ALBERT

Description: The radiation resistance (damping coefficient) and virtual mass for a circular disk that executes small, heaving oscillations at the surface of a semi-infinite body of water, originally calculated by MacCamy (1961a) through the numerical solution of an integral equation, are calculated from a systematic hierarchy of variational approximations. The first member of this hierarchy is based on the exact solution of the boundary-value problem for α = 0 and is in error by less than 2% for 0  α  1, where α = aσ2/g (a = radius of disk, σ = angular frequency, g = gravity). The second approximation provides a variational interpolation between the limiting results for α = 0 and α = ∞ and appears to be in error by less than 2% for all α except in certain narrow intervals, where pseudoresonances pose difficulties. Those difficulties are overcome by local reference to the third approximation. Numerical results are plotted for 0  α  10. Asymptotic results for α ↑ ∞ are developed in an Appendix. The corresponding formulation and the first variational approximation are developed for pitching oscillations of the disk. © 1987, Cambridge University Press. All rights reserved.

Description: The collapse of cavitation bubbles generates microjets which can cause local plastic deformation on neighbouring solid materials. This deformation takes the form of pitting which will eventually lead to large scale material erosion. A model is presented which predicts the relative dimensions of the pits as a function of bubble collapse pressure, the shape of the microjet and the mechanical properties of the solid. The high pressures required to cause material deformation are generated by a water hammer mechanism and the solid is taken to have a simple bilinear elastic plastic response. Measurements on pits produced by both flow and spark generated cavitation are found to lie within the bounds predicted by the model. Both the measurements and the model suggest that there is a threshold microjet velocity below which no damage occurs, although such behaviour is masked, in practice, by statistical variations. © 1987, Cambridge University Press. All rights reserved.

Description: It has been suggested that hairpin vortices may play a key role in developing and sustaining the turbulence process in the near wall region of turbulent boundary layers. To examine this suggestion, a study was done of the hairpin vortices generated by the interaction of a hemisphere protubferancee within a developing laminar boundary layer. Under the proper conditions, hairpin vortices are shed extremely periodically, which allows detailed examination of their behaviour. Shedding characteristics of the hemispheres were determined using hot-film-anemometry techniques. The flow patterns created by the presence of the hairpin vortices have been documented using flow visualization and hot-film-anemometry techniques, and cross-compared with the patterns observed in the near-wall of a fully turbulent boundary layer. In general, it has been observed that many of the visual patterns observed in the near-wall region Of a turbulent boundary layer can also be observed in the wake of the hairpimsheddihg hemisphere, which appears supportive of the importance of hairpin vortices in the near-wall turbulence production process. Furthermore, velocity measurements indicate the presence of strong inflexional profiles just downstream of the hairpin-vortex generation region which evolve into fuller profiles with downstream distance, eventually developing a remarkable similarity to a turbulent-boundary-layer velocity profile. © 1987, Cambridge University Press. All rights reserved.

Description: Liquid flow in a baffled stirred reactor vessel driven by a six blade disk impeller has been investigated experimentally. Laser slit photography provided an overview of the flows which were quantified by measurements of velocity characteristics, obtained with a laser Doppler anemometer, for an impeller rotational speed of 300 r.p.m. and for three impeller clearances from the bottom of the vessel. The mean flow results show an inclination of the impeller stream and the formation of ring vortices above and below the impeller, which depend on the clearance; the flow was strongly three dimensional with large regions having circumferential velocities in a direction opposite to that of the impeller rotation. Impeller induced torque measurements show that the Power number is invariant with clearance for turbulent flow Reynolds numbers (≥40000) and increases with impeller diameter. The flow structure was controlled mainly by convection and pressure forces with turbulent mixing important in the impeller region. © 1987, Cambridge University Press. All rights reserved.

Description: A workshop was held at MIT in February 1986 for specialists performing theoretical research on the interactions of water waves with floating or submerged bodies. The principal applications of this field are related to ship hydrodynamics and to wave loadings on offshore platform. In addition to the traditional approach based on linearization of the wave and body motions, substantial progress has been made on nonlinear problems including both analytical and numerical studies. Subsequent workshops are planned on an annual basis. © 1987, Cambridge University Press. All rights reserved.

Description: Coherent structures of turbulent open-channel flow in the wall region of a channel bed were examined quantitatively using experimental data obtained by flow visualization. Successive pictures of flow patterns in two horizontal cross-sections at different levels near the channel bed were taken, and then were digitized and analysed by a computer. This method of flow visualization and picture processing enabled us to calculate the distributions of the three components of the velocity vectors. The distributions of velocities, streamlines, two-dimensional divergence and three components of vorticity could be calculated and are displayed as graphical output. In our numerical analyses, the idea of a two-dimensional correlation coefficient is introduced, through which the degree of similarity of turbulence structures can be better estimated than with the usual one-dimensional coefficient. Use of the data was based on the premise that the essential element in a turbulence structure is vortex motion. We propose a conceptual model of turbulence structure in which the elementary unit of coherent structure in the buffer layer is presumed to be a horseshoe vortex and in which the characteristics of the multiple structure of turbulence are shown with respect to the scale, arrangement and generating process of horseshoe vortices and longitudinal vortices. Our model clearly explains the generating mechanism and mutual relations of low-speed regions, high-speed regions, ejections, sweeps and localized free-shear layers. © 1987, Cambridge University Press. All rights reserved.

Description: Experiments were performed to investigate the supersonic flow of a turbulent boundary layer over short regions of concave surface curvature. Upstream of each curved wall, the free-stream Mach number was 2.87, and the incoming boundary layer was typical of a two-dimensional, zero-pressure-gradient, high-Reynolds-number flow. Two different curvatures were used, with radii of curvature equal to 10 and 50 initial boundary-layer thicknesses (Models I and II, respectively). The turning angle was 8° in each case. As the boundary layer passed through the curved region, it experienced a strong adverse pressure gradient, as well as the destabilizing influences of bulk compression and concave curvature. Downstream of the curved walls, the flow relaxed on a short plane wall. The mean and turbulent field for each flow was investigated, using normal and inclined hot wires to measure the turbulent fluctuations. Wherever possible, the results were compared with those from a corresponding 8° ramp. The ramp and Model I exhibited a very similar behaviour: turbulence levels increased significantly, and there was a marked increase in structural parameters such as the stress ratio —u'v'/u'— and the length-and timescales of the turbulent motions. Model II behaved quite differently: although the turbulence levels increased, structural parameters were essentially unchanged. The similarities between the ramp and Model I results suggest that the perturbation in both cases is ‘ rapid ’ in that the perturbation can be described in terms of total strains rather than local strains. In contrast, the flow in Model II is sensitive to the local variations in the strain rate. © 1987, Cambridge University Press. All rights reserved.

Description: Using the multiple-scale perturbation method, the diffraction of a nonlinear nearly periodic wavetrain by a vertical circular cylinder is investigated. The envelope of the incident wavetrain is assumed to modulate slowly in the direction of wave propagation. The relationship between the envelopes of incident and scattered waves is derived. It is shown that second-order scattered set-down waves propagate only at the long-wave velocity (gh)½. The formula for low-frequency wave forces acting on the cylinder is presented. The low-frequency wave forces, which are second-order quantities, are caused by set-down waves beneath the wavetrain and the results of the self-interactions of the leading-order first harmonic wave components. Numerical solutions are presented for the case where the wave envelope varies sinusoidally. © 1987, Cambridge University Press. All rights reserved.

Description: The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, is shown to depend on the particle inertia and the free-fall terminal velocity in still fluid. With no inertia the particle settles on average at the same rate as in still fluid, assuming there is no mean flow. Particle inertia produces a bias in each trajectory towards regions of high strain rate or low vorticity, which affects the mean settling velocity. Results from a Gaussian random velocity field show that this produces an increased settling velocity. © 1987, Cambridge University Press. All rights reserved.

Description: G. I. Taylor (1953) gave a simple recipe for the calculation of contaminant dispersion in bounded shear flows at large times after discharge. He decomposed the concentration profile across the flow into a resolved (uniform) part, with an equilibrium (large-time) estimate for the unresolved part. Here an extended recipe is given to include greater resolution and earlier validity. At the two-equation level there is a close similarity to the slow-zone model posed by Chikwendu & Ojiakor (1985). Application is given to Poiseuille pipe flow and to a contraflowing parallel-plate heat exchanger. © 1987, Cambridge University Press. All rights reserved.

Description: The stability properties and stationary statistics of inviscid barotropic flow over topography are examined. Minimum enstrophy states have potential vorticity proportional to the streamfunction and are nonlinearly stable; correspondingly, canonical equilibrium based on energy and enstrophy conservation predicts mean potential vorticity is proportional to the mean streamfunction. It is demonstrated that in the limit of infinite resolution the canonical mean state is statistically sharp, that is, without any eddy energy on any scale, and is identical to the nonlinearly stable minimum enstrophy state. Special attention is given to the interaction between small scales and a dynamically evolving large-scale flow. On the β-plane, these stable flows have a westward large-scale component. Possibilities for a general relation between inviscid statistical equilibrium and nonlinear stability theory are examined. © 1987, Cambridge University Press. All rights reserved.

Description: An investigation of turbulent wakes was conducted and phase-averaged velocity vector fields are presented, as well as phase-averaged and global Reynolds normal and shear stresses. The topology of the phase-averaged velocity fields is discussed in terms of critical point theory. Here in Part 1, the vortex formation process in the cavity region of several nominally two-dimensional bluff bodies is investigated and described using phase-averaged streamlines where the measurements were made in a nominal plane of symmetry. It was found that the flows encountered were always three-dimensional and that the mean-flow patterns in the cavity region were quite different from those expected using classical two-dimensional assumptions. © 1987, Cambridge University Press. All rights reserved.

Description: A solution has been obtained for steady propagation of a two-dimensional fluid fracture driven by buoyancy in an elastic medium. The problem is formulated in terms of an integro-differential equation governing the elastic deformation, coupled with the differential equation of lubrication theory for viscous flow in the crack. The numerical treatment of this system is carried out in terms of an eigenfunction expansion of the cavity shape, in which the coefficients are found by use of a nonlinear constrained optimization technique. When suitably non-dimensionalized, the solution appears to be unique. It exhibits a semi-infinite crack of constant width following the propagating fracture. For each value of the stress intensity factor of the medium, the width and propagation speed are determined. The results are applied to the problem of the vertical ascent of magma through the earth's mantle and crust. Values obtained for the crack width and ascent velocity are in accord with observations. This mechanism can explain the high ascent velocities required to quench diamonds during a Kimberlite eruption. The mechanism can also explain how basaltic eruptions can carry large mantle rocks (xenoliths) to the surface. © 1987, Cambridge University Press. All rights reserved.

Description: The radial dynamics of a spherical bubble in a compressible liquid is studied by means of a rigorous singular-perturbation method to second order in the bubble-wall Mach number. The results of Part 1 (Prosperetti & Lezzi, 1986) are recovered at orders zero and one. At second order the ordinary inner and outer structure of the solution proves inadequate to correctly describe the fields and it is necessary to introduce an intermediate region the characteristic length of which is the geometric mean of the inner and outer lengthscales. The degree of indeterminacy for the radial equation of motion found at first order is significantly increased by going to second order. As in Part 1 we examine several of the possible forms of this equation by comparison with results obtained from the numerical integration of the complete partial-differential-equation formulation. Expressions and results for the pressure and velocity fields in the liquid are also reported. © 1987, Cambridge University Press. All rights reserved.

Description: We consider the problem of global stability of the rigid rotation of two fluids. The realized interfacial configurations minimize a potential. We derive the most general form of the potential in which the working of the contact line may be expressed as a potential. The resulting variational problem for the interfacial potential is solved when the contact-line conditions are prescribed and for coating flows in which the interface makes a tangent contact with the wetted rod. In the former case, good agreement with experiments is obtained except near lines of contact. This shows that a spinning rod interfacial tensiometer is viable. In the latter case of coating flow, we get good agreement with experiments when the effects of gravity are not too large. The problem of bifurcation of coating flow is discussed qualitatively and some experimental results are given. We show how bifurcating sequences fit well into our qualitative description of the solution which must minimize the interfacial potential as the angular velocity is increased. The last bifurcations lead to pendant drops on a rotating ‘ceiling’ under the influence of centripetal forces which replace gravity. The dynamics of rollers of oil in water, or part in water and part in air, are explained in terms of the wavelength dependence of rotating drops. © 1987, Cambridge University Press. All rights reserved.

Description: The effect of vertical throughflow on the onset of convection in a fluid layer, between permeable horizontal boundaries, when heated uniformly from below, is re-examined analytically. It is shown that when the Péclet number Q is large in magnitude, the critical Rayleigh number Rcis proportional to Qnwhere n = 0, 1, 2, 3 or 4, with a coefficient depending on the Prandtl number P, according to the types of boundaries. When the upper and lower boundaries are of different types, the effect of a small amount of throughflow in one direction is to decrease Rc. This is so when the throughflow is away from the more restrictive boundary. Contributions arise from the curvature of the basic temperature profile, and from the vertical transport of perturbation velocity and perturbation temperature. The decrease in Rcis small if P ~ 1 but can be of significant size if P≪1 or P≫1. © 1987, Cambridge University Press. All rights reserved.

Description: A jet is a stream of one fluid entering another at high speed. In the simplest classical model of jet flow, the geometry is two-dimensional, gravity and viscosity are ignored, the moving fluid is a liquid, and the stationary fluid is a gas whose influence is assumed negligible. The description of this idealized flow can be reduced to a problem of complex analysis, but, except for very simple nozzle geometries, that problem cannot be solved analytically. This paper presents an efficient procedure for solving the jet problem numerically in the case of an arbitrary polygonal nozzle. © 1987, Cambridge University Press. All rights reserved.

Description: Intended as a contribution towards understanding the multiple processes entailed in the development of coastal sand bars due to wave action, this theoretical and experimental study deals with the Bragg reflection of long-crested surface waves in a water channel whose bed is corrugated sinusoidally. The present findings complement and in a few respects improve upon those in previous investigations, particularly Davies & Heathershaw (1984). In §2 a linearized theory is presented, being directed to the elucidation of experimental situations where monochromatic waves propagate into a channel with a limited stretch of corrugations on its bed and an imperfectly absorbing beach at its far end. Allowance is made fully for dispersive effects (§2.2) and approximately for small frictional effects (§2.3). Points of interpretation (§2.4) include accounts of degenerate but non-trivial solutions that apply at frequencies terminating the stopping band, wherein the spatial wavefield has an exponential envelope. The experimental results presented in §4 derive from measurements of the wavefield over a stretch of 24 corrugations, at various frequencies both inside and outside the stopping band. Quantitative comparisons (§§4.2 and 4.3) demonstrate close agreements with the theory. © 1987, Cambridge University Press. All rights reserved.

Description: An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave. © 1987, Cambridge University Press. All rights reserved.

Description: Two problems of the stability of ideal fluid flows over an uneven bottom are considered. The first is the study of stratified flow with a rigid lid'. We use the method of multiple scales to derive an equation describing the evolution of internal waves corresponding to different modes and wave vectors. For the case of sinusoidal bottom irregularities we have constructed a solution describing the increase in time of the internal wave field-this proves the instability of the basic flow. The phenomenon is interpreted as a result of interaction (mutual generation) of internal waves with energies of opposite signs. Our consideration is based on the Hamiltonian approach which enables us to prove in the most simple way the existence of waves carrying negative energy. The case of random (not sinusoidal) bottom irregularities is also studied. Using the kinetic equation for the amplitudes of internal waves derived in the paper, we have established that the basic flow remains unstable as well. In the second part of the paper we consider the homogeneous flows with a free upper boundary. It is shown that this problem can be reduced to the previous one, with the only difference being that the role of unstable perturbations is now played by the surface (not internal) gravity waves. The Hamiltonian approach is consistently applied and allows us to take into account the nonlinearity of waves. © 1987, Cambridge University Press. All rights reserved.

Description: The nature of the boundary layer induced by the motion of a three-dimensional vortex loop towards a plane wall is considered. Initially the vortex is taken to be a ring approaching a plane wall at an angle of attack in an otherwise stagnant fluid; the ring rapidly distorts into a loop shape due to the influence of the wall and the trajectory is computed from a numerical solution of the Biot-Savart integral. As the vortex loop moves, an unsteady boundary-layer flow develops on the wall. A method is described which allows the computation of the flow velocities on and near the symmetry plane of the vortex loop within the boundary layer. The computed results show the development of a variety of complex three-dimensional separation phenomena. Some of the solutions ultimately show strong localized boundary-layer growth and are suggestive that a boundary-layer eruption and a strong viscous-inviscid interaction will be induced by the moving vortex. © 1987, Cambridge University Press. All rights reserved.

Description: The channel model of Stocker & Hutter (1986, 1987) is used to construct topographic wave solutions in a rectangular basin on the f-plane with variable but symmetric bathymetry. We show that in a narrow period band three types of eigenmodes can be discerned which exhibit local, midscale and global structure, respectively. Wave motion can be trapped either at the long sides of the elongated basin (channel mode) or at the ends of it (bay mode) or alternatively, a basinwide phase rotation is observed (Ball mode). The new bay modes are explained as resonances of topographic wave reflection in a semi-infinite channel. The influence of the variation of the aspect ratio of the rectangle and the topography parameter on the wave periods is also investigated. © 1987, Cambridge University Press. All rights reserved.

Description: In this paper we consider theoretical and experimental aspects of axisymmetric, swirling flow which is generated in a column of liquid metal by a rotating magnetic field. Two cases are discussed, one in which there is no axial variation in the stirring force, and one where the body force is restricted to a relatively short length of the column. The latter case is of considerable practical interest in continuous casting. One-dimensional stirring, where the swirl is independent of z and 6, is well understood. The magnetic body force is balanced by shear, all inertial forces being zero (except for the centripetal acceleration). However, in two-dimensional axisymmetric stirring, the axial variation in swirl drives a strong secondary poloidal flow. The principal local force balance is between the magnetic torque and inertia. The body force spins up the fluid as it passes through the forced region and the secondary flow sweeps this angular momentum into the unforced region. Consequently, the size and distribution of the swirl is controlled by the secondary flow. The role of wall friction is considered and shown to control the length of the recirculating eddy. An approximate solution of the inviscid equations of motion, based on the angular momentum integral, is derived for the flow in the forced region. This is compared with the results of numerical experiments. The analysis predicts that the swirl velocity scales on {B(σ/pω)1/2}ωR, has a maximum at the bottom of the driven region, and penetrates an axial distance of the order RR away from the forced region. (For turbulent flow the Reynolds number R must be based on an effective eddy viscosity.) All these features were reproduced experimentally. © 1987, Cambridge University Press. All rights reserved.

Description: Axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed numerically as an initial-value problem. The time-independent spherical Couette flows with zero, one and two Taylor vortices computed in our simulations are found to be reflection-symmetric about the equator despite the fact that our pseudospectral numerical method did not impose these properties. Our solutions are examined for self-consistency, compared with other numerical calculations, and tested against laboratory experiments. At present, the most precise laboratory measurements are those that measure Taylor-vortex size as a function of Reynolds number, and our agreement with these results is within a few per cent. We analyse our flows by plotting their meridional circulations, azimuthal angular velocities, and energy spectra. At Reynolds numbers just less than the critical value for the onset of Taylor vortices, we find that pinches develop in the flow in which the meridional velocity redistributes the angular momentum. Taylor vortices are easily differentiated from pinches because the fluid in a Taylor vortex is isolated from the rest of the fluid by a streamline that extends from the inner to the outer sphere, whereas the fluid in a pinch mixes with the rest of the flow. © 1987, Cambridge University Press. All rights reserved.

Description: In this paper, the analysis on secondary flow in curved elliptic pipes of Topakoglu & Ebadian (1985) has been extended up to a point where the rate-of-flow expression is obtained for any value of flatness ratio of the elliptic cross-section. The analysis is based on the double expansion method of Topakoglu (1967). Therefore, no approximation is involved in any step other than the natural limitation of the finite number of calculated terms of the expansions. The obtained results are systematically plotted against the curvature of centreline of the curved pipe for different values of Reynolds number. © 1987, Cambridge University Press. All rights reserved.

Description: The swimming of a flagellar micro-organism by the propagation of helical waves along its flagellum is analysed by a boundary-element method. The method is not restricted to any particular geometry of the organism nor does it assume a specific wave motion for the flagellum. However, only results for an organism with a spherical or ellipsoidal cell body and a helically beating flagellum are presented here. With regard to the flagellum, it is concluded that the optimum helical wave (amplitude a and wavenumber k) has αk « 1 (pitch angle of 45°) and that for the optimum flagellar length L/A = 10 (L being the flagellar length, A being the radius of the assumed spherical cell body) the optimum number of wavelengths NAis about 1.5. Furthermore there appears to be no optimal value for the flagellar radius a, with the thinner flagella being favoured. These conclusions show excellent quantitative agreement with those of slender-body theory. For the case of an ellipsoidal cell body, the optimum aspect ratios B/A and C/A of the ellipsoid are about 0.7 and 0.3 respectively; A, B and C are the principal radii of the ellipsoid. These and all of the above conclusions show good qualitative agreement with experimental observations of efficiently swimming micro-organisms. © 1987, Cambridge University Press. All rights reserved.

Description: Numerical calculations were carried out to study the effect of forced, symmetric, longitudinal flow oscillations on the inherent, strongly antisymmetrical oscillations of a previously studied edgetone flow at a Reynolds number of 450. The flow consists of a two-dimensional jet issuing from a nozzle and impinging on a body with a wedge-shaped leading edge. The flow is assumed to be incompressible, laminar and two-dimensional, and a finite-difference vorticity/stream-function formulation of the Navier-Stokes equations is employed. Three cases were considered with various combinations of forcing frequency and amplitude. It was found that for the two cases with large forcing amplitudes, the naturally dominant flow frequencies lock-in to the forcing frequency and its harmonics. In the third case the forcing amplitude was smaller and lock-in was not observed but the forced oscillations still had a significant impact on the flow. Mode competition between symmetric and antisymmetric modes is discussed for the three cases along with the manner in which the jet vortical structure is altered as a function of time and space. Results for all three cases are presented in the form of computer drawn equivorticity lines and plots of frequency spectra for the jet oscillations and for the pressure on the wedge. © 1987, Cambridge University Press. All rights reserved.

Description: This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial βplane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is 1/2h(u2+v2)). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term Uh'ú). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability. A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously. For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy. © 1987, Cambridge University Press. All rights reserved.

Description: Data which describe the unidirectional spreading of several pure oils and oil-surfactant mixtures on water in the surface-tension regime are reported. Leading-edge position and profiles of velocity, thickness and film tension are given as functions of time. The data are consistent with the numerical similarity solution of Foda & Cox (1980), although the measured dependence of the film tension on the film thickness often differs from the equilibrium relationship. The configuration of the oil film near the spreading origin may be either a coherent multimolecular layer or a multitude of thinning, outward-moving lenses surrounded by monolayer. The pure oils show an acceleration zone connecting the slow-moving inner region to a fast-moving outer region, while the oil-surfactant mixtures show a much more gradual increase in film velocity. © 1987, Cambridge University Press. All rights reserved.

Description: Two approaches have been used to study the torsion effect on the fully developed laminar flow in a helical pipe of constant circular cross-section. The first approach is the series expansion method that perturbs the Poiseuille flow and is valid for low Dean numbers with both the dimensionless curvature and dimensionless torsion being much less than unity. The second is a numerical procedure that solves the complete Navier-Stokes equation and is applicable to intermediate values of the Dean number. The results obtained indicate that, as far as the secondary flow patterns are concerned, the presence of torsion can produce a large effect if the ratio of the curvature to the torsion is of order unity. In these cases the secondary flow, though still consisting of a pair of vortices, can be very much distorted. Under extreme conditions one vortex is so prevalent as to squeeze the second one into a narrow region. However, ordinarily the torsion effect is small and the secondary flow has the usual pattern of a pair of counter-rotating vortices of nearly equal strength. Concerning the flow resistance in the pipe the effect of torsion is always small in all the circumstances that have so far been considered. © 1987, Cambridge University Press. All rights reserved.

Description: Steady and unsteady velocity components over a backward-facing circular arc are measured by laser-Doppler velocimetry. A periodic disturbance is added to the mean flow and the response of unsteady separation is investigated. Special attention is given to the distribution and the flux of vorticity. © 1987, Cambridge University Press. All rights reserved.

Description: A novel primitive model is proposed for the hydrodynamic behaviour of an isolated dissolved polymer molecule in a laminar shear flow. The model, in which inertial effects are neglected, allows for rotation and partial stretching of the molcule, but not for bending. Dilute solutions of flexible long-chain polymers have been experimentally observed to exhibit periodic velocity fluctuations distinct from turbulence over a broad frequency range when flowed in high-shear-rate water-table and pipe configurations. In these experiments, the frequency of the fluctuations does not increase with increasing shear rate; rather, it is lowest in the regions of the flow where the shear is the highest. A manifestation of viscous shear thickening has also been observed in these laminar flows. The proposed polymer representation appears capable of accounting for the salient features of these flows with adjustment of a single dimensionless parameter, a ratio of polymer-spring and solvent-viscosity forces. © 1987, Cambridge University Press. All rights reserved.

Description: It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Q. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness e 〈 1, where e = 2✓2Z/βU in the planar case and e = 2¼ a⅚Z¼/ΩU⅚ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which e decreases in time) seem to evolve into a non-ergodic regime at large scales. © 1987, Cambridge University Press. All rights reserved.

Description: A weakly nonlinear model is developed from the Hamiltonian formulation of water waves, to study the bifurcation structure of gravity-capillary waves on water of finite depth. It is found that, besides a very rich structure of symmetric solutions, non-symmetric Wilton's ripples exist. They appear via a spontaneous symmetrybreaking bifurcation from symmetric solutions. The bifurcation tree is similar to that for gravity waves. The solitary wave with surface tension is studied with the same model close to a critical depth. It is found that the solution is not unique, and that further non-symmetric solitary waves are possible. The bifurcation tree has the same structure as for the case of periodic waves. The possibility of checking these results in low-gravity experiments is postulated. © 1987, Cambridge University Press. All rights reserved.

Description: This is a study of the runup of solitary waves on plane beaches. An approximate theory is presented for non-breaking waves and an asymptotic result is derived for the maximum runup of solitary waves. A series of laboratory experiments is described to support the theory. It is shown that the linear theory predicts the maximum runup satisfactorily, and that the nonlinear theory describes the climb of solitary waves equally well. Different runup regimes are found to exist for the runup of breaking and non-breaking waves. A breaking criterion is derived for determining whether a solitary wave will break as it climbs up a sloping beach, and a different criterion is shown to apply for determining whether a wave will break during rundown. These results are used to explain some of the existing empirical runup relationships. © 1987, Cambridge University Press. All rights reserved.

Description: Laminar-film flow with condensation or evaporation on a vertical fluted cylinder is examined. A kinematic wave equation describing the evolution of the film profile is obtained and solutions presented. The film profile evolves owing to axial, gravity-driven flow and transverse, surface-tension-driven flow from the crests to the valleys of the fluted cylinder. In the case of condensation the majority of the film reaches a uniform thickness and consequently there is a significant improvement in heat transfer compared with the classical unfluted result where the film thickens in an unbounded fashion. For evaporation a critical value of a parameter which involves the ratio of the Weber number and the gradient of the surface curvature is found below which the film totally drys out and above which the fluid funnels into tapering rivulets and only partially drys out. Typical dry-out lines are presented. For short cylinders the evaporative mass transfer for a fluted cylinder is slightly greater than that predicted for an unfluted case. However, when the cylinder is long the mass transfer is far less for a fluted cylinder owing to the reduction in film area associated with partial film dry out. © 1987, Cambridge University Press. All rights reserved.

Description: We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number (N = 0(1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M, and the convergence with N and M is exponentially fast for waves up to approximately 80 % of Stokes limiting steepness (kα ~ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies. © 1987, Cambridge University Press. All rights reserved.

Description: Surface waves in a rectangular container subjected to vertical oscillations are studied. Effects of energy dissipation along the lines of Miles (1967) and the effect of surface tension are included. Sufficient conditions, for two modes to dominate the motion, are given. The analysis is along the lines of Miles (1984a) and Holmes (1986). A complete bifurcation analysis is performed, and the modal amplitudes and phases are shown to have chaotic behaviour. This result is obtained under assumptions different from those of Holmes (1986). The conclusions regarding chaotic motions are based on a theorem of Šilnikov (1970). © 1987, Cambridge University Press. All rights reserved.

Description: In order to elucidate the turbulent structure below a shear-free gas-liquid interface, turbulence measurements were made in a 50 cm square by 40 cm deep tank stirred by a vertically oscillating grid well below the surface, using a split-film anemometer probe rotating in a horizontal circle. This instrument is able to measure both vertical and horizontal velocity fluctuations to within 0.4 mm of the surface, from which spatial spectra and profiles of r.m.s. velocity fluctuations and integral lengthscales can be calculated. The turbulent structure is affected by the presence of the surface within a surface-influenced layer roughly one integral scale, or ten per cent of the distance from the surface to the centre of the grid stroke, in thickness. The shapes of the spectra and profiles within the surface-influenced layer are predicted to a good first approximation by the source theory of Hunt & Graham (1978), which treats the turbulent structure as the superposition of homogeneous turbulence with an irrotational velocity field driven by a source distribution at the surface which cancels the vertical velocity fluctuations there. The magnitudes (as opposed to the shapes) of the profiles scale according to the values that would otherwise occur in the vicinity of the surface-influenced layer were the surface not present. These magnitudes are adequately predicted by the bulk relations determined by Hopfinger & Toly (1976) and Thompson & Turner (1975), with no apparent dependence on turbulent Reynolds number. There are some minor discrepancies between the measured profiles and those of Hunt & Graham. A thin layer of reduced velocity fluctuations below what would be expected from the theory was observed near the surface. Also, anisotropy in the velocity spectra at depths within the surface-influenced layer extended well into the inertial subrange, whereas the Hunt & Graham theory predicts no anisotropy at high wavenumbers. © 1987, Cambridge University Press. All rights reserved.

Description: In displacing a viscous fluid from the gap between two closely spaced parallel plates, a thin film of the original fluid remains on the surface of each plate. Boundary conditions which connect the approximate equations in the region in front of the interface with the approximate solutions in the thin-film region are determined from local solutions of the equations in the vicinity of the interface edge. These interface conditions depend on both b/R (gap half-width/radius of curvature) and μUn/T, where /i is the viscosity of the original fluid, Unis the normal velocity of the interface edge, and T is the interfacial tension. These conditions are determined using perturbation method when μUn/T ⋘1 and numerical methods when μUn/T is 0(1). Though previous theories have shown qualitative agreement with experiments, it is hoped that these new boundary conditions improve the quantitative agreement. © 1987, Cambridge University Press. All rights reserved.

Description: The evolution equation governing wavemaker-generated cross-waves near a cutoff frequency in an infinitely deep, infinitely long channel is shown to be the nonlinear Schrodinger equation with a homogeneous boundary condition at the wavemaker. With the inclusion of an empirically determined damping coefficient, numerical results for growth rate, slow modulation period, and wave amplitude show good agreement with previous experiments. The results also describe observations of trapped and propagating solutions. © 1987, Cambridge University Press. All rights reserved.

Description: An approach utilizing multiple scales and matched asymptotic expansions is developed for the description of small perturbations at large distances from a thin airfoil oscillating harmonically in a uniform supersonic flow. The problem of determining the unsteady perturbation potential is formulated in general, and an analytical solution is derived for an airfoil with parabolic or flat surfaces. The results describe the flow ahead of the region influenced by the trailing edge. The variation in the pressure jump across an attached leading-edge shock wave is also obtained. © 1987, Cambridge University Press. All rights reserved.

Description: Ambient fluid of a submerged water jet was continuously tagged with fluorescent dye at a point outside the turbulent region (at 33 jet nozzle diameters from the jet exit). This made it possible to follow the tagged entrained fluid to 73 jet diameters downstream of the exit, a distance unattainable by other methods. The dispersion of the tagged fluid in a plane containing the jet axis and the tagging source was observed and recorded using photography and simple digital image-processing techniques. Most of the entrainment activity appeared to be the result of engulfment by the large-scale structures over an axial distance of ±1.75 from the source where B is the half-peak velocity radius. The entrained fluid crossed the jet centreline within a downstream distance of Ax = 1.5U. Downstream of the entrainment region, the spread rate of the tagged entrained fluid was close to that of the turbulent jet fluid. However, the peak mean concentration of the tagged entrained fluid was located near the r/x = 0.1 line closest to the tagging source and shifted very slowly towards the jet centreline. A self-preserving distribution of the mean concentration appears to have been approached after a distance of 6B downstream from the tagging source but further verification is needed owing to experimental uncertainties. A small fraction of the tagged entrained fluid was found on the side of the jet remote from the tagging source. On rare occurrences, tagged entrained fluid was observed at the interface most remote from the source. © 1987, Cambridge University Press. All rights reserved.

Description: The three-dimensional nonlinear oscillations of an isolated, inviscid drop with surface tension are studied by a multiple timescale analysis and pre-averaging applied to the variational principle for the appropriate Lagrangian. Amplitude equations are derived which describe the generic cubic resonance caused by the spatial degeneracy of the eigenfrequencies of the linear normal modes. This resonant coupling leads to the instability of the finite amplitude axisymmetric oscillations to small non-axisymmetric perturbations, as is demonstrated here for the three and four-lobed normal modes. Solutions to the interaction equations that describe finite amplitude, non-axisymmetric travelling-wave solutions are also obtained and their stability is investigated. A non-generic cubic resonance between the two-lobed and four-lobed oscillatory modes leads to quasi-periodic motions. © 1987, Cambridge University Press. All rights reserved.

Description: Boundary-layer transition over a stationary disk in rotating flow is studied experimentally. Circular waves are observed in the boundary layer occurring on an end disk of a cylindrical cavity during impulsive spin-down to rest. The transient flow evolves into a quasi-steady regime that exhibits the properties of the Bodewadt flow. The circular waves develop in that flow. The critical Reynolds number Re = r(Ω/v)1/2is determined from frequency and wavelength measurements to be about 25. The corresponding dimensionless wavenumber 2πr/λRe is about 0.6 and the frequency 2πf/ΩiRe about 0.2. © 1987, Cambridge University Press. All rights reserved.

Description: A coherent and stable baroclinic eddy in a rotating fluid was produced on a sloping bottom by releasing a dome of salt water into the ambient fresh water. A strong cyclonic vortex is produced above the heavy dome. The entire eddy system moves north-westward (with the up-slope direction designated north) as a Taylor column. The eddy system displays long lifetimes, but it is shown that a theory of isolated systems cannot account for the experimental observations. Instead, it is demonstrated that the vortex flow above the lens is along the lines of constant depth, producing a net pressure force on the lens, which approximately balances the buoyancy force. When Ekman friction is also included, it accounts for the northward motion of the dome. © 1987, Cambridge University Press. All rights reserved.

Description: We consider the radiation of internal gravity waves from a spherical body oscillating vertically in a stratified incompressible fluid. A near-field solution (under the Boussinesq approximation) is obtained by separation of variables in an elliptic problem, followed by analytic continuation to the frequencies ω〈 N of internal wave radiation. Matched expansions are used to relate this solution to a far-field solution in which non-Boussinesq terms are retained. In the outer near field there are parallel conical wavefronts between characteristic cones tangent to the body, but with a wavelength found to be shorter than that for oscillations of a circular cylinder. It is also found that there are caustic pressure singularities above and below the body where the characteristics intersect. Far from the source, non-Boussinesq effects cause a diffraction of energy out of the cones. The far-field wave fronts are hyperboloidal, with horizontal axes. The case of horizontal oscillations of the sphere is also examined and is shown to give rise to the same basic wave structure. The related problem of a pulsating sphere is then considered, and it is concluded that certain features of the wave pattern, including the caustic singularities near the source, are common to a more general class of oscillating sources. © 1987, Cambridge University Press. All rights reserved.

Description: The growth, breakdown, and transition to turbulence of counter-rotating streamwise vortices, generated via a Görtler instability mechanism, was used to experimentally model the eddy structures found in transitional and turbulent flat-plate boundary layers. The naturally occurring vortices have been studied using smoke-wire visualization and multiple-probe hot-wire rakes. Results show that low-speed regions are formed between the vortices as low-momentum fluid is removed away from the wall. The low-speed regions grow in the normal direction faster than a nominally Blasius boundary layer and create strongly inflexional normal and spanwise profiles of the streamwise velocity component. Instability oscillations develop on these unstable profiles that scale with the local shear-layer thickness and velocity difference. Contrary to expectations however, the spatial scales of the temporal velocity fluctuations correlate better with the velocity gradient in the spanwise direction than with the normal velocity gradient. The nonlinear growth of the oscillations is quite rapid and breakdown into turbulence occurs within a short timescale. © 1987, Cambridge University Press. All rights reserved.

Description: Compressible flows with r. m. s. velocities of the order of the speed of sound are studied with direct numerical simulations using a pseudospectral method. We concentrate on turbulent homogeneous flows in the two-dimensional case. The fluid obeys the Navier-Stokes equations for a perfect gas, and viscous terms are included explicitly. No modelling of small scales is used. We show that the behaviour of the flow differs sharply at low compared with high r. m. s. Mach number Ma, with a transition at Ma—0. 3. In the large scales, temporal exchanges between longitudinal and solenoidal modes of energy retain an acoustical character; they lead to a slowing down of the decrease of the Mach number with time, which occurs with interspersed plateaux corresponding to quiescent periods. When the flow is initially supersonic, the small scales are dominated by shocks behind which vortices form. This vortex production is particularly prominent when two strong shocks collide, with the onset of shear turbulence in the region downstream of the collision. However, at the resolutions reached by our code on a 256x256 uniform grid, this mechanism proves insufficient to bring vortices into equipartition with shocks in the small-scale tail of the energy spectrum. © 1987, Cambridge University Press. All rights reserved.

Description: Simultaneous measurements of time-resolved velocity and temperature have been obtained by laser-Doppler anemometry and numerically compensated fine-wire thermocouples in the near wake of a premixed flame stabilized on a disk baffle located on the axis, and at the exit, of a confining pipe. The diameter of the disk was 0. 056 m, the diameter of the pipe was 0. 080 m, the volumetric equivalence ratio with natural gas as the fuel was 0. 79 and the Reynolds number, based on pipe diameter and upstream pipe bulk velocity of 9 m/s, was 46800. The purpose of the measurements is to quantify the relative magnitudes of terms involving the mean pressure gradient and Reynolds stresses in the balance of turbulent kinetic energy and heat flux in a strongly sheared, high-Reynolds-number, reacting flow. The latter term has been associated with non-gradient diffusion in other flows. Source terms involving the mean pressure gradient are large in the conservation of turbulent heat flux but not in the conservation of Reynolds stress. The thin-flame model of burning suggests that the sign and magnitude of the heat flux is closely related to the conditioned mean velocities. The mean axial velocity of the reactants is larger (by up to 0. 27 of the reference velocity) than that of the products on the low-velocity side of the shear layer that surrounds the recirculation bubble but the reverse is true on the high-velocity side. These observations are related to the sign of the axial pressure gradient, which is associated with the streamline curvature, and the consequent preferential acceleration of the low-density products. Generally, the Reynolds stresses of the products are higher than those of the reactants and, in contrast to previously reported measurements, the contribution to the unconditioned stresses by the difference in the mean velocity between products and reactants, the so-called intermittent contribution, is small. This is a consequence of the high Reynolds number of our flow. © 1987, Cambridge University Press. All rights reserved.

Description: The conical-channel flow of a dilute polymer solution is investigated theoretically. The stress field due to polymer additive is calculated using a new molecular model, based on the physical picture of the polymer molecules unravelling in strong flows and Batchelor's theory for the stress in a suspension of elongated particles. Good agreement is obtained with the experimental results of James & Saringer (1980). The absence of a significant polymer effect in a two-dimensional case (the wedge-channel flow), observed by the same authors (James & Saringer 1982a), is also explained. The fundamental differences between the proposed model and the elastic-dumbbell models are discussed. © 1987, Cambridge University Press. All rights reserved.

Description: Large-amplitude waves can exist on an air-water interface where the air is in steady non-uniform flow and the water is stationary. Computations of such waves are provided here, both for periodic nonlinear Stokes-like waves, and for a specific wave-making configuration in which the periodic solution appears as the downstream far field. The wavemaker geometry chosen here is relevant to the edge region of a hovercraft, and the large-amplitude free-surface disturbance caused by the escaping air is computed as a function of the Froude number based on air-jet velocity and thickness. © 1987, Cambridge University Press. All rights reserved.

Description: A numerical method is developed for nonlinear three-dimensional but axisymmetric free-surface problems using a mixed Eulerian-Lagrangian scheme under the assumption of potential flow. Taking advantage of axisymmetry, Rankine ring sources are used in a Green's theorem boundary-integral formulation to solve the field equation; and the free surface is then updated in time following Lagrangian points. A special treatment of the free surface and body intersection points is generalized to this case which avoids the difficulties associated with the singularity there. To allow for long-time simulations, the nonlinear computational domain is matched to a transient linear wavefield outside. When the matching boundary is placed at a suitable distance (depending on wave amplitude), numerical simulations can, in principle, be continued indefinitely in time. Based on a simple stability argument, a regriding algorithm similar to that of Fink & Soh (1974) for vortex sheets is generalized to free-surface flows, which removes the instabilities experienced by earlier investigators and eliminates the need for artificial smoothing. The resulting scheme is very robust and stable. For illustration, three computational examples are presented: (i) the growth and collapse of a vapour cavity near the free surface; (ii) the heaving of a floating vertical cylinder starting from rest; and (iii) the heaving of an inverted vertical cone. For the cavity problem, there is excellent agreement with available experiments. For the wave-body interaction calculations, we are able to obtain and analyse steady-state (limit-cycle) results for the force and flow field in the vicinity of the body. © 1987, Cambridge University Press. All rights reserved.

Description: A direct numerical simulation of a turbulent channel flow is performed. The unsteady Navier Stokes equations are solved numerically at a Reynolds number of 3300, based on the mean centreline velocity and channel half-width, with about 4x106 grid points (192x129x160 in x, y, z). All essential turbulence scales are resolved on the computational grid and no subgrid model is used. A large number of turbulence statistics are computed and compared with the existing experimental data at comparable Reynolds numbers. Agreements as well as discrepancies are discussed in detail. Particular attention is given to the behaviour of turbulence correlations near the wall. In addition, a number of statistical correlations which are complementary to the existing experimental data are reported for the first time. © 1987, Cambridge University Press. All rights reserved.

Description: The feeding behaviour of many small, free swimming organisms involves the creation of a scanning current by the coordinated movement of a group of appendages. In this paper, we study the generation of scanning currents in Stokes flow in a number of simple models, utilizing the movement of Stokeslets, spheres, or stalks to set up an average scanning drift in a suitable far field formulation. Various mechanisms may then be classified by the rate of decay at infinity of the mean scanning current. In addition, optimal scanning can be investigated by minimizing the mean power required to create a current of prescribed amplitude. The simple mechanisms for scanning described here provide a framework within which the appendage movements of small aquatic organisms can be analysed and the relative merits of scanning and swimming strategies can be investigated. © 1987, Cambridge University Press. All rights reserved.

Description: Hot-film-anemometer measurements were carried out in a shear flow between a flat plate and a moving plate fitted with an array of tall fences. The effect of spatial restriction by the fences on the inner-layer structure of the boundary layer developing on the flat-plate side was investigated. It was revealed that the inner-layer structure was maintained even when the tips of the fences were passing at a distance y+= 45 from the flat plate; the flow did not become laminar-like until the tips reached y+= 25. These results suggested the physical view that the inner layer of wall turbulence has a tough, self-sustaining structure, which is uniquely determined under a given mean wall shear stress and is hardly influenced by outer-layer disturbances provided that its own spatial extent of about 45 v/u* from the wall is maintained. © 1987, Cambridge University Press. All rights reserved.

Description: Experiments on fluidization with water of spherical particles falling against gravity in columns of rectangular cross section are described. All of them are dominated by inertial effects associated with wakes. Two local mechanisms are involved: drafting and kissing and tumbling into stable cross stream arrays. Drafting, kissing and tumbling are rearrangement mechanisms in which one sphere is captured in the wake of the other. The kissing spheres are aligned with the stream. The streamwise alignment is massively unstable and the kissing spheres tumble into more stable cross stream pairs of doublets which can aggregate into larger relatively stable horizontal arrays. Cross stream arrays in beds of spheres constrained to move in two dimensions are remarkable. These arrays may even coalesce into aggregations of close packed spheres separated by regions of clear water. A somewhat weaker form of cooperative motion of cross stream arrays of rising spheres is found in beds of square cross section where the spheres may move freely in three dimensions. Horizontal arrays rise where drafting spheres fall because of greater drag. Aggregation of spheres seems to be associated with relatively stable cooperative motions of horizontal arrays of spheres rising in their own wakes. © 1987, Cambridge University Press. All rights reserved.

Description: To understand the imaging of the sea surface by radar, it is useful to know the theoretical variations in the wavelength and steepness of short gravity waves propagated over the surface of a train of longer gravity waves of finite amplitude. Such variations may be calculated once the orbital accelerations and surface velocities in the longer waves have been accurately determined a non trivial computational task. The results show that the linearized theory used previously for the longer waves is generally inadequate. The fully nonlinear theory used here indicates that for longer waves having a steepness parameter AK = 0.4, for example, the short wave steepness can be increased at the crests of the longer waves by a factor of order 8, compared with its value at the mean level. (Linear theory gives a factor less than 2.) The calculations so far reported are for free, irrotational gravity waves travelling in the same or directly opposite sense to the longer waves. However, the method of calculation could be extended without essential difficulty so as to include effects of surface tension, energy dissipation due to short wave breaking, surface wind drift currents, and to arbitrary angles of wave propagation. © 1987, Cambridge University Press. All rights reserved.

Description: Combustion instability is investigated in the case of a multiple inlet combustor with dump. It is shown that low frequency instabilities are acoustically coupled and occur at the eigenfrequencies of the system. Using spark schlieren and a special phase average imaging of the C2-radical emission, the fluid mechanical processes involved in a vortex driven mode of instability are investigated. The phase average images provide maps of the local non steady heat release. From the data collected on the combustor the processes of vortex shedding, growth, interactions and burning are described. The phases between the pressure, velocity and heat release fluctuations are determined. The implications of the global Rayleigh criterion are verified and a mechanism for low frequency vortex driven instabilities is proposed. © 1987, Cambridge University Press. All rights reserved.

Description: An investigation of an oblique shock wave/laminar boundary layer interaction is presented. The Mach number was 2.15, the Reynolds number was 105 and the overall pressure ratio was 1.55. The interation has been demonstrated to be laminar and nominally two-dimensional. Experimental results include pressure distributions on the plate and single component laser-Doppler velocimetry velocity measurements both in the attached and separated regions. The numerical results have been obtained by solving the full compressible Navier-Stokes equations with the implicit approximate factorization algorithm by Beam & Warming (1980). Comparison with experimental data shows good agreement in terms of pressure distributions, positions of separation and reattachment and velocity profiles. © 1987, Cambridge University Press. All rights reserved.

Description: Laser-Doppler velocity measurements were performed on the entry flow in a 90° bend of circular cross section with a curvature ratio a/R = 1/6. The steady entry velocity profile was parabolic, having a Reynolds number Re = 700, with a corresponding Dean number k = 286. Both axial and secondary velocities were measured, enabling a detailed description of the complete flow field. The secondary flow at the entrance of the bend was measured to be directed completely towards the inner bend. Significant disturbance of the axial velocity field was not measured until a downstream distance (aR)1/2. Maximum secondary velocities were measured at 1.7 (aR)1/2 downstream from the inlet. The development of the axial flow field can be quite well explained from the secondary velocity field. © 1987, Cambridge University Press. All rights reserved.

Description: Density currents and density wedges are two observed manifestations of interactions between an ambient flow and a horizontal buoyant intrusion. In a density current the buoyant pressure force is primarily balanced by the local form drag of the current head which has a blunt shape and abrupt depth change. In a density wedge a distributed interfacial drag is the primary balancing force, leading to a stretched out shape and long distance intrusions. A perturbation analysis of the approach flow to the inclined front of a density current shows that slight momentum changes caused by viscous effects in the ambient flow determine which of these two flow types is established. In a uniform ambient channel flow, any momentum deficit relative to the inviscid case will lead to a local flattening of the front and ultimate breakdown into a density wedge. On the other hand, a momentum surplus will support a steady state density current. Several exploratory experiments on control of the ambient boundary layer through local non uniformities were performed with the objective of achieving stable density current forms with limited intrusion lengths. These methods include a small step, a barrier and suction and are applied for intrusions at either the bottom or surface of an ambient water flow. In all cases, good agreement is found with the force balances predicted by Benjamin's (1968) theory and its extension by Britter & Simpson (1978) which accounts for entrainment in the wake zone of the head. © 1987, Cambridge University Press. All rights reserved.

Description: A novel technique is presented for determining the coefficient of shear induced particle self diffusion in concentrated suspensions of solid spheres, which relies on the fact that this coefficient can be computed from the measured variations in the time taken by a single marked particle in the suspension to complete successive circuits in a Couette device. Since this method does not involve the direct measurement of the lateral position of the marked particle, it requires a much simpler experiment than that used by Eckstein, Bailey & Shapiro (1977) which is shown to be constrained by wall effects at high particle concentration. The diffusion coefficient thus determined was found to be proportional to the product γa2 where γ is the shear rate and a the particle radius, and to have the asymptotic form O.5γa2ϕ2in the dilute limit when the particle concentration ϕ-〉0. © 1987, Cambridge University Press. All rights reserved.

Description: Measurements of surface pressure fluctuation spectra and wave speeds are reported for a well documented separating turbulent boundary layer. Two sensitive instrumentation microphones were used in a new technique to measure pressure fluctuations through pinhole apertures in the flow surface. Because a portion of the acoustic pressure fluctuations is the same across the nominally two dimensional turbulent flow, it is possible to decompose the two microphone signals and obtain the turbulent flow contributions to the surface pressure spectra. In addition, data from several earlier attached flow surface-pressure-fluctuation studies are re examined and compared with the present measurements. The r.m.s. of the surface pressure fluctuation p'increases monotonically through the adverse-pressure-gradient attached flow region and the detached flow zone. Apparently p‘ is proportional to the ratio a of streamwise lengthscale to lengthscales in other directions. For non equilibrium separating turbulent boundary layers, a is as much as 2.5, causing p to be higher than equilibrium layers with lower values of a. The maximum turbulent shearing stress TM appears to be the proper stress on which to scale p ' p'/tm from available data shows much less variation than whenp’ is scaled on the wall shear stress. In the present measurements p'/tMincreases to the detachment location and decreases downstream. This decrease is apparently due to the rapid movement of the pressure-fluctuation-producing motions away from the wall after the beginning of intermittent backflow. A correlation of the detached flow data is given that is derived from velocity and lengthscales of the separated flow. Spectra Φ(ω)for ωδ*/U∞ 〉 0.001 are presented and correlate well when normalized on the maximum shearing stress TM. At lower frequencies, for the attached flow Φ(ω)~ω-0.7while Φ(ω) ~ω)3at higher frequencies in the strong adverse-pressure-gradient region. After the beginning of intermittent backflow, Φ(ω) varies with ω at low frequencies and ω-3 at high frequencies; farther downstream the lower frequency range varies with ω1.4. The celerity of the surface pressure fluctuations for the attached flow increases with frequency to a maximum; at higher frequencies it decreases and agrees with the semi logarithmic overlap equation of Panton & Linebarger. After the beginning of the separation process, the wave speed decreases because of the oscillation of the instantaneous wave speed direction. The streamwise coherence decreases drastically after the beginning of flow reversal. © 1987, Cambridge University Press. All rights reserved.

Description: The three dimensional wave pattern generated by a moving pressure distribution of finite extent acting on the surface of water of depth h is studied. It is shown that, when the pressure distribution travels at a speed near the linear-long-wave speed, the response is governed by a forced nonlinear Kadomtsev Petviashvili (KP) equation, which describes a balance between linear dispersive, nonlinear and three dimensional effects. It is deduced that, in a channel of finite width 2w three dimensional effects are negligible if w ⋘h2/a, a being a typical wave amplitude; in such a case the governing equation reduces to the forced Korteweg de Vries equation derived in previous studies. For aw/h2 = 0(1), however, three dimensional effects are important; numerical calculations based on the KP equation indicate that a series of straight crested solitons are radiated periodically ahead of the source and a three dimensional wave pattern forms behind. The predicted dependencies on channel width of soliton amplitude and period of soliton formation compare favourably with the experimental results of Ertekin, Webster & Wehausen (1984). In a channel for which aw/h2⋙1, three-dimensional, unsteady disturbances appear ahead of the pressure distribution. © 1987, Cambridge University Press. All rights reserved.

Description: The steady state equations of motion are solved for a fluid sphere translating in a quiescent medium. A semi analytical series truncation method is employed in conjunction with a cubic finite element scheme. The range of Reynolds numbers investigated is from 0.5 to 50. The range of viscosity ratios is from 0 (gas bubble) to 107 (solid sphere). The flow structure and the drag coefficients agree closely with the limited available experimental measurements and also compare favourably with published finite difference solutions. The strength of the internal circulation was found to increase with increasing Reynolds number. The flow patterns and the drag coefficient show little variation with the interior Reynolds number. Based on the numerical results, predictive equations for drag coefficients are recommended for both moderate and low-Reynolds-number flows. © 1987, Cambridge University Press. All rights reserved.

Description: Within a granular material stress is transmitted by forces exerted at points of mutual contact between particles. When the particles are close together and deformation of the assembly is slow, contacts are sustained for long times, and these forces consist of normal reactions and the associated tangential forces due to friction. When the particles are widely spaced and deformation is rapid, on the other hand, contacts are brief and may be regarded as collisions, during which momentum is transferred. While constitutive relations are available which model both these situations, in many cases the average contact times lie between the two extremes. The purpose of the present work is to propose constitutive relations and boundary conditions for this intermediate case and to solve the corresponding equations of motion for plane shear of a cohesionless granular material between infinite horizontal plates. It is shown that, in general, not all the material between the plates participates in shearing, and the solutions for the shearing material are coupled to a yield condition for the nonshearing material to give a complete solution of the problem. © 1987, Cambridge University Press. All rights reserved.

Description: This investigation is concerned with the numerical calculation of multiple solutions for a mixed convection flow problem in horizontal rectangular ducts. The numerical results are interpreted in terms of recent observations by Benjamin (1978a) on the bifurcation phenomena for a bounded incompressible fluid. The observed mutations of cellular flows are discussed in terms of dynamic interchange processes. Each cellular flow may be represented by a solution surface in the parametric space of Grashof number Gr and aspect ratio y, which is delimited by stability boundaries. Such a stability map has been generated for each type of cellular flow by a series of numerical experiments. Once these boundaries are crossed one cellular flow mutates into another via a certain dynamical process. Although the nature of the singular points on this map have not been determined precisely, a plausible general structure of the cellular flow exchange process emerges from this map with several features in common with the Taylor Couette flow. The primary modes appear to exchange roles via the formation of tilted cusp. Other salient features such as primary mode hysteresis and quasi critical range for cellular development appear to be present. However no anomolous modes have been observed. © 1987, Cambridge University Press. All rights reserved.

Description: The trajectories of surface and subsurface particles of nonlinear gravity-capillary waves are calculated. Surface tension is found to have a small effect on the trajectories and drift velocity of pure gravity waves (down to 20 cm in length). On the other hand, pure capillary wave trajectories can be considerably altered even when the influence of gravity is small (waves of up to 2 cm in length). When the restoring forces are of comparable magnitude, some remarkable trajectories are possible, containing one or more subloops. Overall, the influence of increased surface tension is to increase the relative horizontal distance travelled by a particle, as well as the magnitude of the time-averaged drift velocity ratio at the surface and, as far as short waves are concerned, its penetration depth. These results can have implications for steep waves where parasitic capillaries are generated and for observations of the wind-drift current. © 1985, Cambridge University Press. All rights reserved.

Description: Experiments in which a long vertical, heated wire is surrounded by concentric annuli of a melt and its crystalline solid show that the convection state changes from a stable unicell surrounded by a stationary cylindrical solid-liquid interface, to a complex time-dependent flow surrounded by a rotating, helical solid-liquid interface. This transition occurs at a Grashof number of approximately 150, which is an order of magnitude less than the critical Grashof number calculated for a liquid annulus surrounded by rigid walls. A linear stability analysis has been carried out for an infinitely tall vertical annulus. When the deformable nature of the crystal-melt interface is taken into account in the boundary conditions, two new modes of instability arise. The most dangerous mode is asymmetrical and corresponds to helical waves travelling vertically upwards. The critical Grashof number and the scaling properties of the eigenstate agree with experiments. The results clearly demonstrate the coupling of convection with crystal-melt interfacial instabilities. © 1985, Cambridge University Press. All rights reserved.

Description: We consider the flow of two immiscible fluids lying between concentric cylinders when the outer cylinder is fixed and the inner one rotates. The interface is assumed to be concentric with the cylinders, and gravitational effects are neglected. We present a numerical study of the effect of different viscosities, different densities and surface tension on the linear stability of the Couette flow. Our results indicate that, with surface tension, a thin layer of the less-viscous fluid next to either cylinder is linearly stable and that it is possible to have stability with the less dense fluid lying outside. The stable configuration with the less-viscous fluid next to the inner cylinder is more stable than the one with the less-viscous fluid next to the outer cylinder. The onset of Taylor instability for one-fluid flow may be delayed by the addition of a thin layer of less-viscous fluid on the inner wall and promoted by a thin layer of more-viscous fluid on the inner wall. © 1985, Cambridge University Press. All rights reserved.

Description: A systematic study is made of the effect of latitude on the linear, normal mode stability characteristics of the laminar barotropic Ekman layer. The outcome depends upon the direction of the geostrophic flow (in the case of flows modelling the atmospheric Ekman layer) or, alternatively, upon the direction of the applied stress (in the case of flows modelling the oceanic Ekman layer). The minimum critical Reynolds number Rcis a function of latitude. For the atmospheric Ekman layer Rc = 30.8 for all latitudes less than 26.2° and increases monotonically with latitude to 54.2. At a latitude of 45° N, Rc is 33.9 and arises for a geostrophic wind directed towards a compass heading 252° (clockwise from north), corresponding to rolls with axes pointing due west and having wavenumber k (with unit of length taken to be the Ekman layer depth) of 0.594. The minimum Rcfor the oceanic boundary layer is 11.6 for latitudes less than 81.1°, and increases with latitude to 11.8. At 45° N latitude, the critical condition arises for a surface-current compass heading of 345.2°, roll axis of 351° and a wavenumber k = 0.33. The results for Rcare all symmetric about the equator, with roll axes and associated basic flow directions rotated by 180°. As the Reynolds number R increases, the effects of the perturbation Coriolis • acceleration on the instability diminish, as has been previously shown, and the error caused by neglect of the horizontal component of angular velocity therefore decreases. The high Reynolds number limit is systematically explored. It is shown that the lower branch of the neutral curve is not inviscid as R→∞; rather kR→ constant. The upper branch is inviscid in the limit R→∞, and corresponds to a regular or singular neutral mode depending on whether the angle ε between the outer geostrophic flow and the roll axis is greater or less than 15.93°. ' Inflectional ' modes, thought to be relevant by some investigators, do not exist for ε〈 15.93°. Lastly, the most unstable inviscid mode corresponding to zero phase speed, a condition to which certain well-known experiments are sensitive, occurs at e = 11.8° with wavenumber k = 0.6. This is in good agreement with published experimental data. © 1985, Cambridge University Press. All rights reserved.

Description: The subcritical transition in plane Poiseuille flow (generated in a long wind channel of rectangular cross-section) was studied experimentally. Cylinder-generated vortical disturbances were introduced into the parabolic flow (in the test section) or the inlet flow. The parabolic flow was also disturbed by a controlled periodic jet from a wall orifice. On the basis of the three kinds of observations, we come to the conclusion that the minimum transition Reynolds number is about 1000 and the related threshold intensity (of the external disturbance triggering the transition) is comparable to the maximum intensity of u-fluctuations in fully turbulent channel flows. © 1985, Cambridge University Press. All rights reserved.

Description: In the two-dimensional sedimentation process beneath an inclined wall, the mixture of the particulate and liquid phases is separated from the wall by a boundary layer of the clear liquid. This paper contains a simple mathematical model giving waves on the interface between the clear liquid and the mixture. These waves are caused by a discontinuity in the gradient of the tangential velocity of the clear liquid, across the interface. In the limiting case of small concentration of the particulate phase in the mixture, the model gives a dispersive wave running upward along the interface in the direction of the flow in the boundary layer. The effect of finite concentration is to introduce a damping. © 1985, Cambridge University Press. All rights reserved.

Description: Research on the abstract properties of the Navier—Stokes equations in three dimensions has cast a new light on the time-asymptotic approximate solutions of those equations. Here heuristic arguments, based on the rigorous results of that research, are used to show the intimate relationship between the sufficient number of degrees of freedom describing fluid flow and the bound on the fractal dimension of the Navier—Stokes attractor. In particular it is demonstrated how the conventional estimate of the number of degrees of freedom, based on purely physical and dimensional arguments, can be obtained from the properties of the Navier—Stokes equation. Also the Reynolds-number dependence of the sufficient number of degrees of freedom and of the dimension of the attractor in function space is elucidated. © 1985, Cambridge University Press. All rights reserved.

Description: Steady finite-amplitude solutions for two-dimensional convection in a layer heated from below with stress-free boundaries are obtained numerically by a Galerkin method. The stability of the steady convection rolls with respect to arbitrary three-dimensional infinitesimal disturbances is investigated. Stability is found only in a small fraction of the Rayleigh-number-wavenumber space where steady solutions exist. The cross-roll instability and the oscillatory and monotonic skewed varicose instabilities are most important in limiting the stability of steady convection rolls. The Prandtl numbers P = 0.71, 7, 104are emphasized, but the stability boundaries are sufficiently smoothly dependent on the parameters of the problem to permit qualitative extrapolations to other Prandtl numbers. © 1985, Cambridge University Press. All rights reserved.

Description: The incompressible three-dimensional Navier-Stokes equations are solved numerically for a fluid-filled cylindrical cannister that is spinning and nutating. The motion of the cannister is characteristic of that experienced by spin-stabilized artillery projectiles. Equations for the internal fluid motion are derived in a non-inertial aeroballistic coordinate system. Steady-state numerical solutions are obtained by an iterative finite-difference procedure. Flow fields and liquid induced moments have been calculated for viscosities in the range of 0.9 x 104—1 x 109 cSt. The nature of the three-dimensional fluid motion inside the cylinder is discussed, and the moments generated by the fluid are explained. The calculated moments generally agree with experimental measurements. © 1985, Cambridge University Press. All rights reserved.

Description: Synthetic turbulent boundary layers were constructed on a flat plate by generating systematicmoving patterns of turbulent spots in a laminar flow. The experiments were carried out in a wind tunnel at a Reynolds number based on plate length of 1.7 x 10. Spots were generated periodically in space and time near the leading edge to form a regular hexagonal pattern. The disturbancemechanism was a camshaft that displaced small pinsmomentarily into the laminar flow at frequencies up to 80 Hz. Themain instrumentation was a rake of 24 single hot wires placed across the flow in a line parallel to the surface. Themainmeasured variable was local intermittency; i.e. the probability of observing turbulent flow at a particular point in space and time. The results are reported in numerous (x, z, t)-diagrams showing the evolution of various synthetic flows along the plate. The dimensionless celerity or phase velocity of the large eddies was found to be very nearly 0.88, independent of eddy scale. All patterns with sufficiently small scales eventually showed loss of coherence as theymoved downstream. A novel phenomenon called eddy transposition was observed in several flows that contained appreciable laminar regions. The original large eddies were replaced by new eddies at new positions, intermediate to the original ones, while preserving the hexagonal pattern. The present results, together with some empirical properties of a turbulent spot, were used to estimate the best choice of scales for constructing a synthetic boundary layer suitable for detailed study as amodel for a natural flow. The values recommended are: spanwise period/thickness≈2.5, streamwise period/thickness ≈ 8.0. © 1985, Cambridge University Press. All rights reserved.

Description: The Stewartson-Warn-Warn (SWW) solution for the time evolution of an inviscid, nonlinear Rossby-wave critical layer, which predicts that the critical layer will alternate between absorbing and over-reflecting states as time goes on, is shown to be hydrodynamically unstable. The instability is a two-dimensional shear instability, owing its existence to a local reversal of the cross-stream absolute vorticity gradient within the long, thin Kelvin cat's eyes of the SWW streamline pattern. The unstable condition first develops while the critical layer is still an absorber, well before the first over-reflecting stage is reached. The exponentially growingmodes have a two-scale cross-stream structure like that of the basic SWW solution. They are found analytically using themethod ofmatched asymptotic expansions, enabling the problem to be reduced to a transcendental equation for the complex eigenvalue. Growth rates are of the order of the inner vorticity scale δq, i.e. the initial absolute vorticity gradient dq0/dy times the critical-layer width scale. This ismuch faster than the time evolution of the SWW solution itself, albeitmuch slower than the shear rate du0/dy of the basic flow. Nonlinear saturation of the growing instability is expected to take place in a central region of width comparable to the width of the SWW cat's-eye pattern, probably leading to chaoticmotion there, with very large 'eddy-viscosity ' values. Those values correspond to critical-layer Reynolds numbers λ-1≪ 1, suggesting that formost initial conditions the time evolution of the critical layer will depart drastically from that predicted by the SWW solution. A companion paper (Haynes 1985) establishes that the instability can, indeed, grow to large enough amplitudes for this to happen. The simplest way in which the instability could affect the time evolution of the critical layer would be to prevent or reduce the oscillations between over-reflecting and absorbing states which, according to the SWW solution, follow the first onset of perfect reflection. The possibility that absorption (or over-reflection)might be prolonged indefinitely is ruled out, inmany cases of interest (even if the 'eddy viscosity' is large), by the existence of a rigorous, general upper bound on themagnitude of the time-integrated absorptivity α(t). The bound is uniformly valid for all time t. The absorptivity α(t) is defined as the integral over all past t of the jump in the wave-induced Reynolds stress across the critical layer. In typical cases the bound implies that, nomatter how large tmay become, | α(t)| cannot greatly exceed the rate of absorption predicted by linear theorymultiplied by the timescale on which linear theory breaks down, say the time for the cat's-eye flow to twist up the absolute vorticity contours by about half a turn. An alternative statement is that| α(t) | cannot greatly exceed the initial absolute vorticity gradient dq0/dy times the cube of the widthscale of the critical layer. In typical cases, therefore, a brief answer to the question posed in the title is that the critical layer absorbs at first, at a rate ∝ dq0/dy, whereas after linear theory breaks down the critical layer becomes a perfect reflector in the long-time average. If absolute vorticity gradients vanish throughout the critical layer then the bound is zero, implying perfect reflection for all t. The general conditions for the bound to apply are that the wave amplitude and critical-layer width are uniformly bounded for all t, themotion is two-dimensional, and vorticity is neither created nor destroyed within the critical layer, nor transported into or out of it by diffusion, by advection, or by othermeans. Vorticitymay, however, be diffused or turbulently transported within the critical layer, provided that the region within which the transport acts is of bounded width and the range of values of vorticity within that region remains bounded. There are no other restrictions on wave amplitude, none on wavelength, and no assumptions about flow details within the critical layer nor about the initial vorticity profile q0(y), apart from an assumption that q0(y) has singularities no worse than a finite number of jump discontinuities. The proof, in itsmost general form,makes use of a new finite-amplitude conservation theorem for disturbances to parallel shear flows, generalizing the classical results of Taylor, Eliassen & Palm, and others. © 1985, Cambridge University Press. All rights reserved.

Description: The three-dimensional steady laminar-boundary-layer equations have been cast in the appropriate form for semisimilar solutions, and it is shown that in this form they have the same structure as the semisimilar form of the two-dimensional unsteady laminar-boundary-layer equations. This similarity suggests that theremay be a new type of singularity in solutions to the three-dimensional equations: a singularity that is the counterpart of the Stewartson singularity in certain solutions to the unsteady boundary-layer equations. A family of simple three-dimensional laminar boundary-layer flows has been devised and numerical solutions for the development of these flows have been obtained in an effort to discover and investigate the new singularity. The numerical results do indeed indicate the existence of such a singularity. A study of the flow approaching the singularity indicates that the singularity is associated with the domain of influence of the flow for given initial (upstream) conditions as is prescribed by the Raetz influence principle. © 1985, Cambridge University Press. All rights reserved.

Description: Measurements of themean and fluctuating velocities have been obtained with pressure and hot-wire probes in the attached boundary layers and wakes of two airfoilmodels at a lowMach number. The firstmodel is a conventional airfoil at zero incidence and the second an advanced supercritical airfoil at an angle of attack of 4°. Themean-flow and Reynolds-stress data and related quantities are presented with emphasis on the trailing-edge region. The results indicate that the flow around the conventional airfoil is aminor perturbation of a symmetric flat-plate flow with small wake curvature and weak viscous-inviscid interaction. The flow around the supercritical airfoil is in considerable contrast with strong streamwise pressure gradients, non-negligible normal pressure gradients, and large surface and streamline curvatures of the trailing-edge flow. The near wake is strongly curved and intensemixing occurs between the retarded upper-surface boundary layer and strongly accelerated lower-surface boundary layer. © 1985, Cambridge University Press. All rights reserved.

Description: Countercurrent flowmay be induced by opposing buoyancy forces associated with compositional gradients and thermal gradients within a fluid. The occurrence and structure of such flows is investigated by solving the double-diffusive boundary-layer equations for steady laminar convection along a vertical wall of finite height. Non-similar solutions are derived using themethod ofmatched asymptotic expansions, under the restriction that the Lewis and Prandtl numbers are both large. Two sets of asymptotic solutions are constructed, assuming dominance of one or the other of the buoyancy forces. The two sets overlap in the central region of the parameter space; each setmatches up with neighbouring unidirectional similarity solutions at the respective borderlines of incipient counterflow. Interaction between the buoyancymechanisms is controlled by their relative strength R and their relative diffusivity Le. Flow in the outer thermal boundary layer deviates from single-diffusive thermal convection, depending upon themagnitude of the parameter RLe. Flow in the inner compositional boundary layer deviates from single-diffusive compositional convection, depending upon themagnitude of RLe1/3. © 1985, Cambridge University Press. All rights reserved.

Description: In this paper the translatorymotion of a compound drop is examined in detail for low-Reynolds-number flow. The compound drop, consisting of a liquid drop or a gas bubble completely coated by another liquid,moves in a third immiscible fluid. An exact solution for the flow field is found in the limit of small capillary numbers by approximating the two interfaces to be spherical. The solution is found for the general case of eccentric configuration withmotion of the inner sphere relative to the outer together with themotion of the system in the continuous phase. The results show that the viscous forces tend tomove the inner-fluid sphere towards the front stagnation point of the compound drop. For equilibrium of the inner sphere with respect to the outer theremust, therefore, be a body force towards the front. This can only be achieved with the necessary condition that there be a buoyant force on the inner sphere, opposite to that of the compound drop in the continuous phase. For a given set of fluids, two or four equilibrium configurationsmay be found. Of these only one or two, respectively, are stable. The others are unstable. For the special case of concentric configuration, the equilibrium is alwaysmetastable. © 1985, Cambridge University Press. All rights reserved.

Description: The convective behaviour of xenon gas in a vertical thermally conducting cylinder (height/radius 6) heated from below was investigated. Convectively induced temperature fluctuations in the gas were analysed with digital signal-processing techniques over a range of Rayleigh number 0 ≤ Ra ≾ 2300. Quiescent, steady-state, periodic and weakly turbulent convective regimes were characterized. Bistability of steady states (mode switching) was observed in the range 400 ≾Ra ≾ 700. At Ra = 1550 a strictly periodic flow developed. With increasing Ra two additional incommensurate frequencies appeared, leading to 'turbulence' at Ra ≈ 2000. This turbulence, characterized by a broadband power spectrum, intermittently showed periodic flow. A periodic window with a period-doubling sequence appeared between 2100 ≾ Ra ≾ 2200. The spectral features of this sequence can be followed into the broad band noise at higher Ra. Although these experiments were conducted quasistatically, a strong hysteresis was observed with decreasing Ra. Furthermore, it was demonstrated that the sequence of convective regimes can be fundamentally altered byminor perturbations (self-heating) from the flow sensors. © 1985, Cambridge University Press. All rights reserved.

Description: The influence of weak periodic wall undulations on the structure of turbulent pipe flow has been studied in three ways: measurements in air flow using pressure probes and hot-wire techniques, visualizations in water flow and numerical predictions based on a turbulence (k-ε) model. The flows at Reynolds numbers of 30000 and 115000 have been particularly investigated. The flow characteristics proved to be very different from those observed in a straight pipe. Calculations and experiments agree well for the mean-and turbulent-energy fields; however the detailed behaviour of some local quantities such as anisotropy of the Reynolds stress is not well predicted particularly in the crest region. So the performances and the limitations of classical closure have been appraised. The existence of an unsteady reverse-flow region downstream of every crest suggested by measurements and calculations has been clearly confirmed by visualizations in water flow. © 1985, Cambridge University Press. All rights reserved.

Description: A rapidly scanning one-velocity-component directionally sensitive fringe-type laser-Doppler anemometer which scans the measurement volume perpendicular to the optical axis of the transmitting optics was used to investigate the flow structure of the steady freestream separated turbulent boundary layer of Simpson, Chew & Shivaprasad (1981a). Space-time correlations were obtained for the first time in a separated turbulent boundary layer and showed that the integral lengthscale Lyfor the large eddies grows in size towards detachment, although the ratio of this lengthscale to the boundary-layer thickness remains constant. Results also indicate local dependence of the backflow on the middle and outer regions of the boundary layer at a given instant in time. © 1985, Cambridge University Press. All rights reserved.

Description: A stratified two-layer fluid is brought to solid-body counterclockwise rotation inside a cylindrical tank having a conical bottom with a radial topographic ridge. A stress is then applied to the top surface by means of a clockwise differentially rotating disk. The resulting Ekman-layer flux causes the top layer to spin-down and the interface to rise near the wall and to descend at the centre of the tank. As this process continues, the interface (front) between the two layers intersects the disk surface, and after that migrates away from the wall and allows the bottom fluid to contact the disk directly. The migration of the front continues until a steady state is reached, the front becomes stationary and the system is in geostrophic balance. The first sign of upwelled flow at the surface always occurs as a high-speed jet-like plume at the bottom topography, and only at a later time does a uniform upwelled flow appear at the surface upstream of the topography. This plume, which always forms near the downstream edge of the topography, migrates in advance of the upstream front to produce an upwelling maximum at the bottom topography. The final width of the upstream flow in steady-state conditions is estimated by a simple theoretical model, which is in good agreement with the experimental results. We have an experimental criterion to predict the occurrence of travelling baroclinic waves on the upstream front. On the downstream side of the ridge, large standing waves are observed, with significant upwelling within them. Under some circumstances cyclones pinch-off from the upstream front, the plume on the ridge and the crests of the large downstream standing waves. We present criteria to predict the occurrence of these pinch-off processes. © 1985, Cambridge University Press. All rights reserved.

Description: This paper analyses effects of chemical reactions on reflected-shock flow fields in shock tubes. The method of linearized characteristics is applied to analyse gasdynamic disturbances due to chemical reactions. The analysis treats cases where combustible gas is highly diluted in inert gas, and assumes that flows are one-dimensional and that upstream flows in front of the reflected-shock waves are in the frozen state. The perturbed gasdynamic properties in the reflected-shock flow fields are shown to be expressible mainly in terms of a heat-release function for combustion process. In particular, simple relations are obtained between the heat-release function and the physical properties at the end wall of a shock tube. As numerical examples of the analysis, the present formulation is applied to calculate gasdynamic properties in the reflected-shock region in a H2-O2-Ar mixture. Procedures are demonstrated for calculation of the heat-release function by numerically integrating rate equations for chemical species. The analytical results are compared with rigorous solutions obtained numerically by use of a finite-difference method. It is shown that the formulation can afford exact solutions in cases where chemical behaviours are not essentially affected by gasdynamic behaviours. When the induction time of the combustion process is reduced to some extent owing to gasdynamic disturbances, some discrepancies appear between analytical results and rigorous solutions. An estimate is made of the induction-time reduction, and a condition is written down for applicability of the analysis. © 1985, Cambridge University Press. All rights reserved.

Description: A simple model is proposed for the two-dimensional injection of irrotational inviscid fluid from a slot into a free stream. In a certain range of values of the ratio of free-stream to injection total heads, the film thickness satisfies a nonlinear integral equation whose solution enables the mass flow in the film to be found. Some experiments are described which both agree with this theory when it is relevant and indicate its deficiencies at other values of the total head ratio. © 1985, Cambridge University Press. All rights reserved.

Description: Temperature measurements taken in association with the velocity measurements described by Gargett, Osborn & Nasmyth (1984) are examined. With careful noise removal the temperature dissipation spectrum is fully resolved and x, the dissipation rate of temperature-fluctuation variance, is determined directly. With directly measured values of x and ε, the turbulent-kinetic-energy dissipation rate per unit mass, the observed temperature spectra are non-dimensionalized by Oboukov-Corrsin-Batchelor scaling. Shapes and levels of the resulting non-dimensional spectra are then examined as functions of the degree of isotropy (measured) in the underlying velocity field. Two limiting cases are identified: Class A, associated with isotropic velocity fields; and Class B, associated with velocity fields which are anisotropic (owing to buoyancy forces repressing vertical relative to horizontal dimensions of energy-containing ‘eddies’). The present observations suggest that the Corrsin—Oboukov—Batchelor theory does not provide a universal description of the spectrum of temperature fluctuations in water. Class A scalar spectra have neither k-5/3nor K-1subranges: a Batchelor-spectrum fit to the high-wavenumber roll-off region yields a value of 12 for the ‘universal' constant q. In striking contrast, the buoyancy-affected Class B spectra exhibit a clear k-5/3subrange, an approach to a k-1subrange, and a value of q ~ 4 which is in rough agreement with most previous estimates. Previous oceanic and atmospheric measurements are re-examined in the light of the present results. It is suggested that these previous results are also affected by vertical scale limitation. Reasons underlying the discrepancies between theories and observations are discussed: these may be different in the two classes presented. © 1985, Cambridge University Press. All rights reserved.

Description: This paper describes an experimental study of the electromagnetic stirring in a mercury induction furnace. The 200 mm-diameter furnace is supplied with a single-phase electric current of frequency 50-4700 Hz. The flow pattern is measured by means of a special two-wire probe, which tracks the thermal wake behind a hot-film probe. The magnitudes of fluctuating velocities are measured by hot-film anemometry. Attention is focused on the influence on the mean and turbulent motion of the electromagnetic-skin depth, which is determined by the supply frequency. The measurements of the mean motion show that, for a fixed magnetic field, stirring is maximum when the value of the skin depth normalized by the pool radius is about 0.2, in agreement with previous theoretical predictions. Two turbulence regimes may be distinguished for different frequency ranges. At low frequency the various properties of the turbulence, such as the mean-square fluctuations, the integral scales and the turbulent dissipation rate, are almost uniform over the whole bath. However, at high frequency the turbulence is non-uniform; there is an increase in the turbulent fluctuations and dissipation rate and a decrease of the integral scale within the electromagnetic-skin depth near the wall. © 1985, Cambridge University Press. All rights reserved.

Description: A new analytic solution is presented for predicting evaporation rates from plane liquid surfaces into a neutral turbulent boundary layer. Conditions of passive dispersion are assumed. Molecular diffusivity is incorporated into the boundary conditions. Both smooth and rough surfaces are considered. A comparison with a wide variety of experimental data is made; this tends to reveal inadequacies and inconsistencies in the data, rather than test the theory. The effects of a roughness change at the boundary of the liquid surface and of high vapour pressures can be included for practical purposes by simple formulae. A criterion is derived for the validity of the neglect of buoyancy effects. © 1985, Cambridge University Press. All rights reserved.

Description: The reflection and transmission of a gravity wave propagating through a jet-type background flow is studied. Only the linear, non-dissipative case is treated, and the hydrostatic approximation used in a stratified non-rotating medium. The behaviour of the gravity wave in the presence of two or one critical levels is investigated. In the first case, i.e. two critical levels, it is found that for high values of the Richardson number the wave is highly attenuated. For sufficiently low values of the Richardson number overreflection and overtransmission occur. It is demonstrated that a wave generated below the jet and propagating upward takes energy from the mean flow at the upper critical level for all values of the Richardson number. The single critical level has been studied as a limiting case of two merging critical levels. In this approach it is found that the wave is not transmitted and no overreflection can occur. © 1985, Cambridge University Press. All rights reserved.

Description: Cavitation-noise measurements from an axisymmetric body with ‘controlled’ generation of cavitation are reported. The control was achieved by seeding artificial nuclei in the boundary layer by electrolysis. It was possible to alter the number density of nuclei by varying the electrolysis voltage, polarity and the geometry of the electrode. From the observed trend of cavitation-noise data it is postulated that there exists an ‘interference effect’ which influences cavitation noise. When the nucleus-number density is high and cavitation numbers are low this effect is strong. Under these conditions the properties of cavitation noise are found to differ considerably from those expected based on theories concerning noise from single-spherical-bubble cavitation. © 1985, Cambridge University Press. All rights reserved.

Description: This paper describes an experimental study, conducted in the I.M.S.T. air-sea interaction tunnel, of waves excited on a water surface by a periodic train of vortices in the air flow above. The water surface, under some conditions, shows a rapidly developing resonant response, while in the non-resonant case waves propagate both upstream and downstream at speeds different from, but dependent upon, the vortex-convection speed. © 1985, Cambridge University Press. All rights reserved.