Analysis of potential flow around two-dimensional body by finite element method

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1 Vol. 7(2), pp. 9-22, May, 2015 DOI: /JMER rticl Numbr: 20E ISSN Copyright 2015 uthor(s) rtain th copyright of this articl Journal of Mchanical Enginring Rsarch Full Lngth Rsarch Papr nalysis of potntial flow around two-dimnsional body by finit lmnt mthod Md. Shahjada arafdr* and Nabila Naz Dpartmnt of Naval rchitctur and Marin Enginring, Bangladsh Univrsity of Enginring and chnology, Dhaka-1000, Bangladsh. Rcivd 24 Novmbr 2014; ccptd 23 March, 2015 h papr prsnts a numrical mthod for analyzing th potntial flow around two dimnsional body such as singl circular cylindr, NC0012 hydrofoil and doubl circular cylindrs by finit lmnt mthod. h numrical tchniqu is basd upon a gnral formulation for th Laplac s quation using Galrkin tchniqu finit lmnt approach. h solution of th systms of algbraic quations is approachd by Gaussian limination schm. Laplac s quation is xprssd in trms of both stam function and vlocity potntial formulation. finit lmnt program is dvlopd in ordr to analyz th rsult. h contours of stram and vlocity potntial function ar drawn. h contour of stram function xhibits th charactristics of potntial flow and dos not intrsct ach othr. h calculatd prssur co-fficint shows th prssur dcrasing around th forwardd fac from th initial total prssur at th stagnation point and raching a minimum prssur at th top of th cylindr. Ky words: Stram function, vlocity potntial, numbr of nods (NDE), numbr of lmnts (NEL). INRODUCION h flow past two dimnsional body such as circular cylindrs and hydrofoil has bn th subjct of numrous xprimntal and numrical studis bcaus this typ of flow xhibits th vry fundamntal mchanisms. h flow fild ovr both th cylindr and hydrofoil is symmtric at low valus of Rynolds numbr. s th Rynolds numbr incrass, flow bgins to sparat bhind th body causing vortx shdding which is an unstady phnomnon. o achiv th goal of obtaining th dtaild information of th flow fild around two dimnsional bodis, Finit Elmnt Mthod (FEM) has bn mrgd as an attractiv, powrful tool in many dsigning procss. h FEM was originatd from th fild of structural calculation in th bginning of th fiftis and was introducd by urnr t al. (1956). h FEM was introducd into th fild of computational fluid dynamics (CFD) by Chung (1977). h first study concrning th stady flow past a circular cylindr was rportd by hom (1933) for Rynolds numbr of 10 and 20. h works of Kawaguti (1953) and Payn (1958) wr rstrictd to low Rynolds numbrs (R = 40) and rlativly low Rynolds numbrs (R = 40~100) rspctivly. Odn (1969) has prsntd a thortical finit lmnt analogu for th Navir- Stoks quations, but without a practical numrical mthod. Dnnis and Chung (1970) introducd finit lmnt mthod into th fild of computational fluid dynamics (CFD) by solving stady flow past a circular cylindr at Rynolds numbr (R 100). *Corrsponding author. mshahjadatarafdr@nam.but.ac.bd. uthor(s) agr that this articl rmain prmanntly opn accss undr th trms of th Crativ Commons ttribution Licns 4.0 Intrnational Licns

2 10 J. Mch. Eng. Rs. ong (1971) prsntd rsults for stady flow using this mthod with prssur and vlocitis as dpndnt variabls. Olson (1974) prsntd a numrical procdur to invstigat stady incomprssibl flow problms using stram function formulation. Hafz (2004) simulats stady an inviscid flows ovr a cylindr using both potntial and stram functions. h objctiv of th prsnt rsarch is to analyz th potntial flow around singl circular cylindr, NC 0012 hydrofoil and doubl circular cylindrs by Galrkin tchniqu of finit lmnt mthod. Du to symmtry of circular cylindrs and NC 0012 hydrofoil, only th uppr half portions hav bn considrd as computational domain. Both stram function and vlocity potntial formulation hav bn usd with dfinit boundary conditions. Contours of stram and vlocity potntial function, vlocity abov crst for both formulations and vlocity and prssur distribution along th surfac for various discrtization ar obtaind which ar compard with th analytical rsult availabl from th litratur and shown graphically. Boundary conditions for vlocity potntial In cas of vlocity formulation, th boundary conditions that nd to b satisfid in ordr to gt th solution of Laplac quation: in Ω as shown in Figur 3: (a) On th boundary a-b, U x (b) On th boundary a--f-g, 0 y (c) On th boundary b-h, 0 y (d) On th boundary g-h, U x Potntial flow around a hydrofoil Lt us considr th flow of an idal fluid around a hydrofoil placd with its axis prpndicular to th plan of th flow as shown Figur 4. MHEMICL FORMULION Potntial flow around circular cylindr Lt us considr th potntial flow of an idal fluid around th circular cylindr placd with its axis prpndicular to th plan of th flow as shown in Figur 1. Now, th potntial flow around circular cylindr can b rprsntd by Laplac quation as: 2 0 Boundary conditions for stram function Now w nd to solv th Laplac quation: shown in Figur 5 with th following boundary conditions: (a) = 0 on th boundary a-b-c-d (b) = yu on th boundary a-f and -d (c) = yu on th boundary f- in Ω as 2 0 (2.1) (1) h vlocity componnts u and v of th flow fild in rlation to stram function or th vlocity potntial ar givn by: u u, v y x or, (2.2), v x x (2) Boundary conditions for vlocity potntial In cas of vlocity potntial formulation, w nd to solv th Laplac quation: following boundary conditions: (a) On th boundary a-f and d-, 1 n (b) On th boundary a-b-c-d and f-, 0 n Potntial flow around two circular cylindrs in Ω as shown in Figur 6 with th Boundary conditions for stram function h half of th fluid domain is takn in th computations as shown in Figur 2 and th boundary conditions that nd to b satisfid in ordr to gt th solution of Laplac quation: givn as follows: (a) = 0 on th boundary a--f-g (b) = yu on th boundary a-b (c) = yu on th boundary b-h (d) = yu on th boundary g-h in Ω ar Lt us considr potntial flow around doubl circular cylindrs as shown in Figur 7. h stram function for th flow can b xprssd as x, y) ( x, y) a ( x, y) b ( x, ) (3) ( y Whr a and b ar th two constants. Now w nd to solv Laplac s quation ; ; with th following boundary conditions:

3 arafdr and Naz 11 Figur 1. Flow around singl circular cylindr. Figur 2. Boundary conditions for th stram function formulation. Figur 3. Boundary conditions for th vlocity potntial function formulation.

4 12 J. Mch. Eng. Rs. Figur 4. Flow around a NC 0012 hydrofoil. Figur 5. Boundary conditions for th stram function formulation for hydrofoil. Figur 6. Boundary conditions for th vlocity potntial function formulation. Figur 7. Flow around doubl circular cylindrs with boundary conditions.

5 arafdr and Naz 13 (a) Ψ 1 = U y on S 1 (b) Ψ 1 = 0 on S 2 and S 3 (c) Ψ 2 = 0 on S 1 and S 3 (d) Ψ 2 = 1 on S 2 () Ψ 3 = 0 on S 1 and S 2 (d) Ψ 3 = 1 on S 3 ( ) [ N] [ N] [ N] [ N] [ k ] ( ) dxdy x x y y (10) th nodal forcs ar rprsntd by th column matrix NUMERICL SOLUION OF POENIL FLOW Numrical solution by stram function mthod h stram function ovr th domain of intrst is discrtizd into finit lmnts having M nods: M ( x, y) Ni( x, y) i [ N]{ } i1 Using th Galrkin mthod, th lmnt rsidual quations ar: i N ( x, y)( ) dxdy 0, i 1, M (5) x y or, [ N] ( ) dxdy 0 (6) x y pplication of th Grn-Gauss thorm givs [ N] [ N] nxds dxdy [ N] nyds x x x y S S [ N] y dxdy (7) y Whr S rprsnts th lmnt boundary and (n x, n y) ar th componnts of th outward unit vctor normal to th boundary. Using Equation (4) in Equation (7) and substituting th vlocity componnts into th boundary intgrals, rsults in: [ N] [ N] [ N] [ N] dxdy { } x x y y S [ N] ( un vn ) ds and this quation is of th form ( ) [ k ]{ } { f } y x (9) (4) (8) ( ) { f } [ N] ( un ) S y vnx ds (11) Numrical solution by vlocity potntial mthod h finit lmnt formulation of potntial flow of an idal fluid in trms of vlocity potntial is quit similar to that of th stram function approach, sinc th govrning quation is Laplac s quation in both cass. By dirct analogy with Equations (4) to (11) it is obtaind as follows: M i i (12) i1 ( x, y) N ( x, y) [ N]{ } Using th Galrkin mthod, th lmnt rsidual quations ar: i N ( x, y)( ) dxdy 0, i 1, M (13) x y or, [ N] ( ) dxdy 0 (14) x y pplication of th Grn-Gauss thorm givs [ N] [ N] nxds dxdy [ N] nyds x x x y S S [ N] y dxdy (15) y Utilizing Equation (12) in th ara intgral of Equation (15) and substituting th vlocity componnts into th boundary intgrals, rsults in: [ N] [ N] [ N] [ N] dxdy { } x x y y S [ N] ( un vn ) ds and this quation is of th form x y (16) h lmnt stiffnss matrix is ( ) [ k ]{ } { f } (17)

6 14 J. Mch. Eng. Rs. RESULS ND DISCUSSION Basd on th prvious mathmatical formulation as outlind as numrical solution by stram function mthod and numrical solution by vlocity potntial mthod a finit lmnt program has bn dvlopd in FORRN 90 for calculating th potntial flow around two dimnsional bodis. For all finit lmnt msh configurations, nods along th vrtical lin abov th crst of th cylindr ar numbrd conscutivly from top to bottom in ordr to b compatibl with vlocity calculations usd in th program. h lmnts ar takn in th form of triangl or quadrilatral for th convninc of discrtization, thus th dg of th body may not b appard as a circl or hydrofoil. Singl circular cylindr Lt us considr th flow around th circular cylindr of unit radius confind btwn two paralll plats having lngth of 7 m and hight 4 m. fluid of uniform vlocity 1.0 m/s is assumd to b flowing from th lft to th right of cylindr as shown in Figur 8. h choic of computational domain in th dirction of flow is arbitrary and th fr stram vlocity is considrd to prvail at distancs sufficintly far from th cylindr. h uppr half of th computational domain surroundd by th path (a-bc-d--f) is takn into account for numrical calculation du to symmtry of flow and is discrtizd by (24 5) triangular lmnts as shown in Figur 9 for stram function formulation. h contour of stram function has bn obtaind from stram function formulation and xhibits th charactristics of potntial flow as shown in Figur 10. h stram lins hav not intrsctd ach othr and man th flow past th cylindr smoothly without any sparation at th trailing dg. h uppr half of th computational domain for vlocity potntial formulation is also discrtizd by (20 7) quadrilatral lmnts as shown in Figur 11. h contours of vlocity potntial hav also bn obtaind from vlocity potntial formulation and xhibit th charactristic of potntial flow i.. no vortics xist at th trailing dg as shown in Figur 12. h vlocitis along th vrtical lin abov th crst of th cylindr ar calculatd and thn compard with th analytical rsult in Figur 13. h avrag dviation for th vlocity profils btwn th two cass is lss than on prcnt. Figur 14 is plottd by calculating th vlocitis abov th crst at two points(x = 3.50, y = 1.00) and (x = 3.50, y = 2.00) against various numbr of nods for stram function formulation which shows that computd vlocitis convrgs to th analytical solution as numbr of nods incrass. Figur 15 is obtaind by plotting th vlocity squar along th surfac of cylindr against th angular coordinats of nodal points. hr ar two typs of curvs of which first typ shows a sinusoidal curv for th whol cylindr obtaind thortically and scond typ consists of four curvs for four diffrnt finit lmnt msh configurations. In Figur 16 th calculatd prssur cofficint (C p ) is compard with th thortical prssur distribution ovr th surfac of th cylindr and th agrmnt is found to b quit satisfactory. h calculatd rsults show th prssur dcrasing around th forwardd fac from th initial total prssur at th stagnation point and raching a minimum prssur at th top of th cylindr. NC 0012 hydrofoil Lt us considr th flow around NC 0012 hydrofoil confind btwn two paralll plats having lngth of 10 m and hight 4m as shown in Figur 17. fluid of uniform vlocity 1 m/s is flowing from th lft to right of th foil. h half of computational domain for stram function formulation is discrtizd by (16 3) lmnts as shown in Figur 18 and th contours of th stram lins ar givn in Figur 19. Similarly, th half of computational domain for vlocity potntial formulation is discrtizd by (16 3) lmnts as shown in Figur 20 and th contours of th stram lins ar givn in Figur 21. Figur 22 dpicts a comparison of prssur distribution prssur ovr th surfac of th foil with th rsults obtaind from constant strngth sourc mthod 11 and shows vry clos agrmnt both at th lading and trailing dg of th foil. Doubl circular cylindrs h flow around two circular cylindrs of unit radius is confind btwn two paralll plats having lngth of 10 m and hight 4m. h distanc btwn th cylindrs is unit lngth and th hight abov th cylindr is also unit lngth. fluid of uniform vlocity 1 m/s is flowing from lft to right of cylindrs as shown in Figur 23. h half of th computational domain is discrtizd by (16 3) lmnts for stram function formulation and (20 7) lmnts for vlocity potntial as shown in Figurs 24 and 25, rspctivly. h contours obtaind from ths two formulations ar drawn in Figurs 26 and 27 rspctivly. Conclusions h papr prsnts a numrical mthod of calculating th potntial flow around two dimnsional bodis by finit lmnt mthod. h following conclusions can b drawn from th prsnt numrical analysis: (i) h prsnt mthod can b an fficint tool for valuating th potntial flow charactristics of two dimnsional body. (ii) h contour of stram function xhibits th charactristics of potntial flow and dos not intrsct ach othr.

7 arafdr and Naz 15 Figur 8. Computational domain for stram function and vlocity potntial formulation. Figur 9. Discrtization of domain by 240 triangular lmnts for stram function formulation. Figur 10. Stram function contours around th half circular cylindr.

8 Y 16 J. Mch. Eng. Rs. Figur 11. Discrtization of computational domain by 140 quadrilatral lmnts for vlocity potntial. Figur 12. Vlocity potntial contours around th half circular cylindr Vlocity distribution abov crst Vlocity Potntial formulation NDE 11 X 8 NDE 11 X 9 NDE 14 X 8 NDE 13 X 9 Stram function formulation NDE 13 X 6 NDE 10 X 9 NDE 12X X-Vlocity Figur 13. Vlocity distributions abov th crst of cylindr.

9 V^2 Vlocity of th crst arafdr and Naz Convrgnc of vlocitis abov crst FEM solution at,x=3.50,y=1.00 FEM solution at,x=3.50,y=2.00 nalytical solution at x=3.50,y=1.00 nalytical solution at x=3.50,y= Numbr of nods Figur 14. Error analysis for Ψ formulation showing convrgnc of vlocitis abov th crst. 4 3 V 2 distribution along cylindr surfac NDE 21 X 8 NDE 21 X 9 NDE 27 X 8 NDE 25X 9 hortical ngl Figur 15. Vlocity profil along cylindr surfac.

10 Cp 18 J. Mch. Eng. Rs. 1 0 Cp Distribution NDE 11 X 8 NDE 11 X 9 NDE 14 X 8 NDE 13 X 9 hortical ngl Figur 16. Distribution of prssur cofficint (C p). Figur 17. Computational domain for flow around NC 0012 hydrofoil. Figur 18. Discrtization of computational domain by 120 triangular lmnts for th hydrofoil.

11 arafdr and Naz 19 Figur 19. Stram function contours for flow around NC 0012 hydrofoil. Figur 20. Discrtization of computational domain by 60 quadrilatral lmnts for th hydrofoil. Figur 21. Contours of vlocity potntial.

12 Cp 20 J. Mch. Eng. Rs Cp distribution around NC0012 hydrofoil Vlocity potntial formulation Constant strngth sourc mthod(rabiul 2008) x/c Figur 22. Chord wis prssur variations. Figur 23. Computational domain for flow around two circular cylindrs.

13 arafdr and Naz 21 Figur 24. Discrtization of domain by 48 triangular lmnts for flow around two circular cylindrs. Figur 25. Msh arrangmnt for vlocity potntial formulation. Figur 26. Stram lins contours from stram function formulation. Figur 27. Vlocity potntial lins.

14 22 J. Mch. Eng. Rs. (iii) h calculatd prssur co-fficint shows th prssur dcrasing around th forwardd fac from th initial total prssur at th stagnation point and raching a minimum prssur at th top of th cylindr. (iv) h calculatd rsults dpnd to a crtain xtnt on th discrtization of th computational domain and accuracy incrass with incras of numbr of lmnts. Nomnclatur: Ψ:Stram function, Φ:Vlocity Potntial function, N : Shap function, U:Fr stram vlocity, k : Elmnt cofficint matrix, f : Elmnt forc vctor. Conflict of Intrst h authors hav not dclard any conflict of intrst. Kawaguti M (1953). Discontinuous flow past a circular cylindr. J. Phys. Soc. Jap. 8: Payn RB (1958). Calculation of Unstady Viscous Flow Past Cylindr. J. Fluid Mch. 4:81. Odn J (1969). Gnral hory of Finit Elmnts. Int. J. Numr. Mthods Eng. 1: Dnnis SCR, Chung GZ (1970). Numrical Solutions for Stady Flow past a Circular Cylindr at Rynolds Numbrs Up to 100. J. Fluid Mch. 42(3): ong P (1971). h Finit Elmnt Mthod for Fluid Flow. In: Gallaghr RH, Odn J, Yamada Y (ds.), Rcnt dvancs in Matrix Mthod of Structural nalysis and Dsign. Univrsity of labama Prss, labama. 904pp. Olson MD (1974). Variational-Finit Elmnt Mthods for wo- Dimnsional and xisymmtric Navir-Stoks Equations. Procdings of th fourth Intrnational Symposium on Finit Elmnt Mthods in Flow Problms, Swansa, UK. Hafz M (2004). Inviscid Flows ovr a Cylindr. Comput. Mthods ppl. Mch. Eng. 193: arafdr MS, Khalil GM, Islam MR (2010). nalysis of potntial flow around two-dimnsional hydrofoil by sourc basd lowr and highr ordr panl mthod. J. Institut. Eng. Malaysia (71):2. REFERENCES urnr MJ, Clough RL, Martin HC, opp LJ (1956). Stiffnss and dflction analysis of complx structurs. J. ro. Sci. 23(9): Chung J (1977). Finit Elmnt nalysis in Fluid Dynamics. McGraw Hill, NY, US. pp hom (1933). h flow past circular cylindrs at low spds. Proc. Royal Soc. 141(845):