Flow-Induced Vibration Analysis of Supported Pipes with a Crack

Transcription

1 Flow-Induced Vibraion Analsis of Suppored Pipes wih a Crack Jin-Huk Lee, Samer Masoud Al-Said Deparmen of Mechanical Engineering American Universi of Sharjah, UAE

2 Ouline Inroducion and Moivaion Aeroacousicall fluid-srucure ineracion Mahemaical Formulaion Fluid loading effec Viscous fricion effec due o flowing fluid inside a pipe FEA (Finie Elemen Analsis) model Preliminar Resuls Conclusion Fuure Work Q & A

3 Inroducion & Moivaion Dubai Oil Mone Deser o Greaes Ci Full Documenar on Dubai ci hps:// 3

4 Inroducion & Moivaion The effec of a crack o he flow-induced vibraion characerisics of suppored pipes is invesigaed based on vibraion mehod. We need o uilize he variaion of he difference beween he naural frequencies of he pipe conveing fluid wih and wihou a crack. The pipe is fluid loaded via ineracion wih he fluid. Fluid loading has wo main effec on vibraing pipes: 1. Fluid mass loads he pipe, i.e., he pipe s naural frequencies are alered. Viscous loading is provided o he pipe near he inner wall due o shear force COMSOL Muliphsics 1. Aeroacousics Module. Srucural Moduel 4

5 5 Mahemaical Model Kinemaic Energ of pipe:, where M = mass per uni lengh of he pipe c = he disance o he crack locaion = he deflecion of he pipe L = oal lengh of he pipe Kineic Energ due o fluid inside he pipe: L pipe c c d d m T 0 1,, 1 d U U M T c fluid d U U L c, where M = mass per uni lengh of he pipe U = fluid veloci

6 6 Mahemaical Model (con d) Poenial Energ of pipe due o srain energ:, where E = modulus of elasici I = area momen of ineria K R = spring coefficien due o crack k = ransverse displacemen (k = 1,) The ransverse displacemen:, where = admissible funcion d = generalized coordinae k = number of divided pipes due o crack d d EI V L pipe c c K c R n i i ki k d 1,

7 Mahemaical Model (con d) Lagrange equaion: M q K q Fe Considering an eernal forcing erm is assumed o be a viscous drag force due o shear sress inside he pipe wall, i can be replaced wih F viscous A, where A = surface area ʋ = average fluid veloci = separaion disance beween he wall and he cener of he pipe F A, where F = force required o mainain he moion 7

8 GEOMETIC MODEL AND BOUNDARY CONTIDIONS Table 1: Srucure and maerial properies of he fluid flow conveing pipe. Maerial Pipe Copper (a) Ouer Diameer Thickness 6e-m 5e-3m fluid PML Modulus of Elasici 110GPa Densi 8700 kg/m 3 Poisson raio 0.35 Pipe lengh 0.5m Fied (b) Beam widh 1.mm Figure 1. (a) Geomeric pipe model, and (b) meshed pipe model wihou crack where blue colored secion represens PML. 8

9 FSI PROBLEM IN FREQUENCY DOMAIN Linearized Navier Sokes (Aeroacousic) + Infinie boundar (absorb boundar): PML Fluid-srucure ineracion (FSI) boundar: The Aeroacousic-Srucure Boundar coupling prescribes coninui in he displacemen field beween wo differen domains, where u fluid = fluid veloci u solid = solid displacemen Srucural Mechanics Module The governing equaions used o solve for he frequenc analsis are he coninui, momenum, and energ equaions + srucural equaions. u i fluid u solid This resuls in he sress being coninuous across he boundar beween wo differen domains. This will pla an imporan role invesigaing he effecs of he fluid o he vibraion mode of he pipe ssem. 9

10 CRACK GEOMETRY The local fleibili in he presence of he crack can be defined as a funcion of he geomer of a crack. a b 5 R o R i Figure. Cross secion of he cuawa cracked pipe and he side view: (a) Geomeric pipe model, and (b) he meshed pipe model wihou crack where blue colored secion represens PML. 10

11 Flow veloci profile Assuming he sead, laminar (Re 300), imcompressible flow of fluid wih consan properies, he full developed veloci profile is chosen. Figure 3. The developmen of he veloci boundar laer in a pipe [10]. u r V ma 1, where V ma is maimum veloci, R o is he inner pipe radius, and z are radial disance from he cener o each ais z Ro 11

12 CRACK MESHING AND FLOW BOUNDARY CONTIDIONS Fluid flow is assumed o be Newonian fluid for laminar case, and he no slip condiion for he flow on a hard wall inside he pipe. The No slip condiion is u 0 Figure 4. Meshed cracked pipe. The maimum elemen size is h ma = 0.λ, where λ is wavelegh. 1

13 Simulaion Resuls: Fluid loading & Crack Effecs Table : Comparison for in vacuo and he pipe filled wih waer for is eigenfrequenc eigenfrequenc eigenfrequenc w/ crack In vacuo Hz Hz Filled wih waer Percenage decrease Hz Hz 11 % 10 % The fluid added mass effec is esimaed b calculaing he frequenc reducion raio δ of each naural frequenc defined as f, where f v and f w are he naural frequencies in vacuo and wih fluid inside he pipe. v f v f w 13

14 Simulaion Resuls: Fluid loading & Crack Effecs (con d) (a) (c) (b) (d) Figure 5. Eigenmodes for he cases in Table where 1,, 3, and 4 from Table denoe (a), (b), (c), and (d) in Fig. 5, respecivel.. 14

15 Simulaion Resuls: Fluid Flow Effecs Table 3: Comparison for differen velociies of he fluid flow inside he pipe for is eigenfrequenc Ma. veloci eigen- frequenc eigenfequenc w/ crack 1 m/s Hz Hz 10 m/s Hz Hz 15

16 Conclusion In his paper, a pipe conveing fluid flow wih crack has been invesigaed hrough numerical simulaion using COMSOL Muliphsics sofware. Vibraional behavior of he pipe ssem has been sudied o show he effec of fluid wihin a pipe as well as ha of crack. Due o he added mass effec induced b he fluid inside he pipe, a significan reducion in naural frequencies is observed (Table ). The veloci of he fluid inside he pipe seems o affec he naural frequenc of he pipe ssem such ha as he veloci increases, is eigenfrequenc decreases. However, here has no been found an specific correlaion beween hem in his work. 16

17 Fuure Work The work o be done in he fuure would be as follows; (1) invesigaion of furher sud of he veloci of he fluid flow in greaer deail, () invesigaion of he crack locaion, (3) sud of dual crack effec raher han single crack, and (4) derivaion of mahemaical model ha corresponds wih he simulaion sud. 17

18 Q & A Thank ou! 18