The wake characteristic of groups of normal flat plates, consisting of two, three, or four plates placed side by side with slits in between, have been investigated experimentally. When the ratio of the slit width to the plate width (slit ratio) was small, the gap flows were observed to be biased either upward or downward in a stable way, leading to multiple, stable flow patterns for a single slit-ratio value. Some regularities were recognized in the gap-flow directions and the appearance of the flow patterns. The plates on the biased side showed high drag and regular vortex shedding, while those on the unbiased side showed the opposite. The origin of the biased flow has also been investigated with water-tank experiments, numerical calculations and wind-tunnel experiments. The-results showed that the origin of biasing is strongly related to the vortex shedding of each plate of a row.

The gravitational exchange of two fluids with different densities between reservoirs connected by a channel of constant depth and slowly varying breadth is analysed as a problem of internal hydraulics. It is shown that maximal two-way exchange with a net barotropic flow requires the presence of two controls, one at the narrrowest section and a second or ‘virtual’ control lying to one side of the narrowest section. The two controls are connected by a subcritical region, but are separated from subcritical conditions in the reservoirs by supercritical flow and stationary internal bores. Solutions are found for maximal exchange without a net barotropic component, in which case the problem is similar to that first examined by Stommel & Farmer (1953). The Stommel & Farmer analysis is shown to be a rather special limiting example of submaximal exchange, not generally applicable to natural flows. The addition of a net barotropic flow yields a range of different flow types, including maximal exchange, one-layer baroclinic flow, one-layer barotropic flow, submaximal flow governed by a reservoir condition and two-layer unidirectional flow. The maximal-exchange solution is integrated for periodic barotropic flow.

The analysis of two-layer exchange flow through contractions with a barotropic component treated by Armi & Farmer (1986) is extended to include exchange flows over sills and through a combination of a sill and contraction. It is shown that exchange over a sill is fundamentally different from exchange through a contraction. Control at the sill crest acts primarily through the deeper layer into which the sill projects and only indirectly controls the surface layer. This asymmetry in the control results in asymmetrical flows. The interface depth above the crest is not one half the total depth, as assumed in other studies by analogy with flow through contractions, but is somewhat deeper; the maximal exchange rate is less than for flow through a contraction of equal depth. When both a sill and a contraction are present, the contraction influences control at the sill crest only if it lies between the sill and the source of denser water. The response to barotropic flow is also asymmetrical: the transition to single-layer flow occurs at much lower speeds for a barotropic component in one direction than the other.

Results of the analysis are applied to exchange flow through the Strait of Gibraltar, which includes both a sill and a contraction. It is shown that maximal exchange conditions apply throughout part of the tidal cycle, and observations illustrate several of the analytical predictions for barotropic flows, including the formation of fronts, single-layer flow, submaximal exchange and reverse flow.

Visualization and measurements of velocity and pressure were made for the flow past prisms of variable length with square cross-section, placed normal to smooth and turbulent approaching flows. Square prisms shed vortices in one of the two fixed wake planes. The plane of shedding is switched irregularly from one to the other. Flow visualization confirms the two main effects of small– and large-scale turbulence on the flow past square prisms that had previously been suggested. In particular, large-scale turbulence intensifies vortex shedding from square prisms through resonant interaction, thereby reducing the base pressure considerably.

A three-dimensional and time-dependent numerical model is used to study the nonlinear interactions between thermal convective motions, rotation, and imposed flows with vertical shear. All cases have Rayleigh numbers of 104 and Prandtl numbers of 1.0. Rotating cases have Taylor numbers of 104.

For the non-rotating cases, the effects of the shear on the convection produce longitudinal rolls aligned with the shear flow and a downgradient flux of momentum. The interaction between the convection and the shear flow decreases the shear in the interior of the fluid layer while adding kinetic energy to the convective motions. For unit Prandtl number the dimensionless flux of momentum is equal to the dimensionless flux of heat.

For rotating cases with vertical rotation vectors, the shear flow favours rolls aligned with the shear and produces a downgradient flux of momentum. However, the Coriolis force turns the flow induced by the convection to produce a more complicated shear that changes direction with height. As in the non-rotating cases, the convective motions become more energetic by extracting energy from the mean flow. For Richardson numbers larger than about − 1.0, the dominant source of eddy kinetic energy is the shear flow rather than buoyancy.

For rotating cases with tilted rotation vectors the results depend upon the direction of the shear. For weak shear, convective rolls aligned with the rotation vector are favoured. When the shear flow is directed to the east along the top, the rolls become broader and the convection weaker. For large shear in this direction, the convective motions are quenched by the competition between the shear flow and the tilted rotation vector. When the shear flow is directed to the west along the top, strong shear produces rolls aligned with the shear. The heat and momentum fluxes become large and can exceed those found in the absence of a tilted rotation vector. Countergradient fluxes of momentum can also be produced.

A computer simulation is used to make a detailed study of the stress tensor in a simple shear flow of two-dimensional disks. The stresses are shown to arise from two momentum-transfer mechanisms: the 'streaming’ or kinetic mode. By which momentum is carried by particles are they move through the bulk material: and the collisional mode, by which momentum is transferred from one point to another in the material by interparticle collisions. As might be expected, the results show that the streaming mode dominates at disperse packings and the collisional mode dominates at dense packings. The friction coefficient, the ratio of shear to normal forces, is shown to decrease at high particle packing for both the collisional and streaming modes of transport. Normal-stress differences are observed within the shear plane and are evident in both the streaming and collisonal parts.

An acoustic and shock-wave theory of the noise generated by advanced turbo-propellers operating at supersonic tip helical velocity and high-subsonic cruise Mach number is developed. The theory includes the thickness and loading noise of the highly swept propeller blades. When operating at their design conditions these propellers radiate extremely intense sound waves. Because of the weakly nonlinear propagation effects these high-intensity acoustic disturbances steepen up quickly to form shock waves. In the present theory advantage is taken of the fact that in the blade fixed-rotating-coordinate system the acoustic and shock-wave fields are time independent. The problem is formulated in this coordinate system as a boundary-value problem. Weakly nonlinear propagation effects are incorporated into the solution following Whitham's nonlinearization procedure (Whitham 1974). The change in the disturbance-propagation velocity due to fluid-particle motion as well as the change in the speed of sound resulting from compression and rarefaction are all taken into account. It is found that the equal-area rule of Whitham's shock-fitting method is also applicable to the present problem. This method permits easy construction of the three-dimensional shock surfaces associated with the acoustic disturbances of these high-speed turbopropellers. Numerical results of the present theory are compared with the measurements of the JETSTAr flight experiment and the United Technology Research Center low-cruise Mach number open-wind-tunnel data. Very favourable overall agreements are found. The comparisons indicate clearly that, when these supersonic turbopropellers are operated at their high subsonic design-cruise Mach number, weakly nonlinear propagation effects must be included in the theory if an accurate prediction of the waveform of the sound wave incident on the design aircraft fuselage is to be obtained. This is especially true for noise radiated in the upstream or forward directions. In the forward directions the effective propagation velocity of the acoustic disturbances is greatly reduced by the convection velocity of the ambient flow. This allows more time for the cumulative nonlinear propagation effects to exert their influence, leading to severe distortion of the waveform and the formation of shock waves.

Our flow-visualization and spectral studies of flow between concentric independently rotating cylinders have revealed a surprisingly large variety of different flow states. (The system studied has radius ratio 0.883, aspect ratios ranging from 20 to 48, and the end boundaries were attached to the outer cylinder.) Different states were distinguished by their symmetry under rotation and reflection, by their azimuthal and axial wavenumbers, and by the rotation frequencies of the azimuthal travelling waves. Transitions between states were determined as functions of the inner- and outer-cylinder Reynolds numbers, Ri and Ro, respectively. The transitions were located by fixing Ro and slowly increasing Ri. Observed states include Taylor vortices, wavy vortices, modulated wavy vortices, vortices with wavy outflow boundaries, vortices with wavy inflow boundaries, vortices with flat boundaries and internal waves (twists), laminar spirals, interpenetrating spirals, waves on interpenetrating spirals, spiral turbulence, a flow with intermittent turbulent spots, turbulent Taylor vortices, a turbulent flow with no large-scale features, and various combinations of these flows. Some of these flow states have not been previously described, and even for those states that were previously described the present work provides the first coherent characterization of the states and the transitions between them. These flow states are all stable to small perturbations, and the transition boundaries between the states are reproducible. These observations can serve as a challenge and test for future analytic and numerical studies, and the map of the transitions provides several possible codimension-2 bifurcations that warrant further study.

Laminar heat- or mass-transfer processes are theoretically investigated for two-dimensional spatially periodic suspensions of circular cylinders, each member of which rotates steadily about its own axis under the influence of an external couple. The novelty of the ensuing convective-diffusion phenomena derives from the absence of convective motion at the suspension lengthscale (the ‘macroscale’), despite its presence at the interstitial or particle lengthscale (the ‘microscale’). The latter fluid motion consists of a cellular vortex-like flow characterized by closed streamlines. These periodically closed streamlines give rise to a situation in which there exists no net flow at the macroscale. The resulting macroscale transport of heat or mass thus proceeds purely by conduction, the rate being characterized by a tensor diffusivity — dependent upon the angular velocity of the cylinders. Matched-asymptotic-expansion methods together with generalized Taylor dispersion theory are used to calculate this macroscale conductivity in the dual limit of large rotary Péclet numbers and small gap widths between adjacent cylinders. This prototype study illustrates the fact that the usual separation of transport processes into distinct convective and conductive contributions is not generally a scale-invariant concept; that is, microscale convec-tional contributions to the transport of heat or mass are not generally representable by corresponding macroscale convectipnal contributions to the transport. Possible applications of the analysis exist in the area of enhanced conduction rates in ferrofluids or other dipolar fluids rotating relative to a fixed external field (or conversely).

In this article, we describe an optical set-up designed to measure directly velocity gradients (strophometry). This strophometer is based on the analysis of the distortions of a fringe pattern ‘written’ instantaneously in a flow field. We apply it to study the transverse velocity-gradient component ∂u/∂y in a plane Poiseuille flow at moderate Reynolds numbers. Mean values and different moments of the fluctuating gradient distribution related to viscous dissipation, vorticity dynamics and intermittency are obtained. These results are interpreted in terms of the large-scale structures which are present in the flow.

A magnetic field suppresses turbulence and thermal convection in a Czochralski crystal puller. The amounts and distributions of dopants and oxygen in the crystal are determined by the motion of the molten silicon during crystal growth. This paper presents analytical solutions for the melt motion in a Czochralski puller with a strong, uniform, axial magnetic field. The relatively small electrical conductivity of the crystal plays a key role in determining the flow. Certain combinations of crystal and crucible rotation rates lead to flow patterns with a large volume of almost stagnant fluid under most of the crystal face. The values of these rotation rates depend on the magnetic field strength.

The neutral curve for stationary vortex disturbances in rotating-disk flow is computed up to a Reynolds number of 107 using the sixth-order system of linear stability equations which includes the effects of streamline curvature and Coriolis force. It is found that the neutral curve has two minima: one at R = 285.36 (upper branch) and the other at R = 440.88 (lower branch). At large Reynolds numbers, the upper branch tends to Stuart's asymptotic solution while the lower branch tends to a solution that is associated with the wave angle corresponding to the direction of zero mean wall shear.

This work is concerned with the parallel-flow shear dispersion of a solute which undergoes a second-order recombination reaction. The primary goal of the work is to provide background theory for the determination of the coefficient of diffusion of reactive-gas-phase species. Approximations are sought for large time and weak combination. At first, dispersion dominates recombination, and a modification of the Chatwin (1970) asymptotic expansion gives the concentration distribution as a regular perturbation expansion in e and t−½, where e is a dimensionless parameter characterizing recombination and t is dimensionless time. The regular expansion breaks down for dimensionless times t of O(ε−2) when dispersion and recombination are of the same importance. At these times, the governing equation is nonlinear and this regime is analysed by a numerical method. Finally, a similarity solution is derived for the concentration of solute when ε2t is large. Overall, the major effect of second-order recombination on dispersion is to flatten the peak of the dominant Gaussian concentration distribution. The speed of the centre of mass of the solute cloud is not affected at leading order by recombination.

A theoretical simulator of immiscible displacement of a non-wetting fluid by a wetting one in a random porous medium is developed. The porous medium is modelled as a network of randomly sized unit cells of the constricted-tube type. Under creeping-flow conditions the problem is reduced to a system of linear equations, the solution of which gives the instantaneous pressures at the nodes and the corresponding flowrates through the unit cells. The pattern and rate of the displacement are obtained by assuming quasi-static flow and taking small time increments. The porous medium adopted for the simulations is a sandpack with porosity 0.395 and grain sizes in the range from 74 to 148 μrn. The effects of the capillary number, Ca, and the viscosity ratio, κ = μo/μw, are studied. The results confirm the importance of the capillary number for displacement, but they also show that for moderate and high Ca values the role of κ is pivotal. When the viscosity ratio is favourable (κ < 1), the microdisplacement efficiency begins to increase rapidly with increasing capillary number for Ca > 10−5, and becomes excellent as Ca → 10−3. On the other hand, when the viscosity ratio is unfavourable (κ > 1), the microdisplacement efficiency begins to improve only for Ca values larger than, say, 5 × 10−4, and is substantially inferior to that achieved with κ < 1 and the same Ca value. In addition to the residual saturation of the non-wetting fluid, the simulator predicts the time required for the displacement, the pattern of the transition zone, the size distribution of the entrapped ganglia, and the acceptance fraction as functions of Ca, κ, and the porous-medium geometry.

The behaviour of non-wetting ganglia undergoing immiscible displacement in a porous medium is studied with the help of a theoretical simulator. The porous medium is represented by a network of randomly sized unit cells of the constricted-tube type. The fluid of a non-wetting ganglion is in contact with the wetting fluid at menisci which are assumed to be spherical cups. The flow in every constricted unit cell occupied by a single fluid is modelled as flow in a sinusoidal tube. The flow in every unit cell that contains a meniscus and portions of both fluids is treated with a combination of a Washburn-type analysis and a lubrication-theory approximation. The flow problem is thus reduced to a system of linear equations the solution of which gives the instantaneous pressures on the nodes, the flowrates through the unit cells, and the velocities of the menisci. The motion of a ganglion is determined by assuming quasi-static flow, taking a small time increment, updating the positions of the menisci, and iterating. The behaviour of solitary ganglia is studied under conditions of quasi-static displacement (Ca slightly larger than critical), as well as dynamic displacement (Ca substantially larger than critical). Shape evolution, rate of flow, mode of break-up, and stranding are examined. The stranding and break-up coefficients are determined as functions of the capillary number and the ganglion size for a 100 × 200 sandpack. The dependence of the average ganglion velocity on ganglion size, capillary number, viscosity ratio and dynamic contact angle is examined for the simple case of motion between straight rows of spheres. It is found, among other things, that when μo < μw the velocity of ganglia can be substantially larger than that of the displacing fluid.

This paper is concerned with oscillations of a body of water in basins of variable depth, employing a system of linearized equations which can be obtained from the theory of a directed fluid sheet for an incompressible, homogeneous, inviscid fluid (Green & Naghdi 1976a, 1977). For free oscillations over a level bottom, an assessment of the range of validity of the linearized theory is made by an appropriate comparison with a corresponding well-known exact solution (Lamb 1932). This assessment indicates an ‘intermediate’ range of validity for the linearized theory not covered by usual classical approximations for long waves. Encouraged by this assessment, we apply the linear theory of a directed fluid sheet to basins of variable depth; and, in particular, consider a class of basin profiles whose equilibrium depth (along its width) varies in one direction only. By a method of asymptotic integration, a general solution is obtained which is relatively simple and accounts for the effect of vertical inertia. The solution is sinusoidal in time, periodic along the breadth direction and involves Bessel functions of the first order in the width direction. For two special basin profiles, detailed comparisons are made between the predictions of the asymptotic solution (i.e. the frequencies in the lowest modes of oscillations) with corresponding results obtained by other procedures.

We develop a time-dependent numerical algorithm, using a boundary-integral approach, to investigate fingering in Hele-Shaw cells. Starting from a sinusoidal variation in the initial interface, stable fingers quickly form for a wide range of the dimensionless surface-tension parameter. For very low values of the parameter, the incipient finger bifurcates. The stable fingers are clearly the same as those obtained by McLean & Saffman (1981) using a steady-state algorithm. These steady-state solutions were found to be linearly unstable. We resolve this apparent discrepancy regarding stability by tracing the fate of small disturbances placed on and about the finger tip. We show that some small disturbances do, indeed, grow initially; however, they reach a maximum amplitude and decay as they convect backward from the tip of the finger to regions where stabilizing surface tension is the major physical force. Relatively large imposed disturbances, on the other hand, cause a finger to bifurcate; the critical disturbance amplitude decreases as the surface tension is reduced

A numerical implementation of the method of matched asymptotic expansions is proposed to analyse two-dimensional uniform streaming flow at low Reynolds number past a straight cylinder (or cylinders) of arbitrary cross-sectional shape. General solutions for both the Stokes and Oseen equations in two dimensions are expressed in terms of a boundary distribution of fundamental single- and double-layer singularities. These general solutions are then converted to integral equations for the unknown distributions of singularity strengths by application of boundary conditions at the cylinder surface, and matching conditions between the Stokes and Oseen solutions. By solving these integral equations, using collocation methods familiar from three-dimensional application of ‘boundary integral’ methods for solutions of Stokes equation, we generate a uniformly valid approximation to the solution for the whole domain.

We demonstrate the method by considering, as numerical examples, uniform flow past an elliptic cylinder, uniform flow past a cylinder of rectangular cross-section, and uniform flow past two parallel cylinders which may be either equal in radius, or of different sizes.

An investigation of the mathematical model of a compacting medium proposed by McKenzie (1984) for the purpose of understanding the migration and segregation of melts in the Earth is presented. The numerical observation that the governing equations admit solutions in the form of nonlinear one-dimensional waves of permanent shape is confirmed analytically. The properties of these solitary waves are presented, namely phase speed as a function of melt content, nonlinear interaction and conservation quantities. The information at hand suggests that these waves are not solitons.

Measurements are reported of the lift on a sphere in oscillatory flow. Two different oscillating-tray rigs were used and measurements were made for a sphere at various distances from both smooth and rough beds. For the rough bed, the roughness consisted of a single close-packed layer of similar spheres glued to the surface of the tray. Both smooth and rough beds showed somewhat similar trends. At sufficiently low Reynolds numbers the lift varied smoothly during the course of the cycle with maxima near the point at which fluid velocity was greatest and minima near velocity reversal. However, at higher Reynolds numbers a secondary peak began to appear near the point of velocity reversal. It is suggested that this secondary peak may be associated with vortex formation around the sphere. When the sphere diameter D was large compared with the viscous-boundary-layer lengthscale 1/β(= (2ν/ω)½), the maximum value of CL during the course of a half-cycle initially increased with increasing Reynolds number, reached a maximum, and then declined steadily. The curves were similar for smaller βD-values except that the tests did not extend to low enough Reynolds numbers to show the initial rise. For the smaller values of βD the lift record became unstable when the Keulegan–Carpenter number exceeded approximately 244. At higher βD-values it was not possible to observe any significant instability for the range of Reynolds numbers investigated.