Volume 12, Issue 4, April 2000
 ARTICLES


The migration of a drop in a uniform temperature gradient at large Marangoni numbers
View Description Hide DescriptionThe steady thermocapillary motion of a spherical drop in a uniform temperature gradient is treated in the situation where convective transport of energy is predominant in the drop phase as well as in the continuous phase, i.e., when the Marangoni numbers are large. It is assumed that the Reynolds numbers in both phases are large as well; to leading order, the velocity fields are given by a potential flow field in the continuous phase and Hill’s vortex inside the drop. The migration velocity of the drop is obtained by equating the rate at which work is done by the thermocapillary stress to the rate of viscous dissipation of energy. The analysis deals with an asymptotic situation wherein convective transport of energy dominates with conduction playing a role only where essential. This leads to thin thermal boundary layers both outside and within the drop. The method of matched asymptotic expansions is employed to solve the conjugate heat transfer problem in the two phases. It is shown that the demand for energy within the drop, necessary to increase its temperature at a steady rate as it moves into warmer surroundings, results in a large temperature difference between the surface of the drop and its interior. The variation of temperature over the drop surface is large as well, and leads to a linear increase of the migration velocity of the drop with increasing Marangoni number. This result is strikingly different from that for the limiting case when the viscosity and thermal conductivity inside the drop become negligible compared to the corresponding properties in the continuous phase. This limit, which holds for a gas bubble, is recovered correctly from the analysis.

Clebsch representation near points where the vorticity vanishes
View Description Hide DescriptionWe demonstrate that there is no Clebsch representation in any neighborhood of a generic vanishing point of the vorticity. This result is placed in the context of the Hamiltonian formulation of fluid mechanics. For stratified fluids, the analogous representation does exist, both locally and globally, under suitable hypotheses.

An investigation of the flow regimes resulting from splashing drops
View Description Hide DescriptionNumerical simulations have been used to investigate the flow regimes resulting from the impact of a 2.9 mm water drop on a deep water pool at velocities in the range 0.8–2.5 m/s. The results were used to identify the conditions leading to the formation of vortex rings, entrapment of a bubble during cavity collapse and the formation of vertical Rayleigh jets. Bubble entrapment and the associated growth of a thin high speed jet were shown to be the result of a capillary wave that propagates down the walls of the crater resulting from dropimpact. Although the existence of a capillary waves is a necessary condition for bubble entrapment, bubbles will only occur when the wave speed and maximum crater size is such that the wave reaches the bottom of the crater before collapse has resulted in the formation of a thick Rayleigh jet. Simulations also clarified the conditions for which dropimpact leads to axisymmetric vortex rings. Results not reported previously, include the observation that a single drop can produce multiple vortex rings and that vortex rings can occur for conditions that lead to broad Rayleigh jets. Based on these results, it was concluded that the formation of vortex rings depends on the time at which vorticity is generated and the nature of its subsequent transport.

Electrohydrodynamics and electrorotation of a drop with fluid less conductive than that of the ambient fluid
View Description Hide DescriptionIn this article, we investigated the electrohydrodynamic responses of a deformable fluid drop in another immiscible fluid under the action of a dcelectric field. Both the ambient and drop fluids considered here were incompressible Newtonian and all of the drop phases were less conductive than their ambient fluids. Under these circumstances, the drops experienced the socalled electrorotation owing to the reverse dipole generated by the external electric field when the electric field strength exceeded a certain threshold value. The experimental observation showed that the threshold electric field strength was dependent on the drop size as well as the viscosity ratio. Also noted was the effect of the electrorotation on both the deformation behavior and the mode of the drop breakup. Specifically, we determined the critical electric capillary number beyond which the steadystate drop shape did not exist and the drop eventually broke up. Finally, the validity of Taylor’s leaky dielectric theory was discussed in the presence of the electrorotation of the drop.

Stability of the shape of a surfactantladen drop translating at low Reynolds number
View Description Hide DescriptionWe examine the effect of surfactants on the evolution of the shape of an initially nonspherical drop translating in an otherwise quiescent fluid at low Reynolds number. A combination of the boundaryintegral method and a finitedifference scheme is used to solve the coupled fluid dynamics and surfactant transport problems, in conjunction with the Frumkin adsorption framework to account for the effects of monolayer saturation and nonideal surfactant interactions. For sufficiently small Bond numbers, the drop achieves a nonspherical steady shape. For large initial deformations or Bond numbers, however, the drop deforms continuously, and eventually breaks up through either the formation of an elongated tail or the development of a reentrant cavity at the trailing end, similar to the mechanisms of drop breakup reported by Koh and Leal [Phys. Fluids A 1, 8 (1989)], and Pozrikidis [J. Fluid Mech. 210, 1 (1990)], for surfactantfree drops.Surfactants are found to have a destabilizing effect on the shape of translating drops. The destabilizing effect is mitigated by the presence of stronglycohesive surfactant interactions, and by surfactant transport between the bulk and the interface.

Experimental investigation of interaction processes between droplets and hot walls
View Description Hide DescriptionA detailed experimental investigation of interaction processes of small liquid droplets with hot walls well above the Leidenfrost temperature has been carried out. The experimental method which uses monodisperse droplet streams in combination with a standard video camera allows very detailed observations and measurements with very high time resolution. The main intent of this paper is to study the mechanical behavior of liquid droplets impacting on hot walls well above the Leidenfrost temperature. A better understanding of this process may lead to a better modeling of twophase flows, especially for applications in fuel preparation processes, combustion processes, and spray cooling. The loss of momentum of the droplets, the droplet deformation, and the onset of droplet disintegration have been investigated. For all experimental results correlations have been developed, which can be used to improve the numerical modeling of twophase flows. Using the correlation for the loss of momentum a theoretical approximation for the maximum droplet deformation has been deduced, which yields a very good agreement with our own measurements as well as with results reported in the literature. A minimum impinging angle for droplet disintegration has been discovered for small impinging angles. Below this impinging angle no droplet disintegration is observed. This phenomenon is directly related to the energy dissipation at the wall during the interaction process. With the presented work the understanding of basic interaction processes between droplets and hot walls may be improved.

Spatial stability of similarity solutions for viscous flows in channels with porous walls
View Description Hide DescriptionThe spatial stability of similarity solutions for an incompressible fluid flowing along a channel with porous walls and driven by constant uniform suction along the walls is analyzed. This work extends the results of Durlofsky and Brady [Phys. Fluids 27, 1068 (1984)] to a wider class of similarity solutions, and examines the spatial stability of small amplitude perturbations of arbitrary shape, generated at the entrance of the channel. It is found that antisymmetric perturbations are the best candidates to destabilize the solutions. Temporally stable asymmetric solutions with flow reversal presented by Zaturska, Drazin, and Banks [Fluid Dyn. Res. 4, 151 (1988)] are found to be spatially unstable. The perturbed similarity solutions are also compared with fully bidimensional ones obtained with a finite difference code. The results confirm the importance of similarity solutions and the validity of the stability analysis in a region whose distance to the center of the channel is more than three times the channel halfwidth.

Anomalous scaling from controlled closure in a shell model of turbulence
View Description Hide DescriptionWe present a model of hydrodynamicturbulence for which the program of computing the scaling exponents from first principles can be developed in a controlled fashion. The model consists of N suitably coupled copies of the “Sabra” shell model of turbulence. The couplings are chosen to include two components: random and deterministic, with a relative importance that is characterized by a parameter called ε. It is demonstrated, using numerical simulations of up to 25 copies and 28 shells that in the limit but for this model exhibits correlation functions whose scaling exponents are anomalous. The theoretical calculation of the scaling exponents follows verbatim the closure procedure suggested recently for the Navier–Stokes problem, with the additional advantage that in the limit the parameter ε can be used to regularize the closure procedure. The main result of this paper is a finite and closed set of scaleinvariant equations for the 2nd and 3rd order statistical objects of the theory. This set of equations takes into account terms up to order and neglects terms of order Preliminary analysis of this set of equations indicates a K41 normal scaling at with a birth of anomalous exponents at larger values of ε, in agreement with the numerical simulations.

Spatially correlated and inhomogeneous random advection
View Description Hide DescriptionWe study the equilibrium statistics of an idealized model of random advection. Theoretical results for largescale homogeneous forcing—notably the shapes of the PDFs—are recovered; they are compared with results for various inhomogeneous and intermittent forcings. The correlation times of the forcings are varied. The PDFs are interpreted in terms of the underlying mixing time scales. Conditional statistics of the tracer field—the conditional diffusion in particular—are computed; they are used to elucidate the influence of smallscale tracer–field correlations. We contrast results for spatially smooth and spatially rough velocity fields.

Experimental study of Taylor’s hypothesis in a turbulent soap film
View Description Hide DescriptionAn experimental study of Taylor’s hypothesis in a quasitwodimensional turbulent soap film is presented. A twoprobe laser Doppler velocimeter enables a nonintrusive simultaneous measurement of the velocity at spatially separated points. Using the cross correlation between a pair of points displaced in both space and time, the velocitycoherence is measured to be better than 90% for scales less than the integral scale. Taylor’s hypothesis is confirmed insofar as the lower moments of the longitudinal velocity difference are equal whether measured with or without invoking the hypothesis. A quantitative study of the decorrelation beyond the integral scale is also presented.

The interaction of vorticity and rateofstrain in homogeneous sheared turbulence
View Description Hide DescriptionThe coupled interaction of vorticity and rateofstrain in homogeneous sheared turbulence is investigated using direct numerical simulation. Conditional sampling and comparison with linear simulations reveal various aspects of the structure and dynamics. Due to the influence of the imposed and distinct directional features develop. Initial stretching of fluctuating by mean extensional strain and the presence of mean vorticity establish a predominant misalignment of with respect to the principal axes of The associated locally induced rotation of the axes results in preferred orientations in and In high amplitude rotationdominated regions of the flow, distinct characteristics are exhibited by the pressure Hessian due to the presence of smallscale spatial structure. Nonlocally induced axes rotation through tends to counteract locally induced rotation in these regions. These features are absent in the linear flow which suggests a lack of spatial coherence in the corresponding intense regions. High amplitude straindominated and comparable rotationstrain regions are also considered. In general, the high amplitude conditional samples capture the main features of the flow. The underlying behavior of and is essentially the same as in isotropic turbulence; the directional preferences observed in shear flow demonstrate the physical implications of the associated mechanisms. Although there is greater directional variation in flows with high results indicate the significance of the persistence of mean shear.

Experimental study of turbulent Poiseuille–Couette flow
View Description Hide DescriptionResults from the experimental study of fully developed turbulent plane Poiseuille–Couette flow (PCF) are presented. This study provides information that improves the physical descriptions of previous experimental works and sheds light on turbulence phenomena reported at low to moderate Reynolds numbers. Molecular tagging velocimetry (MTV) is employed for the first time on gas PCFs over the Poiseuille and Couette flowReynolds number ranges of and respectively. MTV has allowed the measurement of instantaneous velocity profiles at a much higher resolution than previous single point techniques. Results include a locally negative production of streamwise turbulence near the forward moving wall, and a noticeable effect on intensities in the core flow region and near the moving wall. It has been found that the socalled “geometry effect” does play a role in the exchange of fluid motions between the channel walls at low This effect is amplified when one of the walls is put into motion. Results indicate that the exchange of disturbances is mainly driven by the more energetic motions or “active” motions originating at the high stress wall. These disturbances are convected towards the low stress wall and impinge on its surface. This can lead to a localized region of negative turbulence energy production.

Selfdiffusion in freely evolving granular gases
View Description Hide DescriptionA selfdiffusionequation for a freely evolving gas of inelastic hard disks or spheres is derived starting from the Boltzmann–Lorentz equation, by means of a Chapman–Enskog expansion in the density gradient of the tagged particles. The selfdiffusion coefficient depends on the restitution coefficient explicitly, and also implicitly through the temperature of the system. This latter introduces also a time dependence of the coefficient. As in the elastic case, the results are trivially extended to the Enskog equation. The theoretical predictions are compared with numerical solutions of the kinetic equation obtained by the direct simulation Monte Carlo method, and also with molecular dynamics simulations. An excellent agreement is found, providing mutual support to the different approaches.

Planform structure and heat transfer in turbulent free convection over horizontal surfaces
View Description Hide DescriptionThis paper deals with turbulent free convection in a horizontal fluid layer above a heated surface. Experiments have been carried out on a heated surface to obtain and analyze the planform structure and the heat transfer under different conditions. Water is the working fluid and the range of flux Rayleigh numbers (Ra) covered is The different conditions correspond to RayleighBénard convection,convection with either the top watersurface open to atmosphere or covered with an insulating plate, and with an imposed external flow on the heated boundary. Without the external flow the planform is one of randomly oriented line plumes. At large Rayleigh number Ra and small aspect ratio (AR), these line plumes seem to align along the diagonal, presumably due to a large scale flow. The side views show inclined dyelines, again indicating a large scale flow. When the external flow is imposed, the line plumes clearly align in the direction of external flow. The nondimensional average plume spacing, varies between 40 and 90. The heat transfer rate, for all the experiments conducted, represented as where is the conduction layer thickness, varies only between 0.1–0.2, showing that in turbulent convection the heat transfer rates are similar under the different conditions.

Flow structure and modeling issues in the closure region of attached cavitation
View Description Hide DescriptionParticle imagevelocimetry(PIV) and highspeed photography are used to measure the flow structure at the closure region and downstream of sheetcavitation. The experiments are performed in a water tunnel of cross section whose test area contains transparent nozzles with a prescribed pressure distribution. This study presents data on instantaneous and averaged velocity,vorticity and turbulence when the ambient pressure is reduced slightly below the cavitation inception level. The results demonstrate that the collapse of the vapor cavities in the closure region is the primary mechanism of vorticity production. When the cavity is thin there is no reverse flow downstream and below the cavitation, i.e., a reentrant flow does not occur. Instead, the cavities collapse as the vapor condenses, creating in the process hairpinlike vortices with microscopic bubbles in their cores. These hairpin vortices, some of which have sizes as much as three times the height of the stable cavity, dominate the flow downstream of the cavitating region. The averaged velocity distributions show that the unsteady collapse of the cavities in the closure region involves substantial increase in turbulence, momentum, and displacement thickness. Two series of tests performed at the same velocity and pressure, i.e., at the same hydrodynamic conditions, but at different water temperatures, 35 °C and 45 °C, show the effect of small changes in the cavitation index (σ=4.69 vs. σ=4.41). This small decrease causes only a slight increase in the size of the cavity, but has a significant impact on the turbulence level and momentum deficit in the boundary layer downstream. Ensemble averaging of the measured instantaneous velocity distributions is used for estimating the liquid void fraction, average velocities, Reynolds stresses, turbulent kinetic energy and pressure distributions. The results are used to examine the mass and momentum balance downstream of the cavitating region. It is shown that in dealing with the ensembleaveraged flow in the closure region of attached cavitation, one should account for the sharp (but still finite) gradients in the liquid void fraction. The 2D continuity equation can only be satisfied when the gradients in void fraction are included in the analysis. Using the momentum equation it is possible to estimate the magnitude of the “interaction term,” i.e., the impact of the vapor phase on the liquid momentum. It is demonstrated that, at least for the present test conditions, the interaction term can be estimated as the local pressure multiplied by the gradient in void fraction.

Effect of acoustic streaming on the mass transfer from a sublimating sphere
View Description Hide DescriptionThe effect of the acoustic streaming on the mass transfer from the surface of a sphere positioned in an ultrasonic acoustic levitator is studied both experimentally and theoretically. Acoustic levitation using standing ultrasonicwaves is an experimental tool for studying the heat and mass transfer from small solid or liquid samples, because it allows an almost steady positioning of a sample at a fixed location in space. However, the levitator introduces some difficulties. One of the main problems with acoustic levitation is that an acoustic streaming is induced near the sample surface, which affects the heat and mass transfer rates, as characterized by increased Nusselt and Sherwood numbers. The transfer rates are not uniform along the sample surface, and the aim of the present study is to quantify the spatial Sherwood number distribution over the surface of a sphere. The experiments are based on the measurement of the surface shape of a sphere layered with a solid substance as a function of time using a chargecoupled device(CCD) camera with backlighting. The sphere used in this research is a glass sphere layered with a volatile solid substance (naphthalene or camphor). The local mass transfer from the surface both with and without an ultrasonic acoustic field is investigated in order to evaluate the effect of the acoustic streaming. The experimental results are compared with predictions following from the theory outlined [A. L. Yarin, M. Pfaffenlehner, and C. Tropea, J. Fluid Mech. 356, 65 (1998); A. L. Yarin, G. Brenn, O. Kastner, D. Rensink, and C. Tropea, ibid. 399, 151 (1999)] which describes the acoustic field and the resulting acoustic streaming, and the mass transfer at the surface of particles and droplets located in an acoustic levitator. The results are also compared with the experimental data and with the theoretical predictions of Burdukov and Nakoryakov [J. Appl. Mech. Tech. Phys. 6, 51 (1965)], which are valid only in the case of spherical particles much smaller than the sound wavelength. Good agreement between experiment and the theory of Yarin et al. is demonstrated. The timeaveraged heat and mass transfer rates over a sphere surface are greatest at the sphere’s equator and least at its poles in the experiment as predicted by the theory (the ultrasonic standing wave spans the vertical axis passing through the poles). The measured distribution of the mass transfer rate over the sphere surface also agrees with the theoretical predictions, which shows that in strong acoustic fields sublimation (or evaporation) results from the acoustic streaming.

On the characteristics of a spark generated shock wave
View Description Hide DescriptionRecent experiments involving shock waves propagating through weakly ionized plasmas have raised questions regarding interpretation of the experimental results. In aid of analyzing these experimental results, shock waves initiated by a simulated spark and their subsequent propagation in a cylindrical tube containing argon initially at 30 Torr and 300 K, have been analyzed numerically in this paper. Numerical solutions to the compressible Navier–Stokes equations are considered under the four conditions of induced flow (a) without wall friction, (b) with wall friction, (c) with wall friction and purely axial thermal gradients, and (d) with wall friction and both axial and radial thermal gradients. Although plasma processes have not been simulated, it is found that the effects of wall shear and thermal gradients alone are sufficient to explain most of the experimental observations. This work represents a first step in the analysis of this problem before plasmaeffects can be modeled.
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 BRIEF COMMUNICATIONS


Inflow conditions for largeeddy simulations of mixing layers
View Description Hide DescriptionA method to generate inflow boundary conditions for largeeddy simulations(LES) of turbulent free shear layers is presented. A time series of instantaneous velocity planes, with duration approximately equal to the integral time scale of the flow, is extracted from a periodic simulation and saved on disk. This signal is transformed into a periodic one, by using a conventional windowing technique, and is reused in the actual simulation as many times as required to obtain converged statistics. The method is applied in a LES of a spatially developing turbulent mixing layer. It is shown that the periodicity induced by the inflow signal decays rapidly in about 25% of the domain, and that the length of the period has small effect on the statistics, which agree well with the reference experimental data. The method appears to be a costeffective strategy in the generation of inflow data in a large variety of flows.
