ALBERT

Description: It is shown that realizability of second-moment turbulence closure models can be established by finding a Langevin equation for which they are exact. All closure models currently in use can be derived formally from the type of Langevin equation described herein. Under certain circumstances a coefficient in that formalism becomes imaginary. The regime in which models are realizable is, at least, that for which the coefficient is real. The present method does not imply unrealizable solutions when the coefficient is imaginary, but it does guarantee realizability when the coefficient is real; hence, this method provides sufficient, but not necessary, conditions for realizability. Illustrative computations of homogeneous shear flow are presented. It is explained how models can be modified to guarantee realizability in extreme non-equilibrium situations without altering their behaviour in the near-equilibrium regime for which they were formulated. © 1994, Cambridge University Press. All rights reserved.

Description: The flow in a breaking-wave crest is represented by a complex velocity potential on a Riemann surface, satisfying the Bernoulli condition on two free boundaries. The flow is assumed to be stationary in the reference frame which moves with the wave crest, and at large distances approximates Stokes corner flow in the main part of the fluid and a parabolic descending flow in the jet. The interaction of the jet with the rest of the fluid is neglected. The solution is obtained by means of a conformal transformation from a bounded, teardrop-shaped domain, using a Faber polynomial expansion. The Bernoulli condition is applied at a number of discrete points on the boundaries, and the resulting nonlinear equations for the expansion coefficients are solved iteratively. The resulting surface form is similar to that obtained by laboratory experiments and time-dependent numerical simulations of waves up to the point of breaking, with a stagnation point at the top of the crest, an overturning loop with major axis « 8g~? ¥ 5, and a maximum acceleration of « 5.4 g, where g is the gravitational acceleration and ¥ is the flux in the jet. © 1994, Cambridge University Press. All rights reserved.

Description: The equilibrium dynamics in a homogeneous forced-dissipative f-plane shallow-water system is investigated through numerical simulations. In addition to classical two-dimensional turbulence, inertio-gravity waves also exist in this system. The dynamics is examined by decomposing the full flow field into a dynamically balanced potential-vortical component and a residual ‘free’ component. Here the potential-vortical component is defined as part of the flow that satisfies the gradient-wind balance equation and that contains all the linear potential vorticity of the system. The residual component is found to behave very nearly as linear inertio-gravity waves. The forcing employed is a mass and momentum source balanced so that only the large-scale potential-vortical component modes are directly excited. The dissipation is provided by a linear relaxation applied to the large scales and by an eighth-order linear hyperdiffusion. The statistical properties of the potential-vortical component in the fully developed flow were found to be very similar to those of classical two-dimensional turbulence. In particular, the energy spectrum of the potential-vortical component at scales smaller than the forcing is close to the ~ k~3 expected for a purely two-dimensional system. Detailed analysis shows that the downscale enstrophy cascade into any wavenumber is dominated by very elongated triads involving interactions with large scales. Although not directly forced, a substantial amount of energy is found in the inertio-gravity modes and interactions among inertio-gravity modes are principally responsible for transferring energy to the small scales. The contribution of the inertio-gravity modes to the flow leads to a shallow tail at the high-wavenumber end of the total energy spectrum. For parameters roughly appropriate for the midlatitude atmosphere (notably Rossby number ~ 0.5), the break between the roughly — k~3 regime and this shallower regime occurs at scales of a few hundred km. This is similar to the observed mesoscale regime in the atmosphere. The nonlinear interactions among the inertio-gravity modes are extremely broadband in spectral space. The implications of this result for the subgrid-scale closure in the shallow-water model are discussed. © 1994, Cambridge University Press. All rights reserved.

Description: The stability of an axisymmetric vortex with a single radial discontinuity in potential vorticity is investigated in rotating shallow water. It is shown analytically that the vortex is always unstable, using the WKBJ method for instabilities with large azimuthal mode number. The analysis reveals that the instability is of mixed type, involving the interaction of a Rossby wave on the boundary of the vortex and a gravity wave beyond the sonic radius. Numerically, it is demonstrated that the growth rate of the instability is generally small, except when the potential vorticity in the vortex is the opposite sign to the background value, in which case it is shown that inertial instability is likely to be stronger than the present instability. © 1994, Cambridge University Press. All rights reserved.

Description: Intuition and previous results suggest that a peristaltic wave tends to drive the mean flow in the direction of wave propagation. New theoretical results indicate that, when the viscosity of the transported fluid is shear-dependent, the direction of mean flow can oppose the direction of wave propagation even in the presence of a zero or favourable mean pressure gradient. The theory is based on an analysis of lubrication-type flow through an infinitely long, axisymmetric tube subjected to a periodic train of transverse waves. Sample calculations for a shear-thinning fluid illustrate that, for a given waveform, the sense of the mean flow can depend on the rheology of the fluid, and that the mean flow rate need not increase monotonically with wave speed and occlusion. We also show that, in the absence of a mean pressure gradient, positive mean flow is assured only for Newtonian fluids; any deviation from Newtonian behaviour allows one to find at least one non-trivial waveform for which the mean flow rate is zero or negative. Introduction of a class of waves dominated by long, straight sections facilitates the proof of this result and provides a simple tool for understanding viscous effects in peristaltic pumping. © 1994, Cambridge University Press. All rights reserved.

Description: A model for the low-Reynolds-number flow of a capsule through a constriction is developed for either constant-flow-rate or constant-pressure-drop conditions. Such a model is necessary to infer quantitative information on the intrinsic properties of capsules from filtration experiments conducted on a dilute suspension of such particles. A spherical capsule, surrounded by an infinitely thin Mooney—Rivlin membrane, is suspended on the axis of a hyperbolic constriction. This configuration is fully axisymmetric and allows the entry and exit phenomena through the pore to be modelled. An integral formulation of the Stokes equations describing the flow in the internal and external domains is developed. It provides a representation of the velocity at any location in the flow as a function of the unknown forces exerted by the boundaries on the fluids. The problem is solved by a collocation technique in the case where the internal and external viscosities are equal. Microscopic quantities (instantaneous geometry, centre of mass velocity, elastic tensions in the membrane) as well as macroscopic quantities (entry time, additional pressure drop or flow rate reduction) are predicted as a function of the capsule intrinsic properties and flow characteristics. The results obtained for a capsule whose initial diameter is larger than that of the constriction throat show that the maximum energy expenditure occurs when the particle centre of mass is still upstream of the throat (typically I diameter away), and is thus due to the entry process. For large enough or rigid enough capsules, the model predicts entrance or exit plugging, in agreement with experimental observations. It is then possible to correlate the variation of the pore hydraulic resistance to the flow capillary number (ratio of viscous to elastic forces) and to the size ratio between the pore and the capsule. © 1994, Cambridge University Press. All rights reserved.

Description: This paper is concerned with the linear inviscid stability of parallel flow over a compliant or flexible wall. A Fjørtoft-type criterion providing a necessary condition for instability in terms of the basic velocity field and its second-order derivative is established. This criterion assumes a simple form for basic flows with zero velocity at the wall. For the latter flows, another necessary condition for stability is given. The results are helpful in the search for unstable modes in flow over a compliant wall. © 1994, Cambridge University Press. All rights reserved.

Description: A family of Lagrangian stochastic models for the joint motion of particle pairs in isotropic homogeneous stationary turbulence is considered. The Markov assumption and well-mixed criterion of Thomson (1990) are used, and the models have quadratic-form functions of velocity for the particle accelerations. Two constraints are derived which formally require that the correct one-particle statistics are obtained by the models. These constraints involve the Eulerian expectation of the ‘acceleration’ of a fluid particle with conditioned instantaneous velocity, given either at the particle, or at some other particle's position. The Navier—Stokes equations, with Gaussian Eulerian probability distributions, are shown to give quadratic-form conditional accelerations, and models which satisfy these two constraints are found. Dispersion calculations show that the constraints do not always guarantee good one-particle statistics, but it is possible to select a constrained model that does. Thomson's model has good one-particle statistics, but is shown to have unphysical conditional accelerations. Comparisons of relative dispersion for the models are made. © 1994, Cambridge University Press. All rights reserved.

Description: The virtually instantaneous three-dimensional concentration fields in the self-similar region of natural or unexcited, circularly excited and weakly buoyant round jets of Reynolds number based on nozzle diameter of 1000 to 4000 are measured experimentally at a spatial resolution of the order of the Kolmogorov length scale. Isoconcentration surfaces are extracted from the concentration field. These surfaces along with their geometrical parameters are used to deduce the structure and modal composition of the jet. The concentration gradient field is calculated, and its local topology is classified using critical-point concepts. Large-scale structure is evident in the form of ‘clumps’ of higher-concentration jet fluid. The structure, which has a downstream extent of about the local jet diameter, is roughly axisymmetric with a conical downstream end. This structure appears to be present only in fully turbulent jets. The antisymmetric two-dimensional images previously thought to be axial slices of an expanding spiral turn out in our data to instead be slices of a simple sinusoid in three dimensions. This result suggests that the helical mode, when present, is in the form of a pair of counter-rotating spirals, or that the +1 and —1 modes are simultaneously present in the flow, with their relative phase set by initial conditions. In terms of local structure, regions with a large magnitude in concentration gradient are shown to have a local topology which is roughly axisymmetric and compressed along the axis of symmetry. Such regions, which would be locally planar and sheetlike, may correspond to the superposition of several of the layer-like structures which are the basic structure of the fine-scale passive scalar field (Buch & Dahm 1991; Ruetsch & Maxey 1991). © 1994, Cambridge University Press. All rights reserved.

Description: The motion of two equal spherical bubbles moving along their line of centres in a viscous liquid is studied numerically in bispherical coordinates. The unsteady Navier-Stokes equations are solved using a mixed spectral/finite-difference scheme for Reynolds numbers up to 200. Free-slip conditions at the bubble surfaces are imposed, while the normal stress condition is replaced by the sphericity constraint under the assumption of small Weber number. The vorticity shed by the upstream bubble affects the drag on the trailing bubble in a very complex fashion that appears to be quite beyond the power of existing asymptotic analyses. The separation between two equal bubbles rising in line under the action of buoyancy is predicted to reach an equilibrium value dependent on the Reynolds number. This result is at variance with experiment. The explanation offered of this difference casts further doubt on the feasibility of a simplified simulation of bubbly liquid dynamics. © 1994, Cambridge University Press. All rights reserved.

Description: General representations are derived for both the velocity potential and the surface pressure fluctuations induced by an arbitrary distribution of vorticity near a manoeuvring cylinder. The cylinder is inextensible and in unsteady motion. Its axis may be slightly curved, with radius of curvature large in comparison with the cylinder radius. Two model problems are considered in detail to investigate the effect of lateral displacements of a cylinder with an established boundary layer. The boundary layer on the flexible cylinder is found to be shed once the lateral displacement of the cylinder axis exceeds the boundary-layer thickness. The unsteady pressures generated by this vortex shedding are investigated. © 1994, Cambridge University Press. All rights reserved.

Description: In this paper we calculate how a pendant drop evolves at the end of a nozzle when the volume of the drop increases steadily with time. We find that the character of the evolution is strongly dependent on the growth rate of the drop and the radius of the nozzle. Typically we find that once the drop has become unstable, two bifurcations occur shortly after each other when the growth rate of the drop is slow. For large growth rates the bifurcations are well-separated in time. We are able to calculate the volumes of the drops after the bifurcations. A comparison with experimental data shows a satisfactory agreement. © 1994, Cambridge University Press. All rights reserved.

Description: The oscillatory motion of an electrically charged non-spherical colloidal particle in an oscillating electric field is investigated. The particle is immersed in an incompressible viscous fluid and assumed to have a thin electric double layer. For moderate-aspect-ration spheroids and cylinders, a simple algebraic expression is derived that accurately describes oscillatory electrophoretic particle motion in terms of the steady Stokes resistance, added mass, and Basset force. The effects of double-layer conduction and displacement currents within dielectric particles are included. The results indicate that electroacoustic measurements may be able to determine the ζ-potential, dielectric constant, surface conductivity (and microstructural information contained therein), size, density, volume fraction, and possibly shape of non-spherical particles in a dilute suspension. A simple formula is obtained for the high-frequency electrical conductivity of a dilute suspension of colloidal spheroids with arbitrary charge and dielectric constant; only the added mass and Basset force are required and the requisite parameters are given. The result is needed for electroacoustic measurements but it may also be independently useful for determining the dielectric constant, surface conductivity, volume fraction, and possibly the shape of non-spherical particles in a dilute suspension. Electroacoustic energy dissipation is described for a dilute colloidal suspension. It is shown that resistive electrical heating and viscous dissipation occur independently. Electrical and viscous dissipation coefficients that characterize the order volume fraction contributions of the suspended particles are calculated; the electrical dissipation coefficient is O(1) for all oscillation frequencies, whereas the latter vanishes at low- and high-frequencies. The fluid motion is shown to be a superposition of unsteady, viscous and potential flows past an oscillating particle with no applied electric field. The electro-osmotic flow field is insensitive to particle geometry and qualitatively different from the flow past an oscillating particle with no applied field.

Description: The scattering of water waves by a varying bottom topography is considered using two-dimensional linear water-wave theory. A new approach is adopted in which the problem is first transformed into a uniform strip resulting in a variable free-surface boundary condition. This is then approximated by a finite number of sections on which the free-surface boundary condition is assumed to be constant. A transition matrix theory is developed which is used to relate the wave amplitudes at ±∞. The method is checked against examples for which the solution is known, or which can be computed by alternative means. Results show that the method provides a simple accurate technique for scattering problems of this type. © 1994, Cambridge University Press. All rights reserved.

Description: We report the results of an investigation of the weakly nonlinear evolution of a triad of waves, each slightly amplified on a linear basis, that are superimposed on a tanh y mixing layer. The triad consists of a plane wave and a pair of oblique modes that act as a subharmonic of order 1/2. The oblique modes are inclined at approximately±60° to the mean flow direction and because the resonance conditions are satisfied exactly the analysis is entirely self-consistent as an asymptotic theory. The nonlinearity first occurs within the critical layer and the initial interaction is of the parametric resonance type. This produces faster than exponential growth of the oblique waves, behaviour observed recently in the experiments of Corke& Kusek (1993). The critical-layer dynamics lead subsequently to coupled integro-differential equations governing the amplitude evolution and, as first shown in related work by Goldstein& Lee (1992) on boundary layers in an adverse pressure gradient, these equations develop singularities in a finite time. © 1994, Cambridge University Press. All rights reserved.

Description: Rotation strongly affects the stability of turbulent flows in the presence of large eddies. In this paper, we examine the applicability of the classic Bradshaw-Richardson criterion to flows more general than a simple combination of rotation and pure shear. Two approaches are used. Firstly the linearized theory is applied to a class of rotating two-dimensional flows having arbitrary rates of strain and vorticity and streamfunctions that are quadratic. This class includes simple shear and elliptic flows as special cases. Secondly, we describe a large-eddy simulation of initially quasi-homogeneous three-dimensional turbulence superimposed on a periodic array of two-dimensional Taylor-Green vortices in a rotating frame. The results of both approaches indicate that, for a large structure of vorticity W and subject to rotation Ω, maximum destabilization is obtained for zero tilting vorticity (½W + 2Ω = 0) whereas stability occurs for zero absolute vorticity (2Ω = 0) These results are consistent with the Bradshaw-Richardson criterion; however the numerical results show that in other cases the Bradshaw-Richardson number [formula omitted] is not always a good indicator of the flow stability. © 1994, Cambridge University Press. All rights reserved.

Description: By means of direct numerical solution of the kinetic equation for surface gravity waves, it is shown that under certain conditions the constant flux spectra of nonlinear waves, first predicted by Zakharov & Filonenko (1966) for an infinite frequency domain, can be formed in a finite frequency interval. For the case of angular isotropic spectra the conditions and timescales of this flux spectra formation are evaluated. © 1994, Cambridge University Press. All rights reserved.

Description: The boundary-layer flow over a cooled horizontal plate is considered. It is shown that the real part of the spectrum of the evolution operator of the linearized equations is not bounded uniformly from above which explains the difficulties encounterd by a numerical solution. Furthermore it is shown that near the leading edge an asymptotic expansion of the solution is not unique. A one-parametric family of asymptotic expansions of solutions can be constructed. © 1994, Cambridge University Press. All rights reserved.

Description: An exact result is calculated numerically for the dilute limiting, zero shear viscosity of bimodal suspensions of hard spheres. The required hydrodynamic functions are calculated from recent results for the resistivities of unequal spheres. Both the hydrodynamic and Brownian contributions to the Huggins coefficient exhibit a minimum that is symmetric in mixing volume fraction. The resultant minimum deepens with increasing size ratio. The results are discussed in the light of published measurements of the viscosity for bimodal suspensions and previous phenomenological theories. The reduction of viscosity upon mixing is seen to be a result of near-field hydrodynamic shielding of asymmetric particle pairs. It is also shown that the use of far-field hydrodynamic interactions yields qualitatively incorrect results for the viscosity of binary mixtures. A parametrization of the bimodal results allows an estimation of the effects of suspension polydispersity on the Huggins coefficient. For polydispersities of ten percent or less, the Huggins coefficient is essentially unchanged from the value calculated for an equivalent, monodisperse suspension at equal volume fraction. A parametrization of these results is provided for relating the reduction in Huggins coefficient to the polydispersity index. © 1994, Cambridge University Press. All rights reserved.

Description: The effects of crossflow on the growth rate of inviscid Gortler vortices in a hypersonic boundary layer with pressure gradient are studied in this paper. Attention is focused on the inviscid mode trapped in the temperature adjustment layer; this mode has greater growth rate than any other mode at the minimum order of the Gortler number at which Gortler vortices may exist. The eigenvalue problem which governs the relationship between the growth rate, the crossflow amplitude and the wavenumber is solved numerically, and the results are then used to clarify the effects of crossflow on the growth rate of inviscid Gortler vortices. It is shown that crossflow effects stabilize Gortler vortices in different manners for incompressible and hypersonic flows. The neutral mode eigenvalue problem is found to have an exact solution, and as a by-product, we have also found the exact solution to a neutral mode eigenvalue problem which was formulated, but unsolved before, by Bassom & Hall (1991). © 1994, Cambridge University Press. All rights reserved.

Description: The familiar problem of the propagation of surface waves over variable depth is reconsidered. The surface wave is taken to be a slowly evolving nonlinear wave (governed by the Korteweg–de Vries equation) and the depth is also assumed to be slowly varying; the fluid is stationary in its undisturbed state. Two cases are addressed: the first is where the scale of the depth variation is longer than that on which the wave evolves, and the second is where it is shorter (but still long). The first case corresponds to that discussed by a number of previous authors, and is the problem which has been approached through the perturbation of the inverse scattering transform method, a route not followed here. Our more direct methods reveal a new element in the solution: a perturbation of the primary wave, initiated by the depth change, which arises at the same order as the left-going shelf. The resulting leading-order mass balance is described, with more detail than hitherto (made possible by the use of a special depth variation). The second case is briefly presented using the same approach, and some important similarities are noted. © 1994, Cambridge University Press. All rights reserved.

Description: The evolution of spanwise phase variations of nominally two-dimensional instability modes in a plane shear layer is studied in a closed-return water facility using time-harmonic excitation having spanwise-non-uniform phase or frequency distributions. The excitation waveform is synthesized by a linear array of 32 surface film heaters flush-mounted on the flow partition. A spanwise-linear phase distribution leads to the excitation of oblique waves and to the rollup of oblique primary vortices. When the prescribed phase distribution is piecewise-constant and spanwise-periodic, the flow is excited with a linear combination of a two-dimensional wavetrain and pairs of equal and opposite oblique waves, the amplitudes of which depend on the magnitude of the phase variation Φ. As a result of the excitation, the primary vortices undergo spanwise-non-uniform rollup and develop spanwise-periodic deformations that induce cross-shear and secondary vortices in the braid region. The amplitude of the deformations of the primary vortices and the shape and strength of the secondary vortices depend on the magnitude of Φ. When Φis small, the secondary vortices are counter-rotating vortex pairs. As Φ increases, cross-shear induced by oblique segments of the primary vortices in the braid region results in the formation of single secondary vortex strands. The flow is not receptive to spanwise phase variations with wavelengths shorter than the streamwise wavelength of the Kelvin-Helmholtz instability. When the phase variation is Φ. =π, the flow is excited with pairs of oblique waves only and undergoes a double rollup, resulting in the formation of spanwise-deformed vortices at twice the excitation frequency. Measurements of the streamwise velocity component show that the excitation leads to a substantial increase in the cross-stream spreading of the shear layer and that distortions of transverse velocity profiles are accompanied by an increase in the high-frequency content of velocity power spectra. Detailed schlieren visualizations shed light on the nature of ‘vortex dislocations’ previously observed by other investigators. Complex spanwise-non-uniform pairing interactions between the spanwise vortices are forced farther downstream by spanwise-amplitude or phase variations of subharmonic excitation wavetrains. © 1994, Cambridge University Press. All rights reserved.

Description: Direct numerical simulations of a fully developed turbulent channel flow for two relatively small values of the Reynolds number are used to examine its influence on various turbulence quantities in the near-wall region. The limiting wall behaviour of these quantities indicates important increases in the r.m.s. value of the wall pressure fluctuations and its derivatives, the r.m.s. streamwise vorticity and in the average energy dissipation rate and the Reynolds shear stress. If the normalization is based on the wall shear stress and the kinematic viscosity, these changes are shown to be consistent with an increase in strength - but not the average diameter or average location - of the quasi-streamwise vortices in the buffer region. Evidence of this strengthening is provided by the increased sum of the stretching terms for the mean-square streamwise vorticity. It is also shown that a normalization based on Kolmogorov velocity and lengthscales, defined at the wall, is more appropriate in the near-wall region than scaling on the wall shear stress and kinematic viscosity. © 1994, Cambridge University Press. All rights reserved.

Description: This paper is concerned with the stability of steady inviscid flows with closed streamlines. In increasing order of complexity we look at two-dimensional planar flows, poloidal (r, z) flows, and swirling recirculating flows. In each case we examine the relationship between Arnol’d's variational approach to stability, Moffatt's magnetic relaxation technique, and a more recent relaxation procedure developed by Vallis et al. We start with two-dimensional (x, y) flows. Here we show that Moffatt's relaxation procedure will, under a wide range of circumstances, produce Euler flows which are stable. The physical reasons for this are discussed in the context of the well-known membrane analogy. We also show that there is a close relationship between Hamilton's principle and magnetic relaxation. Next, we examine poloidal flows. Here we find that, by and large, our planar results also hold true for axisymmetric flows. In particular, magnetic relaxation once again provides stable Euler flows. Finally, we consider swirling recirculating flows. It transpires that the introduction of swirl has a profound effect on stability. In particular, the flows produced by magnetic relaxation are no longer stable. Indeed, we show that all swirling recirculating Euler flows are potentially unstable to the extent that they fail to satisfy Arnol’d's stability criterion. This is, perhaps, not surprising, as all swirling recirculating flows include regions where the angular momentum decreases with radius and we would intuitively expect such flows to be prone to a centrifugal instability. The paper concludes with a discussion of marginally unstable modes in swirling flows. In particular, we examine the extent to which Rayleigh's original ideas on stability may be generalized, through the use of the Routhian, to include flows with a non-zero recirculation. © 1994, Cambridge University Press. All rights reserved.

Description: Experiments on bifurcation of rotating liquid drops into two-lobed shapes were conducted during a Space shuttle flight. The drops were levitated in air and spinned using acoustic fields in the low-gravity environment. These experiments have successfully resolved the discrepancies existing between the previous experimental results and the theoretical predictions. In the simplest case of a rotating drop that is free from deformation by external forces, the results agree well with the existing theoretical predictions. In the case of a rotating drop subjected to flattening by the acoustic radiation stress, deliberately or otherwise, the experiments suggest the existence of a family of curves, with the free drop as the limiting case. © 1994, Cambridge University Press. All rights reserved.

Description: This paper reports the results of a two-dimensional finite element simulation of the motion of a circular particle in a Couette and a Poiseuille flow. The size of the particle and the Reynolds number are large enough to include fully nonlinear inertial effects and wall effects. Both neutrally buoyant and non-neutrally buoyant particles are studied, and the results are compared with pertinent experimental data and perturbation theories. A neutrally buoyant particle is shown to migrate to the centreline in a Couette flow, and exhibits the Segre-Silberberg effect in a Poiseuille flow. Non-neutrally buoyant particles have more complicated patterns of migration, depending upon the density difference between the fluid and the particle. The driving forces of the migration have been identified as a wall repulsion due to lubrication, an inertial lift related to shear slip, a lift due to particle rotation and, in the case of Poiseuille flow, a lift caused by the velocity profile curvature. These forces are analysed by examining the distributions of pressure and shear stress on the particle. The stagnation pressure on the particle surface are particularly important in determining the direction of migration. © 1994, Cambridge University Press. All rights reserved.

Description: An axisymmetric vortex-sheet model is applied to simulate an experiment of Didden (1979) in which a moving piston forces fluid from a circular tube, leading to the formation of a vortex ring. Comparison between simulation and experiment indicates that the model captures the basic features of the ring formation process. The computed results support the experimental finding that the ring trajectory and the circulation shedding rate do not behave as predicted by similarity theory for starting flow past a sharp edge. The factors responsible for the discrepancy between theory and observation are discussed. © 1994, Cambridge University Press. All rights reserved.

Description: The method of product integration is applied to the vortex dynamics of two-dimensional incompressible viscous media. In the cases of both unbounded and bounded flows under the no-slip boundary condition, the analytic solutions of the Cauchy problem are obtained for the Helmholtz equation in the form of linear and nonlinear product integrals. The application of product integrals allows the generalization in a natural way of the vortex dynamics concept to the case of viscous flows. However, this new approach requires the reconsideration of some traditional notions of vortex dynamics. Two lengthscales are introduced in the form of a micro- and a macro-scale. Elementary ‘vortex objects’ are defined as two types of singular vortex filaments with equal but opposite intensities. The vorticity is considered as the macro-value proportional to the concentration of elementary vortex filaments inhabiting the micro-level. The vortex motion of a viscous medium is represented as the stochastic motion of an infinite set of elementary vortex filaments on the micro-level governed by the stochastic differential equations, where the stochastic velocity component of every filament simulates the viscous diffusion of vorticity, and the regular component is the macro-value induced according to the Biot–Savart law and simulates the convective transfer of vorticity. In flows with boundaries, the production of elementary vortex filaments at the boundary is introduced to satisfy the no-slip condition. This phenomenon is described by the application of the generalized Markov processes theory. The integral equation for the production intensity of elementary vortex filaments is derived and solved using the no-slip condition reformulated in terms of vorticity. Additional conditions on this intensity are determined to avoid the many-valuedness of the pressure in a multi-connected flow domain. This intensity depends on the vorticity in the flow and the boundary velocity at every time instant, together with boundary acceleration. As a result, the successive and accurate application of the product-integral method allows the study of vortex dynamics in a viscous fluid according to the concepts of Helmholtz and Kelvin. © 1994, Cambridge University Press. All rights reserved.

Description: This paper treats a surface-tension-driven liquid-metal flow in a cylinder with a steady externally applied non-uniform axisymmetric magnetic field. The top boundary consists of an annular free surface around a solid disk, modelling the Czochralski growth of silicon crystals. A radial temperature gradient produces a decrease of the surface tension from the disk edge to the vertical cylinder wall. The magnetic flux density is sufficiently large that inertial effects and convective heat transfer are negligible. First we present large-Hartmann-number asymptotic solutions for magnetic fields with either a non-zero or a zero axial component at the free surface. The asymptotic solutions indicate that a purely radial magnetic field at the free surface represents a singular limit of more general magnetic fields. Secondly we present numerical solutions for arbitrary values of the Hartmann number, and we treat the evolution of the thermocapillary convection as the axial magnetic field at the free surface is changed continuously from the full field strength to zero. © 1994, Cambridge University Press. All rights reserved.

Description: The theory of surface gravity waves scattering at vortex flows in the ocean is developed in this paper. A scattering amplitude is found in the Born approximation as a function of vorticity which appears very convenient for investigation of scattering at simple localized flows. It is shown that the wave scattering cross-section is determined by the vertical component of vorticity. For a random (turbulent) vortex field the scattering cross-section per unit voume is determined by a vorticity correlation function. The damping of the coherent wave component and the angular spectrum widening are calculated for multiple scattering by vortex turbulence of drift flows. The spectrum angular width evolution for waves scattered at self-similar vortices of the logarithmic boundary layer is determined only by its dynamical speed and the wave vector. The latter result may be used for a remote sensing of oceanic turbulent drift flows based on observations of surface waves. © 1994, Cambridge University Press. All rights reserved.

Description: This paper examines the dynamics of geostrophic flows with large displacement of isopycnal surfaces. The β-effect is assumed strong i.e. the parameter (Rd cot θ)/Re (where θ is the latitude, Rd is the deformation radius, Re is the Earth's radius) is of the order of, or greater than, the Rossby number. A system of asymptotic equations is derived, with the help of which the stability of an arbitrary zonal flow with both vertical and horizontal shear is proven. It is demonstrated that the horizontal and vertical spatial variables in the asymptotic system are separable, which yields a ‘horizontal’ set of evolutionary equations for the amplitudes of the barotropic and baroclinic modes (the vertical profile of the latter is arbitrary). © 1994, Cambridge University Press. All rights reserved.

Description: Instantaneous and phase averaged pressure distributions in the near field of a jet, and their effects on the conditions for the onset of cavitation are studied in detail. The measurements are performed by using microscopic bubbles as pressure sensors, and holography as a means of detecting them. Experiments are performed at Red exceeding 4 × 105, with and without acoustic excitation. The results show that the highest negative pressure peaks (–0.97) and the resulting cavitation inception occur because of vortex pairing. Prior to pairing the negative peaks are between –0.8 and –0.9. Weak acoustic excitation changes the entire flow structure and the spatial distributions of bubbles, but has little effect on the onset of cavitation. Downstream of the potential core the highest pressure peaks (∼ –0.6) are considerably smaller, in agreement with the occurrence of cavitation there. It is also shown that although the r.m.s. values of pressure fluctuations do not vary with the jet speed, the probability distribution changes significantly, causing a reduction in the inception index with increasing velocity. The probability of cavitation inception is estimated from the distributions of bubbles and pressure peaks. It is shown that the actual, non-uniform bubble distribution increases the probability of inception owning to migration of the bubbles to the low pressure regions. © 1994, Cambridge University Press. All rights reserved.

Description: The objective of this study is to explore concepts for active control of turbulent boundary layers leading to skin-friction reduction using the direct numerical simulation technique. Significant drag reduction is achieved when the surface boundary condition is modified to suppress the dynamically significant coherent structures present in the wall region. The drag reduction is accompanied by significant reduction in the intensity of the wall-layer structures and reductions in the magnitude of Reynolds shear stress throughout the flow. The apparent outward shift of turbulence statistics in the controlled flows indicates a displaced virtual origin of the boundary layer and a thickened sublayer. Time sequences of the flow fields show that there are essentially two drag-reduction mechanisms. Firstly, within a short time after the control is applied, drag is reduced mainly by deterring the sweep motion without modifying the primary streamwise vortices above the wall. Consequently, the high-shear-rate regions on the wall are moved to the interior of the channel by the control schemes. Secondly, the active control changes the evolution of the wall vorticity layer by stabilizing and preventing lifting of the spanwise vorticity near the wall, which may suppress a source of new streamwise vortices above the wall. © 1994, Cambridge University Press. All rights reserved.

Description: A numerical investigation is conducted into the flow of a dilute suspension of rigid rodlike particles between parallel flat plates, driven by a uniform pressure gradient. The particles are assumed to be small relative to lengthscales of the flow with the effect that particle orientations evolve according to the local velocity gradient; the particles are also assumed to be small in an absolute sense, with the consequence that Brownian motions are of consequence. The calculations are performed using a novel approach, with a theoretical basis that has been developed previously in a companion paper (Szeri & Leal 1992). The new approach permits one to solve flow problems of microstructured fluids (such as suspensions, liquid crystals, polymer solutions and melts) without ‘pre-averaging’ or closure approximations. In the present work, the new approach is used to expose previously unknown pathological, non-physical predictions in various constitutive models derived using closure approximations. This appears to have passed unnoticed in prior work. In addition, the new approach is shown to possess several computational advantages. The determination of the orientation distribution of particles is self-adaptive; this leads, in effect, to a very efficient solution of the associated Smoluchowski (or Fokker-Planck) equation. Moreover, the new approach is highly suited to parallel (and vector) implementation on modern computers. These issues are explored in detail in the context of the example flow. © 1994, Cambridge University Press. All rights reserved.

Description: The natural frequencies and damping ratios for surface waves in a circular cylinder are calculated on the assumptions of a fixed contact line, Stokes boundary layers, and either a clean or a fully contaminated surface. These theoretical predictions are compared with the measurements for the first six modes in a brimfull, sharp-edged cylinder of radius 2.77 cm and depth 3.80 cm. The differences between the predicted and observed frequencies were less than 0.5 % for all but the fundamental axisymmetric mode with a clean surface. The difference between the predicted and observed damping ratio for the dominant mode with a clean surface was 20%; this difference was significantly larger for the higher modes with a clean surface and for all of the modes with a contaminated surface. © 1994, Cambridge University Press. All rights reserved.

Description: The motion of a disk rising steadily parallel to the axis of rotation in a uniformly rotating unbounded liquid is considered. In the limit of zero Rossby number the linear viscous equations of motion are reduced to a system of dual integral equations which renders an ‘exact’ solution for arbitrary values of the Taylor number, Ta. The investigation is focused on the drag and the flow field. In the limits of small and large Ta the asymptotic results of the present formulation agree with – and extend – previous investigations by different approaches. A particular novel feature, for large Ta, is the contribution of the Ekman-layer flux to the outer motion. New insight into the structure of the Taylor column is gained; in particular, it is shown that the main part of the column is a ‘bubble’ of recirculating fluid, detached from the body and not communicating with the Ekman layer. However, it turns out that the essential discrepancy in drag between experiments (Maxworthy 1970) and previous theories cannot be attributed to the Ekman-layer suction effect. © 1994, Cambridge University Press. All rights reserved.

Description: Dynamic simulations of the pressure-driven flow in a channel of a non-Brownian suspension at zero Reynolds number were conducted using Stokesian Dynamics. The simulations are for a monolayer of identical particles as a function of the dimensionless channel width and the bulk particle concentration. Starting from a homogeneous dispersion, the particles gradually migrate towards the centre of the channel, resulting in an homogeneous concentration profile and a blunting of the particle velocity profile. The time for achieving steady state scales as (H/a)3a/〈u〉, where H is the channel width, a the radii of the particles, and 〈u〉 the average suspension velocity in the channel. The concentration and velocity profiles determined from the simulations are in qualitative agreement with experiment. A model for suspension flow has been proposed in which macroscopic mass, momentum and energy balances are constructed and solved simultaneously. It is shown that the requirement that the suspension pressure be constant in directions perpendicular to the mean motion leads to particle migration and concentration variations in inhomogeneous flow. The concept of the suspension ‘temperature’ – a measure of the particle velocity fluctuations – is introduced in order to provide a nonlocal description of suspension behaviour. The results of this model for channel flow are in good agreement with the simulations. © 1994, Cambridge University Press. All rights reserved.

Description: We report the results of an experimental study of flow in a Taylor-Couette system where the usual circular outer cylinder is replaced by one with a square cross-section. The objective is to determine the validity of low-dimensional dynamical systems as a descriptive framework for flows in a domain without the special continuous symmetry of the original problem. We focus on a restricted version of the flow, where the steady flow consists of a single cell, thereby minimizing the multiplicity of solutions. The steady-state bifurcation structure is found to be qualitatively unchanged from that of the standard system. A complex but self-consistent bifurcation structure is uncovered for time-dependent flows, culminating in observations of dynamics similar to those of the finite-dimensional Sil'nikov mechanism. Such behaviour has been observed in the standard system with continuous azimuthal symmetry. The present results extend the range of closed-flow problems where there is an apparent connection between the infinite-dimensional Navier-Stokes equations and finite-dimensional dynamical systems. © 1994, Cambridge University Press. All rights reserved.

Description: An instability mechanism that can amplify wind-forced inertial oscillations in the upper ocean is investigated. This forced instability happens because of the phase relationship between the mixed-layer depth and the surface current. It allows the inertial oscillations propagating against the wind to extract energy from it and amplify. The key ingredients for the instability to work are (a) a non-zero mean wind stress, (b) a spatial variability of the oscillations in the direction of the wind stress. The amplification is demonstrated using a simple shallow-water model in a few situations: the dispersion of a localized disturbance with steady and time-varying wind forcing, generation of inertial waves at a coast, and spatial variability induced by mesoscale eddies. Estimates of the growth rate are provided for both dissipative and non-dissipative cases. © 1994, Cambridge University Press. All rights reserved.

Description: An experimental and theoretical investigation of the stability of the viscoelastic flow of a model Boger fluid between rotating cylinders with an applied pressure gradient is presented. In our theoretical study, a linear stability analysis based on the Oldroyd-B fluid model which predicts the critical conditions and the structure of the vortex flow at the onset of instability is developed. Our results reveal that certain non-axisymmetric modes are more unstable than the previously studied axisymmetric mode when the shearing by the cylinder rotation is the dominant flow-driving force. This is consistent with recent results presented by Beris & Avgousti (1992) on the stability of elastic Taylor-Couette flow. On the other hand, the axisymmetric mode is more unstable when the pressure gradient becomes dominant. Furthermore, we investigate the mechanism of purely elastic Taylor-Dean instability with respect to non-axisymmetric disturbances through an examination of the disturbance-energy equation. It is found that the mechanism of the elastic Taylor-Dean instability is associated with the coupling between the disturbance polymeric stresses due to the azimuthal variation of the disturbance flow and the base state velocity gradients. In our experimental study, evidence of non-inertial, cellular instabilities in the Taylor-Dean flow of a well-characterized polyisobutylene/polybutene Boger fluid is presented. A stationary, meridional obstruction is placed between independently rotating, concentric cylinders to generate an azimuthal pressure gradient in opposition to the shearing flow. Flow visualization experiments near the critical conditions show the transition from purely azimuthal flows to secondary vortex flows, and the development of evenly spaced, banded vortex structures. The critical wavenumber obtained from spectral image analysis of the visualizations, and the critical Deborah number are presented for various ratios of the pressure gradient to the shear driving force. Although there is a quantitative discrepancy between data and theory, the qualitative trends in the data are in agreement with our theoretical predictions. In addition, laser-Doppler velocimetry (LDV) measurements show that the instability is a stationary mode when the pressure gradient is the dominant flow-driving force, while it is an oscillatory instability when the shearing is dominant, again as predicted by the theory. © 1994, Cambridge University Press. All rights reserved.

Description: A new mechanism for the creation of structures in two-dimensional turbulence is investigated. The forced Navier-Stokes equations are solved numerically in a periodic square in the limit of zero viscosity. The force is a white-in-time random noise acting in a narrow band of high wavenumbers. The inverse-cascade process and the presence of the boundary lead ultimately to a pile-up of energy in the lowest wavenumber (Bose condensation). In the asymptotic limit where the enstrophy cascade range is negligible, Bose condensation is solely responsible for the generation of coherent vortices and intermittency in the system. We present the evolution of the velocity and vorticity fields through the later stages of the condensate state, and explore the possible implications for atmospheric turbulence constrained by the periodic domain about the earth. © 1994, Cambridge University Press. All rights reserved.

Description: It is shown experimentally that free shear flows can be substantially altered through direct control of the large coherent vortices present in the flow. First, flow-visualization experiments are conducted in Kalliroscope fluid at Reynolds number 550. A foil is placed in the wake of a D-section cylinder, sufficiently far behind the cylinder so that it does not interfere with the vortex formation process. The foil performs a heaving and pitching oscillation at a frequency close to the Strouhal frequency of the cylinder, while cylinder and foil also move forward at constant speed. By varying the phase of the foil oscillation, three basic interaction modes are identified. (i) Formation of a street of pairs of counter-rotating vortices, each pair consisting of one vortex from the initial street of the cylinder and one vortex shed by the foil. The width of the wake is then substantially increased. (ii) Formation of a street of vortices with reduced or even reverse circulation compared to that of oncoming cylinder vortices, through repositioning of cylinder vortices by the foil and interaction with vorticity of the opposite sign shed from the trailing edge of the foil. (iii) Formation of a street of vortices with circulation increased through merging of cylinder vortices with vortices of the same sign shed by the foil. In modes (ii) and (iii) considerable repositioning of the cylinder vortices takes place immediately behind the foil, resulting in a regular or reverse Kármán street. The formation of these three interaction patterns is achieved only for specific parametric values; for different values of the parameters no dominant stable pattern emerges. Subsequently, the experiments are repeated in a different facility at larger scale, resulting in Reynolds number 20000, in order to obtain force and torque measurements. The purpose of the second set of experiments is to assess the impact of flow control on the efficiency of the oscillating foil, and hence investigate the possibility of energy extraction. It is found that the efficiency of the foil depends strongly on the phase difference between the oscillation of the foil and the arrival of cylinder vortices. Peaks in foil efficiency are associated with the formation of a street of weakened vortices and energy extraction by the foil from the vortices of the vortex street. © 1994, Cambridge University Press. All rights reserved.

Description: A boundary integral method is presented for analysing particle motion in a rotating fluid for flows where the Taylor number 2T is arbitrary and the Reynolds number is small. The method determines the surface traction and drag on a particle, and also the velocity field at any location in the fluid. Numerical results show that the dimensionless drag on a spherical particle translating along the rotation axis of an unbounded fluid is determined by the empirical formula D/6n = 1 +(4/7) ^”1/2 +(8/9TI)2T, which incorporates known results for the low and high Taylor number limits. Streamline portraits show that a critical Taylor number c « 50 exists at which the character of the flow changes. For 3 “ 〈 2Tcthe flow field appears as a perturbation of a Stokes flow with a superimposed swirling motion. For T 〈 2TCthe flow field develops two detached recirculating regions of trapped fluid located fore and aft of the particle. The recirculating regions grow in size and move farther from the particle with increasing Taylor number. This recirculation functions to deflect fluid away from the translating particle, thereby generating a columnar flow structure. The flow between the recirculating regions and the particle has a plug-like velocity profile, moving slightly slower than the particle and undergoing a uniform swirling motion. The flow in this region is matched to the particle velocity in a thin Ekman layer adjacent to the particle surface. A further study examines the translation of spheroidal particles. For large Taylor numbers, the drag is determined by the equatorial radius; details of the body shape are less important. © 1994, Cambridge University Press. All rights reserved.

Description: The propagation of weak shock waves in liquids containing a small concentration of gas bubbles is studied theoretically on the basis of a mathematical model that contains all - and only - the effects that contribute to first order in the gas volume fraction. In particular, the thermal exchange between the gas bubbles and the liquid is described accurately. This aspect of the theory emerges as its most significant component, relegating effects such as the relative motion between the phases to roles of minor importance. Comparison with experimental results substantiates the accuracy of the model for shock waves that have had time to broaden from an initial sharp front to a more diffuse profile. For shock waves closer to inception, marked differences are found between theory and experiment. The same problem affects all other published theoretical treatments. It is concluded that some as yet poorly understood mechanism governs the early-time behaviour of shock waves in bubbly liquids. © 1994, Cambridge University Press. All rights reserved.

Description: Solitary waves with constant vorticity in water of finite depth are calculated numerically by a boundary integral equation method. Previous calculations are confirmed and extended. It is shown that there are branches of solutions which bifurcate from a uniform shear current. Some of these branches are characterized by a limiting configuration with a 120° angle at the crest of the wave. Other branches extend for arbitrary large values of the amplitude of the wave. The corresponding solutions ultimately approach closed regions of constant vorticity in contact with the bottom of the channel. A numerical scheme is presented to calculate directly these closed regions of constant vorticity. In addition, it is shown that there are branches of solutions which do not bifurcate from a uniform shear flow. © 1994, Cambridge University Press. All rights reserved.

Description: Dilute aqueous solutions of long alcohol chains were recently found to cause a quadratic dependence of surface tension on the temperature without affecting other bulk properties of the liquid: σ = σ0+ αQ(T—T0)2, σQ0. The impact of such Marangoni instability on the behaviour of a thin liquid layer is studied in this work. We derive an equation describing a nonlinear spatiotemporal evolution of a thin film. The behaviour of the perturbed film in the absence of gravity, critically depends on whether the temperature T0, yielding a minimal surface tension, is attained on the surface of the film. When this is the case, a qualitatively new behaviour is observed: perturbations of the film interface may evolve into continuous steady patterns that do not rupture. Otherwise, the observed patterns due to the linear and quadratic Marangoni effects are qualitatively similar and result in the rupture of the film into separate drops. © 1994, Cambridge University Press. All rights reserved.

Description: Steady two-dimensional flows past a parabolic obstacle lying on the free surface in water of finite depth are considered. The fluid is treated as inviscid and incompressible and the flow is assumed to be irrotational. Gravity is included in the free-surface condition. The problem is solved numerically by using boundary integral equation techniques. It is shown that there are solutions for which the flow is supercritical both upstream and downstream and others for which the flow is subcritical both upstream and downstream. These flows have continuous tangents at both ends of the obstacle at which separation occurs. For supercritical flows, there are up to three solutions corresponding to the same value of the Froude number when the obstacle is concave and up to two solutions when the obstacle is convex. For subcritical flows, there are solutions with waves behind the obstacle. As the Froude number decreases, these waves become steeper and the numerical calculations suggest that they, ultimately, reach limiting configurations with a sharp crest forming a 120° angle. © 1994, Cambridge University Press. All rights reserved.

Description: The evolution of three-dimensional disturbances in an incompressible mixing layer in an inviscid fluid is investigated as an initial-value problem. A Green's function approach is used to obtain a general space–time solution to the problem using a piecewise linear model for the basic flow, thereby making it possible to determine complete and closed-form analytical expressions for the variables with arbitrary input. Structure, kinetic energy, vorticity, and the evolution of material particles can be ascertained in detail. Moreover, these solutions represent the full three-dimensional disturbances that can grow exponentially or algebraically in time. For large time, the behaviour of these disturbances is dominated by the exponentially increasing discrete modes. For the early time, the behaviour is controlled by the algebraic variation due to the continuous spectrum. Contrary to Squire's theorem for normal mode analysis, the early-time behaviour indicates growth at comparable rates for all values of the wavenumbers and the initial growth of these disturbances is shown to rapidly increase. In particular, the disturbance kinetic energy can rise to a level approximately ten times its initial value before the exponentially growing normal mode prevails. As a result, the transient behaviour can trigger the roll-up of the mixing layer and its development into the well-known pattern that has been observed experimentally. © 1994, Cambridge University Press. All rights reserved.

Description: The long-time self-diffusivity in concentrated colloidal dispersions is determined from a consideration of the temporal decay of density fluctuations. For hydrodynamically interacting Brownian particles the long-time self-diffusivity, Ds∞, is shown to be expressible as the product of the hydrodynamically determined short-time self-diffusivity, Ds0(Φ), and a contribution that depends on the distortion of the equilibrium structure caused by a diffusing particle. An argument is advanced to show that as maximum packing is approached the long-time self-diffusivity scales as Ds∞~ Ds0(Φ)/g(2;Φ), whereg(2;Φ), g(2;is the value of the equilibrium radial-distribution function at contact and Φ is the volume fraction of interest. This result predicts that the long-time self-diffusivity vanishes quadratically at random close packing, 〈Φm≈ 0.63, i.e. Ds∞~ D0(1-Φ/Φm)2 as Φ → Φm, where DQ= kT/6Πηa is the diffusivity of a single isolated particle of radius a in a fluid of viscosity y. This scaling occurs because Ds0(Φ) vanishes linearly at random close packing and the radial-distribution function at contact diverges as (1-Φ/Φm)-1 A model is developed to determine the structural deformation for the entire range of volume fractions, and for hard spheres the long-time self-diffusivity can be represented by:Ds∞=Ds∞[1+2∞g(2;∞)]. This formula is in good agreement with experiment. For particles that interact through hard-sphere-like repulsive interparticle forces characterized by a length b(〉 a), the same formula applies with the short-time self-diffusivity still determined by hydrodynamic interactions at the true or ‘hydrodynamic’ volume fraction ∞, but the structural deformation and equilibrium radial-distribution function are now determined by the 4 thermodynamic’ volume fraction Φbbased on the length b. When b〉 a, the long-time self-diffusivity vanishes linearly at random close packing based on the ‘thermodynamic’ volume fraction Φbm. This change in behaviour occurs because the true or ‘hydrodynamic’ volume fraction is so low that the short-time self-diffusivity is given by its infinite-dilution value Φ0. It is also shown that the temporal transition from short- to long-time diffusive behaviour is inversely proportional to the dynamic viscosity and is a universal function for all volume fractions when time is non-dimensionalized by a2/Ds0(Φ)(∞). © 1994, Cambridge University Press. All rights reserved.

Description: The commonly observed phenomenon of steady, viscous, free-surface flow on the outer surface of a rotating cylinder is investigated by means of an iterative, integral-equation formulation applied to the Stokes approximation of the Navier-Stokes equations. The method of solution places no restriction on the thickness of the fluid layer residing on the cylinder surface; indeed, results are presented for cases where the layer thickness is of the same order of magnitude as the cylinder radius. Free-surface profiles and free-surface velocity distributions are presented for a range of flow parameters. Where appropriate, comparisons are made with the results of thin-film theory; excellent agreement is observed. For all film thicknesses and surface tensions, results show a high degree of symmetry about a horizontal axis even though the gravity field is vertical. A proof is presented that, for vanishing surface tension, this is a consequence of the Stokes approximation. © 1994, Cambridge University Press. All rights reserved.

Description: Convex transverse curvature effects in wall-bounded turbulent flows are significant if the boundary-layer thickness is large compared to the radius of curvature (large γ= δ/a). The curvature affects the inner part of the flow if a+, the cylinder radius in wall units, is small. Two direct numerical simulations of a model problem approximating axial flow boundary layers on long cylinders were performed for γ = 5 (a+ ≈43) and γ = 11 (a+≈21). Statistical and structural data were extracted from the computed flow fields. The effects of the transverse curvature were identified by comparing the present results with those of the plane channel simulation of Kim, Moin & Moser (1987), performed at a similar Reynolds number. As the curvature increases, the skin friction increases, the slope of the logarithmic region decreases and turbulence intensities are reduced. Several turbulence statistics are found to scale with a curvature dependent velocity scale derived from the mean momentum equation. Near the wall, the flow is more anisotropic than in the plane channel with a larger percentage of the turbulent kinetic energy resulting from the streamwise velocity fluctuations. As the curvature increases, regions of strong normal vorticity develop near the wall. © 1994, Cambridge University Press. All rights reserved.

Description: A three-dimensional turbulent boundary layer (3DTBL) was generated on the floor of a low-speed wind tunnel by the imposition of a cross-stream pressure gradient using a 30° bend in the horizontal plane. The surface streamlines were deflected by as much as 22° relative to the local tunnel centreline. Downstream of the bend, the 3DTBL gradually relaxed towards a 2DTBL; this was an impulse-and-recovery experiment which focused on the outer layer. Mean velocities were measured with a three-hole yawmeter and turbulence quantities, which included the Reynolds-stress tensor and the triple products, were measured with a cross-wire hot-wire anemometer. The experiment isolated the effects of crossflow from those of adverse streamwise pressure gradients, which may have clouded interpretations of previous 3DTBL experiments. In particular, the detailed developments of the cross-stream shear stress and of the stress/energy ratio become clearer. The shear-stress vector lagged behind the velocity-gradient vector as crossflow developed; however, the two vectors became more closely aligned downstream of the bend. Reductions in the stress/energy ratio implied that crossflow made shear-stress production less efficient. Another effect of three-dimensionality was a change of sign in the vertical transport of turbulent kinetic energy by turbulence, in the inner part of the boundary layer. © 1994, Cambridge University Press. All rights reserved.

Description: The solidification of hot fluid flowing in a thin buoyancy-driven layer between cold solid boundaries is analysed in a series of two papers. As an approximation to flow in a crack in a weakly elastic solid or to free-surface flow beneath a thin solidified crust, the boundaries are considered to be flexible and to exert negligible resistance to lateral deformation. The resultant equations of continuity and motion reduce to a kinematic-wave equation with a loss term corresponding to the accumulation of solidified material at the boundaries. The Stefan problem for the solidification is coupled back to the flow through the advection of heat by the fluid, which competes with lateral heat loss by conduction to the solid. Heat and mass conservation are used to derive boundary conditions at the propagating nose of the flow. In this paper the two-dimensional flow produced by a line release of a given volume of fluid is investigated. It is shown that at short times the flow solidifies completely only near the point of release where the flow is thinnest, at later times complete solidification also occurs near the nose of the flow where the cooling rates are greatest and, eventually, the flow is completely solidified along its depth. Some transient melting of the boundaries can also occur if the fluid is initially above its solidification temperature. The dimensionless equations are parameterized only in terms of a Stefan number S and a dimensionless solidification temperature O. Asymptotic solutions for the flow at short times and near the source are derived by perturbation series and similarity arguments. The general evolution of the flow is calculated numerically, and the scaled time to final solidification, the length and the thickness of the solidified product are determined as functions of S and 0. The theoretical solutions provide simple models of the release of a pulse of magma into a fissure in the Earth's lithosphere or of lava flow on the flanks of a volcano after a brief eruption. Other geological events are better modelled as flows fed by a continual supply of hot fluid. The solidification of such flows will be investigated in Part 2. © 1994, Cambridge University Press. All rights reserved.

Description: The inviscid instability of 0(ɛ) two-dimensional periodic flows to spanwise-periodic longitudinal vortex modes in parallel 0(1) shear flows is considered. In such cases, not only is the effect of fluctuations upon the mean state important but also the influence of the developing mean flow on the fluctuating part of the motion. The former is described by a generalized Lagrangian-mean formulation; the latter by a modified Rayleigh equation. Of specific interest is whether the spanwise distortion of the wave field feeds back to enhance or inhibit instability to longitudinal vortex form. Two cases are considered in detail: uniform shear between wavy walls and non-uniform shear beneath free-surface waves. In both cases wave distortion acts to inhibit, and in some circumstances curtail, instability for all but the shortest waves. © 1994, Cambridge University Press. All rights reserved.

Description: When air blows over water the wind exerts a stress at the interface thereby inducing in the water a sheared turbulent drift current. We present scaling arguments showing that, if a wind suddenly starts blowing, then the sheared drift current grows in depth on a timescale that is larger than the wave period, but smaller than a timescale for wave growth. This argument suggests that the drift current can influence growth of waves of wavelength λ that travel parallel to the wind at speed c. In narrow ‘inner’ regions either side of the interface, turbulence in the air and water flows is close to local equilibrium; whereas above and below, in ‘outer’ regions, the wave alters the turbulence through rapid distortion. The depth scale, la, of the inner region in the air flow increases with c/u*a (u*a is the unperturbed friction velocity in the wind). And so we classify the flow into different regimes according to the ratio la/λ. We show that different turbulence models are appropriate for the different flow regimes. When (u*a + c)/UB(λ) ≪ 1 (UB(z) is the unperturbed wind speed) la is much smaller than λ. In this limit, asymptotic solutions are constructed for the fully coupled turbulent flows in the air and water, thereby extending previous analyses of flow over irrotational water waves. The solutions show that, as in calculations of flow over irrotational waves, the air flow is asymmetrically displaced around the wave by a non-separated sheltering effect, which tends to make the waves grow. But coupling the air flow perturbations to the turbulent flow in the water reduces the growth rate of the waves by a factor of about two. This reduction is caused by two distinct mechanisms. Firstly, wave growth is inhibited because the turbulent water flow is also asymmetrically displaced around the wave by non-separated sheltering. According to our model, this first effect is numerically small, but much larger erroneous values can be obtained if the rapid-distortion mechanism is not accounted for in the outer region of the water flow. (For example, we show that if the mixing-length model is used in the outer region all waves decay!) Secondly, non-separated sheltering in the air flow (and hence the wave growth rate) is reduced by the additional perturbations needed to satisfy the boundary condition that shear stress is continuous across the interface. In a companion paper, we develop a numerical model for the coupled air-water flow with waves of arbitrary speed and in another we examine the detailed energy budget of the wave motions. © 1994, Cambridge University Press. All rights reserved.

Description: The onset of Kármán-vortex shedding is studied experimentally in the wake of different two-dimensional bluff bodies, namely an oblong cylinder, circular cylinders and plates of rectangular cross-section. Different control measures, such as wake heating, transverse body oscillations and base bleed are investigated. As the steady-periodic Kármán shedding has previously been identified as a limit-cycle, i.e. as self-excited oscillations, the experiments are interpreted in the framework of the Stuart–Landau model. The coefficients of the Stuart–Landau equation for the characteristic vortex shedding amplitude, i.e. the linear temporal growth rate, linear frequency and the Landau constant, are fully determined for the two cylinders and in part for the plate. For this purpose transients are generated by suddenly switching transverse body oscillations or base bleed on or off. The analysis of these transients by a refined method based on complex demodulation provides reliable estimates of the model coefficients and yields an experimental validation of the concept that a global instability mode grows or decays as a whole. Also, it is demonstrated that the coefficients of the Stuart–Landau equation are independent of the experimental technique used to produce the transients. © 1994, Cambridge University Press. All rights reserved.

Description: The experimental results for an equilibrium boundary layer in a strong adverse pressure gradient flow are reported. The measurements show that similarity in the mean flow and the turbulent stresses has been achieved over a substantial streamwise distance where the skin friction coefficient is kept at a low, constant level. Although the Reynolds stress distribution across the layer is entirely different from the flow at zero pressure gradient, the ratios between the different turbulent stress components were found to be similar, showing that the mechanism for distributing the turbulent energy between the different components remains unaffected by the mean flow pressure gradient. Close to the surface the gradient of the mixing length was found to increase from Kl≈ 0.41 to Kl≈ 0.78, almost twice as high as for the zero pressure gradient case. Similarity in the triple correlations was also found to be good. The correlations show that there is a considerable diffusion of turbulent energy from the central part of the boundary layer towards the wall. The diffusion mechanism is caused by a second peak in the turbulence production, located at y/δ≈0.45. This production was for the present case almost as strong as the production found near the wall. The energy budget for the turbulent kinetic energy also shows that strong dissipation is not restricted to the wall region, but is significant for most of the layer. © 1994, Cambridge University Press. All rights reserved.

Description: The use of a classical eddy parametrization in the analysis of continental boundary currents leads to the diffusion of momentum and relative vorticity and fails to recognize that the relevant eddies are dominated by the conservation of potential vorticity, which in turn may produce an increase in the mean relative vorticity. To illustrate this effect, we examine a non-inflected barotropic shear flow destabilized by the cross-steam variation in the bottom topògraphy of a continental slope. The finiteamplitude evolution of the waves is analysed in a simple model with a step-like bottom topography and with a piecewise-uniform potential vorticity distribution. The increase in maximum mean vorticity is computed for various values of the Rossby number and the topographic elevation, and it is suggested that a similar effect, taking into account the isopycnal topography as well as the isobaths, could maintain the large inshore shear of the Gulf Stream. Cross-shelf transport of different water ‘types’ (i.e. potential vorticity and passive tracers) are also computed and suggested to be pertinent to the more realistic oceanic problem involving baroclinic effects. The numerical calculation employs the well-known method of contour dynamics, and the Green's function appropriate for the step-like topography is derived. © 1994, Cambridge University Press. All rights reserved.

Description: Steady three-dimensional convection in the form of bimodal cells in a fluid layer heated from below with rigid boundaries is studied through numerical computations for Prandtl numbers in the range 10 〈 P 〈 100. The stability of the steady solutions with respect to disturbances of various symmetries has been analysed. Typically, the range of stable steady bimodal convection is restricted by the transition to oscillatory bimodal convection. The oscillations preserve the spatial symmetry of the steady bimodal convection pattern in the case of high P and higher wavenumbers, but break it in the case of lower P or lower wavenumbers in the range that has been investigated. Some comparisons are made with experimental observations. The transition from bimodal to knot convection has also been studied. © 1994, Cambridge University Press. All rights reserved.

Description: The evolution of three-dimensional disturbances in an incompressible three-dimensional stagnation-point flow in an inviscid fluid is investigated. Since it is not possible to apply classical normal-mode analysis to the disturbance equations for the fully three-dimensional stagnation-point flow to obtain solutions, an initial-value problem is solved instead. The evolution of the disturbances provides the necessary information to determine stability and indeed the complete transient as well. It is found that when considering the disturbance energy, the planar stagnation-point flow, which is independent of one of the transverse coordinates, represents a neutrally stable flow whereas the fully three-dimensional flow is either stable or unstable, depending on whether the flow is away from or towards the stagnation point in the transverse direction that is neglected in the planar stagnation point. © 1994, Cambridge University Press

Description: A global, three-dimensional stability analysis of the steady and the periodic cylinder wake is carried out employing a low-dimensional Galerkin method. The steady flow is found to be asymptotically stable with respect to all perturbations for Re 〉 54. The onset of periodicity is confirmed to be a supercritical Hopf bifurcation which can be modelled by the Landau equations. The periodic solution is observed to be only neutrally stable for 54 〈 Re 〈 170. While two-dimensional perturbations of the vortex street rapidly decay, three-dimensional perturbations with long spanwise wavelengths neither grow nor decay. The periodic solution becomes unstable at Re = 170 by a perturbation with the spanwise wavelength of 1.8 diameters. This instability is shown to be a supercritical Hopf bifurcation in the spanwise coordinate and leads to a three-dimensional periodic flow. Finally the transition scenario for higher Reynolds numbers is discussed. © 1994, Cambridge University Press

Description: A kinematic description is presented of how turbulent diffusion from a continuous source varies with the sampling time in stationary, homogeneous turbulence. Unlike most previous theories, the sampling is assumed to take place at fixed downstream distances from the source. It is shown that the sampling-time effects depend on two-particle velocity statistics. Thus, time-average diffusion at fixed downstream distances is more akin to relative diffusion than to absolute diffusion. Two simple diffusion models are developed from the kinematic equations. These models are in fairly good agreement with diffusion data obtained both in a wind tunnel and in the field. Moreover, these models have significant practical implications. For example, the models indicate that care must be taken when using band-pass spectral filtering as a paradigm for turbulent diffusion. Also, the models show that the mean flow speed U has an important influence on the sampling-time effects. To account for U properly, diffusion measurements with differing sampling times Λ should be compared using the product UΛ, and not just Λ. © 1994, Cambridge University Press. All rights reserved.

Description: We study the advection of a passive scalar by a vortex couple in the small-diffusion (i.e. large Peclet number, Pe) limit. The presence of weak diffusion enhances mixing within the couple and allows the gradual escape of the scalar from the couple into the surrounding flow. An averaging technique is applied to obtain an averaged diffusion equation for the concentration inside the dipole which agrees with earlier results of Rhines & Young for large times. At the outer edge of the dipole, a diffusive boundary layer of width O(Pe-1/2) forms; asymptotic matching to the interior of the dipole yields effective boundary conditions for the averaged diffusion equation. The analysis predicts that first the scalar is homogenized along the streamlines on a timescale O(Pe1/3). The scalar then diffuses across the streamlines on the diffusive timescale, O(Pe). Scalar that diffuses into the boundary layer is swept to the rear stagnation point, and a finite proportion is expelled into the exterior flow. Expulsion occurs on the diffusive timescale at a rate governed by the lowest eigenvalue of the averaged diffusion equation for large times. A split-step particle method is developed and used to verify the asymptotic results. Finally, some speculations are made on the viscous decay of the dipole in which the vorticity plays a role analogous to the passive scalar. © 1994, Cambridge University Press

Description: A low-resolution (643) large-eddy model of forced homogeneous turbulence is numerically simulated using Kraichnan's eddy viscosity. The introduction of a reliable statistical estimate of the Cp exponents allows one to perform a detailed statistical analysis of the velocity field and shows that the probability distribution functions, the structure functions and the power-law exponents p agree with previous numerical and experimental results obtained at much higher effective resolution. This result shows how a simple modelling of the energy transfer produces self-similar dynamics extending to the small scales and obtains the right statistical properties of the velocity field. © 1994, Cambridge University Press

Description: A theoretical study is presented of three-dimensional turbulent flow provoked in a boundary layer by an array of low-profile vortex generators (VGs) on the surface. The typical VG sits in the logarithmic region of the incident boundary layer, and the turbulence model used seems representative in this region. The governing equations yield a forward-marching three-dimensional vortex-type system, which is solved computationally and analytically for spanwise periodic VG arrays. Streamwise vortex patterns of various strengths are produced downstream, owing to three-dimensional distortion of the original logarithmic profile and to the turbulent stresses present. Predictions are given for certain basic VG shapes, e.g. triangular, with various spanwise spacings, and the predictions are found to agree favourably overall with recent experiments. In addition, the analytical formulae obtained prove useful in suggesting designs for favourable VG distributions, based on three factors: close spanwise packing, increased VG length, and suitably non-smooth spanwise shaping. © 1994, Cambridge University Press. All rights reserved.

Description: A complex Stokes flow has several cells, is subject to bifurcation, and its velocity field is, with rare exceptions, only available from numerical computations. We present experimental and computational studies of two new complex Stokes flows: a vortex mixing flow and multicell flows in slender cavities. We develop topological relations between the geometry of the flow domain and the family of physically realizable flows; we study bifurcations and symmetries, in particular to reveal how the forcing protocol's phase hides or reveals symmetries. Using a variety of dynamical tools, comparisons of boundary integral equation numerical computations to dye advection experiments are made throughout. Several findings challenge commonly accepted wisdom. For example, we show that higher-order periodic points can be more important than period-one points in establishing the advection template and extended regions of large stretching. We demonstrate also that a broad class of forcing functions produces the same qualitative mixing patterns. We experimentally verify the existence of potential mixing zones for adiabatic forcing and investigate the crossover from adiabatic to non-adiabatic behaviour. Finally, we use the entire array of tools to address an optimization problem for a complex flow. We conclude that none of the dynamical tools alone can successfully fulfil the role of a merit function; however, the collection of tools can be applied successively as a dynamical sieve to uncover a global optimum. © 1994, Cambridge University Press

Description: The nonlinear problem of the steady-state interaction of a closed fluid-filled cylindrical elastic membrane with a slow viscous shear flow has been solved by a series-expansion technique. The problems of successive orders were both formulated and solved by a symbolic manipulation program, and the calculations were carried to sixth order in a dimensionless parameter related to the applied shear rate. Moderately large deformations (aspect ratios approaching 3) fall within the range of this analysis, which yields the dependences of the following global variables on the system parameters: membrane deformation, orientation, and strain, as well as tank-treading frequency, and mean internal pressure. The solution for the flow field around an isolated capsule is also used to calculate the apparent viscosity of a dilute suspension of flexible cylindrical particles, which yields the paradoxical result that the apparent viscosity decreases as the internal viscosity increases. © 1994, Cambridge University Press

Description: Studies of three-dimensional Stokes flow of two Newtonian fluids that converge in a T-type bifurcation have important applications in polymer coextrusion, blood flow through the venous microcirculation, and other problems of science and technology. This flow problem is simulated numerically by means of the finite element method, and the solution demonstrates that the viscosity ratio between the two fluids critically affects flow behaviour. For the parameters investigated, we find that as the viscosity ratio between the side branch and the main branch increases, the interface between the merging fluids bulges away from the side branch. The viscosity ratio also affects the velocity distribution: at the outlet branch, the largest radial gradients of axial velocity appear in the less-viscous fluid. The distribution of wall shear stress is non-axisymmetric in the outlet branch and may be discontinuous at the interface between the fluids. © 1994, Cambridge University Press

Description: Unsteady flow due to an oscillating sphere with a velocity U0cosωt’, in which U0 and ω are the amplitude and frequency of the oscillation and t’ is time, is investigated at finite Reynolds number. The methods used are: (i) Fourier mode expansion in the frequency domain; (ii) a time-dependent finite difference technique in the time domain; and (iii) a matched asymptotic expansion for high-frequency oscillation. The flow fields of the steady streaming component, the second and third harmonic components are obtained with the fundamental component. The dependence of the unsteady drag on ω is examined at small and finite Reynolds numbers. For large Stokes number, ε = (ωa2/2v)½≫ 1, in which a is the radius of the sphere and v is the kinematic viscosity, the numerical result for the unsteady drag agrees well with the high-frequency asymptotic solution; and the Stokes (1851) solution is valid for finite Re at ε ≫ 1. For small Strouhal number, St = ωa/U0 ≪ 1, the imaginary component of the unsteady drag (Scaled by 6πU0pfva, in which Pf is the fluid density) behaves as Dml ∼ (h0Stlog St–h1St), m = 1,3,5… This is in direct contrast to an earlier result obtained for an unsteady flow over a stationary sphere with a small-amplitude oscillation in the free-stream velocity (hereinafter referred to as the SA case) in which D1∼ –h1 St (Mei, Lawrence & Adrian 1991). Computations for flow over a sphere with a free-stream velocity U0(1–α1+α1cosωt’) at Re = U02a/v = 0.2 and St ≪ 1 show that h0 for the first mode varies from 0 (at α1 = 0) to around 0.5 (at α1 = 1) and that the SA case is a degenerated case in which the logarithmic dependence of the drag in St is suppressed by the strong mean uniform flow. The numerical results for the unsteady drag are used to examine an approximate particle dynamic equation proposed for spherical particles with finite Reynolds number. The equation includes a quasi-steady drag, an added-mass force, and a modified history force. The approximate expression for the history force in the time domain compares very well with the numerical results of the SA case for all frequencies; it compares favourably for the PO case for moderate and high frequencies; it underestimates slightly the history force for the PO case at low frequency. For a solid sphere settling in a stagnant liquid with zero initial velocity, the velocity history is computed using the proposed particle dynamic equation. The results compare very well with experimental data of Moorman (1955) over a large range of Reynolds numbers. The present particle dynamic equation at finite Re performs consistently better than that proposed by Odar & Hamilton (1964) both qualitatively and quantitatively for three different types of spatially uniform unsteady flows. © 1994, Cambridge University Press. All rights reserved.

Description: Weakly nonlinear descriptions of axisymmetric cnoidal and solitary waves in vortices recently have been shown to have strongly nonlinear counterparts. The linear stability of these strongly nonlinear waves to three-dimensional perturbations is studied, motivated by the problem of vortex breakdown in open flows. The basic axisymmetric flow varies both radially and axially, and the linear stability problem is therefore nonseparable. To regularize the generalization of a critical layer, viscosity is introduced in the perturbation problem. In the absence of the waves, the vortex flows are linearly stable. As the amplitude of a wave constituting the basic flow increases owing to variation in the level of swirl, stability is first lost to non-axisymmetric ‘bending’ modes. This instability occurs when the wave amplitude exceeds a critical value, provided that the Reynolds number is larger enough. The critical wave amplitudes for instability typically are large, but not large enough to create regions of closed streamlines. Examination of the most-amplified eigenvectors shows that the perturbations tend to be concentrated downstream of the maximum streamline displacement in the wave, in a position consistent with the observed three-dimensional perturbations in the interior of a bubble type of vortex breakdown. © 1994, Cambridge University Press

Description: Moore's (1978) equation for following the evolution of a thin layer of uniform vorticity in two dimensions is extended to the case of a non-uniform, instantaneously known, vorticity distribution, using the method of matched asymptotic expansions. In general, the vorticity distribution satisfies a boundary-layer equation. This has a similarity solution in the case of a vortex layer of small thickness in a viscous fluid. Using this solution, an equation of motion of a diffusing vortex sheet is obtained. The equation retains the simplicity of BirkhofTs integro-differential equation for a vortex sheet, while incorporating the effect of viscous diffusion approximately. The equation is used to study the growth of long waves on a Rayleigh layer. © 1994, Cambridge University Press

Description: Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimensions, the circulation deposited on a planar interface may be derived analytically using shock polar analysis provided the shock refraction is regular. We present analytical expressions for Γ, the circulation deposited per unit length of the unshocked planar interface, within and beyond the regular refraction regime. To lowest order, F’ scales as where M is the Mach number of the incident shock, η is the density ratio of the gases across the interface, a is the angle between the shock and the interface and γ is the ratio of specific heats for both gases. For a 〉 30°, the error in this approximation is less than 10% for 1.0 〉 M 〉 1.32 for all η 〈 1, and 5.8 〈 r 〈 32.6 for all M. We validate our results by quantification of direct numerical simulations of the compressible Euler equations with a second-order Godunov code. We generalize the results for total circulation on non-planar (sinusoidal and circular) interfaces. For the circular bubble case, we introduce a near-normality’ ansatz and obtain a model for total circulation on the bubble surface that agrees well with results of direct numerical simulations. A comparison with other models in the literature is presented. © 1994, Cambridge University Press

Description: Resonant interactions between internal gravity waves propagating in a stratified shear flow are considered for the case when the background density and shear flow vary slowly with respect to the waves. In Grimshaw (1988) triad resonances were considered, and interaction equations derived for the case when the resonance conditions are met only on certain space-time surfaces, being resonance sites. Here this analysis is extended to include higher-order resonances, with the aim of studying resonant wave interactions near a critical level. It is shown that a secondary resonant interaction between two incoming waves, in which two harmonic components of one incoming wave interact with a single harmonic component of another incoming wave, produces a reflected wave. This result is shown to agree with the study of Brown & Stewartson (1980, 1982 a, b) who obtained this same result by a different approach. © 1994, Cambridge University Press

Description: The instability of a viscous fluid inside a rectangular tank oscillating about an axis parallel to the largest face of the tank is investigated in the linear regime. The flow is shown to be unstable to both longitudinal roll and standing wave instabilities. The particular cases of low and high oscillation frequencies are discussed in detail. The relationship between the roll instability and convective or centrifugal instabilities in unsteady boundary layers is discussed. The eigenvalue problems associated with the roll and standing wave instabilities are solved using Floquet theory and a combination of numerical and asymptotic methods. The results obtained are compared to the recent experimental investigation of Bolton & Maurer (1994) which indeed provided the stimulus for the present investigation. © 1994, Cambridge University Press

Description: The first highly accurate solutions for the resistance tensor of an oblate or prolate spheroid moving near a planar wall obtained by Hsu & Ganatos (1989) are used to compute the translational and angular velocities and trajectories of a neutrally buoyant spheroid in shear flow and the gravitational settling motion of a non-neutrally buoyant spheroid adjacent to an inclined plane. The neutrally buoyant spheroid in shear flow undergoes a periodical motion toward and away from the wall as it continually tumbles forward. For some orientation angles it is found that the wall actually enhances the angular velocity of the particle. For certain inclinations a spheroid settling under gravity near an inclined plane reaches an equilibrium position, after which it translates parallel to the wall without rotation. © 1994, Cambridge University Press

Description: A new roll-type instability has been discovered experimentally. When fluid between two closely spaced, parallel plates is oscillated about an axis midway between the plates, it exhibits an instability that takes the form of longitudinal rolls aligned perpendicular to the axis of rotation. The basic-state oscillatory shear flow, before the onset of rolls, may be viewed as driven by the Q x r term of the Navier-Stokes equation in the oscillatory reference frame. A regime diagram is presented in a parameter space defined by the maximum amplitude of angular oscillation, a, and the non-dimensional frequency, 0 = wd2/v. The equilibrium wavelength of the rolls scales with d, the gap spacing between the plates, and it increases as 0 increases. Supercritical to a weak-roll onset, an abrupt transition to stronger roll amplitude occurs. Photographs of the cell after an impulsive start show the roll development and initial increase in roll wavelength. A variety of phenomena are observed, including wavelength selection via defect creation and elimination, front propagation, secondary wave instabilities, and the transition to turbulence. We also present solutions of the Navier-Stokes equation for the basic-state shear flow in a near-axis approximation. We develop a simple resonance model which shows some promise in understanding the low-a, high-0 behaviour of strong rolls. A theoretical analysis of this instability is presented by Hall (1994). © 1994, Cambridge University Press

Description: The flow field induced by a vertical plate accelerating into a stationary fluid of finite depth with a free surface and a gravitational restoring force is investigated. This is a model problem for some technologically important design issues such as the bow splash of a ship moving at forward speed. Experimentally it is found that a thin jet forms on the plate and rises rapidly upwards. We investigate this jet in the small-time approximation and find an analytical solution for the flow field in which the jet emerges out of a thin region where the horizontal momentum of the main flow is converted by inertial effects into a rising jet. © 1994, Cambridge University Press

Description: The derivation of a new vectorial bedload formulation for the transport of coarse sediment by fluid flow is presented in the first part of the paper. This relation has been developed for slopes up to the angle of repose both in the streamwise and transverse directions. The pressure distribution is assumed to be hydrostatic. The bed shear stress for the onset of particle motion and mean particle velocity are obtained from the mean force balance on a particle. A new generalized Bagnold hypothesis is introduced to calculate the sediment content of the bedload layer. The new formulation possesses two innovative features. It is fully nonlinear and vectorial in nature, in addition, it behaves smoothly up to the angle of repose. A mathematical model of the time evolution of straight river channels is presented in the second half of the paper. This study focuses on the evolution process due to bank erosion in the presence of bedload only. The bed and bank material is taken to be coarse, non-cohesive and uniform in size. The sediment continuity and the fluid momentum conservation equations describe the time evolution of the bed topography and flow field. These equations are coupled through the fluid shear stress acting on the bed. This bed shear stress distribution is predicted with the aid of a simple algebraic turbulent closure model. As regards the computation of the sediment flux, the new fully nonlinear vectorial formulation is found to perform well and renders the evolution model fully mechanistic. The formation of an erosional front in the time development of straight river channels has been so far obscured in physical experiments. Herein, with the help of the new bedload formulation, the existence and migration speed of the front of erosion are inferred from the analysis of the sediment continuity equation. The model successfully describes the time relaxation of an initially trapezoidal channel toward an equilibrium cross-sectional shape, as evidenced by comparison with experimental data. This equilibrium is characterized by a constant width, vanishing sediment transport in the transverse direction, and a small but non-vanishing streamwise transport rate of bed sediment. © 1994, Cambridge University Press. All rights reserved.

Description: Nonlinear stability of a model swept-wing boundary layer, subject to crossflow instability, is investigated by numerically solving the governing partial differential equations. The three-dimensional boundary layer is unstable to both stationary and travelling crossflow disturbances. Nonlinear calculations have been carried out for stationary vortices and the computed wall vorticity pattern results in streamwise streaks which resemble quite well the surface oil-flow visualizations in swept-wing experiments. Other features of the stationary vortex development (half-mushroom structure, inflected velocity profiles, vortex doubling, etc.) are also captured in these calculations. Nonlinear interaction of the stationary and travelling waves is also studied. When initial amplitude of the stationary vortex is larger than the travelling mode, the stationary vortex dominates most of the downstream development. When the two modes have the same initial amplitude, the travelling mode dominates the downstream development owing to its higher growth rate. It is also found that, prior to laminar/turbulent transition, the three-dimensional boundary layer is subject to a high-frequency secondary instability, which is in agreement with the experiments of Poll (1985) and Kohama, Saric & Hoos (1991). The frequency of this secondary instability, which resides on top of the stationary crossflow vortex, is an order of magnitude higher than the frequency of the most-amplified travelling crossflow mode. © 1994, Cambridge University Press

Description: In a two-dimensional incompressible fluid, the barotropic instability of isolated circular vortices can lead to multipole formation. The multipoles we study here are composed of a core vortex surrounded by two or more identical satellite vortices, of opposite-sign vorticity to the core, and the total circulation is zero. First, we present the generation of multipoles from unstable piecewise-constant monopoles perturbed on a monochromatic azimuthal mode. The stationary multipoles formed by this nonlinear evolution retain the same energy, circulation and angular momentum as the original monopoles, but possess a lower enstrophy. These multipolar steady states are then compared to multipolar equilibria of the Euler equation, obtained either analytically by a perturbation expansion or numerically via a relaxation algorithm. Finally the stability of these equilibria is studied. Quadrupoles (one core vortex bound to three satellites) prove relatively robust, whether initially perturbed or not, and resist severe permanent deformations (mode-2 shears or strains of amplitude up to 0.1C„„x)* Amplification of the mode-3 deformation proves more destructive. More complex multipoles degenerate in less than a turnover period into end-products of a lesser complexity, via vortex splitting, pairing or merging. We use the conservation of integral properties to classify the large variety of instability mechanisms along physical guidelines. To conclude, we synthetize the connections between these various vortex forms. © 1994, Cambridge University Press

Description: A series of laboratory experiments was carried out to examine the interaction between stratification and turbulence at an inversion layer, with the objective of gaining insight into certain wave-turbulence encounters in the atmosphere. A three-layer stratified fluid system, consisting of a (thick) strongly stratified inversion layer, sandwiched between an upper turbulent layer and a lower non-turbulent weakly stratified layer, was employed. Oscillating-grid-induced shear-free turbulence was used in the upper layer. During the experiments, a fourth (interfacial) layer developed in the region between the inversion and the turbulent layer, and most of the wave-turbulence interactions were confined to this layer. Detailed measurements of the vertical velocity structure, internal-wave parameters and mixing characteristics were made in the stratified layers and, whenever possible, the results were compared to available theoretical predictions. © 1994, Cambridge University Press