applied
sciences
A icle
Flow Visualiza ion o Spinning and Nonspinning
Socce Balls Using Compu a ional Fluid Dynamics
Takeshi Asai * , Yasumi Nakanishi , Nakaba Akiyama and Sungchan Hong
Facul y o Heal h and Spo s Sciences, Uni e si y o Tsukuba, Tsukuba 305-8574, Japan;
[email p o ec ed] (Y.N.); akiyama.nakaba. @u. sukuba.ac.jp (N.A.);
[email p o ec ed] (S.H.)
*Co espondence: [email p o ec ed]; Tel./Fax: +81-29-853-2711
Recei ed: 5 June 2020; Accep ed: 28 June 2020; Published: 30 June 2020
Fea u ed Applica ion: This s udy indica es ha he ajec o y o a spinning socce ball is egula
and s able. This esul is expec ed o help s ike s o accu a ely aim a cu led sho o pass a
he ou se o he ball’s cou se and help goalkeepe s o easily judge and quickly eac o he
ball’s cou se.
Abs ac :
Va ious s udies ha e been conduc ed on he ae odynamic cha ac e is ics o nonspinning
and spinning socce balls. Howe e , he o ex s uc u es in he wake o he balls a e almos unknown.
One o he main compu a ional luid dynamics me hods used o he analysis o o ex s uc u es
is he la ice Bol zmann me hod as i acili a es high-p ecision analysis. S udies o elucida e he
dominan o ex s uc u e a e impo an because cu led sho s and passes in ol ing spinning balls
a e equen ly used in ac ual socce games. In his s udy, we iden i y he la ge-scale dominan o ex
s uc u e o a socce ball and in es iga e he s abili y o he s uc u e using he la ice Bol zmann
me hod, wind unnel es s, and ee- ligh expe imen s. One o he dominan o ex s uc u es in he
wake o bo h nonspinning and spinning balls is a la ge-scale coun e - o a ing o ex pai . The side
o ce ac ing on a spinning ball s abilizes when he luc ua ion o he sepa a ion poin s o he ball
is supp essed by he o a ion o he ball. Thus, al hough a spinning socce ball is de lec ed by he
Magnus e ec , i s ajec o y is egula and s able, sugges ing ha a spinning ball can be aimed
accu a ely a he ou se o i s cou se.
Keywo ds: ae odynamics; socce ball; o ex; d ag; li
1. In oduc ion
Thebeha io (e.g., he ajec o y)o spo sballsisknown obesigni ican lya ec edbyae odynamic
cha ac e is ics such as d ag and li [
1
,
2
], and a ious s udies (including wind unnel expe imen s)
ha e been conduc ed o in es iga e he e ec o hese cha ac e is ics [
3
–
9
]. Howe e , ela i ely ew
s udies ha e examined he ai low a ound a ball and he o ex s uc u e and dynamics ha gene a e
hese ae odynamic cha ac e is ics [
10
,
11
], al hough analysis o he o ex s uc u e and dynamics
is e y impo an o elucida e he undamen al ae odynamics o spo s balls. Many s udies ha e
p e iously been conduc ed on he o ex s uc u e o a smoo h sphe e [
12
–
14
], and he esul s indica e
ha a ho seshoe o ex, an al e na ing o ex, o a helical o ex is o med depending on he Reynolds
numbe [
15
–
18
]. Howe e , nume ous aspec s ega ding he o ma ion and ansi ion mechanisms o
o ex s uc u es a e s ill unknown.
Va ious s udies ha e been conduc ed on he ae odynamics o socce balls. A d ag c isis occu s
unde ce ain condi ions du ing he d ag and li measu emen s o nonspinning balls o which
wind unnel expe imen s a e ela i ely easy o se up [
19
]. Wind unnel expe imen s ha e also been
Appl. Sci. 2020,10, 4543; doi:10.3390/app10134543 www.mdpi.com/jou nal/applsci
Appl. Sci. 2020,10, 4543 2 o 14
conduc ed on spinning balls o measu e hei Magnus o ce and ae odynamic cha ac e is ics [
20
].
Fu he mo e, a emp s ha e been made o measu e ce ain ae odynamic cha ac e is ics h ough
ee- ligh expe imen s (wi hou using ball-suppo ing de ices), which p o ide condi ions simila o
he ac ual condi ions encoun e ed by a ball [
21
]. Howe e , mos o hese s udies measu ed he d ag
and li ac ing on he ball h ough wind unnel expe imen s, and e y ew s udies ha e isualized and
analyzed he ai low a ound he ball and he dominan o ex s uc u e ha gene a es he d ag and
li o ces [
22
]. In pa icula , he ai low a ound nonspinning (o low-spinning) and spinning balls in
ee ligh and he associa ed dominan o ex s uc u e ha e no ye been examined [
23
]. Mo eo e ,
cu led sho s and passes in ol ing spinning balls a e equen ly used in ac ual socce ball games, and
hence, s udies o elucida e he dominan o ex s uc u e a e impo an .
Compu a ional luid dynamics (CFD) has been con en ionally used o isualize and analyze he
low a ound blu bodies; in gene al, he o ex s uc u e is simula ed and s eady-s a e analysis isca ied
ou using u bulence models such as he ealizable k–
ε
model [
24
,
25
]. Recen ly, he la ice Bol zmann
me hod has become one o he main CFD me hods o uns eady analysis [
26
–
28
]. The simula ion
esul s ob ained om his me hod a e close o he ac ual phenomenon han hose ob ained om
s eady-s a e analysis and, hence, he me hod acili a es he ealiza ion o high- esolu ion isualiza ion
and high-p ecision analysis.
In his s udy, we in es iga ed he ae odynamic cha ac e is ics and la ge-scale dominan o ex
s uc u e o a socce ball using a combina ion o he la ice Bol zmann me hod, wind unnel es s, and
ee- ligh expe imen s. This combina ion enabled us o analyze and isualize he d ag o ce, li o ce,
and side o ce (Magnus o ce) ac ing on nonspinning and spinning balls in ee ligh as well as he
ai low a ound he balls. Ou esul s indica e ha he dominan o ex s uc u e in he wake o a
spinning socce ball is a la ge-scale coun e - o a ing o ex pai [
29
], which is simila o he o ex
o med a ound a wing ip [
30
], and ha he side o ce becomes s able i he luc ua ion o he sepa a ion
poin s o he ball is supp essed by he o a ion o he ball.
2. Ma e ials and Me hods
2.1. CFD Analysis Using he La ice Bol zmann Me hod
A h ee-dimensional socce ball model was cons uc ed (Figu e 1a) om da a ob ained by scanning
a eal socce ball (B azuca, Adidas) using a h ee-dimensional lase scanne (AICON 3D, B euckmann
GmbH, Ge many). Fo a spinning ball, he low speed a he eloci y inle was se o 28 m/s (Reynolds
numbe (Re) =4.25
×
10
5
), and he spinning a es o he ball we e de ined as 25.1, 50.2, and 75.4 ad/s (4,
8, and 12 ps); he cases co esponding o hese spinning a es we e deno ed as Spin A, Spin B, and Spin
C, espec i ely in his s udy. Fo a nonspinning ball, he low speeds a he eloci y inle we e se o 8.2
m/s (Re =1.25
×
10
5
), 19.0 m/s (Re =2.8
×
10
5
), and 27.0 m/s (Re =4.0
×
10
5
); he cases co esponding
o hese low speeds we e deno ed as Nonspin A, Nonspin B, and Nonspin C, espec i ely. The low
speeds (ball speeds) in his expe imen we e based on hose obse ed in ac ual socce games [
23
].
A Ca esian g id was adop ed o gene a e a spa ial g id wi h dimensions o 20 m
×
20 m
×
40 m
(W
×
H
×
L) comp ising nea ly 500 million cells (Figu e 1b). A sec ional g id scale echnique was used
in his s udy wi h a minimum scale o 1 mm and a maximum scale o 4 mm o he cases in ol ing a
spinning ball. Fo he cases in ol ing a nonspinning ball, we de ined minimum scales o 1.63
×
10
-4
mm a Re =1.25
×
10
5
, 7.13
×
10
-5
mm a Re =2.80
×
10
5
, and 5.02
×
10
-5
mm a Re =4.00
×
10
5
,
and a maximum scale o 4 mm. This g id s uc u e could no ep esen de ailed o ex o ma ions
pe ec ly, bu i was used because o compu a ional esou ce cons ain s [
31
]. The ou le p essu e was
de ined as 1013.25 hPa (i.e., a mosphe ic p essu e). The bounda y wall o he socce ball was assumed
o obey a no-slip condi ion, and he ou e walls (including he g ound su ace) we e de ined as slip
walls. The ime s ep o he calcula ion was 1.018
×
10
-5
s o he cases in ol ing a nonspinning ball
and 2.037
×
10
-4
s o he cases in ol ing a spinning ball. In his s udy, ae odynamic simula ions
we e pe o med using he incomp essible low model o comme cial CFD so wa e (Powe FLOW 5.1,
Appl. Sci. 2020,10, 4543 3 o 14
Exa Co p., USA) based on he la ice Bol zmann me hod [
26
]. The beha io o many-pa icle kine ic
sys ems can be exp essed in e ms o he basic mechanical laws go e ning single-pa icle mo ions
a he molecula scale. The Bol zmann equa ion o mula es he p oblem in e ms o a dis ibu ion
unc ion (x, , ), which is he numbe densi y o molecules a posi ion xand speed a ime [
32
].
The equa ion (in he absence o ex e nal o ces) can be w i en as
D
D =∂ (x, , )
∂ + ∇ (x, , )=C(x, , )(1)
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 o 15
He e, ρ is he densi y o ai (1.2 kg/m3), U is he low eloci y (m/s), and A is he p ojec ed a ea
o he socce ball (gi en by πR2, whe e R is he adius o he socce ball). Cd, Cl, and Cs a e measu ed
in he +X, +Z, and +Y di ec ions, espec i ely.
The a io o he pe iphe al eloci y o he eloci y h ough he ai , Sp, was calcula ed as
=
(8)
whe e ω is he angula eloci y o he socce ball ( ad/s) and R is he adius o he socce ball (0.11 m).
One-way analysis o a iance was used o s a is ically es he a e age o he Cs alues o he
spinning and nonspinning balls. Fas Fou ie ans o m (FFT) analysis was employed o compa e he
equency cha ac e is ics o Cs.
.
Figu e 1. (a) Th ee-dimensional socce ball model ob ained by lase scanning and (b) he Ca esian
g id used o gene a e a spa ial g id (W 20 m × H 20 m × L 40 m) o compu a ional luid dynamics
(CFD) analysis. The low is om he le o he igh .
2.2. Wind Tunnel Tes
A low-speed ci cula ing wind unnel wi h a six-componen balance (maximum wind eloci y =
55 m/s; measu ing sec ion = 1.5 m × 1.5 m; u bulence le el = 0.1%) was employed (Figu e 2) o e i y
he d ag o ce coe icien o he nonspinning ball using CFD. The es s we e conduc ed on a he mally
bonded socce ball wi h six panels (B azuca (size 5), Adidas; o icially app o ed o use in
in e na ional games). The ball was suppo ed a he ea using a s ing ( ix u e) i ed o he six-
componen wind unnel balance. The ball was ixed o he s ing using an adhesi e such ha i could
no o a e. The ball panel o ien a ion in he wind unnel expe imen was he same as ha in he CFD
analysis. The low speeds used in he wind unnel expe imen we e in he ange o 7–35 m/s (Re =
1.06 × 105 o 5.31 × 105) o he nonspinning ball [19].
Figu e 1.
(
a
) Th ee-dimensional socce ball model ob ained by lase scanning and (
b
) he Ca esian
g id used o gene a e a spa ial g id (W 20 m
×
H 20 m
×
L 40 m) o compu a ional luid dynamics
(CFD) analysis. The low is om he le o he igh .
He e, he o al de i a i e on he le -hand side ep esen s he con ec i e mo ion o pa icles,
whe eas he igh -hand side exp esses complex in e molecula in e ac ions (collisions). In eg a ion o
he dis ibu ion unc ion makes i possible o ob ain mac oscopic a iables such as luid densi y, speed,
and p essu e.
The collision ope a o ’s main pu pose is o d i e he eloci y dis ibu ion unc ion owa ds i s
equilib ium dis ibu ion. The Bha naga , G oss and K ook (BGK) collision ope a o [
32
] can hen be
de ined as
C(x, , )=−1
τ[ (x, , )− eq(x, , ) ] (2)
whe e τis he elaxa ion ime o he luid, and eq(x, , )is he equilib ium dis ibu ion unc ion.
To sol e hese equa ions e icien ly, we disc e ized hem on a h ee-dimensional cubic la ice using
he D3Q19 model [
32
]. This model disc e ized he eloci y space in o 19 disc e e speeds. The disc e e
LB equa ion, using a speci ic ini e-di e encing o ime (∆ =1), is w i en as
i(x+ci∆ , +∆ )− i(x, )=Ci(x, )(3)
C(x, )=−1
τ[ i(x, )− ieq(x, ) ] (4)
Appl. Sci. 2020,10, 4543 4 o 14
A olume ic bounda y scheme was chosen as he luid-s uc u e in e ac ion me hod. He e,
he pa icle bounda y condi ion was conduc ed a he su ace i sel (i.e., on he ace s ha make up he
geome y desc ip ion). Each o hese ace s had a se o ex uded pa allelog ams co esponding o he
disc e e eloci y di ec ions.
Fo he cases in ol ing a spinning ball, he bounda y laye was simula ed using a sliding
mesh model [
33
]. Tu bulence was modeled acco ding o he e y la ge Eddy simula ion (VLES)
p inciple [
32
], which di ec ly simula es esol able low scales. Un esol ed scales we e modeled
using he eno maliza ion g oup o m o k–
ε
equa ions wi h p op ie a y ex ensions o achie e VLES
ime-accu a e physics. The la ice in his sol e was composed o oxels, which a e h ee-dimensional
cubic cells. The la ice also included su els, which a e su ace elemen s ha occu in a eas whe e he
su ace o a body in e sec s wi h a luid. Fo he cases in ol ing a nonspinning ball, di ec nume ical
simula ion (DNS) was employed wi hou a u bulen model. The a e age d ag, li , and side o ces
ac ing on he socce ball model we e calcula ed om he uns eady d ag, li , and side o ces o e a
pe iod o 1.0 s ( he calcula ion an om 0.2 s o 1.2 s). The ollowing pa ame e s we e u he calcula ed
om he CFD and expe imen al da a collec ed o e a ange o condi ions: wind eloci y (U); o ce
ac ing in he opposi e di ec ion o he wind (i.e., d ag D); o ce ac ing pe pendicula o he wind
di ec ion (i.e., li L); and o ce ac ing sideways (S) (i.e., Magnus o ce) wi h espec o he on al iew.
The ae odynamic o ces de e mined om CFD and h ough he expe imen s we e con e ed in o he
d ag o ce coe icien (Cd), li o ce coe icien (Cl), and side o ce coe icien (Cs) as ollows:
Cd =D
1
2ρU2A(5)
Cl =L
1
2ρU2A(6)
Cs =S
1
2ρU2A(7)
He e,
ρ
is he densi y o ai (1.2 kg/m
3
), Uis he low eloci y (m/s), and Ais he p ojec ed a ea o
he socce ball (gi en by
π
R
2
, whe e Ris he adius o he socce ball). Cd,Cl, and Cs a e measu ed in
he +X,+Z, and +Ydi ec ions, espec i ely.
The a io o he pe iphe al eloci y o he eloci y h ough he ai , Sp, was calcula ed as
Sp =ωR
U(8)
whe e
ω
is he angula eloci y o he socce ball ( ad/s) and Ris he adius o he socce ball (0.11 m).
One-way analysis o a iance was used o s a is ically es he a e age o he Cs alues o he spinning
and nonspinning balls. Fas Fou ie ans o m (FFT) analysis was employed o compa e he equency
cha ac e is ics o Cs.
2.2. Wind Tunnel Tes
A low-speed ci cula ing wind unnel wi h a six-componen balance (maximum wind eloci y =
55 m/s; measu ing sec ion =1.5 m
×
1.5 m; u bulence le el =0.1%) was employed (Figu e 2) o e i y
he d ag o ce coe icien o he nonspinning ball using CFD. The es s we e conduc ed on a he mally
bonded socce ball wi h six panels (B azuca (size 5), Adidas; o icially app o ed o use in in e na ional
games). The ball was suppo ed a he ea using a s ing ( ix u e) i ed o he six-componen wind
unnel balance. The ball was ixed o he s ing using an adhesi e such ha i could no o a e. The ball
panel o ien a ion in he wind unnel expe imen was he same as ha in he CFD analysis. The low
speeds used in he wind unnel expe imen we e in he ange o 7–35 m/s (Re =1.06
×
10
5
o 5.31
×
10
5
)
o he nonspinning ball [19].
Appl. Sci. 2020,10, 4543 5 o 14
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 o 15
Figu e 2. Se up o wind unnel expe imen .
2.3. F ee-Fligh Tes
The low (in he on al c oss-sec ional plane) behind a socce ball du ing ligh was isualized
using i anium e achlo ide. The socce ball was placed di ec ly in on o a socce goal a a dis ance
o 25 m, and a esea ch pa icipan pe o med a nonspinning kick wi h sligh o a ion (knuckle ball)
as well as a side-spinning cu led kick aimed a he goal. Bo h placemen kicks we e deli e ed a
simila eloci ies o mimic he condi ions in a socce game. Two high-speed ideo came as (Pho on
SA2, Pho on Limi ed) we e se up on one side o and behind he ball ajec o y, and pho og aphs
we e aken a 1000 ps. The ball speed and spin a e we e measu ed using high-speed ideo images.
We de ined a ball wi h a o a ion o less han 1 ps as a nonspinning ball and a ball wi h a o a ion o
mo e han 4 ps as a spinning ball [34]. The expe imen al p ocedu e was as ollows. Each socce ball
was b ush pain ed wi h i anium e achlo ide, placed on a designa ed spo , and kicked owa d he
goal. As he ball mo ed owa d he goal, he ai low a ound i was e ealed by whi e smoke
p oduced by he i anium e achlo ide. Pho og aphs we e aken using a high-speed ideo came a.
Finally, he ball was collec ed and cleaned.
3. Resul s and Discussion
3.1. S eady Cd and Cs Ob ained om CFD
The s eady d ag o ce coe icien (a e age Cd) o a nonspinning socce ball ob ained om CFD
was ~0.39 in he subc i ical egime o Re = 1.25 × 105, ~0.19 in he supe c i ical egime o Re = 2.80 ×
105, and ~0.14 a Re = 4.00 × 105 (Figu e 3). The a e age Cd o he nonspinning ball ob ained in wind
unnel es s was ~0.45 in he subc i ical egime o Re = 1.24 × 105 and ~0.18 in he supe c i ical egime
o Re = 2.82 × 105 ( he c i ical Reynolds numbe was 2.67 × 105 (a e age Cd = 0.15)), which we e close
o he a e age Cd alues ob ained om CFD. We obse ed a peculia spike in he subc i ical egime
owing o he e ec o he i egula su ace panel shape on he ball, which can a ec u bulen
ea achmen . The maximum measu emen e o o he a e age Cd o e i ying he measu emen
epea abili y a he same ball o ien a ion was less han 5% in he subc i ical egime, whe eas hose in
he c i ical and supe c i ical egimes we e lowe . Mo eo e , he maximum measu emen e o o Cd
a di e en ball o ien a ions was also less han 5% in he subc i ical egime, whe eas hose in he
c i ical and supe c i ical egimes we e lowe . I is conside ed ha he en i e su ace oughness o he
ball in his wind unnel es was no signi ican ly di e en om ha o a con en ional 32-panel ball
(Van aggio; he c i ical Reynolds numbe is 2.20–2.50 × 105) as he o al bond leng h o he six-panel
ball used in his wind unnel es is simila o ha o he con en ional ball [35,36].
Ou expe imen al se up could no measu e he a e age Cd and Cs o he spinning ball. The e o e,
he a e age Cd and Cs o he spinning ball ob ained om CFD we e e i ied using he expe imen al
esul s epo ed in a p e ious s udy in which a socce ball wi h 14 panels was used (Teamgeis (size
5); he c i ical Reynolds numbe o he ball was 2.6 × 105) [19]. The a e age Cd o a spinning ball
ob ained om CFD was ~0.28 a spin pa ame e (Sp) o 0.1, ~0.34 a Sp o 0.2, and ~0.36 a Sp o 0.3
(Figu e 4a). The a e age Cd ended o inc ease as Sp inc eased. These alues we e sligh ly la ge han
Figu e 2. Se up o wind unnel expe imen .
2.3. F ee-Fligh Tes
The low (in he on al c oss-sec ional plane) behind a socce ball du ing ligh was isualized
using i anium e achlo ide. The socce ball was placed di ec ly in on o a socce goal a a dis ance
o 25 m, and a esea ch pa icipan pe o med a nonspinning kick wi h sligh o a ion (knuckle ball) as
well as a side-spinning cu led kick aimed a he goal. Bo h placemen kicks we e deli e ed a simila
eloci ies o mimic he condi ions in a socce game. Two high-speed ideo came as (Pho on SA2,
Pho on Limi ed) we e se up on one side o and behind he ball ajec o y, and pho og aphs we e aken
a 1000 ps. The ball speed and spin a e we e measu ed using high-speed ideo images. We de ined a
ball wi h a o a ion o less han 1 ps as a nonspinning ball and a ball wi h a o a ion o mo e han 4
ps as a spinning ball [
34
]. The expe imen al p ocedu e was as ollows. Each socce ball was b ush
pain ed wi h i anium e achlo ide, placed on a designa ed spo , and kicked owa d he goal. As he
ball mo ed owa d he goal, he ai low a ound i was e ealed by whi e smoke p oduced by he
i anium e achlo ide. Pho og aphs we e aken using a high-speed ideo came a. Finally, he ball was
collec ed and cleaned.
3. Resul s and Discussion
3.1. S eady Cd and Cs Ob ained om CFD
The s eady d ag o ce coe icien (a e age Cd) o a nonspinning socce ball ob ained om CFD was
~0.39 in he subc i ical egime o Re =1.25
×
10
5
, ~0.19 in he supe c i ical egime o
Re =2.80 ×105
,
and ~0.14 a Re =4.00
×
10
5
(Figu e 3). The a e age Cd o he nonspinning ball ob ained in wind unnel
es s was ~0.45 in he subc i ical egime o
Re =1.24 ×105
and ~0.18 in he supe c i ical egime o
Re =2.82 ×105
( he c i ical Reynolds numbe was 2.67
×
10
5
(a e age Cd =0.15)), which we e close o
he a e age Cd alues ob ained om CFD. We obse ed a peculia spike in he subc i ical egime owing
o he e ec o he i egula su ace panel shape on he ball, which can a ec u bulen ea achmen .
The maximum measu emen e o o he a e age Cd o e i ying he measu emen epea abili y a
he same ball o ien a ion was less han 5% in he subc i ical egime, whe eas hose in he c i ical and
supe c i ical egimes we e lowe . Mo eo e , he maximum measu emen e o o Cd a di e en ball
o ien a ions was also less han 5% in he subc i ical egime, whe eas hose in he c i ical and supe c i ical
egimes we e lowe . I is conside ed ha he en i e su ace oughness o he ball in his wind unnel es
was no signi ican ly di e en om ha o a con en ional 32-panel ball (Van aggio; he c i ical Reynolds
numbe is 2.20–2.50
×
10
5
) as he o al bond leng h o he six-panel ball used in his wind unnel es is
simila o ha o he con en ional ball [35,36].
Ou expe imen al se up could no measu e he a e age Cd and Cs o he spinning ball. The e o e,
he a e age Cd and Cs o he spinning ball ob ained om CFD we e e i ied using he expe imen al
esul s epo ed in a p e ious s udy in which a socce ball wi h 14 panels was used (Teamgeis (size 5);
he c i ical Reynolds numbe o he ball was 2.6
×
10
5
) [
19
]. The a e age Cd o a spinning ball ob ained
Appl. Sci. 2020,10, 4543 6 o 14
om CFD was ~0.28 a spin pa ame e (Sp) o 0.1, ~0.34 a Sp o 0.2, and ~0.36 a Sp o 0.3 (Figu e 4a).
The a e age Cd ended o inc ease as Sp inc eased. These alues we e sligh ly la ge han he a e age
Cd o he spinning ball ob ained in wind unnel es s [
19
]. The a e age side o ce coe icien (a e age
Cs) o he spinning ball ob ained om CFD was ~0.22 a Sp o 0.1, ~0.28 a Sp o 0.2, and ~0.30 a Sp o
0.3 (Figu e 4b). The a e age Cs also ended o inc ease as Sp inc eased, and he alues we e simila o
he a e age Cs o he spinning ball ob ained in wind unnel es s [19].
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 o 15
he a e age Cd o he spinning ball ob ained in wind unnel es s [19]. The a e age side o ce
coe icien (a e age Cs) o he spinning ball ob ained om CFD was ~0.22 a Sp o 0.1, ~0.28 a Sp o
0.2, and ~0.30 a Sp o 0.3 (Figu e 4b). The a e age Cs also ended o inc ease as Sp inc eased, and he
alues we e simila o he a e age Cs o he spinning ball ob ained in wind unnel es s [19].
Figu e 3. Compa ison o he d ag o ce coe icien s o a nonspinning ball de e mined om CFD
analyses and wind unnel es s ( he e o ba s indica e he s anda d de ia ion o Cd).
Figu e 4. Compa ison o (a) d ag o ce coe icien s and (b) side o ce coe icien s o a spinning ball
de e mined om CFD analyses and wind unnel es s.
The a e age Cd o he nonspinning ball ob ained om CFD in his s udy is simila o ha
ob ained om wind unnel es s in he p esen s udy as well as in p e ious s udies [19,20] and, hence,
i is easonable o conclude ha he CFD calcula ions a e alid. Fu he , he a e age Cs o he spinning
Figu e 3.
Compa ison o he d ag o ce coe icien s o a nonspinning ball de e mined om CFD
analyses and wind unnel es s ( he e o ba s indica e he s anda d de ia ion o Cd).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 o 15
he a e age Cd o he spinning ball ob ained in wind unnel es s [19]. The a e age side o ce
coe icien (a e age Cs) o he spinning ball ob ained om CFD was ~0.22 a Sp o 0.1, ~0.28 a Sp o
0.2, and ~0.30 a Sp o 0.3 (Figu e 4b). The a e age Cs also ended o inc ease as Sp inc eased, and he
alues we e simila o he a e age Cs o he spinning ball ob ained in wind unnel es s [19].
Figu e 3. Compa ison o he d ag o ce coe icien s o a nonspinning ball de e mined om CFD
analyses and wind unnel es s ( he e o ba s indica e he s anda d de ia ion o Cd).
Figu e 4. Compa ison o (a) d ag o ce coe icien s and (b) side o ce coe icien s o a spinning ball
de e mined om CFD analyses and wind unnel es s.
The a e age Cd o he nonspinning ball ob ained om CFD in his s udy is simila o ha
ob ained om wind unnel es s in he p esen s udy as well as in p e ious s udies [19,20] and, hence,
i is easonable o conclude ha he CFD calcula ions a e alid. Fu he , he a e age Cs o he spinning
Figu e 4.
Compa ison o (
a
) d ag o ce coe icien s and (
b
) side o ce coe icien s o a spinning ball
de e mined om CFD analyses and wind unnel es s.
Appl. Sci. 2020,10, 4543 7 o 14
The a e age Cd o he nonspinning ball ob ained om CFD in his s udy is simila o ha ob ained
om wind unnel es s in he p esen s udy as well as in p e ious s udies [
19
,
20
] and, hence, i is
easonable o conclude ha he CFD calcula ions a e alid. Fu he , he a e age Cs o he spinning
ball ob ained om CFD is sligh ly la ge han he alue ob ained in wind unnel es s in his s udy,
bu he ends a e simila in ha he a e age Cs inc eases wi h an inc ease in Sp. Some epo s [
34
,
37
]
ha e claimed ha he a e age Cs o he spinning ball inc eases as Re inc eases, and conside ing
ha he a e age Cs depends on he ball speed, hese calcula ions may be deemed o be wi hin he
pe missible ange. Thus, he CFD esul s o he nonspinning and spinning balls in his s udy a e in
b oad ag eemen wi h he esul s o wind unnel es s and can be conside ed eliable.
3.2. Flow Visualiza ion om CFD
The bounda y laye sepa a ion poin s (line) and p essu e dis ibu ion in he case o Nonspin B o
a nonspinning ball (
ω
=0 ad/s, Re =2.80
×
10
5
) luc ua ed i egula ly along he ime axis, and he
accompanying ball wake s eamlines also exhibi ed a de lec ing endency (Figu e 5a). In addi ion,
a la ge-scale coun e - o a ing o ex pai was e y o en o med in he wake o he ball (Figu e 5b,c), and
he posi ion o he o ex pai changed as he sepa a ion poin s and p essu e dis ibu ion luc ua ed.
The o he cases—Nonspin A and Nonspin C—showed simila ends. I is conside ed ha he
simula ions did no accu a ely model he sepa a ion and ea achmen phenomena, al hough he
sepa a ion poin in he case o he subc i ical egime (Nonspin A) shi ed o he s agna ion poin o
he ball.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 o 15
ball ob ained om CFD is sligh ly la ge han he alue ob ained in wind unnel es s in his s udy,
bu he ends a e simila in ha he a e age Cs inc eases wi h an inc ease in Sp. Some epo s [34,37]
ha e claimed ha he a e age Cs o he spinning ball inc eases as Re inc eases, and conside ing ha
he a e age Cs depends on he ball speed, hese calcula ions may be deemed o be wi hin he
pe missible ange. Thus, he CFD esul s o he nonspinning and spinning balls in his s udy a e in
b oad ag eemen wi h he esul s o wind unnel es s and can be conside ed eliable.
3.2. Flow Visualiza ion om CFD
The bounda y laye sepa a ion poin s (line) and p essu e dis ibu ion in he case o Nonspin B
o a nonspinning ball (ω = 0 ad/s, Re = 2.80 × 105) luc ua ed i egula ly along he ime axis, and he
accompanying ball wake s eamlines also exhibi ed a de lec ing endency (Figu e 5a). In addi ion, a
la ge-scale coun e - o a ing o ex pai was e y o en o med in he wake o he ball (Figu e 5b,c),
and he posi ion o he o ex pai changed as he sepa a ion poin s and p essu e dis ibu ion
luc ua ed. The o he cases—Nonspin A and Nonspin C—showed simila ends. I is conside ed ha
he simula ions did no accu a ely model he sepa a ion and ea achmen phenomena, al hough he
sepa a ion poin in he case o he subc i ical egime (Nonspin A) shi ed o he s agna ion poin o
he ball.
Figu e 5. Example o s a ic p essu e and x- o ici y seen on s eamlines a ound a nonspinning socce
ball ob ained using CFD om (a) side iew, (b) op iew, and (c) back iew.
On he o he hand, he bounda y laye sepa a ion poin s and p essu e dis ibu ion in he case o
Spin B o a spinning ball (ω = 50.2 ad/s, Re = 4.25 × 105) de lec ed unde he in luence o o a ion, and
Figu e 5.
Example o s a ic p essu e and x- o ici y seen on s eamlines a ound a nonspinning socce
ball ob ained using CFD om (a) side iew, (b) op iew, and (c) back iew.
Appl. Sci. 2020,10, 4543 8 o 14
On he o he hand, he bounda y laye sepa a ion poin s and p essu e dis ibu ion in he case o
Spin B o a spinning ball (
ω
=50.2 ad/s, Re =4.25
×
10
5
) de lec ed unde he in luence o o a ion, and
he accompanying ball wake s eamlines also exhibi ed a de lec ing endency (Figu e 6a,b). Simila o
he case o a nonspinning ball, a la ge-scale coun e - o a ing o ex pai was obse ed in he wake,
bu no la ge luc ua ions we e seen, and he pai la gely emained a he same posi ion (Figu e 6c).
The o he cases—Spin A and Spin C—exhibi ed simila ends.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 o 15
he accompanying ball wake s eamlines also exhibi ed a de lec ing endency (Figu e 6a,b). Simila
o he case o a nonspinning ball, a la ge-scale coun e - o a ing o ex pai was obse ed in he wake,
bu no la ge luc ua ions we e seen, and he pai la gely emained a he same posi ion (Figu e 6c).
The o he cases—Spin A and Spin C—exhibi ed simila ends.
Figu e 6. Example o s a ic p essu e and x- o ici y obse ed on s eamlines a ound a spinning socce
ball using CFD om (a) side iew, (b) op iew, and (c) back iew.
The asymme y in he low di ec ion o he sepa a ion poin s (line) in he abo emen ioned
bounda y laye s o he ball a ec ed he p essu e dis ibu ion (Figu e 7a,b) and o ex s uc u e,
esul ing in he side and li o ces ac ing on he ball [23]. In addi ion, i is conside ed ha he la ge-
scale coun e - o a ing o ex pai p oduced aces o de lec ion in he sepa a ion poin s and p essu e
dis ibu ion, and he o ex s uc u e may be simila o ha o a wing ip [38,39].
Figu e 6.
Example o s a ic p essu e and x- o ici y obse ed on s eamlines a ound a spinning socce
ball using CFD om (a) side iew, (b) op iew, and (c) back iew.
The asymme y in he low di ec ion o he sepa a ion poin s (line) in he abo emen ioned
bounda y laye s o he ball a ec ed he p essu e dis ibu ion (Figu e 7a,b) and o ex s uc u e,
esul ing in he side and li o ces ac ing on he ball [
23
]. In addi ion, i is conside ed ha he
la ge-scale coun e - o a ing o ex pai p oduced aces o de lec ion in he sepa a ion poin s and
p essu e dis ibu ion, and he o ex s uc u e may be simila o ha o a wing ip [38,39].
In he nea wake s uc u e o he ball ob ained om he back iew using CFD, a la ge-scale
coun e - o a ing o ex pai was o med in he wake o he nonspinning ball (Nonspin B;
ω
=0 ad/s,
Re =2.80
×
10
5
), and uns able mo emen s such as o a ion a ound he axis o a el, b eakdown [
40
,
41
],
and e- o ma ion [
42
] could be obse ed (Figu e 8a–c). Simila ends we e obse ed in he o he
cases in ol ing a nonspinning ball. In he ee- ligh es , a la ge-scale coun e - o a ing o ex pai
was obse ed in he wake o he nonspinning ball (
ω
=5 ad/s, Re =4.10
×
10
5
) (Figu e 8d– )).
The coun e - o a ing o ex pai exhibi ed luc ua ions wi h highly uns able mo emen s such as
o a ion a ound he axis o a el o b eakdown, simila o he cases in he CFD analysis.
Appl. Sci. 2020,10, 4543 9 o 14
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 o 15
Figu e 7. Example o s a ic p essu e obse ed on (a) nonspinning and (b) spinning socce balls using
CFD (back iew). The spin di ec ion o he spinning socce ball in (b) is an iclockwise om he op
iew.
In he nea wake s uc u e o he ball ob ained om he back iew using CFD, a la ge-scale
coun e - o a ing o ex pai was o med in he wake o he nonspinning ball (Nonspin B; ω = 0 ad/s,
Re = 2.80 × 105), and uns able mo emen s such as o a ion a ound he axis o a el, b eakdown
[40,41], and e- o ma ion [42] could be obse ed (Figu e 8a–c). Simila ends we e obse ed in he
o he cases in ol ing a nonspinning ball. In he ee- ligh es , a la ge-scale coun e - o a ing o ex
pai was obse ed in he wake o he nonspinning ball (ω = 5 ad/s, Re = 4.10 × 105) (Figu e 8d– )). The
coun e - o a ing o ex pai exhibi ed luc ua ions wi h highly uns able mo emen s such as o a ion
a ound he axis o a el o b eakdown, simila o he cases in he CFD analysis.
Figu e 7.
Example o s a ic p essu e obse ed on (
a
) nonspinning and (
b
) spinning socce balls using
CFD (back iew). The spin di ec ion o he spinning socce ball in (
b
) is an iclockwise om he op iew.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 o 15
Figu e 8. Flow isualiza ion o nonspinning socce balls using (a–c) CFD and (d– ) ee- ligh es s,
and ha o spinning socce balls using (g–i) CFD and (j–l) ee- ligh es s. The ball spins wi h a
posi i e o a ion abou he Z-axis. The Re alues o he nonspinning and spinning balls a e 2.80 × 105
and 4.25 × 105 (spin a e = 8 ps), espec i ely. In he CFD esul s, he colo o he su ace o he socce
ball a ies acco ding o he p essu e coe icien , whe eas he colo o he s eamlines a ies acco ding
o he o ici y. The iew is om behind he ball (back iew).
In he case o a spinning ball, he CFD esul s (Spin B; ω = 50.2 ad/s, Re = 4.25 × 105) indica ed a
la ge-scale coun e - o a ing o ex pai in he wake downs eam o he o a ing di ec ion ( igh side
o he back iew in Figu e 8), bu he o ex pai emained ela i ely s able and exhibi ed only mino
luc ua ions (Figu e 8g–i). Simila ends we e obse ed in he o he cases in ol ing a spinning ball.
In he ee- ligh es , a la ge-scale coun e - o a ing o ex pai , simila o ha obse ed in he CFD
analysis, was o med in he wake o he spinning ball (ω = 37.7 ad/s, Re = 3.94 × 105) on he
downs eam side o he o a ing di ec ion (coun e clockwise when iewed om he op). Again, his
o ex pai s uc u e was ela i ely s able and had ela i ely mino luc ua ions (Figu e 8j–l).
Va ious o ex s uc u es con aining a la ge-scale coun e - o a ing o ex ha e been obse ed
in he wake o a smoo h sphe e, such as a ho seshoe o ex, an al e na ing o ex, o a hai pin o ex
Figu e 8.
Flow isualiza ion o nonspinning socce balls using (
a–c
) CFD and (
d–
) ee- ligh es s, and
ha o spinning socce balls using (
g–i
) CFD and (
j–l
) ee- ligh es s. The ball spins wi h a posi i e
o a ion abou he Z-axis. The Re alues o he nonspinning and spinning balls a e 2.80
×
10
5
and
4.25
×
10
5
(spin a e =8 ps), espec i ely. In he CFD esul s, he colo o he su ace o he socce ball
a ies acco ding o he p essu e coe icien , whe eas he colo o he s eamlines a ies acco ding o he
o ici y. The iew is om behind he ball (back iew).
