In e na ional Jou nal o Mul iphase Flow 184 (2025) 105113
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Resea ch pape
D op impac on o a mo ing subs a e: Ae odynamic ebound
Bas ian S ump a, Samaneh Abdi Qezeljeh a, Reda Kamal a,b, Fabien Dezi e c,
Alessand o Ma u od, Ilia V. Roisman a, Jeane e Hussong a,∗
aTechnische Uni e si ä Da ms ad , Ins i u e o Fluid Mechanics and Ae odynamics, Da ms ad , 64289, Ge many
bPoli ecnico di Milano, Depa men o Ae ospace Science and Technology, Milano, 20156, I aly
cAi bus Helicop e s S.A.S., Aé opo In e na ional Ma seille P o ence, Ma ignane, 13725, F ance
dAi bus Helicop e s, Ae omechanics & Pe o mance, Ma ignane, 13725, F ance
ARTICLE INFO
Da ase link:h ps://zenodo.o g/doi/10.5281/
zenodo.12684793
Keywo ds:
D op impac
Mul iphase low
D op deposi ion and ebound
Bounda y laye
We ing and icing
Mo ing subs a e
ABSTRACT
The dynamics o d ople s app oaching as mo ing su aces o high su ace- angen ial eloci ies is ele an o
nume ous echnical applica ions, such as icing phenomena in a ia ion. Due o he subs a e mo ion a bounda y
laye is o med which in e ac s wi h impac ing d ople s. In he p esen s udy, he ansi ion om d op impac
and splashing o bounda y laye induced d op ebound is in es iga ed o a ying d op diame e s, d op and
pla e eloci y, as well as impac angles. I is ound ha his ansi ion is s ongly in luenced by he deg ee
o d op de o ma ion ha is induced by ae odynamic o ces ac ing on he d op when i en e s he bounda y
laye . Based on hese conside a ions, a h eshold model is ob ained ha desc ibes he ansi ion om splash
o ae odynamic ebound. I is shown ha he model is alid o a lamina and a u bulen bounda y laye
ag eeing well wi h own and exis ing expe imen al da a.
1. In oduc ion
The in e ac ion o a d op wi h a plana su ace ha exhibi s a as
mo ion angen ial o i s in e ace is ele an o nume ous echnical
applica ions. One example is icing in a ia ion caused by small supe -
cooled d ops in he a mosphe e ha impac on o he ai c a su ace o
on o o a ing helicop e blades (Cao and Chen,2010;Cao e al.,2018).
Ano he example is d ople based cooling o as o a ing componen s,
such as o o windings in elec ic machines. In elec ical machines,
he in e ac ion o he d op wi h he mo ing su ace needs o be ully
unde s ood o be able o eliably p edic he cooling hea lux (Lim
and Kim,2014;Liu e al.,2019). Fo his, i is o pa icula in e es o
know whe he he d ople s a e in di ec con ac wi h he subs a e and
i so, wha he impac ou come will be. Fo he icing, he esul ing ice
laye changes he ae odynamic p ope ies o he ai c a , which leads o
dec eased e iciency and migh lead o haza dous ligh condi ions (Lee
e al.,1984;Kind e al.,1998;Cao e al.,2015;Yamazaki e al.,2021).
Also, he ice acc e ion a e depends, among o he phenomena, on he
impac ou come. Gene ally, a d op impac can be subdi ided in o d op
splash, deposi ion, o ebound. Fo eliable modeling o echnically
ele an p ocesses, i is impo an o de e mine he deposi ed mass a io
and he pa ame e s o he seconda y sp ay o med by ebound and
splash since his seconda y sp ay could again impac on o su aces.
The impac o d ops on o s a iona y we ed and d y su aces has
been s udied ex ensi ely (Ya in,2006;Mo ei a e al.,2010;Josse and
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (J. Hussong).
and Tho oddsen,2016;Ya in e al.,2017). The ou come o d op impac ,
i is bouncing (Sp i les,2024), deposi ion and splash (T opea and
Ma engo,1999) is mainly de e mined by he impac pa ame e s, he
d op eloci y 𝑈𝑑 ,𝑛 no mal o he subs a e, and he d op diame e
𝐷and he ma e ial p ope ies o he liquid, including he kinema ic
iscosi y 𝜈d op, he densi y 𝜌d op and he su ace ension o he d op 𝜎d op.
Co espondingly, he ou come is go e ned by he Reynolds numbe
and Webe numbe , Re=𝑈𝑑 ,𝑛𝐷d op∕𝜈d op and We=𝜌d op𝑈2
𝑑 ,𝑛𝐷d op∕𝜎d op,
espec i ely. I he d op impac s on o a we ed subs a e also he
dimensionless ilm hickness, scaled by he d op diame e 𝐷, becomes
an in luencing pa ame e . Wall-no mal d op impac on o a s a iona y
and plana su ace leads o a adially expanding low in a hin lamella.
The ou come is de ined by he in e ac ion o his low wi h he ou e
wall ilm. I he ine ial e ec s a e dominan in compa ison wi h he
iscous and capilla y o ces, his in e ac ion leads o he eme gence o
a co ona-like liquid je (Ya in and Weiss,1995;Roisman and T opea,
2002). The co ona splash is hen caused by he ins abili y o he Taylo
im (Taylo ,1959;Roisman,2010;Agbaglah e al.,2013;Wang and
Bou ouiba,2021).
D op impac on o a solid d y wall is in luenced also by he con-
di ions a he subs a e su ace, i s mo phology, and we ing p ope -
ies (Mundo e al.,1995;Riboux and Go dillo,2014;Roisman e al.,
2015;Josse and and Tho oddsen,2016). The e olu ion o he d op
h ps://doi.o g/10.1016/j.ijmul iphase low.2024.105113
Recei ed 13 July 2024; Recei ed in e ised o m 8 Oc obe 2024; Accep ed 16 Decembe 2024
In e na ional Jou nal o Mul iphase Flow 184 (2025) 105113
2
B. S ump e al.
diame e is de e mined by he dynamics o he Taylo im o med a
he edge o he sp eading lamella (Roisman e al.,2002). Howe e ,
he ou come o d op impac depends signi ican ly on he ae odynamic
e ec s in he su ounding gas. I is known ha he condi ions leading
o co ona eme gence and hus o he co ona splash a e in luenced
by he p ope ies o he su ounding gas (Xu e al.,2005). Recen ly,
he main mechanisms o he co ona eme gence associa ed wi h he
dynamics o he gas low in a sp eading wedge has been conside ed
and explained (Riboux and Go dillo,2014).
D op bouncing om a d y o e en liquid su aces occu s o impac s
wi h ela i ely low Webe numbe , We ∼(1). I is a ibu ed o a
e y hin ai laye be ween he subs a e and he d op (de Rui e e al.,
2014). Fo he d op impac on o mo ing subs a es (Mundo e al.,1995)
showed ha o he splash deposi ion limi only he no mal componen
o he d op impac eloci y is ele an i he angen ial subs a e eloc-
i y is smalle o compa able wi h he d op impac eloci y. Mos s udies
ha in es iga ed he in e ac ion o a d op wi h a mo ing subs a e
ocused on he hyd odynamics o sp eading on we ing and non-we ing
subs a es (Almohammadi and Ami azli,2017;Mogh ade nejad e al.,
2021). Only a ew s udies in es iga ed pa ame e anges whe e he
ine ia o he gas bounda y laye is su icien ly high o play a signi ican
ole in he d op dynamics. A comple e ae odynamic ebound o d ople s
due o he ai low in he bounda y laye was i s obse ed and ana-
lyzed in Po a o e al. (1976). In his s udy, he h eshold eloci y 𝑈⋆
pla e
o he subs a e, co esponding o he incep ion o he d op ebound, is
ela ed o he hickness 𝛿o he iscous bounda y laye in he gas low.
𝑈⋆
pla e∼𝑈𝑑 ,𝑛√𝜌d op
𝜌gas
𝐷d op
𝛿.(1)
The same ela ion was con i med by Gau hie e al. (2016,2018) o
lamina bounda y laye s.
In he p esen s udy, we show ha he h eshold eloci y (1) a
which d ople ebound can be obse ed is alid o bo h lamina and
u bulen bounda y laye s. Howe e , he coe icien o p opo ionali y
in (1) changes signi ican ly. This means ha no only he bounda y
laye hickness bu also he eloci y p o ile plays a signi ican ole in
he bouncing phenomenon.
2. Expe imen al se up
The expe imen al se up, schema ically shown in Fig. 1(a), consis s o
he o a ing pla e, he d op gene a ion sys em, and he imaging sys em.
The pla e is made o ca bon ibe - ein o ced plas ic (CFRP) and has
a adius o 𝑅= 90 mm. I is a ached o a b ushless DC mo o ha
can accele a e he pla e o an angula eloci y o 𝛺≤17.000 pm
( e olu ions pe minu e). The eloci y is con olled, moni o ed, and
logged by an in-house LabView sc ip . To cap u e he d op impac , a
high-speed came a is used in he imaging sys em. Depending on he
expe imen he u ilized models a e ei he Pho on SA-X2 o Phan om
T3610. A high-speed LED (Cons ella ion 120E) and a di use pla e
p o ide uni o m backg ound illumina ion, and he imaging sys em
can each ame a es o 100,000 ps. The spa ial esolu ion in he
expe imen s a ies in he ange o 9 μm o 15 μm pe pixel.
The d op gene a o is a comme cial mono-dispe se d op chain
gene a o om FMP Technology GmbH which will be e e ed o as
FMP gene a o in he ollowing. I gene a es a mono-dispe se s eam
o d ople s by inducing a Rayleigh–Pla eau ins abili y on o a liquid je
using a piezoelec ic ac ua o (B enn and T opea,1996). The d ople
size can be a ied in he ange 80 μm≤𝐷≤500 μm by ei he changing
he size o he ou le o i ice o he FMP o by al e ing he exci a ion
equency. The adial coo dina e o impac is 88 mm bu can luc ua e
app oxima ely ±1 mm. The absolu e eloci y o he d ople is se by
con olling he p essu e in he p essu ized ank and can be a ied in
he ange 5.5m/s ≤𝑈𝑑 ,abs ≤12 m/s. The eloci y componen 𝑈𝑑 ,𝑛 o
he d op no mal o he pla e can be al e ed by al e ing he impac
angle 10◦≤𝛽≤40◦o he d ople o 𝑈𝑑 ,abs. In Fig. 1(b) and Fig. 1(c),
he geome y o he impac is illus a ed. The no mal and ho izon al
eloci y componen s 𝑈𝑑 ,𝑛 and 𝑈𝑑 ,ℎ can be ob ained di ec ly om he
high-speed eco dings, as will be explained below. A consequence o
he inclined impac is ha he d ople has a eloci y componen in
ci cum e en ial di ec ion 𝑈𝑑 ,𝜑 =𝑈𝑑 ,ℎcos𝛼whe e 𝛼is a cons an o se
angle o 7.1◦, ela i e o he came a axis. To accoun o his he
ela i e eloci y in ci cum e en ial di ec ion 𝑈𝜑, el is conside ed he
cha ac e is ic eloci y which can be o med as
𝑈𝜑, el =𝑈𝜑,pla e−𝑈𝑑 ,𝜑,(2)
whe e 𝑈𝜑,pla e ep esen s he ci cum e en ial eloci y o he pla e. The
pla e Reynolds numbe , Reae o=𝑟2𝜔∕𝜈𝑎, wi h 𝜔being he angula
eloci y o he pla e and 𝜈𝑎 he kinema ic iscosi y o ai , a ies in
he ange 2.5 × 105<Reae o<8.7 × 105. The lowe limi is chosen o
a oid he bounda y laye ’s ansi ional egime, while he uppe limi
aligns wi h ypical alues o helicop e blades and ai c a wings. The
co esponding pla e eloci y a he impac loca ion is in he ange
40 m∕s < 𝑈𝜑,pla e<160 m∕s.
A he uppe end o his ange, weak comp essibili y e ec s may be
expec ed as he Mach numbe app oaches app oxima ely 0.5. Expe -
imen al esul s by Theodo sen and Regie (1944) o Mach numbe s
up o 0.62 indica e ha he heo e ical p edic ions by on Ká mán
(1946), which a e based on he assump ion o incomp essibili y, ex-
hibi s ong ag eemen wi h he expe imen al da a o he momen
coe icien , which in u n is de i ed om he bounda y laye p o ile.
Fu he mo e, he bounda y laye p o ile shows a s eep g adien nea
he wall, con ining any comp essibili y e ec s o a small luid egion.
Consequen ly, wi hin he ange in es iga ed, comp essibili y e ec s a e
negligibly small and a e he e o e no conside ed in he p esen s udy.
Finally he bounda y laye is ully u bulen , as he lamina - o-
u bulen ansi ion o he pla e occu s in he ange 1.8 × 105<Reae o<
3.5 × 105, as shown in She chuk (2009).
2.1. Image p ocessing
The expe imen s a e analyzed in a cus om MATLAB sc ip , u ilizing
he MATLAB image p ocessing oolbox, whe e he d op diame e as
well as he ho izon al and e ical eloci y componen s o he d op be-
o e impac a e measu ed. Fo his, i s , a mo ing backg ound sub ac-
ion wi h a del a o 5 ames ollowed by a bina iza ion is pe o med.
F om he esul ing bina y images, indi idual objec s i.e. d ople s, hei
cen oid, and diame e can be e alua ed. A simple pa icle acking
algo i hm based on he nea es neighbo p inciple is hen used o
ecognize he d ople s in consecu i e ames and hus de e mine he
ajec o ies. To make he e alua ion mo e obus agains alse de ec-
ion, only ajec o ies wi h objec s ha a e ecognized and alloca ed
in 10 consecu i e ames a e conside ed. A median d op diame e is
de e mined o each ajec o y using images o he d op in mul iple
consecu i e ames be o e impac . To add ess po en ial bias due o ou -
lie s, d op images wi h diame e s ha de ia e mo e han wo s anda d
de ia ions om he mean a e conside ed ou lie s and a e excluded.
E en ually, mean alues o he diame e and he espec i e ho izon al
and e ical eloci y componen s a e o med using all ajec o ies in an
expe imen . Fu he mo e, cases in which he d ople diame e o he
d ople impac eloci y sca e by mo e han 20% a e disca ded om
he e alua ion. The mean ela i e s anda d de ia ion o all cases is
𝜎𝐷= 4.49% o he d op diame e , 𝜎𝑈𝑑 𝑛= 4.1% in he no mal di ec ion
and 𝜎𝑈𝜑, el = 0.25% o he ela i e pla e eloci y.
3. Resul s o d op impac and ebound
When consecu i e, mono-dispe se d ople s in e ac wi h a o a ing
pla e, a ious phenomena can be obse ed depending on he impac
pa ame e s. In Fig. 2, a ime se ies o impac ing d ople s o cons an
diame e and eloci y is shown a h ee di e en pla e eloci ies. Fo
In e na ional Jou nal o Mul iphase Flow 184 (2025) 105113
3
B. S ump e al.
Fig. 1. Schema ic ep esen a ion o he expe imen al se up (a) side iew (b) op iew and (c) on iew.
he lowes pla e eloci y, 𝑈𝜑, el = 60 m/s (see Fig. 2(a)) a s ong
in e ac ion o he d ople wi h he pla e can be obse ed. Du ing he
in e ac ion, liquid adhe es o he as -mo ing pla e and ge s d agged
along, while in pa allel, ligamen s o m ha a omize in o seconda y
d ople s and a e ca ied away by he ai low in he bounda y laye .
The liquid ha adhe es o he pla e will each he impac loca ion a e
one ull e olu ion o he o a ing pla e whe e consecu i e d ople s
may impac on o he now we ed su ace. In his case, splashing is
enhanced. S udies on single d op impac on o hin liquid ilms ha e
shown ha e y hin ilms wi h dimensionless ilm hicknesses
ℎ <
0.02 enhance splashing signi ican ly (Geppe ,2019;Zhu e al.,2021;
S ump e al.,2023). In con as , o he highes pla e eloci ies, 𝑈𝜑,𝑟𝑒𝑙 =
140 m/s (see Fig. 2(c)) no splash and also no esidual can be ob-
se ed anymo e, as d ople s unde go a comple e ae odynamic ebound,
p e en ing d op-pla e con ac .
A medium pla e eloci ies, 𝑈𝜑,𝑟𝑒𝑙 = 80 m/s (see Fig. 2(b)) splashing
can only be obse ed occasionally and in a less p onounced o m.
D ople s pa ially ebound while i is e iden in he enla ged sec ion
a 𝑡= 0.1ms ha a small amoun o he d ople esides on he pla e.
The e a e wo easons why he splash migh only occu o some
d ople s:
•The u bulen cha ac e o he ai low leads o small luc ua ions
o he impac loca ion in he adial di ec ion ha esul in a i-
a ions o he ela i e eloci y a impac . The la e ’s a e caused
by he adial bounda y laye o he disk and a e wi hin he dep h
o ield o he came a leading o a possible e o in he impac
posi ion o ± 2 mm. In consequence, when he pla e eloci y is
close o he c i ical eloci y o ebound, some o he d ops will
exceed i and only some d ops will ha e con ac wi h he pla e.
•As depic ed in Fig. 2(b) (see ins an 0.01 ms), he in e ac ion
be ween he d ople and he pla e esul s in he o ma ion o
small pa ches o esidual liquid. A ins an s 0.10 ms and 0.11 ms
i can be obse ed ha such pa ches in e ac wi h a subsequen
d ople a e one e olu ion, esul ing in a splash. I is e iden
ha such pa ches po en ially in luence he ou come o subsequen
d op impac s. Howe e , hese pa ches canno be obse ed in he
ae odynamic ebound egime.
Thus, o inc easing pla e eloci y he ansi ion om splash/con ac
o ae odynamic ebound is con inuous. Fo he ollowing analysis,
h ee ca ego ies o in e ac ion a e de ined. I he ou come o he
impac esul s in a splash hen he expe imen will be classi ied as
‘‘splash/con ac ’’. I splash can only be obse ed o some o he
d ople s hen he expe imen will be ca ego ized as ‘‘ ansi ion’’. I no
splash can be obse ed h oughou he expe imen and he d ople s a e
comple ely epelled om he pla e, we conclude ha he e is no mo e
con ac and hus no liquid esidual on he pla e. This egime is de ined
as ‘‘ae odynamic ebound’’.
3.1. Obse ed impac ou comes
To cha ac e ize he ou come o he d op impac on o a o a ing pla e
he no mal impac eloci y 𝑈𝑑 ,𝑛 as well as he ela i e pla e eloci y
𝑈𝜑, el and he d op diame e a e sys ema ically a ied. In Fig. 3(a) he
ou come o impac is shown o he ela i e angen ial and no mal
impac eloci y. I becomes appa en ha o highe 𝑈𝑑 ,𝑛 also highe
𝑈𝜑,𝑟𝑒𝑙 a e necessa y o achie e ae odynamic ebound. A s aigh line
could sepa a e he ae odynamic ebound om he ansi ion and splash
in he plo . In Fig. 3(b) he ou comes a e shown o a la ge d op
diame e o 𝐷≈ 452 μm. In compa ison o he 𝐷≈ 91.3 μm case o
he la ge d ops, highe pla e eloci ies a e necessa y o achie e an
ae odynamic ebound. I becomes e iden ha he highe he ine ia
o he d op he highe also he ine ia o he ai low needs o be o
e lec he d ople om he pla e.
In p eceding s udies (Po a o e al.,1976;Gau hie e al.,2018)
h eshold pa ame e s, desc ibing he onse o ae odynamic ebound and
based on he ela ion o d op and pla e ine ia a e o mula ed in (1)
assuming a decele a ion o he d ople h oughou he bounda y laye
and possible ajec o y de lec ion due o he ae odynamic d ag. This
assump ion is examined in Fig. 4. In Fig. 4(a) he empo al e olu ion
o he dis ance 𝑦o he cen e o mass o he d op o he pla e and
i s eloci y a e exempli ied o one expe imen in he ae odynamic
ebound egime. The ins an 𝑡= 0is de ined as he ime when he
unde o med (sphe ical) d op wi h an undis u bed ajec o y would
ouch he pla e. In Fig. 4(b) he e olu ion o he co esponding eloci y
is shown. The es ima ed hickness o he u bulen bounda y laye
hickness in his case is 𝛿𝑡≈ 4.5 mm. I can be seen ha he cen e
o mass o he d ople is no decele a ed un il i is in he close icini y
o he pla e su ace (𝑦≈ 100 μm). This sugges s ha a he han he
decele a ion o he whole d ople , he de o ma ion o he d ople is
go e ning he ebound.
3.2. Bounda y laye induced d op de o ma ion: go e ning scales
The dynamics o a d op mo ing h ough an ai low is de e mined
by a balance o he s esses in he ai low and in he de o ming d op.
The main ae odynamic s esses include
𝑝gas ∼𝜌gas𝑈2
gas,(3)
𝑝uns eady ∼𝜌gas𝐷d op
d𝑈gas
d𝑡∼𝜌gas𝐷d op
𝜕 𝑈gas
𝜕 𝑧𝑈d,abs,(4)
In e na ional Jou nal o Mul iphase Flow 184 (2025) 105113
4
B. S ump e al.
Fig. 2. Snapsho s o a s eam o monodispe se d ops impac ing on a pla e a di e en pla e eloci ies 𝑈𝜑, el. The d ople o in e es is highligh ed wi h a g een o e lay. (a)
splash occu ing a 𝑈𝜑, el = 60 m∕s, (b) ansi ion a 𝑈𝜑, el = 80 m/s, (c) ae odynamic ebound a 𝑈𝜑, el = 140 m/s. Impac pa ame e s a e 𝐷≈ 230 μm, 𝑈𝑑 ,𝑛 ≈ 2m/s, 𝛽= 19◦. The
co esponding ideos a e a ailable in he supplemen a y ma e ial. (Fo in e p e a ion o he e e ences o colo in his igu e legend, he eade is e e ed o he web e sion o
his a icle.)
Fig. 3. Veloci y maps showing he d op impac ou comes ‘‘splash’’, ‘‘ ansi ion’’ and ‘‘ ebound’’ o he pla e eloci y 𝑈𝜑, el and he no mal d op impac eloci y 𝑈d,n o (a)
𝐷≈ 91.3 μm, (b) 𝐷≈ 452 μm. The e o ba s show he oo mean squa e e o (RMSE) o he co esponding eloci ies o di e en d ops in one expe imen .
whe e 𝑈gas is he ela i e gas eloci y.
I can be shown ha i he gas eloci y is much highe han he
d op eloci y, 𝑈gas ≫ 𝑈d,abs, he alue o 𝑝uns eady is negligibly small in
compa ison wi h 𝑝gas. Howe e , his is no he case when he hickness
o he bounda y laye 𝛿is much smalle han he ini ial diame e o he
d op, 𝐷d op.
The s esses in he d op a e de ined by he d op de o ma ion eloc-
i y 𝑈de . The p essu e e ms in he d op, a ising om ine ial e ec s
In e na ional Jou nal o Mul iphase Flow 184 (2025) 105113
5
B. S ump e al.
Fig. 4. E alua ion o he d op ajec o y (a) no mal dis ance o he pla e as a unc ion o ime. (b) Veloci y in no mal di ec ion as a unc ion o ime. E o ba s a e he RMSE
be ween he ajec o ies in he expe imen . Impac condi ions: 𝑈𝜑, el = 120 m∕s,𝐷= 218 μm,𝑈𝑑 ,𝑛 = 2.1 m∕s,𝛽= 19.25◦.
due o he de o ma ion low, a e
𝑝d op ∼𝜌d op𝑈2
de , 𝑝d op,uns eady ∼𝜌d op𝐷d op
d𝑈de
d𝑡(5)
The d op de o ma ion eloci y ou side he bounda y laye is negligi-
ble, gi en ha he ae odynamic p essu e associa ed wi h he ai ela i e
eloci y is conside ably smalle han he capilla y p essu e o he d op.
Upon en e ing he bounda y laye , he ela i e gas eloci y inc eases,
esul ing in he de o ma ion o he d op. Fo he modeling o his s age
o d op mo ion, i is su icien o conside only he uns eady p essu e
e m. The alidi y o his assump ion will be examined la e , once he
de o ma ion eloci y has been de e mined. Consequen ly, he balance
o p essu e a he d op in e ace yields:
𝜌gas𝑈2
gas =𝜌d op𝐷d op
d𝑈de
d𝑡.(6)
The ela i e gas eloci y p o ile in he bounda y laye can be
ep esen ed in he o m
𝑈gas =𝑈pla e𝑓(𝜉) −𝑈dcos 𝛽 , 𝜉=𝑧
𝛿,(7)
whe e 𝑓(𝜉)is a dimensionless unc ion o he simila i y a iable 𝜉, and
𝛿is he hickness o he bounda y laye .
The solu ion o he di e en ial Eq. (6) is ob ained in e ms o he
a iable 𝜉using he ans o ma ion o he a iable d𝑡=𝛿d𝜉∕𝑈d op,𝑛
𝑈de (𝜉) =
𝛿 𝜌gas𝑈2
pla e
𝐷d op𝜌d op𝑈d,𝑛 ∫𝜉∗
𝜉
[𝑓(𝜉) −𝑘]2d𝜉 ,(8)
𝑘=𝑈dcos 𝛽
𝑈pla e
, 𝑈d,𝑛 =𝑈dsin 𝛽 ,(9)
whe e 𝜉∗is he dimensionless a iable a which he ela i e gas eloci y
equals ze o.
The ae odynamic ebound, de ined as he onse o he upwa d
mo emen o he d ople wi hou con ac wi h he wall, is d i en by
he ae odynamic o ces expe ienced by he d ople wi hin he bounda y
laye . These o ces, in u n, cause he d ople o de o m. The d ople
ebound occu s i he de o ma ion eloci y a he wall (𝜉= 0) is
equal o he no mal componen o he d ople impac eloci y 𝑈d,𝑛.
This condi ion can now be w i en wi h he help o (8) in he o m
𝐵=𝐵 ebound whe e
𝐵=
𝛿 𝜌gas𝑈2
pla e
𝐷d op𝜌d op𝑈2
d,𝑛 ∫𝜉∗
0
[𝑓(𝜉) −𝑘]2d𝜉 .(10)
𝐵 ebound being an empi ical cons an o o de o uni y.
I is in e es ing ha he scaling (10) has a o m simila o ha de-
ined in (1), ob ained in Gau hie e al. (2016,2018) o lamina ai low
om di e en conside a ions. In his s udy, he scaling is applied o he
u bulen and lamina lows, gene a ed by he o a ing disk.
3.3. Solu ion o u bulen bounda y laye
The egime o he low a ound a o a ing disk is de e mined by
he ae odynamic Reynolds numbe Reae o. As men ioned ea lie , in his
s udy he low is u bulen o all he expe imen al condi ions. An
app oxima e solu ion o he u bulen bounda y laye (She chuk and
Khala o ,1997) is chosen o he conside a ion o d op de o ma ion.
In his solu ion, he in eg al me hod is combined wi h he i ing o
he exis ing expe imen al da a om Cham and Head (1969), I oh and
Hasegawa (1994), Li ell and Ea on (1994).
The p o ile o he hickness o he bounda y laye is exp essed as
𝛿≈ 0.48𝑟Re−1∕6
ae o(11)
In ou expe imen s, he alue o 𝛿is o he o de o 1 mm.
The azimu hal eloci y componen o he gas low in he labo a o y
e e ence ame is assumed in he o m 𝑢𝜑=𝑓(𝜉)wi h
𝑓(𝜉) ≈ 1 −𝜉1∕9, 𝜉=𝑧
𝛿.(12)
The powe -law app oxima ion was i s p oposed by on Ká mán
(1921) using an analogy wi h he u bulen low in a ound pipe and
on a la pla e. While in he o iginal wo k, he exponen is 1/7, om
a compa ison wi h he expe imen al esul a be e ag eemen is ound
wi h a alue o 1/9, (She chuk,2009). Fo mo e de ailed in o ma ion
on he u bulen bounda y laye app oxima ion, he in e es ed eade is
e e ed o comp ehensi e e iews (Kobayashi,1994;C espo del A co
e al.,2005;She chuk,2009;Lingwood and Hen ik Al edsson,2015;
Al edsson e al.,2023).
Now, he coo dina e 𝜉∗co esponding o he posi ion a which he
ela i e gas eloci y equals ze o is
𝜉∗= (1 −𝑘)9.(13)
Now he exp ession o he in eg al in he igh -hand side o (10)
can be de i ed explici ly and he dimensionless ebound h eshold
pa ame e 𝐵 o u bulen low can be exp essed in he o m
𝐵=
𝛿 𝜌gas𝑈2
pla e(1 −𝑘)11
55𝐷d op𝜌d op𝑈2
d,𝑛
.(14)
Exp ession o he h eshold condi ions o ebound (14) has been
de eloped using he heo y which neglec s comple ely he su ace
ension e ec s. These e ec s can be aken in o accoun by conside ing
he ae odynamic Webe numbe
Weae o=
𝜌gas𝐷d op𝑈2
pla e
𝜎(15)
In Fig. 5 he ou come map o d op impac on o a o a ing disk is
shown in e ms o he 𝐵numbe and he Webe numbe Weae o o
u bulen ai low.
In e na ional Jou nal o Mul iphase Flow 184 (2025) 105113
6
B. S ump e al.
Fig. 5. Ou come map o d op impac on o a o a ing disk in e ms o he dimensionless
pa ame e 𝐵, de ined in (10), and Weae o, de ined in (15). The expe imen al pa ame e s
co espond o he u bulen low in he ai low a ound he disk.
Fig. 6. Dependence o he ebound h eshold pa ame e 𝐵 ebound on he geome ical
pa ame e 𝛿∕𝐷d op o he expe imen al da a om Gau hie e al. (2018) wi h ange o
oil d op diame e s 1.4< 𝐷d op <3.1 mm and om Po a o e al. (1976) wi h he ange
o wa e d op diame e s 0.3< 𝐷d op <4 mm.
I becomes e iden ha 𝐵 ebound is well sui ed o desc ibe he onse
o he ae odynamic ebound egime. Fu he mo e, a sligh dependence
o 𝐵 ebound on Weae obecomes appa en a smalle alues o he Webe
numbe Weae o<10. As expec ed, his dependence is mino a high
alues o Weae o o which
𝐵 ebound ≈ 0.5, o Weae o>10.(16)
I is impo an o no e ha he i ed alue o he pa ame e 𝐵 ebound
is o he o de o uni y. This is an impo an indica o o he alidi y o
he model in which he main physical ac o s a e aken in o accoun .
A his s age he alidi y ange o pa ame e s o he solu ion o he
de o ma ion eloci y (8) can be examined. The model used in his s udy
is alid i he a io o he s eady and uns eady p essu es, exp essed in
(5), sa is ies he condi ion 𝑝d op∕𝑝d op,uns eady ≪1. The uppe bound o
his a io is es ima ed using (8):
𝑝d op
𝑝d op,uns eady
≈𝛿2
𝐷2
d op
𝜌gas𝑈2
pla e
𝜌d op𝑈2
d op [∫1
0
𝑓(𝜉)2d𝜉]2
.(17)
In he example 𝑈pla e= 140 m∕s,𝑈d op = 1 m∕s,𝐷d op = 0.5𝛿,
he es ima ion (17) yields 𝑝d op∕𝑝d op,uns eady ≈ 0.03. The expe imen al
pa ame e s in his s udy sa is y he condi ion 𝑝d op∕𝑝d op,uns eady ≪1;
hus he assump ions in his model a e jus i ied.
3.4. Solu ion o lamina bounda y laye , no mal impac (𝑘= 0)
In he case o a lamina bounda y laye a simila i y solu ion o he
eloci y dis ibu ion has been de eloped by on Ká mán (1921) and
sol ed by Coch an (1934). The bounda y laye hickness, de ined as
he no mal dis ance o he pla e om a poin whe e he ai eloci y has
eached 1% o he pla e eloci y, is exp essed (Schlich ing and Ge s en,
2016) in he o m
𝛿= 5.5√𝜈gas∕𝜔, (18)
whe e 𝜔is he angula pla e eloci y, 𝜈gas is he gas kinema ic iscosi y.
The e olu ion o he ai eloci y can be ob ained by nume ically
sol ing he coupled o dina y di e en ial equa ion sys em om on
Ká mán (1921). Co espondingly, he h ee componen s o he gas
eloci y can be exp essed in e ms o he simila i y a iable 𝜉=𝑧∕𝛿.
The nume ical in eg a ion o he exp ession on he igh -hand side o
(10), using he eloci y p o ile 𝑓(𝜉) o lamina low, yields
𝐵=
0.12𝛿 𝜌gas𝑈2
pla e
𝐷d op𝜌d op𝑈2
d,𝑛
,lamina low, 𝑘= 0.(19)
The dependence o he ebound h eshold pa ame e 𝐵 ebound, com-
pu ed using Eq. (19), on he geome ic pa ame e 𝛿∕𝐷d op, as well as he
alues o 𝐵 ebound, compu ed wi h he help o (14), is shown in Fig. 6
using he expe imen al da a om Gau hie e al. (2018) o silicon oil
d ops (𝜈= 100 mm2/s, 𝜌= 960 kg/m3,𝜎= 21 mN/m) and om Po a o
e al. (1976) o wa e .
Mo eo e , he esul s (Po a o e al.,1976) o 𝐵 ebound in he
u bulen egime a e also in he same o de o magni ude, indica ing
a clea imp o emen o he p esen model, which accoun s o he
eloci y p o ile in he ai bounda y laye .
The sca e o he da a o 𝐵 ebound indica es ha he pa ame e 𝐵
is a he sensi i e o he a ia ions o he impac pa ame e s. We can
iden i y a mono onic bu weak g ow h o 𝐵 ebound o highe alues
o 𝛿∕𝐷d op. Ne e heless, he h eshold alues o 𝐵a e in he ange
0.18 < 𝐵 ebound <0.8.
4. Conclusion
In his s udy, d op impac on o a d y o a ing pla e is expe imen ally
in es iga ed using a high-speed ideo sys em. Va ious ypes o impac
ou comes ha e been obse ed o wa e d ops in he diame e ange
o 80 μm< 𝐷 <500 μm and ela i e angen ial eloci ies o 40 m∕s
< 𝑈𝜑, el <160 m∕s. These ou comes include d op comple e o pa ial
ebound, splash and deposi ion.
The h eshold condi ions o he d op ull ebound a e de e mined
om expe imen s and a e modeled heo e ically. In he model, he
d op de o ma ion eloci y caused by he ae odynamic p essu e in he
nea -wall bounda y laye o he ai low is es ima ed om he balance
o he p essu e a he d op su ace. The co esponding dimensionless
pa ame e 𝐵has been o mula ed, which is exp essed no only in e ms
o he impac pa ame e s and densi ies o liquid and gas bu also on he
en i e eloci y p o ile in he bounda y laye . Since he ine ial e ec s
in he gas low and de o ming d op a e dominan , he dependence
o he h eshold alue o he pa ame e 𝐵 ebound, co esponding o he
ebound h eshold, on he ae odynamic Webe numbe , is a he weak.
The h eshold alues o 𝐵=𝐵 ebound a e a he close o lamina
and u bulen low egimes which indica es ha he main physical
ac o s a e conside ed in he model. Fu he mo e, a weak dependence
o 𝐵 ebound on he geome ical pa ame e 𝛿∕𝐷d op is iden i ied.
In e na ional Jou nal o Mul iphase Flow 184 (2025) 105113
7
B. S ump e al.
CRediT au ho ship con ibu ion s a emen
Bas ian S ump : W i ing – e iew & edi ing, W i ing – o iginal
d a , Visualiza ion, Me hodology, In es iga ion, Fo mal analysis, Da a
cu a ion. Samaneh Abdi Qezeljeh: W i ing – e iew & edi ing, W i -
ing – o iginal d a , Visualiza ion, Me hodology, In es iga ion, Fo mal
analysis, Da a cu a ion. Reda Kamal: W i ing – e iew & edi ing,
W i ing – o iginal d a , Visualiza ion, Me hodology, In es iga ion, Fo -
mal analysis, Da a cu a ion. Fabien Dezi e : Supe ision, Funding
acquisi ion. Alessand o Ma u o: Supe ision, Concep ualiza ion. Ilia
V. Roisman: W i ing – e iew & edi ing, Supe ision, Funding acqui-
si ion, Fo mal analysis, Concep ualiza ion. Jeane e Hussong: W i ing
– e iew & edi ing, Supe ision, Funding acquisi ion, Fo mal analysis,
Concep ualiza ion.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Acknowledgmen s
The wo k o Samaneh Abdi Qezeljeh is pa ially suppo ed by
he join DFG/FWF Collabo a i e Resea ch Cen e CREATOR (DFG:
P ojec -ID 492661287/TRR 361; FWF: 10.55776/F90) a TU Da m-
s ad , TU G az and JKU Linz. Fu he mo e, his esea ch p ojec is
unded by he Ge man Fede al Minis y o Economic A ai s and
Ene gy (BMWI) wi hin he amewo k concep ‘‘LuFo VI’’, subp ojec
‘‘NANNY’’. This p ojec has ecei ed unding om he Eu opean Union’s
Ho izon Eu ope esea ch and inno a ion p og am unde he Ma ie
Skłodowska-Cu ie g an ag eemen No 101072551 (TRACES). The au-
ho s g a e ully acknowledge he aluable p elimina y wo k done by
D .-Ing. Ma k Gloe eld and he assis ance in he expe imen s o Osaid
U Rahman Siddiqui.
Da a a ailabili y
We ha e made he da a and he ideos a ailable a he ollowing
link: h ps://zenodo.o g/doi/10.5281/zenodo.12684793.
Re e ences
Agbaglah, G., Josse and, C., Zaleski, S., 2013. Longi udinal ins abili y o a liquid im.
Phys. Fluids 25 (2), h p://dx.doi.o g/10.1063/1.4789971.
Al edsson, P.H., Ka o, K., Lingwood, R., 2023. Flows o e o a ing disks and
cones. Annu. Re . Fluid Mech. 56, 45–68. h p://dx.doi.o g/10.1146/annu e - luid-
121021-043651.
Almohammadi, H., Ami azli, A., 2017. Unde s anding he d op impac on mo ing
hyd ophilic and hyd ophobic su aces. So Ma e 13 (10), 2040–2053. h p:
//dx.doi.o g/10.1039/c6sm02514e.
C espo del A co, E., Se e, E., Bon oux, P., Launde , B.E., 2005. S abili y, ansi ion and
u bulence in o a ing ca i ies. In: Ins abili y o Flows. WIT P ess, pp. 141–195.
h p://dx.doi.o g/10.2495/1-85312-785-X/05.
B enn, G., T opea, C., 1996. Monodispe se sp ays o a ious pu poses — Thei
p oduc ion and cha ac e is ics. Pa . Pa . Sys . Cha ac . 13 (3), 179–185. h p:
//dx.doi.o g/10.1002/ppsc.19960130303.
Cao, Y., Chen, K., 2010. Helicop e icing. Ae onau . J. 114 (1152), 83–90. h p:
//dx.doi.o g/10.1017/S0001924000003559.
Cao, Y., Tan, W., Wu, Z., 2018. Ai c a icing: An ongoing h ea o a ia ion sa e y.
Ae osp. Sci. Technol. 75, 353–385. h p://dx.doi.o g/10.1016/j.as .2017.12.028.
Cao, Y., Wu, Z., Su, Y., Xu, Z., 2015. Ai c a ligh cha ac e is ics in icing condi ions.
P og. Ae osp. Sci. 74, 62–80. h p://dx.doi.o g/10.1016/j.pae osci.2014.12.001.
Cham, T.S., Head, M.R., 1969. Tu bulen bounda y-laye low on a o a ing disk. J.
Fluid Mech. 37 (1), 129–147. h p://dx.doi.o g/10.1017/S0022112069000450.
Coch an, W.G., 1934. The low due o a o a ing disc. Ma h. P oc. Camb idge Philos.
Soc. 30 (3), 365–375. h p://dx.doi.o g/10.1017/S0305004100012561.
de Rui e , J., Lag aauw, R., an den Ende, D., Mugele, F., 2014. We abili y-independen
bouncing on la su aces media ed by hin ai ilms. Na . Phys. 11 (1), 48–53.
h p://dx.doi.o g/10.1038/nphys3145.
Gau hie , A., Bi d, J.C., Clane , C., Qué é, D., 2016. Ae odynamic Leiden os e -
ec . Phys. Re . Fluids 1 (8), 084002. h p://dx.doi.o g/10.1103/PhysRe Fluids.
1.084002,h ps://ui.adsabs.ha a d.edu/link_ga eway/2016PhR F...1h4002G.
Gau hie , A., Bouillan , A., Clane , C., Qué é, D., 2018. Ae odynamic epellency o
impac ing liquids. Phys. Re . Fluids 3 (5), 054002. h p://dx.doi.o g/10.1103/
PhysRe Fluids.3.054002,h ps://ui.adsabs.ha a d.edu/link_ga eway/2018PhR F...
3e4002G.
Geppe , A.K., 2019. Expe imen al In es iga ion o D ople Wall-Film In e ac ion o
Bina y Sys ems (Ph.D. hesis). Ins i u e o Ae ospace The modynamics, Uni e si y
o S u ga , S u ga , h p://dx.doi.o g/10.18419/opus-10577.
I oh, M., Hasegawa, I., 1994. Tu bulen bounda y laye on a o a ing disk in in ini e
quiescen luid. JSME In . J. Se . B 37 (3), 449–456. h p://dx.doi.o g/10.1299/
jsmeb.37.449.
Josse and, C., Tho oddsen, S.T., 2016. D op impac on a solid su ace. Annu. Re . Fluid
Mech. 48, 365–391. h p://dx.doi.o g/10.1146/annu e - luid-122414-034401.
Kind, R., Po apczuk, M., Feo, A., Golia, C., Shah, A., 1998. Expe imen al and
compu a ional simula ion o in- ligh icing phenomena. P og. Ae osp. Sci. 34 (5–6),
257–345. h p://dx.doi.o g/10.1016/S0376-0421(98)80001-8.
Kobayashi, R., 1994. Re iew: Lamina - o- u bulen ansi ion o h ee-dimensional
bounda y laye s on o a ing bodies. J. Fluids Eng. 116 (2), 200–211. h p://dx.
doi.o g/10.1115/1.2910255.
Lee, J.D., Ha ding, R., Palko, R.L., 1984. Documen a ion o Ice Shapes on he Main
Ro o o a UH-1H Helicop e in Ho e . Technical Repo , NASA, URL h ps:
//api.seman icschola .o g/Co pusID:107149015.
Lim, D.H., Kim, S.C., 2014. The mal pe o mance o oil sp ay cooling sys em o
in-wheel mo o in elec ic ehicles. Appl. The m. Eng. 63 (2), 577–587. h p:
//dx.doi.o g/10.1016/j.appl he maleng.2013.11.057.
Lingwood, R.J., Hen ik Al edsson, P., 2015. Ins abili ies o he on Ká mán bounda y
laye . Appl. Mech. Re . 67 (3), 030803. h p://dx.doi.o g/10.1115/1.4029605.
Li ell, H.S., Ea on, J.K., 1994. Tu bulence cha ac e is ics o he bounda y laye
on a o a ing disk. J. Fluid Mech. 266, 175–207. h p://dx.doi.o g/10.1017/
S0022112094000972.
Liu, C., Xu, Z., Ge ada, D., Li, J., Ge ada, C., Chong, Y.C., Popescu, M., Goss, J.,
S a on, D., Zhang, H., 2019. Expe imen al in es iga ion on oil sp ay cooling wi h
hai pin windings. IEEE T ans. Ind. Elec on. 67 (9), 7343–7353. h p://dx.doi.o g/
10.1109/TIE.2019.2942563.
Mogh ade nejad, S., Jadidi, M., Johnson, Z., S olpe, T., Hanson, J., 2021. D ople impac
dynamics on an aluminum spinning disk. Phys. Fluids 33 (7), h p://dx.doi.o g/
10.1063/5.0050997.
Mo ei a, A., Moi a, A.S., Panao, M.R., 2010. Ad ances and challenges in explaining uel
sp ay impingemen : How much o single d ople impac esea ch is use ul? P og.
Ene gy Combus . Sci. 36, 554–580. h p://dx.doi.o g/10.1016/j.pecs.2010.01.002.
Mundo, C., Somme eld, M., T opea, C., 1995. D ople -wall collisions: Expe imen al
s udies o he de o ma ion and b eakup p ocess. In . J. Mul iph. Flow 21 (2),
151–173. h p://dx.doi.o g/10.1016/0301-9322(94)00069-V.
Po a o , O.A., Naza o , O.I., Igna ’e skaya, L.A., Nikol’skii, A.I., 1976. In e ac ion o
d ops wi h bounda y laye on o a ing su ace. J. Eng. Phys. 31 (6), 1453–1456.
h p://dx.doi.o g/10.1007/BF00860580.
Riboux, G., Go dillo, J.M., 2014. Expe imen s o d ops impac ing a smoo h solid
su ace: a model o he c i ical impac speed o d op splashing. Phys. Re .
Le . 113 (2), 024507. h p://dx.doi.o g/10.1103/PhysRe Le .113.024507,h ps:
//ui.adsabs.ha a d.edu/link_ga eway/2014PhR L.113b4507R.
Roisman, I.V., 2010. On he ins abili y o a ee iscous im. J. Fluid Mech. 661,
206–228. h p://dx.doi.o g/10.1017/S0022112010002910.
Roisman, I.V., Lembach, A., T opea, C., 2015. D op splashing induced by a ge
oughness and po osi y: The size plays no ole. Ad . Colloid In e ace Sci. 222,
615–621. h p://dx.doi.o g/10.1016/j.cis.2015.02.004.
Roisman, I.V., Rioboo, R., T opea, C., 2002. No mal impac o a liquid d op on a d y
su ace: model o sp eading and eceding. P oc. R. Soc. Lond. Se . A Ma h. Phys.
Eng. Sci. 458 (2022), 1411–1430. h p://dx.doi.o g/10.1098/ spa.2001.0923.
Roisman, I.V., T opea, C., 2002. Impac o a d op on o a we ed wall: desc ip ion o
c own o ma ion and p opaga ion. J. Fluid Mech. 472, 373–397. h p://dx.doi.o g/
10.1017/S0022112002002434.
Schlich ing, H., Ge s en, K., 2016. Bounda y-Laye Theo y. Sp inge London, Limi ed,
h p://dx.doi.o g/10.1007/978-3-662-52919-5.
She chuk, I.V., 2009. Con ec i e Hea and Mass T ans e in Ro a ing Disk Sys ems.
Sp inge , p. 239. h p://dx.doi.o g/10.1007/978-3-642-00718-7.
She chuk, I.V., Khala o , A.A., 1997. In eg al me hod o calcula ing he cha ac e is ics
o a u bulen bounda y laye on a o a ing disk: Quad a ic app oxima ion o
he angen o he low swi l angle. Hea T ans e Res. 28 (4–6), 402–413. h p:
//dx.doi.o g/10.1615/Hea T ansRes. 28.i4-6.280.
Sp i les, J.E., 2024. Gas mic o ilms in d ople dynamics: When do d ops bounce? Annu.
Re . Fluid Mech. 56 (1), 91–118. h p://dx.doi.o g/10.1146/annu e - luid-121021-
021121.
S ump , B., Roisman, I.V., Ya in, A.L., T opea, C., 2023. D op impac on o a subs a e
we ed by ano he liquid: co ona de achmen om he wall ilm. J. Fluid Mech.
956, 625. h p://dx.doi.o g/10.1017/j m.2022.1060.
Taylo , G.I., 1959. The dynamics o hin shee s o luid II. Wa es on luid shee s.
P oc. R. Soc. Lond. Se . A Ma h. Phys. Eng. Sci. 253 (1274), 296–312. h p:
//dx.doi.o g/10.1098/ spa.1959.0195.
In e na ional Jou nal o Mul iphase Flow 184 (2025) 105113
8
B. S ump e al.
Theodo sen, T., Regie , A., 1944. Expe imen s on d ag o e ol ing disks, cylinde s,
and s eamline ods a high speeds. NACA Techn. Rep. 793 (3), URL h p://hdl.
handle.ne /2060/20050241738.
T opea, C., Ma engo, M., 1999. The impac o d ops on walls and ilms. Mul iphase
Sci. Technol. 11 (1), 19–36. h p://dx.doi.o g/10.1615/Mul ScienTechn. 11.i1.20.
on Ká mán, T.V., 1921. Übe lamina e und u bulen e Reibung. Z. Angew. Ma h. Mech.
1 (4), 233–252. h p://dx.doi.o g/10.1002/zamm.19210010401,h ps://ui.adsabs.
ha a d.edu/link_ga eway/1921ZaMM....1..233K.
on Ká mán, T., 1946. On Lamina and Tu bulen F ic ion. Na ional Ad iso y
Commi ee on Ae onau ics.
Wang, Y., Bou ouiba, L., 2021. G ow h and b eakup o ligamen s in uns eady
agmen a ion. J. Fluid Mech. 910, A39. h p://dx.doi.o g/10.1017/j m.2020.698.
Xu, L., Zhang, W.W., Nagel, S.R., 2005. D op splashing on a d y smoo h su ace. Phys.
Re . Le . 94 (18), 184505. h p://dx.doi.o g/10.1103/phys e le .94.184505.
Yamazaki, M., Jemco , A., Sakaue, H., 2021. A e iew on he cu en s a us o icing
physics and mi iga ion in a ia ion. Ae ospace 8 (7), 188. h p://dx.doi.o g/10.
3390/ae ospace8070188.
Ya in, A.L., 2006. D op impac dynamics: splashing, sp eading, eceding, bouncing. . ..
Annu. Re . Fluid Mech. 2006 (38), 159–192. h p://dx.doi.o g/10.1146/annu e .
luid.38.050304.092144.
Ya in, A.L., Roisman, I.V., T opea, C., 2017. Collision Phenomena in Liquids and Solids.
Camb idge Uni e si y P ess, h p://dx.doi.o g/10.1017/9781316556580.
Ya in, A.L., Weiss, D.A., 1995. Impac o d ops on solid su aces: sel -simila capilla y
wa es, and splashing as a new ype o kinema ic discon inui y. J. Fluid Mech. 283,
141–173. h p://dx.doi.o g/10.1017/S0022112095002266.
Zhu, J., Tu, C., Lu, T., Luo, Y., Zhang, K., Chen, X., 2021. Beha io o a wa e
d ople impac ing a hin wa e ilm. Exp. Fluids 62, 143. h p://dx.doi.o g/10.
1007/s00348-021-03245-0.
