Department of Mechanical Engineering, University of Delaware, Newark, Delaware
WOJCIECH W. GRABOWSKI
Mesoscale and Microscale Meteorology Division, National Center for Atmospheric Research,* Boulder, Colorado
(Manuscript received 20 January 2004, in final form 15 July 2004)
ABSTRACT
Two formulations of an improved superposition method are proposed for studying droplet–droplethydrodynamic interactions. The formulations make explicit use of the boundary conditions on the surfaceof the two interacting droplets. The improved formulations are described through a consistent and rigorousconsideration of the relationship between the drag force and representation of disturbance flows. It isdemonstrated that the improved formulations are much more accurate than the original implementation ofthe superposition method. Specifically, for the case of Stokes disturbance flows, the relative errors on thedrag force can be reduced by one order of magnitude using the improved formulations, when compared withthe original formulation, for situations when the lubrication effect is not dominant. Using the improvedsuperposition method, collision efficiencies of small cloud droplets falling in calm air are also computed andcompared with previously published results.
1. Introduction
The topic of collision coalescence of cloud droplets isof great importance to the understanding and quanti-tative prediction of warm rain formation (Pruppacherand Klett 1997). For droplets of radii less than 60 �m,hydrodynamic interactions between two colliding drop-lets can significantly affect the trajectories of the drop-lets and therefore modify the overall collision efficiency(e.g., Klett and Davis 1973). This may be explainedqualitatively as follows. A small droplet of finite sizemoving in a fluid medium introduces a disturbance flowfield in its neighborhood and as such modifies the flowfield locally. For droplets of radii less than 60 �m, thehydrodynamic interaction time (say, the ratio of colli-sion radius over differential settling velocity) may belarger than the inertial response times of the droplets,so at least one of the droplets has adequate time torespond to the disturbance flow induced by the otherdroplet. The collision radius here is the geometric col-
lision radius defined as the sum of the radii of the twocolliding droplets.
A representation of these disturbance flows is thenneeded for quantitative prediction of collision effi-ciency, which is a required input for the modeling ofsize distribution of cloud droplets through collision co-alescence. One well-known approximate approach isthe superposition method (see, e.g., Pruppacher andKlett 1997). The method was initially designed to studythe hydrodynamic interaction of two spherical dropletsusing the solution of disturbance flow induced by asingle sphere. Basically, it is assumed that each dropletmoves in a flow field generated by the other dropletfalling in isolation. The method, because of its simplic-ity, has been widely used for treating hydrodynamicinteractions of droplets under both the Stokes distur-bance flows (Langmuir 1948; Pinsky et al. 1999) anddisturbance flows at finite Reynolds numbers (Beardand Grover 1974; Neiburger 1967; Shafrir andNeiburger 1963; Lin and Lee 1975).
It is known that the superposition method becomesinaccurate when the separation distance between twodroplets is comparable to the collision radius (Klett1976; Pruppacher and Klett 1997). As we point out inthis paper, this inaccuracy is not a necessary conse-quence of the superposition method. We shall demon-strate that most of this inaccuracy can be removed bysimply revising the way that the perturbation velocitiesat the locations of droplets are computed. Our proposal
* The National Center for Atmospheric Research is sponsoredby the National Science Foundation.
Corresponding author address: Lian-Ping Wang, Department ofMechanical Engineering, 126 Spencer Laboratory, University ofDelaware, Newark, DE 19716-3140.E-mail: [email protected]
is to impose proper boundary conditions at the surfaceof the droplets, which provides a natural optimizationprocedure for the superposition method. The idea issimilar to that of Klett and Davis (1973) who superim-posed two Oseen flows while requiring the compositeflow to satisfy the necessary boundary conditions. Thetheoretical framework discussed in this paper, however,is more consistent as far as Stokes disturbance flows areconsidered and is described more thoroughly than thatof Klett and Davis (1973).
For simplicity, we treat the droplets as solid, nonro-tating spherical particles with nonslip surface boundaryconditions. Only Stokes disturbance flows are consid-ered. It should be pointed out that the no-slip boundarycondition is no longer valid when the minimum sepa-ration between two droplets approaches the mean freepath of air molecules, as the physical assumption thatthe fluid acts as a continuum breaks down (Hocking1973; Jonas 1972). Or alternatively, the lubricationforce between droplets at close separations are typicallyoverestimated by the no-slip boundary condition, lead-ing to underestimation of collision efficiencies andstrong dependence of collision efficiencies on the mini-mum separation gap used to define collisions (Hockingand Jonas 1970; Jonas 1972; Hocking 1973). As will beseen later in this paper, all formulations of the super-position method are unable to handle the lubricationeffect. This, however, may be viewed to our advantagein the sense that our results of collision efficiencies arenot sensitive to the minimum separation gap used forcollision detection.
In section 2, we present a formulation of the im-proved superposition method. In section 3, the resultsof the improved superposition method are then com-pared with those based on the original implementationof the superposition method as well as known analyticalresults of Stimson and Jeffrey (1926) and Davis (1969).These include drag forces on the droplets, flow visual-ization, and collision efficiencies of small cloud dropletsfalling in calm air. Finally, conclusions are drawn insection 4.
2. Formulation of an improved superpositionmethod
We first reexamine the original formulation of thesuperposition method by questioning what is exactlythe representation of the disturbance flows that led tothe modified drag force representation used in themethod. For this purpose, we need to review some basicproperties of a Stokes flow.
a. Basic properties of Stokes disturbance flowinduced by a moving sphere
We first consider the problem of a single dropletmoving at a constant velocity V in still air. The directionof V may not be the same as gravity (Fig. 1). The drop-
let will introduce a disturbance airflow field. Assumingthat the Reynolds number based on |V | and dropletdiameter is very small, the unsteady Stokes disturbanceflow induced by the droplet may be written in a vectorform (Kim and Karrila 1991):
u�x, t� � �34
a
r�
34 �a
r�3� r
r2 �V · r�
� �34
a
r�
14 �a
r�3�V � us�r; a, V�, �1�
where a is the droplet radius, and r is the position vectorrelative to the center of the droplet Y(t) � Y(0) � Vt,
r�x, t� � x � Y�t� � x � Y�0� � Vt. �2�
The above vector form will facilitate the description ofthe two-droplet problem to be discussed later. Equa-tions (1) and (2) imply that
��u��t � V · �u � 0; �3�
namely, the Stokes flow pattern is advected with thedroplet velocity V.
The induced pressure field can be expressed as (Kimand Karrila 1991)
p�x, t� �3a�
2r3 V · r � ps�r; a, V�, �4�
where � is the fluid dynamic viscosity.It is straightforward to show that the above distur-
bance flow field and pressure field satisfy the followingunsteady Stokes equation with a linearized advectionterm:
���u�t
� V · �u�� ���p � p0� � ��2u � �ge3, �5�
where p0(x, t) � �gx3 is the hydrostatic pressure field, is the air density, x3 is the vertical spatial coordinatepointing upward, g is the gravitational acceleration, ande3 is the unit vector in the x3 direction. The linearizedadvection term is a good approximation to the trueadvection term very close to the surface of the droplet.
In fact, the two terms on the left-hand side of the
FIG. 1. Notation for a single droplet.
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above equation sum up exactly to zero due to (3), witheach of them individually scaled as V2/a close to themoving droplet. The terms �p and ��2u are each scaledas �V/a2, and, therefore, are much larger than the termson the left-hand side if the droplet Reynolds numberRe � V(2a)/� is much less than one. Therefore, to theleading order and when (Re r/a) � 1 the followingquasi-steady Stokes equation can replace (5):
���p � p0� � ��2u � �ge3 � 0. �6�
The above quasi-steady Stokes flow representation mayalso be viewed as a leading-order solution to the prob-lem of a droplet moving at time-dependent velocityV(t), provided that, in addition to the condition of avery small droplet Reynolds number, the viscous diffu-sion time scale, a2/�, is much less than the character-istic time scale associated with V(t), namely, the dropletresponse time p � 2pa2/(9�), where p is the dropletdensity. This additional condition amounts to p/ �4.5, which is easily satisfied in the context of cloud mi-crophysics. Therefore, we could extend the use ofquasi-steady Stokes disturbance flow for any time-dependent motion of small cloud droplets. A very simi-lar argument was made in previous publications to ne-glect the fluid acceleration (e.g., Davis 1969).
Of importance is the fact that the disturbance flowgiven by Eq. (1) satisfies the boundary condition on thesurface of the droplet:
u�x, t� | | r|�a � V. �7�
The total force acting on the surface of the droplet canbe calculated as
Fi � �| r |�a
���p � p0��ij � ���ui
�xj�
�uj
�xi��njdS,
�8�
leading to the well-known result,
F � �g4�a3
3e3 � 6��aV, �9�
where the first term is the buoyancy force (typicallyneglected in cloud physics because of the density ratiobetween water and air), and the second term is theStokes drag.
If an imaginary spherical surface SI of center xI andradius b is introduced outside the droplet, then the totalforce acting on this imaginary surface due to the sameStokes disturbance flow can be shown to be
FI � �| x�xI|�b
���p � p0��ij � ���ui
�xj�
�uj
�xi��
� njdS � �g4�b3
3e3. �10�
Namely, the disturbance flow does not result in anyviscous force, since the imaginary surface does not con-
tain any singularity of the disturbance flow field (Kimand Karrila 1991).
b. Superposition method for the two-dropletproblem
Now consider two droplets moving in still air. Thedroplet radii are a1 and a2, center positions are Y1 andY2, and velocities are V1 and V2 (Fig. 2). The inducedflow due to the second droplet moving in isolation isus(r2; a2, V2). The original implementation of the su-perposition method assumes that the net drag force act-ing on the surface of the first droplet is
D1 � �6��a1�V1 � us�Y1 � Y2; a2, V2�. �11�
Similarly, the net drag force on the second droplet is
D2 � �6��a2�V2 � us�Y2 � Y1; a1, V1�. �12�
In other words, the velocity of each droplet relative toair is reduced by the Stokes flow solution due to thepresence of the other droplet.
Based on the properties of the Stokes flow discussedabove, the above drag force representations imply acomposite disturbance flow field as follows:
u�x, t� � us�r1; a1, V1 � us�Y1 � Y2; a2, V2�
� us�r2; a2, V2 � us�Y2 � Y1; a1, V1�,
�13�
where r1 � x � Y1(t) and r2 � x � Y2(t). Although thiscomposite disturbance flow satisfies the Stokes Eq. (6),it does not satisfy the two boundary conditions (BCs)on the surface of the droplets: u(x, t) || r1|�a1
� V1 andu(x, t) || r2|�a2
� V2. (An example of the composite flowis given later in Fig. 4a to show that the airflow cutsthrough the boundaries of the droplets with the above
FIG. 2. Notation for the two-droplet problem.
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original formulation of the superposition method, andfurther that the average air velocity on the surface ofeach droplet is significantly different from the velocityof the droplet.)
c. The improved superposition method
The boundary conditions on the surface of dropletscan be better satisfied by assuming a composite distur-bance flow field:
Here u1 and u2 are the levels of disturbance flow due tothe presence of the other droplets that need to be de-termined. The alignment of streamlines of each distur-bance flow is illustrated in Fig. 2. We will impose thecondition that the average fluid velocity on the surfaceof each droplet is equal to the velocity of that droplet(no slip), namely,
V1 � V1 � u1 �1
4�a12 �
| r1|�a1
us�r2; a2, V2 � u2�ds,
�15�
V2 � V2 � u2 �1
4�a22 �
| r2 |�a2
us�r1; a1, V1 � u1�ds.
�16�
Or equivalently, we have the following integral equa-tions to solve u1 and u2:
u1 �1
4�a12 �
| r1 |�a1
us�r2; a2, V2 � u2�ds, �17�
u2 �1
4�a22 �
| r2 |�a2
us�r1; a1, V1 � u1�ds. �18�
The above integral equation may be solved exactly ifthe integrations can be performed. Alternatively, if weevaluate the integrands using the midpoint values (i.e.,at the center of droplets), we have the following ap-proximations:
u1 � uS�Y1 � Y2; a2, V2 � u2�, �19�
u2 � uS�Y2 � Y1; a1, V1 � u1�. �20�
For the two-droplet problem, u1 and u2 can be solveddirectly since the Stokes flow is linear. The physicalinterpretation of u1 in the integral formulation, (17), isthe average perturbation velocity on the surface of thefirst droplet, induced by the second droplet. In the sec-ond, center-point formulation, (19), u1 is the perturba-tion velocity at the center of droplet 1, induced by drop-let 2.
The composite pressure field is, using the notationgiven by (4),
Equations (14) and (21) together completely specify the
perturbation flow field, and they satisfy the same quasi-steady Stokes Eq. (6).
Now the total surface force on each droplet can becalculated using (8), while observing the results givenby (9) and (10), yielding
F1 � �g4�a1
3
3e3 � 6��a1�V1 � u1, �22�
F2 � �g4�a2
3
3e3 � 6��a2�V2 � u2. �23�
Therefore the drag forces retain the same form exceptthat the perturbation velocities need to be subtracted.
3. Results and discussions
We shall now compare different formulations of thesuperposition method. First, consider two cases withdroplets of equal size touching each other as shown inFig. 3. We denote the vertical touch as case 1 and hori-zontal touch as case 2. At steady state, the symmetry ofthe Stokes flow implies that the two droplets will settleat the same velocity V1 � V2 � V for both cases. Forcase 1, it can be shown that the magnitudes of predicteddrag force based on three different formulations are
D�Original formulation� � 6��a�V �1116
V�,
�24�
D�Integral formulation� � 6��a�V �5
13V�,
�25�
D�Center-point formulation� � 6��a�V �1127
V�.
�26�
For case 2, the results are
D�Original formulation� � 6��a�V �1332
V�,
�27�
FIG. 3. Two cases with droplets touching each other: (a) case1—vertical alignment and (b) case 2—horizontal alignment.
1258 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62
D�Integral formulation� � 6��a�V �7
23V�,
�28�
D�Center-point formulation� � 6��a�V �1345
V�.
�29�
These are compared, in Tables 1 and 2, with the exactStokes flow solutions given by Stimson and Jeffery(1926) and Davis (1969), as well the results of Klett andDavis (1973) and Klett (1976). Several observations canbe made. First the relative error in drag prediction isgreatly reduced with the improved formulations, whencompared to the original formulation. The very largerelative errors in the original formulation of the super-position method were recognized before, and were of-ten criticized in the literature (e.g., Klett 1976; Prup-pacher and Klett 1978). We demonstrate that theselarge errors can be largely removed by simply imposingthe boundary condition constraints in the method. Sec-ond, the integral formulation yields the exact same re-sults as those of Klett and Davis (1973), indicating thelevel of approximations of our improved formulation isvery much similar to that of Klett and Davis (1973).Third, the integral formulation is not necessarily moreaccurate than the center-point formulation; therefore,since the center-point formulation is much simpler toimplement, we recommend the center-point formula-tion as a preferred approach.
Obviously, when the center-point formulation isused, the average fluid velocity on the surface of thedroplet may not satisfy the no-slip boundary condition.For case 1, the center-point formulation yields the fol-lowing average fluid velocity on the surface of the drop-let:
u�x, t� || r1 |�a �2627
V � 0.9630V; �30�
namely, a small positive slip exists between the dropletsurface and the fluid. Since the integral formulation forthis case underpredicts the drag or the relative motion
used to compute the drag, this positive slip worsens thedrag force representation in the center-point formula-tion. On the other hand, for case 2, the implied averagefluid velocity on the surface of the droplet for the cen-ter-point formulation is
u�x, t� || r1 |�a �4645
V � 1.0222V, �31�
giving a 2% negative slip. As in case 1, the integralformulation underpredicts the drag; this negative slipnow improves the drag force representation in the cen-ter-point formulation.
To illustrate the advantages of our improved formu-lations, we compare the vector velocity fields observedwhen moving with the droplets in Fig. 4 for case 1. Inthe original formulation, the disturbance flow is tooweak and the relative velocity field is crossing throughthe droplet surfaces, leading to the severe underpredic-tion of the drag force. On the other hand, in the newformulations, the surface boundary conditions are ap-proximately satisfied (in an average sense), leading tomuch more realistic flow fields. Since the drag force isalso an average flow property, we obtain a rather ac-curate drag even though the nonslip boundary condi-tions are not exactly satisfied at local points on thesurfaces. It should be remembered that the drag force iswhat matters as far as the motion of the droplets isconcerned, so we would expect that our improved su-perposition method will yield reasonably accuratedroplet trajectories for these cases discussed here. Theimproved formulations also show the presence of vor-tex flows in between the two droplets, a feature previ-ously experimentally observed (Kumagai and Muraoka1989).
Next we extend the above comparisons to untouch-ing droplets by varying the separation distance betweendroplets. First, the case of vertical alignment is consid-ered. In Fig. 5a, we compare the normalized drag force,plotted as a function of normalized separation distance,again assuming the droplets are of equal size and settleat the same velocity. The format of this figure is thesame as Fig. 14-2 of Pruppacher and Klett (1997). Thedrag forces predicted using various formulations aregiven here as
TABLE 1. Comparison of predicted drag force for two dropletstouching vertically.
where � � a/(2a � s), and s is the separation distance asshown in the figure. The exact solution of Stimson andJeffery (1926) is also shown:
D
DS�
43
sinh �n�1
� n�n � 1�
�2n � 1��2n � 3�
� �1 �
4 sinh2�n �12� � �2n � 1�2 sinh2
2 sinh�2n � 1� � �2n � 1� sinh2� ,
�35�
FIG. 4. Airflow realized by moving with the droplets: (a) origi-nal formulation; (b) integral formulation; and (c) center-point for-mulation.
FIG. 5. Comparison of various formulations for equal parallelmotion parallel to the line of centers: (a) predicted drag and (b)magnitude of relative error of drag prediction.
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where � and � are related by 2� cosh � � 1. Onceagain, our improved formulations compare well withthe exact result of Stimson and Jeffery (1926). For allformulations, the deviations from the exact solution in-crease with decreasing separation. All formulations un-derpredict the drag force. The integral formulationgives better prediction than the center-point formula-tion for this case. The relative errors are plotted in Fig.5b. The relative error drops very quickly with s/a if s/aexceeds one. All formulations give a relative error ofless than 1% if s/a � 20, indicating the effects of localdisturbance flows are contained within s � 20a. Ourimproved formulations have a relative error of aboutone order of magnitude smaller than the relative errorin the original formulation.
For the case of horizontal alignment and finite sepa-ration, we can obtain
D�Original formulation� �
DS�1 � 0.75 � 0.253, �36�
D�Integral formulation� �
DS� 4�1 � 2�
4 � 3 � 42 � 3 � 25�, �37�
D�Center-point formulation� � DS� 4
4 � 3 � 3�.
�38�
These results are compared in Fig. 6a. Also plotted bysymbols is the numerical result of Davis (1969) basedon exact Stokes solution. For the horizontal alignment,the center-point formulation is better than the integralformulation. The relative errors are displayed in Fig.6b. Again, our improved formulations yield a relativeerror that is one order of magnitude smaller than theoriginal formulation.
The two special cases discussed above have well-defined, finite drag forces when the separation distances approaches zero. It is well known that, for Stokes flowwith no-slip boundary condition, the component of theforce between the droplets acting along the line of cen-ters could increase indefinitely as s approaches zero forconfigurations when the gap is closing due to the rela-tive motion (Davis 1966; Hocking and Jonas 1970; Jo-nas 1972). Such divergent drag force is also known asthe lubrication effect. The case of the antiparallel mo-tion parallel to the line of centers shown in Fig. 7 is suchan example, with the exact force acting on the dropletgiven as (Maude 1961)
D
DS� �
43
sinh �n�1
� n�n � 1�
�2n � 1��2n � 3�
� �1 �
4 cosh2�n �12� � �2n � 1�2 sinh2
2 sinh�2n � 1� � �2n � 1� sinh2� .
�39�
When s → 0, the leading-order expansion of the aboveexpression yields
D
DS� 0.5
a
s, as
s
a→ 0, �40�
which diverges as the gap distance goes to zero becauseof the lubrication effect. In Fig. 7a we compare thenormalized drag force, plotted as a function of normal-ized separation distance, again assuming the dropletsare of equal size and move at the same speed. The solidline is the same as the Stimson–Jeffery–Maude curveshown in Fig. 2 of Davis (1966). The drag forces pre-dicted using various formulations are as follows:
D�Original formulation� �
DS�1 � 1.5 � 0.53, �41�
D�Integral formulation� �
DS� 2�1 � 2�
2 � 3 � 22 � 53 � 25�, �42�
FIG. 6. Comparison of various formulations for equal parallelmotion perpendicular to the line of centers: (a) predicted drag and(b) magnitude of relative error of drag prediction.
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D�Center-point formulation� � DS� 2
2 � 3 � 3�.
�43�
While our improved formulations still perform betterthan the original formulation, all the formulationsbased on the superposition method fail to predict thelubrication effect. The relative errors for all formula-tions are displayed in Fig. 7b, showing that even theimproved formulations incur a large error when s/a � 1.When two droplets move toward each other at smallseparation, the stress (local pressure and viscous shearstress) distributions on the surface of droplets may be-come very nonuniform, with possibly explosive growthof local stress within the region in between the twodroplets, leading to extremely large normal stress act-ing on the droplets. Our approximate formulations willnot handle such a situation accurately, as shown inFig. 7.
Finally, we compare the collision efficiencies of clouddroplets predicted by the original superposition method
and the improved method based on the center-pointformulation in Fig. 8. The gravitational hydrodynamicinteractions in calm air without slip-flow corrections areconsidered here. It was assumed that initially two drop-lets were separated by a center-to-center distance of50(a1 � a2) in the vertical direction. The collision effi-ciencies were computed by finding the far-field, off-center horizontal separation for the grazing trajectorieswith an efficient midpoint algorithm. When estimatesof upper and lower bounds, E1 and E2, were assumed,the algorithm approaches the correct off-center hori-zontal separation for the grazing trajectories by a factorof 1/2 in each iteration. Typically, the code was run forseven iterations, so the accuracy was (E2 � E1)/27 �0.0078(E2 � E1), leading to convergence of the first twodigits in collision efficiency. A second run was thendone with better estimates of the bounds to convergethe value of collision efficiency to four significant digits,provided that the roundoff errors are not important.Since only single precision was used and the roundofferror may become important for small a1 and a2/a1, thedata for a1 � 5 �m and small a2/a1 may not have thesame accuracy. The data based on the center-point for-mulation and the original formulation are also listed inTables 3 and 4, respectively.
Since the level and influence of disturbance flowswere underestimated in the original formulation of thesuperposition method, the collision efficiencies pre-dicted by the original formulation are larger than thosepredicted by the improved formulation. The differencesbetween the two formulations increase with decreasingdroplet size and increasing size ratio. These are ex-pected since the hydrodynamic interaction times aretoo small to influence the motion of droplets whendroplets are large enough. Further, if the size ratio issmall, the disturbance flow due to the larger dropletdominates over that due to the smaller droplet, and inthat case, only the motion of the smaller droplet is af-fected by the hydrodynamic interaction; as such theimproved formulation reduces essentially to the origi-nal formulation.
We also note that the collision efficiencies predictedby the improved formulation compare much better withthe results of Davis and Sartor (1967) based on theStokes flow solution of two-droplet hydrodynamics. InDavis and Sartor (1967), the grazing trajectories wereassumed to be at a finite minimum separation of 0.001times the radius of the larger droplet. We recalculatedthe collision efficiencies with such a small gap, and theresults are displayed in Table 5. We find that allowinga small gap increases the collision efficiencies by lessthan 5% for a2/a1 � 0.10. When a2/a1 � 0.10, the colli-sion efficiencies are small, and the increases could be asmuch as 10%. This dependence of collision efficiencieson the gap size is much weaker than what was reportedby Hocking and Jonas (1970). This weaker dependenceis explained by the lack of the lubrication effect in thesuperposition formulations as shown in Fig. 7.
FIG. 7. Comparison of various formulations for antiparallel mo-tion parallel to the line of centers: (a) predicted drag and (b)magnitude of relative error of drag prediction.
1262 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62
Our improved formulation still overpredicts the col-lision efficiency due to the approximations present inour simple approach, in particular, the lack of the lu-brication effect.
Finally, we note that the collision efficiencies pre-dicted by our improved superposition method comparewell with the results of Klett and Davis (1973) whentaking the zero-Reynolds-number limit in their formu-
FIG. 8. Collision efficiencies predicted by the original superposition method and the im-proved method using the center-point formulation, compared with the results of Davis andSartor (1967) based on Stokes flow solution of two-droplet hydrodynamics.
TABLE 3. Calculated collision efficiencies for hydrodynamic interactions under gravity with the center-point formulation. Zero gapwas used for collision detection.
lation, as shown in Fig. 2 of Klett and Davis (1973).Interestingly, our results are better than those of Klettand Davis (1973) for the a1 � 10 �m case, when com-pared to the prediction of Davis and Sartor (1967).Therefore, it is very important to note that much of thedifferences between the predictions of Klett and Davis(1973) based on an approximate Oseen flow and thoseof Davis and Sartor (1967) based on the Stokes flowhydrodynamics are due to the approximations in Klettand Davis (1973), rather than the true finite-Reynolds-number correction. Similar to our formulation, Klettand Davis (1973) superimposed two Oseen flow solu-tions and required the composite flow to approximatelysatisfy the boundary conditions on the surface of twodroplets. However, the detailed implementation of theboundary conditions were not clearly specified. Fur-thermore, since the two Oseen flows due to the twodroplets satisfy different governing equations, it is notclear what the exact equation is that governs the com-posite flow.
On the other hand, the finite Reynolds number or
finite fluid inertia does play an important role in modi-fying the relative motion [see, e.g., Fig. 3 of Klett andDavis (1973)], and, as shown in Klett and Davis (1973),this can lead to the asymmetry in the disturbance flowsor wake effect, which results in higher collision effi-ciency when the radius ratios are larger than 0.8. Thisphysical effect is not considered at all in this study.
4. Conclusions and remarks
In this paper we demonstrate that the original imple-mentation of the superposition method can be greatlyimproved, using a consistent framework based on therelationship between the disturbance flows and the vis-cous drag forces acting on droplets. Essentially, we re-quire that the composite flow satisfies, in an averagesense, the boundary conditions on the surface of drop-lets. Two formulations—namely, the integral formula-tion, and the center-point formulation—are possible.To illustrate the accuracy of the improved formula-tions, we have considered hydrodynamic interactions
TABLE 4. Calculated collision efficiencies for hydrodynamic interactions under gravity with the original formulation. Zero gap wasused for collision detection.
TABLE 5. Calculated collision efficiencies for hydrodynamic interactions under gravity with the center-point formulation. A smallgap of 0.001a1 was used for collision detection.
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between two droplets under Stokes disturbance flowapproximations. Results on viscous drag force comparefavorably with known exact solutions of Stokes flows.Specifically, the relative errors on the drag force can bereduced by one order of magnitude using the improvedformulations, when compared with the original formu-lation, for situations when the lubrication effect is notdominant.
We also show that the improved formulation resultsin a better prediction of collision efficiencies. However,like previous approximate methods such as Klett andDavis (1973), our improved formulation still overesti-mates the collision efficiencies, by a factor of as much as2 to 3. The level of accuracy of our improved superpo-sition method is comparable to that of Klett and Davis(1973) when the Stokes disturbance flows are consid-ered. The advantage of our formulation is its simplicityof implementation. Another advantage is that ourmethod can be applied to any droplet–droplet, three-dimensional relative configurations in space, while ex-tension of Klett and Davis’s (1973) formulation to gen-eral three-dimensional configurations has not yet beenworked out.
We note that the concepts explored in this papercould be further developed to describe disturbanceflows at finite droplet Reynolds numbers, and we willreport on this aspect in the future. Previous results(Klett and Davis 1973) suggest that Stokes disturbanceflows could underestimate collision efficiencies, par-ticularly when the droplets are of similar size. There-fore, our improved formulations must be further devel-oped to include the finite-Reynolds-number effect inorder to better quantify the collision efficiencies ofcloud droplets. At this stage, the results of Klett andDavis (1973) on collision efficiency could still be moreaccurate (in a qualitative sense) than the Stokes-flow-based formulations discussed here, especially for drop-lets of similar size. The finite-Reynolds-number effecton droplet–droplet relative motion has been demon-strated experimentally (e.g., Steinberger et al. 1968;Kumagai and Muraoka 1989). The work of Klett andDavis (1973) represents an important step toward in-cluding the finite-Reynolds-number effect in the calcu-lation of collision efficiency; however, we feel that theirformulation needs to be reexamined in light of the re-sults discussed in this paper. The ultimate conclusionwill rest on further knowledge based on a first-principle-based solution of droplet–droplet hydrody-namic interactions, such as direct simulations or care-fully designed experimental measurements of two ormore droplets in viscous fluid at finite Reynolds num-bers, as exact analytical solutions have not been provenpossible at finite Reynolds numbers.
Our improved formulations can be easily extended tostudy hydrodynamic interactions among a large numberof droplets in a turbulent flow. This was our motivationto revisit the superposition technique in the first place.
Results from numerical simulations applying the im-proved superposition method for a large number ofdroplets in gravitational settling only, as well as in theturbulent particle-laden flows, are being reported inseparate papers (Wang et al. 2004, manuscript submit-ted to J. Fluid Mech.; Wang et al. 2005).
Finally, we would like to briefly comment on theimplications of the different collision efficiencies on thetime scale of spectral broadening for droplet size dis-tribution. The local time scale for spectral broadening isinversely proportional to collision efficiency. For an ini-tially narrow size distribution of droplets around 10 to20 �m, we have shown here that the calculated collisionefficiencies based on different formulations can differby a factor of 2 to 5, indicating a significant uncertaintyin predicting the time scale for spectral broadening.When the background air turbulence is considered, pre-liminary results Wang et al. (2005) show that air turbu-lence can increase the collision efficiency by as much as50%, leading to shortening of the time needed to pro-duce droplets of 50 �m or larger in radius for which thegravitational coalescence can then become very effec-tive.
Acknowledgments. This study has been supported bythe National Science Foundation through Grant ATM-0114100 and by the National Center for AtmosphericResearch (NCAR). LPW thanks Dr. Charmaine Frank-lin of McGill University for providing us the referenceto the vector form of the Stokes flow solution. OA isgrateful to the additional computing resources providedby the Scientific Computing Division at NCAR.
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