The evolution of the turbulent boundary layer over a bump defined by three tangential circular arcs and swept at 45° was examined. The flat-plate boundary layer approaching the swept bump had a momentum thickness Reynolds number of approximately 3800. The ratios of upstream boundary-layer thickness to bump height and convex radius of curvature were 1.5 and 0.06, respectively. The boundary layer was influenced by alternating signs of streamwise pressure gradient, wall curvature, and mean crossflow, which resulted in a complex boundary-layer flow that grew rapidly on the downstream side of the bump. The mean flow profiles deviated significantly from typical logarithmic layer behaviour, but the flow remained attached. The evolution of the Reynolds stress components was explained by the growth of two internal layers triggered by discontinuities in wall curvature near the leading and trailing edges of the bump. The shear stress vector was found to lag the velocity gradient vector, despite the spanwise flow changing direction above the bump. The measurements were compared to the previous results from a two-dimensional bump with the same profile shape and Reynolds number. Contrary to previous studies, the addition of mean crossflow to this complex flow field did not reduce the vertical mixing relative to the turbulent kinetic energy.

The scalar (concentration) field of two-dimensional liquid mixing layers was investigated at post-mixing-transition conditions. The planar laser-induced fluorescence technique was used for passive scalar measurements, and for chemical product measurements. Following the approach of Koochesfahani & Dimotakis (1986), the chemical product results were used to make resolution-free estimates of mixed-fluid quantities, thus providing a check on the accuracy of the passive scalar measurements. The operating conditions were systematically varied to study the effect of various parameters (Reynolds number, speed ratio, and initial boundary-layer momentum thickness) on the structure of the layer. At conditions which are just past the mixing transition, the study essentially duplicated the results obtained by Koochesfahani & Dimotakis: the chemical-product concentration profiles at high- and low-stoichio-metric ratios (flip experiments) were symmetric and the average concentration of mixed-fluid was uniform across the layer. However, when the layer was pushed to more fully developed conditions, its scalar field evolved to an asymptotic state: the two flip chemical-product concentration profiles developed modest asymmetries, and the average mixed-fluid concentration developed a small variation across the layer, but the change was less than that observed in gases. Based on the chemical reaction data, we infer that the mixture fraction probability density function (p.d.f.) for the fully-developed liquid layer evolves from a ‘non-marching’ type to a ‘tilted’ type. Despite the observed evolution, the average mixed-fluid concentration remained fixed for all the layers past the mixing transition, while the total mixed-fluid probability (the total amount of mixed fluid normalized by the layer's width) showed only a very slight increasing tendency as the layer reached fully developed conditions. The mixture fraction p.d.f., measured by the passive scalar approach, is shown and discussed for a broad range of cases. While it overpredicts the amount of mixing, it showed a qualitatively-correct ‘non-marching’ character initially, but evolved to a qualitatively-incorrect ‘marching’ character at the asymptotic state. The reasons for the poor estimation of the p.d.f. by the passive scalar approach, at fully developed conditions, are attributed to changes in the flow and lack of resolution and suggests caution when using such measures. Furthermore, the study also showed that the Reynolds number alone is inadequate to characterize the state of the layer. A different parameter (the ‘pairing parameter’), which accounts for the initial boundary layers and scales with the number of vortex mergings, was found to better explain the evolution in the structure of the scalar field.

A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.

A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

The structure of an axisymmetric and inviscid swirling flow around a vortex breakdown point is analysed. The model assumes that a free axisymmetric bubble surface is developed in the flow with a stagnation point at its nose. The classical Squire-Long equation for the stream function ψ(x,y) (where y = r2/2) is transformed into a free boundary problem for the solution of y(x, ψ). The development of the flow is studied in three regions: the approaching flow ahead of the bubble, around the bubble nose and around the separated bubble surface. Asymptotic expansions are constructed to describe the flow ahead of and behind the stagnation point in terms of the radial distance from the vortex axis and from the bubble surface, respectively. In the intermediate region around the stagnation point, the flow is approximated by an asymptotic series of similarity terms that match the expansions in the other regions. The analysis results in two possible matching processes. Analytical expressions are given for the leading term of the intermediate expansion for each of these processes. The first solution describes a swirling flow around a constant-pressure bubble surface, over which the flow is stagnant. The second solution represents a swirling flow around a pressure-varying bubble surface, where the flow expands along the bubble nose. In both solutions, the bubble nose has a parabolic shape, and both exist only when H’ > 0 (where H’ is the derivative at the vortex centre of the total head H with the stream function ψ, and can be determined from the inlet conditions). This result is shown to be equivalent to Brown & Lopez's (1990) criterion for vortex breakdown. Good agreement is found in the region around the stagnation point between the pressure-varying bubble solution and available experimental data for axisymmetric vortex breakdown.

The classical fourth-order Orr-Sommerfeld problem which arises from the study of the linear stability of channel flow of a viscous fluid is generalized to include the effects of a temperature-dependent fluid viscosity and heating of the channel walls. The resulting sixth-order eigenvalue problem is solved numerically using high-order finite-difference methods for four different viscosity models. It is found that temperature effects can have a significant influence on the stability of the flow. For all the viscosity models considered a non-uniform increase of the viscosity in the channel always stabilizes the flow whereas a non-uniform decrease of the viscosity in the channel may either destabilize or, more unexpectedly, stabilize the flow. In all the cases investigated the stability of the flow is found to be only weakly dependent on the value of the Péclet number. We discuss our results in terms of three physical effects, namely bulk effects, velocity-profile shape effects and thin-layer effects.

The nonlinear development of a weakly modulated Tollmien-Schlichting wavetrain in a boundary layer is studied theoretically using high-Reynolds-number asymptotic methods. The ‘carrier’ wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise variables. Attention is focused on the scalings appropriate to the so-called ‘upper branch’ and ‘high-frequency lower branch’. The dominant nonlinear effects are found to arise in the critical layer and the surrounding ‘diffusion layer’: nonlinear interactions in these regions can influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by an integro-partial-differential equation, whose nonlinear term is history-dependent and involves the highest derivative with respect to the spanwise variable. Numerical solutions show that a localized singularity can develop at a finite distance downstream. This singularity seems consistent with the experimentally observed focusing of vorticity at certain spanwise locations, although quantitative comparisons have not been attempted.

The accurate measurement of vorticity has proven difficult because of the difficulty of estimating spatial derivatives of velocity fluctuations reliably. A method is proposed for correcting the lateral vorticity spectrum measured using a four-wire probe. The attenuation of the measured spectrum increases as the wavenumber increases but does not vanish when the wavenumber is zero. Although the correction procedure assumes local isotropy, the major contributor to the high-wavenumber part of the vorticity spectrum is the streamwise derivative of the lateral velocity fluctuation, and the correction of this latter quantity does not depend on local isotropy. Satisfactory support for local isotropy is provided by the high-wavenumber parts of the velocity, velocity derivative and vorticity spectra measured on the centreline of a turbulent wake. Second- and fourth-order moments of vorticity show departures from local isotropy but the degree of departure seems unaffected by the turbulence Reynolds number Rλ. The vorticity probability density function is approximately exponential and has tails which stretch out to larger amplitudes as Rλ increases. The vorticity flatness factor, which is appreciably larger than the flatness factor of the streamwise velocity derivative, also increases with Rλ. When Rλ is sufficiently large for velocity structure functions to indicate a r2/3 inertial range, two-point longitudinal correlations of lateral vorticity fluctuations give encouraging support for the theoretical r−4/3 behaviour.

In this paper, we present evidence to show that a smoothly evolving zero-surface tension solution of the Hele-Shaw equations can be singularly perturbed by the presence of arbitrarily small non-zero surface tension in order-one time. These effects are explained by the impact of ‘daughter singularities’ on the physical interface, whose formation was suggested in a prior paper (Tanveer 1993). For the case of finger motion in a channel, it is seen that the daughter singularity effect is strong enough to produce the transition from a finger of arbitrary width to one with the selected steady-state width in O(1) time.

Motivated by the study of blood flow in the major coronary arteries, which are situated on the outer surface of the pumping heart, we analyse flow of an incompressible Newtonian fluid in a tube whose curvature varies both along the tube and with time. Attention is restricted to the case in which the tube radius is fixed and its centreline moves in a plane. Nevertheless, the governing equations are very complicated, because the natural coordinate system involves acceleration, rotation and deformation of the frame of reference, and their derivation forms a major part of the paper. Then they are applied to two particular, relatively simple examples: a tube of uniform but time-dependent curvature; and a sinuous tube, representing a small-amplitude oscillation about a straight pipe. In the former case the curvature is taken to be small and to vary by a small amount, and the solution is developed as a triple power series in mean curvature ratio δ0, curvature variation ε and Dean number D. In the latter case the Reynolds number is taken to be large and a linearized solution for the perturbation to the flow in the boundary layer at the tube wall is obtained, following Smith (1976a). In each case the solution is taken far enough that the first non-trivial effects of the variable curvature can be determined. Results are presented in terms of the oscillatory wall shear stress distribution and, in the uniform curvature case, the contribution of steady streaming to the mean wall shear stress is calculated. Estimation of the parameters for the human heart indicates that the present results are not directly applicable, but point the way for future work.

Competition between conductive cooling and advective heating occurs whenever hot fluid invades a cold environment. Here the solidification of hot viscous flow driven by a fixed pressure drop through an initially planar or cylindrical channel embedded in a cold rigid solid is analysed. At early times, or far from the channel entrance, the flow starts to solidify and block the channel, thus reducing the flow rate. Close to the channel entrance, and at later times, the supply of new hot fluid starts to melt back the initial chill. Eventually, either solidification or meltback becomes dominant throughout the channel, and flow either ceases or continues until the source is exhausted. The evolution of the dimensionless system, which is characterized by the initial Péclet number Pe, the Stefan number S and the dimensionless solidification temperature Θ, is calculated numerically and by a variety of asymptotic schemes. The results show the importance of variations along the channel and caution against models based on a single ‘representative’ width. The critical Péclet number Pec, which marks the boundary between eventual solidification and eventual meltback, is determined for a wide range of parameters and found to be much larger for cylindrical channels than for planar channels, owing to the slower rate of decay of the heat flux into the solid in a cylindrical geometry. For a planar channel Pec is given by the simple algebraic result Pec ∼ 0.46[Θ2/(1 - Θ)(S + 2/π)]3 when (1 - Θ)−1 [Gt ] S [Gt ] 1, but in general it requires numerical solution. Similar analyses, in which there is a spatially varying and time-dependent interaction between the rates of solidification and flow, have a range of applications to geological and industrial processes.

The Betz criterion for minimum induced loss is used to compute the optimal circulation distribution along the span of flapping wings in fast forward flight. In particular, we consider the case where flapping motion is used to generate both lift (weight support) and thrust. The Betz criterion is used to develop two different numerical models of flapping. In the first model, which applies to small-amplitude harmonic flapping motions, the optimality condition is reduced to a one-dimensional integral equation which we solve numerically. In the second model, which applies to large-amplitude periodic flapping motions, the optimal circulation problem is reduced to solving for the flow over an infinitely long wavy sheet translating through an inviscid fluid at rest at infinity. This three-dimensional flow problem is solved using a vortex-lattice technique. Both methods predict that the induced power required to produce thrust decreases with increasing flapping amplitude and frequency. Using the large-amplitude theory, we find that the induced power required to produce lift increases with flapping amplitude and frequency. Therefore, an optimum flapping amplitude exists when the flapping motion of wings must simultaneously produce lift and thrust.

An approach is presented for studying Rossby wave interaction in a shear flow with both regular and singular modes (i.e. those possessing a critical level). The approach relies on a truncated normal mode expansion of the equations of motion. Such an expansion remains valid in the presence of singular modes, provided that these modes are not considered individually, but that complete packets are taken into account in the truncated system. Mathematically, this means that the interaction equations need to be integrated with respect to the phase velocity (or, equivalently, the critical level position) of the singular modes.

The action of two regular modes on a packet of singular modes is treated in detail; in particular, asymptotic results are deduced for the long-term behaviour of the packet. The case of a linear shear is considered as an illustration: analytical expressions are derived for the normal modes and their pseudomomentum, and they are used to present explicit results for the evolution of the packet of singular modes.

A spectral collocation method is used to analyse the linear stability, both viscous and inviscid, of a family of self-similar vortex viscous cores matching external inviscid vortices with velocity u varying as a negative power of the distance r to their axis of symmetry, u ∼ rm−2 (0 < m < 2). Non-parallel effects are shown to contribute at the same order as the viscous terms in the linear governing equations for the perturbations, and are consequently retained. The viscous stability analysis for the particular case m = 1, corresponding to Long's vortex, has recently been performed by Khorrami & Trivedi (1994). In addition to the inviscid non-axisymmetric modes of instability found by these authors, some inviscid axisymmetric unstable modes, and purely viscous unstable modes, both axisymmetric and non-axisymmetric, are also found. It is shown that, while both solution branches (I and II) of Long's vortex are destabilized by perturbations having negative azimuthal wavenumber (n < 0), only the Type II Long's vortex is also unstable for axisymmetric disturbances n = 0, as well as for disturbances with n > 0. Global pictures of instabilities of Long's vortex are given. For m > 1, the vortex cores have the interesting property of losing existence when the swirl number is larger than an m-dependent critical value, in close connection with experimental results on vortex breakdown. The instability pattern for m > 1 is similar to that found for Long's vortex, but with the important difference that the parameter characterizing the different vortices, and therefore their stability, is a swirl parameter, which is precisely the one known to govern the real problem, while this is not the case in the highly degenerate case m = 1.

An efficient method of constructing inviscid Batchelor-model flows is developed. The method is based on an analytic continuation of the potential part of the flow into the closed-streamline vortex region. Numerical solutions are presented for Batchelor-model flows past airfoils with cavities. With the airfoil and dividing streamline shape, the eddy vorticity, and the jump in the Bernoulli constant across the eddy boundary given, the program calculates the corresponding cavity shape and the entire flow.

Most models of gas bubbles in fluidized beds are based on the assumption of an empty central region, the void, surrounded by a ‘cloud’ or ‘shell’ of particles whose voidage is larger than that of the remote emulsion phase. Batchelor & Nitsche (1994) investigated the formation of a void by tracking the paths of particles initially within a buoyant ‘blob’ of gas that has the form of a toroidal vortex. They showed that the particles dropped through the floor of the blob under the influence of gravity, leaving it empty. This paper extends their method to particles initially outside the blob. It is shown that inertia allows these particles to penetrate the blob and it is the extent of this penetration that determines the size of the void. The void is nearly as large as the blob for small, light particles, but becomes smaller relative to the blob with increasing particle size and weight until it disappears altogether. This provides an explanation for experimental observations of voids smaller than the blob (or ‘cloud’ as it is sometimes known), and suggests that when examining bubbles in a gas-fluidized bed the most significant dimension is the diameter of the blob and not that of the void.

Experiments and simulations of a travelling wave state of incompressible Newtonian flow in a curved duct of square cross-section are presented. The travelling wave mode develops from the well-documented steady four-cell flow state and is characterized by oscillations of the two Dean vortices near the centre of the outer wall.

The oscillations were induced by a carefully positioned pin at 5° from the inlet of the curved section along the symmetry line of the cross-section. It was shown that the travelling wave state is characteristic for curved duct flow and that the pin made it possible to observe the oscillations within the 270° long curved duct. Travelling waves were observed at flow rates above Dn = 170 (Dn = Re/(R/a)1/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension. The curvature ratio, R/a, is 15.1).

If no other disturbances are imposed, the oscillations are the result of the selective amplification of random disturbances in the flow, leading to a broad frequency spectrum. The travelling wave was found to lock in to an imposed periodic disturbance at a selected frequency. The flow structure of the locked state was investigated in detail, using flow visualization and a one-component laser Doppler anemometer to measure streamwise or spanwise velocities. Direct numerical simulations using the package CFDS-FLOW3D are in very good agreement with the experiments and confirm the existence of a fully developed, streamwise-periodic travelling wave state.

The inflow region between the two Dean vortices, which transports low-speed fluid away from the outer wall, creates strongly inflectional spanwise profiles of the streamwise velocity. Similarities with twisting vortices in a curved channel and sinuous oscillations of Görtler vortices show that the travelling waves observed here result from a secondary shear instability of these spanwise inflectional profiles.