This investigation is concerned with the numerical calculation of multiple solutions for a mixed-convection flow problem in horizontal rectangular ducts. The numerical results are interpreted in terms of recent observations by Benjamin (1978a) on the bifurcation phenomena for a bounded incompressible fluid. The observed mutations of cellular flows are discussed in terms of dynamic interchange processes. Each cellular flow may be represented by a solution surface in the parametric space of Grashof number Gr and aspect ratio γ, which is delimited by stability boundaries. Such a stability map has been generated for each type of cellular flow by a series of numerical experiments. Once these boundaries are crossed one cellular flow mutates into another via a certain dynamical process. Although the nature of the singular points on this map have not been determined precisely, a plausible general structure of the cellular-flow exchange process emerges from this map with several features in common with the Taylor-Couette flow. The primary modes appear to exchange roles via the formation of tilted cusp. Other salient features such as primary-mode hysteresis and quasi-critical range for cellular development appear to be present. However no anomolous modes have been observed.

Numerical results are presented for the flow field in a rapidly rotating gas. These results are compared with the predictions of asymptotic theories, particularly those of Brouwers concerning viscous effects. The comparison shows that the effect of the viscous core is important and the extent of the different flow regimes is well predicted by the theory of Brouwers.

The motion of a disk rising steadily parallel to the axis of rotation in a uniformly rotating unbounded liquid is considered. In the limit of zero Rossby number the linear viscous equations of motion are reduced to a system of dual integral equations which renders an ‘exact’ solution for arbitrary values of the Taylor number, Ta. The investigation is focused on the drag and the flow field. In the limits of small and large Ta the asymptotic results of the present formulation agree with – and extend – previous investigations by different approaches.

A particular novel feature, for large Ta, is the contribution of the Ekman-layer flux to the outer motion. New insight into the structure of the Taylor column is gained; in particular, it is shown that the main part of the column is a ‘bubble’ of recirculating fluid, detached from the body and not communicating with the Ekman layer. However, it turns out that the essential discrepancy in drag between experiments (Maxworthy 1970) and previous theories cannot be attributed to the Ekman-layer suction effect.

We consider the effect of steady, three-dimensional cellular convective fields impressed upon the moving front of a dilute binary alloy in directional solidification. The flows have length scales longer than the characteristic lengths of the morphological instability. A Floquet problem with multiple degrees of freedom in space governs the interfacial dynamics and determines the morphological patterns and marginal stability boundaries. In the cases of weak flows the induced patterns are superpositions of rolls modulated by the forced flows. When the flows are strong, the instability becomes spatially localized and confined at inward flow-stagnation regions on the front. Numerical computations and the WKB method are used to solve the eigenvalue problems, showing various localized states depending on the structures of the imposed flows.

This paper is concerned with the propagation of small initial disturbances in a conducting gas under the influence of a uniform external magnetic field.

For a perfect conductor, there are three types of plane waves, each of which depends strongly on the angle at which the magnetic field is crossed. The modifying effects of finite conductivity are determined and, in the case of these waves, this is done uniformly for all angles. A general disturbance may be resolved into two parts, one of which satisfies a fourth-order equation and the other a fifth; for a perfect conductor these reduce to second-and fourth-order equations, respectively.

The free oscillations of the gas are examined when it is contained in a rectangular box, and, in particular, when the external field is very weak or very strong. For vanishingly weak fields the idealization of infinite conductivity proves to be inadequate. Finally, the initial-value problem is discussed.

An asymptotic frequency-domain approach is used to describe the radiation from a supersonic swept propeller within the framework of linear acoustics. With this approach the radiation of singularities, their points of origin on the blades, and their relation to blade geometry and loading are easily obtained. In particular, it is shown that a swept propeller with a completely subsonic leading edge can still radiate singularities, if the leading edge is blunt, due to a supersonic edge effect at the blade tips. In addition, the radiation from a family of ‘critical’ swept-blade designs is shown to be more singular than that from a straight-bladed design. Numerical and asymptotic results for such designs show that the peak radiation is, typically, increased by 5–10 dB.

The evolution of a weakly nonlinear, weakly dispersive gravity wave in water of depth d over a bottom of gradual slope δ and Chezy friction coefficient Cf is studied. It is found that an initially sinusoidal wave evolves into a periodic sequence of solitary waves with relative amplitude a/d = α1 = 15δ/4Cf if α1 < αb, where αb is the relative amplitude above which breaking occurs. This prediction is supported by observations (Wells 1978) of the evolution of swell over mudflats.

Because of the associated far-field logarithmic divergence, the transport problem governing two-dimensional phoretic self-propulsion lacks a steady solution when the Péclet number $\mathit{Pe}$ vanishes. This indeterminacy, which has no counterpart in three dimensions, is remedied by introducing a non-zero value of $\mathit{Pe}$, however small. We consider that problem employing a first-order kinetic model of solute absorption, where the ratio of the characteristic magnitudes of reaction and diffusion is quantified by the Damköhler number $\mathit{Da}$. As $\mathit{Pe}\rightarrow 0$ the dominance of diffusion breaks down at distances that scale inversely with $\mathit{Pe}$; at these distances, the leading-order transport represents a two-dimensional point source in a uniform stream. Asymptotic matching between the latter region and the diffusion-dominated near-particle region provides the leading-order particle velocity as an implicit function of $\log \mathit{Pe}$. Another scenario involving weak advection takes place under strong reactions, where $\mathit{Pe}$ and $\mathit{Da}$ are large and comparable. In that limit, the breakdown of diffusive dominance occurs at distances that scale as $\mathit{Da}^{2}/\mathit{Pe}$.

The secondary instability in the wake of a two-dimensional blunt body with a chord to thickness ratio of 46.5 was experimentally investigated for Reynolds numbers of 3500, 5200 and 7000 based on the blunt trailing edge height $h$ . Planar, stereoscopic and high-speed particle image velocimetry (PIV) measurements were performed to characterise the wake and upstream boundary layer. The same mode B secondary instability that is found in the cylinder wake was found to be present in the wake of the elongated body studied here. The most probable wavelength of the secondary instability, defined as the spanwise distance between adjacent streamwise vortex pairs in the wake, was found to range from $0.7h$ to $0.8h$ by applying a spatial autocorrelation to the spanwise–wall-normal instantaneous fields of the $Q$ -criterion. The temporal evolution of the secondary wake vortices was investigated using time-resolved stereoscopic PIV measurements and it was shown that the vortices maintain both their directions of rotation and spanwise positions during the primary vortex shedding cycles. In agreement with previous literature, the secondary instability did not greatly change as the upstream boundary layer transitioned from laminar to turbulent. Moreover, any upstream boundary layer structures were found to rapidly evolve into wake structures just past the blunt trailing edge. The wavelength of the secondary instability was shown to match the spanwise distance between adjacent low-speed zones of streamwise velocity in the wake. These undulating velocity patterns proved to be a viable method for determining the secondary instability wavelength; however, this type of analysis is highly sensitive to the energy content used for data reconstruction when proper orthogonal decomposition is applied beforehand.

Plumes generated by vertically distributed sources of buoyancy have been observed to have substantially lower entrainment coefficients than their equivalent-geometry constant buoyancy flux plumes. Two differences between distributed and localized sources of buoyancy are the presence of a wall shear stress at the source and that non-ideal source conditions are distributed over the whole height of the enclosure for a vertically distributed source. Herein the impact of non-ideal source and boundary conditions on vertically distributed plumes is analysed. It is shown that, at small heights, the plume volume flow rate is significantly influenced by the wall-source volume flux. At larger heights the wall-source buoyancy is greater than the mean plume buoyancy, creating a non-self-similar horizontal buoyancy distribution within the plume. Recent experiments into the behaviour of a vertically distributed source of buoyancy in a confined region have also shown that the plume partially detrains in the stratified region of the enclosure. This detrainment has not been observed for constant buoyancy flux plumes in a confined region. Although models have been proposed to quantify the detrainment process, it is still unclear why vertically distributed buoyancy sources detrain while constant buoyancy flux plumes do not in the same physical geometry. The impact of source and boundary effects on previously published experiments on vertically distributed plumes are reviewed and the possible implications for plume entrainment and detrainment are discussed.

The work concerns the nonlinear dynamics of oceanic internal waves in resonance with a surface shear current. The resonance occurs when the celerity of the wave matches the mean flow speed at the surface. The evolution of weakly nonlinear waves long compared to the thickness of the upper mixed layer is found to be described by two linearly coupled equations (a linearized intermediate long wave equation and the Riemann wave equation). The presence of a pseudodifferential operator leads to qualitatively new features of the wave dynamics compared to the previously studied case of shallow water. The system is investigated primarily by means of numerical analysis. It possesses a variety of both periodic and solitary wave stationary solutions, including ‘delocalized solitons’ with a localized core and very small non-decaying oscillatory tails (throughout the paper we use the term ‘soliton’ as synonymous with ‘solitary wave’ and do not imply any integrability of the system). These ‘solitons’ are in linear resonance with infinitesimal waves, which in the evolutionary problem normally results in radiative damping. However, the rate of the energy losses proves to be so small, that these delocalized radiating solitons can be treated as quasi-stationary, that is, effectively, as true solitons at the characteristic time scales of the system. Moreover, they represent a very important class of intermediate asymptotics in the evolution of initial localized pulses. A typical pulse evolves into a sequence of solitary waves of all kinds, including the ‘delocalized’ ones, plus a decaying train of periodic waves. The remarkable feature of this evolution is that of all the products of the pulse fission (in a wide range of parameters of the initial pulse) the radiating solitons have by far the largest amplitudes. We argue that the radiating solitons acting as intermediate asymptotics of initial-value problems are a generic phenomenon not confined to the particular model under consideration.

The flow of a viscous incompressible fluid in a circular tube generated by a sudden impulse on the axis is studied on the basis of the linearized Navier–Stokes equations. A no-slip boundary condition is assumed to hold on the wall of the tube. At short time the flow is irrotational and may be described by a potential which varies with the square root of time. At later times there is a sequence of moving and decaying vortex rings. At long times the flow velocity decays with an algebraic long-time tail. The impulse generates a time-dependent pressure difference between the ends of the tube.

It is shown that for a sudden uniform discharge at x = 0, t = 0 in a bounded shear flow, the asymptotic concentration distribution at moderately large times can be well approximated by the tilted Gaussian \[ c=\frac{\overline{q}}{(2\pi\langle\sigma^2\rangle)^{\frac{1}{2}}}\exp\left(-\frac{(x-\overline{u}t-g_0(y,z)+2\alpha D)^2}{2\langle \sigma^2\rangle }\right), \] with \begin{eqnarray*} & \langle \sigma^2\rangle = 2Dt + 2\alpha D(x-\overline{u}t)-3\overline{g^2_0}-4\alpha^2 D^2,\\ & \overline{g_0} = 0,\quad D = \overline{ug_0},\quad \alpha = \frac{(\overline{u-\overline{u}})g^2_0}{2D^2}. \end{eqnarray*} Here u(y, z) is the velocity profile, g0(y, z) the centroid displacement function, and the overbars denote cross-sectional averaging. The tilt parameter α makes the concentration distribution suitably skew. The effectiveness of this simple formula is demonstrated for two-layer flows and for plane Poiseuille flow.

This paper presents the results of an analytical investigation of the steady translation of a vertical surface-piercing plate at a small angle of attack. This problem is the antisymmetric equivalent of the symmetric thin-ship problem solved by Michell. The linearized boundary-value problem is transformed into an integral equation of the first kind by the method of Green functions. The Kelvin–Havelock Green function is used to satisfy the linearized free-surface boundary condition and radiation condition. A pressure Kutta condition is imposed at the trailing edge. Effective algorithms are developed to evaluate the hypersingular kernel without recourse to numerical integration. The resulting integral equation is solved by a collocation method with a refined scheme of discretization. After establishing the convergence of the present algorithm, computations are carried out for a surface-piercing rectangular plate of aspect ratio 0.5. The integrated lateral-force and yaw-moment coefficients show good agreement with experimental data. Other parameters of the flow such as pressure distributions, drag, strength of leading-edge singularity and free-surface profiles on the plate are also presented. The incompatibility between the pressure Kutta condition and the linearized free-surface condition does not affect the global solution.

A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant-unlike the deep-water case-in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities ‘contaminate’ most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves ‘close’ to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d= 1; the study is performed down to depth d= 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of short-crested wave fields will not be observable, although most of the short-crested wave fields will be.

A calculation is made of the steady profile adopted by a touching pair of vortex regions with equal and opposite vorticity in a bounded uniform stream. A family of possible solutions is deduced, depending upon the magnitude of a (non-dimensionalized) vorticity parameter. A similar calculation is carried out incorporating a flat plate normal to the stream at the upstream end of the vortex configuration. The requirement of tangential separation at the plate tip selects a unique value of the vorticity. It is found that, as the width of the plate is reduced in relation to that of the channel, the vortex profile asymptotically approaches one member of the above mentioned family. The asymptotic form of the flow in the vicinity of the plate is deduced for this case and compared with a previous calculation.

The coagulation rate of a dilute polydisperse suspension of particles is considered for small Péclet number, which provides a measure of the ratio of the relative gravity-induced motion to Brownian motion between two rigid spheres. In particular, a fourterm expansion for the dimensionless coagulation rate (Nusselt number) as function of the Péclet number is developed by making use of a singular perturbation method. In the limit of the radius of one of the two spheres becoming small, the result agrees with Acrivos & Taylor's (1962) work on mass transfer to spheres at small Péclet number.

Using a hot-wire system developed for this purpose, measurements of the velocity were made in a circular liquid-mercury jet issuing into a low-speed secondary flow and subject to a uniform axial magnetic field. The Reynolds number of the jet was about 10,000 while the magnetic interaction parameter varied from zero to slightly over one. The jet was strongly turbulent under all conditions investigated. The radial distribution of the mean and fluctuating component of the meridional velocity was measured at four axial stations located between 2 and 34 diameters from the nozzle exit. The results indicate that the rate of spreading of the jet is decreased, that the shape of the velocity profile changes and that the turbulent intensity decreases with increasing magnetic-field strength. The high-frequency components of the fluctuations in the second flow seem to be damped more strongly than fluctuations at low frequencies, while the reverse is observed within the core of the jet.

An analysis is made of the two-dimensional flow under gravity of an inviscid non-diffusive stratified fluid into a line sink, involving a velocity discontinuity in the flow field. The fluid above the discontinuity is stagnant and hence is not drawn into the sink. At sufficiently low values of the modified Froude number, this is the only physically possible mode of flow, and is the cause of flow separation in many industrial and natural processes. A proper mathematical solution for flows with a stagnant zone has so far been lacking. This paper presents such a solution, after posing the problem as one involving a free-streamline, which is the line of velocity discontinuity. The solution to be given here is obtained by an inverse method. It is also found herein that the modified Froude number has a value of 0·345 for all separated flows of the kind in question.