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Volume 8, Issue 1, January 1996

Capillary bridge modes driven with modulated ultrasonic radiation pressure
View Description Hide DescriptionThe method of modulated ultrasonic radiation pressure, previously used to drive the capillary modes of drops and bubbles, is used to excite the capillary modes of a cylindrical oil bridge in a Plateau tank. Specific modes may be selected by adjusting the modulation frequency and the location or orientation of the bridge in the ultrasonic field. Mode frequencies were measured as a function of the slenderness for the lowest two axisymmetric modes and two nonaxisymmetric modes. The frequencies of the lowest modes agree with an approximate theory which neglects viscous corrections where the interfacial tension is a fitted parameter.

Hydrodynamic interactions in deep bed filtration
View Description Hide DescriptionDeep bed filtration has been studied experimentally and numerically for small non‐Brownian particles flowing into a random packing of monosize glass spheres at low Reynolds number. It was discovered that packets of particles penetrated further than the same number of particles released one at a time. These collective effects are attributed to hydrodynamic phenomena, one plausible explanation being the existence of relaunchable ‘‘hydrodynamic captures’’ in addition to ‘‘geometric captures.’’

Suppression of coalescence by shear and temperature gradients
View Description Hide DescriptionWe describe laboratory experiments on millimeter‐sized drops of liquid in air which indicate that both thermocapillary and isothermal shear flows are able to prevent the coalescence of bodies of liquid which would occur readily in the absence of such flows. We have also carried out molecular dynamicscomputer simulations of nanometer‐sized drops, which show the same qualitative behavior in the case of an applied shear. At the other extreme, persistent non‐coalescence of larger drops was observed in microgravity conditions in a space shuttle experiment. We give an explanation of the experimental observations based upon lubrication theory and simple continuum hydrodynamics arguments, along with complementary microscopic insight obtained from the molecular simulations.

Head‐on collision of drops—A numerical investigation
View Description Hide DescriptionThe head‐on collision of equal sized drops is studied by full numerical simulations. The Navier–Stokes equations are solved for the fluid motion both inside and outside the drops using a front tracking/finite difference technique. The drops are accelerated toward each other by a body force that is turned off before the drops collide. When the drops collide, the fluid between them is pushed outward leaving a thin layer bounded by the dropsurface. This layer gets progressively thinner as the drops continue to deform, and in several of our calculations we artificially remove this double layer at prescribed times, thus modeling rupture. If no rupture takes place, the drops always rebound, but if the film is ruptured the drops may coalesce permanently or coalesce temporarily and then split again. Although the numerically predicted boundaries between permanent and temporary coalescence are found to be consistent with experimental observations, the exact location of these boundaries in parameter space is found to depend on the time of rupture.

The dynamics of ultrasonically levitated drops in an electric field
View Description Hide DescriptionUltrasonic and electrostatic levitation techniques have allowed the experimental investigation of the nonlinear oscillatorydynamics of free droplets with diameter between 0.1 and 0.4 cm. The measurement of the resonance frequencies of the first three normal modes of large amplitude shape oscillations in an electric field of varying magnitude has been carried out with and without surface charges for weakly conducting liquids in air. These oscillations of nonspherical levitated drops have been driven by either modulating the ultrasonic field or by using a time‐varying electric field, and the free decay from the oscillatory state has been recorded. A decrease in the resonance frequency of the driven fundamental quadrupole mode has been measured for increasing oblate deformation in the absence of an electric field. Similarly, a decrease in this frequency has also been found for increasing DC electric field magnitude. A soft nonlinearity exists in the amplitude dependence of the resonant mode frequencies for freely decaying as well as ultrasonically and electrically driven uncharged drops. This decrease in resonance frequency is accentuated by the presence of free surface charge on the drop. Subharmonic resonance excitation has been observed for drops in a time‐varying electric field, and hysteresis exists for resonant modes driven to large amplitude. Mode coupling from lower‐order resonances to higher‐order modes has been found to be very weak, even for fairly large amplitude shape oscillations. Most of these results are in general agreement with predictions from recent analytical and numerical investigations.

Instabilities in three‐dimensional differentially‐heated cavities with adiabatic horizontal walls
View Description Hide DescriptionConsidered are the transitional instabilities of the flow inside three‐dimensional rectangular cavities that are differentially heated over two opposing vertical walls. The horizontal and lateral walls are adiabatic. Emphasis is on (though not restricted to) the air‐filled, cubical cavity. For this configuration, it was found that the occurrence of unsteady oscillations in the flow was preceded by a steadyinstability (i.e. an instability resulting in a steady solution of the Navier–Stokes equations for large time) which originated in an internal, stratified shear layer that separates from the adiabatic horizontal walls of the cavity. This instability is inherently three‐dimensional and characterized by the presence of streamwise‐oriented, counterrotating vortices. It is probably caused by centrifugal forces. The subsequent, low‐frequency, unsteady instability is strongly influenced by this steady instability and as a result its frequency differs strongly from its counterpart in the two‐dimensional, square cavity. For larger Prandtl numbers, however, the frequencies in the two‐ and three‐dimensional cavities are almost equal since no prior steady instability occurs. The instability mechanism responsible for the unsteady instability is therefore the same in both configurations even though the instability in the three‐dimensional cavity shows a distinct wave‐like modulation in the third direction.

The curvature of material lines in chaotic cavity flows
View Description Hide DescriptionMaterial line folding is studied in two‐dimensional chaotic cavity flows. Line folding is measured by the local curvature k=l×l′/‖l‖^{3}, where l(q) is an infinitesimal vector in the tangential direction of the line, q is a coordinate along the line, and l′ is the derivative of l with respect to q. It is shown both analytically and numerically that folding is always accompanied by compression. The vector l′ plays a crucial role as a driving force for the stretching and folding processes. A material line is stretched when l′ is tangential to the line and it is folded when l′ is normal to the line. The spatial structure of the curvature field is computed numerically. The short‐time structure of the curvature field is similar to the structure of unstable manifolds of periodic hyperbolic points, and closely resembles patterns observed in tracer mixing experiments and in stretching field computations. The long time structure of the field asymptotically approaches an entirely different time‐independent structure. Probability density functions of curvature are independent of both time and initial conditions.

Fluctuating kinetic energy budget during homogeneous flow of a fluid solid mixture
View Description Hide DescriptionEnsemble‐averaging theorems are applied to derive transport equations for the fluctuating kinetic energy of a particulate mixture consisting of a continuous fluid and solid particles. The evolution of fluctuating kinetic energy in a homogeneous flow is examined and discussed.

Phase dynamics of Kármán vortices in cylinder wakes
View Description Hide DescriptionThe temporal evolution of Kármán vortex shedding patterns in the wake of a cylinder placed at right angles to a uniform flow is studied for Reynolds numbers (based on cylinder diameter) between 80 and 140. Focusing on the dynamics of the vortex shedding phase in the wake planview (the plane spanned by the free‐stream direction and the cylinder axis) we study experimentally and model the response of shedding patterns to time‐dependent boundary conditions imposed at the cylinder ends. By appropriate impulsive changes of end conditions, spanwise wave number ‘‘shocks’’ can be produced that travel along the cylinder span. These shock experiments, together with data from steady oblique shedding patterns, are used to determine the parameters for the spanwise Ginzburg–Landau model, which has already been used successfully to describe many of the phenomena observed in cylinder wakes. We then demonstrate experimentally that, in analogy to gasdynamics, it is also possible to produce ‘‘expansion waves’’ of the spanwise wave number, which are well described by the Ginzburg–Landau model without further adjustment of its parameters.

Suppression of the capillary instability in the Rayleigh–Taylor slot problem
View Description Hide DescriptionThe classical Rayleigh–Taylor instability for an interface of finite extent is modified by two independently controlled perturbation effects. A component of gravitational acceleration tangent to the interface is imposed and the interface is subjected to a flow‐induced pressure field. The stability of the flat horizontal interface when surface tension holds heavier liquid above ambient gas is considered in a 2D model problem. The two perturbations are realized by tilting the interface to the horizontal and by inducing a flow with shear. It is found that the effect of tilt angle or shear on its own is destabilizing, while together, in the right combination, they can stabilize. It is thereby shown how to extend the stability limit over the classical Rayleigh–Taylor result. The framework for the analysis is the classical unperturbed pitchfork bifurcation (codimension 2). The coefficients in the unfolding are calculated by applying the Lyapunov–Schmidt technique to a pinned deformableinterface that holds a shear‐induced lubrication flow.

Temporal instability of plane gas sheets in a viscous liquid medium
View Description Hide DescriptionThis paper reports a linear temporal instability analysis of an incompressible plane gas sheet in a quiescent viscousliquid medium of infinite expanse. Results indicate that there exist two unstable modes of disturbance waves, sinuous and varicose, and surface tension always reduces, while the relative velocity between the gas and liquid phases and the gas density always enhance instability development. For both unstable modes, the presence of liquidviscosity increases the instability limit, which is however independent of the absolute value of viscosity. It is also shown that the sinuous mode becomes stable when the gas Weber number, defined as the ratio of aerodynamic forces to surface tension forces, is less than the critical value of one. At slightly larger gas Weber numbers, liquidviscosity exhibits dual effects—it may enhance or suppress the growth of unstable disturbances, depending on specific flow conditions. However, for sinuous mode at high Weber numbers and varicose mode at any Weber numbers, liquidviscosity always reduces disturbance growth rates and dominant wave numbers. Unlike the case for plane liquid sheets, varicose mode controls the instability process for all Weber number ranges and for both inviscid and viscousliquids, and only at high Weber numbers, do varicose and sinuous modes become almost equally important. It is further found that the wave propagation velocity for both unstable modes is much smaller than the gas velocity at the mode of maximum instability, implying that the disturbance waves appear almost stationary rather than travelling‐wave type, in contrast with the plane liquid sheet results.

Melting driven thermohaline convection
View Description Hide DescriptionThis study is concerned with the experimental and numerical analysis of thermohaline natural convection in an aqueous solution, driven by melting of a block of pure ice into the liquid mixture. Experimentally, the flow structure evolution and the heat transfercharacteristics are studied for a binary H_{2}O–Na_{2}CO_{3} mixture in the domain of negative buoyancy ratios and for a wide range of solutal and thermal Rayleigh numbers. The observations are analyzed with the help of the classical results about cell formation in a stratified fluid due to sideways heating. The comparison with the numerical results shows that the mechanisms that define the flow structure are essentially due to thermohaline convection and not to the melting process. Finally, the fusion kinetics is analyzed, and the melting rate is correlated to the governing parameters of natural convection in the liquid.

Rotating flow past a sliced cylinder
View Description Hide DescriptionDepending upon the relative sizes of the parameters of the problem, rotating flow of a vertically confined fluid past an asymmetric object—in this case a circular cylinder with top sliced at an angle—may induce flow inside the Taylor column, driven by the viscous stresses in the column wall. The motion is along the constant‐depth contours, which are not closed in such a situation. We show from theoretical considerations that so long as the angle of the slice is bigger than the one‐quarter power of the Ekman number, E, such an interior motion in the column does occur. In general, the motion consists of two eddies over the obstacle. A series of case study laboratory experiments is presented in support of the analysis, to show the effect of the slice orientation and magnitude on the flow over such a bump, and to illustrate the nature of the flows which are generated when inertial effects are dominant.

Numerical study of nonlinear shallow water waves produced by a submerged moving disturbance in viscous flow
View Description Hide DescriptionTwo‐dimensional solitary waves generated by a submerged body moving near the critical speed in a shallow water channel are studied numerically. The incompressible Navier–Stokes equations in a curvilinear free‐surface‐fitted coordinate system are solved by the finite difference method. The present numerical results are compared with the existing experimental data, and with the numerical solutions of two inviscid‐flow models, i.e. the general Boussinesq equation and the forced Korteweg‐de Vries equation. It is found that the viscous effect in the boundary layer around the body and on the bottom of the channel plays an important role in the generation of solitary waves on the free surface. Hence the Navier–Stokes solutions have a better agreement with the experimental data than those obtained from two inviscid‐flow models. The effect of the submergence depth of the body on the waves generated is also investigated. It reveals that waves are insensitive to the submergence depth of the body, except for a small region quite close to the bottom of the water channel.

The steady boundary layer due to a fast vortex
View Description Hide DescriptionA point vortex located above and convected parallel to a wall is an important model of the process by which a boundary layer becomes unstable due to external disturbances. Often it has been assumed that the boundary layer due to the passage of the vortex is inherently unsteady. Here we show that for a vortex convected by a uniform shear flow, there is a steadysolution when the speed of the vortexc _{ v } is sufficiently fast. The existence of the steady solution is demonstrated analytically in the limit of large vortex velocity (c _{ v }→∞) and numerically at more moderate speeds. This solution may provide a useful base state about which to investigate the stability of a boundary layer induced by external disturbances.

Forced nonlinear disturbances in incompressible boundary layers
View Description Hide DescriptionIn this paper we explore essentially nonlinear disturbances produced in an incompressible boundary layer by a roughness on the wall. The scale of the stationary roughness is supposed to be large enough so that generated waves are governed by the forced Benjamin–Davis–Acrivos (fBDA) equation. The disturbance patterns for a wide range of roughness sizes are analyzed revealing the remarkable phenomenon of bifurcations. A very specific oscillation motion over the obstacle is found. It appears to be the basic mechanism causing the periodic generation of solitary waves upstream and downstream. The general structure of disturbances in space at different values of time is discussed. The asymptotic analysis of the solution, when the intensity of the external agency Q becomes a small parameter, is given. The quadratic term of an expansion based on this parameter is responsible for the redistribution of solution mass between regions located ahead and behind the obstacle, inevitably leading to the gradual growth of nonlinear effects. According to the asymptotic consideration, the time T _{ c } ^{*} determining the onset of the nonlinear stage of disturbance development is O(Q ^{−3}), this estimation correlates well with numerical results.

Investigations of boundary layer transition via Galerkin projections on empirical eigenfunctions
View Description Hide DescriptionIn this paper, Galerkin projections on eigenfunctions as obtained by proper orthogonal decomposition of numerically computed flow fields are used to derive dynamical models for different regions of a transitional boundary layer. The regions investigated cover the stages of the transition process from the evolution of low‐amplitude Tollmien‐Schlichting waves up to the final stages of transition, right at the onset of turbulence. In a first part of the paper, the possibilities and limitations of the approach chosen are investigated in detail, and in a second part the application of the techniques developed before is demonstrated for the case of a spatially evolving boundary layer that is inhomogeneous in all spatial directions. The focus of this work is mainly on how characteristic properties of the dynamics change as transition evolves in the streamwise direction.

The passive scalar spectrum and the Obukhov–Corrsin constant
View Description Hide DescriptionIt is pointed out that, for microscale Reynolds numbers less than about 1000, the passive scalar spectrum in turbulentshear flows is less steep than anticipated and that the Obukhov–Corrsin constant can be defined only if the microscale Reynolds number exceeds this value. In flows where the large‐scale velocity field is essentially isotropic (as in grid turbulence), the expected 5/3 scaling is observed even at modest Reynolds numbers. All known data on the Obukhov–Corrsin constant are collected. The support for the notion of a ‘‘universal’’ constant is shown to be reasonable. Its value is about 0.4.

Small‐scale properties of nonlinear interactions and subgrid‐scale energy transfer in isotropic turbulence
View Description Hide DescriptionUsing results of direct numerical simulations of isotropic turbulence the subgrid‐scale energy transfer in the physical space is calculated exactly employing a spectral decomposition of the velocity field into large (resolved) and small (unresolved) scales. Comparisons with large‐scale quantities reveal large qualitative correlations between regions of subgrid transfer and the boundaries of regions of large vorticity production. This suggests a novel analysis of the nonlinear term, where it is decomposed into four components determined by four combinations of the resolved and unresolved velocity and vorticity fields. It is found that there is a 90% vector‐correlation between the subgrid transfer and the component of the full transfer associated with the resolved velocity and unresolved vorticity, but that 90% of the total subgrid energy production is determined by the component associated with the unresolved velocity and resolved vorticity. These results suggest subgrid‐scale models that have higher correlation values with the exact subgrid scale terms than a number of other physical quantities that are traditionally considered to govern the dynamics of the large scales of turbulence. The distinguishing feature of the new models is that they emphasize the nonlinear dynamics of the smallest resolved scales in the modeling procedures.

Dispersion in a quasi‐two‐dimensional‐turbulent flow: An experimental study
View Description Hide DescriptionThe dispersion of a passive tracer in a quasi‐two‐dimensional turbulent flow and the geometry of corresponding isoconcentration lines are investigated experimentally. The flow consists in an array of 900 vortices, forced in a thin layer and driven in a turbulent regime. Both the instantaneous velocity field and the concentration field are measured. A remarkable regime of anomalous diffusion—characterized by a dispersive front moving like t ^{0.32±0.04}—is observed. Examining the trajectories of individual neutral particles, we reveal the presence of ‘‘traps’’ that control most of the characteristics of this hypodiffusive regime. The fractal dimension of isoconcentration profiles and the exponents of the structure functions of both the velocity and the concentration fields are established. The corresponding values are consistent with mathematical inequalities, recently discovered, but show some disagreements with recent conjectured equalities proposed by Constantin et al. [Nonlinearity 7, 1045 (1994)].