A note on the thrust of airfoils

J. Fluid Mech. (2025), ol. 1012, A6, doi:10.1017/j m.2025.10177
A no e on he h us o ai oils
José M. Go dillo
Á ea de Mecánica de Fluidos, Depa amen o de Ingenie ía Ae oespacial y Mecánica de Fluidos,
Uni e sidad de Se illa, A enida de los Descub imien os s/n 41092, Se illa, Spain
Co esponding au ho : José M. Go dillo, [email p o ec ed]
(Recei ed 4 Oc obe 2024; e ised 1 Ap il 2025; accep ed 22 Ap il 2025)
He e, we show ha he h us o ce o oscilla ing ai oils calcula ed wi hin he linea ised
po en ial low app oach by means o he o ex impulse heo y coincides wi h he one
esul ing om he in eg a ion o he uns eady p essu e dis ibu ion a ound he solid
ob ained by Ga ick (1936) when he e ical componen o he wake eloci y is calcula ed
sel -consis en ly and he analysis e ains he con ibu ion o he lux o ho izon al
momen um induced by he s a ing o ex. The limi a ions o he sel -consis en linea ised
po en ial low app oach o p edic ing he h us o ce o ai oils oscilla ing pe iodically
wi h small ampli udes bu la ge alues o he educed equency a e also discussed, as
well as he easons behind he abili y o o he esul s in he li e a u e o app oxima e
measu emen s be e han Ga ick’s heo y. In ac , o hose cases in which he ai oil
oscilla es pe iodically, he lux o ho izon al momen um induced by he s a ing o ex is
negligible and he o ices in he wake a e con ec ed pa allel o he ee-s eam eloci y,
we ha e deduced an equa ion o he mean h us coe icien which di e s om p e iously
published esul s and is in ag eemen wi h expe imen al and nume ical esul s. In addi ion,
o hose cases in which he ai oil is suddenly se in o mo ion, we ha e also deduced an
equa ion ha e ains he e ec o he s a ing o ex and co ec ly quan i ies he ansien
h us o ce.
Key wo ds: ae odynamics, low-s uc u e in e ac ions
1. In oduc ion
The quan i ica ion o he o ces exe ed o e oscilla ing ai oils wi hin he po en ial
low and slende -body limi s aces back o he classical wo ks o Wagne (1925), who
calcula ed he uns eady li o ce o e an ai oil expe iencing a sudden change in he angle
o a ack, o Theodo sen (1935), who conside ed he analogous case o ai oils pe o ming
pe iodic pi ching and hea ing mo ions, o Ga ick (1936), who calcula ed h us by adding
© The Au ho (s), 2025. Published by Camb idge Uni e si y P ess. This is an Open Access a icle,
dis ibu ed unde he e ms o he C ea i e Commons A ibu ion licence (h ps://c ea i ecommons.o g/
licenses/by/4.0/), which pe mi s un es ic ed e-use, dis ibu ion and ep oduc ion, p o ided he o iginal
a icle is p ope ly ci ed. 1012 A6-1
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J.M. Go dillo
o he suc ion o ce a he leading edge o he ai oil he p ojec ion in he ligh di ec ion o
he li o ce calcula ed by Theodo sen, and o he also seminal con ibu ion o on Ká mán
& Sea s (1938), who ob ained he same esul s p e iously deduced by Wagne (1925)and
Theodo sen (1935) bu making use o a momen um balance i.e. using he o ex impulse
heo y. The esul s o hese classical s udies, which we e o iginally de eloped in he ield
o ae oelas ici y, ha e ecen ly been ex ended o quan i y he uns eady o ces expe ienced
by lying o swimming animals a high alues o he Reynolds numbe (Wu 1961; Smi s
2019).
Expe imen s, see Mackowski & Williamson (2015) and e e ences he ein, as well as
he nume ical simula ions o Young & Lai (2004), e eal ha he classical heo y due
o Ga ick (1936) o e es ima es bo h h us and he p opulsion e iciency o su icien ly
la ge oscilla ion equencies and ampli udes because: (i) he eal wake is non-plana
(Young & Lai 2004; Godoy-Diana, Aide & Wes eid 2008; Mackowski & Williamson
2015), a ac which con as s wi h he app oxima ions made in he linea ised heo y,
(ii) he iscous d ag, which plays an essen ial ole in de e mining he op imal S ouhal
numbe which maximises he p opulsion e iciency (Flo yan, Bu en & Smi s 2018), is
neglec ed in he po en ial low app oach, (iii) he o ices ejec ed om he leading edge
o he ai oil a la ge ampli udes o he hea ing and pi ching mo ions (Young & Lai 2004,
2007), which end o educe h us , a e no cap u ed by he linea ised heo y and (i ) he
h ee-dimensional e ec s associa ed wi h he ini e span o he body (Zu man-Nasu ion,
Ganapa hisub amani & Weymou h 2020) we e no conside ed by Ga ick in his o iginal
con ibu ion.
In spi e o hese d awbacks, a se ies o ecen s udies emphasise ha he linea ised
heo y due o Ga ick is capable o app oxima ing he ime- a ying alue o he h us
o ce o su icien ly small alues o he oscilla ion ampli udes and educed equencies i
he e ec s o s a ic d ag a e aken in o conside a ion in he modelling (Young & Lai 2004,
2007; Mackowski & Williamson 2015; Saada e al. 2017;Flo yane al. 2018). Mo eo e ,
Flo yan e al. (2017) ind ha Ga ick’s esul al eady p o ides he co ec scaling o he
h us o ce e en o la ge ampli udes o he oscilla ions.
In an a emp o imp o e he p edic ions o Ga ick’s heo y o alues o he oscilla ion
equencies la ge han he in e se o he cha ac e is ic esidence ime, a se ies o e y
ecen con ibu ions (Fe nandez-Fe ia 2016,2017; Alaminos-Quesada & Fe nandez-Fe ia
2020; Sanchez-Laulhe, Fe nandez-Fe ia & Olle o 2023), ex ends he linea ised o ex
impulse heo y by on Ká mán & Sea s (1938) wi h he pu pose o calcula ing he
ae odynamic h us . New on’s laws dic a e ha he ae odynamic o ce calcula ed using he
o ex impulse heo y, which esul s om a momen um balance, mus coincide wi h he
one ob ained by in eg a ing he p essu e dis ibu ion a ound he ai oil (Eld edge 2019)
and, hence, he linea ised heo ies by Ga ick (1936) and Fe nandez-Fe ia (2016,2017)
should p o ide iden ical esul s. Howe e , o alues o he educed equency o o de
uni y o la ge , he p edic ions by Fe nandez-Fe ia (2016,2017) a e in be e ag eemen
wi h he expe imen al and nume ical esul s epo ed by Young & Lai (2004,2007)and
Mackowski & Williamson (2015) han he ones deduced using Ga ick’s heo y. Mo i a ed
by he be e ag eemen wi h expe imen al da a, i is explici ly s a ed in Fe nandez-
Fe ia (2016,2017) and Alaminos-Quesada & Fe nandez-Fe ia (2020) ha he o ex
impulse o mula ion o Fe nández-Fe ia co ec s he heo y due o Ga ick, and i is one
o he pu poses o he p esen s udy o ind he o igin o he di e ences be ween he
esul s in Ga ick (1936) and in Fe nandez-Fe ia (2016). Indeed, since he p edic ions in
Fe nandez-Fe ia (2016) do no ep oduce Ga ick’s esul s, one o he wo heo ies is no
sel -consis en because, o he wise, he o ce calcula ed by he di ec in eg a ion o he
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p essu e dis ibu ion a ound he ai oil would be di e en om he alue ob ained h ough
a momen um balance.
I will be shown nex ha he linea ised heo ies due o Ga ick (1936)and he
one deduced using he o ex impulse heo y, which we de elop he e by ex ending he
momen um balance in on Ká mán & Sea s (1938), p o ide iden ical esul s o he
ae odynamic o ce i : (i) he lux o momen um induced by he s a ing o ex emi ed
ini ially om he ailing edge o he ai oil is aken in o accoun and (ii) he e ical
eloci ies o he o ices in he wake a e calcula ed in a sel -consis en manne . Indeed, in
o de o eco e Ga ick’s esul using he o ex impulse o mula ion, i p o es essen ial
ha he e ical eloci y o he o ices in he wake a e calcula ed sel -consis en ly wi hin
he linea ised app oach, namely, as a esul o he e ical eloci ies induced by he o ex
shee ex ending along he ai oil and he wake. In con as , he heo y by Fe nandez-Fe ia
(2016,2017) does no include he lux o momen um induced by he s a ing o ex and,
in addi ion, Fe nandez-Fe ia (2016,2017), Alaminos-Quesada & Fe nandez-Fe ia (2020)
and Sanchez-Laulhe e al. (2023) do no calcula e he e ical eloci ies o he o ices
in he wake in a sel -consis en manne bu , ins ead, impose hei alues: indeed, he
assump ion in equa ion (25) in Fe nandez-Fe ia (2016) implies ha he e ical eloci y
o he o ices in he wake is ze o. Howe e , we show he e ha he e is no need o impose
he alue o he e ical eloci ies o he o ices in he wake because he linea ised
po en ial low heo y al eady pe mi s us o calcula e hese eloci ies in a sel -consis en
manne : in ac , only i his is done does he o ex impulse heo y eco e he esul s
o iginally deduced by Ga ick, consis en ly wi h he ac ha he o ce calcula ed h ough
a momen um balance mus coincide wi h he alue ob ained by di ec in eg a ion o he
p essu e dis ibu ion a ound he ai oil. One o he main conclusions o his s udy is ha
he co ec equa ion o he h us o ce wi hin he linea ised po en ial low app oach is he
one due o Ga ick (1936) o he equa ion deduced he e using he o ex impulse heo y
in a sel -consis en manne , a conclusion ha con adic s he asse ions in Fe nandez-
Fe ia (2016,2017) and Alaminos-Quesada & Fe nandez-Fe ia (2020). Then, he abili y
o Fe nández-Fe ia’s esul s o p edic expe imen al measu emen s does no mean ha
Ga ick’s heo y is inco ec : we show he e ha he success o Fe nández-Fe ia’s esul s
es s on he ac ha he assump ion made in Fe nandez-Fe ia (2016,2017) and Alaminos-
Quesada & Fe nandez-Fe ia (2020) o neglec ing he con ibu ion o he s a ing o ex
and o imposing he e ical eloci ies o he wake o ices o be equal o ze o, e lec s
he ealis ic nonlinea dynamics o he wake o su icien ly la ge alues o he oscilla ion
equency. Clea ly, hese nonlinea e ec s canno be accoun ed o by any sel -consis en
linea heo y. Fo hose cases in which he ai oil oscilla es pe iodically, he lux o
ho izon al momen um induced by he s a ing o ex is negligible and he o ices in he
wake a e con ec ed pa allel o he ee-s eam eloci y, we also deduce he e an equa ion
o he so-called mean h us coe icien . This equa ion di e s om p e iously published
esul s and is in ag eemen wi h expe imen al and nume ical esul s.
Howe e , he esul s in his con ibu ion a e no only limi ed o he s udy o he h us
o ce o pe iodically oscilla ing ai oils: ou esul s also pe mi us o calcula e he h us
o ce in ansien manoeu es, like hose aking place when an ai oil is impulsi ely se in o
mo ion. In ac , we de i e he analy ical exp ession o he h us o ce co esponding o
he so-called Wagne p oblem (Wagne 1925) in wo di e en ways, namely, by he di ec
in eg a ion o he p essu e dis ibu ion a ound he ai oil and by also using he o ex
impulse heo y. We alida e all he analy ical esul s ob ained by means o he nume ical
code de ailed in he Supplemen a y Ma e ial.
This con ibu ion is s uc u ed as ollows: §2is de o ed o showing ha , wi hin he
linea ised po en ial low app oxima ion, he h us o ce calcula ed by means o he o ex
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J.M. Go dillo
z
e2
ρ, p∞,U∞e1e1xe
h( )
α( )
n
c + U∞
e ɛ
0x
Figu e 1. Ske ch o he canonical low conside ed in his s udy.
impulse heo y is iden ical o he classical esul due o Ga ick once he lux o ho izon al
momen um induced by he s a ing o ex is e ained in he analysis and he e ical
componen o he wake eloci y is calcula ed sel -consis en ly. This conclusion will be
illus a ed by he nume ical examples included in §3, whe e we also es ablish he limi s
unde which Ga ick’s heo y can be used o p edic expe imen al measu emen s. Fo hose
cases in which he ai oil oscilla es pe iodically, he lux o ho izon al momen um induced
by he s a ing o ex is negligible and he wake o ices a e con ec ed pa allel o he ee-
s eam eloci y, we deduce in § 4an analy ical equa ion o he mean h us coe icien
which has been alida ed using he esul s o nume ical simula ions ca ied ou using he
o ex-la ice me hod. The main esul s a e summa ised in § 5.
2. Calcula ion o h us h ough a momen um balance
The canonical low o be s udied in wha ollows, which employs he same no a ion and
sign con en ions as hose used in Bisplingho , Ashley & Hal man (1996) excep o he
ac ha , he e, he o igin o he Ca esian coo dina e sys em is loca ed a he leading edge
o he ai oil, is illus a ed igu e 1: an ai oil o cho d cex ending along 0 ⩽x⩽c,wi h
xand zdeno ing he Ca esian ho izon al and e ical coo dina es wi h associa ed uni
ec o s e1and e2, o ms a ime-dependen angle o a ack α( )wi h an inciden uni o m
s eam o densi y ρand eloci y =U∞e1. The o igin o imes is se a =0 and,
hence, wi hin he classical linea ised po en ial low app oach, he ho izon al posi ion o
he s a ing o ex is x=c+U∞ (Wagne 1925; Theodo sen 1935; on Ká mán & Sea s
1938;Glaue 1983; Ashley & Landahl 1985). Mo eo e , he e ical posi ion o he poin
loca ed a a dis ance x=xe<c om he leading edge o he ai oil is za(xe, )=−h( )
and, hence, o he case o a symme ical ai oil wi h ze o hickness conside ed, he e,
za(x, )=−h( )−α( )(x−xe),(2.1)
wi h he subsc ip s aand w e e ing om now on o quan i ies co esponding o ei he he
ai oil o he wake. No ice ha , in his con ibu ion, posi i e li ( )co esponds o a o ce
in he posi i e z-di ec ion, posi i e h us −d( )is posi i e in he nega i e x-di ec ion,
whe eas posi i e h( )co esponds o mo ion in he nega i e z-di ec ion and, simila ly,
posi i e o que m( )is in he coun e clockwise di ec ion, while posi i e α( )gi es
clockwise o a ion. The ae odynamic o ce ( )=( )e2+d( )e1and o que o e such
an ai oil, which possesses he wo deg ees o eedom α( )and h( ), a e calcula ed o he
common case in which he Reynolds numbe e i ies he condi ion Re =ρU∞c/μ 1,
wi h μindica ing he dynamic iscosi y; mo eo e , we will conside ha he ela i e
densi y a ia ions a e negligible and ha α( )1, za,w/c1, h/c1, wi h he e ical
posi ions o he poin s on he ai oil za(x, )de ined in (2.1)andzw(x, ) e e ing
o he e ical posi ion o he poin s in he wake. The e o e, unde hese condi ions,
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he hin bounda y laye o hickness δsuch ha δ/c∝Re−1/21 does no sepa a e and,
hence, he classical, linea ised po en ial low heo y summa ised in e.g. Ashley & Landahl
(1985), is applicable.
Indeed, ou side he hin bounda y laye and he wake, he eloci y ield is i o a ional,
namely, =∇φ,wi hφ=U∞x+φand φindica ing he pe u bed eloci y po en ial
associa ed wi h he pe u bed eloci y ield =∇φ=ue1+we2 e i ying he
condi ion |∇φ|/U∞1 excep , as will become clea in wha ollows, a he leading edge,
x→0 and, o he case o an ai oil which is suddenly se in o mo ion, a x→c+U∞ i.e.
whe e he s a ing o ex is loca ed. Then, by i ue o he con inui y equa ion ∇· =0,
he pe u bed po en ial sa is ies he Laplace equa ion
∇2φ=0,(2.2)
which mus be sol ed subjec o he bounda y condi ion a in ini y φ→0and o he
linea ised impene abili y condi ion, which can be exp essed as
DF
D =∂F
∂ +U∞e1+∇φ·∇F=0wi hF=z−za,w(x, ), (2.3)
and wi h za(x, )gi en in (2.1).
The s anda d linea isa ion o (2.3) yields (Ashley & Landahl 1985)
w
a,w(x,z=0±, )=∂za,w
∂ +U∞
∂za,w
∂x,(2.4)
wi h z=0±indica ing he uppe and lowe sides o he ai oil and he wake and w
a,w
e e ing o he e ical componen o he eloci y on he ai oil o a he wake. In
iew o he linea ised impene abili y bounda y condi ion a 0 ⩽x⩽cgi en by (2.1)
and (2.4), we seek an isymme ic solu ions o he Laplace equa ion (2.2)in he o m
o a o ex shee ex ending along he ai oil and he wake, whose ci cula ion densi y is
he one sa is ying he condi ion exp essed by (2.1)and(2.4). No ice ha , making use
o he no a ion φ± =φ(x,z=0±, ),Γ(x, )=C(U∞e1+∇φ)·d=(φ+−φ−)=
2φ+(x, )and γ(x, )=u+ −u− =∂Γ/∂x, in his con ibu ion
Γ(x>0, )=2φ+(x>0, )=x
−∞
γ(x0, )dx0=x
0
γ(x0, )dx0(2.5)
e e s o he clockwise ci cula ion along any closed loop enci cling he leading edge
o he ai oil and connec ing he poin s (x>0,z=0−)and (x>0,z=0+), whe eas
γ(x, )indica es he ci cula ion densi y. In (2.5) we ha e aken in o accoun ha , since
he o igin o he o ex shee is he leading edge o he ai oil, which is loca ed a x=0,
γ(x<0, )=0, and hence he ci cula ion o x<0 is also ze o because Γ(x<0, )=
x
−∞ γ(x0, )dx0=0. In he ollowing, Γa,w(x, )and γa,w(x, )will indica e he alues
o he ci cula ion and o he ci cula ion densi y on he ai oil, which ex ends along
0⩽x⩽co a he wake, which ex ends along x>c.
The equa ion go e ning he p essu e jump a z=0, namely, p(x,z=0, )=
p(x,z=0−, )−p(x,z=0+, )=p−(x, )−p+(x, ),wi hp=p−p∞indica ing
he pe u bed p essu e, can be deduced om he linea ised Be noulli equa ion
pa icula ised a z=0±
z=0±:ρ∂φ±
∂ +ρU∞
∂φ±
∂x+p± =0.(2.6)
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J.M. Go dillo
Hence, he sub ac ion o he wo equa ions in (2.6) yields
ρ∂Γ
∂ +ρU∞
∂Γ
∂x=ρ∂
∂ x
0
γdx0+ρU∞γ=G(x, )wi h
G(0⩽x⩽c, )=pa(x, )(2.7)
wi h pa he p essu e jump a he ai oil and, since p=0 o x<0andx>c,we
conclude ha G(x<0, )=G(x>c, )=0in(2.7), a ac implying ha he ma e ial
de i a i es o bo h Γand γa e ze o a z=0 o x<0andx>c, namely (Ashley &
Landahl 1985),
DΓ
D =∂
∂xDΓ
D =0⇒∂Γ
∂ +U∞γ=0,∂γ
∂ +U∞
∂γ
∂x=0 o x<0,x>c.
(2.8)
Taking in o accoun ha Γ=0 o x→−∞and also o ins an s <0 and ha he
ci cula ion a he o igin o he wake is p esc ibed by he ci cula ion a ound he ai oil,
namely,
Γw(x=c, )=Γa(x=c, )=Γe( )=c
0
γa(x, )dx,(2.9)
wi h Γe( ) he ci cula ion a ound he ai oil, we deduce om (2.8)and(2.9) ha
Γ(x<0, )=0,Γ(x>c+U∞ , )=0,Γ
w(x=c+U∞( − 0), )=c
0
γa(x, 0)dx,
and γw(x=c+U∞( − 0), )=γw(x=c, 0), (2.10)
wi h γw(x→c, 0)gi en by (2.8)and(2.10) – see also equa ions (13)–(27) in Ashley &
Landahl (1985)
d
d c
0
γa(x, )dx( 0)+U∞γw(c, 0)=0,(2.11)
whe e γw(x→c, 0)=γw(c, 0) om which we conclude ha
γw(x0=c+U∞( − 0), )=γw(x=c, 0)=− 1
U∞
d
d c
0
γa(x, )dx( 0)
=− 1
U∞
dΓe
d ( 0), (2.12)
wi h he ci cula ion a ound he ai oil Γe( )de ined in (2.9). Equa ions (2.7)and(2.12)
indica e ha he uns eady li o ce and he o que
( )=c
0
pa(x, )dxand m( )=c
0
xpa(x, )dx,(2.13)
as well as he densi y o ci cula ion along he wake, γw(x, ), can be exp essed as a
unc ion o γa(x, ).
Finally, he densi y o ci cula ion a he ai oil, γa(x, ), is deduced by imposing ha
he pe u bed e ical eloci y induced by he o ex shee ex ending along z=0, 0 ⩽x⩽
c+U∞ sa is ies he linea ised impene abili y condi ion gi en by (2.1)and(2.4), namely
(Ashley & Landahl 1985),
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w
a(x,z=0±, )=−dh
d −U∞α( )−dα
d (x−xe)
=1
2πc
0
γa(x0, )
x0−xdx0+1
2πc+U∞
c
γw(x0, )
x0−xdx0.(2.14)
In oducing he change o a iables
x0=c+U∞( − 0)⇒dx0=−U∞d 0,(2.15)
and aking in o accoun ha he second in eg al a he igh -hand side o (2.14) can be
exp essed solely in e ms o γaby means o (2.12), he equa ion o γa(x, ) eads
w
a(x,z=0±, )=1
2πc
0
γa(x0, )
x0−xdx0−1
2π
0
dΓe
d 0
c+U∞( − 0)−xd 0
wi h Γe( )=c
0
γa(x, )dx.(2.16)
In o de o sol e he in eg al equa ion (2.16) no ice i s ha , since Γ(x→−∞, )=0
hen, by i ue o (2.8), Γ(x<0, )=0 and hence, φ(z=0±,x<0)=0. Consequen ly,
he local solu ion o he Laplace equa ion (2.2) a he leading edge o he ai oil is he one
co esponding o he low a ound a wedge o angle 2π, namely,
φ=U∞cA0( )
c1/2cos β
2,(2.17)
wi h A0( )a dimensionless ime-dependen cons an , /c1 he adial dis ance o he
leading edge – which is loca ed a z=0, x=0 in he linea ised heo y – and 0 ⩽β⩽2π
indica ing he pola angle measu ed in coun e clockwise manne om he ho izon al axis.
Taking in o accoun : (i) ha he Ku a condi ion ensu es ha γa(x=c, )is ini e in
o de o a oid ha he low u ns a ound he ailing edge o he ai oil and (ii) ha
γa(x/c1, )is gi en by
∂φ
∂ β=0,
c1, =uz=0+,x
c1, =U∞
2A0( )x
c−1/2
⇒γax
c1, =U∞A0( )x
c−1/2
,(2.18)
whe e use o (2.17) has been made, i can be concluded ha he in eg al equa ion (2.16)
can be sol ed using Glaue ’s me hod, which elies on exp essing he unknown unc ion
γa(x, )as he in ini e se ies (Glaue 1983)
γa(x, )
U∞=A0( )





1−x
c
x
c
+∞

n=1
An( )sin(nθ)=A0( )1+cos θ
sin θ+∞

n=1
An( )sin(nθ),
(2.19)
whe e we ha e in oduced he change o a iables
x
c=1−cos θ
2,(2.20)
and, he e o e, θ=0a x=0andθ=πa x=c. No ice ha he expansion (2.19) implies
ha γa(x=c, )=0 and, hence, he densi y o ci cula ion a x=cdoes no sa is y he
1012 A6-7
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J.M. Go dillo
physical condi ion γa(x=c, )=γw(x=c, ). Howe e , he con inui y o he ci cula ion
densi y a he ailing edge could be en o ced ollowing he p ocedu e de ailed in Alben
(2010) and Eld edge (2019) and e e ences he ein by analy ically emo ing he loga i hmic
singula i y a x=cin (2.16). This mo e elabo a e me hod p o ides a as e con e gence o
he solu ion which, howe e , can also be ound using he mo e classical p ocedu e ollowed
he e by simply e aining a la ge numbe o e ms in (2.19), see Alben (2010).
Then, he subs i u ion o he expansion (2.19) in o he in eg al equa ion (2.16) p o ides
us wi h he alues o he ime-dependen coe icien s Ai( )as a unc ion o α( )and
h( ), as de ailed in he Supplemen a y Ma e ial, see also Wagne (1925), Theodo sen
(1935), on Ká mán & Sea s (1938), Wu (1961) and e e ences he ein. Once he alues o
Ai( )a e known, pa,( )and m( )can be de e mined by means o (2.7)and(2.13), as
illus a ed, o ins ance, in he Supplemen a y Ma e ial. Now ha ( )is known, he alue
o he d ag o ce can be ob ained by adding o he p ojec ion in he ligh di ec ion o he
li o ce he esul ing suc ion o ce a he leading edge o he ai oil, yielding
d( )=α( )( )−ρU2
∞πcA2
0( )
4;(2.21)
see Ga ick (1936), Wu (1961) and e e ences he ein, as well as he nex subsec ion.
Hence, he h us o ce ob ained by he di ec in eg a ion o he p essu e dis ibu ion
a ound he ai oil is gi en by
TG( )=−d( )=−α( )( )+ρU2
∞πcA2
0( )
4,(2.22)
whe e we ha e made use o (2.21) and he subsc ip Gindica es Ga ick. Le us poin
ou he e ha he exp ession o A0( )in (2.22) o he case o pe iodic oscilla ions o he
ai oil was p o ided by Ga ick (1936) using he esul s o Theodo sen (1935), whe eas
he co esponding alue o a bi a y hea ing o pi ching mo ions is deduced elsewhe e;
see, o ins ance, he Supplemen a y Ma e ial.
2.1. Fo ces calcula ed h ough a momen um balance
So a we ha e calcula ed he uns eady li and h us o ces on he ai oil as a esul o he
in eg a ion o he p essu e dis ibu ion a ound he solid. I is now ou pu pose o calcula e
he ae odynamic o ce h ough a momen um balance using he con ol olume Ωc( )
limi ed by a ixed su ace Σ∞o dimensionless adius R/c→∞which enci cles bo h he
solid and he wake, by he su ace Σa,w bounding bo h he solid and he wake and by Σ,
which is a ci cle o adius →0 cen ed whe e he s a ing o ex is loca ed, namely, a
x=c+U∞ . The momen um balance applied a he con ol olume Ωc( )de ined abo e
yields
d
d Ωc( )
ρ dω+∂Ωc
ρ ( − c)·ncdσ=∂Ωc
(p−p∞)(
−nc)dσ, (2.23)
wi h nc he uni no mal poin ing ou wa ds he con ol olume, =∇φand cindica ing
he eloci y o he su aces bounding he con ol olume, namely, c=U∞e1a Σand
( − c)·nc=0a Σa,w. Since he e is no ela i e momen um lux ac oss he su aces
Σa,w, and aking in o accoun ha he p essu e o ce a he ai oil is
=Σa( )
(p−p∞)ncdσ, (2.24)
1012 A6-8
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Jou nal o Fluid Mechanics
(2.23) can be w i en, making use o Gauss’ heo em, as
=−d
d ΣaΣw
ρφncdσ+Σ
ρ∇φ(∇φ−U∞e1)·e dσ+Σ
(p−p∞)(
−e )dσ,
(2.25)
whe e we ha e aken in o accoun ha =∇φ=∇φ+U∞e1and e e e s o he uni
ec o in pola coo dina es, see igu e 1.Mo eo e ,in(2.25)weha emadeuseo he
ac ha he in eg als e alua ed a Σ∞ end o ze o because o he Be noulli equa ion
ρ∂φ/∂ +ρ|∇φ|2/2+p=Kand because Σ∞enci cles bo h he ai oil and he wake
and, hence, he ci cula ion a ound Σ∞is ze o, which implies ha he pe u bed eloci y
ield decays as e han U∞c/Ra in ini y, which ensu es ha bo h he lux o momen um
and he in eg al o |∇φ|2along Σ∞ end o ze o. Then, he ae odynamic o ce can be
w i en, in he linea ised app oach, as
=d
d c
0
ρΓandx+d
d c+U∞
c
ρΓwndx
+Σ
ρ∇φ(∇φ−U∞e1)·e dσ+Σ
(p−p∞)(
−e )dσ, (2.26)
wi h n he uni ec o poin ing ou wa ds he side z=0+o Σaand Σw,see igu e 1,
and whe e we ha e aken in o accoun ha he uni no mal poin ing ou wa ds he side
z=0−o Σaand Σwis −n, his being he eason why he in eg and in he i s wo
in eg als in (2.26)isΓ=φ+ −φ−; in addi ion, in (2.26)weha emadeuseo he ac
ha , by i ue o (2.17) and o he pa ag aph p eceding his equa ion, whe e i is shown
ha Γ(x⩽0, )=0, he alue o he in eg al ex ending along a small egion nea he
leading edge ends o ze o. Finally, since Γ(x>c+U∞ , )=0, see (2.10), he leading-
o de equa ion o he pe u bed po en ial a Σ, co esponds o he one cha ac e ising he
low a ound a wedge o angle −π⩽β⩽π, namely,
φ=U∞cC sinβ
2
c1/2
⇒∇φ=C
2U∞sinβ
2
c−1/2e +C
2U∞cosβ
2
c−1/2eβ,(2.27)
wi h e =cos βe1+sin βe2,eβ=−sin βe1+cos βe2, /cindica ing he dimensionless
dis ance o he s a ing o ex loca ed a x=c+U∞ and Cis a dimensionless cons an
which does no depend on ime because, by i ue o (2.8), he alue o he pe u bed
po en ial emains cons an a Σ; hence, Cis ixed a =0+, igh a e he ai oil is
se in mo ion, see Appendix A, whe e Cis calcula ed. No ice ha he las in eg al in
(2.26) is ze o because, by i ue o (2.27), |∇φ|is cons an a Σand hence, by i ue
o he Be noulli equa ion, pis also cons an a Σ. Finally, he hi d in eg al in (2.26),
co esponding o he momen um lux ac oss Σ, is calcula ed using (2.27), which yields
he ollowing exp ession o he ae odynamic o ce:
=d
d c
0
ρΓandx+d
d c+U∞
c
ρΓwndx−ρU2
∞πcC2
4e1.(2.28)
Taking now in o accoun ha α( )1andh( )/c1, he linea isa ion o he no mal
ec o nin equa ion (2.28) yields
n·e2≃1andn·e1=−∂za,w(x, )
∂x.(2.29)
1012 A6-9
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J.M. Go dillo
0510
τ
15 20
1
2
3
4
 G, VI
 G, VI
5
6
7
8
(a)(b)
×10–3
0510
τ
15 20
1
2
3
4
5
6
7
8×10–3
Figu e 2. Dimensionless h us o ces  G(τ) (blue line) and  VI(τ) ( ed line) espec i ely de ined in (3.1)
and (3.2), co esponding o he plunging mo ion p esc ibed by (3.4), o wo di e en alues o he educed
equency: (a)k=2and(b)k=4. No ice ha bo h esul s coincide a e e y ins an o ime, as expec ed om
he esul in (2.52).
05
10
τ
15 20
1
0
–1
–2
2
3
4
 G, FF
5
6
7
8
05
10
τ
15 20
1
0
–1
–2
2
3
4
 G, FF
5
6
7
8
(a)(b)
×10–3 ×10–3
Figu e 3. Dimensionless h us o ces  G(τ) (blue line) and  FF(τ ) (black line) espec i ely de ined in (3.1)
and (3.6), co esponding o he plunging mo ion p esc ibed by (3.4), o wo di e en alues o he educed
equency: (a)k=2and(b)k=4. No ice ha he di e ences be ween he alues o  G(τ) and o  FF(τ)
inc ease wi h he alue o he educed equency k. Mo eo e , he esul s in he igu e show ha he mean h us
co esponding o  FF is smalle han he mean h us co esponding o  G, and also ha he di e ences
be ween he alues o he mean h us become mo e p onounced o he la ge alue o he educed equency.
The esul s depic ed in igu es 2 and 3will be discussed in mo e de ail in § 4, whe e an equa ion o he mean
h us coe icien is deduced.
close o each o he , wi h he iny di e ences be ween he wo esul s a ibu able o he
e ec o he nume ical disc e isa ion in he e alua ion o he in eg als in he de ini ion o
 VI(τ),see(3.2). Nex , igu e 3 compa es he alues o  G(τ) depic ed in igu e 2 wi h
he alues o he dimensionless h us o ce de ined as
 FF(τ) = VI(τ) −1
ρU2
∞c×ρU2
∞cπC
22
−ρc+U∞
c
γww
wdx,(3.6)
1012 A6-16
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Jou nal o Fluid Mechanics
x/c
–0.40123456789
–0.3
–0.2
–0.1
0
0.4
0.3
0.2
0.1
wa,w
Figu e 4. Ve ical eloci ies induced by he o ex shee ex ending along he ai oil and he wake calcula ed
using (2.41). The esul s depic ed in he igu e co espond o he plunging mo ion de ined in (3.4) o h ee
di e en alues o he educed equency: k=8(blackcu e),k=2( edcu e)andk=0.5 (blue cu e).
No ice ha he ampli udes o he e ical eloci ies in he wake o he cases k=2andk=8 a e clea ly la ge
han he ampli ude o he e ical ai oil eloci y, V=5π/180.
wi h  VI(τ) gi en in (3.2) and which ep esen s he dimensionless h us o ce calcula ed
using he o ex impulse heo y unde he assump ions made in Fe nandez-Fe ia (2016),
namely, once he con ibu ions o he s a ing o ex – which is ze o o he pa icula case
o he plunging mo ion conside ed in (3.4) – and o he con ibu ion o he o ex h us
o ce o he o ices in he wake, see (2.40), a e se o ze o. The esul s depic ed in igu e 3
e eal ha he alues o  G(τ) and o  FF(τ ) a e no iden ical o each o he and, in
ac , he di e ences be ween  G(τ) and  FF(τ) become mo e isible as he alue o
he educed equency kis inc eased; no ice ha he di e ences be ween he alues o
he mean h us depic ed in igu es 2 and 3will be discussed in mo e dep h in § 4. Using
he esul s depic ed in igu e 2 andalso hosein(3.2)and(3.6), we can in e ha he
alues o  G(τ) and o  FF(τ ) would be iden ical i w
w(x>c, )=0. Howe e , his
is no he case, as can be app ecia ed in igu e 4, which shows ha , in ac , he ampli udes
o he e ical eloci ies in he wake inc ease wi h he alue o k, despi e he alue o
Vin (3.4) emaining unchanged. No ice also ha igu e 3 illus a es one o he main
esul s deduced by Fe nández-Fe ia: he mean h us p edic ed in Fe nandez-Fe ia (2016)
is simila o he mean h us p edic ed by Ga ick’s heo y o low- o-mode a e alues o
he educed equency, bu i is smalle han he mean h us p edic ed by Ga ick (1936)
o kO(1). In iew o he de ini ions in (3.1), (3.2)and(3.6) and o he conclusions in
§2, he eason o he di e ences depic ed in igu e 3 is ha he esul s in Fe nandez-Fe ia
(2016) we e ob ained neglec ing he e ical eloci ies o he wake o ices i.e. assuming
ha w
w(x, )=0, which implies ha he e m (2.40) is also ze o. Howe e , he alues
o w
w(x, )calcula ed wi hin he linea ised po en ial low heo y by means o (2.41)a e
di e en om ze o, as shown in igu e 4 and, hence, he con ibu ion o h us o he o ex
o ce in (2.40), canno be neglec ed.
Nex , we conside he case in which an oscilla ing ai oil pe o ms a pu ely plunging
mo ion wi h a equency ωand wi h a e ical eloci y gi en by
w
a(0⩽x⩽c, )
U∞=−Vcos (kτ)wi h V=5π
180 .(3.7)
1012 A6-17
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J.M. Go dillo
–6
–4
–2
0
2
4
6
8
1
2
3
4
5
6
7
8
 G, VI
×10
–3
 G
×10
–3
0 0 2 4 6 8 10 12 14 16 18 200.5 1.0 1.5 2.0
ττ
(a)(b)
Figu e 5. (a)Dimensionless h us o ces  G(τ) (blue line) and  VI(τ) (black line) de ined, espec i ely,
in (3.1)and(3.2), co esponding o he plunging mo ion o he ai oil p esc ibed by (3.7) o a alue o he
educed equency k=2. No ice ha bo h esul s coincide a e e y ins an o ime, as expec ed om he esul
exp essedby(2.52). The ed cu e co esponds o he esul s o (3.9), which does no e ain he e ec o he
s a ing o ex. (b)A e a sho ansien , he o ces co esponding o he plunging mo ions de ined in (3.4)–
in blue – and (3.7) – in black – o a alue o he educed equency k=2con e ge o hesame esul wi hjus
a phase shi .
I is deduced in Appendix A ha
C=w
a(x=3c/4, =0+)
U∞=−α0+˙
h0
U∞+˙α0
U∞3c
4−xe,(3.8)
wi h quan i ies wi h he subsc ip 0 indica ing hei co esponding alues a =0+and,
hence, he con ibu ion o he s a ing o ex o he h us o ce canno be neglec ed i
w
a(x=3c/4, =0+)=0. The esul s depic ed in igu e 5(a) con i m his is he case:
indeed, he blue and black cu es ep esen ing he alues o  G(τ) (blue) and o  VI(τ)
(black) de ined in (3.1)and(3.2), a e e y close o each o he . Howe e , he ed cu e,
which ep esen s he nume ical alues o
 VIB = VI(τ) −πC2
4,(3.9)
namely, o he h us o ce calcula ed using he o ex impulse heo y once he con ibu ion
o he s a ing o ex is se o ze o, is well below he black and blue cu es. Due o he
ac ha he alue o he h us o ce calcula ed using he o ex impulse heo y mus be
iden ical o he alue ob ained by di ec in eg a ion o he p essu e dis ibu ion a ound he
ai oil, he esul depic ed in igu e 5(a) illus a es ha he con ibu ion o he s a ing o -
ex canno be neglec ed o hose cases in which w
a(x=3c/4, =0+)=0. Mo eo e , he
esul s in igu e 5(b) illus a e ha , as expec ed, he h us o ces o he plunging mo ions
co esponding o (3.4)and(3.7) con e ge o he same alues wi h jus a phase shi .
Finally, we analyse he case o an ai oil ha is suddenly se in o mo ion, o which
w
a(0⩽x⩽c, )
U∞=−α0+˙
h0
U∞+˙α0
U∞
(x−xe)H( ), (3.10)
wi h H( )indica ing he Hea iside unc ion. The li o ce co esponding o (3.10)was
calcula ed by Wagne (1925) as, see also he Supplemen a y Ma e ial,
1012 A6-18
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Jou nal o Fluid Mechanics
Wagne (τ) =ρU2
∞cπ⎛
⎜
⎜
⎝−
w
ax=3c
4, =0+
U∞⎞
⎟
⎟
⎠
φ(τ), (3.11)
wi h φ(τ) he so-called Wagne unc ion, which is app oxima ed he e using he well-
known equa ion gi en by Jones (1938)
φ(τ)=1−0.165 e−0.0455 τ−0.335 e−0.3τ,(3.12)
whe eas he co esponding exp ession o he h us o ce is deduced in he Supplemen a y
Ma e ial and eads
TG,Wagne (τ) =ρU2
∞cπ
w
ax=3c
4, =0+
U∞
φ(τ)α0
+ρU2
∞cπ⎛
⎜
⎜
⎝
w
ax=3c
4, =0+
U∞
φ(τ)+c
4U∞˙α0⎞
⎟
⎟
⎠
2
.(3.13)
No ice ha a special case o (3.13) is gi en on page 705 o Dono an & Law ence (1957).
Hence, o he case o he so-called Wagne p oblem, cha ac e ised by he linea ised
impene abili y bounda y condi ion gi en by (3.10), we ind ha
 G,Wagne (τ) =TG,Wagne ( )+α0Wagne ( )
ρU2
∞c
=π⎡
⎢
⎢
⎣
w
ax=3c
4, =0+
U∞
φ(τ)+c˙α0
4U∞⎤
⎥
⎥
⎦
2
.(3.14)
Consis en ly wi h he esul s ob ained in § 2no ice ha , since he Wagne unc ion e i ies
φ(τ =0)=1/2, see (3.12) and he Supplemen a y Ma e ial o u he de ails, he alue
o he h us o ce gi en by (3.13)a τ=0, TG,Wagne (τ =0), coincides wi h he alue o
TVI(τ =0)calcula ed using (2.38).
The nume ical esul s in igu e 6, co esponding o he impulsi e mo ion o he ai oil
gi en by
w
a(0⩽x⩽c, )
U∞=−α0H( )wi h α0=5π
180 ,(3.15)
p o ides u he suppo o (3.14) and also illus a es he ele ance o he s a ing o ex
in ei he o (2.37), (2.38), (2.42) o p edic h us when he ai oil is impulsi ely se in o
mo ion.
Howe e , he abili y o he p edic ion in (3.14) o ep oduce expe imen al measu emen s
is limi ed o hose ci cums ances in which he s a ing hypo heses o he linea ised
po en ial low heo y a e no iola ed. Indeed, no ice ha he dynamics o he s a ing
o ex gene a ed when he ai oil is suddenly se in o mo ion is desc ibed by he sel -
simila solu ion ound by Kaden (1931), see also Pullin (1978). The wid h o his
sel -simila egion g ows in ime as ∝(U∞αeq
0c1/2 )2/3,wi hαeq
0indica ing he equi alen
angle o a ack de ined in (A4)o Appendix A. Hence, i is o be expec ed ha , in
1012 A6-19
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J.M. Go dillo
0
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
24
τ
6810
 G, VI
Figu e 6. (a)Dimensionless h us o ces co esponding o he impulsi e mo ion o he ai oil p esc ibed by
(3.15). He e,  VI(τ) (black line) and  G,Wagne (τ) (blue line) ha e been calcula ed using (3.2)and(3.14),
espec i ely. The ed cu e co esponds o he esul s o (3.9), which do no e ain he e ec o he s a ing
o ex.
a eal expe imen , he wake egion su ounding he s a ing o ex expe iences la ge
de o ma ions, causing a clea de ia ion om he linea ised app oxima ion, which assumes
ha he wake is loca ed a z=0. The de ia ions om he linea ised app oxima ion will
ake place o ins an s o ime  ∗,wi h ∗es ima ed om (U∞αeq
0c1/2 ∗)2/3∼cand,
hence, ∗∼c/(U∞αeq
0)c/U∞i αeq
01. The nonlinea olling up o he s a ing
o ex desc ibed abo e will ce ainly modi y he h us o ce p edic ed he e o la ge
imes  ∗; clea ly, he alidi y o ou esul s pe aining o he h us p oduced by he
s a ing o ex, which ha e been deduced wi hin linea ised po en ial low heo y, should
be checked agains de ailed nume ical calcula ions a ini e bu high Reynolds numbe s.
Howe e , his addi ional ask exceeds he limi s o he p esen con ibu ion.
Simila ly, o he cases o ai oils oscilla ing pe iodically which a e no suddenly se
in o mo ion and, hence, he e ec o he s a ing o ex is negligible, he capabili y o
he linea ised po en ial low esul s o p edic he mean h us o ces is also limi ed o
hose cases in which he e ec o nonlinea i ies can be neglec ed. Howe e , he e a e a
numbe o expe imen al condi ions ha igge he de elopmen o nonlinea i ies, limi ing
he applicabili y o Ga ick’s esul s. Fi s , no ice ha he linea ised app oach will only be
alid o p edic expe imen s and nume ical simula ions i H0/c1, wi h H0deno ing he
cha ac e is ic ampli ude o he oscilla ions, bu his necessa y condi ion is no su icien .
Indeed, i has been shown by Ramesh e al. (2014), see also Eld edge (2019) ha , in o de
o p e en low sepa a ion and, hence, he associa ed ejec ion o o ices om he leading
edge, i is necessa y ha he alue o he ad e se p essu e g adien a x=0, which can be
quan i ied h ough he alue o A0in he expansion (2.19), is such ha A0( )α∗,wi h
α∗∼O(0.1)indica ing a alue which can be iewed as a c i ical angle o a ack o low
sepa a ion. The alue o α∗depends on he ype o ai oil and needs o be calcula ed
ei he expe imen ally o by means o ull nume ical simula ions. Fo he case o an
oscilla ing ai oil, he cha ac e is ic alue o A0can be es ima ed as he e ec i e angle o
a ack esul ing om he a io be ween he cha ac e is ic e ical and ho izon al eloci ies,
namely, A0∼ωH0/U∞. Hence, in o de o p e en low sepa a ion a he leading edge, i
is necessa y ha kH
0/cα∗,wi hk he educed equency de ined in (3.5). In addi ion,
1012 A6-20
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Jou nal o Fluid Mechanics
e en in he case ha H0/c1, he sel -induced eloci ies o he o ices ejec ed om
he ailing edge can cause la ge dis o ions in he wake o su icien ly la ge alues o he
educed equency: in ac , he e ical eloci ies in he wake a e app eciably la ge han
he e ical eloci ies o he ai oil o la ge alues o he educed equency, see igu e 4.
Indeed, he ci cula ion a ound he ai oil can be es ima ed as Γe∼U∞cA0∼cH0ω,wi h
A0es ima ed abo e and, mo eo e , he e ical eloci y induced by he o ices ejec ed
du ing a pe iod o oscilla ion on he new o ices lea ing he ailing edge o he ai oil can
be es ima ed, making use o (2.12), as w
0∼1/U∞dΓe/d ∼cH0ω2/U∞. Hence, in o de
o p e en la ge de o ma ions o he wake nea he ailing edge o he ai oil, i is necessa y
ha w
0/U∞O(1), namely, ha (ω H0/U∞)2(H0/c)−1O(1)o , equi alen ly, ha
k2(H0/c)O(1).
The eme gence o nonlinea i ies o alues o he educed equency such ha
k2(H0/c)O(1)is e idenced, o ins ance, when he wake is non-plana , as is clea ly
depic ed in he expe imen s by Godoy-Diana e al. (2008) o in he ull nume ical
simula ions by Young & Lai (2004) and Ma ín-Alcán a a e al. (2015). In hese e e ences,
he o ices in he wake a e a anged in such a way ha he e ical componen o hei sel -
induced eloci ies is negligible, as is e idenced by he ac ha he wake o ices in hese
e e ences a e con ec ed ho izon ally. We hypo hesise ha i is unde hese ci cums ances
ha he p edic ions in Fe nandez-Fe ia (2016,2017) ag ee be e wi h expe imen s han
Ga ick’s heo y. The eason o his is ha , when he o ices a e con ec ed ho izon ally,
w
w≈0 and, hence, he p ojec ion o ρ(∇× )× on he ligh di ec ion is negligible,
making he con ibu ion o he o ex o ce e m (2.40) also negligible in he eal nonlinea
low. Clea ly, hese nonlinea e ec s canno be accoun ed o by he heo y de eloped
by Ga ick (1936) o by he sel -consis en o ex impulse esul s p esen ed he e, which
p edic ha he e ical eloci ies o he wake o ices is di e en om ze o, as can be
app ecia ed in igu e 4.
Then, he capabili y o he esul s in Fe nandez-Fe ia (2016,2017) o p edic bo h
expe imen s and nume ical esul s o alues o he educed equency o o de uni y o
la ge , o which he wake expe iences la ge de o ma ions, seems o be jus i ied by he ac
ha he esul s in Fe nandez-Fe ia (2016,2017) co espond o a linea ised model which
akes in o accoun ealis ic nonlinea e ec s.
The discussion abo e pe mi s us o conclude ha , i he condi ions o a pa icula
expe imen in which he ai oil oscilla es pe iodically a e such ha he e ec s o
nonlinea i ies a e negligible and, hence, he s a ing hypo heses o he linea ised po en ial
low app oach a e alid, he h us o ce should be calcula ed using he sel -consis en
esul s exp essed by (2.22)o byei he o (2.37), (2.38), (2.42). This explains why he
expe imen al measu emen s o he mean h us in Mackowski & Williamson (2015) can
be easonably well p edic ed by Ga ick’s heo y, which should be hen used o small
alues o he educed equency once iscous e ec s a e aken in o conside a ion, see
e.g. Figu e 3(a) in Fe nandez-Fe ia (2017). Howe e , when nonlinea i ies a e igge ed
and he o ices in he wake a e a anged in such a way ha hei e ical sel -induced
eloci ies a e negligible, namely, w
w≈0, he linea ised model in Fe nandez-Fe ia (2016,
2017) and Alaminos-Quesada & Fe nandez-Fe ia (2020) should be used ins ead, once
he modi ica ions o he equa ion o he mean h us coe icien poin ed ou in § 4a e
aken in o accoun , because he con ibu ion o h us o he o ex o ce e m (2.40)
is no calcula ed sel -consis en ly wi hin he linea ised po en ial low app oach in hese
con ibu ions bu , ins ead, i is assumed o be ze o. The alue o he h us o ce deduced
by Fe nandez-Fe ia (2016,2017), once he modi ica ions in he mean h us coe icien
deduced in §4a e aken in o accoun , happens o be a be e app oxima ion o he eal alue
1012 A6-21
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J.M. Go dillo
han he one ob ained by means o he sel -consis en linea ised heo y because, when he
o ices in he wake a e a anged in such a way ha hey a e con ec ed ho izon ally, he
p ojec ion on he ligh di ec ion o ρ(∇× )× in he wake is ze o, which implies ha
he con ibu ion o he o al o ex o ce o he o ices in he wake is also ze o in he eal,
nonlinea case.
4. Mean h us coe icien o he case o ai oils oscilla ing pe iodically when he
wake o ices a e con ec ed downs eam a he ee-s eam eloci y
Fo hose common cases in which he ai oil s a s accele a ing smoo hly, namely, C=0
in (2.37), and he wake o ices a e con ec ed in a di ec ion pa allel o he ee-s eam
eloci y as a consequence o he de elopmen o nonlinea i ies, which implies ha w
w=0,
he h us o ce is (see (2.37))
TVI( )=−α( )( )−ρc
0
γaw
adx,(4.1)
which is iden ical o he one used in Fe nandez-Fe ia (2016), as has been discussed in he
de i a ion o (2.38).
In his sec ion, ou pu pose will be o deduce an equa ion o he h us o ce a e aged
in ime when he ai oil oscilla es pe iodically wi h a equency ω, he wake o ices a e
con ec ed wi h he inciden eloci y and he con ibu ion o he s a ing o ex o h us
is se o ze o. Fo ha pu pose, we i s make use o he ac ha , since w
a(x, )=−˙
h−
U∞α−˙α(x−xe)and
c
0
γa(x, )dx=Γa(x=c, )=Γe( ), ∂
∂xΓaw
a=γaw
a−˙αΓa,(4.2)
hen
−ρc
0
γaw
adx=−ρ−˙
h+U∞α+˙α(c−xe)Γe( )+˙αc
0
Γadx
=ρ˙
h+˙α( )(c−xe)Γe( )−ρd
d αc
0
Γadx
+α( )ρU∞Γe( )+ρd
d c
0
Γadx,(4.3)
whe e we ha e made use o he ac ha Γa(x=0, )=0 and o he iden i y
−˙αc
0
Γadx=−d
d αc
0
Γadx+αd
d c
0
Γadx.(4.4)
The subs i u ion o (4.3)in o(4.1) yields ha
TVI( )=ρ˙
h+˙α( )(c−xe)Γe( )−ρd
d αc
0
Γadx,(4.5)
whe e we ha e made use o he exp ession o he li o ce ( )in (2.32). Consequen ly,
he mean h us o ce can be calcula ed as
TVI( )=ω
2π +2π/ω
TVI( )d =ρω
2π +2π/ω
˙
h( )+˙α( )(c−xe)Γe( )d
=ρU∞
ω
2π +2π/ω
wbs( )Γe( )d ,(4.6)
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Jou nal o Fluid Mechanics
whe e we ha e aken in o accoun ha , since he h us is a pe iodic unc ion o ime,
 +2π/ω
d
d αc
0
Γadxd =0,(4.7)
and
wbs( )=˙
h( )+˙α( )(c−xe)
U∞
(4.8)
deno es minus he dimensionless eloci y o he ailing edge o he ai oil and,
consequen ly,
−w
U∞x
c=3
4, =−w
3/4( )
U∞=wbs( )+αU∞−˙αc
4
U∞
.(4.9)
The esul in (4.6) exp esses ha , o hose cases in which he ai oil oscilla es
pe iodically, he con ibu ion o he s a ing o ex o h us is negligible and he o ices
in he wake a e a anged in such a way ha hey a e con ec ed wi h a eloci y pa allel
o he ee-s eam eloci y, he mean h us is p opo ional o he mean o he p oduc o
minus he e ical eloci y o he ailing edge and he ci cula ion a ound he ai oil. F om
now on, since we a e conside ing he case o ai oils oscilla ing pe iodically, any gene ic
ime-dependen unc ion s( )will be exp essed as he eal pa o
s( )=s∗eikτ,(4.10)
whe e he cons an s∗is a complex numbe , k=ωc/(2U∞)is he educed equency and
τ=2 U∞/c e e s o he dimensionless ime.
Using he esul s in e.g. Ashley & Landahl (1985) o in sec ion V o he Supplemen a y
Ma e ial, he analy ical exp ession o he ime-pe iodic ci cula ion a ound he ai oil,
namely, Γe( )=πU∞c(A0(τ) +A1(τ)/2)/2, is gi en by he eal pa o
Γe( )=πU∞c
2
G∗
ik eikτ,(4.11)
whe e we ha e made use o he no a ion in (4.10). In (4.11), see Ashley & Landahl (1985)
o sec ion V o he Supplemen a y Ma e ial,
G∗=−w
3/4
U∞∗
×2e−ik
∞
1(1+χ)e−ikχ
√χ2−1dχ=−w
3/4
U∞∗
×2e−ik
K0(ik)+K1(ik),(4.12)
whe e Knin (4.12) indica es he modi ied Bessel unc ion o he second kind o o de n.
Now, using he esul s in (4.9), (4.10) and he ac ha d/d =(2U∞/c)d/dτ
−w
3/4
U∞∗
=w∗
bs +a0eiφ1−ik
2,w
∗
bs =2ik h0
c+a0eiφ1−xe
c,
(4.13)
wi h h( )=h0R(eikτ),α( )=a0R(eiφeikτ),h0,a0 eal numbe s indica ing he
ampli udes o he hea ing and he pi ching mo ions and φ he phase shi . The subs i u ion
1012 A6-23
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J.M. Go dillo
10–2 10–1 100
k
101
–
0.2
0
0.2
0.4
0.6
0.8
1.0
Figu e 7. The igu e shows he unc ions GΓ(k)(blue line) and FΓ(k)( ed line) de ined in (4.15), as well as
he unc ion D(k)(black line) de ined in (4.17).
o (4.12)and(4.13)in o(4.11) yields
Γe( )=πU∞cR⎛
⎜
⎜
⎝w∗
bs +a0eiφ1−ik
2×
e−ik
(ik)
K0(ik)+K1(ik)eikτ⎞
⎟
⎟
⎠
=πU∞cRw∗
bs +a0eiφ1−ik
2×(GΓ(k)−iF
Γ(k))eikτ,(4.14)
whe e
e−ik
(ik)
K0(ik)+K1(ik)=GΓ(k)−iF
Γ(k). (4.15)
The unc ions FΓ(k)and GΓ(k)in (4.15) a e ela ed wi h he analogous unc ions F1(k)
and G1(k)de ined in Fe nandez-Fe ia (2016) in he ollowing way:
F1(k)=−π
2FΓ(k), G1(k)=−π
2GΓ(k). (4.16)
Fo ou subsequen pu poses, i p o es con enien o de ine he e he unc ion
D(k)=F(k)−GΓ(k)−kFΓ(k)
2(4.17)
whe e F(k)is he eal pa o he well-known Theodo sen unc ion C(k),de inedas
C(k)=K1(ik)
K0(ik)+K1(ik)=F(k)+iG(k), (4.18)
see, e.g. Ashley & Landahl (1985) o sec ion V o he Supplemen a y Ma e ial. The
di e en unc ions GΓ(k),FΓ(k)and D(k)de ined in (4.15)and(4.17) a e plo ed in
igu e 7.
Now, making use o he ac ha : (i) all ime-dependen unc ions a e o he o m gi en
in (4.10) and (ii) he eal pa o a complex numbe s∗is (s∗+s∗c)/2, wi h he supe sc ip
1012 A6-24
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Jou nal o Fluid Mechanics
ca ec ing he complex a iable s∗indica ing he complex conjuga e o s∗, he subs i u ion
o (4.8), (4.9)and(4.14)in o(4.6) yields
TVI =ρπU2
∞c
4
×w∗
bsw∗c
bs (GΓ(k)+iF
Γ(k))+a0w∗
bse−iφ1+ik
2(GΓ(k)+iF
Γ(k))
+ρπU2
∞c
4×w∗
bsw∗c
bs (GΓ(k)−iF
Γ(k))+a0w∗c
bs eiφ1−ik
2(GΓ(k)−iF
Γ(k))
=2ρπU2
∞ck
2GΓ(k)×h0
c2
+cos φa0h0
c3
4−a
+ρπU2
∞ckG
Γ(k)×k
2a2
0(1−a)1
2−a+a0h0
csin φ
−ρπU2
∞ckF
Γ(k)×a0h0
ccos φ+k
2a0h0
csin φ+a2
0
2(1−a),(4.19)
and, hence, he exp ession o he mean h us coe icien when he con ibu ions o h us
o bo h he s a ing o ex and o he o ices in he wake a e se o ze o, is
CT(k)=TVI
1/2ρU2
∞c=4πk2GΓ(k)
×h0
c2
+a0h0
c3
4−acos φ+a2
0
4(1−a)1
2−a
+2πka
0h0
c
×GΓ(k)sin φ−FΓ(k)cos φ−k
2FΓ(k)sin φ−πkF
Γ(k)a2
0(1−a),
(4.20)
whe e we ha e made use o he equa ion xe=(1+a)c/2 ela ing he posi ion o he
pi ching axis xewi h a. The equa ion o he mean h us coe icien gi en by (4.20),
which does no ake in o accoun nei he he o ex o ce e m (2.40) no he con ibu ion
o he s a ing o ex, di e s om he mean h us coe icien deduced by Ga ick (1936);
see also (B4) in Fe nandez-Fe ia (2016), ep oduced he e o cla i y pu poses
CTG(k)=4πk2F2(k)+G2(k)h0
c2
+πa2
0F2+G21+k21
2−a2
+πa2
0a−1
2F−1
2k2−a+1
2kG −F+πa0h0
c$4kF2+G2sin φ
%
+πa0h0
c4k21
2−aF2+G2cos φ−2k2(Gsin φ+Fcos φ)
+πa0h0
c$2k(Gcos φ−Fsin φ)+k2cos φ%,(4.21)
1012 A6-25
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J.M. Go dillo
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