Volume 12, Issue 11, November 2000
 LETTERS


On the numerical treatment of viscous singularities in wallconfined thermocapillary convection
View Description Hide DescriptionConfined thermocapillary flows often present a viscous singularity where free and solid surfaces meet. Any computational treatment of these flows must filter this singularity, either explicitly by modifying the boundary conditions of the problem, or implicitly by using finite precision methods. The effect of an explicit polynomial filtering is here examined for an axisymmetric sideheated liquid bridge, leading to two results: (i) thermally convective flows converge very slowly with the filtering scale; the numerical experiments must therefore be conducted carefully in order to get physically relevant results; (ii) streamfunction and temperature are not suitable test variables for estimating this convergence.

Symmetry breaking of the flow in a cylinder driven by a rotating end wall
View Description Hide DescriptionThe flow driven by a rotating end wall in a cylindrical container with aspect ratio is time dependent for Reynolds numbers For Reynolds numbers up to 4000 three solution branches have been identified, and we examine a solution on each one. At the flow is axisymmetric and time periodic. At the flow is quasiperiodic with a lowfrequency modulation and supports a modulated rotating wave with azimuthal wave number At the flow is time periodic with a qualitatively different mode of oscillation to that at It also supports a modulated rotating wave, with The peak kinetic energy of the nonaxisymmetric modes is associated with the jetlike azimuthal flow in the interior.
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 ARTICLES


Stability of viscoelastic shear flows subjected to parallel flow superposition
View Description Hide DescriptionSteady viscoelasticshear flows in curved geometries are susceptible to instabilities due to the radial force associated with elastic stresses along curved streamlines. Recent work has shown that the addition of steady or oscillatory shear flow in a transverse direction (orthogonal superposition) can suppress these instabilities. The present work instead investigates the effect of oscillatory parallel superposition, for the particular case of circular Couette flow. For flow of an OldroydB fluid, the oscillation has a weak stabilizing effect if the oscillation amplitude is not too large. If on the other hand, the oscillation amplitude is such that the angular velocity of the moving cylinder changes sign over part of the cycle, the flow is destabilized. However, in the limit where the motion is purely oscillatory (i.e., large amplitude oscillatory shear), the flow is stabilized relative to steady circular Couette flow with the same maximum shear rate. Finally, in the limit where curvature goes to zero—plane Couette flow—parallel superposition makes the already stable flow slightly more stable, increasing the decay rate of fluctuations in the range of parameters studied. Furthermore, twodimensional disturbances are the most slowly decaying; we have extended Squire’s theorem in this case to encompass an arbitrary time dependence.

Viscous drag of a solid sphere straddling a spherical or flat surface
View Description Hide DescriptionThe aim of this paper is to compute the friction felt by a solid particle, of radius a, located across a flat or spherical interface of radius R, and moving parallel to the interface. This spherical interface can be a molecular film around an emulsion or aerosoldroplet, the membrane of a vesicle or the soap film of a foam bubble. For simplicity, the acronym VDB is used to refer to either vesicle, drop, or bubble. The theory is designed as a tool to interpret surface viscosimetry experiments involving spherical probes attached to films or model membranes, taking care of the finitesize effects when the film encompasses a finite fluid volume. The surface of the VDB is a twodimensional fluid, characterized by dilational and shear surfaceviscosities. The particle intercepts a circular disc in the interface, whose size depends on the particle penetration inside the VDB. The threedimensional fluids inside and outside the interface may be different. The analysis holds in the low Reynolds number and low capillary number regime. A toroidal coordinate system is introduced, which considerably simplifies the geometry of the problem. Then the hydrodynamic equations and boundary conditions are written in φ. The solution is searched for the firstorder Fourier component of the velocity field in the radial angle φ. Reformulating the equations in “twovorticityonevelocity” representation, one basically ends up with a set of equations in only. This set is numerically solved by means of the AlternatingDirectionImplicit method. Numerical results show that the particle friction is influenced both by the viscosity and by the finiteness of the VDB volume. Finitesize effects have two origins: a recirculation effect when is not very small, and an overall rotation of the VDBparticle complex when is very large. In principle, the theory allows for a quantitative determination of whatever including the limit (flat interface).

On the Horton–Rogers–Lapwood convective instability with vertical vibration: Onset of convection
View Description Hide DescriptionWe present a numerical and analytical study of diffusive convection in a rectangular saturated porous cell heated from below and subjected to high frequency vibration. The configuration of the Horton–Rogers–Lapwood problem is adopted. The classical Darcy model is shown to be insufficient to describe the vibrational flow correctly. The relevant system is described by timeaveraged Darcy–Boussinesq equations. These equations possess a pure diffusive steady equilibrium solution provided the vibrations are vertical. This solution is linearly stable up to a critical value of the stability parameter depending on the strength of the vibration. The solutions in the neighborhood of the bifurcation point are described analytically as a function of the strength of vibration, and the larger amplitude states are computed numerically using a spectral collocation method. Increasing the vibration amplitude delays the onset of convection and may even create subcritical solutions. The majority of primary bifurcations are of a special type of symmetrybreaking bifurcation even if the system is subjected to vertical vibration.

Electrohydrodynamic flow of a dielectric liquid around a blade electrode
View Description Hide DescriptionInjection of charge into a dielectric liquid, and a Coulomb force that sets the liquid into motion, may be obtained by applying a dc voltage to a bladeshaped, metallic electrode immersed in the liquid. An analysis of this motion and its influence on the transport of electric charge is carried out for a simple charge injection law. It is shown that the liquid motion, the electric field, and the charge distribution in a region around the electrode tip of size of the order of the electrode curvature radius determine the injected current as a function of the far electric potential seen by this region. The current increases exponentially with the potential when the contribution of the space charge to the electric field is negligible and algebraically when it is dominant, and presents a range of multiplicity in between. When the inertia of the liquid matters, the region around the electrode tip is also the origin of an electrohydrodynamic plume. An oscillatory current regime is found in which the space charge in the interelectrode space rearranges into many discrete lumps that, under constant voltage bias and small current, induce oscillations of the electric field at the injecting electrode and thus fire new lumps. An order of magnitude analysis and numerical computations for this regime give results in line with known experimental data. In conjunction with the hydrodynamic instability of the plume, this pulse firing mechanism is seen to lead to more complex, nonperiodic oscillations.

Clepsydrae, from Galilei to Torricelli
View Description Hide DescriptionThe whole article deals with free fall: the free fall of a solid particle carefully studied by Galileo Galilei and the free fall of a fluid particle along a stream line introduced by Evangelista Torricelli. Both limits are brought together in the problem of the vertical emptying of a cylindrical tube of diameter through a hole of diameter and one can move continuously from one limit to the other varying from to The limit, corresponds to Galilei’s problem of the solid free fall, in which the velocity of the upper interface, increases as it comes closer to the hole following the law where is the acceleration due to gravity, is the liquid height above the hole (defined by and is the initial location of the interface. The opposite limit, corresponds to Torricelli’s problem, in which the velocity of the interface slows down as it comes closer to the hole, following the law Theoretically, the problem reduces to the integration of the differential equation: where length and velocity have respectively been reduced by and and the axis oriented from the hole towards the initial interface location With the above equation leads to the solution when and when Galilei’s and Torricelli’s regimes correspond respectively to the limit and These solutions are compared to experimental measurements conducted over a large range of geometrical and physical parameters. In a second stage, the model, developed for nonconstant cross sectional area, is compared to the experimental results obtained in conical Clepsydrae.

Acoustic saturation in bubbly cavitating flow adjacent to an oscillating wall
View Description Hide DescriptionBubbly cavitating flow generated by the normal oscillation of a wall bounding a semiinfinite domain of fluid is computed using a continuum twophase flowmodel.Bubble dynamics are computed, on the microscale, using the Rayleigh–Plesset equation. A Lagrangian finite volume scheme and implicit adaptive time marching are employed to accurately resolve bubblyshock waves and other steep gradients in the flow. The onedimensional, unsteady computations show that when the wall oscillation frequency is much smaller than the bubble natural frequency, the power radiated away from the wall is limited by an acoustic saturation effect (the radiated power becomes independent of the amplitude of vibration), which is similar to that found in a pure gas. That is, for large enough vibration amplitude, nonlinear steepening of the generated waves leads to shocking of the wave train, and the dissipation associated with the jump conditions across each shock limits the radiated power. In the model, damping of the bubble volume oscillations is restricted to a simple “effective” viscosity. For wall oscillation frequency less than the bubble natural frequency, the saturation amplitude of the radiated field is nearly independent of any specific damping mechanism. Finally, implications for noise radiation from cavitating flows are discussed.

Threedimensional instability of a multipolar vortex in a rotating flow
View Description Hide DescriptionIn this paper, the elliptic instability is generalized to account for Coriolis effects and higher order symmetries. We consider, in a frame rotating at the angular frequency Ω, a stationary vortex which is described near its center by the stream function written in polar coordinates where the integer n is the order of the azimuthal symmetry, and p is a small positive parameter which measures the strength of the nonaxisymmetric field. Based on the Lifschitz and Hameiri [Phys. Fluids A 3, 2644–2651 (1991)] theory, the local stability analysis of the streamline is performed in the limit of small p. As for the elliptic instability [Bayly, Phys. Rev. Lett. 57, 2160–2163 (1986)], the instability is shown to be due to a parametric resonance of inertial waves when the inclination angle ξ of their wave vector with respect to the rotation axis takes a particular value given by An explicit formula for the maximum growth rate of the inertial wave is obtained for arbitrary ξ, Ω, and n. As an immediate consequence, it is shown that a vortex core of relative vorticity (assumed positive) is locally unstable if or The predictive power of the local theory is demonstrated on several vortex examples by comparing the local stability predictions with global stability results. For both the Kirchhoff vortex and Moore and Saffman vortex, it is shown how global stability results can be derived from the local stability analysis using the dispersion relation of normal (Kelvin) modes. These results are compared to those obtained by global methods and a surprisingly good agreement is demonstrated. The local results are also applied to rotating Stuart vortices and compared to available numerical data.

Point sink flow in a linearly stratified fluid of finite depth
View Description Hide DescriptionThe evolution of selective withdrawal through a point sink of horizontally unbounded, linearly stratified fluid of finite depth is studied as an initialvalue problem. Following the initiation of discharge from the sink, internal gravity wave modes propagate radially upstream to change the flow pattern. These modes are called cylindrical modes. We first consider the case of (where F is the Froude number) to get linearized governing equations, and seek a linear asymptotic solution for large times t ^{*} after starting the discharge, of the cylindrical modes in a stratified fluid where viscous and diffusive effects are negligible. The obtained solution shows that the strength of the modal front grows like unlike the case of twodimensional modes whose strength at the front is kept constant. Numerical calculations are also performed to study the case of The results then indicate that all the cylindrical modes can propagate upstream for any F except infinity. The steadystate withdrawallayer thickness and time to steady state are also investigated over the full parameter range considering the viscous and diffusive effects. The obtained results are then compared with both the analytical and experimental results of prior works.

Linear and nonlinear Rayleigh–Bénard–Marangoni instability with surface deformations
View Description Hide DescriptionThermoconvective instabilities in a bilayer liquid–gas system with a deformed interface are investigated. In the first part of the work which is devoted to a linear approach, emphasis is put on the role of the upper gas layer on the instability phenomenon. The condition to be satisfied by the gas to remain purely conductive is established. The socalled Oberbeck–Boussinesq approximation is discussed and its range of validity is carefully defined. Instead of the classical Rayleigh, Marangoni, crispation, and Galileo numbers, new dimensionless groups are introduced. A critical comparison with several previous works is made. The nonlinear analysis consists in studying the different convective patterns which can appear above the threshold. Particular attention is devoted to the shape of the interface and the socalled “hybrid” relief. The amplitude of the deformation is also determined and comparison with experimental data is discussed.

Stability of disconnected free surfaces in a cylindrical container under zero gravity: Simple cases
View Description Hide DescriptionThe stability of equilibrium configurations of a capillary liquid in a circular cylindrical container with planar ends is investigated. The liquid is under zero gravity conditions, and its wetting angle is constant over the entire solid surface. Attention is focused on the case for which the free surface consists of two disconnected pieces (connectivity components) that bound the connected liquid domain. First we outline the method used to determine critical states with disconnected free surfaces when each connectivity component is axisymmetric. Then we examine the stability of disconnected surfaces for the simple cases that arise when each connectivity component represents a closed sphere or a part of a sphere. Ten configurations were considered that represent all possible combinations of the following connectivity components: A closed sphere (that bounds a gas bubble), a spherical cap in contact with the lateral wall of a cylinder; a spherical cap in contact with a cylinder endwall, and a portion of a sphere (that does not cross the cylinder’s axis of symmetry) bounded by a cylindrical wall and a flat endwall.

Dynamics of electrohydrodynamic laminar plumes: Scaling analysis and integral model
View Description Hide DescriptionIn this paper electrohydrodynamic plumes are examined in the region far from the injecting electrode and the collector plate, for both twodimensional and axisymmetric geometries. The relative importance of the conduction mechanisms (convection, drift and diffusion of electric charge) is analyzed. Diffusion turns out to be negligible compared to convection and drift for the experimental conditions. But the transverse drift (Coulomb repulsion) is of the same order of magnitude than convection. We find a set of three differential equations giving the evolution of the velocity at the center of the plume and the widths of the plume and the charged core inside.

Green functions for impulsive freesurface flows due to bottom deflections in twodimensional topographies
View Description Hide DescriptionAn analytical investigation is performed of the initial freesurface flow subject to an impulsive concentrated unit flux through an arbitrary point at an otherwise impermeable boundary. The flow is assumed incompressible, inviscid, irrotational and twodimensional. It obeys a zeropotential condition at the horizontal free surface. This problem of impulsive Green functions is solved for various bottom topographies, such as sloping beaches, submerged ridges, and finite basins. The basic result in each case is the normal derivative of the velocity potential along the free surface, which represents the initial surface velocity due to the concentrated impulsive flux. The results have relevance to tsunami modeling.

Spectral decay of a passive scalar in chaotic mixing
View Description Hide DescriptionIn this paper we take a closer look at the decay phase of a passive, diffusing, scalar field undergoing steady, threedimensional chaotic advection. The energy spectrum of the scalar is obtained by numerical simulation of the advection–diffusion equation at high Péclet number. At large times, the spectral decay is found to be exponential and selfsimilar. It is emphasized that the asymptotic decaytime is an important measure of mixing efficiency, alongside the time required for diffusion to first become effective. The largewavenumber spectral form, representing the distribution of scalar energy over small scales, is analyzed. Powerlaw behavior is found at scales intermediate between the large ones, comparable in size with the entire flow volume, and the smallest ones, at which diffusion is effective and the spectrum falls off exponentially with increasing wavenumber. Fitting of the numerical results allows the exponent of the powerlaw to be estimated. It is observed to vary with the parameters of the flow, taking negative values which can be either less than or greater than −1. This implies that the dominant spectral energy at high P may be either at small, large or intermediate scales, depending on the flow. In consequence, the qualitative nature of the scalar field during the decay phase varies from flow to flow, resulting in differing behavior of the predicted decay times in the large P limit obtained by asymptotic analysis. The case in which the spectral exponent exceeds −1 is shown to produce more rapid mixing and the corresponding asymptotic expression for the decay time, independent of P and involving two spectral parameters, is suggested as a quantitative means for optimizing the flow.

Green’s function for steady flow over a small twodimensional topography
View Description Hide DescriptionWe consider steady flow of a thin viscousliquid film over a small twodimensional topography and develop a Green’s function for the linearized problem. The solutions so obtained can be used to analyze the effect of arbitrary small substrate defects on the coating applied to a substrate.

The fluid mechanics of fire whirls: An inviscid model
View Description Hide DescriptionWhirling fire plumes are known to increase the danger of naturally occurring or postdisaster fires. In order for a fire whirl to exist, there must be an organized source of angular momentum to produce the large swirl velocities as air is entrained into the fire plume. These vorticitydriven fires occur over a large range of length and velocity scales, and significantly alter the entrainment and combustion dynamics. A new model is derived for a buoyant plume that incorporates angular rotation and neglects dissipation; the result is a form of the steady state Euler equations. Included is a general solution for large density and temperature variations. Results are presented that identify the mechanisms and their effects toward creating a fire whirl.

New features of swirling jets
View Description Hide DescriptionImportant new features are found for a family of swirling jets with velocity where z is the distance from the jet origin. First, there is a sharp minimum of the pressure coefficient at a certain value of the swirl number which is nearly n independent; this feature can be utilized in technological devices. Second, as increases, a separation zone develops, where the fluid is not at rest in the inviscid limit (contrary to the claims of recent vortex breakdown theories). These results are obtained under the boundary layer approximation for incompressible jets characterized by n and where and are the maximal values of the swirl and longitudinal velocities at Unlike prior results viewed in terms of parameter L (which is the ratio at the outer edge of the jet), the solution dependence on is found similar for both and For any n, (a) the pressure coefficient is minimum at (b) two solutions exist for (fold value), none for (c) as decreases, the jets either consolidate near the axis or separate from it, depending on the solution branch; and (d) the flow in the separation zone tends to become swirlfree and potential.

Largeeddy simulations of turbulent flow in a rotating square duct
View Description Hide DescriptionThe turbulent flow at low Reynolds numbers in a rotating straight square duct was simulated using the largeeddy simulation technique. The rotation axis is parallel to two opposite walls of the duct, and the pressuredriven flow is assumed to be fully developed, isothermal and incompressible. The Reynolds number based on the friction velocity was kept constant in the range of the rotational numbers studied Computations were carried out using a secondorder finite volume code with a localized oneequation dynamic subgrid scale model. Simulations of rotating channel flows were initially carried out and were seen to be in agreement with experiments and direct numerical simulations reported in the literature. The study of the flow in a rotating square duct revealed the influence of the Coriolis force on the spatial distribution of the average velocity fields and Reynolds stresses. At low rotation rates, turbulencedriven secondary flows developed near the corners convect the rotationgenerated crossstream currents. At moderate and high rotation rates, the mean secondary flow structure consists essentially of two large counterrotating cells convecting low/high momentum fluid from the stable/unstable side to the unstable/stable side. Inspection of the terms of the transport equations of the average axial velocity and vorticity components shows the mechanisms responsible for the changes in the average flow structure. Spatial distributions of the Reynolds stresses are mainly influenced by the changes that rotation induces in the main strain rates. It has been found that, globally, at the low Reynolds number studied, rotation tends to significantly reduce the overall turbulence level of the flow.

Vortical turbulence structure and transport mechanism in a homogeneous shear flow
View Description Hide DescriptionDirect numerical simulations (DNS) are carried out to investigate the kinematics of vortical structures in a homogeneous shear flow, and their association with the momentum transfer is studied in detail. Longitudinal streamwise vortices are generated and conditionally averaged over all the computational region. The effects of the nonlinear term on their kinematics are investigated by comparing the DNS and Rapid Distortion Theory (i.e., RDT). As a result, some important similarities are found in the vortical structure between the homogeneous shear flow and nearwall turbulence. It is also found that the strain rate in the vortical structure, which is markedly affected by the nonlinear term, determines the transfer functions associated with the energy cascade of the turbulence.
