ALBERT

Description: This work is aimed at understanding mechanisms which govern the growth of secondary three-dimensional modes of a particular type which feed from a resonant energy exchange with the primary Krármán instability in two-dimensional wakes. Our approach was to introduce controlled time-periodic three-dimensional (oblique) wave pairs of equal but opposite sign, simultaneously with a two-dimensional wave. The waves were introduced by an array of v-component-producing elements on the top and bottom surfaces of the body. These were formed by metallized electrodes which were vapour deposited onto a piezoelectrically active polymer wrapped around the surface. The amplitudes, streamwise and spanwise wavenumbers, and initial phase difference are all individually controllable. The initial work focused on a fundamental/subharmonic interaction, and the dependence on spanwise wavenumber. The results include mode eigenfunction modulus and phase distributions in space, and stream functions for the phase-reconstructed flow field. Analysis of these shows that such a resonance mechanism eXists and its features can account for characteristic changes associated with the growth of three-dimensional structures in the wake of two-dimensional bodies. © 1992, Cambridge University Press. All rights reserved.

Description: Effects of incomplete surface accommodation in rarefied gas flows have been studied using the direct simulation Monte Carlo (DSMC) method in conjunction with the Cercignani—Lampis gas—surface interaction model. Two different flows have been studied, both of which have previously been simulated in DSMC calculations for the case of complete surface accommodation. These are (a) the flow over a sharp, slender circular cone at Mach 5.1 and (b) the flow over a flat-faced circular cylinder at Mach 25. In each case, the gas simulated was nitrogen. It was found that in case (a) the accommodation coefficient of the kinetic energy due to the tangential component of velocity has the greatest influence, whereas in case (b) that of the normal component is also important, having a drastic effect on the rarefied flow field. © 1992, Cambridge University Press. All rights reserved.

Description: Benard convection of a two-component liquid is considered. The liquid displays Soret effects and the boundary temperatures are fiXed to span the solidification temperature of the miXture. Near the lower, heated plate the material is liquid and near the upper cooled plate there is a layer of pure solid solvent; all the solute is rejected during freezing. Linear stability theory is used to determine the effects on the critical conditions for Soret convection in the presence of the solidified layer and the interface between solid and liquid. EXperiments on miXtures of ethyl alcohol and water are performed using interferometry, photography and thermocouple measurements. The measured onset of instability to travelling waves at negative Soret coefficient compares well with those predicted by our linear theory. In the absence of ice the waves develop at finite amplitude to a fiXed-amplitude state. However, when ice is present, these waves fail to persist but evolve to a state of steady finite-amplitude (overturning) convection. These differences are attributed to the presence of the ice and the nonlinear density profile of the basic state, both of which act as sources of non-Boussinesq effects. © 1992, Cambridge University Press. All rights reserved.

Description: This paper describes experiments undertaken to study in detail the control of vortex shedding from circular cylinders at low Reynolds numbers by using feedback to stabilize the wake instability. Experiments have been performed both in a wind tunnel and in an open water channel with flow visualization. It has been found that feedback control is able to delay the onset of the wake instability, rendering the wake stable at Reynolds numbers about 20% higher than otherwise. At higher flow rates, however, it was not possible to use single-channel feedback to stabilize the wake - although, deceptively, it was possible to reduce the unsteadiness recorded by a near-wake sensor. When control is applied to a long span only the region near the control sensor is controlled. The results presented in this paper generally support the analytical results of other researchers. © 1993, Cambridge University Press. All rights reserved.

Description: Solutions of the Poincare equation describing equatorially trapped three-dimensional boundary travelling waves in rotating spherical systems are discussed. It is shown that the combined effects of Coriolis forces and spherical curvature enable the equatorial region to form an equatorial waveguide tube with characteristic latitudinal radius (2/m) and radial radius (1/m), where m is azimuthal wavenumber. Inertial waves with sufficiently simple structure along the axis of rotation and sufficiently small azimuthal wavelength must be trapped in the equatorial waveguide tube. The structure and frequency of the inertial waves are thus hardly affected by the presence of an inner sphere or by the condition of higher latitudes. Further calculations on rotating spherical fluid shells of finite internal viscosity and stress-free boundaries are also discussed. © 1993, Cambridge University Press. All rights reserved.

Description: The interaction of linear, steady, axisymmetric deep-water gravity waves with preexisting large-scale annular currents has been investigated. Waves originating inside the annulus as well as waves approaching the annulus from the outside were studied. Exact linear ray solutions were obtained and involve two non-dimensional parameters, a radius-angle parameter and a velocity parameter. For opposing currents the linear solutions also allow the derivation of radii at which the waves are blocked, reflected at a linear caustic or stopped by the current. Various examples of rays interacting with an annular current are presented to illustrate aspects of the solutions obtained. In particular, the behaviour of the ray solutions at blocking, reflection and stopping is investigated. Linear ray theory is shown to fail at caustics and caustic solutions are briefly discussed. © 1993, Cambridge University Press. All rights reserved.

Description: The efficacy of perturbation approaches for short-long wave interactions is examined by considering a simple case of two interacting wave trains with different wavelengths. Frequency-domain solutions are derived up to third order in wave steepness using two different formulations: one employing conventional wave-mode functions only, and the other introducing a modulated wave-mode representation for the short-wavelength wave. For long-wavelength wave steepness and short-to-long wavelength ratio ϥ1 and ϥ3 respectively, the two results are shown to be identical for ϥ1 ϥ3 〈 0.5. As ϥ1 approaches ϥ3, the conventional wave-mode approach converges slowly and eventually diverges for ϥ1 ⪢ e3. The loss of convergence is because the linear phase of conventional wave-mode functions is ineffective for modelling the modulated phase of the short wave. As expected, this difficulty can be removed by using a modulated wave-mode function for the short wave. On the other hand, for relatively large ϥ3 ~ O(1), the conventional wave-mode approach converges rapidly while the slowly varying interaction between the two waves cannot be accurately predicted by the present modulated wave-mode approach. These findings have important implications to (time-domain) numerical simulations of the nonlinear evolution of ocean wave fields, and suggest that a hybrid wave model employing both conventional (for large-ϥ3 interactions) and modulated (for small-ϥ3 interactions) wave-mode functions should be particularly effective. © 1993, Cambridge University Press. All rights reserved.

Description: We study the flow generated by a small circular cylinder in a mixing layer. The cylinder is executing an oscillatory translation whose frequency is within the range of unstable frequencies of the shear layer. The smallness of the cylinder is measured by the ratio of its radius to the characteristic thickness of the layer. This (small) ratio serves as the expansion parameter for our theory; the flow naturally divides into inner and outer regions. The former is in the immediate vicinity of the cylinder and the latter is the far field which contains the instability waves. The solution to this problem is obtained by the method of matched asymptotic expansion. One objective is to study the dependence of this solution on various parameters such as the frequency of oscillation, velocity ratio, etc., and thus shed light on the associated receptivity, Other objectives deal with a restatement of causality and with the hydrodynamic field near the streamwise location of the cylinder. We find that receptivity is a strong function of frequency and velocity ratio and that the local hydrodynamic field may be quite large. Causality is restated in terms of the well-known exponential integral. © 1993, Cambridge University Press. All rights reserved.

Description: We propose a general geometric method of derivation of invariant relations for hydrodynamic dissipationless media. New dynamic invariants are obtained. General relations between the following three types of invariants are established, valid in all models: Lagrangian invariants, frozen-in vector fields and frozen-in co-vector fields. It is shown that frozen-in integrals form a Lie algebra with respect to the commutator of the frozen fields. The relation between frozen-in integrals derived here can be considered as the Backlund transformation for hydrodynamic-type systems of equations. We derive an infinite family of integral invariants which have either dynamic or topological nature. In particular, we obtain a new type of topological invariant which arises in all hydrodynamic dissipationless models when the well-known Moffatt invariant vanishes. © 1993, Cambridge University Press. All rights reserved.

Description: A theoretical model for infragravity waves generated by incident short-wave groups is developed. Both normal and oblique short-wave incidence is considered. The depth-integrated conservation equations for mass and momentum averaged over a short-wave period are equivalent to the nonlinear shallow-water equations with a forcing term. In linearized form these equations combine to a second-order long-wave equation including forcing, and this is the equation we solve. The forcing term is expressed in terms of the short-wave radiation stress, and the modelling of these short waves in regard to their breaking and dynamic surf zone behaviour is essential. The model takes into account the time-varying position of the initial break point as well as a (partial) transmission of grouping into the surf zone. The former produces a dynamic set-up, while the latter is equivalent to the short-wave forcing that takes place outside the surf zone. These two effects have a mutual dependence which is modelled by a parameter K, and their relative strength is estimated. Before the waves break, the standard assumption of energy conservation leads to a variation of the radiation stress, which causes a bound, long wave, and the shoaling bottom results in a modification of the solution known for constant depth. The respective effects of this incident bound, long wave and of oscillations of the break-point position are shown to be of the same order of magnitude, and they oppose each other to some extent. The transfer of energy from the short waves to waves at infragravity frequencies is analysed using the depth-integrated conservation equation of energy. For the case of normally incident groups a semi-analytical steady-state solution for the infragravity wave motion is given for a plane beach and small primary-wave modulations. Examples of the resulting surface elevation as well as the corresponding particle velocity and mean infragravity-wave energy flux are presented. Also the sensitivity to the variation of input parameters is analysed. The model results are compared with laboratory experiments from the literature. The qualitative agreement is good, but quantitatively the model overestimates the infragravity wave activity. This can, in part, be attributed to the neglect of frictional effects. © 1993, Cambridge University Press. All rights reserved.

Description: The properties of a dilute bidisperse particle-gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit rv4 rc, where rcis the time between successive collisions of a particle, and rv is the viscous relaxation time. In this limit, the particles relax close to their terminal velocity between successive collisions, and we use a perturbation analysis in the small parameter e, which is proportional to tv/tc, about a base state in which all the particles settle at their terminal velocities. The mean velocities of the two species are 0(e) different from their terminal velocities, and the mean-square velocities are 0(e) smaller than the square of the terminal velocity. The distribution functions for the two species, which incorporate the first effects of collisions between particles settling at their terminal velocities, are derived. The velocity distribution is highly anisotropic in this limit, and the mean-square velocity in the vertical direction is twice that in the horizontal plane. The distribution function for each species is singular at its terminal velocity, and the distributions are non-zero in a finite region in velocity space between the two terminal velocities. © 1993, Cambridge University Press. All rights reserved.

Description: The properties of a dilute bidisperse particle—gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit rc rv using a perturbation analysis in the small parameter v, which is proportional to the ratio of timescales rc/rv. Here, rc is the time between successive collisions of a particle, and rv is the viscous relaxation time. The leading-order distribution functions for the two species are isotropic Gaussian distributions, and are identical to the molecular velocity distributions in a two-component gas at equilibrium. Balance equations are written for the mean and mean-square velocities, using a distribution function that is a small perturbation from the isotropic Gaussian. The collisional terms are calculated by performing an ensemble average over the relative configurations of the colliding particles, and the mean velocity and velocity variances are calculated correct to 0(v2) by solving the balance equations. The difference in the mean velocities of the two species is 0(v) smaller than the mean velocity of the suspension, and the fluctuating velocity is 0(vi) smaller than the mean velocity. © 1993, Cambridge University Press. All rights reserved.

Description: This paper considers the dynamics of a gas bubble in response to either a pressure pulse or a pressure step at t = 0, both in the presence and absence of a mean flow. Our work utilizes small-deformation, domain perturbation analysis carried to second and higher order in the amplitude of deformation, e. In the absence of a mean flow, our analysis of the small deformation problem for an initial impulsive perturbation of the bubble volume and shape is closely related to recently published work by Longuet-Higgins on the time-dependent oscillations of an initially deformed bubble in a quiescent fluid. However, in the presence of a mean flow which deforms the bubble, the bubble response to pressure changes is more complex. Specifically, the present analysis identifies a number of different mechanisms for resonant interaction between shape deformation modes and the volume or radial breathing mode of oscillation. This includes not only a fundamental change in the resonant interactions at 0(e2) - where resonant interaction is also found in the absence of mean flow - but resonant interactions also at the level of 0(e*) which are not present without the mean flow. On the other hand, the bubble dynamics in response to a step change in the pressure distribution in a quiescent fluid exhibits similar resonant interactions at 0(e2) to those obtained for a pressure pulse in the presence of mean flow because the bubble oscillates around a non-spherical steady-state shape owing to the nonuniform pressure distribution on the bubble surface in both the cases. © 1993, Cambridge University Press. All rights reserved.

Description: Two-dimensional, unsteady flow of a viscous, incompressible fluid in a stepped channel has been studied by the numerical solution of the Navier-Stokes equation using an accurate finite-difference method. With a sinusoidal mass flow rate, the problem has three governing parameters: the Reynolds number, the Strouhal number, and the step height. The effects on the flow of varying all three parameters has been investigated systematically. In appropriate parameter regimes, a strong vortex wave' is generated during the forward phase when the flow is over the step into the expansion. Secondary effects on the wave can result in a complex flow pattern with each major structure of the flow consisting of an eddy with more than one core. No such wave is found during the reverse phase of the flow. The generation and development of the wave is examined in some detail, and our results are compared and contrasted with those of previous studies, both experimental and theoretical, of flow in non-uniform vessels. © 1993, Cambridge University Press. All rights reserved.

Description: A model inviscid and incompressible flow problem is studied in which an infinite array of equi-spaced identical rectilinear line vortices moves in a uniform stream over a wall in which is embedded an equi-spaced array of discrete line sources of variable strength. It is shown that for a suitable choice of source spacing and strength, a flow that is periodic both in time and in the streamwise direction is possible. The flow is shown to be stable to small two-dimensional disturbances for a range of values of vortex height above the wall and source strength. The implications for the corresponding viscous problem and active flow control are discussed. Crown. © 1993, Cambridge University Press. All rights reserved.

Description: We develop a theory to describe the topographic control of planetary-scale flows resulting from the variation of the Earth's rotation with latitude. We show that on passing over topography, an inertial, zonal current on an equatorial plane may pass through a control at which the flow changes from a subcritical to a supercritical solution branch. Downstream of this control, a transition back to the subcritical solution branch may occur, for example, by the generation of planetary eddies or radiating Rossby waves. We calculate the energy dissipated across such a transition and discuss the relevance of this theory for a number of atmospheric and oceanic phenomena. We also show that this phenomenon is analogous to the hydraulic control of a non-rotating, stratified flow passing through a channel of variable width. © 1993, Cambridge University Press. All rights reserved.

Description: The stability and nonlinear evolution of a ridge of fluid on an inclined plane is investigated. This model was introduced by Hocking (1990). Here we present numerical solutions of the model showing the evolution of the ridge and in some cases the formation of droplets. Also, we investigate the linear stability of the fluid ridge allowing for contact-line motion. We find a preferred wavelength for the linear stability of spanwise disturbances. © 1993, Cambridge University Press. All rights reserved.

Description: The global behaviour of a granular flow is critically dependent on its interaction with whatever solid boundaries with which it comes into contact, whether they be used to drive, retard or simply bound the flow field. This paper describes the results of a computer simulation study of the effects of roughening boundaries by ‘gluing' particles to the surfaces. Roughness is commonly used in experimental devices as a way of approximating a no-slip condition between a granular material and the driving surfaces. On a microscopic level, this produces a boundary that extends out into the flow field to the limit of the roughness elements. This has a strong effect on the way that forces, and, in particular, torque, is transmitted to the particles in the neighbourhood of the boundary. © 1993, Cambridge University Press. All rights reserved.

Description: The phenomenon of vortex reconnection is analysed numerically and the results are compared qualitatively with the predictions of a model of reconnection recently proposed by Saffman. Using spectral methods over both uniform and strained meshes, numerical simulations are performed of two nearly parallel, counter-rotating vortex tubes, over the range of Reynolds numbers Re = 1000–3500. The calculations utilizing a uniform mesh are performed for Re ≤ 1500 with a resolution of 128 points in each direction. The calculations utilizing a stretched mesh are performed for 1500 〈 Re ≤ 3500 with a resolution of up to 160 points in each direction and with a fourfold stretching about the region of reconnection. We present results for the variation of the maximum of vorticity, the time to reconnection, and other diagnostics of this flow as functions of the Reynolds number. From numerical simulation of the model equations, we infer and demonstrate the existence of exact solutions to the model to which its solutions arising from more general initial conditions are attracted at late times. In the limit of infinite Reynolds number, the model predicts eventual saturation of the axial strain, a feature observed in the recent work of Pumir & Siggia and also observed in our full numerical simulations. In this respect the model captures the observed local dynamics of vortex stretching. However, because the global effects of external flows are not included in the model, the model predicts that the axial strain eventually decays and the maximum vorticity grows linearly at late times. In contrast, from the full simulations, we see the possible emergence of the behaviour of the axial strain at infinite Reynolds number. As our simulations are affected by non-local effects, we do observe saturation of the strain but no subsequent decay. It is also shown analytically that the model predicts a reconnection time which varies logarithmically with increasing Reynolds number. Comparison with the full numerical simulations shows a much slower variation of the reconnection time with increasing Reynolds number than predicted by the model. Other points of agreement and disagreement between the Saffman model and the simulations are discussed, Reconnection is also discussed from the point of view of its relation to the possible onset of nearly singular behaviour of the Euler equation. In agreement with the recent numerical results of Pumir & Siggia, our results suggest that no singularity in the vorticity will form in a finite time for this initial condition.

Description: Nonlinear interaction between transverse disturbances and longitudinal rolls has been investigated for flow in an inclined slot with a heated lower wall when both modes of instability occur at nearly the same value of the control parameter. This condition is shown to be possible for a fluid with Prandtl number greater than 0.263897. For slightly supercritical values of the Rayleigh number (R) when the critical Rayleigh number for longitudinal rolls RL C is somewhat less than that for transverse stationary rolls, RS C, and for transverse travelling waves, longitudinal rolls occur first and then remain stable as R is increased beyond RS Cor RT C; no mixed mode state occurs. In contrast, if RS C or RT Cis slightly below RL C, pure transverse modes exist for only a relatively small range of R beyond RS C or RT C. Thereafter, a thee-dimensional mixed mode state occurs well before RL C is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches RL C, the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than RL C Mixed mode convection is also investigated for a special choice of parameters when three modes, namely transverse stationary rolls, transverse travelling waves and longitudinal rolls, become unstable simultaneously. Longitudinal rolls again emerge as the stable supercritical state. © 1993, Cambridge University Press. All rights reserved.

Description: The equivalence of the method of magnetic relaxation to a variational problem with an infinity of constraints is established. This variational problem is solved in principle and approximations to the exact solution are compared to results obtained by numerical relaxation of fields with a single stationary elliptic point. In the case of a finite energy field of the above topology extending to infinity, we show that the minimum energy state is the one in which all field lines are concentric circles and that this state is topologically accessible from the original one. This state is used as a reference state for understanding the relaxation of fields constrained by finite boundaries. We then consider the relaxation of fields containing saddle points and confirm the tendency of the saddle points to collapse and form two Y-points. An infinite family of local equilibrium solutions each describing a Y-point is provided. © 1993, Cambridge University Press. All rights reserved.

Description: The effects of abruptly applied cycles of curvatures and pressure gradients on turbulent boundary layers are examined experimentally. Two two-dimensional curved test surfaces are considered: one has a sequence of concave and convex longitudinal surface curvatures and the other has a sequence of convex and concave curvatures. The choice of the curvature sequences were motivated by a desire to study the asymmetric response of turbulent boundary layers to convex and concave curvatures. The relaxation of a boundary layer from the effects of these two opposite sequences has been compared. The effect of the accompanying sequences of pressure gradient has also been examined but the effect of curvature dominates. The growth of internal layers at the curvature junctions have been studied. Measurements of the Görtler and corner vortex systems have been made. The boundary layer recovering from the sequence of concave to convex curvature has a sustained lower skin friction level than in that recovering from the sequence of convex to concave curvature. The amplification and suppression of turbulence due to the curvature sequences have also been studied. © 1993, Cambridge University Press. All rights reserved.

Description: Bilger, Saetran & Krishnamoorthy (1991) give measured values of the variance, cross-correlation coefficient, autospectra, coherence and phase shift of the reactant concentration fluctuations for an irreversible second-order reaction in an incompressible turbulent scalar mixing layer. The present paper approaches the interpretation of the measured data by evaluating the above quantities in the frozen (slow) and equilibrium (fast) chemistry limits. We assume that the limiting values bracket the corresponding intermediate rate data. The analysis leads to values that correspond with the measured variances and correlation coefficients. The paper offers simple procedures for experimenters to evaluate the fast chemistry limit of the spectral characteristics from the measured mixture fraction fluctuations. The investigation of the limiting spectra suggests that, in the frequency region considered in the Bilger et al. measurements, the shape of the autospectrum is quite insensitive to the chemistry rate. The cross-spectrum is much more sensitive to the chemistry than the autospectrum. The analysis predicts correctly that the coherence decreases with increasing frequency while the phase stays equal to K until the decrease of the coherence leads to indeterminate phase results. © 1993, Cambridge University Press. All rights reserved.

Description: The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t-2. © 1993, Cambridge University Press. All rights reserved.

Description: Compressibility effects within decaying isotropic turbulence and homogeneous turbulent shear flow have been studied using direct numerical simulation. The objective of this work is to increase our understanding of compressible turbulence and to aid the development of turbulence models for compressible flows. The numerical simulations of compressible isotropic turbulence show that compressibility effects are highly dependent on the initial conditions. The shear flow simulations, on the other hand, show that measures of compressibility evolve to become independent of their initial values and are parameterized by the root mean square Mach number. The growth rate of the turbulence in compressible homogeneous shear flow is reduced compared to that in the incompressible case. The reduced growth rate is the result of an increase in the dissipation rate and energy transfer to internal energy by the pressure-dilatation correlation. Examination of the structure of compressible homogeneous shear flow reveals the presence of eddy shocklets, which are important for the increased dissipation rate of compressible turbulence. © 1993, Cambridge University Press. All rights reserved.

Description: We have studied oscillatory flow through a 180° curved tube with the ratio of tube radius to radius of curvature equal to 1/7. The flow rate is constrained to vary sinusoidally about a non-zero mean at a specified period T, and mean flow rate Q. At a certain parameter range of interest Hamakiotes & Berger (1990) predict that fully developed flow undergoes a period-tripling bifurcation. Our measurements using laser-Doppler velocimetry found no bifurcation. An additional experiment was done to ensure that the flow was fully developed. © 1993, Cambridge University Press. All rights reserved.

Description: Three decades have passed since vortex breakdown was first identified as a natural fluid flow phenomenon. Three key theories have been proposed to explain the phenomenon hydrodynamic instability, conjugate states and flow stagnation. Despite a considerable amount of theoretical and experimental investigation, there is still nothing approaching a completely satisfactory theory of vortex breakdown. In addition, there is no agreement on a complete physical description of the structure of vortex breakdown. The present experimental investigation may substantiate a few earlier conjectures. We discuss an experimental finding that might help clarify the phenomenon through the use of flow visualization and laser-Doppler velocimetry. Experimental measurements substantiate earlier measurements and theoretical calculations of the velocity field. The evidence suggests that there is a connection between criticality and instability. © 1993, Cambridge University Press. All rights reserved.

Description: Controlled, amplitude-modulated excitation of a cylinder at low Reynolds number (Re equals; 136) in the cross-stream direction generates several states of response of the near wake including: a locked-in wake structure, which is periodic at the modulation frequency; a period-doubled wake structure, which is periodic at a frequency half the modulation frequency; and a destabilized structure of the wake, which is periodic at the modulation frequency, but involves substantial phase modulations of the vortex formation relative to the cylinder displacement. The occurrence of each of these states depends upon the dimensionless modulation frequency, as well as the nominal frequency and amplitude of excitation. Transition through states of increasing disorder can be attained by either decreasing the modulation frequency or increasing the amplitude of excitation at a constant value of nominal frequency. These states of response in the near wake are crucial in determining whether the far wake is highly organized or incoherent. Both of these extremes are attainable by proper selection of the parameters of excitation.

Description: The effect of a geostrophic boundary current on internal gravity waves is studied with a reduced-gravity model. We found that the boundary current not only modifies the coastal Kelvin wave, but also forms wave guides for short internal gravity waves. The combined effects of current shear, the boundary, and the slope of the interface create the trapping mechanism. These trapped internal gravity waves appear as groups of discrete zonal modes. They have wavelengths comparable to or shorter than the internal Rossby radius of deformation. Their phase speeds are close to that of the internal Kelvin wave. However, they can propagate both in, or opposite to, the direction of the Kelvin wave. The results of the present work suggest the possibility of finding an energetic internal gravity wave phenomenon with near-inertial frequency in a broad geostrophic boundary current. © 1993, Cambridge University Press. All rights reserved.

Description: A fully nonlinear numerical method is developed to study the viscous interactions of a pair of vortex tubes rising toward a free surface. The numerical theory uses Helmholtz's decomposition to treat the irrotational and vortical components of the flow as separate nonlinearly coupled equations. The laminar interactions of a pair of vortex tubes with a clean free surface at intermediate Froude and Weber numbers and a low Reynolds number show two distinct phases. During the rise phase of the vortex pairs, instabilities lead to the formation of helical vorticity. The rotation of the helical vorticity around the primary vortex tubes causes an unsteady oscillation in the free-surface elevation. During the reconnection phase, the helical vortex sheets get narrower and attach themselves to the free surface. The normal connections of cross-axis vorticity with the free surface give whirls. The free-surface elevation is well correlated with the vortical pressure. The numerical results agree qualitatively with experimental measurements. © 1993, Cambridge University Press. All rights reserved.

Description: This paper presents a closed-form solution for the inertial lift force acting on a small rigid sphere that translates parallel to a flat wall in a linear shear flow. The results provide connections between results derived by other workers for various limiting cases. An analytical form for the lift force is derived in the limit of large separations. Some new results are presented for the disturbance flow created by a small rigid sphere translating through an unbounded linear shear flow. © 1993, Cambridge University Press. All rights reserved.

Description: During the day, the shallower regions of a reservoir sidearm absorb more heat per unit volume than the deeper parts, leading to a horizontal pressure gradient that drives a circulation in the sidearm. At night, the shallow regions cool more rapidly, leading to a circulation in the opposite direction. Since the spin-up time of a typical sidearm is at least of the same order as a day, the flow within a diurnally forced sidearm is principally an inertia-buoyancy balance. In this paper, a diurnally forced sidearm is modelled by periodically forced natural convection in a triangular cavity. The periodic forcing enters the model via an internal heating/cooling term in the temperature equation. Reservoir sidearms typically have small bottom slopes and this fact can be exploited to obtain asymptotic solutions of the resulting equations. These solutions clearly demonstrate the transition from the viscous-dominated flow in the shallows to the inertia-dominated flow in the deeper parts. In the inertia-dominated region, the flow response significantly lags the forcing. Numerical solutions of the full nonlinear problem are consistent with the asymptotic solutions. © 1993, Cambridge University Press. All rights reserved.

Description: The transient motion of ordered suspensions of liquid drops, initially arranged on a cubic lattice, is studied as a model of suspension rheology. An asymptotic three-term expansion for the effective stress tensor of a dilute suspension of spherical drops is derived based on the Faxén law for the stresslet. Comparisons with available exact results for cubic lattices suggests that the expansion is remarkably accurate even at concentrations close to maximum packing. The behaviour of suspensions with recurrent structure evolving under the influence of a simple shear flow is investigated, and the results show that the time-averaged behaviour may differ substantially from the instantaneous behaviour. Transient normal stress differences may vanish in the mean, but make appreciable contributions to the instantaneous dynamics. The effect of particle deformation is assessed by numerically computing the motion of initially spherical drops arranged on a cubic lattice. At large times, the suspension is shown to exhibit periodic motions in which the drops oscillate about a mean shape with a phase shift which depends on the geometry of the lattice and the physical properties of the fluids. It is shown that drop deformations cause shear thinning and some type of elastic behaviour, and may lower the effective viscosity of the suspension below that corresponding to the dilute limit. © 1993, Cambridge University Press. All rights reserved.

Description: A turbulent round jet of helium was studied experimentally using a composite probe consisting of an interference probe of the Way-Libby type and an x -probe. Simultaneous measurements of two velocity components and helium mass fraction concentration were made in the x/d range 50–120. These measurements are compared with measurements in an air jet of the same momentum flux reported in Part 1. The jet discharge Froude number was 14000 and the measurement range was in the intermediate region between the non-buoyant jet region and the plume region. The measurements are consistent with earlier studies on helium jets. The mass flux of helium across the jet is within ± 10% of the nozzle input. The mean velocity field along the axis of the jet is consistent with the scaling expressed by the effective diameter but the mean concentration decay constant exhibits a density-ratio dependence. The radial profiles of mean velocity and mean concentration agree with earlier measurements, with the half-widths indicating a turbulent Schmidt number of 0.7. Significantly higher intensities of axial velocity fluctuations are observed in comparison with the air jet, while the intensities of radial and azimuthal velocity fluctuations are virtually identical with the air jet when scaled with the half-widths. Approximate budgets for the turbulent kinetic energy, scalar variance and scalar fluxes are presented. The ratio of mechanical to scalar timescales is found to be close to 1.5 across most of the jet. Current models for triple moments involving scalar fluctuations are compared with measurements. As was observed with the velocity triple moments in Part 1, the performance of the Full model that includes all terms except advection was found to be very good in the fully turbulent region of the jet. © 1993, Cambridge University Press. All rights reserved.

Description: Experiments were performed to study vortex shedding behind a linearly tapered cylinder. Four cylinders were used, with taper ratios varying from 50:1 to 100:1. The cylinders were each run at four different velocities, adjusted to cover the range of laminar vortex shedding for a non-tapered cylinder. The flow was confirmed to consist of discrete shedding cells, each with a constant frequency. For a centrespan Reynolds number greater than 100, the dimensionless mean cell length was found to be a constant. Individual cell size was found to be roughly self-similar. New shedding cells were created on the ends of the cylinders, or in regions adjacent to areas not shedding. Successful scalings were found for both the cell shedding frequencies and their differences, the modulation frequencies. The modulation frequencies were found to be constant along the cylinder span. The shedding frequency Strouhal number versus Reynolds number curve was found to have a slightly steeper slope than the Strouhal number curve for a non-tapered cylinder. Vortex shedding was found to begin at a local Reynolds number of about 60, regardless of any other factors. End effects were found to be of little importance. The vortex splits, which form the links between shedding cells, were found to be similar in some respects to those found by earlier investigators. Amplitude results suggested that the splits at different spanwise locations are temporally sequenced by an overall flow mechanism, a supposition confirmed by flow visualization. Wavelet analysis results showed that while the behaviour of the shedding frequencies in time was relatively unaffected by changing taper ratio, the behaviour of the modulation frequency in time was greatly affected. Comparisons with other experiments point out the universality of vortex splitting phenomena. © 1993, Cambridge University Press. All rights reserved.

Description: We study the acoustic behaviour of a mixture of diethyl-ether bubbles and liquid, at small volume fraction of vapour, contained in a standing-wave tube. We give experimental evidence of the effect of the liquid/vapour phase transition on the positions and amplitudes of the resonances of the tube. The effective-medium theory, which properly describes the behaviour of liquid/gas mixtures, is shown to be inadequate. We find that theoretical studies which take the effect of the phase transition into account do agree with our experimental data. © 1993, Cambridge University Press. All rights reserved.

Description: A turbulent round jet of air discharging into quiescent air was studied experimentally. Some x -wire hot-wire probes mounted on a moving shuttle were used to eliminate rectification errors due to flow reversals in the intermittent region of the jet. Moments of velocity fluctuations up to fourth order were measured to characterize turbulent transport in the jet and to evaluate current models for triple moments that occur in the Reynolds stress equations. Fourth moments were very well described in terms of second moments by the quasi-Gaussian approximation across the entire jet including the intermittent region. Profiles of third moments were found to be significantly different from earlier measurements: 2, (uw2) and 2 are found to be negative near the axis of the jet. The Basic triple moment model that included turbulent production and models for the dissipation and the return-to-isotropy part of the pressure correlations was found to be unsatisfactory. When mean-strain production and a model for rapid pressure correlations were also included, predictions were satisfactory in the fully turbulent region. The consistency of the measurements with the equations of motion was assessed: momentum flux across the jet was found to be within + 5 % of the nozzle input and the integral of radial diffusive flux of turbulent kinetic energy across the jet calculated from the measured third moments was found to be close to zero. © 1993, Cambridge University Press. All rights reserved.

Description: The spin-down of a barotropic axisymmetrie vortex, such as observed in laboratory models, is examined analytically. In addition to the classical, self-similar Ekman decay due to viscous effects (Greenspan &Howard 1963), which is characterized by an azimuthal velocity profile with the position of its maximum velocity fixed and a decay time equal to the Ekman timescale, the effects of nonlinearity and a free surface are considered separately. The Ekman circulation in the radial and vertical planes whose strength is determined by the vorticity of the overlying fluid, leads to radial advection of the azimuthal velocity. This nonlinearity results in a nonlinear kinematic wave equation for the circulation and leads to the outward/inward propagation of the position of maximum azimuthal velocity for cyclonic/anticyclonic vortices. The associated steepening of the azimuthal velocity profile may lead to a shock formation when the absolute vorticity of the initial profile is negative at a certain radius. For anticyclonic vortices having a monotonically increasing angular velocity profile this shock formation occurs at the core. For such vortices (or arbitrary cyclonic vortices) this dynamical ‘breaking’ criterion is, despite significant differences in the physics concerned, identical to Rayleigh's kinematical criterion for the onset of centrifugal instability. For a dynamically active free-surface fluid the spin-down of a decaying vortex is prolonged by a radially dependent factor proportional to the Froude number. This conclusion holds both in a cylinder with a parabolic bottom (mimicking the shape of the free surface of a fluid in solid-body rotation) and in a flat-bottomed cylinder. In view of the constancy of background vorticiity the former geometry is relevant for a comparison to geophysical f-plane vortices. The latter geometry, however, is more easily established in a laboratory experiment, but the evolution of the azimuthal velocity profile is much more complicated and depends on the initial azimuthal velocity profile in a highly convoluted way. © 1993, Cambridge University Press. All rights reserved.

Description: Electrophoresis in an alternating electric field is the basis for electroacoustical measurements. These measurements provide new means of investigating the electrokinetic properties of colloidal systems. In order to relate electroacoustical signals to the charge and the size of colloidal particles, an expression is required for the dynamic electrophoretic mobility of colloidal particles in a continuous fluid. In this paper, an exact analytical solution to the problem is given for an arbitrary ratio between the particle radius and the electric double-layer thickness, in the case where the electrokinetic potential of the uniformly charged particle is small and unaffected by the alternating field. © 1993, Cambridge University Press. All rights reserved.

Description: In this paper, we present detailed and systematic experimental results on the sedimentation of solid particles in aqueous solutions of polyox and polyacrylamide, and in solutions of polyox in glycerin and water. The tilt angles of long cylinders and flat plates falling in these viscoelastic liquids were measured. The effects of particle length, particle weight, particle shape, liquid properties and liquid temperature were determined. In some experiments, the cylinders fall under gravity in a bed with closely spaced walls. No matter how or where a cylinder is released the axis of the cylinder centres itself between the close walls and falls steadily at a fixed angle of tilt with the horizontal. A discussion of tilt angle may be framed in terms of competition between viscous effects, viscoelastic effects and inertia. When inertia is small, viscoelasticity dominates and the particles settle with their broadside parallel or nearly parallel to the direction of fall. Normal stresses acting at the corners of rectangular plates and squared-off cylinders with flat ends cause shape tilting from the vertical. Cylinders with round ends and cone ends tilt much less in the regime of slow flow. Shape tilting is smaller and is caused by a different mechanism to tilting due to inertia. When inertia is large the particles settle with their broadside perpendicular to the direction of fall. The tilt angle varies continuously from 90° when viscoelasticity dominates to 0° when inertia dominates. The balance between inertia and viscoelasticity was controlled by systematic variation of the weight of the particles and the composition and temperature of the solution. Particles will turn broadside-on when the inertia forces are larger than viscous and viscoelastic forces. This orientation occurred when the Reynolds number Re was greater than some number not much greater than one in any case, and less than 0.1 in Newtonian liquids and very dilute solutions. In principle, a long particle will eventually turn its broadside perpendicular to the stream in a Newtonian liquid for any Re 〉 0, but in a viscoelastic liquid this turning cannot occur unless Re 〉 1. Another condition for inertial tilting is that the elastic length XU should be longer than the viscous length v/U where U is the terminal velocity, v is the kinematic viscosity and A = v/c2is a relaxation time where c is the shear wave speed measured with the shear wave speed meter (Joseph 1990). The condition M=U/c 〉 1 is provisionally interpreted as a hyperbolic transition of solutions of the vorticity equation analogous to transonic flow. Strong departure of the tilt angle from 6 = 90° begins at about M = 1 and ends with 6 = 0° when 1 〈 M 〈 4. © 1993, Cambridge University Press. All rights reserved.

Description: An experimental and numerical investigation of the density distribution produced in a container by a negatively buoyant jet has been undertaken to evaluate the effect of the forced vertical motion of the environment. Vertical motion results from inflows and exhausts above and below the jet. Three distinct cases were identified. In the first, a velocity in the environment opposed the jet and produced a steady flow. This configuration was used to measure the entrainment flux along the length of the fountain. This configuration is similar to a jet impinging on an interface for which the entrainment depends on the local Froude number. The experiments covered a wider range of local Froude numbers than previously published and have produced results which are different from those in the literature. In the second case, the environment was at rest except for the motion induced by the fountain. An interface formed at the base of the fountain and moved quickly to the top. Once there, it advanced slowly due to entrainment through the end of the fountain and the length of the fountain increased. The final case is a co-flowing environment. No interface formed if the environment velocity was greater than the advance velocity of the end of the fountain. However, one formed for a smaller environment velocity and the end of the fountain was observed to undergo a quasi-periodic jump phenomenon. The top of the fountain would advance with the environment particles for a short time and then snap back to the elevation of a fountain in an infinite environment. A new interface formed and the cycle repeated. © 1993, Cambridge University Press. All rights reserved.

Description: In this investigation we explore the effect of unsteady vortical structures on the adiabatic wall temperature distribution in an impinging jet. Treating first the simpler case of a free jet, we introduce a conceptual model for the separation of the total temperature, appealing to the dynamics of particle pathlines and vortex rings in the jet. The presence of a region of higher total temperature on the inside of the jet and a region of lower total temperature toward the jet periphery, predicted by the model, exhibits good agreement with the experimental data taken at high subsonic Mach number. The results from a numerical simulation further confirm the theoretical expectations. Through a similar argument, we show that when a thermally insulated flat plate is inserted into the jet, the wall temperature distribution is modified by the presence of secondary vortical structures, which are induced near, and swept over, the plate surface. When the plate is near the jet nozzle, a region of lower wall temperature, attributable to these additional vortices, is observed in the experimental data. When the plate is further from the nozzle, no secondary vortices are formed and no region of lowered wall temperature is measured. Self-sustaining acoustic resonance, when it occurs, is found to alter significantly this picture of the wall temperature distribution. Although the scope of this work is limited to free and impinging jets, this present topic, along with the previously reported mechanism of the Eckert-Weise effect, exemplifies the wider family of problems in which unsteady vortical structure strongly affects the wall temperature and heat transfer. © 1993, Cambridge University Press. All rights reserved.

Description: In certain geophysical contexts such as lava lakes and mantle convection, a cold, viscous boundary layer forms over a deep pool. The following model problem investigates the buoyant instability of the layer. Beneath a shear-free horizontal boundary, a thin layer (thickness d1) of very viscous fluid overlies a deep layer of less dense, much less viscous fluid; inertia and surface tension are negligible. After the initial unstable equilibrium is perturbed, a long-wave analysis describes the growth of the disturbance, including the nonlinear effects of large amplitude. The results show that nonlinear effects greatly enhance growth, so that initial local maxima in the thickness of the viscous film grow to infinite thickness in finite time, with a timescale 8μ/∆pgd1. In the final catastrophic growth the peak thickness is inversely proportional to the remaining time. (A parallel analysis for fluids with power-law rheology shows similar catastrophic growth.) While the small-slope approximation must fail before this singular time, the failure is only local, and a similarity solution describes how the peaks become downwelling plumes as the viscous film drains away. © 1993, Cambridge University Press. All rights reserved.

Description: The stability of the viscometric motion of a viscoelastic fluid held between rotating parallel disks with large radii to small-amplitude perturbations is studied for the Oldroyd-B constitutive model. The disturbances are assumed to be radially localized and are expressed in Fourier form so that a separable eigenvalue problem results; these disturbances describe either axisymmetric or spiral vortices, depending on whether the most dangerous disturbance has zero or non-zero azimuthal wavenumber, respectively. The critical value of the dimensionless radius R* for the onset of the instability is computed as a function of the Deborah number De, a dimensionless time constant of the fluid, the azimuthal and radial wavenumbers, and the ratio of the viscosities of the solvent to the polymer solution. Calculations meant to match the experiments of McKinley et al. (1991) for a Boger fluid show that the most dangerous instabilities are spiral vortices with positive and negative angle that start at the same critical radius and travel outward and inward toward the centre of the disk; the axisymmetric mode also becomes unstable at only slightly greater values of R*, or De for fixed R*. The predicted dependence of the value of De for a fixed R* on the gap between the disks agrees quantitatively with the measurements of McKinley et al., when the longest relaxation time for the fluid at the shear rate corresponding to the maximum value of R* is used to define the time constant in the Oldroyd-B model. © 1993, Cambridge University Press. All rights reserved.

Description: A two-dimensional hot-air jet is investigated experimentally in the transitional regime. The density effect on the near-field behaviour of the initially laminar jet is explored by flow visualization, mean flow measurements and spectral analysis of fluctuating data. It is shown that the broadband amplitude spectra which characterize cold jets become line-dominated for hot jets when the ratio of the jet-exit to the ambient density is below approximately 0.9. Below this critical density ratio the oscillations in the hot jet are shown to be self-excited. That is, the onset of the global oscillations is identified as a Hopf bifurcation and the critical parameter is determined from amplitude spectra and autobicoherence, with the latter proving to be more reliable. Furthermore, the development of three-dimensional structures, which contribute to the jet spreading, is revealed by flow visualization. It is found that, for the parameters investigated, the spreading of the two-dimensional hot jet is not as spectacular as in the axisymmetry case. © 1993, Cambridge University Press. All rights reserved.

Description: A stability analysis is undertaken to theoretically study the effects of gravity modulation and cross-diffusion on the onset of convection in horizontally unbounded doubly diffusive fluid layers. We investigate the stability of doubly stratified incompressible Boussinesq fluid layers with stress-free and rigid boundaries when the stratification is either imposed or induced by Soret separation. The stability criteria are established by way of Floquet multipliers of the amplitude equations. The topology of neutral curves and stability boundaries exhibits features not found in modulated singly diffusive or unmodulated multiply diffusive fluid layers. A striking feature in gravity-modulated doubly cross-diffusive layers is the existence of bifurcating neutral curves with double minima, one of which corresponds to a quasi-periodic asymptotically stable branch and the other to a subharmonic neutral solution. As a consequence, a temporally and spatially quasi-periodic bifurcation from the basic state is possible, in which case there are two incommensurate critical wavenumbers at two incommensurate onset frequencies at the same Rayleigh number. In some instances, the minimum of the subharmonic branch is more sensitive to small parameter variations than that of the quasi-periodic branch, thus affecting the stability criteria in a way that differs substantially from that of unmodulated layers. © 1993, Cambridge University Press. All rights reserved.

Description: This paper complements the instability theory of frontal waves investigated by Orlanski (1968), and reinterprets the unstable modes obtained. First, the stability of a frontal model is reconsidered by using a matrix method. The major part of Orlanski's (1968) result is verified but some flaws are found in some parameter regions: unstable modes do not exist over the entire Ri-Ro region. Also, the features of the neutral waves in the one-layer subsystems are studied, in order to determine the instability of the full two-layer system. As a result, the unstable mode called the (B)-mode by Orlanski (1968) and suggested by Sakai (1989) to be Rossby-Kelvin instability caused by a resonance between a Rossby wave and a gravity wave, proves to be a geostrophic unstable mode caused by resonance between a Rossby wave and the Rossby-gravity mixed mode. In addition, some of the analytical conclusions about the stability of this frontal model are explained by the features of the neutral waves in the one-layer subsystem. © 1993, Cambridge University Press. All rights reserved.

Description: The flow of a periodic suspension of two-dimensional viscous drops in a closed channel that is bounded by two parallel plane walls executing relative motion is studied numerically using the method of interfacial dynamics. Ordered suspensions where at the initial instant the drops are arranged in several layers on a hexagonal lattice are considered for a variety of physical conditions and geometrical configurations. It is found that there exists a critical capillary number below which the suspensions exhibit stable periodic motion, and above which the drops elongate and tend to coalesce, altering the topology of the initial configuration. At sufficiently large volume fractions, a minimum drop capillary number exists below which periodic motion is suppressed owing to the inability of the drops to deform and bypass other neighbouring drops in adjacent layers. This feature distinguishes the motion of dense emulsions from that of foam. The effects of capillary number, viscosity ratio, volume fraction of the dispersed phase, lattice geometry, and instantaneous drop shape, on the effective stress tensor of the suspension are illustrated and the results are discussed with reference to theories of foam. Two simulations of a random suspension with 12 drops per periodic cell are performed, and the salient features of the motion are identified and discussed. These include pairing, tripling, and higher-order interactions among intercepting drops, cluster formation and destruction, and drop migrations away from the walls. The macroscopic features of the flow of random suspensions are shown to be significantly different from those of ordered suspensions and quite independent of the initial condition. The general behaviour of suspensions of liquid drops is compared to that of suspensions of rigid spherical particles, and some differences are discussed. © 1993, Cambridge University Press. All rights reserved.

Description: Asymptotic expressions for the onset of convection in a horizontal fluid layer of finite extent heated from below and rotating about a vertical axis are derived in the limit of large rotation rates in the case of stress-free upper and lower boundaries. In the presence of vertical sidewalls, the critical Rayleigh number Rc is much lower than the classical value for an infinitely extended layer. In particular, we find that Rc grows in proportion to τ when the sidewall is insulating, where τ is the dimensionless rotation rate. When the sidewall is infinitely conducting, Rcgrows in proportion to τ4/3as in the case of an infinitely extended layer but with a lower coefficient of proportionality. Numerical results obtained at finite values of τ show good agreement with the asymptotic formulae. © 1993, Cambridge University Press. All rights reserved.

Description: An experimental investigation of forced and free oscillations of liquid bridges positioned between two rods of equal diameter is presented. Both the resonance frequencies and damping rates for different aspect ratios of the bridge are reported. The damping rate data of the liquid bridges are obtained by high-speed videography and are the first ever reported. Resonance frequencies for the three modified Reynolds numbers of 14, 295 and 1654, and damping rates for the two modified Reynolds numbers of 14 and 295 are reported. These values of modified Reynolds numbers are generated by using ethylene glycol, distilled water, and mercury in small bridges. Gravitational effects are kept small by reducing the size of the capillary bridge. The internal flow fields of several bridges for different modified Reynolds numbers are described based on high-speed visualization. Experimental results show good agreement with results of linear and nonlinear theory. © 1993, Cambridge University Press. All rights reserved.

Description: Motivated by the observed potential vorticity structure of the stratospheric polar vortex, we study the dynamics of linear and nonlinear waves on a zonal vorticity interface in a two-dimensional barotropic flow on the surface of a sphere (interfacial Rossby waves). After reviewing the linear problem, we determine, with the help of an iterative scheme, the shapes of steadily propagating nonlinear waves; a stability analysis reveals that they are (nonlinearly) stable up to very large amplitude. We also consider multi-vortex equilibria on a sphere: we extend the results of Thompson (1883) and show that a (latitudinal) ring of point vortices is more unstable on the sphere than in the plane; notably, no more than three point vortices on the equator can be stable. We also determine the shapes of finite-area multi-vortex equilibria, and reveal additional modes of instability feeding off shape deformations which ultimately result in the complex merger of some or all of the vortices. We discuss two specific applications to geophysical flows: for conditions similar to those of the wintertime terrestrial stratosphere, we show that perturbations to a polar vortex with azimuthal wavenumber 3 are close to being stationary, and hence are likely to be resonant with the tropospheric wave forcing; this is often observed in high-resolution numerical simulations as well as in the ozone data. Secondly, we show that the linear dispersion relation for interfacial Rossby waves yields a good fit to the phase velocity of the waves observed on Saturn's ‘ribbon’. © 1993, Cambridge University Press. All rights reserved.

Description: We present experimental results for the wake structure of spheres moving in homogeneous and stratified fluid. In homogeneous fluid, the results of Kim & Durbin (1988) are confirmed and it is shown that the two characteristic frequencies of the wake correspond to two instability modes, the Kelvin-Helmholtz instability and a spiral instability. For the stratified wake four general regimes have been identified, depending principally on the Froude number F. For F 〉 4.5 the near wake is similar to the homogeneous case, and for F 〈 0.8 it corresponds to a triple-layer flow with two lee waves, of amplitude linear in F, surrounding a layer dominated by quasi-two-dimensional motion. Froude numbers close to one (F∈]0.8, 1.5[) give rise to a saturated lee wave of amplitude equal to half the sphere radius, which suppresses the separation region or splits it into two. Between F = 1.5 and 4.5 a more complex regime exists where the wake recovers progressively its behaviour in homogeneous fluid: the axisymmetry of the recirculating zone, the Kelvin-Helmholtz instability and, finally, the spiral instability. © 1993, Cambridge University Press. All rights reserved.

Description: Direct numerical simulations of turbulent flows over riblet-mounted surfaces are performed to educe the mechanism of drag reduction by riblets. The computed drag on the riblet surfaces is in good agreement with the existing experimental data. The mean-velocity profiles show upward and downward shifts in the log—law for drag-decreasing and drag-increasing cases, respectively. Turbulence statistics above the riblets are computed and compared with those above a flat plate. Differences in the mean-velocity profile and turbulence quantities are found to be limited to the inner region of the boundary layer. Velocity and vorticity fluctuations as well as the Reynolds shear stresses above the riblets are reduced in drag-reducing configurations. Quadrant analysis indicates that riblets mitigate the positive Reynolds-shear-stress-producing events in drag-reducing configurations. From examination of the instantaneous flow fields, a drag reduction mechanism by riblets is proposed: riblets with small spacings reduce viscous drag by restricting the location of the streamwise vortices above the wetted surface such that only a limited area of the riblets is exposed to the downwash of high-speed fluid that the vortices induce. © 1993, Cambridge University Press. All rights reserved.

Description: The role of acoustics in flame/vortex interactions is examined via asymptotic analysis and numerical simulation. The model consists of a one-step, irreversible Arrhenius reaction between initially unmixed species occupying adjacent half-planes which are allowed to mix and react by convection and diffusion in the presence of an acoustic field or a time-varying pressure field of small amplitude. The main emphasis is on the influence of the acoustics on the ignition time and flame structure as a function of vortex Reynolds number and initial temperature differences of the reactants. © 1993, Cambridge University Press. All rights reserved.

Description: This work is an attempt to explain observations of vortices in experiments with shallow water in rotating paraboloidal vessels. The most long-lived vortices are invariably anticyclones, while cyclones quickly disperse, and they are larger than the Rossby radius. These experiments are designed to simulate geophysical flows, where large, long-lived, anticyclonic vortices are common. The general condition for vortices to be steady is that they propagate faster than linear Rossby waves, so that the vortex energy is not dispersed by coupling to linear waves. The propagation velocity is determined by a general integral relation that gives the velocity of the centre of mass. In geophysical flows, to lowest order in the Rossby number, the difference between the centre-of-mass velocity and the maximum phase velocity of the Rossby waves is proportional to the relative perturbation of the fluid depth. Since for anticyclones the difference is positive they may be steady, whereas cyclones cannot be. In the laboratory experiments this velocity difference is absent because of the latitudinal dependence of the effective gravity caused by the centrifugal force. However, to the next order in the Rossby number, there is another nonlinear contribution, so that anticyclones (but not cyclones) still propagate faster than the linear Rossby waves, and may thus be steady. The velocity difference is smaller than for geophysical flows, and vanishes in the limit of small Rossby number. The existence conditions also show that we can expect the experimental vortices to be smaller (as measured by the Rossby radius) than the planetary vortices. The theory does not apply to vortices that are much smaller than the Rossby radius. © 1993, Cambridge University Press. All rights reserved.

Description: A substantial amount of drilling fluid can invade a permeable bed during the drilling of an oil well. The presence of this fluid, often referred to as filtrate, can greatly influence the performance of instruments lowered into the wellbore for the purpose of locating these permeable beds. The invaded filtrate can also substantially alter the physical properties of the porous rock. For these reasons, it is of great interest to known where the filtrate goes upon entering the bed. The objective of this study is to quantify the influence of the difference in density between the filtrate and the naturally occurring formation fluid on the shape of the filtrate front as the filtrate invades the formation. This type of phenomenon is often referred to as buoyancy or gravity segregation. In this study, Part 1, we determine the behaviour of the filtrate as it accumulates (and spreads out) at a horizontal impermeable barrier within the formation. This is a combined theoretical and experimental study in which an X-ray CT scanner is extensively used to determine the appropriateness and limitations of the simplifying assumptions used in the theory. In Part 2, the flow of the invading filtrate within the entire bed will be presented. The problem addressed in Part 1 may be viewed from the broader, more fundamental, perspective, as a well-defined model fluid mechanics problem for flow in porous media. One fundamental issue infrequently addressed concerns the consequence on the dynamics of the fluids of heterogeneities, always present to some degree, in consolidated porous solids. The X-ray CT scanner enables the assessment of the appropriateness of modelling such porous solids as spatially homogeneous, a very popular assumption. This study also addresses the limitation of the small-slope approximation when a fluid-fluid interface occurs in a porous solid, an approximation which has enjoyed great success in free-surface fluid mechanics problems when no porous media is present. © 1993, Cambridge University Press. All rights reserved.

Description: The results of an investigation on the effect of a weak heterogeneity of a porous medium on natural convection are presented. A medium heterogeneity is represented by spatial variations of the permeability and of the effective thermal conductivity. As a general rule the existence of horizontal thermal gradients in heterogeneous porous media provides a sufficient condition for the occurrence of natural convection. The implications of this condition are investigated for horizontal layers or rectangular domains subject to isothermal top and bottom boundary conditions. Results lead to a restriction on the classes of thermal conductivity functions which allow a motionless solution. Analytical solutions for rectangular weak heterogeneous porous domains heated from below, consistent with a basic motionless solution, are obtained by applying the weak nonlinear theory. The amplitude of the convection is obtained from an ordinary non-homogeneous differential equation, with a forcing term representative of the medium heterogeneity with respect to the effective thermal conductivity. A smooth transition through the critical Rayleigh number is obtained, thus removing a bifurcation which usually appears in homogeneous domains with perfect boundaries, at the critical value of the Rayleigh number. Within a certain range of slightly supercritical Rayleigh numbers, a symmetric thermal conductivity function is shown to reinforce a symmetrical flow while antisymmetric functions favour an antisymmetric flow. Except for the higher-order solutions, the weak heterogeneity with respect to permeability plays a relatively passive role and does not affect the solutions at the leading order. In contrast, the weak heterogeneity with respect to the effective thermal conductivity does have a significant effect on the resulting flow pattern. © 1993, Cambridge University Press. All rights reserved.

Description: We consider a floating or submerged body in deep water translating parallel to the undisturbed free surface with a steady velocity U while undergoing small oscillations at frequency a〉. It is known that for a single source, the solution becomes singular at the resonant frequency given by t == Uco/g=¼, where g is the gravitational acceleration. In this paper, we show that for a general body, a finite solution exists as T → ¼ if and only if a certain geometric condition (which depends only on the frequency co but not on U) is satisfied. For a submerged body, a necessary and sufficient condition is that the body must have non-zero volume. For a surface-piercing body, a sufficient condition is derived which has a geometric interpretation similar to that of John (1950). As an illustration, we provide an analytic (closed-form) solution for the case of a submerged circular cylinder oscillating near t = ¼. Finally, we identify the underlying difficulties of existing approximate theories and numerical computations near t = j, and offer a simple remedy for the latter. © 1993, Cambridge University Press. All rights reserved.

Description: Turbulent vortices in the wake of a circular cylinder have been examined in some detail following their detection by a method based on vorticity concentration and circulation. The experimental data were obtained at a Reynolds number Re = 5600 (based on the free-stream velocity and cylinder diameter) with an array of eight X-wires aligned in the plane of mean shear. Distributions of conditionally averaged vorticity and circumferential velocity within vortices are in reasonable agreement with those inferred from an Oseen vortex. The conditionally averaged streamwise velocity distribution through the vortex centre has a maximum at the centre, implying a vortex convection velocity greater than the local mean velocity. An Oseen vortex model reproduces the measured mean velocity and Reynolds stresses reasonably well. Results are also given for the streamwise variations of vorticity concentration, circulation and size of the vortices. © 1993, Cambridge University Press. All rights reserved.

Description: The two-dimensional wave instability of convection induced by a semi-infinite heated surface embedded in a fluid-saturated porous medium is studied. Owing to the inadequacy of parallel-flow theories and the inaccuracy of the leading-order boundary-layer approximation at the point of incipient instability given by these theories, the problem has been re-examined using numerical simulations of the full time-dependent nonlinear equations of motion. Small-amplitude localized disturbances placed in the steady boundary layer are shown to propagate upstream much faster than they advect downstream. There seems to be a preferred wavelength for the evolving disturbance while it is in the linear regime, but the local growth rate depends on the distance downstream and there is a smooth, rather than an abrupt, spatial transition to convection. The starting problem, where the temperature of the horizontal surface is instantaneously raised from the ambient, is found to give rise to a particularly violent fluid motion near the leading edge. A strong thermal plume is generated which is eventually advected downstream. The long-term evolution of the instability is computed. The flow does not settle down to a steady or a time-periodic state, and evidence is presented which suggests that it is inherently chaotic. The evolving flow field exhibits a wide range of dynamical behaviour including cell merging, the ejection of hot fluid from the boundary layer, and short periods of relatively intense fluid motion accompanied by boundary-layer thinning and short-wavelength waves. © 1993, Cambridge University Press. All rights reserved.

Description: A phenomenological study of the processes occurring when a shock wave interacts with porous polyester and polyether foams has been undertaken. Plane shock waves generated in a shock tube were reflected off a slab of foam mounted against the back wall of the tube. Tests were conducted with an initial shock wave Mach number of 1.4 and a 70 mm thick slab of foam. The reduction in reflected shock wave strength and substantial increase in the back wall pressure over that for rigid wall reflection, found by other workers, were confirmed. Piezoelectric pressure transducers were used to record the pressure before, alongside and behind the foam specimen. Schlieren photographs of the flow were made and showed some features not previously reported. In particular it is shown that there is a flow of gas across the face of the foam at some point of the process. Previous investigations of this interaction process have assumed that the face of the foam is a contact surface. Short duration photographs of the distortion of the foam were taken, enabling the wave propagation in the foam material itself to be studied. It is established that the front of this compaction wave in the foam material moves at considerably lower velocity (∼ 90 m/s) than the gas wave as detected by the pressure transducers (∼ 200 m/s). This result contrasts with the assumption made in previous work that the two-phase medium behaves essentially as a homogeneous substance. A simple physical model based on a zone of compacted material in the foam acting as a high-resistance flow barrier, is proposed to explain the observed phenomena. © 1993, Cambridge University Press. All rights reserved.

Description: The leading-order fluid motions and frequencies in resonance tubes coupled to a combustion-driven flow source, such as occurs in various types of pulse combustors, are usually strongly related to those predicted by linear acoustics. However, in order to determine the amplitudes of the infinite number of classical acoustic modes predicted by linear theory alone, and hence the complete solution, a nonlinear analysis is required. In the present work, we adopt a formal perturbation approach based on the smallness of the mean-flow Mach number which, as a consequence of solvability conditions at higher orders in the analysis, results in an infinitely coupled system of nonlinear evolution equations for the amplitudes of the linear acoustic modes. An analysis of these amplitude equations then shows that the combination of driving processes, such as combustion, that supply energy to the acoustic oscillations and those, such as viscous effects, that dampen such motions, in conjunction with the manner in which the resonance tube is coupled to its flow source, provides an effective mode-selection mechanism that inhibits the (linear) growth of all but a few of the lower-frequency modes. For the common case of long resonance tubes, the lowest frequencies correspond to purely longitudinal modes, and we analyse in detail the solution behaviour for a typical situation in which only the first of these has a positive linear growth rate. Basic truncation strategies for the infinitely coupled amplitudes are discussed, and we demonstrate, based on analyses with both two and three modes, the stable bifurcation of an acoustic oscillation, or limit cycle, at a critical value of an appropriate bifurcation parameter. In addition, we show that the bifurcated solution branch has a turning point at a second critical value of the bifurcation parameter beyond which no stable bounded solutions exist. © 1993, Cambridge University Press. All rights reserved.

Description: The generation of a standing wave of vortices in thin channels has been experimentally observed and discussed in the literature for the last several years. The specific cause of the wave and its response to various conditions remains largely unexplored. In this paper we model pulsatile flow through thin channels with inserted deflectors to generate the vortex wave, and we examine various measures to quantify its effects. We focus on the numerical solution of the transient vortex wave phenomenon and its response to a superimposed bulk flow, variations of pulsation, deflector spacing and shape as well as transverse suction. The quantifying measures are mapped over a Reynolds’ number-Strouhal number domain. © 1993, Cambridge University Press. All rights reserved.

Description: A two-layer planetary geostrophic model is adopted to study the breaking of planetary waves in the presence of Ekman pumping and the associated-mean flow. The governing equation for the interface is a quasi-linear equation, which is solved analytically by the method of characteristics. The waves are forced by annual or interannual upwelling or downwelling along the eastern boundary of a subtropical gyre. It is found that the time and position at which breaking occurs is mainly determined by the speed and depth of the eastern boundary perturbation, while the intensity of a breaking front is mainly determined by the amplitude of the perturbation. The breaking of a planetary wave is affected significantly by Ekman pumping and the associated mean flow, particularly for annual and interannual forcing. Downward Ekman pumping, as in a subtropical gyre, suppresses breaking in downwelling waves caused by eastern boundary upwelling, but enhances breaking in upwelling waves caused by eastern boundary downwelling. In the presence of steady downward Ekman pumping, downwelling breaking will not occur except for interfaces near the surface. The structure and intensity of a breaking front is also discussed. © 1993, Cambridge University Press. All rights reserved.

Description: An exact analytic solution of the shallow water equations for the motion of a bore over a uniformly sloping beach is derived. This solution is valid only when the initial bore is supercritical. The results agree very well with those of Keller, Levine Whitham (1960). © 1993, Cambridge University Press. All rights reserved.

Description: An experimental investigation was designed to test the hypothesis that all axisymmetric turbulent free jets become asymptotically independent of the source conditions and may be described by classical similarity analysis. Effects of initial conditions were studied by varying jet exit boundary conditions and the global density ratio. The exit velocity profile and turbulence level was changed by using both pipe and nozzle flow hardware. Initial density differences were imposed by using three gases: helium, methane, and propane. The scalar field (concentration) in the momentum-dominated regime of the far field (10 to 60 jet exit diameters downstream) of turbulent free jets was characterized using Rayleigh light scattering as the diagnostic. The results show that regardless of the initial conditions axisymmetric turbulent free jets decay at the same rate, spread at the same angle, and both the mean and r.m.s. values collapse in a form consistent with full self-preservation. The means and fluctuations follow a law of full self-preservation in which two virtual origins must be specified. The two displacements are required to account for the effects of a finite source of momentum and different development of the velocity and mass distributions in the near fields of the jets. The memory of the jet is embodied in these two virtual origins. © 1993, Cambridge University Press. All rights reserved.

Description: In the present work, we apply new tools from the field of adiabatic dynamical systems theory to make quantitative predictions of various important mixing quantities in quasi-steady Stokes flows which possess slowly varying saddle stagnation points. Many of these quantities can be obtained before experiments or numerical simulations are performed using only knowledge of the streamlines in steady-state flows and the externally determined flow parameters. The location and size of the main region in which mixing occurs is determined to leading order by the slowly sweeping instantaneous stagnation streamlines. Tracer patches get stretched by an amount 0(1/) during each period, and the average striation thickness of the patch decreases by a factor of e in the same time. Also, the rate of stretching of material interfaces is bounded from below with an analytically obtained exponentially growing lower bound. Finally, the highly regular appearance of islands in quasi-steady Stokes’ flows is explained using an extension of the KAM theory. As an example to illustrate these results, we study the transport of passive scalars in a low Reynolds number flow in the two-dimensional eccentric journal bearing when the angular velocities of the cylinders are slowly modulated, continuously and periodically in time, with frequency. In contrast to the flows usually studied with dynamical systems, these slowly varying systems are singular perturbation (apparently far from integrable) problems exhibiting large-scale chaos, in which the non-integrability is due to the slow, continuous 0(1) modulation of the position of the saddle stagnation point and the two streamlines stagnating on it. © 1993, Cambridge University Press. All rights reserved.

Description: This study involves a numerical simulation of spatially evolving secondary instability in plane channel flow. The computational algorithm integrates the time-dependent, three-dimensional, incompressible Navier-Stokes equations by a mixed finite-difference/spectral technique. In particular, we are interested in the differences between instabilities instigated by Klebanoff (K-) type and Herbert (H-) type inflow conditions, and in comparing the present spatial results with previous temporal models. It is found that for the present inflow conditions, H-type instability is biased towards one of the channel walls, while K-type instability evolves on both walls. For low initial perturbation amplitudes, H-type instability exhibits higher growth rates than K-type instability while higher initial amplitudes lead to comparable growth rates of both Hand K-type instability. In H-type instability, spectral analysis reveals the presence of the subharmonic two-dimensional mode which promotes the growth of the three-dimensional spanwise and fundamental modes through nonlinear interactions. An intermodal energy transfer study demonstrates that there is a net energy transfer from the three-dimensional modes to the two-dimensional mode. This analysis also indicates that the mean mode transfers net energy to the two-dimensional subharmonic mode and to the three-dimensional modes. © 1993, Cambridge University Press. All rights reserved.

Description: Stokesian dynamics is used to simulate the dynamics of a monolayer of a suspension of bimodally distributed spherical particles subjected to simple shearing flow. Hydrodynamic forces only are considered. Many-body far-held effects are calculated using the inverse of the grand mobility matrix. Near-held effects are calculated from the exact equations for the interaction between two unequal-sized spheres. Both the detailed microstructure (e.g. pair-distribution function and cluster formation) and the relative viscosity are determined for bimodal suspensions having particle size ratios of 2 and 4. The flow of an 4 infinite’ suspension is simulated by considering 25, 49, 64, and 100 particles to be 'one’ cell of an inhnite periodic array. The effects of both the size ratio and the relative fractions of the different-sized particles are examined. When the area fraction, Oa, is less than 0.4 the particle size distribution does not affect the calculated viscosity. For (fa 0.4, and for a hxed fraction of small spheres, the bimodal suspensions generally have lower viscosities than monodispersed suspensions, with the size of this effect increasing with fa. These results compare favourably with experiment when (j)aand the volume fraction, fv, are normalized by the maximum packing values in two and three dimensions, respectively. At the microstructural level, viscosity reduction is related to the influence of particle size distribution on the average number of particles in clusters. At a fixed area fraction, the presence of smaller particles tends to reduce average cluster size, particularly at larger fa9where significant viscosity reductions are observed. Since the presence of large clusters in monodispersed suspensions has been directly linked to high viscosities, this provides a dynamic mechanism for the viscosity reduction in bimodal suspensions. © 1993, Cambridge University Press. All rights reserved.

Description: The temporal development of two-dimensional viscous incompressible flow generated by a circular cylinder impulsively started into steady rotatory and rectilinear motion at Re = 200 (based on the cylinder diameter 2a and the magnitude U of the rectilinear velocity) is studied computationally. We use an explicit finite-difference/pseudospectral technique and a new implementation of the Biot-Savart law to integrate a velocity/vorticity formulation of the Navier-Stokes equations. Results are presented for the four angular: rectilinear speed ratios α = Ωa/U (where Ω is the angular speed) considered experimentally by Coutanceau & Ménard (1985). For α ≤ 1, extension of the computations to dimensionless times larger than achieved either in the experimental work or in the computations of Badr & Dennis (1985) allows for a more complete discussion of the temporal development of the wake. Using the frame-invariant vorticity distribution, we discuss several aspects of the vortex kinematics and dynamics not revealed by the earlier work, in which vortex cores were identified from frame-dependent streamline and streamfunction information. Consideration of the flow in the absence of sidewalls confirms the artifactual nature of the trajectory of the first vortex reported by Coutanceau & Ménard for α = 3.25. For α greater than unity (the largest value considered by Badr & Dennis), our results indicate that at Re = 200 shedding of more than one vortex does indeed occur for α = 3.25 (and possibly for larger α), in contrast to the conclusion of Coutanceau & Ménard. Moreover, the shedding process is very different from that associated with the usual Kármán vortex street for α = 0. Specifically, consecutive vortices can be shed from one side of the cylinder and be of the same sense, in contrast to the non-rotating case, in which mirror-image vortices of opposite sense are shed alternately from opposite sides of the cylinder. The results are discussed in relation to the possibility of suppressing vortex shedding by open- or closed-loop control of the rotation rate. © 1993, Cambridge University Press. All rights reserved.

Description: The process of flux rope formation in a convecting cell is studied. The magnetic field has both a meridional and an azimuthal component, and so corresponds to a twisted field. Convection occurs in this cylindrical cell because of heating from below, and is assumed to take an axisymmetric form. Only the Boussinesq problem is studied here, but both the kinematic and the dynamic regimes are considered. The two cases where the twisted field is due to (a) an imposed flux of vertical current and (b) an imposed flux of vertical vorticity are considered. Strongly twisted ropes can be generated more easily in case (b) than in case (a). We show that convection can produce ropes twisted in the opposite direction from that of the initial field. We also find that solutions can be oscillatory even when linear theory predicts steady solutions. © 1993, Cambridge University Press. All rights reserved.

Description: The existence of steady solitary waves on deep water was suggested on physical grounds by Longuet-Higgins (1988) and later confirmed by numerical computation (Longuet-Higgins 1989; Vanden-Broeck & Dias 1992). Their numerical methods are accurate only for waves of finite amplitude. In this paper we show that solitary capillary-gravity waves of small amplitude are in fact a special case of envelope solitons, namely those having a carrier wave of length 2n{T/pg) (g = gravity, T = surface tension, p = density). The dispersion relation c2= 2(1 — Mamax) between the speed c and the maximum surface slope amaxis derived from the nonlinear Schrodinger equation for deep-water solitons (Djordjevik & Redekopp 1977) and is found to provide a good asymptote for the numerical calculations. © 1993, Cambridge University Press. All rights reserved.

Description: In recent work it has been shown that there can be substantial transient growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds number is below the critical value predicted by linear stability analysis. This growth, which may be as large as 0(1000), occurs in the absence of nonlinear effects and can be explained by the non-normality of the governing linear operator - that is, the nonorthogonality of the associated eigenfunctions. In this paper we study various aspects of this energy growth for two- and three-dimensional Poiseuille and Couette flows using energy methods, linear stability analysis, and a direct numerical procedure for computing the transient growth. We examine conditions for no energy growth, the dependence of the growth on the streamwise and spanwise wavenumbers, the time dependence of the growth, and the effects of degenerate eigenvalues. We show that the maximum transient growth behaves like 0(R2), where R is the Reynolds number. We derive conditions for no energy growth by applying the Hille-Yosida theorem to the governing linear operator and show that these conditions yield the same results as those derived by energy methods, which can be applied to perturbations of arbitrary amplitude. These results emphasize the fact that subcritical transition can occur for Poiseuille and Couette flows because the governing linear operator is non-normal. © 1993, Cambridge University Press

Description: The dynamics of an asymmetric annulus of vorticity in an incompressible, inviscid two-dimensional fluid are experimentally studied using a pure electron plasma. A strict fluid analogy requires that the plasma system behave like an ideal fluid in a frictionless cylindrical container. For certain parameters the asymmetric annulus undergoes a complex evolution which is quite different from that of a symmetric annulus. During the first ‘active’, phase the symmetries grow until the annulus collapses, leaving a large vortex at the device centre. In the second, ‘passive’, phase the remainder of the annulus winds around this central vortex into an ever tighter spiral. Finally, slow shear instabilities destroy the structure of the highly evolved spiral. © 1993, Cambridge University Press. All rights reserved.

Description: The fluid flow induced by a cascade of circular cylinders which oscillates harmonically in an unbounded, incompressible, viscous fluid which is otherwise at rest is investigated both numerically and experimentally. Attention in this paper is mainly concentrated on the induced steady streaming flow which occurs when the ratio of the amplitude of the oscillation of the cascade to the size of the cylinder, e, is very small. The leading-order flow is then governed by the steady Navier-Stokes equations. In order to solve these equations numerically we first generate numerically a grid system using the boundary element method and then use a finite-difference scheme on the newly generated rectangular grid system. Numerical results show that for small values of the streaming Reynolds number Rs there are four recirculating flows of equal strength around each circular cylinder of the cascade. At large values of Rs symmetry breaks down and numerical solutions are found for asymmetrical flows. Numerically, a critical value of Rs, Rs0say, is identified such that the flow is symmetrical when Rs 〈 Rs0and asymmetrical when Rs 〉 Rs0and these results are in reasonable agreement with experimental results, which are also presented in this paper. © 1993, Cambridge University Press

Description: The development of disturbances (two-dimensional non-linear and three-dimensional linear) in the entrance region of a circular pipe is studied in the limit of Reynolds number R → ∞ in the framework of triple-deck theory. It is found that lower-branch axisymmetric disturbances can interact in the resonant manner. Numerical calculations show that a two-dimensional nonlinear wave packet grows much more rapidly than that in the boundary layer on a flat plate, producing a spike-like solution which seems to become singular at a finite time. Large-sized, short-scaled disturbances are also studied. In this case the development of axisymmetric disturbances is governed by single one-dimensional equation in the form of the Korteweg-de Vries and Benjamin-Ono equations in the long- and short-wave limits respectively. The nonlinear interactions of these disturbances lead to the formation of solitons which can run both upstream and downstream. Linear three-dimensional wave packets are also calculated. © 1993, Cambridge University Press. All rights reserved.

Description: An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp = 1/CpO + k/ωmax + ∈(k) where cp is the wave phase speed for a particular mode, CpO is the phase speed at k = 0, ωmax is the maximum possible wave angular frequency and ωmax ≤ Nmax where Nmax is the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) = o(k) for k large, and is bounded for finite k. In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp ≈ (∫∞0N2(y)ydy)-½ + k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functions N2(x) which is taken to be a class of realvalued functions of a real variable x where O ≤ x ∞ such that N2(x) = O(e-βx) as x → ∞ and 1/β is an arbitrary length scale. It is demonstrated that N2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax = Nmax. More generally, it is demonstrated that when k → ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that if N2(x) is O(e-βx) as x → ∞ and has a maximum value N2max then a sufficient condition for 1/cp ∼ k/Nmax to hold for large k for the lowest mode is that N2(t)/t is convex for O ≤ t ≤ 1 where t = e-βx.

Description: Interaction of isotropic quasi-incompressible turbulence with a weak shock wave was studied by direct numerical simulations. The effects of the fluctuation Mach number Mtof the upstream turbulence and the shock strength M— 1 on the turbulence statistics were investigated. The ranges investigated were 0.0567 〈 Mt 0.110 and 1.05 Mx 〈 1.20. A linear analysis of the interaction of isotropic turbulence with a normal shock wave was adopted for comparisons with the simulations. Both numerical simulations and the linear analysis of the interaction show that turbulence is enhanced during the interaction with a shock wave. Turbulent kinetic energy and transverse vorticity components are amplified, and turbulent lengthscales are decreased. The predictions of the linear analysis compare favourably with simulation results for flows with M 〈 a(M — 1) with a « 0.1, which suggests that the amplification mechanism is primarily linear. Simulations also showed a rapid evolution of turbulent kinetic energy just downstream of the shock, a behaviour not reproduced by the linear analysis. Investigation of the budget of the turbulent kinetic energy transport equation shows that this behaviour can be attributed to the pressure transport term. Shock waves were found to be distorted by the upstream turbulence, but still had a well-defined shock front for Mt2 〈 a(M— 1) with a « 0.1). In this regime, the statistics of shock front distortions compare favourably with the linear analysis predictions. For flows with Ml 〉 a(M— 1) with a«0.1, shock waves no longer had well-defined fronts: Shock wave thickness and strength varied widely along the transverse directions. Multiple compression peaks were found along the mean streamlines at locations where the local shock thickness had increased significantly. © 1993, Cambridge University Press

Description: Steady axially invariant (fully developed) incompressible laminar flow of a Newtonian fluid in helical pipes of constant circular cross-section with arbitrary pitch and arbitrary radius of coil is studied. A loose-coiling analysis leads to two dominant parameters, namely Dean number, Dn = Reλ½, and Germano number, Gn = Reη, where Re is the Reynolds number, λ is the normalized curvature ratio and η is the normalized torsion. The Germano number is embedded in the body-centred azimuthal velocity which appears as a group in the governing equations. When studying Gn effects on the helical flow in terms of the secondary flow pattern or the secondary flow structure viewed in the generic (non-orthogonal) coordinate system of large Dn, a third dimensionless group emerges, γ = η/(λDn)½. For Dn 〈 20, the group γ* = Gn Dn-2 = η/(λRe) takes the place of γ. Numerical simulations with the full Navier-Stokes equations confirmed the theoretical findings. It is revealed that the effect of torsion on the helical flow can be neglected when γ ≤ 0.01 for moderate Dn. The critical value for which the secondary flow pattern changes from two vortices to one vortex is γ* 〉 0.039 for Dn 〈 20 and γ 〉 0.2 for Dn ≥ 20. For flows with fixed high Dean number and A, increasing the torsion has the effect of changing the relative position of the secondary flow vortices and the eventual formation of a flow having a Poiseuille-type axial velocity with a superimposed swirling flow. In the orthogonal coordinate system, however, the secondary flow generally has two vortices with sources and sinks. In the small-γ limit or when Dn is very small, the secondary flow is of the usual two-vortex type when viewed in the orthogonal coordinate system. In the large-γ limit, the appearance of the secondary flow in the orthogonal coordinate system is also two-vortex like but its orientation is inclined towards the upper wall. The flow friction factor is correlated to account for Dn, A and γ effects for Dn ≤ 5000 and γ 〈 0.1. © 1993, Cambridge University Press. All rights reserved.

Description: Upward flow of air can support a thin layer of liquid on a plane wall against gravity. Such apparently stationary layers are for example sometimes seen on the windscreen of a car travelling at high speed in rain. We solve here the two-dimensional case of a layer whose length is finite, but significantly greater than the meniscus length. The flow is steady, with a fixed layer boundary, inside which there is a steadily circulating viscous liquid, and outside which the air exerts a traction which is assumed to have a known (small) constant drag coefficient CD. The air also exerts a non-uniform pressure on the liquid layer, of a magnitude determined by the shape of the layer, and the relationship between these two quantities can be obtained by thin-airfoil theory. In the lubrication approximation, the problem can be reduced to a nonlinear singular integrodifferential equation to determine the unknown shape of the layer boundary. This equation is solved numerically for various (small) wall angles, for cases where the effect of surface tension is confined to a small meniscus region near the layer's leading edge. The numerical results indicate that solutions exist only for walls whose inclination is less than 0.70 C, and, for a range of inclinations below that maximum value, that two distinct steady solutions can exist at each inclination. © 1993, Cambridge University Press

Description: We present novel measurements of the primary instabilities of thin liquid films flowing down an incline. A fluorescence imaging method allows accurate measurements of film thickness h(x, y, t) in real time with a sensitivity of several microns, and laser beam deflection yields local measurements with a sensitivity of less than one micron. We locate the instability with good accuracy despite the fact that it occurs (asymptotically) at zero wavenumber, and determine the critical Reynolds number Rcfor the onset of waves as a function of angle /?. The measurements of Rc(fi) are found to be in good agreement with calculations, as are the growth rates and wave velocities. We show experimentally that the initial instability is convective and that the waves are noise-sustained. This means that the waveform and its amplitude are strongly affected by external noise at the source. We investigate the role of noise by varying the level of periodic external forcing. The nonlinear evolution of the waves depends strongly on the initial wavenumber (or the frequency f). A new phase boundary f(R) is measured, which separates the regimes of saturated finite amplitude waves (at high f) from multipeaked solitary waves (at low f). This boundary probably corresponds approximately to the sign reversal of the third Landau coefficient in weakly nonlinear theory. Finally, we show that periodic waves are unstable over a wide frequency band with respect to a convective subharmonic instability. This instability leads to disordered two-dimensional waves. © 1993, Cambridge University Press

Description: A microburst can be modelled by releasing a volume of fluid that is slightly heavier than the ambient fluid, allowing it to fall onto a horizontal surface. Vorticity develops on the sides of this parcel as it descends and causes it to roll up into a turbulent vortex ring which impinges on the ground. Such a model exhibits many of the features of naturally occurring microbursts which are a hazard to aviation. In this paper this model is achieved experimentally by releasing a volume of salt water into fresh water from a cylindrical dispenser. When care is taken with the release the spreading rate of the surface outflow is measurable and quite repeatable despite the fact that the flow is turbulent. An elementary numerical approximation to this model, based on in viscid vortex dynamics, has also been developed. A scaling law is proposed which allows experiments with different fluid densities to be compared with each other and with the numerical results. More importantly the scaling law allows us to compare the model results with real microbursts. © 1992, Cambridge University Press. All rights reserved.

Description: By using asymptotic analysis, an eigensolution technique has been developed for predicting the flow of gas contained in a pie-shaped cylinder of finite length rotating rapidly about its verteX. This problem has application to aconventional cylindrical gas centrifuge with radial walls. Three different types of boundary layers eXist in the flow:Ekman layers on the top and bottom, buoyancy layers on the radial walls, and a cylindrical ‘pancake’ layer on the outer wall of the cylinder. A single siXth-order partial differential equation is obtained for the aXial velocity in the cylindrical layer, and the other layers provide matching conditions. The problem is formulated for no-slip and prescribed temperature conditions on the solid surfaces and for adiabatic no shear stress with zero pressure at the inner freesurface. Eigenvalues are computed for this problem and compared with those for the open cylinder, and solutions are presented for flows induced by mass throughput and by differential temperature conditions. © 1992, Cambridge University Press. All rights reserved.