Volume 12, Issue 6, June 2000
 LETTERS


The coalescence cascade of a drop
View Description Hide DescriptionWhen a drop is deposited gently onto the surface of a layer of the same liquid, it sits momentarily before coalescing into the bottom layer. Highspeed video imaging reveals that the coalescence process is not instantaneous, but rather takes place in a cascade where each step generates a smaller drop. This cascade is selfsimilar and we have observed up to six steps. The time associated with each partial coalescence scales with the surface tension time scale. The cascade will, however, not proceed ad infinitum due to viscous effects, as the Reynolds number of the process is proportional to the square root of the drop diameter. Viscous effects will therefore begin to be important for the very smallest drops. This cascade is very similar to the one observed previously by Charles and Mason [J. Colloid Sci. 15, 105 (1960)] for two immiscible liquids, where one of the liquids replaces the air in our setup.
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 ARTICLES


Growth and collapse of a vapor bubble in a narrow tube
View Description Hide DescriptionThe fluid mechanical aspects of the axisymmetric growth and collapse of a bubble in a narrow tube filled with a viscous liquid are studied numerically. The tube is open at both ends and connects two liquid reservoirs at constant pressure. The bubble is initially a small sphere and growth is triggered by a large internal pressure applied for a short time. After this initial phase, the motion proceeds by inertia. This model simulates the effect of an intense, localized, brief heating of the liquid which leads to the nucleation and growth of a bubble. The dimensionless parameters governing the problem are discussed and their effects illustrated with several examples. It is also shown that, when the bubble is not located at the midpoint of the tube, a net flow develops capable of pumping fluid from one reservoir to the other. The motivation for this work is offered by the possibility to use vapor bubbles as actuators in fluidhandling microdevices.

Fingering phenomena for driven coating films
View Description Hide DescriptionA theoretical and numerical model is formulated to describe the instability and the longtime evolution of both gravitydriven and surfaceshearstressdriven thin coating films. A single evolution equation, of higherorder diffusive type, models the flow for either problem. It is derived using the lubrication approximation. For partially wetting systems, the effect of finite contact angle is incorporated in the equation using a particular disjoining pressure model. The base state, in each case, is a twodimensional steadily propagating capillary front. Slight perturbations of the base state, applied along the front, initiate the fingering instability. Earlytime results accurately reproduce the wavelengths of fastest growth and the corresponding eigenmodes as reported in published linear stability analyses. As time proceeds, depending on parameter values, various fingering patterns arise. For conditions of perfect wetting with the substrate downstream of the moving front covered with a thin precursor layer, predicted nonlinear finger evolution agrees well with published experiments. The ultimate pattern, in this case, is a steadily translating pattern of wedgeshaped fingers. Alternatively, for partially wetting systems that exhibit sufficiently large static contact angles, long straightsided fingers or rivulets are formed. Finally, for larger contact angles, or at relatively low speeds, we predict that the flowing rivulets will become unstable and break up into strings of isolated droplets.

Extensional viscosity of dilute polystyrene solutions: Effect of concentration and molecular weight
View Description Hide DescriptionThis paper reports a detailed investigation of the steady shear and extensional properties of monodisperse polystyrene solutions for a range of molecular weights from 1.95 to 20 million and a range of concentrations from 69 ppm to 777 ppm. The steady shear and dynamic properties are reasonably well described by the Zimm model. The relaxation time and polymerviscosity evaluated using the Zimm model exhibit the expected scaling with concentration and molecular weight for theta solutions. In an extensional flow, these solutions show strain hardening and need about 5.5 to 6 strain units to reach steady state. The stress growth depends both on strain rate and strain. However, at moderate values of strain and when the Weissenberg number exceeds 6, the extensional stress growth depends only on total strain. The steady state extensional viscosity for each fluid was observed to depend on the magnitude of Weissenberg number (Wi). For the steady state extensional viscosity was observed to be an increasing function of the strain rate. At a the steady state extensional viscosity exhibits a maximum and surprisingly, for the steady state extensional viscosity is a decreasing function of the strain rate. The results indicate that even dilute solutions, with concentrations as low as 0.21 times the critical concentration, do show extension thinning under certain conditions. The transient and steady state extensional viscosities are found to be proportional to molecular weight and concentration (c). This result is rather unexpected as the Zimm model would predict a scaling with c and for the transient viscosity and scaling with c and for the steady state viscosity. On the other hand, the Rouse model predicts a scaling with c and (for transient extensional viscosity) and with c and (for steady state extensional viscosity). Hence the data appear to follow a Rouselike behavior for the transient viscosity. This implies that the advent of the stretching reduces the hydrodynamic interaction and a free draining behavior is obtained. As a result, predictions using the Zimm model parameters, estimated from shear data, are unable to predict the transient extensional viscosity. The data are analyzed using a constitutive equation that incorporates an anisotropic drag coefficient along with a Rouse spectrum of relaxation time. Such a model captures the extensional behavior of the solutions.

Stability of twodimensional strip casting processes
View Description Hide DescriptionThe solidification of molten materials is of great significance in modern metallurgical engineering. We study disturbances of a process that is characterized by three disparate lengths: the solidification length L, the wavelength Λ, and the depth of the slab The present analysis is motivated by the relation which is always true in practical strip casting processes. This justifies the use of an asymptotic expansion based on shallow water equations for long waves to describe the linear stability of disturbances. The leadingorder equations govern a quasiparallel flow. In this limit we found two different types of disturbances: a weakly damped stable mode that runs downstream and a strongly damped perturbation traveling upstream. We focus on the downstream moving mode and show that this disturbance is strongly frequency dependent. Although the velocity disturbances are damped, there is a regime of parameters where the perturbations of the displacement grow in the horizontal direction. In our analyses we found a region of preferred frequencies. The displacement grows at these frequencies faster than for neighboring frequencies. The wavelengths of disturbances oscillating at these preferred frequencies are in qualitative agreement with the experimental observation of the wavelengths of harmonically varying grooves in the completely solidified material.

Length scales of turbulence in stably stratified mixing layers
View Description Hide DescriptionTurbulence resulting from Kelvin–Helmholtz instability in layers of localized stratification and shear is studied by means of direct numerical simulation. Our objective is to present a comprehensive description of the turbulence evolution in terms of simple, conceptual pictures of shear–buoyancy interaction that have been developed previously based on assumptions of spatially uniform stratification and shear. To this end, we examine the evolution of various length scales that are commonly used to characterize the physical state of a turbulent flow. Evolving layer thicknesses and overturning scales are described, as are the Ozmidov, Corrsin, and Kolmogorov scales. These considerations enable us to provide an enhanced understanding of the relationships between uniformgradient and localizedgradient models for sheared, stratified turbulence. We show that the ratio of the Ozmidov scale to the Thorpe scale provides a useful indicator of the age of a turbulent event resulting from Kelvin–Helmholtz instability.

Anisotropy of turbulence in stably stratified mixing layers
View Description Hide DescriptionDirect numerical simulations of turbulence resulting from Kelvin–Helmholtz instability in stably stratified shear flow are used to study sources of anisotropy in various spectral ranges. The set of simulations includes various values of the initial Richardson and Reynolds numbers, as well as Prandtl numbers ranging from 1 to 7. We demonstrate that smallscale anisotropy is determined almost entirely by the spectral separation between the small scales and the larger scales on which background shear and stratification act, as quantified by the buoyancy Reynolds number. Extrapolation of our results suggests that the dissipation range becomes isotropic at buoyancy Reynolds numbers of order although we cannot rule out the possibility that smallscale anisotropy persists at arbitrarily high Reynolds numbers, as some investigators have suggested. Correlationcoefficient spectra reveal the existence of anisotropic flux reversals in the dissipation subrange whose magnitude decreases with increasing Reynolds number. The scalar concentration field tends to be more anisotropic than the velocity field. Estimates of the dissipation rates of kinetic energy and scalar variance based on the assumption of isotropy are shown to be accurate for buoyancy Reynolds numbers greater than Such estimates are therefore reliable for use in the interpretation of most geophysicalturbulence data, but may give misleading results when applied to smallerscale flows.

Direct numerical simulation of the flow in a liddriven cubical cavity
View Description Hide DescriptionDirect numerical simulation of the flow in a liddriven cubical cavity has been carried out at a Reynolds number above 10 000. Both transient and steadyinthemean states of the flow posses long time scales requiring long integration times. A large fraction of the total kinetic energy and dissipation is concentrated in the nearlid mean flow. The flow over most of the domain is laminar with distinct walljet profiles found in three of the walls. The high momentum fluid near the lid transmits its energy into a downflowing nonparallel wall jet which separates ahead of the bottom wall. From the collision of this separated layer against the bottom wall two wall jets emerge. In this process the energy lost to turbulence by the impingement is partly recovered by the emerging wall jets.

Crosschannel advective–diffusive transport by a monochromatic traveling wave
View Description Hide DescriptionThe crosschannel tracer flux due to the combined effects of advection and diffusion is considered for twodimensional incompressible flow in a channel, where the flow is that due to a monochromatic traveling wave and the boundary conditions at the walls are fixed tracer concentration. The tracer flux is computed numerically over a wide range of the parameters and with U the maximum fluid velocity, c the wave phase speed, K the tracer diffusivity, and L the channel width. Prior work has used analytical methods to obtain solutions for δ either infinite (stationary overturning cells) or small. In addition to the full numerical solutions,solutions obtained using mean field theory are presented, as well as a new asymptotic solution for small ε, and one for small δ due to Flierl and Dewar. The various approximations are compared with each other and with the numerical solutions, and the domain of validity of each is shown. Mean field theory is fairly accurate compared to the full numerical solutions for small δ, but tends to underpredict the tracer flux by 30–50% for larger δ. The asymptotic solution derived by Flierl and Dewar for small δ is found to break down when rather than when as suggested by the original derivation, and a scaling argument is presented which explains this.

Bragg scattering of surface waves over permeable rippled beds with current
View Description Hide DescriptionIn this study we develop a timedependent wave equation for waves propagating with a current over permeable rippled beds. As well known, Bragg resonance occurs when the incident wavelength is twice the wavelength of the bottom ripple undulation and no current is present. However, the current in the nearshore region changes the resonance condition. A onedimensional wave field is solved numerically based on the derived equation to study the effect of current on the Bragg resonance condition. Nonlinear wave–wave resonant interactiontheory provides an explanation of the effect on Bragg resonance. Numerical results also indicate that the maximum reflection coefficient increases as current velocity increases from a negative to a positive value. Furthermore, the velocity of the current affects the position of the maximum reflection coefficient.

Numerical simulation of pattern formation in the Bénard–Marangoni convection
View Description Hide DescriptionThe Bénard–Marangoni convection is numerically simulated using the full threedimensional Navier–Stokes equation. It is shown that hexagonal convection appears when the surfacetension effect dominates over the buoyancy effect. When two effects are comparable to each other, a mixed type of roll and hexagonal convections is stably obtained. It is found that the kinetic energy production by buoyancy does not depend on the convectionpattern, whereas that by surfacetension strongly depends on the convectionpattern. When the surfacetension effect is dominant, the kinetic energy production by surfacetension in the hexagonal convection is larger than that in the roll convection. Hexagonal convection gives a vertical heat transfer larger than that of roll convection.

On the relationship of effective Reynolds number and Strouhal number for the laminar vortex shedding of a heated circular cylinder
View Description Hide DescriptionThe laminar vortex shedding of airflow behind a circular cylinder with different heating temperatures was experimentally investigated with emphasis on the relationship of wake frequency and the Reynolds number. A new method to generate the twodimensional parallel vortex shedding for the heated cylinder was developed and tested. An “effective Reynolds number” that employs a kinematicviscosity computed from an “effective temperature” is used to account for the temperature effects on the vortex shedding frequency. The present result shows that the frequency data could be successfully collapsed with the effective temperature computed by for a wide range of cylinder temperatures, and being the freestream temperature and cylinder surface temperature, respectively. Moreover, the relationship between Strouhal number and effective Reynolds number was found to be “universal.” The physical interpretation of and the applicable region of the curve are discussed.

Mixing of viscous polymer liquids
View Description Hide DescriptionA torsionally driven cavity is proposed as a suitable mixing device for viscoelastic fluids. “Chaotic” behavior occurs in the secondary flow plane using polyacrylamide Boger fluids due to the inducement of elasticflow instabilities in situations where inertial forces are small. The torsionally driven cavity is also a suitable welldefined geometry for testing the ability of nonNewtonian constitutive models to describe the behavior of elastic fluids in threedimensional flow as a prelude to solving more difficult mixing problems.

Axisymmetric capillary waves on thin annular liquid sheets. I. Temporal stability
View Description Hide DescriptionA reduceddimension approach is employed to analyze the nonlinear distortion and disintegration of axisymmetric thin inviscid annular liquid sheets in a surrounding void with nonzero gascore pressure at zero gravity. Linear and nonlinear solutions for the free motion of periodically disturbed infinite linearly stable and unstable sheets are obtained and compared in this first paper. (The forced motion of semiinfinite annular sheets exiting from a nozzle or atomizer is considered in the second paper.) Both sinuous and dilational modes are studied. Both modes are dispersive unlike the planar case where only the dilational mode is dispersive. These modes are coupled even in the linear representation although for sufficiently large annular radius, a pure dilational linear oscillation is found. The sinuous oscillation always excites the dilational mode. Nonlinear effects can modify the wave shapes substantially, causing an increase in breakup time for the dilational mode and a decrease in breakup time for the sinuous mode. The capillary sheet instability due to the nonlinear interaction of harmonic and subharmonic dilational disturbances, originally observed on planar sheets, is also observed and analyzed for the annular geometry. Parametric studies on the influence of annular radius, disturbance wavelengths, and their ratios are reported.

Axisymmetric capillary waves on thin annular liquid sheets. II. Spatial development
View Description Hide DescriptionThe forced motion of semiinfinite axisymmetric thin inviscid annular liquid sheets, exiting from a nozzle or atomizer into a surrounding void under zero gravity but with constant gascore pressure is analyzed by means of the reduceddimension approach described in C. Mehring and W. A. Sirignano [Phys. Fluids 12, 1417 (2000)]. Linear analytical timedependent (“limitcycle”) solutions to the pure boundaryvalue problem are presented as well as linear and nonlinear numerical (transient) solutions to the mixed boundary and initialvalue problem of initially undisturbed sheets harmonically forced at the orifice or nozzle exit. Group velocities for the six independent solutions to the linear boundaryvalue problem are used to determine the location of boundary conditions. Numerical simulations of the linear transient problem are employed to validate these predictions. Parameter studies on sheet breakup and collapse lengths as well as on breakup and collapse times are reported. The dependence on modulation frequency, modulated disturbance amplitude, Weber number, and annular radius is presented for various cases of the mixed problem, i.e., for linearly or nonlinearly stable and unstable, dilationally or sinusoidally forced sheets. Nonlinear effects often have significant effects on breakup times and lengths or on collapse times and lengths. Nonlinear wave forms can deviate substantially from linear predictions resulting in major impacts on the size of the rings and shells that will remain after breakup.

Interaction between Görtler vortices and twodimensional Tollmien–Schlichting waves
View Description Hide DescriptionThe nonlinear interaction between Görtler vortices and twodimensional Tollmien–Schlichting (TS) waves is studied with a spatial, nonparallel model based on the parabolized stability equations. The effect of the TS waves on the development of the vortices is accounted for, showing that TS wave amplitudes of the same order of magnitude as the vortices result in significant nonlinear interaction. The range of governing parameters that has been studied so far is extended and the main effects of Görtler number, spanwise wave number, and initial amplitudes are identified. The study shows that the relative growth rates and initial amplitudes are the two most significant parameters.

A closure method for random advection of a passive scalar
View Description Hide DescriptionA novel functional method is applied to calculate the statistics of a passive scalar in an isotropic turbulent velocity field. The method yields asymptotic series expansions for small velocity correlation time from which approximate closure equations are derived. The closure method admits a diagram expansion, and is implemented as a Mathematica program. Padé summation of the asymptotic series yields accurate values for the effective diffusivity and gives formulas expressing the Lagrangian correlation of the velocity in terms of the Eulerian correlation. The approximations compare very favorably with numerical simulations of advection by a Gaussian velocity field.

Using cavitation to measure statistics of lowpressure events in largeReynoldsnumber turbulence
View Description Hide DescriptionThe structure of the pressure field of a turbulent water flow between counterrotating disks is studied using cavitation. The flow is seeded with microscopic gas bubbles and the hydrostaticpressure is reduced until large negative pressure fluctuations trigger cavitation.Cavitation is detected via light scattering from cavitating bubbles. The spatial structure of the lowpressure events are visualized using a highspeed video system. A fast photo detector is used to measure the scaling of the cavitation statistics with the pressure. This data is used to determine the shape of the tail of the probability density function for the pressure. The tail is found to be exponential and scales more rapidly with Reynolds number than the standard deviation of the pressure. This may indicate the influence of internal intermittency.

Temporal and spatial unmixedness downstream of a plate array
View Description Hide DescriptionThe effect of a plate array on a turbulent velocity and turbulent concentration field is determined. Profiles of mean and rootmeansquare velocity and concentration, profiles of temporal and spatial unmixedness, and profiles of the variance of the gradient of velocity and the variance of the gradient of concentration are presented. Velocity and concentration integral length scales are compared. A biplane injection grid is used to produce the turbulent concentration and turbulent velocity field. Helium is injected through the jets at the grid nodes as air passes through the grid. The timeresolved velocity and concentration data are obtained using a twosensor probe that consists of a hot wire and a TSI 144020 aspirating concentration probe. The addition of a plate array is shown to decrease the spatial unmixedness to a nearly zero value in about half the downstream distance observed without plates. Further, an increase in dissipation is shown with the array in place that reduces the temporal unmixedness to a value less than the value observed without the plates in about onethird the downstream distance.

Secondorder temperature and velocity structure functions: Reynolds number dependence
View Description Hide DescriptionAn interpolation relation is used to fit secondorder moments of temperature and velocity fluctuation increments which have been measured in three types of flows (decaying grid turbulence, cylinder wake, and circular jet) for values of the Taylor microscale Reynolds number in the range 30 (grid turbulence) to about 500 (jet). Several checks confirm the analytical framework underpinning the fit. The magnitude of the resulting scaling exponents increases with the longitudinal one being first to asymptote to a constant value. The scaling exponent for the temperature increment is generally smaller than that for the longitudinal velocity increment but larger than that for the transverse velocity increment when For the magnitude of the scaling exponents of the temperature and transverse velocity increments are nearly equal. Within the framework of smallscale intermittency, the magnitude of the Obukhov–Corrsin “constant” increases at small in similar manner to that of the longitudinal and transverse velocity Kolmogorov constants.
