Embedded LES of a turbulent thermal boundary layer over ice roughness

Con en s lis s a ailable a ScienceDi ec
Compu e s and Fluids
jou nal homepage: www.else ie .com/loca e/comp luid
Embedded LES o a u bulen he mal bounda y laye o e ice oughness
Denis So omayo -Zakha o a,∗,1, Ricca do Gaudioso a,b,2, Ma iachia a Gallia a,3
aIns i u e o Fluid Mechanics, TU B aunschweig, He mann-Blenk-S aße 37, B aunschweig, 38108, Niede sachsen, Ge many
bDepa men o Ae ospace Science and Technology, Poli ecnico di Milano, Via La Masa 34, Milano, 20156, I aly
A R T I C L E I N F O
Keywo ds:
La ge eddy simula ion
Bounda y laye
Hea ans e
The mal
Passi e scala
Roughness
Non-homogeneous
Icing
A B S T R A C T
The nume ical p edic ion o ice acc e ion ia icing codes elies on he p ope es ima ion o he hea ans e
coe icien on ough ice geome ies. To unde s and he hea ans e physics a play, di ec nume ical
simula ions (DNS) on ough su aces can be pe o med, al hough his esul s in a e y expensi e op ion i
geome ies ob ained om di e en icing condi ions wan o be analyzed. La ge eddy simula ion (LES) p esen s
i sel as a less expensi e op ion o pe o m such s udies, gi ing insigh in o he physics o u bulence, as
well as opening he possibili y o calib a ion o oughness models. The p esen s udy e i ies and alida es a
se up o pe o m embedded LES (ELES) o a ze o p essu e g adien incomp essible low o e a la pla e wi h
ice oughness hea ed o a cons an wall empe a u e. An expe imen al da abase is used, which p o ides he
geome ies o ough pla es ob ained om unw apped scans o ice shapes gene a ed on a NACA0012 ai oil.
The low-speed low o e he la pla e p esen s a 𝑅𝑒𝐿= 3.85⋅105 and 𝑃 𝑟 = 0.729. The e i ica ion is ca ied ou
by analyzing he e ec s o he mesh esolu ion and he domain span on wall p ope ies such as he skin ic ion
coe icien and S an on numbe . Addi ionally, an analysis o u bulence- ela ed low s a is ics is pe o med o
gua an ee he p ope de elopmen o u bulence. The alida ion shows good ag eemen be ween ELES esul s
and expe imen al da a, especially o he S an on numbe dis ibu ions, showcasing ha his se up can be used
o he s udy o hea ans e on ice oughness.
1. In oduc ion
Icing codes a e CFD ools ha p edic he loca ion and shape o ice
acc e ion on ai c a componen s exposed o icing condi ions. Se e al
icing codes exis such as DICEPS [1], FENSAP-ICE [2], LEWICE [3],
PoliMIce [4] and IGLOO2D [5], which e ec i ely p edic ice shapes
o med by he impingemen o supe cooled wa e d ople s in ime ice
condi ions [6], wi h comple e eezing o impinged wa e d ople s. On
he o he hand, glaze ice condi ions do no a o comple e eezing
igh upon impac due o he lack o low enough ees eam empe -
a u es o due o la ge amoun s o d ople s impinging on he su ace,
esul ing in he o ma ion o hin wa e ilms. The calcula ion o he
a e o ice acc e ion on hese wa e ilms equi es pe o ming a local
mass and he mal balance on he su ace [7], om which accu acy
issues a ise due o he necessi y o p ope ly p edic ing he con ec i e
hea lux in he p esence o ough ice su aces. S udies such as he one
o Hansman [8] no e ha he local a e o ice acc e ion on glaze ice
condi ions is almos p opo ional o he su ace con ec i e hea ans e
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (D. So omayo -Zakha o ).
1Ph.D. Resea ch assis an , Ins i u e o Fluid Mechanics, TU B aunschweig, He mann-Blenk-S aße 37, 38108, B aunschweig, Ge many.
2Ph.D. S uden , Ins i u e o Fluid Mechanics, TU B aunschweig, He mann-Blenk-S aße 37, 38108, B aunschweig, Ge many. Depa men o Ae ospace Science
and Technology, Poli ecnico di Milano, Via La Masa 34, 20156 Milan, I aly.
3Senio scien is , Ins i u e o Fluid Mechanics, TU B aunschweig, He mann-Blenk-S aße 37, 38108, B aunschweig, Ge many.
coe icien 𝐻𝑇 𝐶, meaning ha icing codes need o be able o es ima e
his quan i y wi h ela i ely high accu acy. This can be done h ough
he use o RANS models ha can ake in o accoun he ice oughness
e ec s on he he mo-ae odynamic low ields and su ace p ope ies
such as he 𝐻𝑇 𝐶, al hough hese ice oughness e ec s a e ye no well
unde s ood.
Resea ch on he physics o lows o e ough su aces has led o
expe imen al and nume ical s udies ha look in o co ela ing he s a-
is ical pa ame e s o ough su aces o a oughness- ep esen a i e
quan i y, such as he equi alen sand-g ain oughness heigh 𝑘𝑠, which
is commonly used in oughness s udies [9]. Said s a is ical pa ame-
e s can be he oo -mean-squa e o he su ace luc ua ions 𝑅𝑞, hei
skewness 𝑆𝑘, o an e ec i e slope pa ame e , among o he pa ame e s
ha can be used o cha ac e ize oughness. Co ela ions such as he
ones o Bons [10] and Flack and Schul z [11] ha e been elabo a ed
and calib a ed based on limi ed expe imen al da abases. On he o he
hand, co ela ions such as he one o Fo ooghi e al. [12] a e based
on esul s o di ec nume ical simula ions (DNS) o lows o e ough
h ps://doi.o g/10.1016/j.comp luid.2025.106652
Recei ed 20 Decembe 2024; Recei ed in e ised o m 30 Ma ch 2025; Accep ed 23 Ap il 2025
Compu e s and Fluids 297 (2025) 106652
A ailable online 5 May 2025
0045-7930/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
D. So omayo -Zakha o e al.
su aces. I is no ed ha in con as o expe imen s, DNS p o ides a
deepe insigh in o he physics o u bulence and hea ans e , which
can be use ul in he design o oughness co ela ions as e idenced in
he s udies o Thakka e al. [13],Fo ooghi e al. [14],MacDonald e al.
[15],Yang e al. [16,17], which ha e pe o med DNS on plane channel
con igu a ions ea u ing syn he ic o expe imen al oughness. Plane
channels cons i u e simpli ied domains wi h spanwise and s eamwise
pe iodici y, which esul s in he analysis o a su ace wi h a unique
homogeneous 𝑘𝑠 alue. Ice oughness is inhe en ly non-homogeneous
as shown in se e al s udies [18], i.e., 𝑘𝑠 and s a is ical pa ame e s
a y s eamwise and he su ace is exposed o g owing bounda y lay-
e s. This lack o s eamwise pe iodici y can d as ically inc ease he
equi ed compu a ional esou ces o nume ical s udies. Ca dillo e al.
[19],Yuan and Piomelli [20] ha e pe o med DNS on bounda y laye s
o e homogeneous oughness, while no s udy has been ca ied ou
on non-homogeneous oughness su aces. Since ice oughness p esen s
di e en non-homogeneous dis ibu ions based on icing condi ions and
he geome y o he iced ai c a componen , ca ying a DNS o he
s udy o se e al speci ic ice oughness cases could esul in he usage
o as amoun s o compu a ional esou ces, making he elabo a ion
and calib a ion o oughness co ela ions o icing applica ions e y
cons ained.
La ge eddy simula ion (LES) p esen s an op ion o d as ically educe
he equi ed compu a ional esou ces ela i e o DNS, opening he pos-
sibili y o s udying di e en ice oughness cases. LES has been applied
o lows o e ai oils wi h acc e ed ice, being se e al o hese wo ks
e iewed by S ebbins e al. [21], while mo e ecen s udies a e ound
also in Wong e al. [22],Sheidani e al. [23], and Bo nho e al. [24].
Ne e heless, hese s udies look owa ds he impac o ice acc e ion
o e ae odynamic o ces and a e no mean o calib a ion o oughness
models o he calcula ion o 𝐻𝑇 𝐶 on ice oughness. Rela ed o his las
opic, he s udy o So omayo -Zakha o [25] p esen s a me hodology
o pe o m embedded LES (ELES) on scanned ice shapes ob ained om
icing expe imen s on an ai oil. In his nume ical in es iga ion, a LES
egion was c ea ed a ound he ice ough su ace loca ed a he leading
edge o he ai oil in o de o esol e he u bulen low, while he
es o he low domain was simula ed ia RANS. The expec ed en-
hancemen o 𝐻𝑇 𝐶 due o oughness was ob ained om he nume ical
simula ions, al hough wi hou any expe imen al 𝐻𝑇 𝐶 measu emen s
o alida e he app oach. This p omp ed a high in e es in alida ing
his app oach using an al eady a ailable expe imen al da abase, which
may include 𝐻𝑇 𝐶 measu emen s.
The objec i e o he p esen s udy is o e i y and alida e an ELES
se up o he simula ion o bounda y laye s o e non-homogeneous ice
oughness. The expe imen al da abase o McCa ell e al. [26] is used as
a e e ence o he nume ical simula ions since i p esen s unw apped
scans o ice shapes ob ained on a NACA0012 ai oil, which esul in
la pla es wi h a non-homogeneous ough egion. Addi ionally, he
da abase p esen s measu emen s o skin ic ion 𝑐𝑓 and S an on numbe
𝑆𝑡 (p ope y analogous o 𝐻𝑇 𝐶) pe o med on hese ough pla es,
which a e used o alida ion pu poses. The ELES a e ca ied ou using
he so wa e ANSYS FLUENT 21R2 and a e based on he geome y
and low condi ions o he expe imen s, esul ing in he simula ion
o an incomp essible ze o p essu e g adien (ZPG) low o e a hea ed
ough pla e. The empe a u e is ea ed as a passi e scala , he e o e,
i does no ha e any in luence on he momen um anspo , i.e., no
e ec on he eloci y ield o in luid p ope ies such as densi y o
dynamic iscosi y. The p esen manusc ip is o ganized as ollows: a
de ailed desc ip ion o he nume ical se up o he ELES is p esen ed
in Sec ion 2, ollowed by a desc ip ion o he es ed cases and low
condi ions o he s udy included in Sec ion 3. Resul s ela ed o he
e i ica ion o he se up a e p esen ed in Sec ion 4 which desc ibe he
e ec s o domain size and mesh esolu ion, while a compa ison wi h he
expe imen al measu emen s and alida ion o he se up is p esen ed in
Sec ion 5. The conclusions o he s udy a e p esen ed in Sec ion 6.
Fig. 1. Expe imen ally ob ained ough pla es by McCa ell e al. [26]. Colo ing displays
local heigh s ℎ o oughness elemen s. Geome ies: Le : 113 012.05 (low oughness).
Righ : 113012.04 (high oughness).
Fig. 2. Schema ic o he expe imen al se up used by McCa ell e al. [26] in he hea ed
ough pla e expe imen s a he BSWT.
2. Nume ical se up
2.1. Expe imen al da a
The in es iga ed su aces we e aken om he s udy o McClain
e al. [27], which con ains digi al geome ies o ough pla es based on
he unw apped scans o ice shapes gene a ed on a NACA0012 unde
icing condi ions. Fo he expe imen , hese ough pla es we e scaled up
10 imes and 3D p in ed in bo h plas ic and aluminum. As an example,
he 2 selec ed ough pla es o he p esen nume ical s udy a e shown
in Fig. 1, which p esen a size o 952.5 mm o s eamwise leng h and
203.2 mm o span leng h. The s eamwise non-homogeneous na u e
o ice oughness can be clea ly obse ed, wi h geome y 113 012.05
p esen ing lowe oughness le els han geome y 113012.04.
In he expe imen o McCa ell e al. [26], he ough pla es we e
ins umen ed o pe o m measu emen s o he 𝐻𝑇 𝐶 ia in a ed cam-
e as and su ace hea ing, in addi ion o skin ic ion measu emen s,
using a low eloci y 10 imes slowe o ma ch he Reynolds numbe o
he icing condi ions a e he scaling up p ocedu e. The measu emen s
we e pe o med a he Baylo Uni e si y Subsonic Wind Tunnel (BSWT)
ollowing he se up schema ized in Fig. 2.
The ob ained 𝐻𝑇 𝐶 measu emen s a e p esen ed in he o m o
S an on numbe 𝑆𝑡 s. Reynolds numbe 𝑅𝑒𝑥 in Fig. 3 o he plas ic and
aluminum su aces, being 𝑆𝑡 de ined by Eq. (1), while 𝑅𝑒𝑥 is de ined
by Eq. (2).
𝑆𝑡 =𝐻𝑇 𝐶
𝜌 𝑐𝑃𝑈∞
(1)
𝑅𝑒𝑥=𝜌 𝑈∞𝑥
𝜇(2)
Compu e s and Fluids 297 (2025) 106652
2
D. So omayo -Zakha o e al.
Fig. 3. Measu emen s o 𝑆𝑡 s. 𝑅𝑒𝑥 by McCa ell e al. [26] using geome ies
113012.05 (low oughness) and 113012.04 (high oughness). Le : Plas ic su ace.
Righ : Aluminium su ace.
Fig. 4. Flow domain o he nume ical simula ions.
Table 1
Condi ions and low p ope ies o nume ical simula ions.
𝑈∞𝜌 𝜇 𝐿 𝑅𝑒𝐿
[m/s] [kg/m3] [Pa s] [mm] [–]
6.672 1.124 1.858 ⋅10−5 952.5 3.85 ⋅105
𝑇𝑏𝛥𝑇 𝑐𝑃𝜆 𝑃 𝑟
[K] [K] [J/kg K] [W/m K] [–]
300 10 1007 2.565 ⋅10−2 0.729
He e, 𝜌 is he luid densi y, 𝑈∞ is he ees eam eloci y, 𝑐𝑃 is he
luid-speci ic hea capaci y a cons an p essu e, 𝜇 is he luid dynamic
iscosi y and 𝑥 is he s eamwise posi ion, wi h 𝑥= 0 mm being a
he on o he ough pla e. I is no ed ha he su ace p esen s an
unhea ed egion ex ending up o 𝑥= 44 mm, equi alen o a 𝑅𝑒𝑥=
0.18 ⋅105, implying a delayed o ma ion o he he mal bounda y laye
ela i e o he eloci y bounda y laye . The ob ained da a is compa ed
wi h he heo e ical alues o 𝑆𝑡 o a lamina and u bulen bounda y
laye on a smoo h su ace wi h a cons an hea lux and an unhea ed
su ace leng h 𝜉= 44 mm, being hei de ini ion p esen ed in Eqs. (3)
and (4), espec i ely [28], whe e 𝑃 𝑟 =𝑐𝑃𝜇∕𝜆 is he P and l numbe ,
wi h 𝜆 being he luid he mal conduc i i y. The alues o he luid
p ope ies ep oduce he ai condi ions o he expe imen , lis ed in
Table 1 in Sec ion 3.1.
𝑆𝑡lam,HF = 0.453 𝑅𝑒−1∕2
𝑥𝑃 𝑟−0.66 [1 − (𝜉
𝑥)3∕4]−1∕3
(3)
𝑆𝑡 u b,HF = 0.030 𝑅𝑒−1∕5
𝑥𝑃 𝑟−0.40 [1 − (𝜉
𝑥)9∕10]−1∕9
(4)
Fig. 3 shows ha he expe imen al da a ma ch he lamina co ela-
ion while p esen ing alues abo e he smoo h u bulen one a e he
oughness-induced ansi ion akes place in he egions be ween 𝑅𝑒𝑥=
1⋅105 and 𝑅𝑒𝑥= 2 ⋅105. None heless, in he cases wi h he aluminum
su ace, a sudden inc ease o 𝑆𝑡 can be obse ed a 𝑅𝑒𝑥= 1 ⋅105.
I was discussed wi h he au ho s o he expe imen al wo k ha he
eason behind his was he high conduc i i y ha aluminum p esen s,
con eying hea ho izon ally owa ds he on egions o he ough
pla e. This is no an e ec o an appa en bounda y laye ansi ion, bu
a he he mal conduc ion, dis ega ded in he p esen s udy. The e o e,
se e al di e ences in he expe imen al measu emen s be ween bo h
ypes o su aces a e a ibu ed o his conduc i e e ec being s onge
in he aluminium su ace in con as o he plas ic one.
2.2. Nume ical domain
The low domain o he nume ical simula ions is depic ed in Fig.
4, based on he e e ence expe imen al se up p esen ed in he p e ious
sec ion. The domain heigh 𝛥𝐻 = 304.8 mm is p ese ed om he
expe imen s, while he domain span 𝛥𝑍 is a ied as a pa ame e in he
p esen simula ions, ex ending om 𝑧max =𝛥𝑍∕2 o 𝑧min = −𝛥𝑍∕2.
The domain is di ided in o h ee subdomains: a Lamina subdomain,
a LES subdomain, and a RANS subdomain, wi h in e aces be ween
subdomains a loca ions 𝑥= 101.6 mm (𝑅𝑒𝑥= 0.41 ⋅105) and a
𝑥= 1117.6 mm (𝑅𝑒𝑥= 4.51 ⋅105). An analysis o he e ec o he
loca ion o he in e aces on he simula ion esul s is p esen ed in
Appendix. I can be seen ha he 𝑥 coo dina e is se o be ze o a
he beginning o he ough pla e, which in he expe imen s ex ends
up o 𝑥= 952.5 mm (𝑅𝑒𝑥= 3.85 ⋅105). This measu e is aken as he
e e ence leng h scale 𝐿= 952.5 mm. Ne e heless, he la pla e is
ex ended wi h a smoo h zone om 𝑥= 952.5 up o 𝑥= 4064.0 mm
(𝑅𝑒𝑥= 16.41 ⋅105) o nume ical easons. The ough pla e and i s
ex ension a e no-slip walls se a a cons an wall empe a u e 𝑇𝑤. This
app oach simpli ies he analysis compa ed o applying a p esc ibed
hea lux o he base o he ough pla e, as in he expe imen , which
would in ol e modeling hea conduc i i y in he solid body. Al hough
modeling solid body conduc ion in he simula ion could cap u e he
di e ences exhibi ed by he ma e ials used in he expe imen s in he
lamina egion, his is ou o he scope o he p esen s udy. Mo eo e , i
is expec ed ha conside ing an iso he mal bounda y condi ion ins ead
o a cons an hea lux one may ha e a s ong in luence on he esul s in
egions whe e he bounda y laye is lamina while p esen ing minimum
in luence on he u bulen bounda y laye [28], which de elops on he
ough egion. This is suppo ed by compa ing Eqs. (3) and (4) wi h
hei iso he mal coun e pa , de ined in Eqs. (5) and (6). The 𝑆𝑡 in
he lamina egion di e s by ∼36%, while in he u bulen egion he
di e ence alls down o ∼4%. The iso he mal and cons an hea lux
ends a e compa ed wi h expe imen al da a and nume ical esul s in
Sec ion 5.
𝑆𝑡lam,T = 0.332 𝑅𝑒−1∕2
𝑥𝑃 𝑟−0.66 [1 − (𝜉
𝑥)3∕4]−1∕3
(5)
𝑆𝑡 u b,T = 0.0287 𝑅𝑒−1∕5
𝑥𝑃 𝑟−0.40 [1 − (𝜉
𝑥)9∕10]−1∕9
(6)
A slip and adiaba ic egion be o e he ough pla e is inco po a ed
om 𝑥= −1016 mm up o 𝑥= 0 mm. The op wall o he en i e low
domain is se as a symme y condi ion (slip and adiaba ic wall), as well
as he side walls o he Lamina and he RANS subdomains, while he
side walls o he LES subdomain a e se as spanwise pe iodic in e aces.4
A homogeneous low wi h eloci y 𝑈∞, empe a u e 𝑇∞< 𝑇𝑤, and no
u bulen con en en e s he low domain ia he Inle . Resul s ha e
4Se ing he side walls o he Lamina and RANS subdomains o spanwise
pe iodic in e aces as wi h he LES does no ha e any impac on he nume ical
esul s in he LES subdomain, which co e s he egion o in e es .
Compu e s and Fluids 297 (2025) 106652
3
D. So omayo -Zakha o e al.
Fig. 5. 𝛿lam and 𝛿 u b s. 𝑅𝑒𝑥. Ve ical dashed lines ( ) co espond o he loca ions
o he in e aces.
shown ha u bulence e en ually de elops due o oughness inside he
LES subdomain due o a oughness-induced ansi ion o he bounda y
laye . Since he low p esen s a ZPG, he Ou le is se a ze o ela i e
s a ic p essu e as he Inle .
A bounda y laye hickness 𝛿 is es ima ed ollowing Eqs. (7) and
Eqs. (8) o a lamina and u bulen bounda y laye o e a smoo h
pla e [29], espec i ely. The hickness e olu ion along he pla e leng h
is p esen ed in Fig. 5. The es ima ion deli e s a 𝛿 u b = 31 mm a he
LES-RANS in e ace, implying a 𝛥𝐻∕𝛿 u b ≈ 9.7 a his posi ion, and
indica ing ha he 𝛥𝐻 om he expe imen al se up is su icien o
he simula ions o a oid bounda y laye cons aining. In addi ion, 𝛿 u b
needs o be conside ed when choosing 𝛥𝑍, and his aspec is analyzed
in de ail in Sec ion 3.2.
𝛿lam = 4.91 𝑅𝑒−1∕2
𝑥𝑥(7)
𝛿 u b = 0.38 𝑅𝑒−1∕5
𝑥𝑥(8)
By subdi iding he ough pla e in s eamwise sub- egions o ex en-
sion 𝛥𝑥di = 25.4 mm (𝛥𝑅𝑒𝑥= 0.103 ⋅105), he .m.s. o he spa ial
luc ua ions o he ough geome y 𝑅𝑞 was ob ained a each sub-
egion as desc ibed in So omayo -Zakha o e al. [18]. Summa izing
his me hod, 𝑅𝑞 is compu ed a he 𝑥 cen e posi ion 𝑥𝑝 o a speci ic
sub- egion ia Eq. (9), whe e ℎ is he loca ion in he 𝑦 coo dina e o
he ough su ace and ℎ𝑚 is he mean loca ion in he 𝑦 coo dina e o
he ough su ace in he cu en sub- egion, compu ed ia Eq. (10).
𝑅𝑞(𝑥𝑝) = ⎡⎢⎢⎢⎢⎣
1
𝛥𝑥di 𝛥𝑍
𝑥𝑝+𝛥𝑥di
2
∫
𝑥𝑝−𝛥𝑥di
2
𝑧max
∫
𝑧min (ℎ(𝑥, 𝑧) − ℎ𝑚(𝑥𝑝))2𝑑𝑥 𝑑𝑧⎤⎥⎥⎥⎥⎦
1
2
(9)
ℎ𝑚(𝑥𝑝) = 1
𝛥𝑥di 𝛥𝑍
𝑥𝑝+𝛥𝑥di
2
∫
𝑥𝑝−𝛥𝑥di
2
𝑧max
∫
𝑧min
ℎ(𝑥, 𝑧)𝑑𝑥 𝑑𝑧 (10)
Fig. 6 p esen s he 𝑅𝑞 s. 𝑅𝑒𝑥 dis ibu ions o e he ough pla e
o he in es iga ed su aces. By using 𝑅𝑞 o quan i y oughness le els,
i can be obse ed ha he oughness s a s o sligh ly mani es a a
loca ion o 𝑅𝑒𝑥≈ 0.7⋅105, being his a dis ance o mo e han 20 𝛿lam
om he Lamina -LES in e ace. On he o he hand, oughness almos
anishes a 𝑅𝑒𝑥≈ 4 ⋅105, being his posi ion loca ed a sligh ly mo e
han 4𝛿 u b om he LES-RANS in e ace. Also, su ace 113012.05
displays lowe oughness le els in con as o su ace 113 012.04, ex-
plaining he di e ences ha hea ans e measu emen s p esen in Fig.
3. Indeed, a mo e ough su ace would imply a highe hea ans e
enhancemen in con as o a smoo h su ace.
2.3. Mesh cha ac e is ics
An adap i e ca esian mesh is gene a ed in he LES subdomain, wi h
cubic elemen s ha p og essi ely hal e hei edge size as hey app oach
he bounda y co esponding o he ough pla e, esul ing in di e en
mesh le el esolu ions. The maximum mesh le el s a ing a he op
wall p esen s an edge size in 𝑥, 𝑦, and 𝑧 di ec ions o 𝛥𝑙max = 6.4 mm
Fig. 6. 𝑅𝑞 s. 𝑅𝑒𝑥 o in es iga ed su aces. Ve ical dashed lines ( ) co espond o
he loca ions o he in e aces.
Fig. 7. Mesh s uc u e used o he LES subdomain. The mesh ge s e ined as i
app oaches he ough pla e.
(=𝛥𝑥max =𝛥𝑦max =𝛥𝑧max). A he ough wall, an edge size o 𝛥𝑙 es =
0.4 mm (=𝛥𝑥 es =𝛥𝑦 es =𝛥𝑧 es) was selec ed o he minimum mesh
le el, ac ing as he scale o he su ace disc e iza ion. This esul s in 5
mesh le els, as shown in Fig. 7. Close o he ough pla e, he ca esian
mesh me ges wi h a body- i ed bounda y laye mesh, which p esen s
a i s laye hickness 𝛥𝑦min = 0.02 mm and a geome ical g ow h a io
o 1.15. The numbe o laye s was selec ed by ying o ma ch he las
laye o he bounda y laye mesh o hal o 𝛥𝑙 es.
Addi ionally, while mo ing s eamwise, he ine le els o he ca e-
sian mesh end o co e a bigge po ion o he subdomain, accoun ing
o he bounda y laye g ow h, as shown in Fig. 8, which p esen s
con ou s o ime-a e aged non-dimensional eloci y |𝑢|∕𝑈∞ on an 𝑥-𝑦
plane cu o he LES subdomain o he case wi h he highes oughness
le els (case 2, see Sec ion 3.2). I can be seen ha he bounda y laye
mainly esides in he 4 h and 5 h mesh le els, he ines o he LES
subdomain.
The Lamina and RANS subdomains a e disc e ized wi h hexahed al
elemen s, which g ow geome ically in he 𝑦 di ec ion om a i s laye
hickness 𝛥𝑦min = 0.02 mm a he slip egion, ough pla e and pla e
ex ension un il a 𝛥𝑦 = 6.4 mm, keeping his size un il eaching he op
wall. Bo h domains p esen a spanwise disc e iza ion o 𝛥𝑧 = 6.25 mm.
Also, in he Lamina subdomain, he mesh g ows geome ically s eam-
wise om 𝑥= 0 mm, om a alue o 𝛥𝑥 = 0.8 mm un il eaching
𝛥𝑥 = 6.4 mm a he Lamina -LES in e ace. On he o he di ec ion,
om he in e ace owa ds he inle , i g ows om 𝛥𝑥 = 0.8 mm o
alues o 𝛥𝑥 ≈ 50 mm. In he RANS subdomain, he mesh g ows
geome ically s eamwise om he LES-RANS in e ace s a ing wi h a
alue o 𝛥𝑥 = 6.4 mm owa ds a alue o 𝛥𝑥 ≈ 150 mm a he ou le .
I is no ed ha an analysis o he e ec o he LES-RANS in e ace
on he esul s is p esen ed in Appendix. Bo h he Lamina and RANS
subdomains p esen a ound 105 mesh elemen s, his being 1% o he
amoun o mesh elemen s p esen in he LES subdomain.
The lis ed quan i ies a e scaled in wall uni s ela i e o he case in
analysis, being hese p esen ed in Table 2 in Sec ion 3.2.
Compu e s and Fluids 297 (2025) 106652
4
D. So omayo -Zakha o e al.
Fig. 8. Con ou s o |𝑢|∕𝑈∞, along wi h mesh le el dis ibu ion o he LES subdomain o case 2 (see Sec ion 3.2). Black lines indica e he limi s be ween mesh le els. The ine
mesh le els no mally g ow om he ough pla e while mo ing s eamwise o accoun o he bounda y laye g ow h.
Table 2
Tes cases o nume ical simula ions.
ID Elemen s 𝛥𝑙 es (Le els) 𝛥𝑙+
es 𝛥𝑦+
min 𝑅+
𝑞,max 𝛥𝑍 (𝑆𝐹 )
1 11.0 ⋅1060.4 mm (5) 4–20 0.1–0.5 70 50 mm (24%)
1-Ex . 22.4 ⋅1060.4 mm (5) 4–20 0.1–0.5 70 100 mm (12%)
1-Doub. 22.0 ⋅1060.4 mm (5) 4–20 0.1–0.5 70 100 mm (24%)
1-Coa . 3.3 ⋅1060.8 mm (4) 9–40 0.1–0.5 70 50 mm (24%)
1-Fine 36.6 ⋅1060.2 mm (6) 2–10 0.1–0.5 70 50 mm (24%)
2 11.8 ⋅1060.4 mm (5) 4–30 0.1–0.8 265 50 mm (24%)
1-𝜉11.0 ⋅1060.4 mm (5) 4–20 0.1–0.5 70 50 mm (24%)
2-𝜉11.8 ⋅1060.4 mm (5) 4–30 0.1–0.8 265 50 mm (24%)
2.4. Nume ical schemes
The ELES was pe o med conside ing an incomp essible low wi h
cons an p ope ies, using he so wa e ANSYS FLUENT 21R2. The
in ended physics o be sol ed a e modeled ia he Na ie –S okes equa-
ions p esen ed in Eq. (11) o he momen um anspo and in Eq. (12)
o he he mal anspo , whe e 𝑢𝑖 is he low eloci y in he 𝑖 di ec ion,
𝑝 is he low s a ic p essu e and 𝜃=𝑇𝑤−𝑇 is he low ela i e
empe a u e. This las p ope y is ea ed as a passi e scala , he e o e,
ha ing no in luence on he momen um anspo .
𝜕𝑢𝑖
𝜕𝑡 +𝜕𝑢𝑖𝑢𝑗
𝜕𝑥𝑗
= − 1
𝜌
𝜕𝑝
𝜕𝑥𝑖
+𝜇
𝜌
𝜕2𝑢𝑖
𝜕𝑥2
𝑗
(11)
𝜕𝜃
𝜕𝑡 +𝜕𝑢𝑗𝜃
𝜕𝑥𝑗
=𝜆
𝜌 𝑐𝑃
𝜕2𝜃
𝜕𝑥2
𝑗
(12)
The Lamina subdomain does no p esen any u bulence model,
and, gi en he absence o Inle u bulence con en , he bounda y laye
will emain lamina up o he Lamina -LES in e ace (𝑅𝑒𝑥= 0.41 ⋅105),
sol ing Eqs. (11) and (12) in hei cu en o m. The RANS subdomain
uses he SST k-𝜔 u bulence model [30] he e o e modi ying Eqs. (11)
and (12) in o RANS o m. A cons an u bulen P and l numbe o 𝑃 𝑟𝑡=
𝑐𝑃𝜇𝑡∕𝜆𝑡= 0.85 is used o calcula e he u bulen he mal conduc i i y 𝜆𝑡
om he eddy iscosi y 𝜇𝑡. The LES subdomain uses he WALE subg id-
scale (SGS) model o Nicoud and Duc os [31], he e o e modi ying
Eqs. (11) and (12) in o explici LES o m, allowing he solu ion o he
low ield down o he su ace wi hou any ex a oughness models,
esul ing in a wall- esol ed LES. The s udy o So omayo -Zakha o
[25] ca ies a nume ical scheme analysis and alida ion on canonical
cases, om which a subg id-scale cons an 𝐶𝑤= 0.325 is selec ed,
as well as a SGS P and l numbe 𝑃 𝑟sgs =𝑐𝑃𝜇sgs∕𝜆sgs = 0.85 o
calcula e he SGS he mal conduc i i y 𝜆sgs om he SGS iscosi y 𝜇sgs.
This nume ical schemes analysis es ablishes as well he usage o he
non-i e a i e ime ad ancemen (NITA) sol e wi h 2nd o de implici
ansien scheme [32], p essu e-implici wi h spli ing ope a o s (PISO)
scheme o p essu e- eloci y coupling wi h neighbo co ec ion [32],
a 2nd o de p essu e scheme and a bounded cen al di e ence (BCD)
scheme o con ec i e e ms, wi h a BCD boundedness pa ame e o
Fig. 9. Va ia ion o mean alues o 𝑐𝑓 and 𝑆𝑡 s. 𝑡′ inside he LES subdomain o
case 1 (see Sec ion 3.2). Ve ical dashed lines ( ) co espond o he ini ial ime 𝑡′
𝑙 o
sampling o low s a is ics.
0.7 [33]. Las ly, he leas -squa es cell-based me hod is employed o
he compu a ion o spa ial g adien s.
The ELES simula ion is ini ialized om he esul s o a RANS simu-
la ion using he SST k-𝜔 𝛾-𝑅𝑒𝜃 ansi ion model [34] in he whole low
domain.5 The con e ged RANS solu ion is pe u bed o exci e he ini ial
de elopmen o u bulen con en . This s a egy is based on a syn he ic
u bulence gene a ion echnique p o ided by he sol e [35], which
gene a es a pe u bed eloci y ield ou o s eady-s a e RANS esul s
and i is used o educe he ime needed o he simula ion o each
a s a is ically s able s a e. The ELES simula ions a e compu ed wi h
a ime-s ep 𝛥𝑡 ha gua an ees a maximum CFL <0.5, co esponding
o a non-dimensional ime-s ep 𝛥𝑡′=𝑈∞𝛥𝑡∕𝐿≈ 1.75 ⋅10−5. The
compu a ions we e ca ied wi h a low- ime 𝑡𝐼′=𝑈∞𝑡𝐼∕𝐿≈ 2, which
was long enough un il a e ageable beha io could be obse ed on low
p ope ies, such as on he mean alues o 𝑐𝑓 and 𝑆𝑡 compu ed om
he LES subdomain, as shown in Fig. 9 (ob ained om case 1, see
Sec ion 3.2). Once his s a e was eached, low s a is ics s a ed o be
compu ed du ing an ex a amoun o low- ime 𝑡′
𝑆=𝑈∞𝑡𝑆∕𝐿≈ 2.5.
2.5. Analysis o wall p ope ies
The wall p ope ies a e compu ed om ime-a e aged low p op-
e ies a he ough su ace. These quan i ies a e ob ained h ough he
ope a ion p esen ed in Eq. (13), whe e 𝜓 is he ime-a e age du ing a
pe iod 𝑡𝑆 o an a bi a y p ope y 𝜓.
𝜓(𝑥, 𝑦, 𝑧) = 1
𝑡𝑠
𝑡𝐼+𝑡𝑆
∫
𝑡𝐼
𝜓(𝑥, 𝑦, 𝑧, 𝑡)𝑑𝑡 (13)
The e o e, he ime-a e aged p essu e 𝑝, shea s ess 𝜏, and hea lux
𝑞 a e ex ac ed on he ough su ace and hen applied in Eq. (14) o
5The ansi ion model is exploi ed o ini ialize he ELES wi h an al eady
sol ed lamina bounda y laye up o he posi ion whe e oughness s a s o
appea .
Compu e s and Fluids 297 (2025) 106652
5

D. So omayo -Zakha o e al.
Fig. 10. 𝑅𝑞 s. 𝑅𝑒𝑥 o cases based on 𝛥𝑍. Ve ical dashed lines ( ) co espond o
he loca ions o he in e aces.
compu e 𝜏𝑤 and in Eq. (15) o compu e 𝑞𝑤. He e, 𝐴𝑤 co esponds o
he we ed a ea o he ough su ace in he cu en sub- egion, 𝜏𝑥 is
he 𝑥 di ec ion componen o 𝜏, and 𝜂𝑥 is he 𝑥 di ec ion componen o
he no mal o 𝑑𝐴𝑤.
𝜏𝑤(𝑥𝑝) = 1
𝛥𝑥di 𝛥𝑍 ∫
𝐴𝑤(𝜏𝑥−𝑝 𝜂𝑥)𝑑𝐴𝑤(14)
𝑞𝑤(𝑥𝑝) = 1
𝛥𝑥di 𝛥𝑍 ∫
𝐴𝑤
𝑞 𝑑𝐴𝑤(15)
Finally, 𝑐𝑓 and 𝑆𝑡 a e calcula ed o each sub- egion ia Eqs. (16)
and (17), espec i ely, whe e 𝛥𝑇 =𝑇𝑤−𝑇∞. Following Eq. (1), he
de ini ion o 𝐻𝑇 𝐶 =𝑞𝑤∕𝛥𝑇 can be e ie ed.
𝑐𝑓=2𝜏𝑤
𝜌∞𝑈2
∞
(16)
𝑆𝑡 =𝑞𝑤
𝜌 𝑐𝑃𝑈∞𝛥𝑇 (17)
2.6. Analysis o low s a is ics
The analysis o he low s a is ics is pe o med h ough he ex ac-
ion o bounda y laye p o iles in he 𝑦 di ec ion a loca ions 𝑥𝑝 wi h he
in en ion o ma ching he p o ile wi h ex ac ed wall p ope ies o scal-
ing a each sub- egion. The e o e, he ope a ion ⟨𝜓⟩ is p esen ed o an
a bi a y ime-a e aged p ope y 𝜓, which s ands o plane-a e aging
in he 𝑥 and 𝑧 di ec ions.
⟨𝜓(𝑥𝑝, 𝑦)⟩=1
𝛥𝑥di 𝛥𝑍
𝑥𝑝+𝛥𝑥di
2
∫
𝑥𝑝−𝛥𝑥di
2
𝑧max
∫
𝑧min
𝜓(𝑥, 𝑦, 𝑧)𝑑𝑥 𝑑𝑧 (18)
P ope ies on he bounda y laye p o iles such as he eloci y ⟨𝑢⟩+,
empe a u e ⟨𝜃⟩+, and Reynolds and he mal s esses a e scaled in wall
uni s, indica ed by he supe sc ip (+), by means o he ic ion eloci y
𝑢𝜏 and empe a u e 𝑇𝜏, which a e compu ed a each sub- egion ia Eqs.
(19) and (20), espec i ely.
𝑢𝜏=√𝜏𝑤
𝜌(19)
𝑇𝜏=𝑞𝑤
𝜌𝑐𝑃𝑢𝜏
(20)
The wall-dis ance 𝑦+ is scaled ia Eq. (21), no ing ha ℎ𝑚 is aken
in o accoun o ep esen ha he p o ile begins a he mean oughness
heigh o he sub- egion.
𝑦+=𝜌 𝑢𝜏(𝑦−ℎ𝑚)
𝜇(21)
3. Tes cases
3.1. Flow p ope ies
Based on he da a o McCa ell e al. [26], empe a u es o 𝑇∞= 295
K a he ees eam and 𝑇𝑤= 305 K a he wall we e selec ed o he
cu en s udy. This leads o a bulk empe a u e 𝑇𝑏= 0.5 (𝑇𝑤+𝑇∞) = 300
K, om which he ai p ope ies a e es ima ed and p esen ed in Table
1. Such selec ion o pa ame e s implies a empe a u e di e ence o
𝛥𝑇 = 10 K a he wall. P ope ies as 𝜌, 𝑐𝑃, 𝜇 and 𝜆 a e kep cons an a e
e i ying ha hei change is a ound ±1.7%, ±0.0%, ±1.3% and ±1.5%
o a ±5 K a ia ion o 𝑇𝑏, espec i ely. Addi ionally, a P and l numbe
𝑃 𝑟 = 0.729 is ob ained, as well as a Reynolds numbe based on he
e e ence leng h o 𝑅𝑒𝐿= 3.85 ⋅105 and a Mach numbe o 𝑀𝑎∞= 0.02
jus i ying he incomp essible low assump ion.
3.2. Selec ed cases
The es cases conside ed o he p esen in es iga ion a e summa-
ized in Table 2, whe e hei designa ion (ID) is displayed. Case 1 is
based on he expe imen al geome y 113 012.05, case 2 is based on
113012.04, and hese a e conside ed he e e ence cases. The selec ion
o a small 𝛥𝑍 is encou aged o spa e compu a ional cos s. Ini ially, he
e e ence cases 1 and 2 a e selec ed wi h a 𝛥𝑍 = 50 mm, co e ing a
po ion o he o iginal geome y om 𝑧max = 25 mm o 𝑧min = −25 mm.
Howe e , wo impo an aspec s mus be aken in o conside a ion and
a e subjec ed o analysis.
The i s aspec ela es o he e ec o 𝛥𝑍 o e 𝑅𝑞 dis ibu ions.
Since he o iginal expe imen al geome ies p esen a o al span o
𝛥𝑍 = 203.2 mm (see Sec ion 2.1), he selec ion o a smalle 𝛥𝑍 has
he po en ial o a ec ing low p ope ies i ce ain su ace ea u es
a e emo ed by his educ ion o smoo hed by any o he addi ional
p ocedu e. This emo al o smoo hing can be obse ed by compa ing
he 𝑅𝑞 dis ibu ions o he o al span geome y wi h he one wi h a
smalle 𝛥𝑍. To analyze he e ec ha he selec ion o 𝛥𝑍 has on
he low p ope ies, case 1-Ex . (Ex ended) wi h a 𝛥𝑍 = 100 mm is
es ablished (𝑧max = 50 mm o 𝑧min = −50 mm), e ec i ely p esen ing
wice he span han he e e ence case. Fig. 10 is p esen ed, showing
he e ec o he choice o 𝛥𝑍 on he dis ibu ions o 𝑅𝑞 ac oss 𝑅𝑒𝑥.
I can be seen ha he e e ence cases p esen lowe alues o 𝑅𝑞,
while case 1-Ex . p esen s alues close o he end o he o iginal
geome y. The main eason behind he di e ences be ween 𝑅𝑞 o he
cases and he o iginal geome y is an a i icial p ocedu e pe o med on
he low domain cons uc ion o ensu e ha he side walls o he LES
subdomain would ma ch o pe iodici y: he ough geome y close o
he side walls is smoo hed owa ds alues o he local mean geome ical
heigh in he 𝑦 di ec ion using a sinusoidal unc ion as a il e in he 𝑧
di ec ion. The size o he il e a each side is 𝛥𝑍 il e = 6 mm, which
showed p ope smoo hing and good geome ical beha io . This implies
ha a ac ion o he span o 𝑆𝐹 = 2 𝛥𝑍 il e ∕𝛥𝑍 is a ec ed by he
il e , being 𝑆𝐹 = 24% o 𝛥𝑍 = 50, esul ing in a smoo hing e ec
close o he side walls, and ul ima ely educing 𝑅𝑞. The e ec o he
il e is mi iga ed as a la ge 𝛥𝑍 = 100 mm is chosen, esul ing in a
𝑆𝐹 = 12%. This leads o lesse 𝑅𝑞 decay, which is analyzed in case
1-Ex . On he o he hand, u he educ ion o he span below 𝛥𝑍 = 50
would imply an inc ease on 𝑆𝐹 , which, o example, could each alues
o 48% o a 𝛥𝑍 = 25 mm, ha ing almos hal o he domain il e ed.
I is no ed ha a second eason o he 𝑅𝑞 di e ence could be due o a
s ong educ ion o 𝛥𝑍 below ep esen a i e spa ial wa eleng hs ha
o m he oughness on he su ace. The p e iously selec ed alues o
𝛥𝑍 a e no small enough o his e ec o be p ominen in compa ison
o he smoo hing e ec , al hough bo h a ec he inal 𝑅𝑞 dis ibu ion,
po en ially p esen ing an e ec on low p ope ies such as he 𝐻𝑇 𝐶.
The second aspec ela es o he e ec o 𝛥𝑍 o e u bulence since
small cons aining span alues may no allow he p ope de elopmen
o u bulen con en on he bounda y laye . The e e ence cases p esen
Compu e s and Fluids 297 (2025) 106652
6
D. So omayo -Zakha o e al.
Fig. 11. Cases wi h di e en mesh esolu ion 𝛥𝑙 es. (a) Case 1-Coa . (b) Case 1 ( e e ence). (c) Case 1-Fine.
𝛥𝑍 = 50 mm, which esul s in a 𝛥𝑍∕𝛿 u b ≈ 1.6 a he LES-RANS
in e ace, and a 𝛥𝑍∕𝛿 u b ≈ 2.5 a he middle o he LES subdomain,
whe e oughness is p ominen . This 𝛥𝑍 is expec ed o be su icien o
cap u e he de elopmen o spanwise u bulence s uc u es acco ding
o p e ious nume ical and expe imen al s udies. Fo example, Lund
e al. [36] used a 𝛥𝑍∕𝛿 u b ≈𝜋∕2 a he cen e o a LES ZPG u bulen
bounda y laye , while expe imen s om Tomkins and Ad ian [37]
ound e idence o la ge s uc u es in he loga i hmic laye ex ending up
o 0.6𝛿 u b spanwise. Case 1-Ex . wi h a 𝛥𝑍 = 100 mm (𝛥𝑍∕𝛿 u b ≈ 3.2)
could be used o p o e his aspec by compa ing i o Case 1. Howe e ,
he p e iously s a ed change on 𝑅𝑞 in oduced by he spanwise ex en-
sion could mislead he compa ison o he esul ing low p ope ies and
s a is ics be ween case 1-Ex . and case 1. The e o e, o isola e he e ec
o 𝛥𝑍 on u bulen low s a is ics, case 1-Doub. (Double) is in oduced,
esul ing om he pai ing o wo iden ical case 1 geome ies nex o
each o he spanwise. This implies a 𝛥𝑍 = 100, which p ese es he
same 𝑅𝑞 s. 𝑅𝑒𝑥 dis ibu ion as case 1, allowing he compa ison o
u bulen p ope ies unde domains wi h a di e en span bu wi h he
same oughness e ec s.
In addi ion o he span analysis, a mesh s udy is ca ied ou wi h a
coa se and e ined e sion o case 1, designa ed as case 1-Coa . and
case 1-Fine, espec i ely. The coa sening/ e inemen is con olled by
modi ying he 𝛥𝑙 es o wice/hal i s o iginal alue, esul ing in he
meshes shown in Fig. 11, which show he di e en ca esian mesh
e inemen le els. I was analyzed ha a hese esolu ions, he choice
o 𝛥𝑙 es does no ha e any no able e ec o e 𝑅𝑞 dis ibu ions due o
he e ec o su ace meshing.
Finally, all p e iously s a ed cases conside ha he en i e ough
pla e and i s ex ension a e a 𝑇𝑤. Ne e heless, cases 1-𝜉 and 2-𝜉 a e
in oduced wi h an unhea ed leng h o 𝜉= 44 mm as in he expe imen s
o assess i s e ec on he esul s, ea ing he egion be ween 𝑥= 0 mm
and 𝑥= 44 mm as adiaba ic.
4. Ve i ica ion
4.1. E ec o mesh esolu ion 𝛥𝑙𝑟𝑒𝑠
The in luence o 𝛥𝑙𝑟𝑒𝑠 on he esul s is analyzed by compa ing case 1
wi h case 1-Coa . and case 1-Fine. Fig. 12 p esen s compu ed alues o
𝑐𝑓 and 𝑆𝑡 s. 𝑅𝑒𝑥 ac oss cases. As a e e ence, he 𝑐𝑓 s. 𝑅𝑒𝑥 dis ibu ion
o a lamina (Eq. (22)) and u bulen (Eq. (23)) bounda y laye o e a
smoo h pla e is p esen ed, as well as he 𝑆𝑡 s. 𝑅𝑒𝑥 dis ibu ions o a
lamina (Eq. (5)) and u bulen (Eq. (6)) bounda y laye o e a smoo h
pla e a a cons an empe a u e. I can be seen ha as 𝛥𝑙𝑟𝑒𝑠 is e ined,
he dis ibu ions o 𝑐𝑓 and 𝑆𝑡 end o con e ge, being he g ea es
di e ences be ween mesh le els loca ed a 𝑅𝑒𝑥≈ 1.8⋅105, co esponding
o he loca ion o he highes alues o 𝑅𝑞 (see Fig. 6). All cu es show
ha a lamina bounda y laye is ini ially o med, which ma ches he
heo e ical smoo h wall alues and manages o c oss he Lamina -LES
in e ace (𝑅𝑒𝑥= 0.41 ⋅105). T ansi ion o u bulence is induced by
oughness inside he LES subdomain a e 𝑅𝑒𝑥= 1 ⋅105, esul ing in
𝑐𝑓 and 𝑆𝑡 dis ibu ions wi h alues abo e heo e ical smoo h ones o
u bulen low. As 𝑅𝑞 dec eases app oaching he LES-RANS in e ace,
Fig. 12. In luence o he mesh esolu ion 𝛥𝑙𝑟𝑒𝑠 on 𝑐𝑓 and 𝑆𝑡. Ve ical dashed lines (
) co espond o he loca ions o he in e aces.
Table 3
Va ia ion o 𝑐𝑓 and 𝑆𝑡 wi h mesh e inemen .
ID Mean(𝑐𝑓) Mean(𝑆𝑡)
1-Coa . 0.00717 0.00320
1 0.00737 0.00328
Va .(%) 2.76 2.45
1-Fine 0.00750 0.00334
Va .(%) 1.68 1.71
he u bulen low eco e s i s smoo h s a us, and wall p ope ies ma ch
again he expec ed smoo h dis ibu ion.
𝑐𝑓,lam = 0.664 𝑅𝑒−1∕2
𝑥(22)
𝑐𝑓, u b = 0.0574 𝑅𝑒−1∕5
𝑥(23)
A quan i ica ion o he a ia ion o he wall p ope ies be ween
a e ined le el wi h i s espec i e coa se le el is p esen ed in Table
3, showing he mean alues o 𝑐𝑓 and 𝑆𝑡 compu ed inside he LES
subdomain. The a ia ions o he mean alues dec ease as he mesh
esolu ion is e ined, demons a ing ha he esolu ion o case 1 is
app op ia e o he s udy o he o e all oughness e ec s on he low.
A quali a i e analysis o he u bulen low beha io is ca ied
on be ween loca ions 𝑅𝑒𝑥= 1.8⋅105 and 𝑅𝑒𝑥= 2.0⋅105. Fig. 13
displays con ou s o non-dimensional ins an aneous eloci y |𝑢|∕𝑈∞
and ime-a e aged eloci y |𝑢|∕𝑈∞ a he mid-plane o he low domain
(𝑧= 0 mm). I can be obse ed h ough |𝑢|∕𝑈∞ ha as he mesh
esolu ion is inc eased, mo e de ailed luc ua ing beha io is cap u ed.
Sepa a ion egions a e obse ed a e each oughness peak, and he
highly packed oughness elemen s seem o no allow he low o eco e
a e sepa a ion, a beha io epo ed o d- ype oughness [38–40].
F om |𝑢|∕𝑈∞, i can be seen ha he le sepa a ion bubble is a ec ed
Compu e s and Fluids 297 (2025) 106652
7
D. So omayo -Zakha o e al.
Fig. 13. Con ou s o non-dimensional low eloci y o di e en mesh esolu ions in
he egion o 𝑅𝑒𝑥= 1.8⋅105 o 𝑅𝑒𝑥= 2.0⋅105 and 𝑧= 0 mm. Le column: Ins an aneous
alues |𝑢|∕𝑈∞. Righ column: Time-a e aged alues |𝑢|∕𝑈∞.
Fig. 14. Con ou s o non-dimensional low empe a u e o di e en mesh esolu ions
in he egion o 𝑅𝑒𝑥= 1.8⋅105 o 𝑅𝑒𝑥= 2.0⋅105 and 𝑧= 0 mm. Le column: Ins an aneous
alues 𝜃∕𝜃∞. Righ column: Time-a e aged alues 𝜃∕𝜃∞.
Table 4
Loca ions o he ex ac ion o bounda y laye p o iles.
ID 𝑥 [mm] 𝑅𝑒𝑥⋅10−5 [–]
1 444.5 1.8
2 622.3 2.5
3 800.1 3.2
4 977.9 4.0
5 1079.5 4.4
by he mesh esolu ion be ween case 1-Coa . and case 1, jus i ying he
di e ence o 𝑐𝑓 a 𝑅𝑒𝑥= 1.8⋅105 be ween hese cases. In con as , case
1 and case 1-Fine p esen simila low cha ac e is ics, which esul s in
he lowe di e ences o 𝑐𝑓 be ween hese cases obse ed in Fig. 12 and
in Table 3.
To analyze he empe a u e ields be ween 𝑅𝑒𝑥= 1.8⋅105 and 𝑅𝑒𝑥=
2.0⋅105, Fig. 14 is p esen ed, displaying con ou s o non-dimensional
ins an aneous ela i e empe a u e 𝜃∕𝜃∞ and ime-a e aged ela i e
empe a u e 𝜃∕𝜃∞ (𝜃∞=𝑇𝑤−𝑇∞) a he mid-plane o he low domain
(𝑧= 0 mm). As obse ed o he eloci y ields, an inc ease in mesh
esolu ion manages o cap u e mo e de ailed beha io o spa ial luc-
ua ions o 𝜃∕𝜃∞. S ill, con ou s o 𝜃∕𝜃∞ display a consis en beha io
a di e en mesh esolu ions, indica ing small empe a u e g adien s
inside sepa a ion egions a e oughness elemen s, while he highes
empe a u e g adien s a e obse ed a he oughness peaks.
By pe o ming an analysis o he low s a is ics, he bounda y laye
p o iles in he 𝑦 di ec ion ac oss he low domain a e ex ac ed a he
𝑥 loca ions lis ed in Table 4.
Fig. 15. Mean u bulen p o iles a di e en mesh esolu ions. Dashed lines ( )
indica e he heo e ical smoo h pla e eloci y and empe a u e log-laye s.
P o iles o ⟨𝑢⟩+ and ⟨𝜃⟩+ a e displayed in Fig. 15 o he h ee
conside ed mesh esolu ions, showing no majo e ec s on almos all
he p o iles excep on he 1s one, which p esen s highe alues o
⟨𝑢⟩+ o case 1-Coa . This can be ela ed o he lowe 𝑐𝑓 p edic ion a
𝑅𝑒𝑥= 1.8⋅105 associa ed wi h such case. S ill, he eloci y and he mal
shi s wi h espec o he log-law a e clea ly isible on he 2nd and
3 d p o iles, as e idence o he oughness in luence o e he bounda y
laye , being his shi simila ac oss mesh esolu ions.
The mean Reynolds and he mal s esses a e analyzed in Fig. 16.
He e, only he 2nd and 5 h p o iles a e displayed, loca ed in he
ough zone and smoo h zone, espec i ely, allowing he analysis o
he impac o oughness on he low s a is ics. The no mal s esses
⟨𝑢′2⟩+, ⟨𝑣′2⟩+, ⟨𝑤′2⟩+ and ⟨𝜃′2⟩+ display a simila beha io a di e en
esolu ions o bo h p o iles. Ne e heless, he 5 h p o ile exhibi s
he expec ed aniso opic beha io close o he wall on smoo h su -
aces, whe e ⟨𝑢′2⟩+>⟨𝑤′2⟩+>⟨𝑣′2⟩+, epo ed by se e al s udies as
in Jiménez [39]. The 2nd p o ile exhibi s he endency o iso opy ha
oughness o ces in o he u bulen ield, wi h ⟨𝑢′2⟩+≈⟨𝑤′2⟩+≈⟨𝑣′2⟩+.
Speci ically, he damping o he peak o ⟨𝑢′2⟩ is p ominen due o he
cons aining na u e o he oughness o e he spanwise luc ua ions,
which is associa ed wi h he des uc ion o he bu e laye caused by
he in usion o la ge oughness elemen s, as also s a ed by Yuan and
Piomelli [41]. Simila ends ha e been obse ed in he li e a u e on
sand-g ain oughness by Ca dillo e al. [19] and Yuan and Piomelli
[20]. In he case o ⟨𝜃′2⟩+, a simila damping o he peak alue due o
oughness is obse ed, al hough i s alue inc eases a ound 𝑦∕𝛿𝜃= 0.5.
The shea componen ⟨𝑢′𝑣′⟩+ e eals sensi i i y o he mesh on he 5 h
p o ile, possibly indica ing a sligh ly di e en endency o eco e a e
he ough zone dic a ed by he di e en mesh esolu ions. In pa icula ,
⟨𝑢′𝑣′⟩+ is uni o mly educed ac oss he bounda y laye a he 2nd
p o ile in con as o he 5 h one, mani es ing once again he in luence
o inc eased local ic ion due o oughness. On he o he hand, ⟨𝑣′𝜃′⟩+
shows simila peak alues be ween he 2nd and 5 h p o ile, al hough
wi h a shi ed peak loca ion away om he wall due o oughness
e ec s.
O e all, he mesh esolu ion o case 1 manages o disc e ize he
ele an spa ial wa eleng hs o he ough su ace and cap u e i s in e -
ac ions wi h he low ield, as well as no p esen ing s ong a ia ions
on low p ope ies such as 𝑐𝑓 and 𝑆𝑡 dis ibu ions and u bulence
s a is ics h ough u he mesh e inemen . This implies ha his mesh
le el can be used o u he analysis in he cu en s udy.
4.2. E ec o domain span 𝛥𝑍
The e ec o 𝛥𝑍 is analyzed by compa ing he esul s o case 1 wi h
case 1-Ex . and case 1-Doub. I is wo h eminding ha 𝛥𝑍 can ha e
mainly wo e ec s on he esul s, namely he cons ic ion o u bulen
Compu e s and Fluids 297 (2025) 106652
8
D. So omayo -Zakha o e al.
Fig. 16. Mean Reynolds and he mal s esses a di e en mesh esolu ions. Only he
2nd ( ) and 5 h p o ile ( ) a e displayed. The wall dis ance is scaled wi h espec o
he local bounda y laye hickness, 𝛿99 o 𝛿, and he mal bounda y laye hickness, 𝛿𝜃.
Fig. 17. In luence o he domain span 𝛥𝑍 on 𝑐𝑓 and 𝑆𝑡. Ve ical dashed lines ( )
co espond o he loca ions o he in e aces.
con en due o a small 𝛥𝑍 in ela ion o 𝛿, and a ia ion on he alues
o 𝑐𝑓 o 𝑆𝑡 due o di e en 𝑅𝑞 dis ibu ions. Fig. 17 p esen s he esul s
o 𝑐𝑓 and 𝑆𝑡 s. 𝑅𝑒𝑥 dis ibu ions o case 1, case 1-Doub. and case 1-
Ex ., oge he wi h heo e ical alues desc ibed by Eqs. (22), (23), (5)
and (6). Case 1 and case 1-Doub. do no exhibi any no able di e ences,
implying ha 𝛥𝑍 = 50 mm allows he co ec de elopmen o u bulen
low ields, al hough a de ailed analysis o low s a is ics is necessa y
o ully suppo his s a emen . On he o he hand, case 1 and case 1-
Ex . display di e ences due o hei di e en 𝑅𝑞 dis ibu ions, mo e
p ominen o 𝑐𝑓 in con as o 𝑆𝑡, al hough he ends and o de s
o magni ude emain simila . Maximum peak alues can be obse ed
a ound 𝑅𝑒𝑥= 1.8⋅105 o all cases, wi h a eco e y owa ds a smoo h
beha io a ound 𝑅𝑒𝑥= 4 ⋅105, whe e 𝛥𝑍∕𝛿≈ 2 and s ill inside he LES
subdomain.
Fig. 18. Mean u bulen p o iles o di e en domain spans. Dashed lines ( ) indica e
he heo e ical smoo h pla e eloci y and empe a u e log-laye s.
Fo a deepe analysis, he low s a is ics o case 1, case 1-Doub. and
case 1-Ex . a e conside ed. P o iles o ⟨𝑢⟩+ and ⟨𝜃⟩+ a e p esen ed in Fig.
18. Case 1 and case 1-Doub. p esen simila p o iles. The only sensible
di e ence be ween hese cases and case 1-Ex . is p esen ed by he 1s
p o ile o ⟨𝑢⟩+, co esponding o he zone be o e he i s oughness
peak. This is indeed e lec ed in he di e ence o he 𝑐𝑓 p edic ion a
𝑅𝑒𝑥= 1.8⋅105 showed in Fig. 17, whe e he highe ic ion cap u ed by
inc easing he span is esul ing in a lowe ⟨𝑢⟩+ es ima ion. The be e
ma ch in 𝑆𝑡 be ween he cases is ound h ough he good ag eemen o
he he mal p o iles.
Reynolds and he mal s esses a e depic ed in Fig. 19 o he 2nd
and 5 h p o iles. In gene al, case 1 and case 1-Doub. display simila
alues ac oss he s esses, suppo ing he claim ha 𝛥𝑍 = 50 mm is
la ge enough o sus ain uncons ained u bulence de elopmen . The
di e ences be ween case 1 and case 1-Ex . a e mo e signi ican o
he Reynolds s esses compa ed o he he mal ones. The la ges dis-
c epancies a e obse ed in he quan i ies ⟨𝑢′2⟩+ and ⟨𝑢′𝑣′⟩+, whe e
he p edic ions o case 1 and case 1-Doub. show no able a ia ions.
The mal s esses ma ch ai ly well, al hough p esen ing he same be-
ha io desc ibed in he mesh analysis, whe e oughness induces ⟨𝜃′2⟩+
o p esen educed peak alues nex o he wall bu la ge alues a ound
𝑦∕𝛿𝜃= 0.5, while ⟨𝑣′𝜃′⟩+ is shi ed away om he wall. Once again,
hese insigh s jus i y he use o he nume ical se up used o case 1 o
u he analysis and o alida ion.
5. Valida ion
5.1. Compa ison wi h expe imen al da a o 𝑐𝑓
Fig. 20 p esen s a compa ison be ween he 𝑐𝑓 s. 𝑅𝑒𝑥 dis ibu ions
ob ained om case 1 and case 1-Ex . wi h expe imen ally measu ed
𝑐𝑓 alues,6 as well as he heo e ical cu es desc ibed by Eqs. (22)
and (23). Quali a i e esemblance can be obse ed be ween he ELES
esul s, whe e he wo peaks be o e and a e 𝑅𝑒𝑥= 2⋅105 a e cap u ed
by he simula ions bu o e es ima ing hei magni udes. Ne e heless,
case 1-Ex . seems o app oach expe imen al esul s in a be e way,
being jus i ied by he be e desc ip ion o 𝑅𝑞 dis ibu ions. The e o e, i
can be s a ed ha a main sou ce o inaccu acy o nume ical esul s wi h
espec o he expe imen al da a in ol es he usage o a 𝛥𝑍 ha does
no co e he en i e expe imen al span o 𝛥𝑍 = 203.2 mm, al hough
his esul would emain alid espec o i s own opog aphy and 𝑅𝑞
dis ibu ion.
6In he esul s igu e, he expe imen al da a o 𝑐𝑓 was shi ed in he 𝑥
di ec ion an amoun o −50.8 mm (2 in), based on discussions wi h he au ho s
o he expe imen al s udy, who indica ed ha such a shi could exis e en
hough i was no eco ded in he pape . Indeed, his adjus men enhanced
he quali a i e alignmen wi h he nume ical esul s o bo h analyzed cases.
Compu e s and Fluids 297 (2025) 106652
9