Volume 12, Issue 10, October 2000
 ARTICLES


Quick deposition of a fluid on the wall of a tube
View Description Hide DescriptionWe are interested in the amount of fluid left behind a drop moved inside a capillary tube. Long ago, Taylor showed that for very viscous liquids moved at small velocities, the film thickness is a monotonic increasing function of the capillary number. New data obtained with liquids of low viscosity are reported here and compared with Taylor’s law. Two successive effects are observed: above a threshold in capillary number, the film is thicker than a Taylor film; at a very high speed, the deposition law becomes a decreasing function of the drop velocity. Both behaviors are analyzed thanks to scaling arguments and shown to be consequences of inertia.

A corotating tworoll mill for studies of twodimensional elongational flows with vorticity
View Description Hide DescriptionA closedform analytical solution for the Stokes flow generated by a corotating tworoll mill is used to generate twodimensional elongational flows with welldefined amounts of vorticity. These flow devices can generate conditions found between simple shear and purely elongational flows, and are among the few systems that can take into account the presence of nearby boundaries. In the laboratory, these devices have been used for studies of the microstructural dynamics of complex fluids such as polymeric solutions,colloids, bubbles and drops, etc., and allow pointwise optical studies of the flow parameters. Hence, a comparison is presented of available experimental and numerical results for the stagnation point versus the analytical solution presented here. Based upon this solution, a new design for the tworoll mill is presented for the purpose of minimizing the disadvantages of current mills.

Compound droplet in extensional and paraboloidal flows
View Description Hide DescriptionExact analytical solutions are found for the steady state creeping flow in and around a vapor–liquid compound droplet, consisting of two orthogonally intersecting spheres of arbitrary radii (a and b), submerged in axisymmetric extensional and paraboloidal flows of fluid with viscosity The solutions are presented in singularity form with the images located at three points: the two centers of the spheres and their common inverse point. The important results of physical interest such as drag force and stresslet coefficient are derived and discussed. These flow properties are characterized by two parameters, namely the dimensionless viscosity parameter: and the dimensionless parameter: where is the viscosity of the liquid in the sphere (part of the compound droplet) with radius b. We find that for some extensional flows, there exists a critical value of for each choice of Λ in the interval 0⩽Λ⩽1 such that the drag force is negative, zero or positive depending on whether or respectively. For other extensional flows, the drag force is always positive. The realization of these various extensional flows by simply changing the choice of the origin in our description of the undisturbed flow field is also discussed. In extensional flows where the drag force is always positive, we notice that this drag force for vapor–liquid compound droplet is maximum when β≈1 (i.e., two spheres have almost the same radii). Moreover, we find the drag force is a monotonic function of Λ, i.e., the drag force for vapor–liquid compound droplet lies between vapor–vapor and vaporrigid assembly limits. We also find that the maximum value of the drag in paraboloidal flow depends on the viscosity ratio Λ and significantly on the liquid volume in the dispersed phase.

Coalescence and capillary breakup of liquid volumes
View Description Hide DescriptionThe problem of the mathematical modeling of coalescence and breakup of liquid volumes surrounded by an inviscid gas is considered. As is shown, an unphysical singularity in the known selfsimilar solutions of the Navier–Stokes equations intended to describe the topological transition of the flow domain arises as a consequence of the assumption that the free surface becomes smooth immediately after the onset of coalescence or remains so up to the very moment of breakup. Then the standard kinematicboundary condition prescribes that fluid particles belonging to the free surface remain there at all times and thus couples the scales for lengths and velocities in a selfsimilar solution leading to the singularity. An alternative approach allowing one to remove the singularity at a macroscopic level is formulated. Its key idea is that the topological transition, being a particular case of an interface formation/disappearance process, is associated with a freesurface cusp either propagating away from the point of initial contact of two volumes leading to their coalescence or “severing” a liquid thread connecting them in the case of breakup. The interface becomes (or, in a reverse flow, ceases to be) smooth at a finite distance from the point where, in the standard approach, a singularity would have taken place. An earlier developed macroscopic theory of interface formation/disappearance is applied without any ad hoc changes.

Inviscid damping of asymmetries on a twodimensional vortex
View Description Hide DescriptionThe inviscid damping of an asymmetric perturbation on a twodimensional circular vortex is examined theoretically, and with an electron plasma experiment. In the experiment, an elliptical perturbation is created by an external impulse. After the impulse, the ellipticity(quadrupole moment) of the vortex exhibits an early stage of exponential decay. The measured decay rate is in good agreement with theory, in which the perturbation is governed by the linearized Euler equations. Often, the exponential decay of ellipticity is slow compared to a vortex rotation period, due to the excitation of a quasimode. A quasimode is a vorticity perturbation that behaves like a single azimuthally propagating wave, which is weakly damped by a resonant interaction with corotating fluid. Analytically, the quasimode appears as a wave packet of undamped continuum modes, with a sharply peaked frequency spectrum, and it decays through interference as the modes disperse. When the exponential decay rate of ellipticity is comparable to the vortex rotation frequency, the vorticity perturbation does not resemble a quasimode; rather, it is rapidly dominated by spiral filaments. Over longer times, linear theory predicts algebraic decay of ellipticity; however, nonlinear oscillations of ellipticity emerge in the experiment before a transition to algebraic decay would occur.

Statistical representation of a spray as a point process
View Description Hide DescriptionThe statistical representation of a spray as a finite point process is investigated. One objective is to develop a better understanding of how singlepoint statistical information contained in descriptions such as the dropletdistribution function (ddf), relates to the probability density functions(pdfs) associated with the droplets themselves. Singlepoint statistical information contained in the dropletdistribution function (ddf) is shown to be related to a sequence of single surrogatedroplet pdfs, which are in general different from the physical singledroplet pdfs. It is shown that the ddf contains less information than the fundamental singlepoint statistical representation of the spray, which is also described. The analysis shows which events associated with the ensemble of spray droplets can be characterized by the ddf, and which cannot. The implications of these findings for the ddf approach to spray modeling are discussed. The results of this study also have important consequences for the initialization and evolution of direct numerical simulations (DNS) of multiphase flows, which are usually initialized on the basis of singlepoint statistics such as the dropletnumber density in physical space. If multiphase DNS are initialized in this way, this implies that even the initial representation contains certain implicit assumptions concerning the complete ensemble of realizations, which are invalid for general multiphase flows. Also the evolution of a DNS initialized in this manner is shown to be valid only if an as yet unproven commutation hypothesis holds true. Therefore, it is questionable to what extent DNS that are initialized in this manner constitute a direct simulation of the physical droplets. Implications of these findings for large eddy simulations of multiphase flows are also discussed.

On weakly nonlinear modulation of waves on deep water
View Description Hide DescriptionWe propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger (NLS) equation for deepwater waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear fourwave resonance (the “figure 8” of Phillips) even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to highwavenumber modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

Stability of a fourvortex aircraft wake model
View Description Hide DescriptionThe stability of an aircraft wake model composed of an external vortex pair (modeling the wing tip vortices) and an internal vortex pair rotating in the opposite direction (modeling the vortices generated by the fuselage and the horizontal tail) in a stationary configuration is investigated with the vortex filament stability method used by Crow [AIAA J. 8, 2172 (1970)] and Crouch [J. Fluid Mech. 350, 311 (1997)]. It is shown that this configuration is unstable with respect to twodimensional and threedimensional disturbances. For long wavelength threedimensional symmetric perturbations, the rapid growth observed in the numerical simulations of Rennich and Lele [J. Air. 36, 398 (1999)] is found. Moreover, the analysis allows one to show that without an excitation of the longwave mode, the system will naturally develop short wavelength instabilities localized within the inner vortices which do not affect the outer vortices. Inspection of the initial value problem shows that the longwave modes can be efficiently initiated by the introduction of perturbations on the internal vortices.

Numerical study of cavitation inception in the near field of an axisymmetric jet at high Reynolds number
View Description Hide DescriptionCavitation inception in the near field of high Reynolds number axisymmetric jets is analyzed using a simplified computational model. The model combines a vorticity–streamfunction finitedifference scheme for the simulation of the unsteady flow field with a simplified representation for microscopic bubbles that are injected at the jet inlet. The motion of the bubbles is tracked in a Lagrangian reference frame by integrating a semiempirical dynamical equation which accounts for pressure, drag, and lift forces. The likelihood of cavitation inception is estimated based on the distributions of pressure and microscopic bubbles. The computations are used to examine the role of jet slenderness ratio, Reynolds number, bubble size, and bubble injection location on the cavitation inception indices. The results indicate that, for all bubble sizes considered, the cavitation inception index increases as the jet slenderness ratio decreases. Larger bubbles entrain more rapidly into the cores of concentrated vortices than smaller bubbles, and the corresponding inception indices are generally higher than those of smaller bubbles. The inception indices for larger bubbles are insensitive to the injection location, while the inception indices of smaller bubbles tend to increase when they are injected inside the shear layer near the nozzle lip. Although it affects the bubble distributions, variation of the Reynolds number leads to insignificant changes in pressure minima and in the inception indices of larger bubbles, having noticeable effect only on the inception indices of smaller bubbles. Computed results are consistent with, and provide plausible explanations for, several trends observed in recent jet cavitation experiments.

Twophase flow analysis of unstable fluid mixing in onedimensional geometry
View Description Hide DescriptionA twophase flow model for an accelerationdriven compressible fluid mixing layer is applied to an initially planar/cylindrical/spherical fluid configuration. A conservative form of the onedimensional compressible equations is derived under the assumption that the fluid concentration is continuous. With a hyperbolic conservation law for the concentration gradient, the model supports traveling discontinuities in this quantity. The primary examples of this wave type are the moving boundaries of a finite mixing layer, which determine the instability growth rate. Constitutive laws for interfacial averages, previously derived for planar incompressible mixing, are reinterpreted and shown to be applicable to other onedimensional mixing problems of interest. The equations of motion for an incompressible mixing layer in planar, cylindrical, or spherical geometry are solved exactly, up to a history integral of a function of the edge trajectories, and without assuming incompressible flow outside the layer. Full solutions are obtained by numerically integrating a coupled system of ordinary differential equations for the volume fraction characteristics. Results for selfsimilar Rayleigh–Taylor mixing in planar geometry are compared to the work of others. This comparison suggests that the shape of the fluid concentration profile is primarily a consequence of mass conservation, parameterized by the expansion ratio of the mixing zone edges.

The velocity distribution of barotropic turbulence
View Description Hide DescriptionWe study the statistical properties of the velocity and velocity gradient distributions in barotropic turbulence. At large enough Reynolds number, the velocity distribution becomes nonGaussian outside the vortex cores, and its characteristics are completely determined by the properties of the far field induced by the coherent vortices. The velocity gradients are always nonGaussian inside coherent vortices, due to the spatial velocity correlations associated with the ordered flow in the vortex cores, and become nonGaussian also in the background turbulence at large enough Reynolds number.

The baroclinic secondary instability of the twodimensional shear layer
View Description Hide DescriptionThe focus of this study is on the numerical investigation of twodimensional, isovolume, high Reynolds and Froude numbers, variabledensity mixing layers. Lagrangian simulations, of both the temporal and the spatial models, are performed. They reveal the breakingup of the strained vorticity and densitygradient braids, connecting two neighboring primary structures. The secondary instability arises where the vorticity has been intensified by the baroclinic torque. A simplified model of the braid of the variabledensity mixing layer, consisting of a strained vorticity and densitygradient filament, is analyzed. It is concluded that the physical mechanism responsible for the secondary instability is the forcing of the vorticity field by the baroclinic torque, itself sensitive to perturbations. This mechanism suggests a rapid route to turbulence for the variabledensity mixing layer.

The effect of forcing on the spatial structure and spectra of chaotically advected passive scalars
View Description Hide DescriptionThe stationary distribution of passive tracers chaotically advected by a twodimensional largescale flow is investigated. The value of the tracer is conserved following each fluid element except when the element enters certain localized regions. The tracer value is then instantenously reset to a value associated with the region entered. This resetting acts as a forcing for the tracer field. This problem is mathematically equivalent to advection in open flows and results in a fractal tracer structure. The spectral exponent of the tracer field is different from that for a passive tracer with the usual additive forcing (the socalled Batchelor spectrum) and is related to the fractal dimension of the set of points that have never visited the forcing regions. We illustrate this behavior by considering a timeperiodic flow whose effect is equivalent to a simple twodimensional areapreserving map. We also show that similar structure in the tracer field is found when the flow is aperiodic in time.

Evolution of the vorticityarea density during the formation of coherent structures in twodimensional flows
View Description Hide DescriptionIt is shown: (1) that in twodimensional, incompressible, viscous flows the vorticityarea distribution evolves according to an advectiondiffusion equation with a negative, time dependent diffusion coefficient and (2) how to use the vorticitystream function relations, i.e., the socalled scatterplots, of the quasistationary coherent structures in order to quantify the experimentally observed changes of the vorticity distribution moments leading to the formation of these structures.

An evaluation of the assumed beta probability density function subgridscale model for large eddy simulation of nonpremixed, turbulent combustion with heat release
View Description Hide DescriptionThe assumed beta distribution model for the subgridscale probability density function (PDF) of the mixture fraction in large eddy simulation of nonpremixed, turbulent combustion is tested, a priori, for a reacting jet having significant heat release (density ratio of 5). The assumed beta distribution is tested as a model for both the subgridscale PDF and the subgridscale Favre PDF of the mixture fraction. The beta model is successful in approximating both types of PDF but is slightly more accurate in approximating the normal (nonFavre) PDF. To estimate the subgridscale variance of mixture fraction, which is required by the beta model, both a scale similarity model and a dynamic model are used. Predictions using the dynamic model are found to be more accurate. The beta model is used to predict the filtered value of a function chosen to resemble the reaction rate. When no model is used, errors in the predicted value are of the same order as the actual value. The beta model is found to reduce this error by about a factor of two, providing a significant improvement.

Effect of preferential concentration on turbulent collision rates
View Description Hide DescriptionThe effect of particle inertia on the interparticle collision rates of a turbulentaerosol was investigated recently by Sundaram and Collins (1997) using direct numerical simulation (DNS). They observed that for values of the particle Stokes number (here defined as the ratio of the particle response time to Kolmogorov time scale) near unity, the collision frequency was enhanced by between one and two orders of magnitude. This enhancement was attributed in part to the local enrichment of the particle concentration in lowvorticity regions of the flow due to the centrifuge effect commonly referred to as preferential concentration (Eaton and Fessler 1994). Sundaram and Collins (1997) showed that the correction factor for the collision kernel in a preferentially concentrated system is where is the particle radial distribution function and σ is the collision diameter. This paper uses DNS, in combination with statistical analysis, to study the dependence of the radial distribution function on the turbulence and particle parameters. A curve fit of the results over a broad range of the relevant dimensionless parameters enables easy estimation of The effect of system Reynolds number over the limited range accessible by DNS is also presented. In general, the degree of preferential concentration increases with increasing Reynolds number.

Largeeddy simulation of a turbulent piloted methane/air diffusion flame (Sandia flame D)
View Description Hide DescriptionThe Lagrangian Flamelet Model is formulated as a combustionmodel for largeeddy simulations of turbulent jet diffusionflames. The model is applied in a largeeddy simulation of a piloted partially premixed methane/air diffusionflame (Sandia flame D). The results of the simulation are compared to experimental data of the mean and RMS of the axial velocity and the mixture fraction and the unconditional and conditional averages of temperature and various species mass fractions, including CO and NO. All quantities are in good agreement with the experiments. The results indicate in accordance with experimental findings that regions of high strain appear in layer like structures, which are directed inwards and tend to align with the reaction zone, where the turbulence is fully developed. The analysis of the conditional temperature and mass fractions reveals a strong influence of the partial premixing of the fuel.

Direct numerical simulation of turbulent thermal boundary layers
View Description Hide DescriptionIn this paper, a method of generating realistic turbulent temperature fluctuations at a computational inlet is proposed and direct numerical simulations of turbulent thermal boundary layers developing on a flat plate with isothermal and isoflux wall boundary conditions are carried out. Governing equations are integrated using a fully implicit fractionalstep method with grids for the Reynolds number of 300, based on the freestream velocity and the inlet momentum thickness, and the Prandtl number of 0.71. The computed Stanton numbers for the isothermal and isoflux walls are in good agreement with powerlaw relations without transient region from the inlet. The mean statistical quantities including rootmeansquare temperature fluctuations,turbulent heat fluxes, turbulent Prandtl number, and skewness and flatness of temperature fluctuations agree well with existing experimental and numerical data. A quadrant analysis is performed to investigate the coherence between the velocity and temperature fluctuations. It is shown that the behavior of the wallnormal heat flux is similar to that of the Reynolds shear stress, indicating close correlation between the streamwise velocity and temperature. The effect of different thermal boundary conditions at the wall on the nearwall turbulence statistics is also discussed.

Stably stratified turbulent channel flows. I. Stratification regimes and turbulence suppression mechanism
View Description Hide DescriptionOur objective in this study is to study inhomogeneous stratified shear flows using large eddy simulation; stratified pressuregradientdriven channel flow was selected. The flows can be separated into three regimes: buoyancy affected, buoyancy controlled, and buoyancy dominated flows. The regime boundaries are defined by Richardson and Reynolds numbers based on the friction velocity. Buoyancy affected flows remain actively turbulent and attain a statistical steady state that resembles a lower Reynolds number unstratified flow.Flows in the buoyancy controlled regime are not in equilibrium. In the cases studied, an asymmetry develops with respect to the channel centerline leading to onesided turbulence. Eventually, the “inactive” half undergoes a transition initiated by the active half and symmetry is restored. At higher Richardson numbers, the flows are buoyancy dominated, the nearwall burstsweep process is completely disrupted and turbulence production ceases, leading to relaminarization. In relaminarizing flows, the inner and outer regions behave nearly independently. While the inner region turbulence decays monotonically, largescale restratification, internal waves, and potential energydriven motions are observed in the outer region. The simulation results are used to construct a physical model of stratified wallbounded flows. Stable stratification weakens the interaction between the inner and outer regions by decreasing the vertical transport, leading to neardecoupling of the two layers at strong enough stratification. The notion that the disappearance of the log region marks the onset of buoyancy control provides a criterion for estimating the Richardson number delineating the transition from buoyancy affected to buoyancy controlled flows. Data that should be useful for creating parametrizations for prediction of stratified flows are also presented.

Aeroacoustics of a stagnation flow near a rigid wall
View Description Hide DescriptionPropagation of disturbances in a nonuniform mean flow is investigated by highorder numerical simulation for various initial conditions. Monopole and dipole acoustic, vortical and entropy pulses are embedded in an incompressible stagnation flow, which is taken as a prototype of a nonuniform low Mach number mean flow near a rigid wall at high angle of attack. Numerical results are discussed in terms of baroclinic generation of disturbance vorticity that appear to be a key process in energy transfer between a nonuniform mean flow and a propagating disturbance. These phenomena lead to amplification of sound waves originated from an acoustic pulse. Vorticity generation governs wave radiation of a nearwall entropy pulse and makes the radiated waves similar to those from a vortical dipole. Interaction of initial pulse vorticity with generated vorticity leads to various radiated wave patterns discussed here.
