Volume 11, Issue 6, June 1999
 LETTERS


Motion of a finger with bubbles in a HeleShaw cell: An exact solution
View Description Hide DescriptionExact solutions are reported for the problem of a finger moving steadily with bubbles at the tip in a HeleShaw cell in the absence of surface tension.

Intermittency and scaling laws for wall bounded turbulence
View Description Hide DescriptionWell defined scaling laws clearly appear in wall bounded turbulence, very close to the wall, where a distinct violation of the refined Kolmogorov similarity hypothesis (RKSH) occurs together with the simultaneous persistence of scaling laws. A new form of RKSH for the wall region is here proposed in terms of the structure functions of order two which, in physical terms, confirms the prevailing role of the momentum transfer towards the wall in the near wall dynamics.
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 ARTICLES


Extremum principles for selection in the Saffman–Taylor finger and Taylor–Saffman bubble problems
View Description Hide DescriptionWe consider the Saffman–Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele–Shaw configuration, and the Taylor–Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of traveling wave solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the channel through which it propagates. To select the “correct” member of the family of solutions we propose two related extremum principles. Employing these principles for a symmetric family with zero surface tension selects the solution (finger or bubble,viscous or inviscid) which happens to be the same as that obtained by taking the limit of the nonzero isotropic surface tension problem. For other problems, e.g., perturbation by anisotropicsurface tension, the fingers selected by the two approaches are not necessarily the same. We claim that the finger selected by our criteria describes what will be observed on an intermediate time scale in which the effect of perturbations is neglected, whereas the finger selected by taking the limit of vanishingly small perturbations describes what will be observed on the asymptotic time scale for the problem with the perturbation included. The intermediate time scale is much shorter than the asymptotic time scale. For infinitesimal the asymptotic time scale may be well beyond any reasonable observational time scale.

Deformation of a liquid drop adhering to a plane wall: Significance of the drop viscosity and the effect of an insoluble surfactant
View Description Hide DescriptionThe transient deformation of a viscousdrop attached to a plane wall and subjected to an overpassing simple shear flow is studied in the limit of a vanishing Reynolds number. The buoyancy force is negligible, the contact line has and retains a prescribed circular shape, and the interface is either devoid of surfactants, in which case it exhibits constant surface tension, or it is occupied by an insoluble surfactant, in which case the surface tension is related to the surfactant concentration by means of a linear constitutive equation. In the numerical procedure, the interfacial velocity is computed by solving the equations of Stokes flow using a boundaryelement method in which the interface is discretized into sixnode curved triangles. The convection–diffusion equation for the concentration of the surfactant is integrated in time over the evolving interface using a semiimplicit finitevolume method. In the numerical investigations, the deformation of a drop is simulated from the initial state, where the interface has the prescribed shape of a section of a sphere and a uniform surfactant distribution, to either a steady deformed shape, or up to the point where evidence for continued deformation is obtained. The results show that the presence of surfactant promotes the deformation of drops whose viscosity is low compared to that of the ambient fluid, but has a small influence on the deformation of drops whose viscosity is comparable to, or higher than, that of the ambient fluid. At a fixed capillary number, the deformation of the drop equilibrium shape for constant or varying surface tension increases monotonically with raising the dropviscosity because of the increasing importance of the image flow due to the wall. Thus, whereas a viscousdrop suspended in an infinite shear flow extends indefinitely into a slender ligament only when its viscosity is less than about four times the viscosity of the ambient fluid and the shear rate is sufficiently high, an analogous threshold for a drop attached to a wall does not seem to arise.

The effect of viscosity on the spherical stability of oscillating gas bubbles
View Description Hide DescriptionGas bubbles driven in radial oscillations are subject to an instability of the spherical shape that is opposed by surface tension and viscosity. An exact linear formulation for the study of the phenomenon has been available for many years, but its complexity has discouraged a detailed investigation. With the recent theory of sonoluminescence of Lohse and coworkers [Hilgenfeldt et al., Phys. Fluids, 8, 2808 (1996)], there has been a renewed interest in the problem and new data have become available. This paper presents a numerical method for the solution of the pertinent equations and compares the theory with these new data. The coupling of the strong nonlinearity of the bubble radial oscillations with the parametric mechanism of the surface instability results in a very complex structure for the stability boundary. Nevertheless, a good agreement between theory and data is found. A comparison with earlier approximate models is also made.

Sonoluminescence light emission
View Description Hide DescriptionSingle bubblesonoluminescence is not an exotic phenomenon but can quantitatively be accounted for by applying a few wellknown, simple concepts: the Rayleigh–Plesset dynamics of the bubble’s radius, polytropic uniform heating of the gas inside the bubble during collapse, the dissociation of molecular gases, and thermal radiation of the remaining hot noble gas, where its finiteopacity (transparency for its own radiation) is essential. A system of equations based on these ingredients correctly describes the widths, shapes, intensities, and spectra of the emitted light pulses, all as a function of the experimentally adjustable parameters, namely, driving pressure, driving frequency, water temperature, and the concentration and type of the dissolved gas. The theory predicts that the pulse width of strongly forced xenon bubbles should show a wavelength dependence, in contrast to argon bubbles.

Flow visualization of Taylormode breakup of a viscous liquid jet
View Description Hide DescriptionWe recently reported a new spray technique called ultrasoundmodulated twofluid (UMTF) atomization and the pertinent “resonant liquid capillary wave (RLCW) theory” based on linear models of Taylormode breakup of capillary waves. In this article, flow visualizations of liquid jets near the nozzle tip are presented to verify the central assumption of the RLCW theory that the resonant liquid capillary wave in UMTF atomization is initiated by the ultrasound at the nozzle tip. Specifically, a bright band beneath the nozzle tip was seen in ultrasonic and UMTF atomization separately, but not in twofluid atomization. The bright band can be attributed to scattering of laser light sheet by the capillary waves generated by the ultrasound on the intact liquid jet. As the capillary wave travels downstream in the direction of airflow, it is amplified by the air blowing around it and eventually collapsed into drops. Therefore, the jet breakup time can be determined by dividing the measured band length with the capillary wave velocity. The breakup times thus determined for water and glycerol/water jets are twice the values predicted by the modified Taylor’s model with a sheltering parameter, and are one order of magnitude shorter than those in conventional twofluid atomization. Furthermore, the images of the spray in the proximity of the nozzle tip obtained by 30 ns laser pulses are consistent with the drop sizes obtained 2.3–6 cm downstream from the nozzle tip by 13 s time average of continuous laser light. Also reported in this article is the good agreement between the measuredviscosity effects on the dropsize and size distribution in UMTF atomization and those on the relative amplitude growth rates of capillary waves at different wavelengths predicted by Taylor’s model as a result of its inclusion of higher order terms other than the first in viscosity. These new findings have led to the conclusion that UMTF atomization occurs via Taylormode breakup of capillary waves; secondary atomization and drop coalescence are negligible. Further, UMTF atomization offers a means to control the dropsize and size distribution of twofluid atomization for uniform drop formation.

Measurement of the nonlinear behavior of acoustical rigid porous materials
View Description Hide DescriptionThe measurement of the flow resistivity of porous materials shows that two types of behavior can be observed, depending on the value of a Reynolds number based on the porous material microgeometry. For Reynolds numbers (Re) smaller than a critical Reynolds number, the increase of the resistivity is quadratic in Re. For Re larger than this value the increase is linear in Re (Forchheimer’s law). A comparison between acoustic measurements and an equivalent fluid model shows that the main effect of high sound level on sound propagation through rigid porous materials is the variation of the flow resistivity.

Onset of stationary and oscillatory convection in a tilted porous cavity saturated with a binary fluid: Linear stability analysis
View Description Hide DescriptionIn the present work, we study the onset of doublediffusive convective regimes in a tilted rectangular cavity, filled with a porous medium, saturated by a binary fluid. Two opposite walls are maintained at different but uniform temperatures and concentrations while the two other walls are impermeable and adiabatic. When the thermal and solutal buoyancy forces are comparable in intensity but have opposite signs, the motionless doublediffusive regime with linear temperature and concentration profiles is a solution of the problem. The first part of the study consists of a linear stability analysis of the motionless regime. We determine the critical thermal Rayleigh number for the onset of stationary and oscillatory convection. Indeed, we point out that there exist primary Hopf bifurcations for the studied problem in porous medium, while in the same configuration with a fluid medium only primary stationary bifurcations exist. When the first primary bifurcation creates a steady state branch of solutions, the bifurcation is either transcritical or pitchfork depending on the aspect ratio, A and the tilt, φ of the cavity. The onset of oscillatory convection (Hopf bifurcation) depends not only on A and φ but also on the Lewis number, Le and the normalized porosity, ε. Then, we determine the parts of the (Le, ε) parameter space for which the first primary bifurcation is stationary or oscillatory. In particular, it is found that in the case and for the first primary bifurcation is always a Hopf bifurcation for any A and φ except for For only stationary primary bifurcations exist. In the case zones where stationary and oscillatory primary bifurcations exist are separated by a curve depending on A and φ. The last part of this work consists of a series of numerical simulations. The onset of stationary and oscillatory convection is obtained numerically at the critical Rayleigh number predicted by linear analysis. We also verified the frequency of oscillations for several sets of dimensionless parameters. The numerical simulations show multiple subcritical solutions.

Routes to chaos in widegap spherical Couette flow
View Description Hide DescriptionThe dynamical behavior of widegap instabilities in spherical Couette flow is investigated experimentally with chaosanalyzing techniques applied on time series from LaserDopplerVelocimetry (LDV) measurements. With an increasing Reynolds number of the rotating inner sphere, the flow undergoes two Hopf bifurcations and several mode changes. The transition scenarios for two different gap widths investigated can be described primarily as the Ruelle–Takens–Newhouse type and show a strong dependence on the meridional coordinate. The transition processes are accompanied by phenomena such as a locked torus in reconstructed phase space and periodic long wave modulations of chaotic flows. Although the investigated gap widths are all classified as wide, a comparison exhibits significant differences in the development of chaotic motion.

Particle dispersion in variable density and viscosity shear flows
View Description Hide DescriptionThe dispersion of small dense particles by an unsteady planar shear layer formed between two streams of different velocity, density, and viscosity is investigated numerically. The twophase flow is assumed to be in the dilute regime. The Lagrangian transport element method is employed to provide twodimensional unaveraged simulations of the carrier flow. The evolution of the particle field is captured by computing the trajectories of individual particles using a reduced form of the equation of particle motion. The analysis focuses on the impact of the densities and viscosities of the two streams on the dispersion of particles of different Stokes numbers (St). The results verify the well established behavior between St and dispersion: maximized dispersion for intermediate St due to centrifugation of particles by the vortices into the irrotational streams, reduced dispersion at low St with particles following the flow and minimal dispersion for large St with particles remaining unaffected by the flow. Variation of the viscosities of the irrotational streams does not substantially alter the above trends except for a shift in the value of St where different behaviors are experienced. This shift is linked to the existence of an effective viscosity that controls the dispersion process. This viscosity is the one characterizing the well mixed vortical structures. Variations in the densities of the two streams decreases the dispersion of intermediate St particles as compared to the uniform density case. This behavior is due to the impact of baroclinic vorticity generation that creates asymmetries in the overall vorticity field that diminish the centrifugation of the particles into one of the two streams.

Competition of mixing and segregation in rotating cylinders
View Description Hide DescriptionUsing discrete element methods, we study numerically the dynamics of the size segregation process of binary particle mixtures in threedimensional rotating drums, operated in the continuous flow regime. Particle rotations are included and we focus on different volume filling fractions of the drum to study the interplay between the competing phenomena of mixing and segregation. It is found that segregation is best for a more than halffilled drum due to the nonzero width of the fluidized layer. For different particle size ratios, it is found that radial segregation occurs for any arbitrary small particle size difference and the final amount of segregation shows a linear dependence on the size ratio of the two particle species. To quantify the interplay between segregation and mixing, we investigate the dynamics of the center of mass positions for each particle component. Starting with initially separated particle groups we find that no mixing of the component is necessary in order to obtain a radially segregated core.

Computation of flow through a fluidsediment interface in a benthic chamber
View Description Hide DescriptionIn this paper we present a singledomain approach for the solution of flow in a composite region made up of a pure fluid layer and an underlying saturated porous layer. As an example, we compute the unsteady, axisymmetric flow and scalar transport in a stationary cylindrical container with a rotating lid, filled to the midheight with a porous material and to the top with water. A generalized equation known as the Brinkmanextended Darcy equation is solved inside the porous medium, along with the incompressible Navier–Stokes equations in the upper fluid layer. Comparisons with experimental data previously obtained by the authors for flow in the same geometry show good agreement, thus verifying the accuracy of the present computations. The results indicate that a singledomain approach can provide good predictions of interfacial flow, thereby obviating the need for ad hoc interface conditions. The existence of a thin Brinkman layer below the interface is observed. Radial profiles of computed velocity components adjacent to the interface show remarkable similarity, despite vast differences in magnitudes, showing that good matching between the two different flows has been achieved by the present singledomain approach.

On a threedimensional volume tracking model of droplet impact
View Description Hide DescriptionA threedimensional model has been developed of droplet impact onto asymmetric surface geometries. The model is based on RIPPLE, and combines a fixedgrid control volume discretization of the flowequations with a volume tracking algorithm to track the dropletfree surface.Surface tension is modeled as a volume force acting on fluid near the free surface. Contact angles are applied as a boundary condition at the contact line. The results of two scenarios are presented, of the oblique impact of a 2 mm water droplet at 1 m/sec onto a 45° incline, and of a similar impact of a droplet onto a sharp edge. Photographs are presented of such impacts, against which the numerical results are compared. The contact angle boundary condition is applied in one of two ways. For the impact onto an incline, the temporal variation of contact angles at the leading and trailing edges of the droplet was measured from photographs. This data is applied as a boundary condition to the simulation, and an interpolation scheme proposed to evaluate contact angles between the leading and trailing edges. A simpler model is then proposed, for contact angle as a function of contact line velocity, and applied to both geometries. The model requires values of only two contact angles, at a rapidly advancing and a rapidly receding contact line. Simulation results compare well with photographic data.

A Lagrangian analysis of advectiondiffusion equation for a three dimensional chaotic flow
View Description Hide DescriptionThe advectiondiffusion equation is studied via a global Lagrangian coordinate transformation. The metric tensor of the Lagrangian coordinates couples the dynamical system theory rigorously into the solution of this class of partial differential equations. If the flow has chaotic streamlines, the diffusion will dominate the solution at a critical time, which scales logarithmically with the diffusivity. The subsequent rapid diffusive relaxation is completed on the order of a few Lyapunov times, and it becomes more anisotropic the smaller the diffusivity. The local Lyapunov time of the flow is the inverse of the finite time Lyapunov exponent. A finite time Lyapunov exponent can be expressed in terms of two convergence functions which are responsible for the spatiotemporal complexity of both the advective and diffusive transports. This complexity gives a new class of diffusion barrier in the chaotic region and a fractallike behavior in both space and time. In an integrable flow with shear, there also exist fast and slow diffusion. But unlike that in a chaotic flow, a large gradient of the scalar field across the KAM surfaces can be maintained since the fast diffusion in an integrable flow is strictly confined within the KAM surfaces.

Optimal and adaptive control of chaotic convection—Theory and experiments
View Description Hide DescriptionIn theory and experiments, optimal and adaptive control strategies are employed to suppress chaotic convection in a thermal convection loop. The thermal convection loop is a relatively simple experimental paradigm that exhibits complex dynamic behavior and provides a convenient platform for evaluating and comparing various control strategies. The objective of this study is to evaluate the feasibility of employing optimal control and nonlinear estimator to alter naturally occurring flow patterns and to compare the performance of the optimal controller with that of other controllers such as neural network controllers. It is demonstrated that when the system’s model is not known, experimental data alone can be utilized for the construction of a proportional controller.

Loworder parabolic theory for 2D boundarylayer stability
View Description Hide DescriptionWe formulate here a lowest order parabolic (LOP) theory for investigating the stability of twodimensional spatially developing boundary layer flows. Adopting a transformation earlier proposed by the authors, and including terms of order where R is the local boundarylayer thickness Reynolds number, we derive a minimal composite equation that contains only those terms necessary to describe the dynamics of the disturbance velocity field in the bulk of the flow as well as in the critical and wall layers. This equation completes a hierarchy of three equations, with an ordinary differential equation correct to (similar to but different from the Orr–Sommerfeld) at one end, and a “full” nonparallel equation nominally correct to at the other (although the latter can legitimately claim higher accuracy only when the mean flow in the boundary layer is computed using higher order theory). The LOP equation is shown to give results close to the full nonparallel theory, and is the highestorder stability theory that is justifiable with the lowestorder mean velocity profiles for the boundary layer.

Symmetric and nonsymmetric Holmboe instabilities in an inviscid flow
View Description Hide DescriptionUsing linear stability analysis we studied the effect of displacing a thin density interface with respect to the center of the shear layer on the stability of an inviscid, stably stratified, parallel flow. When no interface displacement is present and the flow is unbounded, pure Holmboe instabilities exist at all bulk Richardson numbers and are the most unstable instabilities for values of the bulk Richardson number greater than 0.046. When the interface displacement is nonzero the two modes of a Holmboe instability split into a stronger and a weaker mode. As the height of the vertical domain decreases the roles of the two modes switch with the originally weaker mode becoming the stronger mode and vice versa. The importance of including the height of the vertical domain in the stability analysis was illustrated by comparing theoretical results with the field data of Yoshida et al. [Yoshida, Ohtani, Nishida, and Linden, in Physical Processes in Lakes and Oceans, edited by J. Imberger (American Geophysical Union, Washington, DC, 1998), pp. 389–400]. The assumption that the instabilities are initially twodimensional is examined. When the flow is unbounded, both symmetric and nonsymmetric Holmboe instabilities are initially twodimensional. When boundaries are included, the twodimensional assumption is valid except when the total vertical domain is small in which case threedimensional primary instabilities are possible.

Fully nonlinear threedimensional convection in a rapidly rotating layer
View Description Hide DescriptionFully nonlinear threedimensional convection in a rotating layer is studied for large Taylor numbers. In this regime, the leading order nonlinearity arises from the distortion of the horizontally averaged temperature profile. As a result, steady rolls, squares, hexagons, triangles, and a pattern called patchwork quilt all have identical Nusselt numbers. A similar degeneracy is present in overstable convection with six patterns having identical timeaveraged Nusselt numbers and oscillation frequencies. These results are obtained via an asymptotic expansion in the Taylor number that determines, for each Rayleigh number, the timeaveraged Nusselt number and oscillation frequency from the solution of a nonlinear eigenvalue problem for the vertical temperature profile. A number of other patterns are determined by a weakly nonlinear analysis that cannot be extended into the fully nonlinear regime by the present methods, but these patterns are necessarily unstable.

Cascade of structures in longwavelength Marangoni instability
View Description Hide DescriptionThe longwavelength instability for thermocapillarydriven convection in two dimensions is studied numerically. The system under consideration consists of a horizontal periodical liquid layer bounded from below by a rigid wall and from above by a deformable free surface. The liquid is heated from the bottom wall and cooled from above. The problem can be approximated by the Stokes equation and has two dimensionless parameters. One parameter is the dynamic Bond number which is the ratio between gravity and thermocapillary force. The other is the static Bond number, which describes the ratio between gravity and the surface tension. Using the boundary integral method we present fullscale direct numerical simulations of the longwavelength Marangoni instability in two dimensions. The time evolution of the free surface leads to the formation of drained regions (socalled “dry spots”). The simulations demonstrate a remarkable complexity of the touchdown process, involving a deep cascade from large to increasingly small structures. In the behavior of the minimum height of the interface at large time simple scaling dependence on time was not observed. Extrapolation of exhibits infinitetime singularity. The dependence of the size of drained region on the parameters is discussed.
