MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
_________________
DOI: 10.2478/ms -2020-0013
104
FSI Compu a ion and Expe imen al Ve i ica ion o Fluid Flow in
Flexible Tubes
Dominik Šedi ý
1
, Simona Fialo á
1
, Roman Klas
1
, Michal Ko ek
2
1
B no Uni e si y o Technology, Vik o Kaplan Depa men o Fluids Enginee ing, Technicka 2896/2, 61669 B no,
Czech Republic, co esponding au ho email: [email p o ec ed] b .cz
2
Depa men o Physical Measu emen , Technical Uni e si y o Libe ec, The Ins i u e o Nanoma e ials, Ad anced
Technology and Inno a ion, S uden ska 1402/2 Libe ec 1 461 17, Czech Republic
P esen ed pape is ocused on he expe imen al and compu a ional s udy o luid low in pipes wi h lexible walls. One possible eal
example o his phenomenon is he blood low in a e ies o hei subs i u es in he human body. The a e y ma e ial i sel should be
unde s ood as aniso opic and he e ogeneous. The e o e, he expe imen was ca ied ou on he de o ming ube, made o silicone
(polydime hylsiloxane - PDMS). Ob ained esul s and obse ed e en s we e e i ied by nume ical FSI simula ions. Due o he la ge
de o ma ions occu ing du ing loading o he ube, i was necessa y o wo k wi h a dynamic mesh in he CFD pa . Based on expe imen al
es ing o he ube ma e ial, a non-Hookean and Mooney-Ri lin ma e ial model we e conside ed. Blood lowing in essels is a
he e ogeneous liquid and exhibi s non-New onian p ope ies. In he eal expe imen al s and has been somewha simpli ied. Wa e , chosen
as he liquid, belongs o he New onian liquids. The esul s show mainly compa isons o uns eady eloci y p o iles be ween he expe imen
and he nume ical model.
Keywo ds: Fluid S uc u e In e ac ion, Pa icle Image Velocime y, simula ions, expe imen al e i ica ion, lexible ubes.
1.
I
NTRODUCTION
P esen ed pape is ocused on he s udy o luid low in
pipes wi h lexible walls. One possible eal example o his
phenomenon is he blood low in blood essels (especially
la ge a e ies) in he human body o hei subs i u es in he
o m o a i icial blood essels o allo ansplan s o
xeno ansplan s. Degene a i e ca dio ascula disease,
accompanied by a he oscle o ic mani es a ions in dila o y o
obli e a e o m, leads o blood ci cula ion diso de s o
aneu ysms, which, in pa icula ly la ge a e ies such as he
ao a, may cause hei up u e and subsequen bleeding [1],
[2]. Degene a i e ascula disease is gene ally a he op o
he causes o dea h. The e o e, i is necessa y o add ess his
p oblem om he enginee ing poin o iew, which akes in o
accoun he beha io o eal essels o bioine polyme s
loaded by pulsa ing luid low, based on expe imen and
p edic i e ma hema ical-physical models.
Al hough he p esen ime o e s ex ensi e possibili ies in
he ield o nume ical me hods, i is no easy o ealize a
ai h ul simula ion o luid low h ough a lexible ube in a
luid-s uc u e in e ac ion (FSI) ask [3]. In mos cases he
esul s a e limi ed o an expe imen o o a nume ical
simula ion wi hou compa ison. In wo ks [4] and [5] we e
done expe imen al measu emen s wi h pulsa ile low and
s aigh elas ic ube. Nex s ep was expe imen al simula ion
o low in cu ed elas ic ube [6]. All hese wo ks we e done
wi hou any nume ical compa ison. On he o he side he e
a e pape s like [7] and [8], which sol ed low in mo e
complica ed geome ies only by nume ical me hods. Due o
he e y low luid lows, he non-s a iona y na u e o
physical quan i ies, he size o he simula ed a ea, and he
sensi i i y o bounda y and ini ial condi ions, he deg ee o
ideli y o he esul s achie ed by nume ical simula ion can
be p oblema ic. On he o he hand, in he expe imen i is
p ac ically impossible o achie e he same condi ions as in
he human body, no is i possible o ob ain all he necessa y
da a in his way. The aim o he s udy was he e o e p ima ily
o compa e he expe imen al pa ealized by pa icle image
elocime y (PIV) wi h nume ical esul s o luid-s uc u e
in e ac ion (FSI) simula ion, which in addi ion o p edic ion
allows a ela i ely la ge deg ee o e alua ion and da a
collec ion o u he s udy and analysis.
2. E
XPERIMENT
The a e y ma e ial i sel should be unde s ood as
aniso opic and he e ogeneous. I exhibi s la ge elas ic
de o ma ions and is classi ied as so-called hype elas ic
ma e ials [9], which a e cha ac e ized by non-linea
Jou nal homepage: h ps://con en .sciendo.com
MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
105
dependence be ween s ess and s ain. I is impossible o
de ine a e age ma e ial p ope ies o a e ies. The p ope ies
depend on ype o a e y, gende , age and s a e o heal h [10].
Blood lowing in blood essels is a he e ogeneous liquid and
exhibi s non-New onian p ope ies [11]. As i was men ioned,
in he eal expe imen i is p ac ically impossible o simula e
he same condi ions as in he human body, and he e o e he
expe imen al s and has been somewha simpli ied (see
Fig.1.). Wa e , chosen as he liquid, belongs o he New onian
liquids. Fo he pu poses o basic esea ch, he g adual s eps
in selec ing a mo e common luid we e delibe a e wi h a iew
o de ining he e ec s o indi idual componen s, ma e ials
and models so ha he indi idual e ec s could be sepa a ed
om each o he and assessed well. Fo his eason, only low
in a s aigh unb anched ube was also add essed.
The de o ming ube i sel was made o silicone
(polydime hylsiloxane - PDMS), which as subsequen ly
conside ed hype elas ic and homogeneous [12], [13]. The
expe imen al ci cui was u he composed om a diaph agm
pump as a sou ce o pulsa ing low, a ela i ely igid pipeline
( ela i e o silicone), a en al e, wo p essu e senso s
(loca ed in on o and behind he ube), and a high speed
came a. In his case i was a 2D PIV, in which only one plane
is being sensed.
To p e en ube oscilla ion, he ube was p eloaded in he
axial di ec ion by 10 mm and ixed a bo h ends by clamps o
simula e he ixed suppo . This also pa ially educed
unwan ed de lec ion o he ube om he s aigh di ec ion.
Ne e heless, he e was some sligh de lec ion o he ube o
he sides. The main p ope ies o he expe imen al equipmen
a e lis ed in Table 1. I is ob ious ha he equency o
diaph agm pump is e y low. This is essen ial o make a
decision i he ma e ial beha io is iscoelas ic o no [12].
Since he equency is low, he ma e ial was conside ed only
as hype elas ic wi hou any losses.
In he ollowing Fig.2. shows he s a ic p essu e a he inle
and ou le o he ube and he mean c oss-sec ional eloci y
o one pe iod. The measu ed poin in he case o eloci ies
o cou se co esponds o hal he leng h o he expe imen al
ube, see Table 1. I is appa en om he esul s ha he
eloci y ield a ies conside ably in spi e o he ela i ely
calm p essu e cou se. This phenomenon is mainly caused by
luc ua ing p essu e di e en ial, which is no a i s sigh in
Fig.2.
The e ec o he elas ic de o ma ions o he ube on he
eloci y o he liquid is pa ly e iden a he inle and ou le
o he ube, which o cou se canno be s udied by he
expe imen . I is e y p oblema ic o place a high-speed
came a a he same ime on he p essu e sampling posi ion o
posi ional and op ical easons. Da a om Fig.2. was hen used
o de ine bounda y condi ions o nume ical simula ion. The
cou se o eloci ies o he nex pe iods is shown in Fig.3. I
is ob ious om he ime eco d o indi idual pe iods ha he
plo is no pe ec ly pe iodic. This is due o he inaccu acies
o measu emen and especially he cha ac e is ics o he
expe imen al ci cui , which is concei ed as closed.
Fig.1. Expe imen al s and.
Fig.2. Expe imen al esul s o one pe iod.
Fig.3. Mean axial eloci y o 3 pe iods.
-0,1
0
0,1
0,2
0,3
0,4
0
0,5
1
1,5
2
0,0 0,2 0,4 0,6 0,8 1,0
[m ∙ s-1]
p [ba ]
[s]
-0,1
0,4
02
[
s
]
p2 p1 mean
-0,1
0
0,1
0,2
0,3
0,4
0,0 1,0 2,0 3,0
U mean [m ∙ s-1]
[s]
MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
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Table 1. Expe imen al s and p ope ies.
Pipe p ope ies
Leng h
Inne diame e
Wall hickness
Ma e ial
500 mm
18.1 mm
2.2 mm
PDMS
Pump equency
~ 1 Hz (0.968 Hz)
P essu e senso s
Range
Accu acy
Senso s d
is ance
F equency
0
–
2 ba
0.35 % max.
ange
630 mm
500 Hz
2D PIV
Sensing equency o came a
Sensing posi ion
1000 Hz
Middle o he pipe leng h
3.
N
UMERICAL
FSI
SIMULATIONS
As men ioned in he in oduc o y passage, he main means
o e i ying and inducing he e en s obse ed du ing he
expe imen we e nume ical FSI simula ions. These
simula ions we e pe o med in ANSYS Wo kbench 17.2,
whe e Sys em Coupling was used o bind ini e elemen
me hod (FEM - solid phase) and ini e olume me hod (CFD
- liquid phase) [14]. S ess-s ain analysis u ilized T ansien
S uc u al and luid low simula ion was pe o med using
ANSYS Fluen sol e . The geome y co esponded o he
expe imen , bu due o he ime-consuming simula ion, only
a qua e o he ube was conside ed (see Fig.4. and Fig.5.).
In o he es s es ed, i was e i ied ha he da a ob ained o
a qua e and he en i e c oss-sec ion o he ube showed
absolu ely minimal de ia ions in he s a ic p essu e
o e p essu e ange. Unde he acuum esul ing om he
implemen a ion o he closed expe imen al ci cui , he
in e nal c oss-sec ion o he solid ube canno be educed o a
le el less han he ini ial unloaded con igu a ion. This esul s
in de o ma ion o he shape o he ci cula c oss-sec ion o he
ube and consequen signi ican de lec ion o he ube om
he di ec di ec ion due o addi ional s esses.
Fig.4. Scheme o luid and s uc u al domain geome ies and
bounda y condi ions.
Fig.5. Dimensions o luid domain.
Table 2. Lis o symbols.
A
, A
NH
,
A
MR
[N
-1
m
3
] T ans o m ela ion
C
1
, C
2
[Pa]
Mooney-Ri lin ma e ial
cons an s
d, d
1
, d
2
, d
3
[m]
Inne diame e s o he
simula ed domain
E
[Pa]
Young modulus
[
-
]
Da cy ic ion ac o
g
[m s
-2
]
G a i y accele a ion
G
[Pa]
Ini ial shea modulus
I


[
-]
Fi s in a ian o he igh
Cauchy-G een de o ma ion
enso
I


,
I


[
-]
Fi s and second in a ian o
he igh Cauchy-G een
de o ma ion enso
K
[Pa]
Bulk modulus
L
1
, L
2
, L
3
[m]
Leng h pa s o simula ed
domain
p, p
1
, p
2
[Pa]
[ba ]
S a ic p essu e and absolu e
s a ic p essu es o domain
inle and ou le
31
= d
3
/2,
32
[m]
Va iable inne and ou e adius
o he ube
310
,
320
[m]
S a ing inne and ou e adius
o he ube
s
3
[m]
Wall hickness o es ed ube
[s]
Time
U
[m s
-1
] Expe imen al eloci y alue in
he ube
[m s
-1
]
Flow eloci y in he ube
W
[Pa]
S ain ene gy densi y
x
[m]
x
-
coo dina e, axis o he ube
μ
[Pa s]
Dynamic iscosi y
-
liquid
ν
[
-
]
Poisson's a io
ρ, ρ
F
[kg m
-3
]
Tube and liquid densi y
Table 3. Range o he calcula ed domain.
L
1
= 25 mm d
1
= 25 mm
s
3
= 2.2 mm
L
2
= 40 mm d
2
= 15 mm
L
3
= 500 mm d
3
= 15 mm
MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
107
Table 4. Desc ip ion o nume ical model.
Simula ion FEM CFD
Ma e ial neo-Hookean wa e
Ma e ial cons an s
ρ = 1210 kg.m
-3
E = 1.9 MPa
ν = 0.5
G = 0.633 MPa
ρ
F
= 998.2
kg.m
-3
μ = 0.001 Pa.s
N . o cells 10 000 249 000
Type o elemen s quad a ic, Hex
20 linea , Hex 8
Max. skewness 0.03 0.50
Max. Aspec a io 2.81 3.92
Bounda y
condi ions
F ic ionless,
Fixed, Fluid
solid in e ace
Veloci y inle
(p essu e inle )
P essu e ou le
Damping Con ol
and dynamic mesh
upda e me hod
Nume ical
Damping: 0.1
Di usi i y
Based on Cell
Volume
Di usion
Pa ame e : 0
Solu ion
S abiliza ion -
Coe icien -
based
Scale Fac o :
0.002
Calcula ion mode
Uns eady, ime s ep = 0.005 s,
incomp essible and lamina low,
incomp essible solid
Reynolds numbe in he
middle o he ube ~ 0 - 8000
This phenomenon can be coun e ed as in he expe imen by
p eloading he ube. In he case o hype elas ic Neo-Hookean
ma e ials, he p es essing alone does no ha e a signi ican
e ec on he magni ude o he adial de o ma ions o he ube,
i a pu ely ci cula c oss-sec ion is conside ed, he same alue
o axial s esses on he inne and ou e adius o he ube and
he ixed ube leng h. The calcula ion ime o he bound ask
(Sys em Coupling mode) is a he limi o accep abili y e en
wi h such a simple ask due o possible es ing and
e i ica ion o he esul s. Simula ions we e done on
compu e s a ion wi h ollowing ha dwa e: AMD Ryzen
5 2600x Six-Co e P ocesso , RAM 32 GB. One pe iod o
simula ion (200 coupling s eps) ook abou 20 hou s o
compu ing ime. The ac ha he ull ube conside ed would
be, like i s qua e o a pu ely ci cula c oss sec ion, speaks o
he educ ion o he ube o a qua e . Howe e , he ac ual
ube exhibi s some a ia ions in ci cula i y and wall
hickness. In he ollowing Table 2. o Table 4. a e used
ma kings, compu e domain dimensions and nume ical model
se ings. Sensi i i y analysis is a pa o all nume ical
simula ions. I expe imen al da a exis s, he main c i e ion is
he ma ch o he nume ical simula ion esul wi h he
expe imen . In his pa icula case, he numbe o solid and
liquid phase compu a ional cells in he adial and axial
di ec ion was he e o e moni o ed wi h espec o hei
minimum numbe .
Due o he la ge de o ma ions occu ing du ing loading o
he ube, i was necessa y o wo k wi h a dynamic mesh in he
CFD pa o he ask. The e o e, Dynamic Mesh Upda e
Me hods in Di usion-Based Smoo hing we e chosen in he
de o ming egions, which seemed o be he mos sui able o
he a ailable op ions, see Table 4. All zones o he luid
domain excep o he liquid-elas ic wall in e ace (wall
de o ming, see Fig.4.) we e hen de ined as de o ming.
Howe e , he posi ion o hese zones has been ixed by
coo dina es and ec o s because he posi ion o hese en i ies
is in ac cons an o ixed. The wall de o ming in e ace was
subjec o he Sys em Coupling se ing wi hin which da a a e
sha ed and ans e ed be ween he s uc u al and luid sol e .
Due o he p oblems wi h he con e gence o he simula ion,
he Solu ion S abiliza ion was also used, see Table 4. The
se ing o he s abiliza ion coe icien can signi ican ly
in luence he simula ion, e.g. in he acuum a ea.
The liquid in he CFD simula ion made up o pu e wa e did
no con ain o he pa icles ha we e, by na u e, pa o he
expe imen . Howe e , he concen a ion o hese pa icles
mus be less han 2 % a a e y low le el wi h a densi y close
o wa e . The e o e, hese pa icles ha e no signi ican e ec
on he esul s ob ained. The ube low egime was conside ed
lamina . This ac , which a i s glance con adic s a
meaning ul concep ion o he ask, because in he cen e o
he ube eaches Re alue up o 8000, has i s eason. In he
ollow-up wo k should be es ed liquid, which alls in o he
ca ego y o non-New onian simila o blood. Howe e , he
u bulen beha io o non-New onian luids is no ye
su icien ly desc ibed [14], [15], [16]. F om his poin o
iew, i is mo e ce ain o conside only lamina low, e en
wi h ega d o he expe ience gained o he ollow-up wo k.
The second eason in using he lamina low egime is he
bounda y condi ions. I is known ha i is necessa y ei he o
impo he inpu eloci y p o ile (in his case uns eady) in o
CFD sol e , bu his is no known due o he loca ion o he
p essu e sampling, o o ex end he a ea by a ee sec ion o
he pipeline whe e he co esponding eloci y p o ile is
gene a ed. Howe e , he leng hening o he compu a ional
domain b ings a conside able dis o ion o he esul s
achie ed, since he compu a ional domain i sel is ela i ely
sho . In he inpu c oss sec ion de ined by he diame e d1 in
Fig.5., he Reynolds numbe is e en g ea e han hal he
leng h o he ube. This is one eason why he expec ed
uns eady u bulen eloci y p o ile has been eplaced by a
non-s a iona y pis on p o ile ha co esponds o he eloci y
inle condi ion wi h a cons an eloci y dis ibu ion ac oss he
c oss-sec ion.
In addi ion o he de ini ion o bounda y condi ions, he
se ing o he s uc u al sol e consis s in p esc ibing a
sui able hype elas ic ma e ial. Based on expe imen al es ing
o he ube ma e ial, a non-Hookean (1) and Mooney -Ri lin
(2) ma e ial model was conside ed o be he mos sui able
neo-Hookean wi h ega d o he less p oblema ic con e gence
o he p oblem.
MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
108
=



−3 (1)
=



−3+



−3 (2)
The ensile modulus o PDSM o ma e ial E was de e mined
by a uniaxial ensile es and i s p ope ies a e lis ed in
Table 4. In e ms o ideli y o nume ical simula ion, he
choice o bounda y condi ions is essen ial. The inpu and
ou pu alues o s a ic p essu es a e known om he
expe imen , see Fig.2., bu which do no ep esen a comple e
speci ica ion o he bounda y condi ions o he luidic pa o
he simula ion. A common combina ion o bounda y
condi ions in a CFD simula ion is a combina ion o eloci y
and s a ic p essu e condi ions, o a combina ion o wo
p essu e condi ions. In he la e case, howe e , one o he
p essu es shall be he o al p essu e con aining he dynamic
componen . The dynamic p essu e componen is ela i ely
small and is o en neglec ed in some applica ions. Howe e ,
due o he ela i ely small dimensions o he ube, dynamic
p essu e is impo an . In he case o an absolu ely igid ube,
i would o cou se be possible o use he magni ude o he
mean eloci ies ob ained by he 2D PIV me hod a hal he
ube leng h o de e mine he dynamic componen . Howe e ,
in he case o a lexible ube, he dynamic p essu e a he inle
o ou le o he compu ing a ea will di e conside ably
compa ed o he dynamic p essu e a hal he leng h o he
ube. The e o e, bounda y condi ions a e equi ed in
connec ion wi h Fig.2.
The e a e wo ways o deal wi h he p oblem. The i s
a ian assumes ha he ube is hal ed, he mean speed o
Fig.2. a he ou le o he ube while using he expe imen ally
de e mined s a ic p essu e a he inle o he ube. I is ac ually
a combina ion o speed and s a ic p essu e condi ions. A e
pe o ming his FSI simula ion, he app oxima e size o he
dynamic p essu e componen a he inle o he ube can be
g adually de e mined. The basic c i e ion o he co ec ness
o such a p ocedu e, bu no he only one, is, o cou se, he
FSI simula ion o he o iginal ube wi h he p essu e ou pu
condi ion co esponding o he expe imen , while main aining
he mean eloci y a he cen e o he ube, which also
co esponds o he expe imen . The second way o de ine he
dynamic componen o he inle p essu e is a one-dimensional
FSI simula ion ep esen ing a combina ion o a luid equa ion
o mo ion (3), a con inui y equa ion (4) and a leas a simple
solid phase equa ion o mo ion (5) assuming a ixed leng h
ube. The densi y o he ube ma e ial is close o wa e , see
Table 4. and he e o e i makes no sense o conside i s
ine ial e ec s. The ques ion o he iscous beha io o he
ma e ial has al eady been answe ed in ela ion o he
equency o p essu e pulses in he expe imen al ci cui . I
should u he be no ed ha al hough in he ela ionship (4)
he e is a modulus o liquid comp essibili y K, he liquid is
ne e heless unde s ood o be incomp essible. Howe e , he
con e gence o he 1D model is mo e a o able wi h his
se ing. The equa ion o mo ion o he ube (5) can easily be
ex ended by 1D models wi h neo-Hookean ma e ial (6) o
Mooney-Ri lin can be es ed wi h wo pa ame ic ma e ial
models (7) wi h cons an s C
1
= 292308 Pa and C
2
= 24359 Pa.

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


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

||

(3)

=!"
#$
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*
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
(4)

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(5)
+=+,-=
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
.
*/'
*0
!
*'/'
*'0
1
(6)
+=+23=
4

5

.
*/'
*0
!
*'/'
*'0
1&
&
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.
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16
(7)
Fig.6. Bounda y condi ions o CFD and 1D simula ions o he
inle eloci y.
Bo h ways we e used o ealize he inal FSI simula ion o
he es ube. In connec ion wi h he abo e, i is necessa y o
men ion one mo e ac , which is based on he possibili ies o
he expe imen . In Table 1., he accu acy o p essu e
ansduce s a 0.35 % o hei maximum measu able ange o
2 ba , i.e. 700 Pa, is men ioned. The absolu e p essu e sizes
a e ela i ely la ge, bu he p essu e di e en ial o he s a ic
p essu es be ween he inle and he ou le o he ube akes
subs an ially smalle alues. The ube i sel is only 500 mm
long, so he p essu e d op is minimal and he hyd os a ic
p essu e is also small. E en when using ano he ype o
encode wi h a highe accu acy class, a measu emen e o o
a simila ask canno simply be neglec ed. I is he e o e
ad isable o elimina e undesi ed p essu e inaccu acies o a
leas he s a ic ou le p essu e o Fig.2. is a simple cu e, see
Fig.6. The cou se o s a ic p essu e was he e o e eplaced by
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0,8
1
0
10000
20000
30000
40000
50000
60000
70000
80000
0 0,2 0,4 0,6 0,8 1
[m∙s-1]
p [Pa]
[s]
p essu e ou le
eloci y inle
eloci y inle 1D Neo-Hook
eloci y inle 1D Mooney-Ri lin

MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
109
he polynomial o he 6 h deg ee and in he s eady-s a e a eas
by a s aigh line. In Fig.6., he esul s o 1D FSI simula ions
ob ained h ough ela ionships (3) o (7) o he neo-Hookean
and Mooney-Ri lin ma e ial models a e also eco ded. I
seems ha 1D simula ions a e pa icula ly sui able o
checking he esul s ob ained o as a s a ing poin o e ining
bounda y condi ions. Be e ma ch o 1D Mooney-Ri lin
ma e ial model o e 1D neo-Hookean ma e ial in Fig.6.
compa ed o he inal o m o he inle eloci y inle bounda y
condi ion in ANSYS FSI simula ion is no c i ical. The eason
is o cou se he e m in he denomina o o he ela ion (7),
which s ands a he cons an C2, and which is i sel a cons an .
I is known ha he sum o C1 and C2 co esponds o G / 2 i
ν = 0.5 (Table 4.). The la ge ampli ude magni ude o he 1D
neo-Hookean model in Fig.6. is due o he na u e o he
nume ical 1D model and he inpu bounda y condi ions,
which a e in pa s by con inuous unc ions, de ined by he
measu emen in e al, o which his model can espond wo se
han he Mooney-Ri lin model. O cou se, i is ideal o i
bounda y condi ions wi h one con inuous unc ion, i
possible. Howe e , by de aul , pu ely expe imen al da a was
used in 1D simula ions.
4.
R
ESULTS
As i was men ioned be o e, one o he FSI simula ion
accu acy c i e ia was he ma ch o mean eloci ies a hal he
leng h o he ube compa ed o he expe imen al da a. The
compa ison is shown in Fig.7.
Fig.7. Compa ison o he a e age eloci y alues du ing one
pe iod.
The wa e o ms o bo h cu es a e ai ly consis en , bu i is
e iden ha he nume ical calcula ion did no a ec all he
dynamics o he expe imen al da a. This can be caused by
di e en size o ime s ep in FSI simula ion, bu also by
di e en asks. We a e no able o co e all he cha ac e is ics
o he expe imen al ack in he simula ion. The size o he
ime s ep in FSI simula ion, in u n, is ela ed o he o al
compu a ion ime, see Table 1. and Table 4., which, i he
ime scale is he same as in he expe imen , would g ow i e
imes. Equally impo an c i e ia o simula ion ideli y a e
he ime sequences o eloci y p o iles se a hal he ube
leng h. I we we e o conside a eal applica ion in he o m
o blood low in a e ies, we would ge a mo e o less ai h ul
pic u e o he dynamics o blood ci cula ion in he a ec ed
a ea. Fo example, i is no longe di icul o analyze he
ele an ube s esses o a e ies in he FSI simula ion.
Howe e , since expe imen al da a is missing o ob ious
easons, he co esponding s esses will no be plo ed.
Indeed, s ess analysis o o a ionally symme ic bodies is
well desc ibed in li e a u e [5], [17], [18]. O he wo ks [3],
[7], [19], which we e ca ied ou a his opic, show ha he
link be ween he s a ic p essu e in he ube and he ac ual o
simula ed de o ma ion o he ube is less p one o possible
inaccu acies han is he case wi h pu ely hyd aulic
pa ame e s. By his is mean he coupling be ween he s a ic
p essu e and he co esponding low eloci y, which can be
ai h ully simula ed in he lexible ube subs an ially mo e
di icul . The e o e, he ollowing igu es (Fig.8. o Fig.24.)
show mainly compa isons o eloci y p o iles be ween he
expe imen and he nume ical model. The cu e
co esponding o he FSI simula ion only ex ends o hal he
p o ile as i co esponds o a qua e o he simula ed ube.
The ac ual wid h o he plana expe imen al eloci y p o ile
may a y because he hose, in con as o he simula ion,
swung sideways while he high-speed came a posi ion was
s a iona y.
Fig.8. = 0.05 s.
Fig.9. = 0.1 s.
-0,1
0
0,1
0,2
0,3
0,4
0 0,2 0,4 0,6 0,8 1
[m∙s-1]
[s]
0 0,2 0,4 0,6 0,8 1
Nume ical Expe imen al
MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
110
Fig.10. = 0.15 s.
Fig.11. = 0.2 s.
Fig.12. = 0.25 s.
Fig.13. = 0.3 s.
Fig.14. = 0.35 s.
Fig.15. = 0.4 s.
Fig.16. = 0.45 s.
Fig.17. = 0.5 s.
MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
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Fig.18. = 0.55 s.
Fig.19. = 0.6 s.
Fig.20. = 0.65 s.
Fig.21. = 0.7 s.
Fig.22. = 0.75 s.
Fig.23. = 0.8 s.
Fig.24. = 0.85 s.
Ob iously, he expe imen al eloci y p o iles exhibi a
g ea e deg ee o ins abili y and asymme y due o he size o
he ime s ep and he con inuous ope a ion o he pump, which
pulsa es cons an ly ib a es he ube ega dless o he
heo e ical beginning and end o he simula ed pe iod. I is
also known ha any such ask depends g ea ly on he
ini ializa ion condi ions. In p inciple, howe e , in he i s
hal o he simula ed pe iod, he speed p o iles ob ained
h ough he FSI simula ion in shape co espond qui e well o
he expe imen . In he second pa o he es ed pe iod, he
asymme y o he eloci y p o ile u he inc eases and he
de ia ion be ween he expe imen al da a and he nume ical
simula ion inc eases. I can be u he no ed ha i we assume
MEASUREMENT SCIENCE REVIEW, 20, (2020), No. 3, 104-114
112
a symme ic eloci y p o ile a he beginning o he simula ed
pe iod (Fig.8. o Fig.11.), his symme y is p og essi ely
dis u bed du ing in la ion o he hose due o p essu e build-
up in he sys em. The si ua ion wo sens when he s a ic
p essu e eaches i s maximum and he ube begins o sh ink,
which akes place in he second hal o he pe iod examined.
Fo his, i is necessa y o conside a sligh ly a iable wall
hickness o he ac ual ube, which leads o he o ma ion o
o als and une en de o ma ion o he o iginally almos
ci cula c oss-sec ion. In Table 3. i is s a ed ha he hickness
o he ube is ela i ely hin, co esponding o app oxima ely
2.2 mm. Changes in hickness uni o mi y a he le el o i e
o en hund ed hs o millime e s will signi ican ly a ec he
size o he adial de o ma ions o he ube. The wo s si ua ion
hen occu s a he e y end o he simula ed pe iod, when pa
o he ube ge s unde acuum (see Fig.2.) and he eloci y
p o ile shows de ia ions om he p o ile simula ed in size
and shape. As men ioned p e iously, du ing he acuum
mode, he ube ends o de ia e om he s aigh di ec ion.
5.
C
ONCLUSION
Al hough he es ed domain is ela i ely simply shaped, FSI
simula ion o a cylind ical ube p esen s a numbe o
p oblems wi h espec o he ideli y o he cha ac e is ic
alues o he low ield in he ube. We could conside and
hen simula e a b anched ube simila o iden ical o he ac ual
bloods eam, bu such a p oblem has wo weak poin s. The
i s is he ue alue o he expe imen and he second is he
ideli y o he FSI simula ion. Bo h o hese sho comings a e
la gely based on he ele ance o bounda y condi ions and he
ideli y o he expe imen al model, eal and i ual.
The expe imen desc ibed in his wo k mos closely
app oxima es he simula ion o blood low h ough la ge
essels wi h a high elas in con en . The ask o he la ge
essels is, among o he hings, o con e he pulse low in o
a con inuous low. F om his poin o iew, he
expe imen ally measu ed low in he ube, de e mined a hal
i s leng h and desc ibed by he mean eloci y o he c oss-
sec ion, can be conside ed as accep able.
In he expe imen al pa o he wo k, i would be wo h
conside ing he implemen a ion o an open hyd aulic ci cui ,
which would limi he e ec o iming o he suc ion and
discha ge al es o he diaph agm pump. Thus, i would be
possible o imp o e he compliance o he non-s a iona y
cou se o physical quan i ies in indi idual pe iods and o
educe he acuum a ea. Rega ding he bounda y condi ions
and hus he posi ion o he measu emen si es,
econ igu a ion o he expe imen al s and a e he
expe imen al da a om he s a ic p essu e ansduce s could
be use ul. I has been said ha i is ela i ely p oblema ic o
de e mine bo h s a ic p essu es and he cha ac e o an
uns eady eloci y ield a he same measu ed posi ion.
The e o e, changing he p essu e apping poin o he posi ion
o he high-speed came a seems o be an ad an ageous way,
while main aining he equi ed op ical condi ions. Al hough
i is s ill no possible o link he s a ic p essu e magni udes o
he co esponding eloci y p o ile o eloci y p o ile a he
inle o he ube and a hal i s leng h a one ime, his would
be bene icial as a guide o u he adjus ing he bounda y
condi ions. O cou se, wo high-speed came as supplemen ed
by a p essu e senso would be op imal. In connec ion wi h he
sensing equency o non-s a iona y s a ic p essu es and he
choice o p essu e ansduce s, we can ce ainly ecommend
wo k dealing wi h he dependence o he sensing equency
on he magni ude o he measu ed p essu e ampli ude wi h
ega d o i s dec ease due o damping.
In he ield o FSI simula ions implemen ed using
comme cial so wa e ANSYS wi h CFD sol e ANSYS
Fluen , besides he men ioned bounda y condi ions, he so-
called solu ion s abiliza ion seems o be an impo an op ion.
The size o he Scale Fac o wi h he coe icien -based
me hod chosen can signi ican ly a ec he simula ed cu en
ield and un o una ely no clea opinion can be gi en wi h
espec o i s size. The alue o he simula ions was 0.002. As
he alue o his ac o inc eased, he esponse o he change
in eloci y ield o he change in s a ic p essu e in he sys em
dec eased.
The ques ion o bounda y condi ions was al eady
men ioned in connec ion wi h he e alua ion o he
expe imen . Fo u he de e mina ion in e ms o
simula ions, he e is no choice bu o use he di ision o he
ube in o smalle uni s so ha i is possible o use he
expe imen ally de e mined eloci y ield a a pa icula
loca ion o he ube and make i a bounda y condi ion. The
combina ion o eloci y and p essu e condi ions on he
sho ened ube e en ually allows he necessa y amoun o
dynamic p essu e in he ube o a i e a he posi ion o he
p essu e senso . The disad an age o such an app oach is, o
cou se, he ime consuming, which is gene ally associa ed
wi h FSI simula ions.
Auxilia y one-dimensional FSI simula ions a e able o
quali a i ely cap u e he basic end o mean eloci ies in he
c oss-sec ion a he ube en y, bu he e o in he
de e mina ion o mean eloci y appea ed qui e signi ican . A
disad an age is also he na u e o he ube ma e ial i sel wi h
espec o he equency o p essu e pulsa ions. Pu ely elas ic,
mo e p ecisely, hype elas ic ube beha io combined wi h
expe imen al da a yields mo e no iceable a ia ions in mean
eloci y ampli ude in 1D FSI simula ions. O cou se, in e nal
damping can be in oduced in o one-dimensional models, bu
will esul in unwan ed a enua ion o some equencies ha
does no co espond o eali y. In his con ex , he wo-
pa ame e Mooney-Ri lin ma e ial model appea s o be mo e
a o able han he neo-Hookean, which in 1D yields be e
esul s wi h espec o he concep o he ela ionship be ween
adial de o ma ion o he ube wall and he in e nal p essu e
in he ube. Conside ing he con inuous unc ions o he inpu
and ini ial condi ions, he di e ences be ween he neo-
Hookean and he wo-pa ame e Mooney-Ri lin ma e ial
model o he same ube in 1D FSI simula ions would be small.
By his is mean in pa icula he magni ude o he adial
de o ma ions o he ube and he eloci y in he cen e o he
ube. Howe e , adial de o ma ions o he ube a e no pa o
he p esen ed wo k. The es ima ion o he inpu mean eloci y
in he c oss-sec ion o he ube by 1D FSI simula ions can
hen se e as a s a ing alue o he subsequen e inemen
o he bounda y condi ion, o as a con ol o he FSI
simula ion o he sho ened ube, which uses he basic da a