ENGINEERING FOR RURAL DEVELOPMENT Jelgava, FLOW-INDUCED VIBRATION OF CYLINDRICAL STRUCTURES.

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1 FLOW-INDUCED VIBRATION OF CYLINDRICAL STRUCTURES Sabine Upnere 1,2, Normund Jekabon 2 1 Riga Technical Univerity, Latvia; 2 Ventpil Univerity College, Latvia upnere@gmail.com, normundj@venta.lv Abtract. Two-dimenional numerical imulation o the cro-low around circular cylinder in taggered arrangement have been perormed. The part o the water-cooling ytem rod bundle i invetigated. One o the cylinder i ree to ocillate in the tranvere and longitudinal direction, in repone to the luid load. The moving cylinder i modelled a a imple ma-pring-damper ytem. Cylinder are rigid object. The Navier- Stoke equation and continuity equation are ued to decribe a luid. Dynamic o the tructure are decribed by equation o motion. Turbulent incompreible low with Reynold number i imulated uing the Unteady Reynold Averaged Navier-Stoke type Low-Re turbulence model. Fluid-tructure interaction i modelled by Arbitrary Lagrangian Eulerian ramework, which combine the luid low ormulated uing an Eulerian decription and a patial rame with olid mechanic ormulated uing the Lagrangian decription and a material rame. Commercial olver i ued or analyzed the proce imulation. The numerical reult o benchmark cae are compared with literature to check the ued methodology. Keyword: luid-tructure interaction, low-induced vibration, circular cylinder. Introduction The high velocity coolant lowing through the water-cooling ytem o dierent indutrial and nuclear plant i a ource that can induce component vibration and intability, which lead to the maintenance and operational problem, a well a ecurity problem. In thi paper the low through circular cylinder rod bundle o the taggered arrangement i tudied. Generally, there are our main low-induced mechanim cauing tructural vibration: luid-elatic intability, periodic wake hedding, turbulence-induced excitation and acoutic reonance [1]. With the greatet potential or low-induced vibration damage to the rod bundle ha been luid-elatic intability [1; 2]. Many preview tudie have been devoted to enhancing undertanding o procee in the rod bundle ytem ince the work o Connor (1970, identiied and evaluated a baic luid-elatic diplacement mechanim), Belvin (1974, recat the obervation in term o coupled vibration o an ininite tube row) or Tanaka and Takahara (1980, carried out an experimental program to meaure the unteady luid dynamic orce coeicient or in-line tube array). However, numerical calculation o the multi-cylinder ytem i challenging due to complex interaction between the hare layer and vortice, turbulence and other phenomena. Beide, luid-tructure interaction (FSI) problem have trong nonlinearity and multidiciplinary nature [3]. Modelling actuality o low-induced vibration i till on top and there are new publication, or example [4; 5]. In the preent tudy a luid ha been decribed by the Navier-Stoke equation and continuity equation o ma in the Eulerian ramework. Although cylinder are rigid bodie, due to otware eature moving cylinder are decribed a linear elatic object governed by equation o motion in Lagrangian ormulation. A a coupling algorithm between Eulerian and Lagrangian ormulation the Arbitrary Lagrangian-Eulerian (ALE) method [6] ha been ued. In order to preent large unteady tructure o vortice in the low ield and to conume computational cot the o-called Unteady Reynold Averaged Navier-Stoke (URANS) equation have been calculated. Finite Element analyi i done uing commercial otware Comol Multiphyic. Thi reearch contain a erie o two-dimenional numerical tudie that analye turbulent inglephae low in the part o a multi-cylinder ytem. Mathematical ormulation The luid problem i governed by the Reynold Averaged Navier-Stoke equation (1) and conervation o ma (2) in Eulerian orm or incompreible low. With ubcript and denoted variable are related with luid and olid domain, repectively. u ρ + ρu t ( u) + p-µ u + ρ τ = in Ω or t > 0 t (1) 1244

2 where ū Reynold-averaged velocity, m -1 ; p preure o theluid, Pa; µ t luid dynamicvicoity, N m -2 ; Ω bounded luid domain; orce, N; τ Reynold tre tenor, N m -2. u = 0 inω or t > 0 (2) Since luid i incompreible, thermal eect are neglected. The denity i contant in pace and time due to the aumption o homogeneity and incompreibility o the luid. The problem mut be upplemented with initial (3), Dirichlettype boundary (Γ D, ) condition or velocity (4) and Neumann type o boundary (Γ N, ) condition or normal tree (5): σ u = u in Ω or t 0, (3) 0 = u = g on Γ D, or t > 0, (4) n = d on Γ or t >, (5) N, 0 where u 0 initial velocity, m -1 ; n normal vector; σ tre tenor, N m -2 ; g impoed luid velocity on boundary, m -1 ; d impoed normal tree on boundary, N m -2. Dynamic o the tructure i decribed in Lagrangian coordinate; equation are written in the initial coniguration Ω 0. Moving tructure are decribed with ield equation(6) and Hook law (7), which are valid or mall diplacementη. Boundary condition have been deined by Dirichlet and Neumanntype condition in olid domain with (8) and (9), repectively. ρ 2 η - σ 2 t = 1245 inω ( η) I + µε, where λ, µ Lamé contant; ε train tenor; I econd rank unity tenor; g impoed olid diplacement on boundary, m; ρ denity o olid,kg m -3. 0, σ = λ 2 (7) σ η = g on Γ D,, (8) n = d onγ, (9) On the interace (I(t)) boundary between luid (F(t)) and olid-domain (S(t)) kinematic (10), dynamic (11) and geometric (12) coupling condition have been applied. η u = on I( t), (10) t σ n = σ n on I( t), (11) ( t) F( t) I( t) S( t), F( t) S( t) =. N, Ω = 0 (12) Equation et o (1), (2), (6) and (10)-(12) ormulate a coupled luid-tructure problem. The luid equation are deined in a moving domain Ω t. One option to deal with the motion o the luid-domain i to ue the ALE ramework, which i a well-etablihed technique to model FSI.ALE introduce a (6)

3 ixed reerence rame, which i mapped at every tep to the deired phyical domain and equation o motion are then recat in the reerence rame. Numerical modelling technique Due to mall diplacement, the ixed geometry approach (the luid domain i conidered a ixed during the iteration) ha been applied to couple Computational Fluid Dynamic (CFD) and Solid Mechanic interace. Segregated olver which equentially olve or luid low, olid diplacement, and moving meh i ued to reduce the computational cot. Backward Dierence Formula ha been choen or the time tepping. The luid low in thi reearch i turbulent; correponding Reynold number R e = For the approximation o the turbulent low in the utilized otware (Comol Multiphyic) the RANS typea pproach ha been ued. Low Reynold turbulence model have been invetigated becaue o the importance o the low near boundarie to model lit and drag orce and low eparation rom urace with higher accuracy. Two dierent cloure model have been teted to evaluate the mot convenient one: low Reynold number k-epilon and SST (Shear-Stre Tranport) k-omega [7]. The low Reynold number k-epilon i imilar to the k-epilon model alone without wall unction; the low i olved near the wall, too. The SST k-omega model combine the bet o two approache: the k-epilon in the ree-tream and the k-omega model near the wall. The model doe not ue the wall unction. According to the comparion o convergence tability (ee Fig. 1), the SST k-omega model ha been choen a the mot uitable cloure cheme or the tudied cae. Fig. 1. Comparion o convergence tability o two turbulence model: let low Reynold number k-epilon model; right SST k-omega model The ocillating lexibly-mounted rigid cylinder i modelled a a imple ma-damper-pring ytem. The equation o motion or the cylinder in the cro low direction i: m & η + c & η + kη = (t) (13) where m matrix o total ocillating ma o the ytem, kg; c damping matrix, kg -1 ; k tine matrix, N m -1 ; (t) orce acting on the cylinder in the cro low direction, N. Firtly, there are calculated the lit and drag orce acting on the tationary cylinder in the rod bundle. Ater that, pring tine i deined uing the previouly calculated aerodynamic orce and determined maximum diplacement. In order to compute high ocillation amplitude, the tructural damping coeicient in treamline and tranvere direction are et to zero. The imulation ha been perormed in 2D pace domain uing the Finite Element method. The invetigated geometry, domain dimenion and boundary condition can be een in Fig. 2, where D i the diameter o the cylinder. Contant velocity peciied by an equation u = u 0 i deined a the inlow boundary condition o velocity. The outlet boundary condition i determined by the rule that the outlow o luid mut be perpendicular to the boundary. At the wall and the moving wall, no-lip condition i ued: the tangential luid velocity i equivalent with the velocity o rigid boundary and the normal component o velocity i zero. 1246

4 Fig. 2. Computational domain, dimenion and boundary condition: D i a diameter o the cylinder and u i the ree-tream velocity Tet cae Two tet cae, one or CFD and one or FSI,were done to etimate the ued approach, olver, meh ize and other convergence criterion. The reult o the tet cae have been compared with the data rom literature. Claical CFD benchmark cae with R e = ha been choen to evaluate turbulence model. Single tationary circular cylinder in cro-low i imulated. Stable olution i achieved ater approx. 0.5 econd. Error o the calculated drag coeicient compared with the data in literature i 6.7 %. To check the ued FSI methodology, two cylinder in tandem have been analyzed. The irt cylinder i tationary, while the econd one ha one degree o reedom to ocillate in the tranvere direction. In thi cae R e = The ketch o the FSI computational domain and boundary condition are hown in Fig. 3. Fig. 3. Computational domain and boundary condition o FSI tet cae: D i the diameter o the cylinder, k i the pring tine coeicient, c i the damping coeicient and u i the ree-tream velocity The calculated reult have been compared with the reult rom [8] and [9]. Comparion o the ocillation amplitude ratio (y max /D) with the numerical and experimental data ha been ummarized in Table 1. Table 1 Comparion o the maximum amplitude o the ocillating cylinder Cae Amplitude ratio Error (y max /D) (e) Preent tudy % Derakhandeh, [8] % Experimental value, [9] From Table 1 it can be een that the value o the numerical calculation are larger than the experimental value. 1247

5 Reult and dicuion Two cae have been invetigatedto analye the multi-cylinder ytem (ee Fig. 4): moving cylinder ha 1DOF, the movement i allowed in tranvere direction (y-direction); moving cylinder ha 2DOF (x and y direction). The cylinder i modelled rom teel and the ued luid i water. In order to get a ater convergence, a tationary cae ha been imulated irt. The reult o the tationary cae have been applied a initial condition in the dynamic cae. Fig. 4. Computational domain and dimenion: let 1DOF model; right 2DOF model The cylinder doe not ocillate in the cae o 1DOF at the given initial and boundary condition. Ater approximately 0.085, the moving cylinder reache it maximum diplacement at D. Ditribution o the low ield (velocity magnitude, m -1 ) around the rod bundle can be een in Fig. 5. Fig. 5. Ditribution o the velocity magnitude (m -1 ) around the rod bundle o 1DOF cae The lexibly-mounted cylinder ocillate, when it ha 2DOF. A major impact on the total movement i y component (lit direction) o the diplacement. Maximum diplacement in y direction i 0.1 D, while in x direction it i D. The hedding requency (Hz) o the ocillating cylinder ha been diplayedin Fig. 6. Fig. 6. Lit orce o the moving cylinder in the requency domain uing Fat Fourier Tranorm in 2DOF cae 1248

6 At the given condition the irt expected requency o the ingle cylinder i 31 Hz, moving the cylinder in the rod bundle the irt requency i higher 57.5 Hz. Concluion 1. Two-dimenional numerical imulation o low-induced vibration have been done to etimate the ued FSI calculation methodology. Due to turbulent low and tructural element interaction it i a challenging tak. 2. There i imulated a part o the cooling ytem with taggered arrangement rod. The middle circular cylinder o the rod bundle, which i mounted on the elatic ytem, can be ocillated in longitudinal and tranvere direction. The cylinder motion ha been modelled a a imple madamper-pring ytem. 3. Two Low-Re turbulence model have been analyzed. Baed on the characteritic o convergence the SST k-omega turbulence model ha been choen a more uitable in the analyed cae. 4. The numerical imulation reult o CFD (e = 6.7 %) and FSI (e = 8.2 %) benchmark cae have a uicient agreement with the experimental data rom literature. 5. Further ull rod bundle imulation will be required to compare the modelling reult with the experimental meaurement. Reerence 1. Pettigrew M.J. et al. Vibration analyi and vibration damage aement in nuclear and proce equipment. Proceeding o CORENDE: Regional congre on nondetructive and tructural evaluation, 1997, Bueno Aire, Argentina, pp Lowdon A., Tonk N., Wilkinon T.S. Fluid-elatic intability o in-line tube array in cro-low. Appl. Math. Modelling, vol 4, 1990, pp Chakrabarti S.K. (Ed.) Numerical Model in Fluid Structure Interaction, Advance in Fluid Mechanic, vol. 42, WIT Pre, Yu K.R. et al. Flow-induced vibration o in-line cylinder arrangement at low Reynold number. J. o Fluid and Structure, vol. 60, 2016, pp Nguyen V.-T., Nguyen H.H. Detached eddy imulation o low induced vibration o circular cylinder at high Reynold number.j. o Fluid and Structure, vol. 63, 2016, pp Arbitrary Lagrangian-Eulerian and Fluid-Structure Interaction. Numerical imulation. Edited by Souli M., Benon J. Eatbourne: John Wilew & Son, 2010, 306 p. 7. Davidon L. An introduction to turbulence model. Goteborg: Chalmer Univerity o Technology, p. 8. Derakhhhandeh J.F. et. al. Numerical imulation o vortex-induced vibration o elatic cylinder. Proceeding o International conerence 18th Autralaian Fluid Mechanic Conerence, December 3-7, 2012, Launceton, Autralia. 9. Ai G.R.S. Experimental Mechanim or low-induced vibration o interering blu bodie. London: Imperial College London, p. 1249