Journal of Fluid Science and Technology

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1 Journal of Flud Scence and Technology Numercal Smulaton of Incompressble Flows around a Fsh Model at Low Reynolds Number Usng Seamless Vrtual Boundary Method * Hdetosh NISHIDA ** and Kyohe TAJIRI ** **Department of Mechancal and System Engneerng, Graduate School of Scence and Technology, Kyoto Insttute of Technology, Matsugasak, Sakyo-ku, Kyoto , Japan E-mal: nshda@kt.ac.jp Abstract In ths paper, the numercal smulaton of ncompressble flows around a fsh model at low Reynolds number usng the seamless vrtual boundary method s presented. In order to satsfy the velocty condtons on the vrtual boundary ponts, the forcng term s added not only on the grd ponts near the boundary but also on the grd ponts nsde the boundary n the seamless vrtual boundary method, so that the smooth physcal quanttes can be obtaned. The present approach s verfed by the flows around a 2D oscllatng crcular cylnder and a sphere. The flows around the statonary and swmmng fsh models are smulated for applcaton to the more complcated flow geometry wth movng boundary. These results show that the flows wth the statonary and movng bodes can be predcted precsely. Then, t s concluded that the present seamless vrtual boundary method s very promsng for the numercal smulaton of ncompressble flows wth the complcated geometry and the movng boundary. Key words: CFD, Cartesan Approach, Seamless Vrtual Boundary Method, Movng Boundary, Incompressble Flow 1. Introducton *Receved 1 Apr., 2009 (No ) [DOI: /jfst.4.500] For the practcal flow smulatons, the boundary ftted coordnates (BFC) s usually adopted. Ths BFC approach has the hgh adaptablty to the boundary confguraton. However, n the complcated flow geometry, t s dffcult to generate the computatonal grd. Moreover, n the BFC approach, t s necessary that the governng equatons are transformed from the physcal plane to the computatonal plane, so that the transformed governng equatons have more terms than the orgnal equatons n the Cartesan coordnates. Therefore, the computatonal effort s larger than the Cartesan grd approach. Then, n recent years, the Cartesan grd approach s hghlghted agan for the numercal flow smulatons. In the Cartesan grd approaches, especally, the mmersed boundary method s appled to many smulatons of ncompressble flow. One of ths mmersed boundary method s the vrtual boundary method (1)-(4). In order to satsfy the velocty condtons on the (vrtual) boundary ponts, the vrtual boundary method requres the external forcng term added to the momentum equatons. Ths forcng term s usually estmated by two ways,.e., feedback (2)-(4) and drect (1) procedures. Also, n both forcng term estmatons, the grd ponts added the forcng term are restrcted only near the boundary. Therefore, the unphyscal oscllatons near the boundary appear n the pressure feld. In order to mprove ths weak 500

2 Journal of Flud Scence and Technology pont, Tyag and Acharya (5) ntroduced the forcng terms on both sdes of the mmersed boundary,.e., two-sde forcng scheme. Ye et al. (6) proposed the reshaped cell method n whch the spatal dervatves of the governng equatons were dscretzed n cell that was cut by the boundares. Tseng and Ferzger (7) developed the mmersed boundary method wth second order ghost cell reconstructon to enforce the boundary condtons. Km et al. (8) added the mass source/snk term to the contnuty equaton n order to satsfy the mass conservaton at cells contanng the mmersed boundary. Nshda and Sasao (9) proposed the seamless vrtual boundary method. In ths method, the forcng term s added not only on the grd ponts near the boundary but also n the regon nsde the boundary, so that the unphyscal pressure oscllatons are removed. In ths paper, we try to apply the seamless vrtual boundary method to the complcated flow smulaton wth movng boundary. Frst, n order to valdate the present seamless vrtual boundary method, the numercal smulatons of ncompressble flows around a 2D oscllatng crcular cylnder and a sphere are consdered. Fnally, the numercal smulatons of flow around the statonary and swmmng fsh models are carred out. The effect of fns,.e., the breast fns, abdomen fns, hp fn, back fn and tal fn, s dscussed. 2. Seamless Vrtual Boundary Method 2.1 Governng equatons In the vrtual boundary method, the velocty condtons on the vrtual boundary are satsfed by addng the forcng term to the momentum equatons. Then, the ncompressble Naver-Stokes equatons can be wrtten n non-dmensonal form by u x = 0, (1) 2 u u p 1 u + u j = + + G, t x x Re x x j j j (2) where Re denotes the Reynolds number defned by Re=UL/ν. U, L and ν are the reference velocty, the reference length and the knematc vscosty, respectvely. The last term n Eq.(2), G, denotes the addtonal forcng term. 2.2 Forcng term estmaton In order to estmate the addtonal forcng term n the governng equatons, G, there are manly two ways, that s, the feedback (2)-(4) and drect (1) forcng term estmatons. In ths paper, the drect forcng term estmaton shown n Fg.1 s adopted. We explan n 2D but the extenson to 3D s straghtforward. U u=u u+1 Fg. 1 Drect forcng terms estmaton. 501

3 Journal of Flud Scence and Technology For the forward Euler tme ntegraton, the forcng term can be determned by G = u j u x j p + x 2 1 u Re x j x j n U + n+1 u t n, (3) n+1 where U denotes the nterpolated velocty by lnear nterpolaton. Namely, the external force s specfed as the velocty components at next tme step satsfy the relaton, n + 1 n+ 1 u = U. In ths forcng term estmaton, the grd ponts added forcng term are restrcted near the boundary. Then, the pressure dstrbutons near the boundary show unphyscal oscllatons. 2.3 Seamless vrtual boundary method In order to remove aforementoned unphyscal oscllatons near the boundary, the seamless vrtual boundary method (9) was proposed. In the seamless vrtual boundary method, the forcng term s added not only near the boundary but also n the regon nsde the boundary shown n Fg.2. On the grd ponts near the boundary, the addtonal forcng term s estmated by the same procedure as the usual drect forcng term estmaton, Eq.(3). In the regon nsde the boundary, the forcng term s determned by satsfyng the relaton, n+1 U = U, where U b s the specfed velocty, e.g., U b =0 n statonary sold meda. b :Forcng ponts :Vrtual Boundary ponts :Forcng ponts :Vrtual Boundary ponts (a) Usual vrtual boundary method. (b) Seamless vrtual boundary method. Fg. 2 Grd ponts added forcng terms. 2.4 Numercal method The ncompressble Naver-Stokes equatons (2) are solved by the second order fnte dfference method on the collocated grd arrangement. The convectve terms are dscretzed by the second order fully conservatve fnte dfference method (10). The dffuson and pressure terms are dscretzed by the usual second order centered fnte dfference method. For the tme ntegraton, the fractonal step approach (11) based on the two-step Runge-Kutta scheme (12) s appled. For the usual ncompressble Naver-Stokes equatons, the two-step Runge-Kutta scheme s wrtten by u n+ 1 n n 1 n n n 1 n 1 = ( 1 γ )u + γu + t[ α( F p ) + β( F p )], (4) where α, β, and γ are the parameters of two-step Runge-Kutta scheme and F denotes the convectve and dffuson terms. In ths paper, the parameters are set as α=9/8, β=-7/8, γ=-3/4 n order to satsfy the second order of tme accuracy. The fractonal step approach based on the two-step Runge-Kutta scheme can be wrtten by 502

4 Journal of Flud Scence and Technology u * n n 1 n n 1 n 1 = ( 1 γ )u + γu + t[ αf + β( F p )], (5) u n+ 1 = u * n t( α p G ), (6) where u * denotes the fractonal step velocty. The resultng pressure equaton s solved by the SOR method. The convergence crteron of the momentum equatons and pressure equaton s set as φ L2 < , where φ L2 s the L-2 resdual of physcal quanttes φ,.e., the velocty or pressure. 3. Valdaton of Seamless Vrtual Boundary Method 3.1 Flow around a 2D oscllatng crcular cylnder The computatonal doman s shown n Fg.3. The crcular cylnder n the unform flow moves vertcally as, y( t ) = y y + amp 0 sn( 2πft ), 2 (7) where the ampltude s y amp /2=0.2D and the non-dmensonal frequency s f=0.2. The ntal locaton of crcular cylnder s (x 0,y 0 )=(5.5D,5.5D). The grd resoluton s wth non-unform grd system. The mpulsve start determned by the unform flow s adopted. On the nflow boundary, the velocty s fxed by the unform flow and the pressure s mposed by the Neumann condton obtaned by the normal momentum equaton. The velocty s extrapolated from the nner ponts and the pressure s obtaned by the Sommerfeld radaton condton (13) on the outflow and sde boundares. On the vrtual boundary and nsde the boundary, the movng crcular cylnder velocty s mposed. Fgure 4 shows the pressure contours wth Re=200 at top and bottom locatons. These results show that the vortex sheddng becomes larger than the flow around a statonary crcular cylnder. In the close-up vews, the smooth pressure felds near the crcular cylnder can be obtaned. The present drag and lft coeffcents and the Strouhal number shown n Table 1 are n very good agreement wth the reference result (14). Fg. 3 Computatonal doman. Table 1 Comparson of flow characterstcs. C D C L St Present 1.59 ± 0.17 ± Wu et al. (14) 1.58 ± 0.20 ±

5 Journal of Flud Scence and Technology (a) Overall vew (t=211.25). (b) Overall vew (t=215). (c) Close-up vew (t=211.25). (d) Close-up vew (t=215). Fg. 4 Pressure contours. 3.2 Flow around a sphere Fgure 5 shows the computatonal doman. The ntal and boundary condtons are the same as the prevous smulaton. On the vrtual boundary and nsde the boundary, the non-slp velocty condton (u =0) s mposed. In order to reduce the number of grd ponts, the herarchcal Cartesan grd wth 5 levels s ntroduced. The body shape s determned by the trangular polygon, so that the arbtrary body surface can be represented. The sphere surface generated by the 4900 trangular polygons s shown n Fg.6. The steady flow (Re=100) and unsteady flow (Re=300) are smulated wth about1.8 mllon grd ponts. In Fg.7 the pressure contours wth Re=100 at the steady state s shown. The unphyscal oscllatons do not appear, so that the smooth pressure feld can be obtaned. Also, the symmetrcal flow feld can be reappeared. Fgure 8 shows the so-surface of second nvarant of velocty gradent tensor Q' wth Re=300. The characterstc perodcal harpn vortex s observed clearly. Table 2 shows the drag coeffcent and vortex length n comparson wth the reference soluton (15). It s confrmed that the approprate solutons can be obtaned. Then, the present approach gves the accurate solutons n the 3D smulaton. Fg. 5 Computatonal doman. Fg. 6 Surface mesh and herarchcal grd. 504

6 Journal of Flud Scence and Technology Table 2 Comparson of flow characterstcs (Re=100). C D L V/D Present Shrayama (15) Fg. 7 Pressure contours. (a) t=122. (b) t=124. (c) t=126. (d) t=128. Fg. 8 Iso-surface of second nvarant of velocty gradent tensor. 4. Applcaton to Flow around a Fsh Model 4.1 Statonary fsh model In ths paper, we consder the achelognathus morokae (Japanese btterlng) n the unform flow. The surface shape generated by the trangular polygon s shown n Fg.9. The reference length D s body length of a fsh and the total number of trangular polygons s The characterstc shape of a fsh,.e., the breast fns, abdomen fns, hp fn, back fn and tal fn, can be reappeared. Fgures10 and 11 show the computatonal doman and the used 5 leveled herarchcal grd wth about 3.8 mllon grd ponts. Fg. 9 Surface shape of a fsh model. 505

7 Journal of Flud Scence and Technology Fg. 10 Computatonal doman. Fg. 11 Herarchcal grd wth 5 levels. The ntal and boundary condtons are the same as the flow around a sphere. On the vrtual boundary and nsde the boundary, the non-slp velocty condton (u =0) s mposed. The smulaton s carred out for steady flow wth Re=100. The pressure contours and ts drawng plane are shown n Fg.12. The smooth pressure dstrbutons can be obtaned n any plane. In x-y plane, the pressure feld represents the symmetrc dstrbuton comparable to the model symmetry. Also, the pressure dstrbuton n the detal,.e., around the breast fns, abdomen fns, hp fn and back fn, can be observed clearly. In ths case, the non-dmensonal drag force s The flow rate error s 0.024%, so that the reasonable soluton can be obtaned qualtatvely. (a) Drawng plane. (b) Pressure contours on y=5.5d and y=5.35d. (c) Drawng plane. (d) Pressure contours on z=5.4d and z=5.25d. Fg. 12 Pressure dstrbutons. 506

8 Journal of Flud Scence and Technology 4.2 Swmmng fsh model In ths secton, the swmmng fsh based on the prevous statonary model s controlled by 2D rotaton on four rotatng axes (S1, S2, S3, S4) shown n Fg.13, n order to model the practcal swmmng fsh. S1 s fxed at (x,y)=(5.2d,5.5d). The rotatng angle on each rotatng axs s defned by (8) where n f s the frame number and N f denotes the total number of frames (N f =100). ψ s the maxmum rotatng angle,.e., ψ 1=π /90, ψ 2=2π /45, ψ 3=π /18, and ψ 4= π /12 n ths paper. The swmmng behavor s shown n Fg.14. Fg. 13 Fsh shape and rotatng axs. 5/100 frame 15/100 frame 25/100 frame 35/100 frame 45/100 frame 55/100 frame 65/100 frame 75/100 frame 85/100 frame 95/100 frame Fg. 14 Fsh shape for swmmng model. 507

9 Journal of Flud Scence and Technology (a) 20/100 frame. (b) 45/100 frame. (c) 70/100 frame. (d) 95/100 frame. Fg. 15 Pressure dstrbutons on x-y plane. (a) 5/100 frame. (b) 30/100 frame. (c) 55/100 frame. (d) 80/100 frame. Fg. 16 Pressure dstrbutons on y-z plane. 508

10 Journal of Flud Scence and Technology The ntal and boundary condtons are the same as the statonary case. On the vrtual boundary and nsde the boundary, the movng velocty condton (u = u move ) s mposed. u move s the movng velocty determned by Eq.(8). The smulaton wth Re=100 s executed. Fgures 15 and 16 show the pressure feld wth the moton n one perod. Fgure 15 s the pressure on x-y plane (z=5.5d), and Fg.16 s one on y-z plane (x=6.0d). At the tal fn, t can be confrmed that the hgh pressure regon appears n the moton drecton sde and the low pressure regon appears n the moton drecton reverse sde. The second nvarant of velocty gradent tensor Q' near a swmmng fsh s shown n Fg.17. In ths swmmng model, the breast fns are always open, so that t s remarkable to form the large eddy. The comparatvely small eddy wth the moton of the abdomen, hp and back fns can be observed. The separaton vortces from the breast, abdomen, hp and back fns are greatly shoved by the tal fn, so that the complcated vortex structures are generated near the tal fn. However, these vortces near the tal fn are transported rear and the large eddy s formed, because of the effect of large vscosty,.e., low Reynolds number (Re=100). (a) 5/100 frame. (b) 15/100 frame. (c) 25/100 frame. (d) 35/100 frame. (e) 45/100 frame. (f) 55/100 frame. Fg. 17 Iso-surface of second nvarant of velocty gradent tensor. 509

11 Journal of Flud Scence and Technology (g) 65/100 frame. (h) 75/100 frame. () 85/100 frame. Fg. 17 -Contnued. (j) 95/100 frame. Fg. 8 (a) Overall hstory. (b) Hstory between t=6.0 and 6.1. Fg. 18 Tme hstory of non-dmensonal drag force. Fgure 18 shows the tme hstory of non-dmensonal drag force. The non-dmensonal drag force wth small fluctuatons greatly changes wth the tme. These small fluctuatons are correspondent to the moton wth the perodcal swmmng model. After the ntal dsturbance comes out from the computatonal regon, the non-dmensonal drag force becomes negatve value. In comparson wth the statonary case, the non-dmensonal drag force becomes smaller. Then, t can be confrmed that the some thrust s arsng. 5. Concludng Remarks In ths paper, the seamless vrtual boundary method s appled to the numercal smulaton of ncompressble flows wth complcated geometry and movng boundary. In the flows around a 2D oscllatng crcular cylnder and a sphere, the present method gves the approprate flow felds wth the smooth pressure dstrbutons and the specfed velocty 510

12 Journal of Flud Scence and Technology condtons on the vrtual boundary. For the flow around a statonary fsh models, the characterstc surface shape of a fsh,.e., breast fns, abdomen fns, hp fn, back fn and tal fn, can be reappeared by usng the trangular polygon. The flows around these fns can be resolved qualtatvely. Fnally, the swmmng fsh model s constructed by 2D rotaton on four rotatng axes. The flow wth the motons of each fn can be clarfed and the non-dmensonal drag force becomes smaller than the flow around a statonary fsh model. Then, t s concluded that the present seamless vrtual boundary method s very promsng for the numercal smulaton of ncompressble flows wth the complcated geometry and the movng boundary. References (1) E.A. Fadlun, R. Verzcco, P. Orland, and J. Mohd-Yosof, Combned mmersed-boundary fnte-dfference methods for three-dmensonal complex smulatons, J. Comput. Phys., 161, (2000), pp (2) D. Goldsten, R. Handler, and L. Srovch, Modelng a no-slp flow boundary wth an external force feld, J. Comput. Phys., 105, (1993), pp (3) H. Nshda, Cartesan grd approach wth vrtual boundary method and ts applcatons, Notes on Numercal Flud Mechancs (Sprnger), 78, (2001), pp (4) E.M. Sak and S. Brngen, Numercal smulaton of a cylnder n unform flow: applcaton of a vrtual boundary method, J. Comput. Phys., 123, (1996), pp (5) M. Tyag and S. Acharya, Large eddy smulaton of turbulent flows n complex and movng rgd geometres usng the mmersed boundary method, Int. J. Numer. Meth. Fluds, 48, (2005), pp (6) T. Ye, R. Mttal, H.S. Udaykumar, and W. Shyy, An accurate Cartesan grd method for vscous ncompressble flows wth complex mmersed boundares, J. Comput. Phys., 156, (1999), pp (7) Y.-H. Tseng and J. H. Ferzger, A ghost-cell mmersed boundary method for flow n complex geometry, J. Comput. Phys., 192, 593(2003), pp (8) J. Km, D. Km, and H. Cho, An mmersed-boundary fnte-volume method for smulatons of flow n complex geometres, J. Comput. Phys., 171, (2001), pp (9) H. Nshda and K. Sasao, Incompressble flow smulatons usng vrtual boundary method wth new drect forcng terms estmaton, Proc. Internatonal Conference on Computatonal Flud Dynamcs 2006, (2006), pp (10) Y. Mornsh, T.S. Lund, O.V. Vaslyev, and P. Mon, Fully conservatve hgher order fnte dfference schemes for ncompressble flow, J. Comput. Phys., 143, 90(1998), pp (11) C.H. Rhe and W.L. Chow, Numercal study of the turbulent flow past an arfol wth tralng edge separaton, AIAA Journal, 21-11, (1983), pp (12) R.A. Renaut, Two-step Runge-Kutta method and hyperbolc partal dfferental equaton, Math. Compt., , (1990), pp (13) K. Kawakam, H. Nshda, and N. Satofuka, An open boundary condton for the numercal analyss of unsteady ncompressble flow usng the vortcty-streamfuncton formulaton, Trans. JSME Ser. B, , (1994), pp (n Japanese) (14) G. Wu, B. Pao, and S. Kuroda, Development of cut cell method for ncompressble flud flows, Proc. 18th Symp. on Computatonal Flud Dynamcs, D9-2 (CD-ROM), (2004), pp.1-5. (n Japanese) (15) S. Shrayama, Flow past a sphere: topologcal transtons of the vortcty feld, AIAA Journal, 30-2, (1992), pp