ARTICLES
Nonpa allel local spa ial s abili y analysis o pipe en ance
swi ling lows
M. A. He ada, M. Pe
´ ez-Sabo id, and A. Ba e o
Escuela Supe io de Ingenie os, Uni e sidad de Se illa, 41092 Se illa, Spain
共Recei ed 29 Oc obe 2003; accep ed 9 Ma ch 2004; published online 12 May 2004兲
A spa ial local iscous s abili y analysis o a swi ling low de eloping in a cylind ical pipe has been
ca ied ou nume ically. E en a mode a ely low swi l s eng hs, we ha e ound he exis ence o
cen i ugal modes in addi ion o he shea ones ound in p e ious s abili y analysis o nonswi ling
lows de eloping in pipes. I is ound ha hese cen i ugal ins abili ies de elop a Reynolds
numbe s ha a e much lowe han hose equi ed o he g owing o he shea ins abili y. Mo eo e ,
he ex en o he egion whe e cen i ugal ins abili ies appea is much la ge han ha whe e he
shea laye ins abili y g ows. We ha e ound om he analysis ha he mos uns able mode was he
coun e - o a ing one (n⫽⫺1). The c i ical Reynolds numbe o which linea analysis p edic s he
g ow h o he con ec i e ins abili ies is o he cen i ugal modes one hund ed imes smalle han o
he shea laye ones. © 2004 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.1728158兴
I. INTRODUCTION
The e ec o he swi l s eng h on he s abili y o de el-
oping swi ling lows in pipes has no been su icien ly in es-
iga ed ye . I is well known ha in he absence o swi l, he
ully de eloped Poiseuille low in a pipe is s able o in ini-
esimal axisymme ic and nonaxisymme ic dis u bances un-
de bo h empo al and spa ial analyses. This ac mo i a ed
he s udy o he linea s abili y o he de eloping low in he
pipe en ance.1–5 E en hough he la e analyses succeed in
asce aining egions o ins abili y wi hin he lamina en ance
egion, hey yield c i ical Reynolds numbe s o ins abili y
whose alue di e s om one s udy o ano he . The disc ep-
ancies be ween he exis ing analyses may be a ibu ed o he
calcula ion o he basic low, o he di e en app oaches
共spa ial o empo al兲 o he s abili y p oblem, and also o he
di e en ea men o he nonpa allel e ms in he linea ized
equa ions. E en wo se is he disag eemen be ween he he-
o e ical esul s and he expe imen al ones,6 he ini e ampli-
ude na u e o he applied dis u bances and possible bypass
mechanisms we e sugges ed as a possible explana ion o
hese disc epancies.5
The linea s abili y esul s o he ully de eloped Poi-
seuille low in pipes a e modi ied signi ican ly in he p es-
ence o swi l. In e ec , he empo al s abili y analysis i s
ca ied ou by Pedley7,8 showed ha Poiseuille low wi h
supe imposed solid body o a ion becomes uns able o non-
axisymme ic dis u bances, while i emains s able o axi-
symme ic ones. These esul s we e la e con i med and ex-
ended by mo e comple e empo al s abili y s udies.9–12 I
was ound he e ha he mos uns able dis u bance co e-
sponded o he mode wi h azimu hal wa e numbe n⫽⫺1.
Spa ial s abili y analyses13,14 o he same basic low ha e
shown ha he low was con ec i ely uns able unde nonaxi-
symme ic pe u ba ions; he mos uns able mode being p e-
cisely he n⫽⫺1 one p e iously ound by empo al s abili y
analyses. In addi ion, a ansi ion om con ec i e o abso-
lu e ins abili y was ound o su icien ly la ge alues o he
Reynolds numbe and swi l s eng h.14
Despi e he a o emen ioned s udies, which deal wi h
ully de eloped swi ling low in a o a ing pipe, he di e en
p oblem o he s abili y o a swi ling low de eloping in he
en ance egion o a pipe a es has no been in es iga ed
ye . In his pape , we add ess his p oblem conside ing a
basic low a he pipe en ance wi h an uni o m axial eloci y
p o ile and wi h an azimu hal eloci y p o ile o he Bu ge
ype. The de elopmen o he esul ing swi ling basic low in
he pipe en ance egion has been compu ed in a sel -
consis en manne by sol ing he s eady Na ie –S okes
equa ions in he slende app oxima ion using a s anda d
me hod o lines. Then, he comple e se o Na ie –S okes
equa ions a e linea ized a ound ha basic solu ion in o de o
pe o m a local, nonpa allel, spa ial s abili y analysis a di -
e en pipe sec ions, which pe mi s one o de e mine he
c i ical Reynolds numbe beyond which con ec i ely un-
s able pe u ba ions de elop. The esul s show ha e en a
small amoun o swi l modi ies comple ely he pic u e o he
low wi h espec o he nonswi ling case. In pa icula , he
c i ical Reynolds numbe s ound o swi ling lows a e much
smalle han hose o nonswi ling ones. Also he ex en o
he egion whe e he low may become uns able is much
la ge o he swi ling case han o he nonswi ling one.
The gene al p oblem o mula ion and he nume ical
scheme a e desc ibed in Sec. II. Nume ical esul s on he
low s abili y a e gi en and discussed in Sec. III. Finally, a
summa y o he esul s is gi en in Sec. IV.
PHYSICS OF FLUIDS VOLUME 16, NUMBER 7 JULY 2004
21471070-6631/2004/16(7)/2147/7/$22.00 © 2004 Ame ican Ins i u e o Physics
II. FORMULATION OF THE PROBLEM
A. The basic o ex
Le us in es iga e he s eady, axisymme ic de eloping
swi ling low a he en ance egion o a cylind ical pipe o
adius R. The a iables ha e been made dimensionless using
Rand RRe as adial and axial leng h scales, W0as cha ac-
e is ic eloci y in he axial and azimu hal di ec ions,
/Ras
cha ac e is ic eloci y in he adial di ec ion, and
W0
2as
cha ac e is ic p essu e; W0is he cha ac e is ic axial eloci y
a he en ance and
and
a e he densi y and kinema ic
iscosi y o he luid, espec i ely. I he cha ac e is ic Rey-
nolds numbe , Re⫽W0R/
is la ge enough, he eloci y com-
ponen s u0, 0,w0in cylind ical coo dina es ( ,
,z) and
p essu e, p0, o he basic swi ling low sa is y, up o o de o
Re⫺2, he ollowing:
1
共 u0兲⫹
w0
z⫽0, 共1兲
0
2
⫽
p0
,共2兲
u0
0
⫹w0
0
z⫹
0u0
⫽
冋
1
冉
0
冊
⫺
0
2
册
,共3兲
u0
w0
⫹w0
w0
z⫽⫺
p0
z⫹
冋
1
冉
w0
冊
册
.共4兲
Equa ions 共1兲–共4兲ha e been sol ed subjec o he ol-
lowing bounda y condi ions.
A he axis, ⫽0,
u0⫽ 0
⫽
w0
⫽0共 egula i y and symme ic condi ions兲.
共5兲
A he pipe wall, ⫽1, he impe meabili y and nonslip
condi ions ead
0⫽w0⫽u0⫽0. 共6兲
Since we do no ha e any in o ma ion abou he low
ups eam o he ini ial s a ion, an uni o m axial eloci y and
an azimu hal eloci y p o ile o he Bu ge s o ex ype ha e
been assumed a he pipe en ance z⫽0,
w0⫽1, 0⫽S
␦
关1⫺exp共⫺共 /
␦
兲2兲兴.共7兲
␦
⬍1 is he dimensionless o ex co e leng h and Sis a swi l
pa ame e which cha ac e izes he s eng h o he o ex a
he en ance. Condi ions 共7兲a e s anda d in he swi ling low
li e a u e and hei use is jus i ied in linea s abili y analyses
o de eloping swi ling lows since he condi ions o he low
a he pipe en ance should no a ec signi ican ly he de el-
opmen o ins abili ies due o he p esence o a swi l.
To sol e Eqs. 共1兲–共4兲wi h bounda y condi ions 共5兲–共7兲,
we ha e used a s anda d explici me hod o lines. Fo ha
pu pose, we ha e i s elimina ed he p essu e in Eqs. 共1兲–共4兲
by adding he esul o aking de i a i es wi h espec o zin
Eq. 共2兲 o ha ob ained by aking de i a i es wi h espec o
in Eq. 共4兲. The esul ing sys em o equa ions is hen dis-
c e ized in he adial di ec ion a gi en poin s, i⫽(i
⫺1)/(N⫺1), i⫽1,...,Nusing second-o de cen al di e -
ences, and a e some algeb a, one can ob ain a simple idi-
agonal sys em which yields he alues o he adial eloci y
a he di e en lines, (u0
i)i⫽1
N, in e ms o (w0
i)i⫽1
Nand
( 0
i)i⫽1
N. In his manne , one a i es a a sys em o 2No di-
na y di e en ial equa ions o (
w/
z)i⫽1
Nand (
/
z)i⫽1
N
which can be sol ed, o example, wi h a s anda d a iable
s ep-size ou h-o de Runge–Ku a me hod. In his p oblem,
we ha e conside ed N⫽400 lines, and a ela i e e o ole -
ance o 10⫺6 o he Runge–Ku a sol e , which allow o a
e y s able and accu a e downs eam in eg a ion o he equa-
ions. I should be ema ked ha he use o he me hod o
lines a oids complica ed and expensi e nonlinea i e a ions
equi ed, o example, by implici ini e di e ence me hods.
Figu es 1 and 2 show he adial p o iles o bo h he axial
eloci y 共1兲and he azimu hal eloci y 共2兲o a basic low
FIG. 1. Radial dis ibu ion o he axial eloci y o he basic low a se e al
zs a ions o he pipe; S⫽0.2 and
␦
⫽0.5.
FIG. 2. Radial dis ibu ion o he azimu hal eloci y o he basic low a
se e al zs a ions o he pipe; S⫽0.2 and
␦
⫽0.5.
2148 Phys. Fluids, Vol. 16, No. 7, July 2004 He ada, Pe
´ ez-Sabo id, and Ba e o
(S⫽0.2 and
␦
⫽0.5兲a se e al zs a ions o he pipe. No e
ha he axial eloci y e ol es downs eam owa d a p o ile
o he Poiseuille ype while he swi l decays.
Figu e 3 shows he eloci y a he axis, wc⬅w(
⫽0,z), as a unc ion o dimensionless pipe axial dis ance, z,
o se e al alues o he swi l s eng h, S, and o
␦
⫽0.5. I
can be obse ed ha in all cases shown in Fig. 3, he classi-
cal 共nonswi ling兲Poiseuille low (wc→2) is eached su i-
cien ly a om he pipe en ance; he azimu hal eloci ies
being e en ually damped by iscous di usion. Fo mode -
a ely small swi l s eng hs, he de elopmen o he swi ling
low does no di e much om he nonswi ling one. How-
e e o la ge swi l s eng hs (S⫽1.5 in Fig. 3兲,wcp esen s
a nonmono onic e olu ion along he pipe and he dis ance
equi ed o each he pu e Poiseuille low inc eases. This
inc ease in he en ance leng h may be due o he p essu e
g adien associa ed wi h cen i ugal o ces which lowe s he
g ow h a es o he wall bounda y laye s. Finally, o a swi l
s eng h S⬃1.6 共no shown in he igu e兲 he downs eam
in eg a ion o he basic low e en ually ails a some zs a ion
close o he pipe en ance. Some au ho s15–18 ha e in e -
p e ed his ailu e o he slende 共QC兲app oxima ion o de-
sc ibe he basic low as he occu ence o o ex b eakdown.
Howe e , in his s udy, we a e no in e es ed in he o ex
b eakdown phenomenon 共which equi es azimu hal eloci-
ies o he same o de as he axial ones兲, bu on he linea
s abili y analysis o a de eloping slende and axisymme ic
swi ling low wi h mode a ely low S alues.
B. Nonpa allel local linea s abili y o mula ion
As usual in s abili y analysis, we spli he eloci y and
p essu e ields (u, ,w,p) in o wo p oblems: he basic p ob-
lem (u0, 0,w0,p0) and a pe u ba ion one (u
¯
,
¯
,w
¯
,p
¯
),
u⫽u0/Re⫹u
¯
, ⫽ 0⫹
¯
,w⫽w0⫹w
¯
,p⫽p0⫹p
¯
.
共8兲
The p esence o he Reynolds numbe in he e m ep esen -
ing he basic adial eloci y in Eq. 共8兲is due o he use o W0
in he de ini ion o he nondimensional eloci y ield
(u, ,w).
The pe u ba ion ec o s⬅关u
¯
,
¯
,w
¯
,p
¯
兴is usually w i en
in he s anda d o m:
s共 ,z,
, 兲⫽S共 兲X共z,
, 兲,共9兲
whe e he complex eigen unc ions only depend on he adial
coo dina e
S共 兲⬅关u1共 兲, 1共 兲,w1共 兲,p1共 兲兴,共10兲
while he exponen ial pa o he pe u ba ions desc ibes he
wa e-like na u e o he dis u bance,
X共z,
, 兲⬅exp关Rekz⫹i共n
⫺
兲兴.共11兲
⬅⍀R/W0and k⬅Rk*a e he nondimensional equency
and he axial wa e numbe , espec i ely, and ⍀and k*a e
he co esponding dimensional alues. The eal and he
imagina y pa s
␥
and
␣
o he complex axial wa e numbe
k⬅
␥
⫹i
␣
,共12兲
a e he exponen ial g ow h a e and he axial wa e numbe ,
espec i ely.
In oducing 共8兲in o he Na ie –S okes equa ions and
neglec ing e ms o o de Re⫺2and highe , one a i es a he
ollowing se o nonpa allel, linea , s abili y equa ions:
w1k⫹
u1
⫹u1
⫹in 1
⫽0, 共13兲
i
u1⫺kw0u1⫺u0
Re
u1
⫺u1
Re
u0
⫹2 0 1
⫺in 0 1
⫺
p1
⫹1
Re
冋
2u1
2⫹1
u1
⫺
冉
⫺k2⫹n2
2
冊
u1⫺u1
2
⫺2in 1
2
册
⫽0, 共14兲
i
1⫺kw0 1⫺u0
Re
1
⫺w1
Re
0
z⫺u0
Re
1
⫺in 0 1
⫺u1
冉
0
⫹
0
冊
⫺in p1
⫹1
Re
冋
2 1
2⫹1
1
⫺
冉
⫺k2⫹n2
2
冊
1⫹2inu1
2⫺
1
2
册
⫽0, 共15兲
i
w1⫺kw0w1⫺u0
Re
w1
⫺w1
Re
w0
z⫺u1
w0
⫺in 0w1
⫺kp1⫹1
Re
冋
2w1
2⫹1
w1
⫺
冉
⫺k2⫹n2
2
冊
w1
册
⫽0. 共16兲
Equa ions 共13兲–共16兲mus be sol ed subjec o he ol-
lowing adial bounda y condi ions:19
⫽1: u1⫽ 1⫽w1⫽0; 共17兲
FIG. 3. Veloci y a he axis wcas unc ion o z o
␦
⫽0.5 and di e en
alues o he swi l pa ame e .
2149Phys. Fluids, Vol. 16, No. 7, July 2004 Nonpa allel spa ial s abili y analysis o pipe
⫽0:
再
u1⫽ 1⫽0,
w1/
⫽0, 共n⫽0兲,
u1⫾i 1⫽0,
u1/
⫽0, w1⫽0, 共n⫽⫾1兲
u1⫽ 1⫽w1⫽0, 共
兩
n
兩
⬎1兲.
,
共18兲
C. Nume ical scheme
I p o es con enien o ew i e Eqs. 共13兲–共16兲in he
o m
0⫽
冋
L1⫹1
ReL2⫹kL3⫹k2
ReL4
册
S,共19兲
whe e L1,L2,L3, and L4a e complex ma ices which de-
pend on zand . To sol e 共19兲nume ically, he equa ions a e
disc e ized in he di ec ion using a s agge ed Chebyshe
spec al colloca ion echnique.20 No e ha no bounda y con-
di ions o p essu e a e needed when using s agge ed collo-
ca ion poin s o p essu e.
Le us map he in e al 0⭐ ⭐1 in o he Chebyshe
polynomial domain ⫺1⭐
⭐1 using he algeb aic ans o -
ma ion
⫽1
2共1⫹
兲.共20兲
Then, he
coo dina e is disc e ized in Ncolloca ion poin s.
N, which coincides wi h he numbe o Chebyshe polyno-
mials con ained in he spec al ep esen a ion o S, anges
he e be ween 40 and 50.
The sys em is sol ed using he linea companion ma ix
me hod desc ibed in Re . 21. The esul ing linea eigen alue
p oblem is sol ed using an eigen alue sol e sub ou ine
DGVCCG om he IMSL lib a y, which p o ides he en i e
spec um o eigen alues and eigen unc ions. Spu ious eigen-
alues we e uled ou by compa ing he compu ed spec ums
ob ained o di e en alues o he numbe No colloca ion
poin s.
III. RESULTS
P e iously o s udy he e ec o he swi l on he s abili y
o he low a he en ance egion o he pipe, we ha e ana-
lyzed he nonswi ling case, S⫽0. In his case, which has
been widely in es iga ed in he pas ,1–5 he e is only a el-
e an physical mode 共we call his mode he shea mode兲 ha
i he Reynolds numbe is su icien ly high becomes con ec-
i ely uns able 共
␥
⬎0 and cg⫽
/
␣
⬎0),22 wi hin a ce ain
egion close o he pipe inle . The c i ical Reynolds numbe ,
Re*, a each zs a ion is de ined as he minimum Reynolds
numbe o which he shea mode becomes neu ally s able
共
␥
⫽0兲a a ce ain equency
*共c i ical equency兲, while i
emains s able 共
␥
⬍0兲 o any o he equency. The a ia ion
o he c i ical Reynolds numbe wi h z o bo h axisymme ic
n⫽0 and nonaxisymme ic pe u ba ion n⫽1 is plo ed in
Fig. 4. The co esponding a ia ion o he c i ical equency
*wi h zis plo ed in Fig. 5. No e ha , o he nonswi ling
case conside ed in hese igu es, he axisymme ic mode is
he mos uns able one 共absolu e minimum alue o Re*兲.
Also, o Reynolds numbe s la ge han he minimum alue
o Re*, he egion whe e he low is uns able is e y close o
he pipe en ance; obse e ha he dimensionless leng h o
he egion is less han 0.01 while he en ance leng h o he
basic low is oughly hi y imes la ge , see Fig. 3. These
esul s a e in ag eemen wi h hose ob ained in p e ious lin-
ea s abili y analyses. I should also be poin ed ou ha ou
nume ical esul s ag ee p e y well wi h hose ob ained by
Ga g, who also used a nonpa allel app oxima ion o desc ibe
he s abili y o he low.2
The s abili y esul s ge comple ely modi ied when lows
wi h a small amoun o swi l a e conside ed, e en hough he
downs eam e olu ion o some quan i ies o he basic low
such as he eloci y a he axis 共see Fig. 3兲, he p essu e, and
he s ess a he wall pipe a e nea ly he same in nonswi ling
and swi ling lows. This is he case o a low wi h S⫽0.2 and
␦
⫽0.5 o which ou nume ical esul s showed he exis ence
o new modes 共cen i ugal modes兲in addi ion o he shea
FIG. 4. C i ical Reynolds numbe o nonswi ling lows as a unc ion o z:
axisymme ic n⫽0 mode 共b oken line兲; nonaxisymme ic n⫽1 mode 共solid
line兲.
FIG. 5. C i ical equency o nonswi ling lows as a unc ion o z: axisym-
me ic n⫽0 mode 共b oken line兲; nonaxisymme ic n⫽1 mode 共solid line兲.
2150 Phys. Fluids, Vol. 16, No. 7, July 2004 He ada, Pe
´ ez-Sabo id, and Ba e o
modes ound in he S⫽0 case. The shea modes become
con ec i ely uns able o e y high Reynolds numbe
共Re⬃14000兲while he cen i ugal ones become con ec-
i ely uns able a much smalle Reynolds numbe s. In addi-
ion, we ha e ound ha he coun e - o a ing pe u ba ions
(n⬍0) become con ec i ely uns able a Reynolds numbe s
o he o de o 200. Le us poin ou ha cen i ugal ins a-
bili ies a ela i ely low Reynolds numbe s ha e also been
ound in he s abili y analyses o o he di e en swi ling
lows such as columna ee o ex lows o he Bu ge s
ype.23
To in es iga e he appea ance o hese modes, we ha e
i s s udied he beha io o he swi ling basic low unde
axisymme ic pe u ba ions n⫽0. In his case, only shea
modes a e ound, which become uns able a alues o Re
sligh ly smalle han hose ound in he nonswi ling case, see
Fig. 6. Obse e ha he s abili y cha ac e is ics o he axi-
symme ic modes do no change signi ican ly by he p es-
ence o a small amoun o swi l in he basic low. The si u-
a ion looks qui e simila o he case o co- o a ing
pe u ba ions (n⬎0), see Fig. 7; he e o e, we conclude ha
bo h axisymme ic and co- o a ing modes a e o he shea
ype. A comple ely di e en si ua ion p esen s i sel when
coun e - o a ing pe u ba ions (n⬍0) a e conside ed. In e -
ec , in addi ion o he shea modes, we ha e ound in his
case a new se o modes, which we ha e called cen i ugal
modes. These modes become uns able a much smalle Rey-
nolds numbe s han hose o he shea ype. Figu e 8 shows
he a ia ion o he c i ical Reynolds numbe wi h z o cen-
i ugal modes wi h azimu hal wa e numbe s n⫽⫺1, n
⫽⫺2, and n⫽⫺3, espec i ely. No e in Fig. 8 ha he c i i-
cal Reynolds numbe s a e no only much smalle han hose
in he case o shea modes bu he ex en o he ins abili y
egion is o he o de o he en ance leng h o he pipe. Fo
example, o Re⫽250 he low is uns able unde coun e -
o a ing pe u ba ions wi h n⫽⫺1 in a la ge po ion (z
⬃0.15) o he en ance egion, which ex ends up o z⬃0.3.
In addi ion, i can be obse ed in Fig. 8 ha he highes o de
coun e - o a ing modes a e he mos uns able ones in a e-
gion e y close o he pipe en ance, bu his endency
quickly changes as we mo e downs eam whe e he lowes
o de coun e - o a ing mode n⫽⫺1 has he smalles c i ical
Reynolds numbe s in mos o he en ance egion. The e o e,
n⫽⫺1 seems o be he mos dange ous mode o he g ow h
o cen i ugal ins abili ies in de eloping swi ling lows. I is
wo h no ing ha in he case o a Poiseuille low wi h supe -
imposed solid body o a ion, he n⫽⫺1 modes we e also he
mos uns able ones.
Le us now discuss he e ec on he s abili y analysis o
␦
, which measu es he cha ac e is ic adius o he o ex co e
a he pipe en ance 关see Eq. 共7兲兴. Thus, a small alue o
␦
co esponds o he case o a e y concen a ed nea -axis o -
ex low, while a alue
␦
⬃1 co esponds o he case o a
swi ling low a he pipe en ance wi h nea ly solid body
FIG. 6. C i ical Reynolds numbe s o nonswi ling, S⫽0, and swi ling, S
⫽0.2, lows o axisymme ic n⫽0 modes and
␦
⫽0.5. FIG. 7. C i ical Reynolds numbe s o nonswi ling, S⫽0, and swi ling, S
⫽0.2, lows o co- o a ing n⫽1 modes and
␦
⫽0.5.
FIG. 8. C i ical Reynolds numbe o he cen i ugal modes n⬍0共coun e -
o a ing modes兲 o S⫽0.2 and
␦
⫽0.5.
2151Phys. Fluids, Vol. 16, No. 7, July 2004 Nonpa allel spa ial s abili y analysis o pipe
o a ion. Fo he case n⫽⫺1, Fig. 9 shows he c i ical Rey-
nolds numbe as a unc ion o
␦
calcula ed a he pipe sec ion
z⫽0.08, which is he sec ion a which he minimum c i ical
Reynolds numbe is eached, see Fig. 8. Obse e ha he
e olu ion o c i ical Reynolds numbe wi h
␦
is nonmono-
onic, i i s inc eases and hen dec eases up o eaching i s
minimum alue o he case o almos solid body o a ion
lows 共
␦
⫽1兲; ne e heless, independen o he concen a ion
o o ex, he low become uns able o Reynolds numbe s
la ge han 290.
Finally, we ha e also in es iga ed he e ec o he swi l
s eng h Son he low s abili y. Again, we ha e ound ha
he smalles c i ical Reynolds numbe s a e ob ained o mode
n⫽⫺1 in a pipe sec ion nea z⫽0.08. Figu e 10 shows he
c i ical Reynolds numbe as a unc ion o Sa z⫽0.08 o
n⫽⫺1 and
␦
⫽1. Obse e ha Re*dec eases mono onically
wi h S, bu he dependence is much s onge o small S
alues, o which Re*becomes e y high, in ag eemen wi h
he esul s ob ained o he case o nonswi ling lows.
IV. SUMMARY AND CONCLUSIONS
We ha e ca ied ou a spa ial, local, iscous, linea s a-
bili y analysis o de eloping swi l lows in he en ance e-
gion o a cylind ical pipe a es . The nume ical esul s o
he nonswi ling case (S⫽0) con i m he esul s o p e ious
in es iga ions. Only lows a e y la ge Reynolds numbe s
de elop axisymme ic and nonaxisymme ic shea laye in-
s abili ies in a egion o he pipe which is loca ed e y close
o he pipe en ance and whose ex en is sho compa ed o
he o al en ance leng h. Fo swi ling lows, e en a small
alues o S, he s abili y esul s a e qui e di e en . In his
case, we ha e ound bo h axisymme ic n⫽0 and co- o a ing
n⬎0 pe u ba ions, which beha e much in he same way as
he shea laye modes o nonswi ling case. On he o he
hand, a new ype o uns able modes associa ed wi h coun e -
o a ing pe u ba ions n⬍0 appea . These modes, which we
ha e called cen i ugal modes, become uns able a much
smalle Reynolds numbe s and, con a y o he shea modes,
he egion o low ins abili y ex ends o a la ge po ion o he
en ance egion. Nume ical esul s show ha coun e - o a ing
mode n⫽⫺1 is he mos uns able one. I has also been ound
ha e en o swi l s eng h ha is no oo la ge, S⫽0.2, he
c i ical Reynolds numbe o which linea analysis p edic s
g ow h o con ec i e ins abili ies is o he cen i ugal
modes abou 100 imes smalle han o he shea ones.
ACKNOWLEDGMENT
This wo k has been pa ially suppo ed by he Di ec io
´n
Gene al de Ensen
˜
anza Supe io o Spain, P ojec No.
BMF2000-0528.
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