Bellhouse et al. (1973) have developed a high-efficiency membrane oxygenator which utilizes pulsatile flow through furrowed channels to achieve high mass transfer rates. We present numerical solutions of the time-dependent two-dimensional Navier–Stokes equations in order to show the structure of the flow. Experimental observations which support this work are presented in a companion paper (Stephanoff, Sobey & Bellhouse 1980).

Steady flow through a furrowed channel will separate provided the Reynolds number is sufficiently large. The effect of varying the Reynolds number and the geometric parameters is given and comparisons with solutions calculated using the modern boundary-layer theory of Smith (1976) show excellent agreement. Unsteady flow solutions are given as the physical and geometric parameters are varied. The structure of the flow patterns leads to an explanation of the high efficiency of the devices of Bellhouse.

The multipole representation technique of Gluckman, Pfeffer & Weinbaum for slow, viscous, axisymmetric motion has been applied to a system of two spheroids. The two particles may have different shapes and volumes. The limitations of this method have been studied through calculations of the drag forces and velocities of particles with aspect ratios from 0·1 to 10 and relative volumes V from 1 to 103. The multipole-expansion convergence is quite slow for large relative volumes and requires on the order of $4 \times V^{\frac{1}{3}}$ multipoles. Wild fluctuations in the drag as the number of multipoles increases were found for large oblate spheroids. Relative velocities are given quite accurately for separations of the centres greater than 1·02 times the minimum separation. Comparisons are made with previous results and approximate theories for two spheres, for non-identical particles at large separation, and for a sphere near a large spheroid.

This paper continues an investigation of the effects of surface tension on steep water waves in deep water begun in Hogan (1979a). A Stokes-type expansion method is given which can be applied to most wavelengths. For capillary waves (2 cm or less) it is found that the surface of the highest wave encloses a bubble of air, as was found for pure capillary waves by Crapper (1957). For intermediate waves (20 cm) the wave profiles are similar to those of pure gravity waves and the wave properties increase monotonically. For gravity waves (200 cm) the wave properties all exhibit a maximum just short of the maximum wave height obtained by the method. The integral properties for all the waves are drawn and given in numerical form in the appendix.

A grid-generated ‘isotropic’ turbulent flow has been subjected to a spectrally local perturbation in the form of a high wavenumber, sinusoidal ripple in the mean velocity. It is introduced as the wake of a fine wire, low solidity screen (a ‘zither’), operating below its vortex-shedding Reynolds numbers. The perturbation appears most clearly downstream as a strongly periodic component in the transverse correlation of streamwise turbulent velocity, R11(r2). Its Fourier transform, E11(k2, a ‘one-dimensional spectrum’, shows a corresponding local ‘spike’. The downstream evolution of this perturbation has been chronicled for four different spectral locations. Their decays are approximately exponential. Measurements of the one-dimensional spectrum E11(k1) at several downstream stations show evidence of energy transfer to other wavenumber regions.

The decay of the velocity correlation ripple is approximately consistent with the decrease in time of the narrow-band correlation function presented by Comte-Bellot & Corrsin (1971). It is also found that the linear perturbation response calculated by Kraichnan (1959) for a disturbance wavenumber larger than those of the incident turbulence shows fair agreement with this decay.

This paper describes the flow structure observed over a bell-shaped hill with height h (the profile of which is the reciprocal of a fourth-order polynomial) when it was placed first in a large towing tank containing stratified saline solutions with uniform stable density gradients and second in an unstratified wind tunnel. (A similarly shaped model hill was also studied in a small towing tank.) Observations were made at values of the Froude number F (≃ U/Nh) in the range 0·1 to 1·7 and at F = ∞, where U is the towing speed and N is the Brunt-Väisälä frequency, and at values of the Reynolds number from 400 to 275000. For F ≲ 0·4, the observations verify Drazin's (1961) theory for low-Froude-number flow over three-dimensional obstacles and establish limits of applicability. For Froude numbers of the order of unity, it is found that a classification of the lee-wave patterns and separated-flow regions observed in two-dimensional flows also appears to apply to three-dimensional hills.

Flow-visualization techniques were used extensively in obtaining both qualitative and quantitative information on the flow structure around the hill. Representative photographs of dye tracers, potassium permanganate dye streaks, shadowgraphs, surface dye smears, and hydrogen-bubble patterns are included here. While emphasis is centred on obtaining a basic understanding of the flow around three-dimensional hills, the results are applicable to the estimation of air pollutant dispersion around hills.

A general class of solutions is studied describing three-dimensional steady convection flows in a fluid layer heated from below with boundaries of low thermal conductivity. Non-linear properties of the solutions are analysed and the physically realizable convection flow is determined by a stability analysis with respect to arbitrary three-dimensional disturbances. The most surprising result is that square-pattern convection is preferred in contrast to two-dimensional rolls that represent the only form of stable convection in a symmetric layer with highly conducting boundaries. The analysis is carried out in the limit of infinite Prandtl number and for a particular boundary configuration. But it is shown that the results hold for arbitrary Prandtl number to the order to which they have been derived and that other assumptions about the boundaries require only minor modifications as long as their thermal conductance remains low.

Observations of flow in furrowed channels support the calculations of part 1 (Sobey 1980). If the mainstream flow is steady there is a critical Reynolds number below which separation does not occur. Above that Reynolds number vortices form and fill the furrow. When the mainstream is oscillatory, the flow may separate during the acceleration to form strong vortices. During the deceleration the vortices grow to fill the furrow and channel. As the mainstream reverses the vortices are ejected from the furrows as the fluid flows between the wall and the vortex. Photographs show that this pattern occurs for sinusoidally varying walls, furrows that are arcs of circles and rectangular hollows.

Two models are discussed to account for the motion of the concentration interface in turbulent mixing layers. In the first one the interface is treated as a vortex sheet and its roll-up is studied. It is argued that this situation represents only the first stages of layer growth and another model is studied in detail in which a row of vortex cores entrains an essentially passive concentration interface with no vorticity. Both models give values of the spreading rate in approximate agreement with observations, and their relation is discussed.

The Lagally theorem for unsteady flow expresses the forces and moments acting on a rigid body moving in an inviscid and incompressible fluid in terms of the singularities of the analytically continued flow within the body. Previous generalizations of the Lagally theorem, originally given by Lagally (1922) for steady flows, are due to Cummins (1957) and Landweber & Yih (1956), who consider the effect of flow unsteadiness on the forces and moments. In these, the system of image singularities within the body was assumed to consist of isolated or continuous (surface or volume) distributions of sources and doublets. A further extension of Lagally's theorem ie due to Landweber (1967), who derived expressions for the steady forces and moments acting on a rigid body generated by isolated or a continuous distribution of multipoles. The purpose of the present paper is to generalize the Lagally theorem so as to include the effects of multipoles in unsteady flow, and deformability of the body, as well as to present a briefer derivation of the resulting formulae. Two examples, illustrating the application of the force and moment formulae, will be presented.

Howells’ model of the scalar spectrum for isotropic turbulent flow is simplified using a restriction to large Péclet number and statistically stationary turbulence at high wavenumbers, and is generalized by introducing Batchelor's constant as a free parameter. The resulting model is compared with data from the atmospheric surface layer, ocean, and liquid mercury. It is found that Howells’ model, which is applicable for arbitrary Prandtl number (Pr), does not compare well with data for large and intermediate Pr at wavenumbers higher than those in the inertial-convective range. The model implies that the inertial-diffusive range asymptotic form, , cannot appear even for Pr as small as that of mercury (Pr = 0·018).

The heat transfer and free convective motion, in inclined air layers heated from below, for angles of incidence 0 [les ] ϕ [les ] 30°, and Rayleigh numbers 100 < Ra cos ϕ < 10000, are studied experimentally. Results of both heat-transfer measurements and flow-visualization studies are reported. The purpose of the study was to investigate the fact, first noted by Hollands et al. (1976), that the experimental heat-transfer data, for ϕ > 20°, is not a function of the product Ra cos ϕ only, as expected from theoretical consideration. This discrepancy between theory and experiment is here attributed to a hypothesized secondary transition in the convective motion, due primarily to perturbation velocities in the upslope direction. This secondary transition appears to be the same as that predicted theoretically by Clever & Busse (1977); qualitative agreement with their theory is observed.

Nonlinear capillary instability of an axisymmetric infinite liquid column is investigated with an initial velocity disturbance consisting of a fundamental and one harmonic component. A third-order solution is developed using the method of strained co-ordinates. For the fundamental disturbance alone, the solution shows that a cut-off zone of wavenumbers (k) exists such that the surface waves grow exponentially below the cut-off zone, linearly in the middle of the zone (near k = 1), and an oscillatory solution exists for wavenumbers above the boundary of the zone. For an input including both the fundamental and a harmonic, all wave components grow exponentially when the fundamental is below the cut-off zone. Using a Galilean transformation, the solution is applied to a progressive jet issuing from a nozzle. The jet breaks into drops interspersed with smaller (satellite) drops for k < 0·65; no satellites exist for k > 0·65. It is shown theoretically that the formation of satellites can be controlled by forcing the jet with a suitable harmonic added to the fundamental.

A physical explanation is suggested for how the instability of certain fluid systems may be provoked by the addition of a ‘bottom-heavy’ density gradient. It is shown that in all the recent examples of this behaviour the stratification shifts the oscillation frequency at the marginal state towards the diffusion rate associated with the driving mechanism for the instability, and this allows a more effective release of the available energy. When the driving mechanism is an adverse temperature gradient, for example, the frequency shift induced by a bottom-heavy solute distribution can increase the temperature change that a vertically-displaced fluid parcel acquires during each halfcycle, thereby enhancing the thermal buoyancy forces which drive the instability.

We show how trains of nonlinear, dispersive wavesIn some of the caaes to be described these are in fact sequences of solitary waves which are ordered by amplitude and which separate in space, as they propagate. In other cases the wave amplitude decreases as the wave propagates, but since the essential balance is between nonlinear steepening and frequency dispersion we feel justified in using the adjective ‘solitary’ to describe them though they violate the clessical description of such waves. can be produced by allowing a region of mixed fluid, with a potential energy greater than its surroundings, to collapse towards its equilibrium state. The number of waves and their amplitude depend on the properties of the mixed region and of the ambient stratification. Three different geometrical configurations have been chosen and while each gives qualitatively the same results the form taken by the generated waves and the final equilibrium shape of the mixed region depend critically on these geometrical factors. We relate the internal waves produced by this mechanism to waves produced in natural systems and show that our observations support at least one possible explanation for those found in the oceans and planetary atmospheres.

The heat transfer across an inclined air layer in the longitudinal roll regime often falls below expected values. A companion paper (Ruth et al. 1980) presents the heat transfer measurement and flow visualization observations; the present paper discusses the mechanisms which are thought to be responsible for this anomalous behaviour and formulates a simple model which correlates the results of our previous study and that of Hart (1971). The observed suppression of heat transfer below values expected for pure longitudinal roll motion is explained in terms of a shear instability. A stabilizing mechansim, which would restore the longitudinal rolls at higher Rayleigh numbers, is described; the return of the heat transfer to expected levels at higher Rayleigh numbers is ascribed to this mechanism. Comparisons are made between the results of the present model and the analyses of Clever & Busse (1977).

The behaviour of a perturbed capillary jet is experimentally determined by studying the jet from the time it emerges from a small hole to the point at which individual droplets of fluid begin to form. Under any particular set of externally applied experimental parameters, i.e. jet velocity, disturbance wavenumber and amplitude, there is a unique, minimum time to this breakup into drops. This characteristic is needed to relate the magnitude of the input voltage to the modulating device and the output velocity perturbation that it produces. Using this relationship we then compare the experimentally produced profiles of jet shape to corresponding ones calculated by using the theory of Chaudhary & Redekopp (1980). The agreement is satisfactory, especially for small values of input voltage. Then, in the neighbourhood of the cutoff wavenumber, we show that the predicted linear growth of the jet profile is also reproduced in the experimental model.

Interferometric data were obtained in the 10 cm × 18 cm hypervelocity shock tube of oblique shock-wave reflexions in argon at initial temperatures and pressures of nearly 300 °K and 15 Torr. The shock Mach-number range covered was 2 ⩽ Ms ⩽ 8 over a series of wedge angles 2° ⩽ θw ⩽ 60°. Dual-wavelength laser interferograms were obtained by using a 23 cm diameter field of view Mach-Zehnder interferometer. In addition to our numerous results, the available data for argon and helium obtained over the last two decades were also utilized. It is shown analytically and experimentally that in non-stationary flows six domains exist in the (Ms, θw) plane where regular reflexion (RR), single-Mach reflexion (SMR), complex-Mach reflexion (CMR) and double-Mach reflexion (DMR) can occur. The transition boundaries between these regions were all established analytically. The experimental results from different sources substantiate the present analysis, and areas of disagreement which existed in the literature are now clarified and resolved. It is shown that real-gas effects have a significant influence on the size of the regions and their boundaries. In addition, isopycnics (constant density lines) are given for the four types of reflexion, as well as the density distribution along the wedge surface. This data should provide a solid base for computational fluid dynamicists in comparing numerical techniques with actual experimental results.

We continue the experiments of Chaudhary & Maxworthy (1980) to include the behaviour at and beyond the point along the jet at which individual drops interspersed with small satellite drops begin to form. In particular, we show how the perturbation wavenumber and amplitude for a fundamental disturbance alone combine to give a variety of modes of breakup into drops and a variety of patterns of drop recombination further downstream. We then show how the addition of a third harmonic to this fundamental can be used in a variety of parameter combinations to control the behaviour of the satellite drops and in some cases eliminate them completely.

Compressibility is added to Busse's (1976) study of convection in a rotating, electrically conducting layer of fluid, of varying depth, which is permeated by an azimuthal magnetic field that is orthogonal to both the rotation vector Ω and the gravitational acceleration g. On the basis of a linear theory, we investigate the kinds of infinitesimal wave motion which the fluid can support when the dominant balance of forces is between pressure gradient and Coriolis force, so that the Proudman–Taylor theorem holds. Unlike Busse, however, we assume that the fluid is statically stable in the sense that the temperature gradient is sub-adiabatic.

In the absence of diffusion, these waves are dynamically neutral and take the form of Rossby waves modified by compressibility and magnetic field. The waves are examined in two limits, the adiabatic and the isothermal, and we define two distinct frequencies at which a pure Rossby wave can oscillate. When diffusion is restored, the disparity between these frequencies makes the fluid susceptible to overstability. We prove that all such amplifying waves must propagate eastward, i.e. in the direction of g ∧ Ω, irrespective of the sign of the depth gradient or the magnitudes of the diffusivities. The theorem does not apply if the fluid is incompressible or convectively unstable at the outset, and therefore does not contradict Busse's result that the direction of azimuthal propagation can be altered by the diffusivities. Nevertheless, we suggest reasons why Busse's method of regarding the imaginary part of the marginal stability equation as a dispersion relation is not in general a reliable one.

We examine the instabilities by subjecting the neutral waves to a weakly diffusive perturbation. We discover, in particular, a new kind of magnetic instability which is crucially dependent upon both compressibility and the depth gradient. In agreement with the general result described above, the instability takes the form of a slow, eastward propagating, amplifying wave.

The principal source for all instabilities is elastic energy, which cannot be tapped in a Boussinesq fluid, since the work done by compression is neglected under that approximation.

A theoretical and experimental investigation of the collapse of a mixed region into a linearly stratified fluid is presented. An analytical model is developed for the motion of the nose of the mixed region, valid for small dimensionless times and it compares well with the experimental results of the present study and previous investigators. For longer times the length of the mixed region did not increase as the square root of the dimensionless time, as previously reported (Wu 1969), for an aspect ratio (half the initial vertical extent divided by the initial horizontal extent) equal to one. In our experiments the length did increase as a power of the dimensionless time also, but with values of the exponent between 0·89 and 0·38, depending on the values of the various parameters. It is concluded that the motion cannot be described by a simple quasi-steady buoyancy-inertia balance only. We argue that the production of internal solitary waves and their interaction with the mixed region are critical to a complete understanding of the experimental results.