ALBERT

Description: A pattern-recognition procedure designed to extract footprints of organized structures from turbulent signals is developed and used to analyse the large-eddy organization of several turbulent wake flows. The pattern-recognition technique is intended to be a general-purpose analytical tool that makes no use of specific flow characteristics, and that can be implemented as a computer code independent of the types of signals to be processed. The technique is applied to analyse the wake generated by a single cylinder at downstream positions ranging from x/D = 10 to x/D = 220. Also the structural features of the wakes behind a rotating cylinder, two cylinders of unequal diameters and two cylinders of equal diameter, one rotating, are examined at x/D= 140. In the near wake the large-scale motions detected are Karman vortices, whose periodic activity persists up to 60 diameters. Further downstream the most significant coherent structures detected are single and double rollers with shear-aligned vorticity, whose dimensions and velocity intensities are properly scaled by the half-width of the wake and the local r.m.s. values, respectively. The similarities observed in the organized motions identified in the different wakes bxx/d — 140, suggest that the roller organization may be an intrinsic characteristic of fully developed turbulent plane wake flows, irrespective of initial conditions. © Copyright Cambridge University Press 1989. All rights reserved.

Description: In this joint theoretical, numerical and experimental study, we investigate the phenomenon of forced generation of nonlinear waves by disturbances moving steadily with a transcritical velocity through a layer of shallow water. The plane motion considered here is modelled by the generalized Boussinesq equations and the forced Korteweg-de Vries (fKdV) equation, both of which admit two types of forcing agencies in the form of an external surface pressure and a bottom topography. Numerical results are obtained using both theoretical models for the two types of forcings. These results illustrate that within a transcritical speed range, a succession of solitary waves are generated, periodically and indefinitely, to form a procession advancing upstream of the disturbance, while a train of weakly nonlinear and weakly dispersive waves develops downstream of an ever elongating stretch of a uniformly depressed water surface immediately behind the disturbance. This is a beautiful example showing that the response of a dynamic system to steady forcing need not asymptotically tend to a steady state, but can be conspicuously periodic, after an impulsive start, when the system is being forced at resonance.A series of laboratory experiments was conducted with a cambered bottom topography impulsively started from rest to a constant transcritical velocity U, the corresponding depth Froude number F = U/(gh0)½ (g being the gravitational constant and h0 the original uniform water depth) being nearly the critical value of unity. For the two types of forcing, the generalized Boussinesq model indicates that the surface pressure can be more effective in generating the precursor solitary waves than the submerged topography of the same normalized spatial distribution. However, according to the fKdV model, these two types of forcing are entirely equivalent. Besides these and some other rather refined differences, a broad agreement is found between theory and experiment, both in respect of the amplitudes and phases of the waves generated, when the speed is nearly critical (0.9 〈 F 〈 1.1) and when the forcing is sufficiently weak (the topography-height to water-depth ratio less than 0.15) to avoid breaking. Experimentally, wave breaking was observed to occur in the precursor solitary waves at low supercritical speeds (about 1.1 〈 F 〈 1.2) and in the first few trailing waves at high subcritical speeds (about 0.8 〈 F 〈 0.9), when sufficiently forced. For still lower subcritical speeds, the trailing waves behaved more like sinusoidal waves as found in the classical case and the forward-running solitary waves, while still experimentally discernible and numerically predicted for 0.6 〉 F 〉 0.2, finally disappear at F ≈ 0.2. In the other direction, as the Froude number is increased beyond F ≈ 1.2, the precursor soliton phenomenon was found also to evanesce as no finite-amplitude solitary waves can outrun, nor can any two-dimensional waves continue to follow, the rapidly moving disturbance. In this supercritical range and for asymptotically large times, all the effects remain only local to the disturbance. Thus, the criterion of the fascinating phenomenon of the generation of precursor solitons is ascertained.

Description: When fluid is withdrawn from a body of stratified fluid the surfaces of constant density are deformed towards the region of withdrawal. The equations describing the flow caused by withdrawal through a point sink in a two-layer unbounded system in which viscous forces dominate are formulated using the boundary-integral representation of Stokes flow. It is shown by dimensional and analytic arguments that surface tension between the layers is a necessary condition for the stability of an interfacial equilibrium in which only one fluid is withdrawn. The critical flow rate above which both fluids are withdrawn is determined numerically as a function of the capillary number. When the flow is supercritical a small adaptation of the numerical scheme allows the proportion of fluid withdrawn from each layer to be found. The various analyses and conclusions further our understanding of the physical processes that determine the compositional output of volcanic eruptions that tap an underlying stratified reservoir of magma. © Copyright Cambridge University Press 1989. All rights reserved.

Description: The effects of bottom friction on coastal trapped waves were investigated using an /-plane, two-layer model including shelf-slope topography. At a change-over latitude where the phase speeds of the internal Kelvin wave and the continental-shelf wave coincide, there are two types of behaviour of the ‘frictional’ eigenvalue (the phase speed and the damping rate) and the eigenfunctions, in terms of the inertial frequency/: if the frequency o) is large, wave characteristics change from one wave to another with/ (Case I); while if w is small enough, the wave characteristics do not change (Case II). In actual environments, it is predicted that the weather-band phenomena (period 2 days to 2 weeks) correspond to Case I, and very low-frequency (VLF) phenomena such as signals of El Nino along the American Continent correspond to Case II. Further, for baroclinic VLF waves, it is found that bottom friction retards the lower-layer velocity, which causes a decrease in damping. Therefore, the VLF signals caused by El Nino can travel far from their origin, overcoming the effect of bottom friction. A bottom-intensified structure in barotropic VLF waves, due to bottom friction, has also been found. © Copyright Cambridge University Press 1989. All rights reserved.

Description: In this paper we examine some general features of the time-dependent dynamics of drop deformation and breakup at low Reynolds numbers. The first aspect of our study is a detailed numerical investigation of the ‘ end-pinching’ behaviour reported in a previous experimental study. The numerics illustrate the effects of viscosity ratio and initial drop shape on the relaxation and/or breakup of highly elongated droplets in an otherwise quiescent fluid. In addition, the numerical procedure is used to study the simultaneous development of capillary-wave instabilities at the fluid-fluid interface of a very long, cylindrically shaped droplet with bulbous ends. Initially small disturbances evolve to finite amplitude and produce very regular drop breakup. The formation of satellite droplets, a nonlinear phenomenon, is also observed. © Copyright Cambridge University Press 1989. All rights reserved.

Description: We study the efficiencies of swimming motions due to small deformations of spherical and cylindrical bodies at low Reynolds number. A notion of efficiency is defined and used to determine optimal swimming strokes. These strokes are composed of propagating waves, symmetric about the axis of propulsion. © Copyright Cambridge University Press 1989. All rights reserved.

Description: Measurements of velocity and temperature fluctuations are made in a turbulent boundary layer with nominally zero pressure gradient for two different slightly heated wall conditions: impermeable and porous surfaces. The temperature fluctuations are measured at three points in the flow to permit the identification of two spatially coherent events: coolings and heatings. Conditional velocity vectors in the plane of mean shear are viewed in a reference frame which translates at a constant velocity. Conditioning is on coolings, heatings or a combination of these events. Sectional streamlines, derived from the velocity vector data, show a succession of critical points: saddles and unstable foci. Coolings are aligned with diverging separatrices through the saddles whereas heatings are identified with the foci. Coolings are associated with a large strain rate and also a large spanwise vorticity: this result seems consistent with the presence of hairpin vortices which extend to different distances from the wall. In contrast, the strain rate and spanwise vorticity are small at the foci. The stabilizing influence of suction is observed in the topology of the organized motion and in the contribution from this motion to the conventional stresses, temperature variance and heat fluxes. © Copyright Cambridge University Press 1989. All rights reserved.

Description: In this paper the Dean (1928) equations are extended to the case of a helical pipe flow, and it is shown that they depend not only on the Dean number K but also on a new parameter λ/ where λ is the ratio of the torsion τ to the curvature κ of the pipe axis and the Reynolds number referred in the usual way to the pipe radius a and to the equivalent maximum speed in a straight pipe under the same axial pressure gradient. The fact that the torsion has no first-order effect on the flow is confirmed, but it is shown that this is peculiar to a circular cross-section. In the case of an elliptical cross-section there is a first-order effect of the torsion on the secondary flow, and in the limit λ/ → ∞ (twisted pipes, provided only with torsion), the first-order ‘displacement’ effect of the walls on the secondary flow, analysed in detail by Choi (1988), is recovered. Different systems of coordinates and different orders of approximations have recently been adopted in the study of the flow in a helical pipe. Thus comparisons between the equations and the results presented in different reports are in some cases difficult and uneasy. In this paper the extended Dean equations for a helical pipe flow recently derived by Kao (1987) are converted to a simpler form by introducing an appropriate modified stream function, and their equivalence with the present set of equations is recovered. Finally, the first-order equivalence of this set of equations with the equations obtained by Murata et al. (1981) is discussed. © 1989, Cambridge University Press. All rights reserved.

Description: Preliminary tests have been carried out on short circular cylinders with both ends free. Drag force is measured across the range 6 × 104 〈 Re 〈 2.6 × 105 for cylinders of length to diameter ratio L/D between 1 and 10. The effect of hemispherical ends is also investigated. A kind of periodic vortex shedding is found in the range 2 〈 L/D 〈 8. The oil-film surface flow visualization shows that the ‘eyes’ near the free ends (regions of low pressure) gradually disappear as L/D is reduced to 3. An asymmetric flow pattern is established for very short cylinders (L/D 〈 3). The detailed measurements of pressure distribution along and across models shows asymmetries of minimum and base pressures along the span. The asymmetric flow produces yawing and rolling moments which are also measured. © 1989, Cambridge University Press. All rights reserved.

Description: A photochromic tracer method has been used to record pulsatile flow velocity profiles simultaneously at three axial locations along a flow channel. Two major advantages of this multiple-trace method are that it enables velocity data to be acquired in an efficient non-invasive manner and that it provides a detailed description of the spatial relationship of the flow field. The latter is found to be particularly useful in the investigation of transitional type flows; for example, in describing coherent flow structures. Studies of the flow patterns in tubes with mild to moderate degrees of vessel constriction were performed using a 2.9 Hz sinusoidal flow superimposed on a steady flow (frequency parameter of 7.5; mean and modulation Reynolds numbers of 575 and 360, respectively). With mild constrictions (〈50% area reduction), isolated regions of vortical and helical structures were observed primarily during the deceleration phase of the flow cycle and in the vicinity of the reattachment point. As expected, these effects were accentuated when the constriction was asymmetric. For moderate constrictions (50%-80%), transition to turbulence was triggered just before peak flow through the breakdown of waves and streamwise vortices that were shed in the high-shear layer. During this vortex generation phase of the flow cycle, the wall shear stress fluctuated quite intensely, especially in the vicinity of the reattachment point, and its instantaneous value increased by at least a factor of eight. Such detailed descriptions of the transition to turbulence and of the spatial and temporal variation of the wall shear stress, particularly near the reattachment point, have not been previously reported for pulsatile flow through constricted tubes. The observed wall shear stress variations support a proposal by Mao & Hanratty (1986) of an interaction of the imposed flow oscillation with the turbulent fluctuations within the viscous sublayer. © 1989, Cambridge University Press. All rights reserved.

Description: The Orr-Sommerfeld equation admits two solution modes for the two-dimensional plane wake. These are the sinuous mode with antisymmetric stream wise fluctuations and the varicose mode with symmetric streamwise fluctuations. The varicose mode is often ignored because its amplification rates are considerably less than those of the sinuous mode. An experimental investigation of the varicose mode in a two-dimensional turbulent wake was undertaken to determine if this mode of instability agrees as well with linear stability theory, as did the sinuous mode in previous experiments (Wygnanski, Champagne & Marasli 1986). The experiments demonstrated that, although it is possible to generate a nearly pure symmetric disturbance wave, it is very difficult to do as the flow is very sensitive to the slightest asymmetries which might be present in the experiments. These asymmetries are preferentially amplified, resulting in the eventual distortion of an initially prominent symmetric wave. It was therefore necessary to decompose phase-averaged measurements of the streamwise component of the velocity fluctuations into their symmetric and antisymmetric parts, and the results were compared with the appropriate theoretical eigenfunctions from linear stability theory. The lateral distribution of the amplitude and the phase of each mode agree reasonably well with their theoretical counterparts from the Orr-Sommerfeld equation. Slowly diverging linear theory predicts the streamwise variation of the sinuous mode quite well, but fails to do so for the varicose mode. An eddy-viscosity model, coupled with the slowly diverging linear equations, predicts the streamwise variation of both modes reasonably well and describes the transverse distributions of the perturbation amplitudes for both modes, but it fails to predict the distribution of phase for the varicose mode. © Copyright Cambridge University Press 1989. All rights reserved.

Description: Two-dimensional motions generated by Langmuir circulation instability of stratified layers of water of finite depth are studied under a simplifying assumption making it mathematically analogous to double-diffusive thermosolutal convection with constant solute concentration and constant heat flux at the boundaries. The nature of possible motions is mapped over a significant region in (S, R) parameter space, where S and R are parameters measuring, respectively, the stabilizing and destabilizing agencies in the problem. In the Langmuir circulation problem R measures the effects of wind and surface wave action, and S measures the stabilizing effect of buoyancy: in the thermosolutal problem, R measures the destabilizing effects of heating, while S measures the stabilizing effect of solute concentration. Effects of lateral boundary or symmetry conditions are found to be crucial in determining the qualitative behaviour. Complex temporal behaviour, including intermittently chaotic flows, are found under suitably constrained (no flux) lateral conditions but are unstable and not realized when these constraints are relaxed and replaced by periodic lateral conditions. Multiple steady states also arise, with those found under constrained lateral conditions losing stability either to travelling waves, or to other steady states when the lateral boundary conditions are relaxed. In some regions of the parameter space, multiple stable nonlinear motions have been found under periodic boundary conditions. The multiple stable states may either be coexisting travelling waves and steady states (different from those found under the constrained lateral conditions). The existence of robust travelling waves may explain some field observations of laterally drifting windrows associated with Langmuir circulations. © Copyright Cambridge University Press 1989. All rights reserved.

Description: We consider the centrifugal separation of an initially homogeneous mixture in an asymmetric geometry. The mixture is shown to acquire a uniform and negative relative vorticity, manifested as a retrograde circulation, as the heavier phase is forced outwards by the centrifugal acceleration. The perturbation theory formulated accounts for inertial effects and endwall boundary layers as well as slight deviations of endwall shape from a level plane. The time-dependent separation process is described and the flow field, including kinematic shocks, is calculated in some cases of technological interest. © Copyright Cambridge University Press 1989. All rights reserved.

Description: The steady motion of a flat surfboard propelled by a solitary wave is considered. The shape of the free surface and the flow of the fluid are determined numerically by series truncation for flows without spray or splash. These flows all bifurcate from the uniform horizontal flow at the critical value of the Froude number. Various limiting cases of these special flows are described analytically. Flows past submerged hydrofoils are discussed also. © Copyright Cambridge University Press 1989. All rights reserved.

Description: A calculation is made of the turbulent transport terms (third moments) that occur in the Reynolds stress equation for buoyant and/or sheared fluids. This calculation is based on neglect of a two-time fourth-order cumulant – a weaker approximation than neglect of the usual single-time fourth-order cumulant. The previously used eddy-damping assumption for single point moments is avoided. This assumption is then examined critically. Comparison is afterward made between the turbulent transport terms derived here and those derived previously by the eddy-damping method, and between the respective derivations. Also the dissipation of third moments is calculated. The calculation is formally limited to mean quantities which vary but slowly in space and time, and to small anisotropy.

Description: An effluent outlet within thirty channel breadths upstream of a branching in a river can lead to high shoreline pollution levels along one of the branches, or near the tip of the central region. This paper identifies an optimal splitting of a steady discharge that minimizes the shoreline pollution. © 1989, Cambridge University Press. All rights reserved.

Description: The Marginal Ice Zone includes wide areas covered by dispersed ice floes in which wave conditions are significantly affected by the ice. When the wind blows from the solid ice pack, towards the open sea, growing waves are scattered by the floes, and their spectral characteristics modified. To further understand this problem, a model for the evolution of wind waves in a sparse field of ice floes has been developed. The sea state is described by a two-dimensional discrete spectrum. Time-limited wave growth is obtained by numerical integration of the energy balance equation using the exact nonlinear transfer integral. Wave scattering by a single floe is represented in terms of far-field expressions of the diffracted and forced potentials obtained numerically by the Green function method. The combined effect of a homogeneous field of floes on the wave spectrum is expressed in terms of the Foldy–Twersky integral equations under the assumption of single scattering. The results show a strong dependence of the spectrum amplitude and directional properties on the ratio of the ice floe diameter to the wavelength. For a certain range of this parameter, the ice cover appears to be very effective in dispersing the energy; the wave spectrum rapidly tends to isotropy, a tendency which prevents the normal growth of wave energy and the decrease in peak frequency. Therefore, in the Marginal Ice Zone, the ability of an offshore wind to generate a significant wave field is severely limited. © 1989, Cambridge University Press. All rights reserved.

Description: A method of using local wave properties to provide a detailed description of breakings in a random wave field is developed. These properties, derived through the Hilbert transform, include the angular frequency, phase velocity, and surface-velocity components. The breaking characteristics are presented, including the probability of breaking, its time- and lengthscales, its intensity, and the phase of its inception. The time- and lengthscales, of breaking events were found to be linearly proportional to the corresponding scales of underlying waves, and to indicate that the breaking region is geometrically similar. Consistent results were obtained from temporal and spatial measurements. Finally, on the basis of these results we have evaluated geometric and kinematic criteria for identifying breaking waves. © 1989, Cambridge University Press. All rights reserved.

Description: The development of the Reynolds stress field was studied for flows in which an initially two-dimensional boundary layer was skewed sideways by a spanwise pressure gradient ahead of an upstream-facing wedge. Two different wedges were used, providing a variation in the boundary-layer skewing. Measurements of all components of the Reynolds stress tensor and all ten triple products were measured using a rotatable cross-wire anemometer. The results show the expected lag of the shear stress vector behind the strain rate. Comparison of the two present experiments with previous data suggests that the lag can be estimated if the radius of curvature of the free-stream streamline is known. The magnitude of the shear stress vector in the plane of the wall is seen to decrease rapidly as the boundary-layer skewing increases. The amount of decrease is apparently related to the skewing angle between the wall and the free stream. The triple products evolve rapidly and profiles in the three-dimensional boundary layer are considerably different than two-dimensional profiles, leaving little hope for gradient transport models for the Reynolds stresses. The simplified model presented by Rotta (1979) performs reasonably well providing that an appropriate value of the T-parameter is chosen. © 1989, Cambridge University Press. All rights reserved.

Description: Viscous oscillatory flow past particles, governed by the unsteady Stokes equation, is considered. The problem is addressed in its general form for arbitrary flows and particle shapes using the boundary-integral method. It is shown that the leading-order correction to the force exerted on a particle in unsteady flow may be inferred directly from the drag in steady translational motion. For axisymmetric flow, a numerical procedure for solving the boundary-integral equation is developed, and is applied to study streaming oscillatory flow past spheroids, dumbbells, and biconcave disks. The effect of the particle geometry on the structure of the flow is studied by comparing the streamline pattern associated with these particles to that for the sphere. The results reveal the existence of travelling stagnation points on the surface of non-spherical particles, and the formation of unsteady viscous eddies in the interior of the flow. These eddies grow during the decelerating flow period, and shrink during the accelerating flow period. For particles with concave boundaries, unsteady free eddies may originate from an expansion of wall eddies that reside within the concave regions. © 1989, Cambridge University Press. All rights reserved.

Description: We present results of numerical simulations of stably stratified, randomly forced turbulence. The selection of forcing and damping are designed to give insight into the question of whether cascade of energy to large scales is possible for strongly stratified three-dimensional turbulence in a manner similar to two-dimensional turbulence. We consider narrow-band wavenumber forcing, whose angular distribution ranges from two-dimensional to three-dimensional isotropic. Our principal results are as follows; for two-dimensional forcing, and for sufficiently small Froude number, the statistically steady state is characterized by a weakly inverse-cascading horizontal- velocity variance field. The vertical variability of the horizontal-velocity field is pronounced, but seems to approach a limit independent of the Brunt-Väisäläfrequency N, as N→ ∞. If, on the other hand, the Froude number exceeds a critical value, the vertical variability is weak, and the statistics of the scales larger than the forcing scale is near that predicted by inviscid equipartitioning. For all forcing functions considered the vertical motion and temperature field (w, T), centred at smaller scales, are more three-dimensionally isotropic, with no large-scale organization. At large N, (small Froude number) the w-field scales as 1/N, with horizontal motion field nearly independent of N. Furthermore, at large N and for horizontal forcing, the horizontal motion field is consistent with the condition that a substantial fraction of the total dissipation is attributable to an effective drag acting upon all horizontal scales of motion, which in turn flattens the slope of the energy spectrum in the inverse-cascade range, and increases it in the enstrophy-cascade range. © 1989, Cambridge University Press. All rights reserved.

Description: We develop a simple model in which longitudinal, compressible, unsteady heat transfer between heater and gas is computed in the small-Mach-number limit. This calculation is used to determine the transfer function of the heater, which plays an important role in the stability limits of the thermoacoustic instability of the Rijke tube. The transfer function is determined analytically in the limit of small expansion parameter y, and numerically for y of order unity. In the case pμ/cp= constant, an analytical solution can be found. © 1989, Cambridge University Press. All rights reserved.

Description: In this paper the normal-mode small-amplitude waves of high-speed jets are investigated analytically and computationally. Three families of instability waves, each having a distinct wave pattern and propagation characteristics, have been found. One of the families of waves is the familiar Kelvin-Helmholtz instability wave. The other two families of waves do not appear to have been clearly identified and systematically studied before. Waves of one of the new wave family propagate with supersonic phase velocities relative to the ambient gas. They are, therefore, referred to as supersonic instability waves. Waves of the other family have subsonic phase velocities. Accordingly they are called subsonic waves. The subsonic waves have the unusual property that they are unstable only for jets with finite thickness mixing layers. They are neutral waves when calculated by a vortex-sheet jet model. Earlier Oertel (1979, 1980, 1982) using a novel optical technique observed in a series of experiments three sets of waves in high-speed jets. The origin of these waves, however, remains so far unexplained and a theory has yet to be developed. In the present study it will be shown that the computed wave patterns and propagation characteristics of the Kelvin-Helmholtz, the supersonic and the subsonic instability waves match essentially those observed by Oertel. The physical mechanisms which give rise to the three families of waves as well as some of the most salient characteristic features of each set of waves are discussed and reported here. © 1989, Cambridge University Press. All rights reserved.

Description: Convection rolls in a fluid layer heated from below become unstable to disturbances in the form of waves travelling along the axis of the rolls when the Rayleigh number exceeds a critical value i?n. This transition to a time-dependent form of convection also occurs in the presence of a vertical magnetic field when the fluid is electrically conducting. In this paper the finite-amplitude properties of these waves are investigated for the values 0.1 and 0.025 of the Prandtl number. It is shown that the onset of oscillations reduces the heat transport by convection and that a mean flow in the direction of propagation is associated with the waves. Although the magnetic field has an inhibiting influence on steady convection, the inhibiting influence on the onset of oscillations is even stronger such that in some cases a higher heat transport is obtained in the presence of a magnetic field than in its absence. For similar reasons subcritical finite-amplitude onset of travelling-wave convection occurs for sufficiently large magnetic field strengths. Finally the stability of travelling-wave convection is investigated and the Rayleigh number RU1for the transition to asymmetric wave convection is determined. © 1989, Cambridge University Press. All rights reserved.

Description: The relative viscosities of suspensions of randomly oriented rods in a Newtonian fluid were measured using falling-ball rheometry. The rods were monodisperse and sufficiently large to render colloidal and Brownian forces negligible. Steel and brass ball bearings were dropped along the centreline of cylindrical columns containing the suspensions. The terminal velocities of the falling balls were measured and used to determine the average viscosities of the suspensions. The suspensions behaved as Newtonian fluids in that they were characterized by a constant viscosity. They exhibited a linear relative viscosity-volume fraction relationship for volume fractions less than 0.125, and, for volume fractions between 0.125 and 0.2315, the specific viscosity increased with the cube of the volume fraction. The relative viscosity was found to be independent of falling-ball size for a ratio of falling ball to fibre length greater than 0.3. It was found to be independent of the diameter of the containing cylindrical column for a ratio of column diameter to fibre length greater than 3.2. The value determined for the intrinsic viscosity is in good agreement with theoretical predictions for suspensions of randomly oriented rods. © 1989, Cambridge University Press. All rights reserved.

Description: A wide-angle model for water-wave propagation on an irregular bathymetry is developed based on the linear mild-slope equation. The spectral model decomposes the incident wave wavetrain into directional modes, or an angular spectrum. The effect of the bottom topography is shown to force the generation of additional directional wave modes. Nonlinearity is incorporated in the model by correcting the wave parameters iteratively using an empirical nonlinear dispersion relationship which is approximately valid over the entire range of water depths. Numerical examples are presented for waves incident on a transverse bar field, a laboratory experiment involving wave focusing over an elliptic shoal on a sloping beach for which detailed measurements are available and for waves focusing behind a circular shoal resting on a flat bottom. The application of the model is limited to cases in which the model domain is rectangular and the depth variation in the lateral direction is small if waves of large incident angle are modelled. © 1989, Cambridge University Press. All rights reserved.

Description: Three-dimensional numerical calculations of an inviscid gas past scoops in a gas centrifuge are performed by solving the Euler equations using the explicit Roe upwind scheme with a second-order of accuracy. The scoop is modelled as a cylindrical or a wing-shaped rod attached to a central tube and extending radially outwards, and no inlet flow to the scoop is assumed. The scoops are placed close to the bottom end plate, and there is no covering baffle plate. The numerical grid employed is of the multibox type. The main results are as follows. For a cylindrical scoop, a detached bow shock is formed in front of the scoop. Behind the shock, strong radially inward motion of gas towards the central axis is induced, and it excites an upward flow which becomes a countercurrent. The inward flow just in front of the scoop produces a vortex column in the upstream region of the scoop. For a wing-shaped scoop, an oblique shock attached to the scoop is formed, and an inward flow is induced behind the shock. The shock is not so strong as that in the case of a cylindrical scoop model. The drag coefficient of the wing-shaped scoop is almost one-fourth of that of the cylindrical scoop. The addition theorem of the scoop drag is verified for the wing-shaped scoop. © 1989, Cambridge University Press. All rights reserved.

Description: Two-component LDV measurements are obtained over an ellipse at an angle of attack. Detailed information is provided for the stagnation region, the two separation regions and the two free shear layers. Steady and unsteady flow measurements are presented. Periodic disturbances in the oncoming stream are introduced and ensemble-averaged unsteady data are obtained. An integral picture of laminar flow over a lifting body with no sharp edges is thus presented, which is readily available for comparison with asymptotic or numerical calculations. © 1989, Cambridge University Press. All rights reserved.

Description: An experimental study has been carried out to examine double-diffusive convection in a porous medium. The experiments were performed in a horizontal layer of porous medium consisting of 3 mm diameter glass beads contained in a box 24 cm x 12 cm x 4 cm high. The top and bottom walls were made of brass and were kept at different constant temperatures by separate baths, with the bottom temperature higher than that of the top. The onset of convection was detected by a heat flux sensor and by the temperature distribution in the porous medium. When the porous medium was saturated with distilled water, the onset of convection was marked by a change in slope of the heat flux curve. The temperature distribution in the longitudinal direction in the middle of the layer indicated a convection pattern consisting of two-dimensional rolls with axes parallel to the short side. This pattern was confirmed by flow visualization. When the porous medium was saturated with salinity gradients of 0.15% cm 1 and 0.225% cm -1 , the onset of convection was marked by a dramatic increase in heat flux at the critical AT, and the convection pattern was three-dimensional. When the temperature difference was reduced from supercritical to suberitical values, the heat flux curve established a hysteresis loop. Results from linear stability theory, taking into account effects of temperature-dependent viscosity, volumetric expansion coefficients, and a nonlinear basic state salinity profile, are discussed. © 1989, Cambridge University Press. All rights reserved.

Description: The problem of calculating nonlinear two-dimensional free-surface potential flow about a pair of counter-rotating point vortices rising under their own influence towards a free surface is solved numerically. The two vortices are inserted into fluid which is initially at rest. A boundary/integral-equation method is used to obtain free-surface elevations and streamlines about the rising pair of vortices for several vortex strengths. The paths of the two vortices are compared with those of a counter-rotating vortex pair under a rigid wall. © 1989, Cambridge University Press. All rights reserved.

Description: The linear stability characteristics of non-isothermal flow in vertical annuli has been determined for two geometries. The analysis demonstrates that a fully developed mixed-convection flow in a vertical annulus is unstable in certain regions of an appropriate parameter space. Consequently, parallel countercurrent flows, predicted by previous numerical models and commonly used by engineers, are often physically unrealizable and can be observed experimentally only in special circumstances. In addition, it is found that the most unstable disturbances are often asymmetric in the parameter range of practical interest. The instability behaviour was also found to depend on whether the inner or outer cylinder was heated. © 1989, Cambridge University Press. All rights reserved.

Description: In the interior of an extended liquid a cyclic cavitation process can be set up by a focused acoustic wave field. A cluster of supercritical cavities is developed when the tensile strength of the liquid is exceeded. This occurs almost simultaneously in a confined region around the focal point where the sound speed rapidly vanishes when the cavities approach their critical size. As a consequence, the inner boundary condition for the acoustic field, carried by the single-phase liquid, changes from total reflection at the focal point without phase shift, as at a rigid wall, to reflection at the boundary of the cluster, which now forms a low-pressure two-phase kernel embedded in the single-phase liquid. The cluster is compliant to expansion waves but essentially rigid to compression waves. As soon as the cluster is formed its further development is determined jointly by the sound field and by the far field pressure of the liquid. The former first makes the cavities in the cluster grow and then contributes to its collapse, while the latter tends to bring about its collapse from the moment of cluster formation. © 1989, Cambridge University Press. All rights reserved.

Description: Effects of product dissociation and preferential diffusion on the structure, propagation and diffusional-thermal instability of the classical one-dimensional laminar flame have been studied by using activation energy asymptotics. Analytical expressions as functions of dissociation and diffusion parameters have been obtained for such bulk flame parameters as the flame temperature reduction and the propagation rate eigenvalue, and for the dispersion relation governing flame stability. Results on flame propagation show that while under most situations the flame speed is reduced due to product dissociation, it can attain values in excess of the non-dissociative limit for highly mobile product species which can preferentially back diffuse to the upstream portion of the reaction zone where they recombine and release the associated recombination heat. Results on flame stability show that it is promoted in the presence of product dissociation which has a moderating influence on the flame temperature fluctuations, and for highly-mobile product species because of the enhanced burning rate and curvature-induced concentration modification. © 1989, Cambridge University Press. All rights reserved.

Description: On physical grounds it was recently suggested that limiting capillary–gravity waves of solitary type may exist on the surface of deep water (Longuet-Higgins 1988). This paper describes accurate numerical calculations which support the conjecture. The limiting wave has a phase speed c = 0.9267 (gτ)¼. It is one of a family of solitary waves having speeds c  1.30 (gτ)¼. The maximum angle of inclination αmax of the free surface is a monotonically decreasing function of the speed c. Physical arguments suggest that αmax has a positive lower bound. © 1989, Cambridge University Press. All rights reserved.

Description: The proper orthogonal decomposition technique (Lumley's decomposition) is applied to the turbulent flow in a channel, to extract coherent structures by decomposing the velocity field into characteristic eddies with random coefficients. In the homogeneous spatial directions a generalization of the shot-noise expansion is used to determine the characteristic eddies. In this expansion the Fourier coefficients of the characteristic eddy cannot be obtained from second-order statistics. Three different techniques are used to determine the phases of these coefficients: (i) a technique based on the bispectrum, (ii) a spatial compactness requirement, and (iii) a functional continuity argument. Results from these three techniques are found to be very similar. The implications of these techniques and the shot-noise expansion are discussed in the Appendix. The dominant eddy is found to contribute as much as 76% to the turbulent kinetic energy. In two and three dimensions, the characteristic eddies consist of an ejection region straddled by streamwise vortices which leave the wall in a very short streamwise distance of approximately 100 wall units.

Description: In this paper, we study the evolution of strongly three-dimensional disturbances which are generated by a point force in a parallel mixing layer. When the input force is a pulse, a wave packet develops whose wavefronts are approximately parallel to the spanwise direction. This is in sharp contrast to a wave packet in a wall boundary layer for which the wavefronts are strongly curved. On the other hand, when the input disturbance is oscillating harmonically in time, a spatially growing instability wave develops in a downstream wedge of (#, z)-space. The size of this wedge, as a function of excitation frequency and velocity ratio, is determined. The receptivity of the shear layer to pulse-type and harmonic excitation is also studied. It is found that the shear layer is especially sensitive to relatively high-frequency forcing on its centreline. © 1989, Cambridge University Press. All rights reserved.

Description: In this paper we use a boundary-integral technique to numerically investigate the motion of a viscous drop toward a fluid-fluid interface at low Reynolds number. We consider only the case of a drop moving toward its homophase. The solutions include large deformations of both the drop and interface for capillary numbers in the range 0.2≤Ca ≤10 and the viscosity ratios between 0.1 ≤ λ ≤10, and illustrate the approach toward a film-drainage geometry for a drop which starts at a large distance from an initially undeformed, flat interface. We also consider briefly the effect of starting the drop closer to the interface. © 1989, Cambridge University Press. All rights reserved.

Description: Degenerating stable temporal Orr-Sommerfeld eigenmodes are studied for plane Poiseuille flow. The discrete spectrum of the eigenmodes is shown to possess infinitely many degeneracies, each appearing at a certain combination of k (the modulus of resultant wavenumber) and αR (the streamwise wavenumber time the Reynolds number). The eigenmodes are found to degenerate in a specific manner which confines the streamwise phase velocities of the degeneracies to be around § of the centreline velocity. The responses of the degeneracies are investigated through the initial-value problem. The responses of the first four symmetric and the first two antisymmetric degeneracies are evaluated numerically for arbitrary initial disturbances expanded in terms of Chebyshev polynomials. The first symmetric and the first antisymmetric degeneracies exhibit temporal growth of the amplitudes in the wavenumber space. The maximum amplitudes are at most 7 times larger than the corresponding initial amplitudes. The amplitudes of the responses of the other four degeneracies decay rapidly owing to their higher damping rates. The time for which the degeneracy-response is in the growing phase is shown to be stretched with increasing Reynolds number. The degeneracies can therefore be active for longer periods of time at larger Reynolds numbers. © 1989, Cambridge University Press. All rights reserved.

Description: 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 Ω(r)= 1/2(Ω1 + Ω2)tanh (1/D ln r/R), where r is the distance from the rotation axis; and R,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 〈 Z)erlt= f. In a weakly supercritical flow (|D—Z〉crit| 4 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 £2 coincides with the rotation velocity Q(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 AD =| D—Dcrlt|, but the character of this dependence is different in different regions of the supercriticality parameter AD. 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. © 1989, Cambridge University Press. All rights reserved.

Description: The exact time-dependent three-dimensional Navier-Stokes and temperature equations are integrated numerically to simulate stably stratified homogeneous turbulent shear flows at moderate Reynolds numbers whose horizontal mean velocity and mean temperature have uniform vertical gradients. The method uses shear-periodic boundary conditions and a combination of finite-difference and pseudospectral approximations. The gradient Richardson number Ri is varied between 0 and 1. The simulations start from isotropic Gaussian fields for velocity and temperature both having the same variances. The simulations represent approximately the conditions of the experiment by Komori et al. (1983) who studied stably stratified flows in a water channel (molecular Prandtl number Pr = 5). In these flows internal gravity waves build up, superposed by hot cells leading to a persistent counter-gradient heat-flux (CGHF) in the vertical direction, i.e. heat is transported from lower-temperature to higher-temperature regions. Further, simulations with Pr = 0.7 for air have been carried out in order to investigate the influence of the molecular Prandtl number. In these cases, no persistent CGHF occurred. This confirms our general conclusion that the counter-gradient heat flux develops for strongly stable flows (Ri ≈ 0.5–1.0) at sufficiently large Prandtl numbers (Pr = 5). The flux is carried by hot ascending, as well as cold descending turbulent cells which form at places where the highest positive and negative temperature fluctuations initially existed. Buoyancy forces suppress vertical motions so that the cells degenerate to two-dimensional fossil turbulence. The counter-gradient heat flux acts to enforce a quasi-static equilibrium between potential and kinetic energy. Previously derived turbulence closure models for the pressure-strain and pressure-temperature gradients in the equations for the Reynolds stress and turbulent heat flux are tested for moderate-Reynolds-number flows with strongly stable stratification (Ri = 1). These models overestimate the turbulent interactions and underestimate the buoyancy contributions. The dissipative timescale ratio for stably stratified turbulence is a strong function of the Richardson number and is inversely proportional to the molecular Prandtl number of the fluid. © 1989, Cambridge University Press. All rights reserved.

Description: A series of experiments are performed in a Hele-Shaw cell, consisting of two parallel closely spaced glass plates. A liquid (oil or water, both of viscosity of 1.0 cP) is injected at a constant volumetric flow rate, q, to radially displace a much more viscous liquid (glycerine, 1050 cP) in the cell. Oil is immiscible with and water is miscible with glycerine. The data presented in this paper are taken mostly at late stages of the fingering process, when the pattern has multiple generations of splitting. Correlations with time are obtained for the finger length and the overall pattern density. The time- and lengthscales have been found for the immiscible case. At the same dimensionless time, immiscible patterns are similar and have the same generation of splitting. The overall density of each pattern decreases with time. The pattern shows fractal behaviour only after a certain number of generations of splitting. The fractal dimension of the immiscible pattern decreases from 1.9 to 1.82 when the pattern goes from the third to the sixth generation of splitting. The fractal dimension of the miscible pattern reaches a constant value after about ten generations of splitting and the fractal dimension ranges from 1.50 to 1.69 for q/Db = 4.8 x 105-7.0 x 106. The miscible patterns are insensitive to dispersion for large q/Db. For immiscible fingers A/6 scales with Ca~0.31〈sup/〉forcapillary number Ca ranging from about 8 x 10~4to 0.05. For miscible fingers, A/6 is insensitive to dispersion and ranges from 5 to 10 for large q/Db. Here D is the molecular diffusion coefficient in glycerine, 6 the cell gap width and A the splitting wavelength. © 1989, Cambridge University Press. All rights reserved.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: 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.

Description: 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.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: In this paper we investigate the effect of a weak vertical magnetic field on the boundary-layer flow of an electrically conducting fluid past a vertical heated wall. We derive similarity solutions for the flow and temperature and show that the flow is composed of three regions: an inner region where the flow is a regular perturbation of the classical boundary-layer flow due to a heated semi-infinite vertical plate; an inviscid outer region where fluid is entrained from downwards towards the plate; and beyond this a quiescent region, separated from the outer region by a free shear layer. Thus the effect of the magnetic field is to inhibit the entrainment of fluid across the magnetic field lines in the whole region and confine it to an outer boundary layer. © Copyright Cambridge University Press 1989. All rights reserved.

Description: We consider single-phase gases in which the fundamental derivative is negative over a finite range of pressures and temperatures and show that inadmissible discontinuities give rise to shock splitting. The precise conditions under which splitting occurs are delineated and the formation of the split-shock configuration from smooth initial conditions is described. Specific numerical examples of shock splitting are also provided through use of exact inverse solutions. © Copyright Cambridge University Press 1989. All rights reserved.

Description: Weakly nonlinear theory is used to study the porous-medium analogue of the classical Rayleigh-Bénard problem, i.e. Lapwood convection in a saturated porous layer heated from below. Two particular aspects of the problem are focused upon: (i) the effect of thermal imperfections on the stability characteristics of steady rolls near onset; and (ii) the evolution of unstable rolls.For Rayleigh-Bénard convection it is well known (see Busse and co-workers 1974, 1979, 1986) that the stability of steady two-dimensional rolls near onset is limited by the presence of cross-roll, zigzag and sideband disturbances; this is shown to be true also in Lapwood convection. We further determine the modifications to the stability boundaries when small-amplitude imperfections in the boundary temperatures are present. In practice imperfections would usually consist of broadband thermal noise, but it is the Fourier component with wavenumber close to the critical wavenumber for the perfect problem (i.e. in the absence of imperfections) which, when present, has the greatest effect due to resonant forcing. This particular case is the sole concern of the present paper; other resonances are considered in a complementary study (Rees & Riley 1989).For the case when the modulations on the upper and lower boundaries are in phase, asymptotic analysis and a spectral method are used to determine the stability of roll solutions and to calculate the evolution of the unstable flows. It is shown that steady rolls with spatially deformed axes or spatially varying wavenumbers evolve. The evolution of the flow that is unstable to sideband disturbances is also calculated when the modulations are π out of phase. Again rolls with a spatially varying wavenumber result.

Description: Edge waves of frequency o) and longshore wavenumber k in water of depth h(y) = h1H(〈ry/h1}i0 ^ y 〈 oo, are calculated through an asymptotic expansion in crjkhx on the assumptions that er 1 and khx = 0(1). Approximations to the tree-surface displacement in an inner domain that includes the singular point at h ~ 0 and the turning point near gh % o)2/k2 and to the eigenvalue A = (s)2/crgh are obtained for the complete set of modes on the assumption that h(y) is analytic. A uniformly valid approximation for the free-surface displacement and a variational approximation to A are obtained for the dominant mode. The results are compared with the shallow-water approximations of Ball (1967) for a slope that decays exponentially from er to 0 as h increases from 0 to h, and of Minzoni (1976) for a uniform slope that joins h = 0 to a flat bottom at h = and with the geometrical-optics approximation of Shen, Meyer & Keller (1968). © Copyright Cambridge University Press 1989. All rights reserved.

Description: The present work investigates the dynamics of the one-dimensional, unsteady flow of a spherical bubble cloud subject to harmonic far-field pressure excitation. Bubble dynamics effects and energy dissipation due to viscosity, heat transfer, liquid compressibility and relative motion of the two phases are included. The equations of motion for the average flow and the bubble radius are linearized and a closed-form solution is obtained. The results are then generalized by means of Fourier synthesis to the case of arbitrary far-field pressure excitation. The flow displays various regimes (sub-resonant, trans-resonant and super-resonant) with different properties depending on the value of the relevant flow parameters. Examples are discussed in order to show the effects of the inclusion of the various energy dissipation mechanisms. Finally the results for the case of Gaussian-shaped far-field pressure change are presented and the most important limitations of the theory are briefly discussed. The simple linearized dynamical analysis developed so far clearly demonstrates the importance of the complex phenomena connected to the interaction of the dynamics of the bubbles with the flow and provides an introduction to the more realistic study of the same flows with nonlinear bubble dynamics.

Description: It is well-known that laminar flow of a liquid in a duct is predicted to choke if the viscosity of the liquid increases exponentially with increasing pressure. In other words, the pressure drop in the duct is predicted to become unbounded when the volumetric flow rate reaches a critical finite value. Choking is not observed in practice, however: the reason why is investigated here. It is shown that choking is always predicted to occur if the viscosity is independent of temperature or heat generation by viscous dissipation is neglected. If the viscosity decreases exponentially with increasing temperature and heat generation is not neglected, however, and if the temperature field is fully developed or if the flow is adiabatic, it is shown that choking is predicted not to occur. © 1989, Cambridge University Press. All rights reserved.

Description: The paper formulates the concept of a semi-preserving developing of a free non-isothermal swirling jet. On the basis of experimental data it may be estimated that the semi-preserving conditions are established at the distance x—x0 « (15–20) D. Also, it has been pointed out that self-preservation, which requires the turbulence structure to be similar during decay, may be considered as an asymptotic state of the semi-preserving development achieved in practice at a distance where it,. Njux ur0.0. © Copyright Cambridge University Press 1989. All rights reserved.

Description: The effects of chemical heat release on the large-scale structure in a chemically reacting, turbulent mixing layer are investigated using direct numerical simulations. Three-dimensional, time-dependent simulations are performed for a binary, single-step chemical reaction occurring across a temporally developing turbulent mixing layer. It is found that moderate heat release slows the development of the large-scale structures and shifts their wavelengths to larger scales. The resulting entrainment of reactants is reduced, decreasing the overall chemical product formation rate. The simulation results are interpreted in terms of turbulence energetics, vorticity dynamics, and stability theory. The baroclinic torque and thermal expansion in the mixing layer produce changes in the flame vortex structure that result in more diffuse vortices than in the constant-density case, resulting in lower rotation rates of the large-scale structures. Previously unexplained anomalies observed in the mean velocity profiles of reacting jets and mixing layers are shown to result from vorticity generation by baroclinic torques.

Description: Experiments were conducted to study the effects of heat release in a planar, gas-phase, reacting mixing layer formed between two free streams, one containing hydrogen in an inert diluent, the other, fluorine in an inert diluent. Sufficiently high concentrations of reactants were utilized to produce adiabatic flame temperature rises of up to 940 K (corresponding to 1240 K absolute). The temperature field was measured at eight fixed points across the layer. Flow visualization was accomplished by schlieren spark and motion picture photography. Mean velocity information was extracted from Pitot-probe dynamic pressure measurements. The results showed that the growth rate of the layer, for conditions of zero streamwise pressure gradient, decreased slightly with increasing heat release. The overall entrainment into the layer was substantially reduced as a consequence of heat release. A posteriori calculations suggest that the decrease in layer growth rate is consistent with a corresponding reduction in turbulent shear stress. Large-scale coherent structures were observed at all levels of heat release in this investigation. The mean structure spacing decreased with increasing temperature. This decrease was more than the corresponding decrease in shear-layer growth rate, and suggests that the mechanisms of vortex amalgamation are, in some manner, inhibited by heat release. The mean temperature rise profiles, normalized by the adiabatic flame temperature rise, were not greatly changed in shape over the range of heat release of this investigation. A small decrease in normalized mean temperature rise with heat release was however observed. Imposition of a favourable pressure gradient in a mixing layer with heat release resulted in an additional decrease in layer growth rate, and caused only a very slight increase in the mixing and amount of chemical product formation. The additional decrease in layer growth rate is shown to be accounted for in terms of the change in free-stream velocity ratio induced by the pressure gradient.

Description: The problems of dynamical onset of convection, textural transitions and chaotic dynamics in a two-dimensional, rectangular Rayleigh-Bénard system have been investigated using well-resolved, pseudo-spectral simulations. All boundary conditions are taken to be no-slip. It is shown that the process of creating the temperature gradient in the system, is responsible for roll creation at the side boundaries. These rolls either induce new rolls or move into the interior of the cell, depending on the rate of heating. Complicated flow patterns and textural transitions are observed in both non-chaotic and chaotic flow regimes. Multistability is frequently observed. Intermediate-Prandtl-number fluids (e.g. 0.71) have a quasiperiodic time dependence up to Rayleigh numbers of order 106. When the Prandtl number is raised to 6.8, one observes aperiodic (chaotic) flows of non-integer dimension. In this case roll merging and separation is observed to be an important feature of the dynamics. In some cases corner rolls are observed to migrate into the interior of the cell and to grow into regular rolls; the large rolls may shrink and retreat into corners. The basic flow patterns observed do not change qualitatively when the chaotic regime is entered.

Description: The method of contour dynamics (CD) is applied to several inviscid prototype flows typical of the motions found in the transition region of the free shear layer. Examples of the interaction between the fundamental stream wise-layer perturbation and its first subharmonic are presented that illustrate the events of pairing and tearing of two rolled-up cores and also the coalescence of three rolled-up cores. The present simulations of the temporally unstable two-dimensional layer, at effectively infinite Reynolds number, support the hypothesis that the dynamics of the large-scale roll-up is only weakly dependent on Reynolds number. However, we find fine-scale structure that is not apparent in previous simulations at moderate Reynolds number. Spiral filaments of rotational fluid wrap around the rolled-up vortex cores producing ‘spiky’ vorticity distributions together with the entanglement of large quantities of irrotational fluid into the layer. Simulations proceeded only until the first such event because we were unable to resolve the fine detail generated subsequently. The inclusion of prescribed vortex stretching parallel to the vortex lines is found to accelerate the initial roll-up and to enhance the production of spiral vortex filaments. In the fundamental-subharmonic interaction, vortex stretching slows but does not prevent pairing. © Copyright Cambridge University Press 1989. All rights reserved.

Description: The initial unsteady two-dimensional flow around an impulsively started circular cylinder is investigated using the random-vortex numerical method. To understand the mechanisms of the primary and secondary vortex formation, we investigate the relationship between the creation, diffusion and convection of vorticity to the genesis and evolution of the complex and£unstable ’ flow structures of the recirculating zone behind the cylinder. Our simulation reveals detailed geometric features of the wake which are in agreement with experimental observations and with other numerical calculations. Numerical calculation at Reynolds numbers R = 3000, 9500 and 104show that the numerical method is appropriately sensitive to changes of the Reynolds number. Numerical functionals such as the length of the wake, velocity on the wake axis and the angle of separation for our calculations are in satisfactory agreement with known experimental and numerical results. This numerical method gives results comparable to those of a previously published method but does so using much less memory and computer time. © 1989, Cambridge University Press. All rights reserved.

Description: The distortion by large-scale random motions of small-scale turbulence is investigated by examining a model problem. The changes in energy spectra, velocity and vorticity moments, and anisotropy of small-scale turbulence are calculated over timescales short compared with the timescale of small-scale turbulence by applying rapid distortion theory with a random distortion matrix for different initial conditions irrotational or rotational, and isotropic or anisotropic large-scale turbulence with or without mean strain, and isotropic or anisotropic small-scale turbulence. We have obtained the following results (1) Irrotational random strains broaden the small-scale energy spectrum and transfer energy to higher wavenumbers. (2) The rotational part of the large-scale strain is important for reducing anisotropy of turbulence rather than transferring energy to higher wavenumbers. (3) Anisotropy in small-scale turbulence is reduced by large-scale isotropic turbulence. The reduction of anisotropy of the velocity field depends on the initial value of the velocity anisotropy tensor of the small-scale velocity field ui defined by u5/u1 —and ui Ai, also on the anisotropy of the distribution of the energy spectrum in wavenumber space. The reduction in anisotropy of the vorticity field o)t depends only on the vorticity anisotropy tensor. (4) The pressure-strain correlation is calculated for the change in Reynolds stress of the anisotropic small-scale turbulence. The correlation is proportional to time and depends on the difference between the velocity and wavenumber anisotropy tensors. These results (which are exact for small time) differ significantly from current turbulence models. (5) The effect of large-scale anisotropic turbulence on isotropic small-scale turbulence is calculated in general. Results are given for the case of axisymmetric large scales and are compared with the observed behaviour of small-scale turbulence near interfaces. (6) When a mean irrotational straining motion is applied to turbulence with distinct large-scale and small-scale components in their velocity field, the large-scale irrotational motions combine with the mean straining to increase further the anisotropy of the vorticity of the small scales, but the large-scale rotational motions reduce the small-scale anisotropy. For isotropic straining motion, the latter is weaker than the former. After the mean distortion ceases, both kinds of large-scale straining tend to reduce the anisotropy. This also has implications for modelling the rate of reduction of anisotropy. © 1989, Cambridge University Press. All rights reserved.

Description: Eastward-flowing zonal currents on a thin rotating shell, such as a planetary atmosphere or ocean, have integral properties analogous to open channel flows, the latitudinal width of the zonal current being the analogue of the depth of an open channel flow. The purpose here is to apply the formalism and some of the concepts of open channel flow hydraulics to zonal flows and demonstrate the results with laboratory experiments. In particular a critical relationship is found between a representative zonal velocity, U, and the half-width of the current, a. A dimensionless parameter (U/βa2), the Froude/Rossby number, is found analogous to the Froude number of open channel flow. Westward-flowing currents do not have an equivalent analogue. © 1989, Cambridge University Press. All rights reserved.

Description: The stability of core-annular flow (CAF) in pipes is analysed using the linear theory of stability. Attention is confined to the potentially stable case of lubricated pipelining with the less viscous liquid, say water, in the annulus. The effects of surface tension and density are included, but gravity is excluded. We find upper and lower branches of the neutral curve in a Reynolds number (IR) Vs wavenumber (a) plane. A window of parameters is identified in which CAF is stable to small disturbances. When R is below the lower critical value, CAF is destabilized by surface tension and long waves break up into slugs and bubbles. The sizes of slugs and bubbles of oil in water observed by Charles, Govier & Hodgson (1961) are given by the wavelength of the fastest growing long wave. This long-wave instability is a capillary instability, modified by shear, which reduces to Rayleigh's instability in the appropriate limit. At higher R, the capillary instability is stabilized by shear. At yet higher R, above the upper critical value, the flow is unstable to generally shorter waves which leads to emulsification, water droplets in oil. The theory agrees with experiments. The analysis seems to be applicable to the design of lubricated pipelines; for example, there is an optimum viscosity ratio for stability, greater stability can be obtained by using heavy liquid as a lubricant when the flow is unstable to capillary modes on the lower branch and by using light liquids when the flow is unstable to emulsifying disturbances on the upper branch. © 1989, Cambridge University Press. All rights reserved.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: Turbulence in the convective boundary layer (CBL) uniformly heated from below and topped by a layer of uniformly stratified fluid is investigated for zero mean horizontal flow using large-eddy simulations (LES). The Rayleigh number is effectively infinite, the Froude number of the stable layer is 0.09 and the surface roughness height relative to the height of the convective layer is varied between 10−6 and 10−2. The LES uses a finite-difference method to integrate the three-dimensional grid-volume-averaged Navier–Stokes equations for a Boussinesq fluid. Subgrid-scale (SGS) fluxes are determined from algebraically approximated second-order closure (SOC) transport equations for which all essential coefficients are determined from the inertial-range theory. The surface boundary condition uses the Monin–Obukhov relationships. A radiation boundary condition at the top of the computational domain prevents spurious reflections of gravity waves. The simulation uses 160 × 160 × 48 grid cells. In the asymptotic state, the results in terms of vertical mean profiles of turbulence statistics generally agree very well with results available from laboratory and atmospheric field experiments. We found less agreement with respect to horizontal velocity fluctuations, pressure fluctuations and dissipation rates, which previous investigations tend to overestimate. Horizontal spectra exhibit an inertial subrange. The entrainment heat flux at the top of the CBL is carried by cold updraughts and warm downdraughts in the form of wisps at scales comparable with the height of the boundary layer. Plots of instantaneous flow fields show a spoke pattern in the lower quarter of the CBL which feeds large-scale updraughts penetrating into the stable layer aloft. The spoke pattern has also been found in a few previous investigations. Small-scale plumes near the surface and remote from strong updraughts do not merge together but decay while rising through large-scale downdraughts. The structure of updraughts and downdraughts is identified by three-dimensional correlation functions and conditionally averaged fields. The mean circulation extends vertically over the whole boundary layer. We find that updraughts are composed of quasi-steady large-scale plumes together with transient rising thermals which grow in size by lateral entrainment. The skewness of the vertical velocity fluctuations is generally positive but becomes negative in the lowest mesh cells when the dissipation rate exceeds the production rate due to buoyancy near the surface, as is the case for very rough surfaces. The LES results are used to determine the root-mean-square value of the surface friction velocity and the mean temperature difference between the surface and the mixed layer as a function of the roughness height. The results corroborate a simple model of the heat transfer in the surface layer. © 1989, Cambridge University Press. All rights reserved.

Description: 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.

Description: 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.

Description: 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’.

Description: 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.

Description: 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 x 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. © 1989, Cambridge University Press. All rights reserved.

Description: 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 (Prandt1 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. © 1989, Cambridge University Press. All rights reserved.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: 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.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: 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 Peclet 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 U212/Dwhen 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. © 1989, Cambridge University Press. All rights reserved.

Description: The onset of thermal convection in a double layer of two superimposed immiscible fluids heated from below is investigated. The linearized equations of the problem are analysed in a much wider region of the parameter space than has been studied before. It is shown that the onset of steady convection in the two layers may occur in the form of either viscously or thermally coupled motions. In addition to the oscillatory interfacial instability, which depends on a non-vanishing distortion of the interface, there exists another oscillatory instability which corresponds to a cyclic variation between viscous and thermal coupling. Conditions for the onset of this instability are outlined and its connections with the other modes of the system are demonstrated in bifurcation diagrams. In the experiments the shadowgraph method is used for the visualization of the onset of convection and for the measurement of its wavelength. Changeovers between viscous and thermal coupling can be identified, but the experimental realization of an oscillatory onset has been elusive so far. © 1989, Cambridge University Press. All rights reserved.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: A two-dimensional Galerkin formulation of the three-dimensional Oberbeck-Boussinesq equations is used to describe the onset of convection in an infinite rigid horizontal channel uniformly heated from below. The dependence of the critical Rayleigh number on the channel aspect ratio is determined and results are compared with those of an idealized model studied by Davies-Jones (1970). Asymptotic results are derived for both narrow and wide channels, corresponding to limits of small and large aspect ratios respectively. In the latter case the main core flow, consisting of two-dimensional rolls with axes perpendicular to the vertical walls of the channel, can be represented by the solution of an amplitude equation. Close to the walls, however, the motion remains fully three-dimensional and a reversal of the vertical flow is associated with a local subdivision of each main roll into a pair of co-rotating rolls. © Copyright Cambridge University Press 1989. All rights reserved.

Description: The compositional stratification in solid formed by cooling a binary alloy from below is investigated theoretically and experimentally. It is shown that in order to grow composite solid the boundary temperature needs to be below the eutetic temperature. Two separate cases are considered. In the first, heavy fluid is released on solidification. The solid growth is then governed by the diffusive transport of heat and composition. The resultant solid is shown to have a fixed composition until the far-field conditions change. In the second case, light fluid is released on solidification. This generates turbulent compositional convection in the melt which significantly increases the transport of heat and composition across the solid/melt interface. As a result, the fraction of heavy component in the solid initially increases, but subsequently decreases to conserve mass. A simple theoretical model, using the approximation of a flat solid/melt interface is developed; this predicts differences in the thermal flux in saturated and undersaturated melts. Laboratory experiments involving aqueous solutions of sodium carbonate cooled from below which released light fluid displayed compositional convection and stratification of the solid as predicted. © Copyright Cambridge University Press 1989. All rights reserved.