sus ainabili y
A icle
Nume ical Analysis o Flow A ound a Cylinde in C i ical and
Subc i ical Regime
I an Kološ * , Vladimí a Michalco áand Lenka Lauso á


Ci a ion: Kološ, I.; Michalco á, V.;
Lauso á, L. Nume ical Analysis o
Flow A ound a Cylinde in C i ical
and Subc i ical Regime. Sus ainabili y
2021,13, 2048. h ps://doi.o g/
10.3390/su13042048
Academic Edi o s: Ma c A. Rosen
and JoséAl a ez
Recei ed: 30 No embe 2020
Accep ed: 10 Feb ua y 2021
Published: 14 Feb ua y 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
Depa men o S uc u al Mechanics, Facul y o Ci il Enginee ing, VŠB-Technical Uni e si y o Os a a,
708 00 Os a a, Czech Republic; [email p o ec ed] (V.M.); [email p o ec ed] (L.L.)
*Co espondence: [email p o ec ed]
Abs ac :
Modeling he wind low a ound cylind ical buildings is one o he p oblems wi hin u ban
physics. Despi e he simple geome y o he cylinde , i is an in e es ing physical phenomenon.
Pa ial knowledge o low ield p ope ies can be ound in he li e a u e, bu in e ms o hei use o
p ac ical asks, he da a a e s ill incomple e. The au ho s pe o med a nume ical analysis o he low
a ound he smoo h cylinde in he subc i ical and c i ical egime o Reynolds numbe s in he ange
o Re = 2.3
×
10
3
o 4
×
10
5
. Tu bulen low was sol ed using LES model and he nume ical solu ion
was compa ed wi h a ailable da a om expe imen s o s anda d. Analysis o he mean s eam
eloci y showed he elonga ion o he co e o he wake wi h dec easing Re. The p essu e coe icien
e alua ion showed a big di e ence be ween i s dis ibu ion in he subc i ical and c i ical egime.
In he subc i ical egime, a signi ican inc ease in he minimum alue and a shi o he ex eme close
o he axis o he cylinde is p o en. The esul s o he d ag coe icien con i m a signi ican dec ease
in he ansi ion om subc i ical o c i ical egime, which is indica ed in he ci ed expe imen s.
Keywo ds:
CFD; LES; ci cula cylinde ; subc i ical and c i ical egime; d ag coe icien ; li coe icien ;
p essu e coe icien ; no malized mean s eam eloci y
1. In oduc ion
Ci il enginee ing and a chi ec u al enginee ing a e s ongly associa ed wi h he
apidly de eloping applied scien i ic discipline o u ban physics, which o e s a wide
ange o a eas o in e es [
1
]. Resea che s in es iga e he complex ela ionship be ween
spa ial composi ion and building ypology on he one hand and he mal and clima ic
condi ions wi hin and be ween buildings on he o he hand in [2].
A holis ic iew o he sys em in he building indus y, in e ms o sus ainabili y,
akes in o conside a ion en i onmen al, economic, cul u al, and social issues. The use
o compu a ional luid dynamics (CFD) can signi ican ly help o design a s uc u e in
speci ic si ua ions, e alua e he equi emen s o load-bea ing capaci y and eliabili y, and
e i y i s p ope ies. The common goal o bo h sec o s, u ban and ci il enginee ing, is
o design ecological and ene gy e icien buildings, as well as o adap o he na u al and
cul u al en i onmen .
Flow a ound a cylind ical objec is one o he equen ly sol ed p oblems; he simple
geome y o he cylinde is an in e es ing physical phenomenon. I conce ns a ious ypes
o s uc u es such as cooling owe s, chimneys [
3
], buildings o ci cula shape (Figu e 1), py-
lons o cable ca s, b idge s uc u es [
4
–
6
], o sho e s uc u es, ai -cooled hea exchanges [
7
],
s o age anks, and o he indus ial buildings [
8
] and hei s uc u al componen s. Mo e
de ailed knowledge o he low ield and he e ec s o wind low on objec s inds i s appli-
ca ion no only in ci il enginee ing [
9
], bu also in wind enginee ing [
10
]. P oblems o wind
low can ela e o he layou o buildings and he shapes o he buildings hemsel es [
11
,
12
],
as well as he ype o cladding [3], he shape o balconies [13], and he like.
Sus ainabili y 2021,13, 2048. h ps://doi.o g/10.3390/su13042048 h ps://www.mdpi.com/jou nal/sus ainabili y
Sus ainabili y 2021,13, 2048 2 o 13
Figu e 1. Buildings o ci cula c oss-sec ion: (a) [14], (b) design o a complex o esiden ial buildings [13].
The na u e o he wind low is de ined by he Reynolds numbe (1), which is a
dimensionless pa ame e ep esen ing he a io o ine ia o ce o iscous o ce in a low [
15
].
Re =u·D
υ, (1)
whe e u[m
·
s
−1
] ep esen s he low eloci y,
ν
[m
2·
s
−1
] is he kinema ic iscosi y o he
unning ai , and D[m] is he diame e o he cylinde .
The Reynolds numbe a ec s he alue o he d ag coe icien c
d
. I de ines he deg ee
o d ag o ce ha ac s in a di ec ion ha is opposi e he ela i e low eloci y (ho izon al
di ec ion). This is an impo an quan i y in he dimensioning o s uc u es o ci cula c oss-
sec ion, and i de ines he deg ee o loading o he s uc u e due o wind. The cou se o c
d
depending on Re ob ained om expe imen al measu emen s [16] is shown in Figu e 2.
Figu e 2. Va ia ion o cdand Flow T ansi ions o Single Cylinde Flow [16,17].
I is clea om he g aph ha he e is a signi ican dec ease in c
d
in he egion o
he ansi ion be ween subc i ical and c i ical egimes and ha he esea ch is no ully
cla i ied in his zone. The e is a change in he bounda y laye o he cylinde which
signi ican ly inc eases he complexi y o nume ical simula ions. The lack o expe imen al
da a is associa ed wi h he equen p oblem o achie ing high Re numbe s in he wind
unnel. Al hough pa ial esul s o expe imen s o some o he Re alues o subc i ical
and c i ical egions can be ound in he scien i ic li e a u e, hey do no gi e a su icien ly
de ailed pic u e o he cdin his egion.
Ano he undamen al quan i y used in ci il enginee ing is he p essu e coe icien c
p
,
which de ines he dis ibu ion o he p essu e load on he cylinde [18–20]. The bounda y
laye and he s uc u e o he nea wake behind a cylinde ha e always a ac ed a en ion
o heo e ical easons and consequen ly o p ac ical applica ions [7,21–23].
Sus ainabili y 2021,13, 2048 3 o 13
The objec i e o his wo k is o examine he sui abili y o using LES u bulen model
o he e ec i e calcula ion o u bulen cha ac e is ics o he low a ound a cylind i-
cal objec and o con ibu e o he addi ion o in o ma ion abou he low ield p op-
e ies. The e a e in es iga ed d ag coe icien c
d
, p essu e coe icien c
p
[
18
–
20
], and
eloci y p o ile in he wake behind he cylinde [
7
,
21
–
23
], hese quan i ies a e essen ial
o wind enginee ing. Speci ically, he low a ound he cylinde in his esea ch is done
o
Re = 2.3 ×103 o 4 ×105
, which is an in e al ha includes he subc i ical and c i ical
egion [
16
]. Nume ical simula ions a e ca ied ou in ANSYS Fluen so wa e by using high
pe o mance compu e s o he Na ional Supe compu e Cen e IT4Inno a ions and a e
e i ied on he basis o a ailable expe imen al da a [
24
–
29
]. The abili y o ob ain ele an
and easonably eliable in o ma ion ega ding he cou se o c
d
in he c i ical a ea by he
me hod o nume ical modeling would be o g ea impo ance o u he speci ica ion o
wind load on s uc u es.
2. Me hods
2.1. Task Desc ip ion
Iso he mal low a ound a smoo h cylinde wi h he na u e o he low o eigh
di e en Re numbe s, Re
∈
(2.3
×
10
3
; 4
×
10
5
), is modeled acco ding o Table 1. In all
cases he cylinde diame e D= 0.1 m is iden ical, he Reynolds numbe change is ensu ed
by he change in eloci y o he low. The basic pa ame e s o he calcula ion a e gi en in
Table 2.
Table 1. Selec ed Re and co esponding eloci ies.
Subc i ical Re = 2.3 ×103Re = 4 ×103Re = 2 ×104
u0= 0.35 m·s−1u0= 0.6 m·s−1u0=3m·s−1
C i ical Re = 1 ×105Re = 1.4 ×105Re = 2 ×105Re = 3 ×105Re = 4 ×105
u0= 15 m·s−1u0= 21 m·s−1u0= 30 m·s−1u0= 45 m·s−1u0= 60 m·s−1
Table 2. Basic calcula ion pa ame e s.
Cylinde D= 0.1 m
Flowing medium—ai
densi y (cons an )
kinema ic iscosi y
dynamic iscosi y
ρ= 1.225 kg·m−3
ν= 1.5 ×10−5m2·s−1
µ=ν·ρ= 1.8 ×10−5
kg·(m·s)−1
2.2. Nume ical Model and Bounda y Condi ions
The ask is sol ed in he academic e sion o ANSYS Fluen so wa e ( e sion 2020 R2)
using he nume ical La ge eddy simula ion u bulence model (LES model). The la ge eddy
me hod is a simula ion echnique based on il e ing a low ield in o mac o- and mic o-scale
eddy s uc u es. La ge-scale eddy s uc u es a e simula ed di ec ly. Tu bulen s uc u es
o mic oscales, which a e gene ally iso opic, a e exp essed using subg id-scale models.
In addi ion, hese small o ices con ibu e li le o he momen um ans e (and o hea
ans e in aniso opic asks), he e o e hey a e exp essed by pa ame e iza ion schemes
embedded in he equa ions o la ge o ices.
Fil a ion o he con inui y Equa ion (2) and he momen um Equa ion (3) yields ini ial
ela ions o he ma hema ical desc ip ion o he p esen iso opic p ocess by LES me hod.
Con inui y equa ion:
∂ρ
∂ +∂(ρˆ
ui)
∂xi
=0, (2)
Sus ainabili y 2021,13, 2048 4 o 13
momen um equa ion (Na ie –S okes):
∂(ρˆ
ui)
∂ +∂ρˆ
uiˆ
uj
∂xj
−∂σij
∂xj
+∂ˆ
p
∂xi
=∂τij
∂xj
, (3)
whe e
ρ
[kg
·
m
−3
] is he densi y o he lowing medium, see Table 2, [s] is ime,
ˆ
ui,j
[m
·
s
−1
]
ep esen s he ime a e age o he eloci y componen s,
σij
is s ess enso ,
ˆ
p
is he ime
a e age o he s a ic p essu e,
τij
[Pa] is he enso o he esidual subg id s ess, which
a ises due o he il a ion o la ge o ices.
The le sides in (2) o (3) desc ibe mac o o ex s uc u es ( a iables deno ed by
he canopy), he subg id e m o mic os uc u es is exp essed on he igh side o he
Na ie –S okes Equa ion (3) and is exp essed using subg id-scale models in which subg id
u bulen iscosi y is de ined. Indi idual subg id-scale models di e om each o he in he
desc ip ion o u bulen iscosi y. The p esen ed p oblem is sol ed by wall-modeled la ge
eddy simula ion (WMLES) subg id model [
30
], in which subg id u bulen eddy iscosi y
is calcula ed wi h he use o a hyb id leng h scale
ν =minh(κdw)2,CSmag∆2i·S· {1−exp[−y+/253]}, (4)
whe e
dw
is he wall dis ance,
S
is he s ain a e,
κ=
0.4187, and
CSmag =
0.2 a e cons an s,
and
y+
is he no mal o he wall inne scaling. The LES model is based on a modi ied g id
scale o accoun o he g id aniso opies in wall-modeled lows:
∆=min(max(Cw·dw;Cw·hmax,hwn);hmax), (5)
whe e
hmax
is he maximum edge leng h o a cell,
hwn
is he wall-no mal g id spacing, and
Cw=0.15 is a cons an .
The bounda y condi ions o all simula ions a e gi en in Table 3.
Table 3. Bounda y condi ions.
Bounda y Type o Bounda y Condi ion
en y in o compu ing a ea: eloci y inle
ou pu om compu ing a ea: p essu e ou le
e ical side walls:
symme y: he e is no ic ion on he side walls,
he sides o he compu a ional a ea do no
a ec he longi udinal eloci y, he no mal
eloci y and he low o quan i ies ac oss he
bo de a e ze o
uppe and lowe ho izon al walls: wall: co esponds o wind unnel condi ions
2.3. Meshing
Uns uc u ed e ahed al mesh (Figu e 3) wi h hexahed al p isma ic cells (Figu e 4)
co e ing he bounda y laye was used o nume ical simula ions. Ex e nal dimensions o
he mesh a e 7
×
1.8
×
0.2 m (x
×
y
×
z), which co esponds o mul iples o he cylinde
diame e 70 D
×
18 D
×
2D(Figu e 5). The dimensions o he compu a ional domain we e
chosen o app oxima ely ma ch o he bounda y condi ions in he compa ed expe imen s.
Because he expe imen al da a come om di e en au ho s, di e en coun ies, and di e -
en pe iods, he compu a ional a ea is no a model o a speci ic wind unnel bu ies o
espec he gene al p inciples o he modeling o luids. These a e mainly: gi ing he luid
enough space o be a ec ed as li le as possible by he shea o ces a he wind unnel walls,
and o comply wi h he ecommended maximum o blockage a io ( a io o he windwa d
a ea o he objec o he c oss-sec ion o he unnel) o app oxima ely 5%.
Sus ainabili y 2021,13, 2048 5 o 13
Figu e 3. Uns uc u ed e ahed al mesh— he size o he cells p og essi ely dec eases owa ds he wall.
Figu e 4. Dimensions o cells in he p isma ic laye a he wall o he cylinde o Re ≥1×105.
Figu e 5. Dimensions o he compu a ional domain.
In o al, wo a ian s o he mesh wi h di e en numbe and a e age size o cells
we e used o he eason o he a he big ange be ween minimal and maximal Re and
because o he highe equi emen s o he LES model on he esolu ion o he mesh o high
eloci ies.
The mesh o he lowe eloci ies (u
0
= 0.35–15 m
·
s
−1
) consis ed o 1,138,960 cells.
I has heigh o he i s cell a he wall 1
×
10
−5
m, he hexahed al p isma ic laye has
20 laye s wi h he o al hickness 0.0035 m. P isma ic laye is ollowed by e ahed al cells
ha inc easingly g ow ou wa d om he wall o he cylinde om he size 0.0015 o 0.03 m.
The mesh o he highe eloci ies (u
0
= 21–60 m
·
s
−1
) consis ed o 9,111,680 cells.
These pa ame e s a e di e en compa ed o he p e iously desc ibed mesh: heigh o he
i s cell a he wall 3.5
×
10
−6
m, 40 laye s o he p isma ic laye , he size o e ahed al
cells g ows om he size 0.00075 o 0.015 m.
Bo h g ids mee he condi ion o nea wall modeling y
+≤
1. This is a dimensionless
quan i y, dependen on he ype o low and o he pa ame e s, gi en by he o mula:
y+=ρ·yp·u∗
µ, (6)

Sus ainabili y 2021,13, 2048 6 o 13
whe e
ρ
is densi y o he lowing medium [kg
·
m
−3
], y
p
is dis ance o he i s cell poin
om he wall [m], u* is ic ion eloci y [m·s−1], µis dynamic iscosi y [kg·(m·s)−1].
I ollows om (6) ha , in keeping y
+≤
1, he e is an in e se ela ion be ween he
i s cell size a he wall and he eloci y o he low, which is why he mesh o all Re is no
iden ical. To main ain he condi ion y
+≤
1, i is necessa y o c ea e a e y small i s cell
nea he wall o highe eloci ies.
3. Resul s
3.1. Flow Field Cha ac e is ics
Based on he a ailable expe imen al da a, i can gene ally be assumed ha o he
low a ound he cylinde up o Re = 1.4
×
10
5
, he dis ance o he minimum mean low
eloci y u
x
in he wake inc eases om he axis o he cylinde wi h dec easing Re numbe .
The leng h o he so-called co e o he wake also inc eases wi h dec easing Re (co e o
he wake means an a ea o swi ling wake wi h a nega i e alue o u
x
). This is ob ious
om he illus a i e pic u es o nume ical simula ions in Figu e 6. Di e en eloci y
ield dis ibu ions o wo di e en low egimes: subc i ical, Re = 4
×
10
3
and c i ical,
Re = 1.4 ×105
a e shown in Figu e 6a,b. The oo mean squa e (RMS) o mean s eam
eloci y luc ua ion is shown in Figu e 6c,d o he same low egimes.
Figu e 6.
Veloci y ield o subc i ical and c i ical egimes. Mean s eam eloci y: (
a
)Re=4
×
10
3
;u
0
= 0.6 m
·
s
−1
;
(b) Re = 1.4 ×105
;u
0
=3m
·
s
−1
. RMS o mean s eam eloci y luc ua ions: (
c
)Re=4
×
10
3
;u
0
= 0.6 m
·
s
−1
;
(d) Re = 1.4 ×105;u0=3m·s−1.
Based on he s eamwise eloci y in he wake behind a cylinde axis, he eloci y
ield cha ac e is ics a e e alua ed in his pape . The s eamwise eloci y is de ined as a
dimensionless quan i y, so-called no malized mean s eam eloci y, and is exp essed by he
a io u
x
/u
0
.The dis ance om he cylinde axis is de ined by he a io x/D, whe e
x/D= 0
applies o he cylinde axis. The no malized mean s eam eloci y p o iles in he wake a
he le el o he cylinde axis ob ained o simila Re om physical measu emen s, as well
as om nume ical simula ions in subc i ical and c i ical egimes, a e eco ded in Figu e 7.
Sus ainabili y 2021,13, 2048 7 o 13
Figu e 7.
No malized mean s eam eloci y o selec ed Re; CFD and expe imen al da a (F ohlich [
22
],
Beudan [21], Khashehchi [7], Chao Fu [23], Can well [18]), subc i ical and c i ical egimes.
3.1.1. No malized Mean S eam Veloci y in he Subc i ical Region
Expe imen al da a o smoo h cylinde s in he subc i ical egion a e aken om [
21
]
and [
22
] o Re = 3.9
×
10
3
and om he newe [
7
] o Re = 4
×
10
3
o [
23
] o
Re = 5 ×103
.
Figu e 7shows CFD da a o Re numbe s close o he expe imen s. The minimum alue o
he no malized mean s eam eloci y in all cases is wi hin he ange u
x
/u
0∈
(
−
0,2;
−
0.3)
and i s dis ance o he cylinde axis is x/D = 2. The leng h o he co e o he wake ( he
ansi ion om nega i e o posi i e alues u
x
/u
0
)is in hese cases abou x/D = 2.5. Only he
sou ce [22] p esen s a di e en cou se o ux/u0.
Howe e , he a ea in he immedia e icini y o he cylinde emains unclea (app ox-
ima ely x/D o 1.2). Acco ding o [
23
], he s eam eloci y is also nega i e in his pa ,
while all nume ical simula ions show posi i e u
x
alues. In [
7
], he esul s o
Re = 4 ×103
a e p esen ed up o he dis ance x/D = 1.5 and he alues in he immedia e icini y o he
cylinde a e no p esen ed he e. Howe e , he p o ile da a u
x
/u
0
o he lowe Re numbe s,
which a e gi en in [
7
], p o e he di icul y o he objec i e desc ip ion o he eloci y ield
and indica e he possibili y o a sho ange wi h posi i e alues u
x
in he immedia e
icini y behind he cylinde .
Ano he si ua ion is om he poin o iew o de e mining u
x
a a g ea e dis ance om
he axis o he cylinde and de e mining i s maximum alue. In his case, bo h nume ical
simula ions coincide (Re = 2. 3
×
10
3
and Re = 4
×
10
3
). A he dis ance x/D = 3.5, whe e he
da a om [
23
] end, he no malized mean s eam eloci y alues based on CFD calcula ions
a e in he ange 0.57–0.7, while [
23
] p esen s u
x
/u
0
=0.46 and [
7
] e en u
x
/u
0
=0.40. I can
be said ha all nume ical simula ions o Re = 4
×
10
3
show o he dis ance x/D
≥
3.5 a
be e ag eemen wi h [22], when he no malized mean s eam eloci y is abou he alue
ux/u0=0.7.
3.1.2. No malized Mean S eam Veloci y in he C i ical Region
CFD simula ions in Figu e 7ha e shown a sho ening o he co e o he wake wi h
inc easing Re numbe . The minimum alue o no malized mean s eam eloci y dec eases
o he alue u
x
/u
0
=
−
0.31 and hei dis ance om he axis o he cylinde by assump-
ion is sho ened by inc easing Re up o he dis ance x/D
≈
1. Howe e , o lows wi h
Re = 1.4 ×105 he expe imen al da a [18,22] di e om CFD calcula ions (Figu e 7).
F om he poin o iew o de ining he maximum alue o u
x
a a g ea e dis ance om
he cylinde axis, he nume ical simula ions o he low wi h Re = 1.4
×
10
5
a e compa ed
wi h expe imen al da a a a dis ance app oxima ely x/D = 4.5 a he alue u
x
/u
0
= 0.75.
Sus ainabili y 2021,13, 2048 8 o 13
Fo lows wi h he highes p esen ed Re in CFD, he mean s eam eloci y s abilizes a a
dis ance x/D ≈3 wi h he alue ux/u0= 0.81.
3.2. P essu e Load on he Cylinde Ci cum e ence—P essu e Coe icien cp
The p essu e load on he cylinde ci cum e ence o all p esen ed simula ions was
e alua ed using a dimensionless c
p
coe icien . I is gi en by he a io o he s a ic p essu e
p
i
and he dynamic p essu e p
dyn
ela i e o he e e ence poin . The e e ence poin was
se 0.2 m behind he en ance o he a ea, which is 0.8Din on o he axis o he cylinde ,
when he low ield is no ye a ec ed by he lowing obs acle.
P essu e coe icien o a cons an ai densi y a iso he mal low ρis de ined
cp=pi
pdyn
=pci −p e
1/2·ρ·u2
e
, (7)
whe e p
e
is he s a ic p essu e a he e e ence poin [Pa], p
ci
is he esul ing s a ic p essu e
on he cylinde su ace a he i-poin [Pa], u
e
is he mean s eamwise eloci y a he
e e ence poin [m·s−1].
All da a o he e i ica ion o nume ical simula ions ob ained om expe imen al
esea ch a e shown in Figu e 8. They a e aken mainly om [
19
], whe e he au ho ocuses
on he posi ion o he minimum c
p
(maximum comp essi e load) o Re
∈
(1.3
×
10
2
;
2.1 ×105
). Resul s o hese expe imen s all in o bo h he subc i ical and c i ical egimes.
Fu he esul s om he expe imen s used o compa ison in his a icle all in o he c i ical
egime and hey a e aken om [29] o Re = 1.5 ×105and om [27] o Re = 1.5 ×105.
Figu e 8.
Mean p essu e dis ibu ion—expe imen al da a; subc i ical and c i ical egime (No -
be g [19], Tani [29], James [27]).
One o he o he goals o his wo k is o compa e he p essu e coe icien dis ibu ion
c
p
o di e en Re wi h a ailable expe imen al esul s. Due o he complexi y o desc ibing
he p oblem o highe Re and due o he limi a ion o he numbe o a ailable ele an
expe imen al esul s, he esul ing analyzes o he p essu e coe icien c
p
in his a icle a e
di ided sepa a ely o he subc i ical and c i ical egime.
3.2.1. P essu e Coe icien cp—Subc i ical Region
I can be seen in Figu es 8and 9 ha in he subc i ical egion, he alue o he minimum
o he p essu e coe icien is in he ange o c
p,min ≈
(
−
1.15 o
−
1.25) bo h in he esul s o
he expe imen s and in he esul s o he pe o med simula ions. Ve y sligh di e ences
depending on Re a e seen o c
p
dis ibu ion o Re > 8
×
10
3
. Howe e , he di e ence
Sus ainabili y 2021,13, 2048 9 o 13
be ween he minimum alue o c
p
o Re = 3
×
10
3
and Re = 8
×
10
3
is e iden . The
CFD esul s also co espond o hese esul s, he e is a di e en minimum o c
p
a ound
Re = 3 ×103and Re = 2 ×104.
Figu e 9.
Mean p essu e dis ibu ion, subc i ical egime, CFD and expe imen al da a (No be g [
19
]).
The posi ion o c
p,min
o he p esen ed Re coincides (
ϕmin ≈
70
◦
) and wi h inc easing
Re is shi ing sligh ly nea e o he e ical axis o he cylinde (abou 5
◦
,
ϕmin ≈
75
◦
).
Fo he lowes Re, p esen ed in he subc i ical egion (Re = 3
×
10
3
), he esul s o he
p essu e coe icien o he expe imen and he CFD (Re = 2.3
×
10
3
) a e close and ha e
he alue o c
p,min ≈ −
1.15. All o he expe imen al esul s o highe Re in his egion a e
cp,min ≈ −1.25, which is again in ag eemen wi h he esul s o nume ical simula ions.
On he leewa d side o he cylinde , he esul s o expe imen s and nume ical sim-
ula ions sligh ly di e . The bes ag eemen wi h he expe imen al esul s is shown by
he simula ion o he alue Re = 2.3
×
10
3
(expe imen Re = 3
×
10
3
), dis ibu ions a e
almos iden ical and he alue c
p≈ −
0.93 is on he whole back side o he cylinde in
he ange
ϕ∈(90◦,180◦).
In gene al, no alues o c
p
on he leewa d side exceed he alue
c
p,min ≈ −
1 in nume ical simula ions. Con e sely, he p essu e coe icien eco ded in he
expe imen s on he back o he cylinde (
ϕ
> 110
◦
) showed wi h he inc ease in Re also
inc ease in comp essi e load up o he alue o cp≈ −1.2.
3.2.2. P essu e Coe icien cp—C i ical Region, Re ≥1×105
As men ioned abo e, om he a ailable expe imen al da a, he desc ip ion o he
low ield in he c i ical egion is no ye ully unde s ood. The complexi y o he si u-
a ion is p o ed by he esul s o he mean p essu e dis ibu ion esul s ob ained om
he expe imen al esea ch in Figu es 8and 10, which di e signi ican ly om each o he ,
and he CFD esul s also di e om hem. The e is also an in e es ing compa ison o c
p
wi h s anda d esul s [
31
]. The minimum posi ion o Re = 5
×
10
5
acco ding o [
31
] and
CFD is
ϕmin ≈85◦
, howe e , hey di e in he alue. CFD calcula ions end o o e es i-
ma e he ex eme p essu e on he cylinde , o Re = 2.4
×
10
5
is c
p,min,CFD
=
−
2.6, while
cp,min,s anda d =−2.2
. The posi ion o he c
p
minimum in expe imen s o di e en Re is
app oxima ely
ϕmin ≈
65
◦
, bu he alues o c
p,min
di e signi ican ly om each o he . The
CFD esul s a e close o s anda d on he leewa d side o ϕ> 135◦.