The off-center collision of binary bouncing droplets of equal size was studied numerically by a volume-of-fluid method with two marker functions, which has been justified and validated by comparing with available experimental results. A nonmonotonic kinetic energy (KE) recovery with varying impact parameters was discovered. This can be explained by the prolonged entanglement time and the enhanced internal-flow-induced viscous dissipation for bouncing droplets at intermediate impact parameters, compared with those at smaller or larger impact parameters. The distribution of the local viscous dissipation rate (VDR) in the droplet interior shows two major concentration areas, and the competition between these two concentration areas accounts for the nonmonotonic viscous dissipation with varying impact parameters. The nonmonotonic KE recovery with varying impact parameters can also be attributed to the competition between the VDR induced by normal strains and shear strains. The nonmonotonicity was further numerically verified for wider ranges of parameters, and a practically useful formula was proposed to correlate the KE dissipation factor with the impact parameter for various Weber numbers and Ohnesorge numbers.
REFERENCES
1.
N.
Ashgriz
and J.
Poo
, “Coalescence and separation in binary collisions of liquid drops
,” J. Fluid Mech.
221
, 183
(1990
).2.
G.
Brenn
, “Droplet collision
,” in Handbook of Atomization and Sprays
(Springer
, Berlin
, 2011
).3.
O.
Jayaratne
and B. J.
Mason
, “The coalescence and bouncing of water drops at an air/water interface
,” Proc. R. Soc. London, Ser. A
280
, 545
(1964
).4.
Y.
Jiang
, A.
Umemura
, and C.
Law
, “An experimental investigation on the collision behaviour of hydrocarbon droplets
,” J. Fluid Mech.
234
, 171
(1992
).5.
M.
Orme
, “Experiments on droplet collisions, bounce, coalescence and disruption
,” Prog. Energy Combust. Sci.
23
, 65
(1997
).6.
K.-L.
Pan
, C. K.
Law
, and B.
Zhou
, “Experimental and mechanistic description of merging and bouncing in head-on binary droplet collision
,” J. Appl. Phys.
103
, 064901
(2008
).7.
J.
Qian
and C. K.
Law
, “Regimes of coalescence and separation in droplet collision
,” J. Fluid Mech.
331
, 59
(1997
).8.
C.
Tang
, P.
Zhang
, and C. K.
Law
, “Bouncing, coalescence, and separation in head-on collision of unequal-size droplets
,” Phys. Fluids
24
, 022101
(2012
).9.
E.
Villermaux
and B.
Bossa
, “Single-drop fragmentation determines size distribution of raindrops
,” Nat. Phys.
5
, 697
(2009
).10.
E.
Villermaux
and B.
Bossa
, “Size distribution of raindrops
,” Nat. Phys.
6
, 232
(2010
).11.
12.
W. A.
Sirignano
, Fluid Dynamics and Transport of Droplets and Sprays
(Cambridge University Press
, 1999
).13.
S. L.
Anna
, “Droplets and bubbles in microfluidic devices
,” Annu. Rev. Fluid Mech.
48
, 285
(2016
).14.
S.-Y.
Teh
, R.
Lin
, L.-H.
Hung
, and A. P.
Lee
, “Droplet microfluidics
,” Lab Chip
8
, 198
(2008
).15.
J. C.
Bird
, R.
Dhiman
, H.-M.
Kwon
, and K. K.
Varanasi
, “Reducing the contact time of a bouncing drop
,” Nature
503
, 385
(2013
).16.
T.
Liu
and C.-J.
Kim
, “Turning a surface superrepellent even to completely wetting liquids
,” Science
346
, 1096
(2014
).17.
Y.
Liu
, M.
Andrew
, J.
Li
, J. M.
Yeomans
, and Z.
Wang
, “Symmetry breaking in drop bouncing on curved surfaces
,” Nat. Commun.
6
, 10034
(2015
).18.
Y.
Liu
, L.
Moevius
, X.
Xu
, T.
Qian
, J. M.
Yeomans
, and Z.
Wang
, “Pancake bouncing on superhydrophobic surfaces
,” Nat. Phys.
10
, 515
(2014
).19.
Y.
Lu
, S.
Sathasivam
, J.
Song
, C. R.
Crick
, C. J.
Carmalt
, and I. P.
Parkin
, “Robust self-cleaning surfaces that function when exposed to either air or oil
,” Science
347
, 1132
(2015
).20.
D.
Richard
, C.
Clanet
, and D.
Quéré
, “Surface phenomena: Contact time of a bouncing drop
,” Nature
417
, 811
(2002
).21.
W.
Ristenpart
, J.
Bird
, A.
Belmonte
, F.
Dollar
, and H.
Stone
, “Non-coalescence of oppositely charged drops
,” Nature
461
, 377
(2009
).22.
H. P.
Kavehpour
, “Coalescence of drops
,” Annu. Rev. Fluid Mech.
47
, 245
(2015
).23.
X.
Chen
, D.
Ma
, and V.
Yang
, “Collision outcome and mass transfer of unequal-sized droplet collision
,” in 50th AIAA Aerospace Sciences Meeting
(AIAA
, 2012
).24.
D.
Liu
, P.
Zhang
, C. K.
Law
, and Y.
Guo
, “Collision dynamics and mixing of unequal-size droplets
,” Int. J. Heat Mass Transfer
57
, 421
(2013
).25.
K.
Sun
, P.
Zhang
, M.
Jia
, and T.
Wang
, “Collision-induced jet-like mixing for droplets of unequal-sizes
,” Int. J. Heat Mass Transfer
120
, 218
(2018
).26.
C.
Tang
, J.
Zhao
, P.
Zhang
, C. K.
Law
, and Z.
Huang
, “Dynamics of internal jets in the merging of two droplets of unequal sizes
,” J. Fluid Mech.
795
, 671
(2016
).27.
X.
Xia
, C.
He
, D.
Yu
, J.
Zhao
, and P.
Zhang
, “Vortex-ring-induced internal mixing upon the coalescence of initially stationary droplets
,” Phys. Rev. Fluids
2
, 113607
(2017
).28.
K.
Krishnan
and E.
Loth
, “Effects of gas and droplet characteristics on drop-drop collision outcome regimes
,” Int. J. Multiphase Flow
77
, 171
(2015
).29.
L.
Reitter
, M.
Liu
, J.
Breitenbach
, K.-L.
Huang
, D.
Bothe
, G.
Brenn
, K.-L.
Pan
, I.
Roisman
, and C.
Tropea
, “Experimental and computational investigation of binary drop collisions under elevated pressure
,” in 28th European Conference on Liquid Atomization and Spray System
, 2017
.30.
Z.
Zhang
, Y.
Chi
, L.
Shang
, P.
Zhang
, and Z.
Zhao
, “On the role of droplet bouncing in modeling impinging sprays under elevated pressures
,” Int. J. Heat Mass Transfer
102
, 657
(2016
).31.
Z.
Zhang
and P.
Zhang
, “Cross-impingement and combustion of sprays in high-pressure chamber and opposed-piston compression ignition engine
,” Appl. Therm. Eng.
144
, 137
(2018
).32.
Z.
Zhang
and P.
Zhang
, “Modeling kinetic energy dissipation of bouncing droplets for Lagrangian simulation of impinging sprays under high ambient pressure
,” Atomization Sprays
28
, 673
(2018
).33.
K. H.
Al-Dirawi
and A. E.
Bayly
, “A new model for the bouncing regime boundary in binary droplet collisions
,” Phys. Fluids
31
, 027105
(2019
).34.
S.
Farokhirad
, J. F.
Morris
, and T.
Lee
, “Coalescence-induced jumping of droplet: Inertia and viscosity effects
,” Phys. Fluids
27
, 102102
(2015
).35.
R.
Attarzadeh
and A.
Dolatabadi
, “Coalescence-induced jumping of micro-droplets on heterogeneous superhydrophobic surfaces
,” Phys. Fluids
29
, 012104
(2017
).36.
X.
Tang
, A.
Saha
, C. K.
Law
, and C.
Sun
, “Bouncing drop on liquid film: Dynamics of interfacial gas layer
,” Phys. Fluids
31
, 013304
(2019
).37.
X.
Tang
, A.
Saha
, C. K.
Law
, and C.
Sun
, “Bouncing-to-merging transition in drop impact on liquid film: Role of liquid viscosity
,” Langmuir
34
, 2654
(2018
).38.
F.
Blanchette
, “Modeling the vertical motion of drops bouncing on a bounded fluid reservoir
,” Phys. Fluids
28
, 032104
(2016
).39.
X.
Chen
and V.
Yang
, “Thickness-based adaptive mesh refinement methods for multi-phase flow simulations with thin regions
,” J. Comput. Phys.
269
, 22
(2014
).40.
C.
Hu
, S.
Xia
, C.
Li
, and G.
Wu
, “Three-dimensional numerical investigation and modeling of binary alumina droplet collisions
,” Int. J. Heat Mass Transfer
113
, 569
(2017
).41.
B.
Sakakibara
and T.
Inamuro
, “Lattice Boltzmann simulation of collision dynamics of two unequal-size droplets
,” Int. J. Heat Mass Transfer
51
, 3207
(2008
).42.
F. H.
Zhang
, E. Q.
Li
, and S. T.
Thoroddsen
, “Satellite formation during coalescence of unequal size drops
,” Phys. Rev. Lett.
102
, 104502
(2009
).43.
S.
Popinet
, “Gerris: A tree-based adaptive solver for the incompressible Euler equations in complex geometries
,” J. Comput. Phys.
190
, 572
(2003
).44.
S.
Popinet
, “An accurate adaptive solver for surface-tension-driven interfacial flows
,” J. Comput. Phys.
228
, 5838
(2009
).45.
E.
Coyajee
and B. J.
Boersma
, “Numerical simulation of drop impact on a liquid–liquid interface with a multiple marker front-capturing method
,” J. Comput. Phys.
228
, 4444
(2009
).46.
P.
Zhang
and C. K.
Law
, “An analysis of head-on droplet collision with large deformation in gaseous medium
,” Phys. Fluids
23
, 042102
(2011
).47.
J.
Li
, “Macroscopic model for head-on binary droplet collisions in a gaseous medium
,” Phys. Rev. Lett.
117
, 214502
(2016
).48.
R.
Aumann
, M.
McCracken
, and J.
Abraham
, “An evaluation of a composite model for predicting drop-drop collision outcomes in multidimensional spray computations
,” in (SAE International
, 2002
).49.
P.
Brazier-Smith
, S.
Jennings
, and J.
Latham
, “The interaction of falling water drops: Coalescence
,” Proc. R. Soc. London, Ser. A
326
, 393
(1972
).50.
J.-P.
Estrade
, H.
Carentz
, G.
Lavergne
, and Y.
Biscos
, “Experimental investigation of dynamic binary collision of ethanol droplets—A model for droplet coalescence and bouncing
,” Int. J. Heat Fluid Flow
20
, 486
(1999
).51.
P. J.
O’Rourke
, “Collective drop effects on vaporizing liquid sprays
,” Ph.D. thesis, Princeton University
, 1981
.52.
S. L.
Post
and J.
Abraham
, “Modeling the outcome of drop–drop collisions in Diesel sprays
,” Int. J. Multiphase Flow
28
, 997
(2002
).53.
A.
Munnannur
and R. D.
Reitz
, “Comprehensive collision model for multidimensional engine spray computations
,” Atomization Sprays
19
, 597
(2009
).54.
X.
Chen
, D.
Ma
, P.
Khare
, and V.
Yang
, “Energy and mass transfer during binary droplet collision
,” in 49th AIAA Aerospace Sciences Meeting
(AIAA
, 2011
).55.
Z.
Zhang
and P.
Zhang
, “Kinetic energy recovery and interface hysteresis of bouncing droplets after inelastic head-on collision
,” Phys. Fluids
29
, 103306
(2017
).56.
Z.
Mohamed-Kassim
and E. K.
Longmire
, “Drop impact on a liquid–liquid interface
,” Phys. Fluids
15
, 3263
(2003
).57.
M.
Kwakkel
, W.-P.
Breugem
, and B. J.
Boersma
, “Extension of a CLSVOF method for droplet-laden flows with a coalescence/breakup model
,” J. Comput. Phys.
253
, 166
(2013
).58.
C.
Gotaas
, P.
Havelka
, H. A.
Jakobsen
, and H. F.
Svendsen
, “Evaluation of the impact parameter in droplet-droplet collision experiments by the aliasing method
,” Phys. Fluids
19
, 102105
(2007
).59.
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