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Volume 9, Issue 2, February 1997

Hierarchy of transverse structure functions
View Description Hide DescriptionAn experimental validation of a recent statistical model [Phys. Rev. Lett. 72, 336 (1994)] is reported. The analysis is focused on the transverse components of the velocity structure functions, and extend previous tests conducted only on longitudinal velocity components.

Correlation between experimental results and numerical solutions of the Navier–Stokes problem for noncoalescing liquid drops with Marangoni effects
View Description Hide DescriptionExperiments show that, when two drops are brought in contact and pushed toward each other in the presence of temperature differences, coalescence is inhibited. Axisymmetric numerical solutions of the Navier–Stokes problem have been obtained to give an explanation of the phenomenon, assuming that a thin air film exists between the contacting drops and that, provided that welldefined dynamic conditions prevail on the liquid surfaces, the film experiences a suitable pressure that balances the pressure in the drops. The numerical results agree with the experimental ones, qualitatively explain why an air film between the drops could be created by Marangoni effects and show that the suppression of coalescence is obtained as long as films sufficiently large exhibit excess pressures of the same order of magnitude of the pressure needed to deform the drops.

The nature of inviscid vortex breakdown
View Description Hide DescriptionWe show that vortex breakdown appears as a jump bifurcation due to structural instability of swirling flows when their solution locally fails to exist and the flow transits to another stable or metastable state. The flow pattern inside the steady separation zone and, consequently, the vortex breakdown features depend on the flow history. Therefore, we take into account the flow pattern inside the separation zone. The stagnation zone model (without velocity jump) excels the traditional analytic continuation method (leading to a recirculation zone) in that solutions always exist, and, for large enough inflow swirl, exhibit nonuniqueness and folds due to smooth variations of flow parameters, thus predicting the experimentally observed hysteretic jump transitions.

Spreading of drops on solid surfaces in a quasistatic regime
View Description Hide DescriptionThe problem of interaction of a drop with a solid boundary is formulated in the framework of a recently developed theory of the threephase contact line motion and analyzed in the case of finite Bond and small capillary and Weber numbers. Evolution of the freesurface shape in a quasistatic regime of the drop spreading under gravity on a horizontal plane and on the surface of a rotating disk is investigated. In the considered regime, the freesurface shape deformation in time is independent of the initial conditions of the drop deposition onto the solid surface, while the threephase contactline motion is described by the same equations as in a general case. This feature makes the quasistatic regime informative and desirable from the point of view of investigation of the wetting phenomenon. Accuracy of the socalled “spherical cap approximation’’ often used in experimental studies of wetting is discussed. The theory describes both the “spontaneous” and “forced” regimes of the drop spreading and the transition between them. The results are compared with experimental data.

Linear nonaxisymmetric oscillations of nearly inviscid liquid bridges
View Description Hide DescriptionLinear nonaxisymmetric oscillations of liquid bridges are analyzed in the limit of large capillary Reynolds numberC ^{−1}. A boundary layer analysis is used that yields a second approximation of the damping rate and frequency in terms of the capillary Reynolds number, the slenderness of the bridge and the axial and azimuthal wave numbers of the mode being excited. Very good agreement with previous numerical results is obtained for C≲0.01, while the first correction of the damping rate gives a poor approximation, except for unrealistically small values of C (C≲10^{−5}).

Asymptotic solutions of miscible displacements in geometries of large aspect ratio
View Description Hide DescriptionAsymptotic solutions are developed for miscible displacements at Stokes flow conditions between parallel plates or in a cylindrical capillary, at large values of the geometric aspect ratio. The single integrodifferential equation obtained is solved numerically for different values of the Péclet number and the viscosity ratio. At large values of the latter, the solution consists of a symmetric finger propagating in the middle of the gap or the capillary. Constraints on conventional convectiondispersionequation approach for studying miscible instabilities in planar Hele–Shaw cells are obtained. The asymptotic formalism is next used to derive—in the limit of zero diffusion— a hyperbolic equation for the crosssectionally averaged concentration, the solution of which is obtained by analytical means. This solution is valid as long as sharp shock fronts do not form. The results are compared with recent numerical simulations of the full problem and experiments of miscible displacement in a narrow capillary.

Dynamics of small, spherical particles in vortical and stagnation point flow fields
View Description Hide DescriptionThe transport in vortical and stagnation point flow fields is analyzed for particles across the entire range of density ratios, based on the Maxey–Riley equation [Phys. Fluids 26, 883 (1983)] without history effects. For these elementary flow fields, the governing equations simplify substantially, so that analytical progress can be made towards quantifying ejection/entrapment trends and accumulation behavior. For a solid body vortex, the analysis shows that optimal ejection or entrapment occurs for all density ratios, as the difference between inward and outward forces reaches a maximum for intermediate values of the Stokes number. The optimal Stokes number value is provided as a function of the density ratio. Gravity is shown to shift accumulation regions, without affecting the entrapment or ejection rates. For a point vortexflow, the existence of up to three different regimes is demonstrated, which are characterized by different force balances and ejection rates. For this flow, optimal accumulation is demonstrated for intermediate Stokes numbers. The stagnation point flow gives rise to optimal accumulation for heavy particles, whereas light particles do not exhibit optimal behavior. The analysis furthermore indicates that nonvanishing density ratios give rise to a finite Stokes number regime in which the particle motion is oscillatory. Above and below this regime, the motion is overdamped.

The dynamics of a concentration interface in a dilute suspension of solid heavy particles
View Description Hide DescriptionGravitational settling of solid heavy particles in a dilute suspension is studied analytically and numerically. The particle Reynolds number is assumed to be less than unity, for which the viscous drag force on the particle is well approximated by the linear Stokes law. The particulate volume fraction (or concentration) is assumed to be small enough for the effects of particle–particle interactions to be negligible. The ratio of the particle and fluid densities is considered large enough however, so that the momentum exchange between the two phases caused by the viscous drag forces (which is of the order of the particulate mass loading factor is significant. The particulate base concentration, , is assumed to be a smooth function of the vertical coordinate (hence, a stratified suspension) and a perturbation of the initially stationary settling regime is considered in the form of a horizontally propagating monochromatic wave with wavenumber and frequency . Analytical solutions for the perturbations in the limit of small particle inertia (such that , where is the particle response time) are found to be similar to those for internal waves propagating in a stratified fluid with effective density . On the other hand, it is found that in the opposite limit of large particle inertia the perturbations are damped. As an example, we consider a suspension consisting of two layers with uniform concentrations of particles (for and (for separated by the interface layer of thickness , where the concentration gradient is substantial. The solutions obtained in the longwave limit show that if the concentration in the lower layer exceeds that in the upper layer , the disturbance of the interface brings about wavy motions analogous to internal waves in a twolayer fluid. In the case of inverse stratification the disturbance grows exponentially and generates plumelike “bubbles,’’ similar to those produced due to the Rayleigh–Taylor instability in a twolayer fluid. The results of the numerical simulations show that, as expected, the waves are damped and the instability growth rate is reduced for particles having larger inertia.

Some spinup effects on the geostrophic and quasigeostrophic drag on a slowly rising particle or drop in a rotating fluid
View Description Hide DescriptionIn this paper the initial “spinup” stage of the flow field generated by the slow axial motion of a symmetric particle or drop in a rotating fluid in a “short cylinder” of length is considered for small Rossby and Ekman numbers, and . The motion starts from solid body rotation. Attention is focused on timedependent effects in the extended core (outside the Ekman and Stewartson layers) and their influence on the behaviour of the drag force. When the particle velocity is established by an impulsive start, the flowfield and the drag build up on the spinup time scale; the particle advances during the transient stage. Results for the timedependent geostrophic case, and also for various values of , for both sphere and disk particles, are presented. It is shown that when the particle velocity is established by release under a constant axial (say, buoyant) force, during the spinup some rapid inertial oscillations may appear which are inconsistent with some of the assumption of the present analysis and require a separate investigation. The present theory is in qualitative agreement with available experiments, but a quantitative comparison requires new experiments.

Twodimensional numerical analysis of the Poiseuille–Bénard flow in a rectangular channel heated from below
View Description Hide DescriptionThe Poiseuille–Bénard flow (PBF) is studied by a twodimensional numerical simulation for a Prandtl number equal to 6.4 (that of water at 23 °C) and for a wide range of Rayleigh (Ra) and Reynolds (Re) numbers: Ra⩽6000 and Re⩽3. The two observed flow configurations are (1) thermally stratified Poiseuille flow and (2) thermoconvective transversal rolls superimposed to the basic Poiseuille flow. The time evolution of the velocity components, the spatial development of the transversal rolls, their frequency, wavelength and velocity, the Nusselt number, together with the stability map in the Ra–Re plane, are studied in detail. Whenever possible, quantitative comparisons are made with published results: most of the experimental data, based on laserDoppler anemometry (LDA), are recovered with amazing accuracy; a good agreement with results of convective stability deduced from a weakly nonlinear Ginzburg–Landau theory is also obtained.

Investigation of the stochastic collisions of drops produced by Rayleigh breakup of two laminar liquid jets
View Description Hide DescriptionThe stochastic collisions of drops of two intersecting streams were investigated experimentally. The drop streams were produced by Rayleigh breakup of two laminar jets of propanol2 and were arranged spatially so that they lie in one plane and intersect at an angle which was varied in the experiments. The collisional interactions of the drops were visualized using video equipment. In the zone between the drop streams downstream of the intersection point new drops occur which are formed by the collisions. The visualization showed that these new drops may be produced either by the merging of two colliding drops or by the breakup of liquid bridges formed between drops after offcenter collisions. Measurements of velocity and size of the drops in the flow field were carried out using a phaseDoppler anemometer (PDA). These data and the frequency of drop arrival in the measurementcontrol volume of the PDA give insight into the drop formation processes caused by the collisions and enable the computation of the collision frequency.

First transitions in circular Couette flow with axial stratification
View Description Hide DescriptionThe first flow regimes which have been observed experimentally for a circular Couette flow with a stable, axial stratification in density are investigated through direct numerical simulations of the threedimensional NavierStokes equations for a Boussinesq fluid. The setup of two concentric cylinders has a nondimensional gap width of the outer cylinder is fixed and the stratification in density in the axial direction is linear. The main effect of an axial density stratification is to reduce the height of the Taylor vortices and to cause the formation of density layers of small aspect ratio. For large enough Prandtl number, the primary bifurcation from circular Couette flow is found to be axisymmetric and of Hopftype in the direct numerical simulations. An analytical solution for onset of instability and slightly different boundary conditions from the experimental ones agrees within 0.6% with numerical simulations at a Prandtl number of 700. The experimental flow regimes with welldefined density layers are well reproduced by the numerical simulations in the appropriate range of relative Reynolds number where denotes the critical Reynolds number for the primary bifurcation from circular Couette flow. However, the increase of axial scale with is found to be continuous, whereas it is quantized in the laboratory experiments. Numerical results reveal that the first two transitions between the flow regimes are primarily due to the temporal behavior of the axially symmetric part of the flow. Onset of nonaxisymmetric motions appears at the same as in the homogeneous fluid case at the same Stratification precludes large axial displacements and the azimuthal modes patterns have a quite distinct appearance from the homogeneous wavy modes. At large enough Re, a destabilization of the jetlike outflow between pairs of vortices causes the suppression of the density front which is located at the same axial height. This nonaxisymmetric flow regime presents common features with the wavy outflow boundary (WOB) pattern, which is commonly observed in the homogeneous CouetteTaylor case.

Nonsolenoidal flow in a liquid diffusion couple
View Description Hide DescriptionAdvective nonsolenoidal (∇⋅v≠0) flow driven by diffusioninduced density changes in strictly zero gravity is studied in a twodimensional rectangular box. Our model, which is more general than the Oberbeck–Boussinesq model, is a precursor for the study of fluid flow that occurs due to density changes during isothermal interdiffusion in a binary liquid under the influence of stochastic microgravity (gjitter). We consider perturbation expansions of mass fraction (w) of the second chemical component of a binary solution, pressure (p), velocity (v), and chemical flux (j) with respect to a small parameter α [=ρ_{0}∂(1/ρ)/∂w], where ρ is the density and ρ_{0} is its value for some average composition. The total barycentric velocity field is given by the sum of an average flow, having a nonzero divergence, and a solenoidal flow derived from a pseudostreamfunction. At first order in α, we obtain a fourth order partial differential equation for this pseudostreamfunction. We solve this equation analytically in a quasisteadystate approximation for an infinitely long diffusion couple by using transform techniques. We also solve it numerically for the full timedependent problem for a finite domain. We conclude that such nonsolenoidal flows will dominate for sufficiently small gravity, for which the Oberbeck–Boussinesq approximation will certainly not be valid.

Stability characteristics of a periodically unsteady mixing layer
View Description Hide DescriptionIn nature, in many technological applications and in some laboratory experiments, the basic state of shear flows can be timevarying. The effects of such variations on the stability characteristics of these flows are not well understood. In previous work, Miksad et al. [J. Fluid Mech. 123, 1 (1982)] and Hajj et al. [J. Fluid Mech. 256, 385 (1992)], it has been shown that lowfrequency components, generated by nonlinear difference interactions, play an important role in the redistribution of energy among spectral components. In particular, phase modulation was found to be the most effective mechanism in energy transfer to the sidebands of unstable modes. In this work, the effects of smallamplitude lowfrequency mean flow unsteadiness on the stability of a plane mixing layer are determined. By extending earlier analytical arguments, it is shown that periodicity in the mean flow causes modulations of the most unstable modes. The analysis is then verified experimentally by comparing levels of amplitude and phase modulations in mixing layers with steady and unsteady basic flows. The results show that smallamplitude lowfrequency unsteadiness results in enhanced modulations of the fundamental mode. These modulations cause variations in the growth rates of the unstable modes and energy redistribution among them.

Destabilization of plane Poiseuille flow of insulating liquids by unipolar charge injection
View Description Hide DescriptionThe electrohydrodynamicinstabilities of plane Poiseuille flow when subjected to an orthogonal arbitrary unipolar injection of charge has been studied. Appropriate boundary conditions for the perturbed charge density in the space charge limited current regime in the presence of forced flows has been derived. The effect of the injection level on the stability of the flow is analyzed, as well as the role played by the ratio of the hydrodynamic to the true ionic mobility. It is shown that in the low Reynolds number region traverse rolls are destabilized more strongly as the ratio of mobilities decreases.

Traveling wave instability in helical coil flow
View Description Hide DescriptionComplementary flow visualization photographs and numerical calculations are presented for the transitional state between the laminar and turbulent flow regimes in a helically coiled pipe. The flow visualization covers a Reynolds number range from 3800 to 8650 (890<De<2030, where De is the Dean number). Estimates of the wavelength and wave speed of a traveling wave instability are made from photographs and video recordings at Re=5060 and 5480 (De=1190 and 1280). The unsteady threedimensional finite difference approximations of the Navier–Stokes equations formulated for the toroidal coordinate system are solved numerically. The calculations are performed in a curved pipe with a radius of curvature to pipe radius ratio equal to 18.2 and Re=5480 (De=1280). These test conditions match the flow visualization and previously reported laser Doppler velocimetrymeasurements. The calculations reveal a complex interaction between the centrifugal force and the crossstream velocity, hence explaining the mechanism for maintaining the traveling wave. An analogy is made with known centrifugal instabilities to explain the character of the motion observed in the inner half of the pipe along planes defined by the radial and streamwise coordinate directions. Simple considerations show that the crossstream flow has the potential for a centrifugal instability.

Dynamics of baroclinic vortices in a rotating, stratified fluid: A numerical study
View Description Hide DescriptionThis study deals with the instabilities that arise in the flow generated in a rotating tank by the evolution of a twolayer density stratified fluid. Numerical investigations have been performed by direct simulation of the NavierStokes equations for axisymmetric and fully threedimensional flows. In the former case results have shown the attainment, in a very short time, of an equilibrium position and the formation of an anticyclonic structure in the upper light layer and a cyclonic one in the lower layer, consistently with the observation of Griffiths and Linden. In the long term, however, the Ekman layer at the bottom damps out the cyclone and a steady state with only an anticyclone in the upper layer is reached. In threedimensions the flow is unstable to azimuthal disturbances and the steady state is no longer achieved. In particular a ring of cyclonic vorticity, surrounding the anticyclone, by the combined effects of baroclinic and barotropic processes, breaks, entrains vorticity from the anticyclone and eventually forms vortex pairs. As observed by Griffiths and Linden the azimuthal wave number of the instability depends on the Richardson number and the ratio between the depth of the light fluid and the total depth . However, since several modes, in addition to the most unstable, are amplified an initial perturbation whose energy is not equidistributed among the modes can lead to an instability with wave number different from the expected . Finally, the analysis of the equation for the energy of the instability has shown that the instability is initially driven by baroclinic effects, even for low values of . The barotropic source, in contrast, sets in only in the largeamplitude phase of the instability and its effect is larger when is small.

Lagrangian and Eulerian view of the bursting period
View Description Hide DescriptionLowdimensional models for the turbulent wall layer display an intermittent phenomenon with an ejection phase and a sweep phase that strongly resembles the bursting phenomenon observed in experimental flows. The probability distribution of interburst times has the observed shape [E. Stone and P. J. Holmes, Physica D 37, 20 (1989); SIAM J. Appl. Math. 50, 726 (1990); Phys. Lett. A 5, 29 (1991); P. J. Holmes and E. Stone, in Studies in Turbulence, edited by T. B. Gatski, S. Sarkar, and C. G. Speziale (Springer, Heidelberg, 1992)]. However, the time scales both for bursts and interburst durations are unrealistically long, a fact that was not appreciated until recently. We believe that the long time scales are due to the model’s inclusion of only a single coherent structure, when in fact a succession of quasiindependent structures are being swept past the sensor in an experiment. A simple statistical model of this situation restores the magnitude of the observed bursting period, although there is a great deal of flexibility in the various parameters involved.

Largeeddy simulation of highSchmidt number mass transfer in a turbulent channel flow
View Description Hide DescriptionMass transfer through the solid boundary of a turbulent channel flow is analyzed by means of largeeddy simulation(LES) for Schmidt numbers Sc=1, 100, and 200. For that purpose the subgrid stresses and fluxes are closed using the Dynamic Mixed Model proposed by Zang et al. [Phys. Fluids A 5, 3186 (1993)]. At each Schmidt number the mass transfer coefficient given by the LES is found to be in very good quantitative agreement with that measured in the experiments. At high Schmidt number this coefficient behaves like Sc^{−2/3}, as predicted by standard theory and observed in most experiments. The main statistical characteristics of the fluctuating concentration field are analyzed in connection with the welldocumented statistics of the turbulent motions. It is observed that concentration fluctuations have a significant intensity throughout the channel at Sc=1 while they are negligible out of the wall region at Sc=200. The maximum intensity of these fluctuations depends on both the Schmidt and Reynolds numbers and is especially influenced by the intensity of the velocity fluctuations present in the buffer layer of the concentration field. At Sc=1, strong similarities are observed between the various terms contributing to the turbulent kinetic energy budget and their counterpart in the budget of the variance of concentration fluctuations. At high Schmidt number, the latter budget is much more influenced by the small turbulent structures subsisting in the viscous sublayer. The instantaneous correlation between the spatial characteristics of the concentration field and those of the velocity field is clearly demonstrated by the presence of low and highconcentration streaks close to the wall. The geometrical characteristics of these structures are found to be highly Sc dependent. In particular their spanwise wavelength is identical to that of the streamwise velocity streaks at Sc=1 while it is reduced by half at Sc=200. Analysis of the cospectra between concentration and normal velocity fluctuations emphasizes the fact that the largescale structures play an essential role in the turbulent mass transfer process at high Schmidt number. Overall the picture that emerges from this investigation fully confirms the conclusions of Campbell and Hanratty [AIChE J. 29, 221 (1983)]: highSchmidtnumber mass transfer at a solid wall is governed by the lowfrequency part of the normal velocity fluctuation gradient at the wall, i.e., by the largescale structures observed in planes parallel to the wall in the viscous sublayer.

The effect of threedimensional freestream disturbances on the supersonic flow past a wedge
View Description Hide DescriptionThe interaction between a shock wave (attached to a wedge) and small amplitude, threedimensional disturbances of a uniform, supersonic, freestream flow are investigated. The paper extends the twodimensional study of Duck et al. [P W. Duck, D. G. Lasseigne, and M. Y. Hussaini, “On the interaction between the shock wave attached to a wedge and freestream disturbances,” Theor. Comput. Fluid Dyn. 7, 119 (1995) (also ICASE Report No. 9361)] through the use of vector potentials, which render the problem tractable by the same techniques as in the twodimensional case, in particular by expansion of the solution by means of a FourierBessel series, in appropriately chosen coordinates. Results are presented for specific classes of freestream disturbances, and the study shows conclusively that the shock is stable to all classes of disturbances (i.e., time periodic perturbations to the shock do not grow downstream), provided the flow downstream of the shock is supersonic (loosely corresponding to the weak shock solution). This is shown from our numerical results and also by asymptotic analysis of the FourierBessel series, valid far downstream of the shock.