NASA-TM- 106751

NASA Technical Memorandum 106751 I€l_ _000 _1 _ICOMP-94-23; AIAA-94--0835

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Numerical Simulations of Drop Collisions

M.R.H. NobariUniversity of MichiganAnn Arbor, Michigan

and

G. TryggvasonInstitutefor ComputationalMechanics in PropulsionLewis Research CenterCleveland, Ohio

and University of MichiganAnn Arbor, Michigan

Prepared for the32nd Aerospace Sciences Meeting and Exhibitsponsored by the American Institute of Aeronautics and AstronauticsReno, Nevada, January 10-13, 1994

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NUMERICAL SIMULATIONS OF DROP COLLISIONS

M.R.H. NobanThe Universityof Michigan

• AnnArbor, MIand

G. Tryggvasont, Institute for Computational Mechanics in Propulsion,

Lewis Research Center,Cleveland,OHand

The University of MichiganAnn Arbor, MI

Abstract incorporated into the equations used to predict thelarge scale behavior. Many spray models (see

Three-dimensional simulations of the off-axis Heywood l, for a discussion and references) usecollisions of two drops are presented. The full point particles to represent the drops. The dropNavier-Stokes equations are solved by a Front- motion is related to the fluid flow by empiricalTracking/Finite-Difference method that allows a laws for drag, heat transfer and combustion. Oftenfully deformable fluid interface and the inclusion of it is possible to focus on the dynamic of a singlesurface tension. The drops are accelerated towards drop and how it interacts with the surroundingeach other by a body force that is turned off before flow. When the number of drops per unit volume isthe drops collide. Depending on whether the high, however, it is necessary to account for theinterface between the drops is ruptured or not, the interactions between the drops and their collectivedrops either bounce or coalesce. For drops that effect on the flow. To account for drop collisions,coalesce, the impact parameter, which measures models must contain "collision rules" thathow far the drops are off the symmetry line, determine whether the drops coalesce or not. Thesedetermines the eventual outcome of the collision, rules are usually based on experimentalFor low impact parameters, the drops coalesce investigations of binary collisions of drops, but thepermanently, but for higher impact parameters, a small spatial and temporal scales make detailedgrazingcollision, where the drops coalesce and then experimentalmeasurementsdifficult and usually thestretch apart again is observed. The results are in record consist of little more than photographs or aagreement with experimental observations, video tape. Since the collision process generally

involves large drop deformation and rupture of theinterface separating the drops, it has not been

Introduction amenable to detailed theoretical analysis. Previousstudies are therefore mostly experimental, but

The dynamic of fluid drops is of considerable sometimes supplemented by greatly simplifiedimportance in a number of engineering applications theoretical argument.and natural processes. The combustion of fuelsprays, spray painting, various coating processes, Two recent experimental investigations of dropas well as rain, are only a few of the more common collisions can be found in Azhgriz and Poo2, andexamples. While it is usually the collective Jiang, Umemura and Law3 who show severalbehavior of many drops that is of interest, often it photographs of the various collision modes foris the motion of individual drops that determines both waterand hydrocarbon drops. These, and otherthe large scale properties of the system. Thus, for experimental investigations have providedexample, the total surface area of sprays depends on considerable information and, in particular, it isthe size of the individual drops as well as their now understoodthat the outcome of a collision cannumber density. Computational models for be classified into about five main categories. Forengineering predictions of spray combustion head-on collisions we have four main categories:generally do not resolve the motion of individual bouncing collision, where the drops collide anddrops and must rely on "subgfid" models where the separate, retaining their identity; coalescenceaverage effects of the unresolved scales are "-collision, where two drops become one; separation

collision, where the drops temporarily become one

*Graduate Student,Department of Mechanical but then break up again; and shattering collision,Engineering where the impact is so strong that the drops break

up into several smaller drops. These categoriestAssociate Professor, Department of Mechanical survive for off-axis collisions, but a fifth one,Engineering

grazing or stretching collision, appears. Here, the rectangular box and the drops are initially placeddrops coalesce upon contact, but are sufficiently far near each end of the domain. A force that is turnedapart so that they continue along the original path off before the drops collide, is applied to drive themand separate again. The form of the collision together initially. Generally, the density anddepends on the size of the drops, their relative viscosity of the ambient fluid are much smallervelocities, their off-axis position and the physical than of the drop fluid and thus have only a smallproperties of the fluids involved. For a given fluid, effect on the results. While it is therefore oftensome of these collision regimes are not observer, sufficient to solve only for the fluid motion insideWater drops, for example, do not show bouncing, the drop, here we solve for the motion everywhere,(Jiang et al. 3, state that they also did not find both inside and outside the drops. The Navier-reflective collision for water drops. This is Stokes equations are valid for both fluids, and aapparently due to a limited parameter range studied single set of equations can be written for the wholeby them as the experiments by Azhgriz and Poo2, domain as long as the jump in viscosity andshow.) Other investigations of drop collisions may density is correctly accounted for and surfacebe found in Bradley and Stow4, and Podvysotsky tensionis included:

and Shraiber5, for example. The major goal of these apt + V. p_ = -Vp + ffxinvestigations has been to clarify the boundaries atbetween the major collision categories and explain

how they depend on the parameters of the problem. + V.I_(V_ + v_T)+ "ffa&(2- _f).Simple models used to rationalize experimentalfindings have been presented by Park and Blair6, Here, _ is the velocity, p is the pressure, and /9Ryley and Bennett-Cowell 7, Brazier-Smith et al.8,_ and/.t are the discontinuous density and viscosityAzhgriz and Poo2,and Jiang, Umemura and Law3. fields, respectively. F"-a is the surface tension force

and J_xis a body force used to give the drops theirIn principle, numerical solutions of the Navier-Strokes equations, where all scales of motion are initial velocity. Notice that the surface tensionfully resolved, can provide the missing force has been added as a delta function, onlyinformation, but various numerical difficulties affecting the equations_where the interface is. Theassociated with moving boundaries between two detailed form of Fo will be discussed below. Thefluids have made detailed simulations difficult in above equations are supplemented by thethe past. Nevertheless, several authors have incompressibility conditionscomputed the axisymmetric head-on collision of V. _"= 0drops with a wall. The earliest work is Foote9who which, when combined with the momentumfollowed the evolution of a rebounding equations leads to a non-separable elliptic equationaxisymmetricdrops at low Weber number using the for the pressure. We also have equationsof state forMAC method. More recent computations work can the density and viscosity:

be found in Fukai et al1° who use a moving finite _pelement method. We have recently conducted a + _'"Vp = 0dtnumerical study of the head-on collision of twoaxisymmetric drops, see Nobari, Jan and 9/'t+_.V/t=0.Tryggvason II, where we examined the boundary _gtbetween coalescing and reflecting collision for equal These last two equations simply state that densitysize drops. Here, we present numerical simulations and viscosity within each fluid remains constant.of three-dimensional, off-axis collisions, where thefull Navier Stokes equations are solved to give a Nondimensionalization gives a Weber and adetailed picture of the flow during collision. Reynoldsnumberdefinedby:

We = PdDU2 Re = PdUD

Formulation and Numerical Method o" 12dIn addition, the density ratio r = Pd I Po and the

The numerical technique used for the simulations viscosity ratio 2, =/.td //.to must be specified.presented in this paper is the Front Tracking/Finite Here, the subscript d denotes the drop fluid and oDifference method of Unverdi and Tryggvason12'13. the ambient fluid. In off-axis collisions, the dropsSince the procedure has bee described in detail approach each other along parallel lines that arebefore, we only outline it brieflyhere. some distance apart. If this distance is greater than '

the drop diameter, D, the drops never touch and noThe physical problem and the computational collision takes place. If this distance is zero, wedomain is sketched in Figure 1. The domain is a have a head-on collision. To describe off-centered

• X

U/2X

Figure 1.The computational domain and the initial conditions. The dropsare initiallytwo and a half diameter apart.

collision a new nondimensional parameter, usually similar to the one discussed by Unverdi andcalled the impact parameter, is required in addition Tryggvason12,that spreads the density jump to theto the Weber and the Reynolds number defined grid points next to the front and generates a smoothearlier. This parameter is usually defined as density field that changes from one density to the

I- Z other over two to three grid spaces. While this- _ replaces the sharp interface by a slightly smoother

where X is the perpendicular distance between the grid interface, all numerical diffusion is eliminatedsince the grid-field is reconstructedat each step. The

lines that the drops move along before collision, surface tension forces are computed from the

The force used to drive the drops together initially geometryof the interface and distributed to the gridin the same manner as the density jump. Generally,is taken as curvature is very sensitive to minor irregularity in

fx =C(p-Po)(X-Xc) the interface shape and it is difficult toachieveso the force acts only on the drops. Here C is an accuracy and robustness at the same time. However,adjustable constant and Xc is midway between the by computing the surface tensionforces directly by

drops. This force is turned off before the actual ff_ = a_ ? x _dscollision takes place. Initially, the drops are placewith their centers two and a half diameter between we ensure that the net surface tension force is zero,them, and C is varied to give different collision or:

velocities. __ax_ds = 0

To solve the Navier Stokes equations we use a Here, _is the outward normal, ? a tangent vectorfixed, regular, staggered grid and discretize the to the boundary curve for each element and rc ismomentum equations using a conservative, second twice the mean curvature. This is important fororder centered difference scheme for the spatial long time simulations since even small errors canvariables and an explicit second order time lead to a net force that moves the drop in anintegration method. The pressure equation, which is unphysical way.non-separable due to the difference in densitybetween the drops and the ambient fluid, is solved As the drops move and deform, it is necessary toby a Black and Red SOR scheme. Other versions of add and delete points at the front and to modify theour code use a multigrid iteration. The novelty of connectivity of the points, to keep the frontthe scheme is the way the boundary, or the front, elements of approximately equal size and as "wellbetween the drops and the ambient fluid is tracked, shaped" as possible. This is described in UnverdiThe front is represented by separate computational and Tryggvason. 12When the drops are close, wepoints that are moved by interpolating their rupture the interface, in several of ourvelocity from the grid. These points are connected computations, by removing surface elements thatby triangular elements to form a front that is used are nearly parallel and reconnecting the remainingto keep the density and viscosity stratification sharp ones to form a single surface. Here, thisand to calculate surface tension forces. At each time restructuring of the interface is done at prescribedstep information must be passed between the front time if the interfaces are close enough. While thisand the stationary grid. This is done by a method rather arbitrary (and we have simply selected the

Figure 2. Comparisonbetweena fully three-dimensionalsimulation(righ0and resultsobtained by an axisymmetric code (left).Theinitial conditions areshown at the top of

each column and the solutionis then shownat threeequispacedtimes for each run.

restructuring of the interface is done at prescribed Tryggvason 18 presented simulation of thermaltime if the interfaces are close enough. While this migrationof many two dimensional bubbles.rather arbitrary (and we have simply selected thetime when the drops look close enough)this allows Results and Discussionssome control over the dynamic of the rupture, ascompared with numerical methods where the front For the computations presented here, We=23,is not tracked and the film would always rupture Re=68, r=40,and _,=20, but the impact parameter,once it is thinner than a few grid spaces. For a I, is varied. The computational domain is resolvedmore detailed discussion of this point see Nobari, by a 32 by 32 by 64 cubic mesh and the dropJan, and Tryggvasona. diameteris 0.4 times the shorterdimension.

The method and the code has been tested in various While we have done extensive checks of theways, such as by extensive grid ref'mementstudies, accuracy of our axisymmetric code, the three-comparison with other published work and dimensionalcede has not been tested as thoroughly.analytical solutions. It has also been used to We have therefore conducted a few calculations ofinvestigate a number of other multifluid problems, head-oncollisions where the results from the three-In addition to the computations of head-on dimensional simulations can be compared with thecollisions of drops by Nobari, Jan and axisymmetric results. Figure 2 shows thisTryggvason4, Unverdi and Tryggvason13simulated comparison. The axisymmetric results are to thethe collision of fully three dimensional bubbles, left and the fully three dimensional results to theErvin 14investigated the lift of deformable bubbles right. The initial conditionsare shown at the top ofrising in a shear flow (see also Esmaeeli, Ervin, eachcolumn and the drops are then shown below atand Tryggvason 15,Jan and Tryggvason16examined equispaced times. The force that acts on the dropsthe effect of contaminants on the rise of buoyant initially is turned off before impact (just before thebubbles and Nobari and Tryggvason17followed the second frame). As the drops collide they becomecoalescence of drops of different sizes. Nas and

-.40 .oo t/(8}ou,) .8o 1.2o

Figure 3. The x-position of the center of mass ofone drop versus time, as computed by both thefully three-dimensional code and an axisymmetricone, for two different resolutions.

flatter, and the ambient fluid between them ispushed away, leaving a thin film of fluid betweenthe drops. Here, this film is not removed and thedrops therefore rebound, recovering their sphericalshape. Obviously, the results are in goodagreement. Figure 3 shows a more quantitativecomparison, where we plot the x-position of thecenter of mass for the drops in figure 2, as well asfor drops computed on a coarser grid (16 by 16 by32). The agreement is reasonably good, althoughthe coarse grid results are not in as good agreementwith each other as the finergrid results are.

In figure 4, the off-axis collision of two drops, forI=0.75, is shown. The pair is shown at severalequispaced times, beginning with the initialposition at the top of the figure. Once the dropshave the desired velocity, around the third framefrom the top, the force that is applied to drive thedrops together is turned off. The drops continue tomove together, and in the fourth frame they havecollided, deforming as they do so. Since thecollision parameter is relatively high, the dropsslide past each other and continue along theiroriginal path. The bottom four frames show themotion of the drops after the collision. During thecollision the drops become nearly flat where theyface each other, and as the drops slide past eachother the fluid layer between the drops becomesprogressively thinner. If it becomes thin enough itshould rupture, but here we have not allowed thatto happen. (As seen in figure 6, rupture of this filmwill change the resulting evolution considerably.)In figure 5, the velocity components of the center Figure 4. Bouncing collision. Here I=0.75 and theof mass of one of the drops, (a), and the kinetic and drops are not allowed to coalesce. The initialthe surface tension energy, (b), is plotted versus conditions are shown at the top and the drops aretime. The solid curve in (a) is the velocity in the then shown every 0.42 time unit.horizontal direction. It increases as the forceaccelerates the drops together, and then decreasesslightly due to the drag from the outer fluid after

1.25"

__ u coalesce.Allparametersare the sameas in figure2,Ca) ___v......w except that in the left column I=0.50, and in the

rightcolumnI=0.825.The film between the drops.,s is ruptured at time 0.46 for both runs. In these

computations we put t=0.0 when the distancebetweenthecenterof thedrops is onediameter.Forthe low impactparametercase, the drops deformconsiderablyduringthe initial impact, as observed

.2_ for head-oncollisions,but the impactparameteris ,sufficientlylarge so the drops still slidepast eachother.Asthe f'dmis rupturedand thedropscoalescethe momentumof eachdropis sufficientlylargeso

-.2__1.so -:50 ' .go z.;o 2.so the large combined drop continues to elongate.._o Eventually,however,surfacetensionovercomesthe

__ K.m. stretchingand the drop is pulled into a spherical---- S,E.

shape. Due to the velocity of the drops thatcoalesced,thecombineddroprotates.

" Whilethe low impactparameterdrops are in manyr.,_ way similar to drops undergoing a head-on

/ collision,thehighimpactparameterdrops in figure/ 6b deformonly slightlyas they collide.When the

.2o / interfacebetweenthemis ruptured,theyhaveheady/ passedeachotherandafterrupturetheirmomentum

is sufficientlylarge so they continue along theiroriginal path and stretch the fluid column.® ....... "'-- ......... =----;--' connectingthemuntilit is near breaking.We have-,._o -:5o ,/(_/o ,._ 2.50 not writtenthe softwarenecessaryfor rupturingthe

Figure 5. (a) Centerof mass velocityof onedrop filamentconnectingthe drops and therefor mustfrom the computationin figure4. (b) Kineticand stop the computationsat this point. Notice, thatsurfacetensionenergyof onedrop. the coalesced drop rotates, as the low impact

parameteronedid,althoughmuchless.the force is turned off. When the drops actuallycollide, it is reducedmore rapidly,but eventually In figure7, the surfacetensionenergy, the kineticresumes a nearly constant decay rate after the energyandthe totalenergyof the dropsfrom figurecollision is over. The velocitycomponentin the 6 are plotted versus time. Initially, the kineticvertical direction (shortdashes)is non zero only energyis increasedby the force thatacceleratestheduring the actual collision.The kineticenergyin dropstogether.Since this force is not constant (it(b) shows similar behavior as the velocity: it increaseslinearlywith distancefrom the centerofdecreasesslowlyafter the forceis turnedoff,more thecomputationalbox)the increaseis not quadraticrapidly during collision and then resumes slow asfor thecomputationsreportedin Nobari,Jan, anddecay.The surface tensionenergyrisesduringthe TryggvasonIx.Afterthe force has beenturnedoff,collisionas the drop deforms,thus contributingto the dropsmove a short distancebefore colliding.the reduction in the kineticenergy.Noticethatthe Since the ambient fluid has a finite viscosity,droposcillatesslightlyafterthe collisionas seenin kineticenergyis dissipateddue to frictionand thethe surfacetensionenergyplot. dropsslowdown.Asthe dropscomein contact,the

kinetic energy of the low impact number dropsAlthoughbouncingis observedforreal drops,it is decreasesrapidly,butthehighimpactnumberdropsactually a relatively rare outcomeof a collision, arenotaffectedtoanysignificantdegree.Similarly,only seen when the drop deform and trap fluid the surface tension energy of the low impactbetweenthem and the velocityis sufficientlylarge numberdrops increasesand the dropsdeform, butso the film does not have time to drain beforethe the surfacetensionenergyof theother dropshardlydropsrebound.To investigatethebehaviorofdrops increases at all since the drops remain almostthat coalesce, we have written software to spherical. When the film between the drops isautomaticallyremovethe front boundingthe thin ruptured,part of the drops surfaceis removedandfilm between the drops at a prescribedtime and the surfaceenergyreduced.This reductionis larger 'allow the drops to coalesce.Figure6 shows the for the low impactnumber drops since the arearesults of two computations where the drops removedis larger. Initially, the kinetic energy of

Figure 6. Coalescing collisions. The initial conditions are shown at the top of the figure and the pair is thenshown every 0.42 time units. The film is ruptured at t=0.46 in both cases, but the impact parameter is differentfor the two runs. In the left columnI=0.5, and I---0.825in the right one. For the low impact parameter, the dropscoalesce permanently, but for the higher impact parameter they separateagain.

,.8o. drop or not. Figure8 shows our computationsin(a) the I-Re plane. In addition to the computations

shownin figure 5, we have conductedtwo othercalculations.at different impact parameters. The

,.2_ runs that lead to a coalesceddrops are shownby,_,,v blacksquaresandthoseleadingto grazingcollision "

as open squares. We have also plotted theexperimental results of Jiang et al3, for the

.,_ boundarybetween these two collision modes for ,We=23.Their results do not extend down to theReynoldsnumber simulated here, but since the

ko. boundaryisonlyweaklydependenton theReynoldsnumberit seemssafe to extrapolatetheirresults to

,. ,.oo our Reynoldsnumber.The dashed line showsthis-_.oo -zo . t/(a_,) 2.5o

extrapolation,showingthat the numericalresults...... are consistentwith theexperiments.

(b) !t_../ _-----_ Conclusion

// s.e,

_.2o____ s_ ......... The purposeof this paper is to demonstrate thet'rutrglt

feasibilityof accuratenumericalpredictionsof fullythree-dimensionaloff-axiscollisionsof two drops.

._ To doso,wehavesimulateda fewcases,both with

andwithoutrupturingofthe interfaceseparatingthe

drops.Althoughtheruptureof thefilmbetweenthedropsis donein an ad hoc, way, the resultsare in

.0, ...... reasonably good agreement with experimental-5.® -_o t/?_,) 2.50 ,.oo observations_Forexactpredictionsof the boundary,

Figure7. The energiesfor the dropsin figure6. (a) amoreaccuratecriteriafor the rupturetime 19wouldI=0.5, (b) I=0.825. The total energy, the surface havetobe used.Thesecomputations,whichrequireenergy and the kinetic energy of the drops are aboutten hours on a CRAY-XMP,are done on aplottedversustime. relativelycoarsemeshand are thereforelimited to

relatively small Reynolds and Weber numbers.the high impactnumber drops is nearly unaffected Nevertheless, they do demonstrate well the(and continuesto be dissipatedat the samerate as capabilityofthe method.before the dropscollide),but as the coalesceddropstarts to stretch and the surface tensionenergytoincrease, the kinetic energydrops sharply.As the Acknowledgmentfilamentbetweenthe drops startsto neckdown,theincreasein surfacearea stops andthekineticenergy Wewouldliketo acknowledgediscussionswithDr.levels off. For the low impact numberdrops, the D. Jacqminat the NASALewisResearch Center.rupture takes place near the point of maximum Partof thisworkwasdonewhileoneof the authorsdeformation and surface energy is initially converted (GT) was visiting the Institute for Computationalintokinetic energyas the drop adjuststo the new Mechanicsin Propulsionat NASA Lewis. Thisshape.The momentumof the drop beforeimpact work is supportedin part by NASA grant NAG3-is, however, sufficiently large so the drop is 1317,and NSF grant CTS-913214. Some of thestretchedas the fluidof the originaldropscontinue computations were done at the San Diegoalong the paths they were following before SupercomputingCenter which is funded by thecollision. This leads to an increase in surface NationalScienceFoundation.tensionenergyanddecreasein kineticenergy.Whenthe surface tension energy reaches maximumthekinetic energy is not zero due to the finite References ._rotationalmotionof the coalesceddrop.Eventually,thecoalesceddroposcillates. 1.J. B. Heywood.InternalCombustionengine .

Fundamentals.McG-raw-Hiil,(1988).930 pages.For modeling of droplet collisions, the majorquestionis whetherthe collisionresultsin a single

1.00 Computational Results,Re---68,We=23

[]Grazing Collision

,_ 0.75 - []Q

E,row

_. 0.50 • _1,

PermanentCoalescence • ExperimentsD.I= ' We=23

0.25 •

0.00 i i 0 i0 100 200 300 400 500

Reynolds Number

Figure 8. The boundaries between coalescing and grazing collisions in the Re-I plane for W_23. The solidcircles, connected by a line, are experimentaldata fromJiang et a!3"The squaresare computed results.

2. N. Ashgriz and J.Y. Poo. Coalescence and 11. M.1LH.Nobari, Y.-J. Jan, and G.separation in binary collisions of liquid drops. J. ° Tryggvason.Head-on collision of drops---AFluid Mech. 221 (1990), 183-204. numerical investigation. Submitted for publication3. Y.J. Jiang, A. Umemura, and C.K. Law. An (1993).experimental investigation on the collision 12. S.O. Unverdi and G. Tryggvason. A Frontbehavior of hydrocarbon droplets. J. Fluid Mech. Tracking Methodfor Viscous Incompressible234 (1992), 171-190. Flows. J. Comput. Phys., 100 (1992) 25-37.4. S.G. Bradley and C.D. Stow. Collision between 13. S.O. Unverdi and G. Tryggvason. Multifluidliquid drops. Phil. Trans. R.Soc. Lond. A 287 flows. Physica D 60 (1992) 70-83:(1978), 635-678. 14. E.A. Ervin: Full Numerical Simulations of5. A.M. Podvysotsky and A.A. Shraiber. Bubbles and Drops in Shear Flow, Ph.D. Thesis,Coalescence and breakup of drops in two phase The University of Michigan, 1993.flows. Intl. J. Multiphase Flow 10 (1984), 195- 15. A. Esmaeeli, E.A. Ervin, and G. Tryggvason:209. Numerical Simulationsof Rising Bubbles. To6. J.Y. Park and L_M. Blair. The effect of appear in Proceedingsof the IUTAM Conference oncoalescence on drop size distribution in an agitated BubbleDynamicsand lnterfacial Phenomena (Ed.:liquid-liquid dispersion. Chem. Engng. Sci. 30, J.R. Blake)1057-1064. 16. Y.-J. Jan, and G. Tryggvason: Computational7. DJ. Ryley and B.N. Bennett-Cowell. The Studies of ContaminatedBubbles. Submitted forcollision behavior of steam-borne water drops. Int. publication (1993).J. Mech. Sci. 9 (1967), 817-833. 17. M.R.H. Nobari and G. Tryggvason:8. P.R. Brazier-Smith, S.G. Jennings and J. Coalescenceof Initially Stationary Drops.Latham. The interaction of falling water drops: Submittedfor publication (1993)coalescence Proc. R. Soc. Lond. A 326 (1972), 18. S. Nas and G. Tryggvason: Computational393-408. Investigationof the Thermal Migration of Bubbles

• 9. G.B. Foote. The Water Drop Rebound Problem: and Drops. 1993ASME Winter Annual Meeting.Dynamics of Collision. J. Amaos.Sci. 32 (1975), 19. D. Jacqmin and M.R. Foster. The evolution of390-402. thin fdms generated by the collision of highly

• 10. J. Fukai, Z. Zhao, D. Poulikakos, C.M. deforming droplets. Submitted for publicationMegaridis, and O. Miyatake. Modelingof the (1993).deformation of a liquid droplet impinging uponafiat surface. Phys. Fluids A, 5 (1993), 2589-2599.

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October 1994 Technical Memorandum4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

NumericalSimulations of Drop Collisions

6. AUTHOR(S) WU-505-90-5K

M.R.H. Nobari and G. Tryggvason

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATIONREPORT NUMBER

National Aeronautics and SpaceAdministrationLewis Research Center E-9167Cleveland, Ohio 44135-3191

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National Aeronautics and SpaceAdministration NASA TM- 106751Washington,D.C. 20546-0001 ICOMP-94-23

AIAA-94-0835

11. SUPPLEMENTARY NOTES

Prepared for the 32ndAerospace Sciences Meeting and Exhibitsponsoredby the American Institute forAeronautics andAstronautics, Reno, Nevada,January 10-13, 1994. M.R.H. Nobari, University of Michigan,Ann Arbor, Michigan 48823; and G. Tryggvason,Institute for Computational Mechanicsin Propulsion (work fundedunder NASACooperativeAgreement NCC3-233), and University of Michigan, Ann Arbor, Michigan 48823. ICOMPProgram Director, Louis A. Povinelli. o_anization code 2600, (216) 433-5818.

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13. ABSTRACT (Maximum 200 words)

Three-dimensional simulations of the off-axiscollisions of two drops are presented. The full Navier-Stokes equations aresolved by a Front-Tracking/Finite-Differencemethodthat allows a fully deformable fluid interface and the inclusion ofsurface tension. The drops are acceleratedtowards each other by a body force that is turned off before the drops collide.Depending on whether the interfacebetween the drops is ruptured or not, the drops either bounce or coalesce. For dropsthat coalesce, the impact parameter, which measures how far the drops are off the symmetry line, determines the eventualoutcome of the collision. For low impact parameters, the drops coalescepermanently, but for higher impact parameters, agrazing collision, where the drops coalesce and then stretch apart again is observed. The results are in agreement withexperimental observations.

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Drop collisions; Front tracking 16.PRICECODEA03

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