In this paper, vortex-dynamical perspectives were adopted to interpret the recently reported observation that the total viscous dissipation of off-center droplet bouncing varies nonmonotonically with the impact parameter [C. He, X. Xia, and P. Zhang, “Non-monotonic viscous dissipation of bouncing droplets undergoing off-center collision,” Phys. Fluids 31, 052004 (2019)]. The particular interest of this study is on analyzing the velocity and vorticity vector fields and their correlations, such as helicity and enstrophy. The helicity analysis identifies a strong interaction between the “ring-shaped” vortices and the “line-shaped” shear layers in the non-axisymmetric droplet internal flow. A general relation between the total enstrophy and the total viscous dissipation rate for an unsteady free-surface flow was theoretically derived and numerically verified. It shows that the equality between the total enstrophy and the total viscous dissipation rate holds for a single-phase flow confined by stationary boundaries but is not satisfied for a gas–liquid two-phase flow due to the interfacial movement. Both the total enstrophy and a defined “half-domain” helicity show the nonmonotonic variation with the impact parameter, implying their interrelation with the nonmonotonic viscous dissipation.
REFERENCES
1.
P.
Brazier-Smith
, S.
Jennings
, and J.
Latham
, “The interaction of falling water drops: Coalescence
,” Proc. R. Soc. London, Ser. A
326
, 393
(1972
).2.
S.
Bradley
and C.
Stow
, “Collisions between liquid drops
,” Proc. R. Soc. London, Ser. A
287
, 635
(1978
).3.
N.
Ashgriz
and J.
Poo
, “Coalescence and separation in binary collisions of liquid drops
,” J. Fluid Mech.
221
, 183
(1990
).4.
Y. J.
Jiang
, A.
Umemura
, and C. K.
Law
, “An experimental investigation on the collision behavior of hydrocarbon droplets
,” J. Fluid Mech.
234
, 171
(1992
).5.
J.
Qian
and C. K.
Law
, “Regimes of coalescence and separation in droplet collision
,” J. Fluid Mech.
331
, 59
(1997
).6.
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
).7.
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
).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.
M.
Sommerfeld
and M.
Kuschel
, “Modelling droplet collision outcomes for different substances and viscosities
,” Exp. Fluids
57
, 187
(2016
).10.
G.
Finotello
, R. F.
Kooiman
, J. T.
Padding
, K. A.
Buist
, A.
Jongsma
, F.
Innings
, and J.
Kuipers
, “The dynamics of milk droplet–droplet collisions
,” Exp. Fluids
59
, 17
(2018
).11.
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
).12.
M.
Orme
, “Experiments on droplet collisions, bounce, coalescence and disruption
,” Prog. Energy Combust. Sci.
23
, 65
(1997
).13.
G.
Brenn
, “Droplet collision
,” in Handbook of Atomization and Sprays
(Springer
, Berlin
, 2011
).14.
H. P.
Kavehpour
, “Coalescence of drops
,” Annu. Rev. Fluid Mech.
47
, 245
(2015
).15.
N.
Roth
, C.
Rabe
, B.
Weigand
, F.
Feuillebois
, and J.
Malet
, “Droplet collision outcomes at high Weber number
,” in Proceedings of the 21st Conference
(Institute for Liquid Atomization and Spray Systems
, 2007
).16.
K.-L.
Pan
, P.-C.
Chou
, and Y.-J.
Tseng
, “Binary droplet collision at high Weber number
,” Phys. Rev. E
80
, 036301
(2009
).17.
G.
Finotello
, J. T.
Padding
, N. G.
Deen
, A.
Jongsma
, F.
Innings
, and J.
Kuipers
, “Effect of viscosity on droplet-droplet collisional interaction
,” Phys. Fluids
29
, 067102
(2017
).18.
C.
Gotaas
, P.
Havelka
, H. A.
Jakobsen
, H. F.
Svendsen
, M.
Hase
, N.
Roth
, and B.
Weigand
, “Effect of viscosity on droplet-droplet collision outcome: Experimental study and numerical comparison
,” Phys. Fluids
19
, 102106
(2007
).19.
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
).20.
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
).21.
H.
Deka
, G.
Biswas
, S.
Chakraborty
, and A.
Dalal
, “Coalescence dynamics of unequal sized drops
,” Phys. Fluids
31
, 012105
(2019
).22.
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
).23.
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
).24.
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
).25.
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
).26.
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
).27.
C.
He
, X.
Xia
, and P.
Zhang
, “Non-monotonic viscous dissipation of bouncing droplets undergoing off-center collision
,” Phys. Fluids
31
, 052004
(2019
).28.
S.
Farokhirad
, J. F.
Morris
, and T.
Lee
, “Coalescence-induced jumping of droplet: Inertia and viscosity effects
,” Phys. Fluids
27
, 102102
(2015
).29.
R.
Attarzadeh
and A.
Dolatabadi
, “Coalescence-induced jumping of micro-droplets on heterogeneous superhydrophobic surfaces
,” Phys. Fluids
29
, 012104
(2017
).30.
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
).31.
X.
Tang
, A.
Saha
, C. K.
Law
, and C. J. L.
Sun
, “Bouncing-to-merging transition in drop impact on liquid film: Role of liquid viscosity
,” Langmuir
34
, 2654
(2018
).32.
F.
Blanchette
, “Modeling the vertical motion of drops bouncing on a bounded fluid reservoir
,” Phys. Fluids
28
, 032104
(2016
).33.
M. R.
Behera
, A.
Dasgupta
, and S.
Chakraborty
, “On the generation of vorticity and hydrodynamics of vortex ring during liquid drop impingement
,” Phys. Fluids
31
, 082108
(2019
).34.
R.
Cresswell
and B.
Morton
, “Drop-formed vortex rings—The generation of vorticity
,” Phys. Fluids
7
, 1363
(1995
).35.
T.
Lundgren
and P.
Koumoutsakos
, “On the generation of vorticity at a free surface
,” J. Fluid Mech.
382
, 351
(1999
).36.
P.
Shankar
and M.
Kumar
, “Vortex rings generated by drops just coalescing with a pool
,” Phys. Fluids
7
, 737
(1995
).37.
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
).38.
M.-H.
Tsai
, K. L.
Law
, and H.-Y.
Chu
, “Asymmetric vortexes induced traveling drop on an oscillatory liquid bath
,” Phys. Fluids
31
, 102102
(2019
).39.
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
).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.
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
).42.
S.
Popinet
, “An accurate adaptive solver for surface-tension-driven interfacial flows
,” J. Comput. Phys.
228
, 5838
(2009
).43.
S.
Popinet
, “Numerical models of surface tension
,” Annu. Rev. Fluid Mech.
50
, 49
(2018
).44.
X.
Xia
, C.
He
, and P.
Zhang
, “Universality in the viscous-to-inertial coalescence of liquid droplets
,” Proc. Natl. Acad. Sci. U. S. A.
116
, 23467
(2019
).45.
N.
Smirnov
, V.
Betelin
, V.
Nikitin
, L.
Stamov
, and D.
Altoukhov
, “Accumulation of errors in numerical simulations of chemically reacting gas dynamics
,” Acta Astronaut.
117
, 338
(2015
).46.
N.
Smirnov
, V.
Betelin
, R.
Shagaliev
, V.
Nikitin
, I.
Belyakov
, Y. N.
Deryuguin
, S.
Aksenov
, and D.
Korchazhkin
, “Hydrogen fuel rocket engines simulation using LOGOS code
,” Int. J. Hydrogen Energy
39
, 10748
(2014
).47.
48.
K.-L.
Pan
, K.-L.
Huang
, W.-T.
Hsieh
, and C.-R.
Lu
, “Rotational separation after temporary coalescence in binary droplet collisions
,” Phys. Rev. Fluids
4
, 123602
(2019
).49.
P. A.
Davidson
, Turbulence: An Introduction for Scientists and Engineers
(Oxford University Press
, 2015
).50.
C. R.
Doering
, J. D.
Gibbon
, and J.
Gibbon
, Applied Analysis of the Navier-Stokes Equations
(Cambridge University Press
, 1995
).51.
G.
Tryggvason
, “Direct numerical simulations of gas–liquid flows
,” in Multiphase Flow Handbook
(CRC Press
, Boca Raton, FL
, 2016
), Chap. 2.2, p. 95
.© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
