FSI Computation and Experimental Verification of Fluid Flow in Flexible Tubes

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, 104-114 106 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.  + =−    −   ||  (3)  =!" #$ #% & '()* * −  (4)    =+  (5) +=+,-=   . */' *0 ! *'/' *'0 1 (6) +=+23= 4  5  . */' *0 ! *'/' *'0 1& & ' .  */' !  *'/' 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, 104-114 111 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