscous effects, the initial region of a strong jet issuing into a uniform crossflow is modelled as an entraining three-dimensional vortex sheet, which acts like a sheet of vortices and sinks, redistributing the vorticity in the bounding shear layer and inducing non-symmetrical deformations of the cross-section of the jet. This leads to a deflection of the jet in the direction of the stream, and the loci of the centroids of the cross-sections of the jet describe a quadratic curve.

Deformations predicted by each of the three models are compared with measurements obtained from photographs of the cross-sections of a jet of air emerging into a uniform crossflow in a wind tunnel. Mean velocity measurements around the jet made with a hot-wire anemometer agree with the theory; they clearly invalidate models of jets based on ‘pressure drag’.

A series of two- and three-dimensional numerical simulations of transient flow in a side-heated cavity has been conducted. The motivation for the work has been to resolve discrepancies between a flow description based on scaling arguments and one based on laboratory experiments, and to provide a more detailed description of the approach to steady state. All simulations were for a Rayleigh number of 2 × 109, and a water-filled cavity of aspect ratio 1. The simulations (beginning with an isothermal fluid at rest) generally agree with the results of the scaling arguments. In addition, the experimental observations are entirely accounted for by the position of the measurement instruments and the presence of an extremely weak, stabilizing temperature gradient in the vertical.

An experimental investigation of the flow around smooth circular cylinders in the Reynolds number range 0.8 × 105 < Re < 2 × 105 is presented. Measured quantities include spectra, spanwise correlations and cross correlations of cylinder pressures and wake-velocity fluctuations, and low-frequency boundary-layer flow direction reversals near separation. The flow motion in the critical range is found to be characterized by intermittent, symmetric boundary-layer reattachments, occurring in cells with a well-defined spanwise structure, accompanying a significant decrease in drag coefficient and a weakening of the vortex shedding.

The effect of spatial periodicity in grain structure on the average transport properties resulting from flow through porous media are derived from the basic conservation equations. At high Péclet number, the mechanical dispersion that is induced by the stochastic fluid velocity field in disordered media and is independent of the molecular diffusivity is absent in periodic media where the velocity field is deterministic. Instead, the fluid motion enhances diffusion by an amount proportional to U2l2/D when the bulk flow is in certain directions (of which there are an infinite number), and to D otherwise. The non-mechanical dispersion mechanisms associated with the zero velocity of the fixed grains is qualitatively similar in ordered and disordered media.

The multicellular flow between two vertical parallel plates is numerically simulated using a time-splitting pseudospectral method. The steady flow of air, and the time-periodic flow of oil (Prandtl numbers of 0.71 and 1000, respectively) are investigated and descriptions of these flows using both physical and spectral approaches are presented. The details of the time dependency of the flow and temperature fields of oil are shown, and the dynamics of the process is discussed. The spectral transfer of energy among the axial modes comprising the flow is explored. The spectra of kinetic energy and thermal variance for air are found to be smooth and viscously dominated. Similar spectra for oil are bumpier, and the dynamics of the time-dependent flow are determined to be confined to the lower end of the spectrum alone.

The three-dimensional linear stability of the multicellular flow of air is parametrically studied. The domain of stable two-dimensional cellular motion was found to be constrained by the Eckhaus instability and by two types of monotone instability. The two-dimensional multicellular flow is unstable above a Grashof number of about 8550 (with the critical Grashof number for the base flow being 8037). Therefore the flow of air in a sufficiently tall vertical enclosure should be considered to be three-dimensional for most practical applications.

The title problem, which is an important early element of the mechanism of the development of a surface oil flow visualization picture, is studied by a linear stability analysis that predicts the most strongly amplified wavenumber of transverse surface waves as a function of Reynolds number, surface tension, film viscosity and initial thickness of the film. Wall scaling virtually removes the additional dependence on Reynolds number. In a simple experiment the observed wavenumber agrees with the predicted most strongly amplified value to within the experimental accuracy.

We study the diffraction, to second order, of plane monochromatic incident gravity waves by a vertically axisymmetric body. The second-order double-frequency diffraction potential is obtained explicitly. A sequence of one-dimensional integral equations along the generator of the body involving free-surface ring sources of general order are formulated and solved for the circumferential components of the second-order potential. The solution is expedited by analytic integration in the entire local-wave-free outer field of a requisite free-surface integral. The method is validated by extensive convergence tests and comparisons to semi-analytic results for the second-order forces and moments on a uniform vertical circular cylinder. Complete second-order forces, moments, surface pressures and run-up on the vertical cylinder as well as a truncated vertical cone are presented. A summary of the important findings is given in §5.

Motivated by a problem in turbine blade cooling, we consider suction from an inviscid channel flow into a slot in the channel wall. The flow is assumed to separate smoothly from the leading edge of the slot and the pressure in the stagnant separated region controls the suction. The mass flux into the slot is found in terms of the pressure; for small values of this flux the predicted flow pattern is found to be quite different from that which would result if there were no separated region. In particular, the stagnation point never penetrates more than approximately 0.05 slot widths into the slot.

Equations which modify those derived by Widnall & Bliss (1971) and Moore & Saffman (1972) are presented in which jet-like flow along the axis of a vortex tube interacts with the motion of the tube. The equations describe two major effects. The first is the propagation of axial waves along the vortex tube which is similar to the flow of shallow water. A local decrease in cross-section area of the vortex tube produces higher swirling velocity and lower pressure. The resulting axial pressure gradient causes a propagating wave of area and axial velocity in order to move fluid into the region of smaller area. The second effect is instability to helical disturbances when the jet-like axial velocity is high enough to overcome the stabilizing effect of the swirling motion. An elementary nonlinear theory of vortex breakdown is presented which has an analogy with the formation of bores in shallow-water theory. A numerical example shows the growth of a helical disturbance behind a vortex breakdown front.

Experiments have been carried out with the objective of studying the relationship between flow structure, flow excitation and the reaction process in the near field of a low-speed coflowing jet diffusion flame. The effect of axial forcing and increasing pressure on the structure and controllability of the flame has been studied in an attempt to elucidate some of the underlying mechanisms of control. The experiments were conducted in a variable-pressure flow facility which permits the study of reacting flows at pressures ranging from 10 to 1000 kPa (0.1 to 10 atm.). The flame was excited by adding a small-amplitude, periodic fluctuation to the central fuel jet exit velocity. The flow was visualized using an optical scheme which superimposes the luminous image of the flame on its schlieren image, giving a useful picture of the relationship between the luminous soot-laden core flow and the edge of the surrounding hot-gas envelope. Phase-conditioned velocity measurements were made with a one-component laser Doppler anemometer. The excitation frequency was varied, and it was found that a narrow band of frequencies exists in which several of the instabilities of the flow seem to be in coincidence, causing the flame to break up periodically into a series of distinct eddies. Hereafter this will be called the strongly coupled state. Maps of the one-dimensional velocity vector field, viewed in a frame of reference convecting with the large eddies, are used to study the topology of the flow. When the excitation frequency lies above the strongly coupled range, the flow pattern is found to contain stagnation points which straddle the axis of the jet. When the excitation frequency is reduced to a point where strong coupling occurs the stagnation points move onto the axis promoting breakup of the flame. As the pressure is increased, the relative role of diffusion is decreased and the flame becomes highly three-dimensional. In the strongly coupled state, the flow continues to be very periodic, even to the extent that much of the three-dimensional structure is repeatable from cycle to cycle.

The effect of interaction on the boundary layer induced by a convected rectilinear vortex is considered. Two schemes are employed in the numerical discretization of the edge interaction condition; the first, developed by Veldman (1981) is useful at larger Reynolds numbers but fails to capture the interactive phase of the motion for Reynolds numbers less than 8 × 104. A scheme devised by Napolitano, Werlé & Davis (1978) is employed at smaller Reynolds numbers and yields similar results to Veldman's scheme at higher Reynolds numbers, while exhibiting greater numerical stability during the interactive phase of the motion. The effect of interaction is found to be negligible during much of the motion, even for a strong vortex, but during the latter stages of the calculations, interaction appears to round off the top of the eddy and delays breakdown for all Reynolds numbers studied when compared with the non-interactive results of Doligalski & Walker (1984). In addition, in the latter stages of the calculations, and during the early stages of the interactive phase, a third eddy is formed with vorticity of the same sign as the main eddy spawned deep within the boundary layer. Such a tertiary eddy has been observed in the experimental work of Walker et al. (1987) in their study of the boundary layer induced by a vortex ring. During the interactive phase of the motion a streamwise lengthscale emerges whose length is approximately $O(Re^{-\frac{3}{11}})$, broadly in line with the analytical predictions of Elliott, Cowley & Smith (1983). A novel feature of the computations is the use of a pseudospectral method (Burggraf & Duck 1982) in the streamwise direction which requires no special coding in reversed-flow regions.

The relaxation of a reattached turbulent boundary layer downstream of a wall fence has been investigated. The boundary layer has an adverse pressure gradient imposed upon it which is adjusted in an attempt to bring the boundary layer into equilibrium. This is done by adjusting the pressure gradient so as to bring the Clauser parameter (G) down to a value of about 11.4 and then maintain it constant. In the region from the reattachment point to 2 or 3 reattachment lengths downstream, the boundary layer recovers from the initial major effects of reattachment. Farther downstream (where G is constant) the pressure-gradient parameter changes very slowly and profiles of non-dimensionalized eddy viscosity appear self-similar. However, pressure gradient and eddy viscosity are both roughly twice as large as expected on the basis of previous studies of equilibrium turbulent boundary layers. It is not known whether equilibrium has been achieved in this downstream region. This is another illustration of the great sensitivity of boundary-layer structure to perturbations.

For a reactive solute, with weak second-order recombination, an investigation is made of the near-source behaviour (where concentrations are high), and of the far field (where the recombination has an accumulative effect). Despite the loss of material and increased spread due to recombination, the far-field concentration distribution is shown to be nearly Gaussian. This permits a simplified (Gaussian) treatment of the chemical nonlinearity. Explicit solutions are given for the total amount of solute, variance and kurtosis for solutes with no first-order reactions.

The mechanisms associated with the formation and growth of water droplets in the large low-pressure turbines used for electrical power generation are poorly understood and recent measurements have indicated that an unusually high loss is associated with the initial nucleation of these droplets. In order to gain an insight into the phenomena which arise in the turbine situation, some experiments were performed to investigate the behaviour of condensing steam flows in a blade passage. This study has revealed the fundamental significance of droplet nucleation in modifying the single-phase flow structure and results are presented which show the change in shock wave pattern when inlet superheat and outlet Mach number are varied. The trailing-edge shock wave structure appears considerably more robust towards variations of inlet superheat than purely one-dimensional considerations may suggest and the inadequacies of adopting a one-dimensional theory to analyse multi-dimensional condensing flows are demonstrated. Over a certain range of outlet Mach numbers an oscillating shock wave will establish in the throat region of the blade passage and this has been shown to interact strongly with droplet nucleation, resulting in a considerably increased mean droplet size. The possible implications of these results for turbine performance are also discussed.

A model has been constructed for a mixing layer of a rotating fluid with a large Reynolds number which is an analogue of a mixing-layer model for a plane flow widely used in the literature. The angular velocity profile in such a model has the form: \[ \Omega(r) = {\textstyle\frac{1}{2}}(\Omega_1 + \Omega_2)-{\textstyle\frac{1}{2}}(\Omega_1 - \Omega_2)\tan h\left(\frac{1}{D}\ln\frac{r}{R}\right), \] where r is the distance from the rotation axis; and R, Ω1,2, and D are the model's parameters. The model permits a relatively simple analytical study of the stability for two-dimensional disturbances. It is shown that the stability is defined by the ‘shear-width’ parameter D, namely the model is unstable when D < Dcrit = ½. In a weakly supercritical flow (|D − Dcrit| [Lt ] 1), one mode with azimuthal number m = 2 develops. In this case two vortices are produced in the vicinity of a critical layer (CL), i.e. a radius where the wave's azimuthal velocity Ωp coincides with the rotation velocity Ω(r). A study is made of their nonlinear evolution corresponding to different CL regimes: viscous, nonlinear, and unsteady. It is found that the instability saturates at a low enough level and the equilibrium amplitude depends on the degree of supercriticality ΔD = |D − Dcrit|, but the character of this dependence is different in different regions of the supercriticality parameter ΔD.

It is shown that, despite the specific form of the velocity profile in the model under consideration, results concerning the critical-layer dynamics have a high degree of universality. In particular, it becomes possible to formulate the criterion that the instability will be saturated at a low level for an arbitrary weakly supercritical flow.