Calculation of Potential Flow Around An Elliptic Cylinder Using Boundary Element Method

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Calculation of Potential Flow Around An Elliptic Cylinder Using Boundary Element Method M. Mushtaq Saima Nazir N. A. Shah, Ph.D. G. Muhammad Abstract In this paper, a direct boundary element method is applied for calculating the incompressible potential flow field (i.e. velocity distribution) around an elliptic cylinder. To check the accuracy of the method, the computed flow velocity is compared with the analytical solution for the flow over the boundary of an elliptic cylinder. It can be seen that the output given by the boundary element method is approximately the same as the analytical result obtained for an elliptic cylinder. Elliptic Coordinates Let z = c cosh (1) where z = x + i y and = + i be a transformation from the plane to the z plane. Then (1) implies the transformation equations x = c cosh cos (2) y = c sinh sin (3) Squaring and adding (2) and (3), we get x 2 c 2 cosh 2 + y 2 c 2 sinh 2 = 1 (4) From (4), it is clear that if has the constant value, the point ( x, y ) lies 0 on an ellipse whose semi axes a and b are given by a = c cosh 0 and b = c sinh 0 (5) and therefore a 2 b 2 = c 2 ( cosh 2 sinh 2 ) = c 2 (6) 0 0 Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 37

The ellipses corresponding to constant values of are therefore confocal, the distance between the foci being 2 c. Now for the ellipse =, (5) gives 0 = 0 tanh 1 b a = 1 2 ln a + b a b Equation (7) determines the parameter 0 (7) in terms of semi axes a and b. Flow Past An Elliptic Cylinder Let there be a circular cylinder of radius = a + b 2 with centre at the origin of the plane. Let a uniform stream of velocity U in the positive direction of axis be flowing past the cylinder (Figure I). Figure I We know that the complex velocity potential for the flow past a circular cylinder of radius = a + b 2 in the plane is given by circle theorem w ( ) = U + ( a + b )2 4 (8) The Joukowski transformation z = + c 2 4, c 2 = a 2 b 2 (9) transforms the circle = a + b 2 in the plane onto an ellipse of semi axes a and b, with centre at the origin in the z plane. Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 38

Since z = x + i y and = + i Then c 2 x = 1 + 4 ( 2 + 2 ) c 2, y = 1 4 ( 2 + 2 ) are the transformation equations from the plane to z plane. The complex velocity potential w ( z ) for flow past an elliptic cylinder x 2 having cross section a 2 + y 2 2 b = 1 in the z plane is found by eliminating from (8) and (9). Solving (9), we get 4 2 4 z + c 2 = 0 Then = z + z 2 c 2 2 We take the positive sign with the radical so as to obtain the transformation which maps the region outside the circle in the plane onto the region outside the ellipse in the z plane. Thus = 1 2 ( z + z 2 c 2 ) (10) From (8) and (10), we get w ( z ) = U a + b 2 z + z 2 c 2 a + b + z z 2 c 2 a b (11) is the required complex velocity potential. The form of complex velocity potential can, however, be simplified by means of elliptic s, which takes the form w ( ) = U c e 0 cosh ( 0 ) (12) To find the speed V, the complex velocity is given by d w d z = d w d d d z which in elliptic s becomes d w d z = U e 0 sinh ( ) cos + i cosh ( ) sin 0 0 sinh cos + i cosh sin Thus Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 39

V = d w d z = U e 0 sinh 2 ( ) cos 2 + cosh 2 ( ) sin 2 0 0 sinh 2 cos 2 + cosh 2 sin 2 (13) Squaring and after simplification, (13) takes the form V 2 = U 2 e 2 0 sinh 2 ( ) + sin 2 0 sinh 2 + sin 2 Since e 2 0 = a + b a b, therefore V 2 = U 2 ( a + b ) 2 sin 2 b 2 + ( a 2 b 2 ) sin 2 Dividing (3) by (2) (14) y x = tanh tan (15) On the surface of the cylinder, = 0, (7) gives tanh 0 = b a From (15), it follows that sin = and cos = a y b 2 x 2 + a 2 y 2 (16) b x b 2 x 2 + a 2 y 2 (17) Using (16), after simplification (14) takes the form V = U ( a + b ) a y b 4 x 2 + a 4 y 2 (18) which is the analytical formula for calculating the velocity distribution past an elliptic cylinder in terms of a and b. Distribution Boundary element methods can be used to solve two-dimensional interior or exterior flow problems. Consider the flow past an elliptic cylinder of Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 40

semi axes a and b with center at the origin and let the onset flow be the uniform stream with velocity U in the positive direction of the x-axis as shown in Figure II. Figure II The condition to be satisfied on the boundary of an elliptic cylinder is e.c n = u.s n (19) where u.s is the velocity potential of uniform stream and e.c is the velocity potential at the surface of the elliptic cylinder. But the velocity potential of the uniform stream is given as u.s = U x Then u.s n = U x n = U ( ^n. ^i ) (20) Thus from (19) and (20), we get e.c n = U ( ^n. ^i ) (21) The equation of the boundary of the elliptic cylinder is x 2 a 2 + y2 b 2 = 1 (22) Thus from (21), we get e.c n = U x b 2 b 4 x 2 + a 4 y 2 (23) Equation (23) is the boundary condition which must be satisfied over the boundary of an elliptic cylinder. Now for the approximation of the boundary of the elliptic cylinder, the s of the extreme points of the boundary elements can be generated within the computer program as follows: Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 41

Divide the boundary of the elliptic cylinder into m elements in the clockwise direction by using the formula k = [(m + 3) 2k] / m, k = 1, 2,, m (24) Then the s of the extreme points of these m elements are calculated from x k = a cos k, k = 1, 2,, m (25) y k = b sin k Take m = 8, a = 2 and b = 1. First consider the case of constant boundary elements where there is only one node at the middle of the element and the potential and the potential derivative are constant over each element and equal to the value at the n middle node of the element. Figure III Figure III shows the discretization of an elliptic cylinder into 8 constant boundary element. The s of the middle node of each boundary element are given by x m = (x k + x k+1 ) / 2 y m = (y k + y k+1 ) / 2 k, m = 1, 2,, 8 (26) and therefore the boundary condition (23) in this case takes the form e.c n = U x m b 2 b 4 x 2 m + a 4 y 2 m. The velocity U of the uniform stream is also taken as unity. Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 42

Table I The comparison of the analytical and computed velocity distributions over the boundary of the elliptic cylinder for 8 constant boundary elements. Element x y R = x 2 + y 2 Computed Analytical 1-1.58.33.16106E+01.10777E+01.95693E+00 2 -.65.79.10240E+01.14989E+01.14688E+01 3.65.79.10240E+01.14989E+01.14688E+01 4 1.58.33.16106E+01.10777E+01.95693E+00 5 1.58 -.33.16106E+01.10777E+01.95693E+00 6.65 -.79.10240E+01.14989E+01.14688E+01 7 -.65 -.79.10240E+01.14989E+01.14688E+01 8-1.58 -.33.16106E+01.10777E+01.95693E+00 Table II The improvement gained by using 16 constant boundary elements Element x y R = x 2 + y 2 Computed Analytical 1-1.89.19.18962E+01.58244E+00.55447E+00 2-1.60.53.16866E+01.12226E+01.12010E+01 3-1.07.80.13350E+01.14319E+01.14227E+01 4 -.38.94.10154E+01.14984E+01.14926E+01 5.38.94.10154E+01.14984E+01.14926E+01 6 1.07.80.13350E+01.14319E+01.14227E+01 7 1.60.53.16866E+01.12226E+01.12010E+01 8 1.89.19.18962E+01.58244E+00.55447E+00 9 1.89 -.19.18962E+01.58244E+00.55447E+00 10 1.60 -.53.16866E+01.12226E+01.12010E+01 11 1.07 -.80.13350E+01.14319E+01.14227E+01 12.38 -.94.10154E+01.14984E+01.14926E+01 13 -.38 -.94.10154E+01.14984E+01.14926E+01 14-1.07 -.80.13350E+01.14319E+01.14227E+01 15-1.60 -.53.16866E+01.12226E+01.12010E+01 16-1.89 -.19.18962E+01.58244E+00.55447E+00 Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 43

Table III The improvement gained by using 32 constant boundary elements Element x y R = x 2 + y 2 Comp. Ana. 1-1.97.10.19736E+01.29449E+00.28991E+00 2-1.90.29.19172E+01.78642E+00.77805E+00 3-1.75.47.18082E+01.11020E+01.10954E+01 4-1.53.63.16551E+01.12852E+01.12810E+01 5-1.26.77.14714E+01.13904E+01.13877E+01 6 -.93.87.12786E+01.14510E+01.14491E+01 7 -.57.95.11085E+01.14845E+01.14830E+01 8 -.19.99.10046E+01.14995E+01.14982E+01 9.19.99.10046E+01.14995E+01.14982E+01 10.57.95.11085E+01.14845E+01.14830E+01 11.93.87.12786E+01.14510E+01.14491E+01 12 1.26.77.14714E+01.13904E+01.13877E+01 13 1.53.63.16551E+01.12852E+01.12810E+01 14 1.75.47.18082E+01.11020E+01.10954E+01 15 1.90.29.19172E+01.78641E+00.77805E+00 16 1.97.10.19736E+01.29449E+00.28990E+00 17 1.97 -.10.19736E+01.29449E+00.28990E+00 18 1.90 -.29.19172E+01.78641E+00.77805E+00 19 1.75 -.47.18082E+01.11020E+01.10954E+01 20 1.53 -.63.16551E+01.12852E+01.12810E+01 21 1.26 -.77.14714E+01.13904E+01.13877E+01 22.93 -.87.12786E+01.14510E+01.14491E+01 23.57 -.95.11085E+01.14845E+01.14830E+01 24.19 -.99.10046E+01.14995E+01.14982E+01 25 -.19 -.99.10046E+01.14995E+01.14982E+01 26 -.57 -.95.11085E+01.14845E+01.14830E+01 27 -.93 -.87.12786E+01.14510E+01.14491E+01 28-1.26 -.77.14714E+01.13904E+01.13877E+01 29-1.53 -.63.16551E+01.12852E+01.12810E+01 30-1.75 -.47.18082E+01.11020E+01.10954E+01 31-1.90 -.29.19172E+01.78642E+00.77805E+00 32-1.97 -.10.19736E+01.29448E+00.28990E+00 Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 44

Table IV The comparison of the analytical and computed velocities at the mid points of 8 linear boundary elements over an elliptic cylinder. Element x y R = x 2 + y 2 Computed Analytical 1-1.71.35.17433E+01.87357E+00.95693E+00 2 -.71.85.11084E+01.14583E+01.14688E+01 3.71.85.11084E+01.14583E+01.14688E+01 4 1.71.35.17433E+01.87357E+00.95693E+00 5 1.71 -.35.17433E+01.87357E+00.95693E+00 6.71 -.85.11084E+01.14583E+01.14688E+01 7 -.71 -.85.11084E+01.14583E+01.14688E+01 8-1.71 -.35.17433E+01.87357E+00.95693E+00 Table V The improvement gained by using 16 linear boundary elements y Computed Element x R = x 2 + y 2 Analytical 1-1.92.19.19334E+01.51847E+00.55447E+00 2-1.63.54.17196E+01.11710E+01.12010E+01 3-1.09.82.13611E+01.14180E+01.14227E+01 4 -.38.96.10353E+01.14950E+01.14926E+01 5.38.96.10353E+01.14950E+01.14926E+01 6 1.09.82.13611E+01.14180E+01.14227E+01 7 1.63.54.17196E+01.11710E+01.12010E+01 8 1.92.19.19334E+01.51848E+00.55447E+00 9 1.92 -.19.19334E+01.51847E+00.55447E+00 10 1.63 -.54.17196E+01.11710E+01.12010E+01 11 1.09 -.82.13611E+01.14180E+01.14227E+01 12.38 -.96.10353E+01.14950E+01.14926E+01 13 -.38 -.96.10353E+01.14950E+01.14926E+01 14-1.09 -.82.13611E+01.14180E+01.14227E+01 15-1.63 -.54.17196E+01.11710E+01.12010E+01 16-1.92 -.19.19334E+01.51847E+00.55447E+00 Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 45

Table VI The improvement gained by using 32 linear boundary elements y Computed Element x R = x 2 + y 2 Analytical 1-1.98.10.19832E+01.28201E+00.28991E+00 2-1.90.29.19264E+01.76380E+00.77805E+00 3-1.76.47.18170E+01.10855E+01.10954E+01 4-1.54.63.16631E+01.12762E+01.12810E+01 5-1.26.77.14786E+01.13860E+01.13877E+01 6 -.94.88.12848E+01.14490E+01.14491E+01 7 -.58.95.11139E+01.14836E+01.14830E+01 8 -.20.99.10094E+01.14990E+01.14982E+01 9.20.99.10094E+01.14990E+01.14982E+01 10.58.95.11139E+01.14836E+01.14830E+01 11.94.88.12848E+01.14490E+01.14491E+01 12 1.26.77.14786E+01.13860E+01.13877E+01 13 1.54.63.16631E+01.12762E+01.12810E+01 14 1.76.47.18170E+01.10855E+01.10954E+01 15 1.90.29.19264E+01.76380E+00.77805E+00 16 1.98.10.19832E+01.28201E+00.28990E+00 17 1.98 -.10.19832E+01.28201E+00.28990E+00 18 1.90 -.29.19264E+01.76380E+00.77805E+00 19 1.76 -.47.18170E+01.10855E+01.10954E+01 20 1.54 -.63.16631E+01.12762E+01.12810E+01 21 1.26 -.77.14786E+01.13860E+01.13877E+01 22.94 -.88.12848E+01.14490E+01.14491E+01 23.58 -.95.11139E+01.14836E+01.14830E+01 24.20 -.99.10094E+01.14990E+01.14982E+01 25 -.20 -.99.10094E+01.14990E+01.14982E+01 26 -.58 -.95.11139E+01.14836E+01.14830E+01 27 -.94 -.88.12848E+01.14490E+01.14491E+01 28-1.26 -.77.14786E+01.13860E+01.13877E+01 29-1.54 -.63.16631E+01.12762E+01.12810E+01 30-1.76 -.47.18170E+01.10855E+01.10954E+01 31-1.90 -.29.19264E+01.76381E+00.77805E+00 32-1.98 -.10.19832E+01.28201E+00.28990E+00 Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 46

Figure IV: Comparison of computed and analytical velocity distributions over the surface of an elliptic cylinder using 8 boundary elements with constant variation. Figure V: Comparison of computed and analytical velocity distributions over the surface of an elliptic cylinder using 16 boundary elements with constant variation. Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 47

Figure VI: Comparison of computed and analytical velocity distributions over the surface of an elliptic cylinder using 32 boundary elements with constant variation. Consider next the case where the boundary of an elliptic cylinder is divided into linear elements. In this case the nodes where the boundary conditions are specified are at the intersection of the elements. The boundary of the cylinder can be divided into elements by the formula k = [(m + 2) 2k] / m, k = 1, 2,, m (27) Figure VII shows the discretization of an elliptic cylinder into 8 linear boundary elements. Figure VII Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 48

The s of the extreme points of the elements and the s of the mid-point of each element where the velocity will be calculated can be found from (23) and (24) respectively. The reason for distributing the elements in this way is such that the velocities are calculated at the same values of in both figures III and IV, so that the calculated results could be easily compared. Figure VIII: Comparison of computed and analytical velocity distributions over the surface of an elliptic cylinder using 8 boundary elements with linear variation. Figure IX: Comparison of computed and analytical velocity distributions over the surface of an elliptic cylinder using 16 boundary elements with linear variation. Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 49

Figure X: Comparison of computed and analytical velocity distributions over the surface of an elliptic cylinder using 32 boundary elements with linear variation. Conclusion A direct boundary element method has been used for the calculation of incompressible potential flow around two dimensional body i.e. an elliptic cylinder. The computed velocity distribution obtained using this method is compared with the analytical solutions for flows over the boundary of an elliptic cylinder using 8,16 and 32 constant and linear boundary elements. It is found that the results obtained with the direct boundary element method for the flow field calculations are in excellent agreement with the analytical results for the bodies under consideration. M. Mushtaq, University of Engineering & Technology. Lahore, Pakistan Saima Nazir, University of Engineering & Technology. Lahore, Pakistan N. A. Shah, Ph.D., University of Engineering & Technology. Lahore, Pakistan G. Muhammad, University of Engineering & Technology. Lahore, Pakistan References 1. Hess, J.L. and Smith, A.M.O.: Calculation of potential flow about arbitrary bodies, Progress in Aeronautical Sciences, Vol. 8, pp 1-158, Pergamon Press 1967. 2. Hess, J.L.: Higher order numerical solutions of the integral equation for the two-dimensional Neumann problem, Computer Methods in Applied Mechanics and Engineering, Vol. 2, 1973 pp 1-15. Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 50

3. Morino, L., Chen, Lee-Tzong and Suciu,E.O.: A steady and oscillatory subsonic and supersonic aerodynamics around complex configuration, AIAA Journal, Vol. 13, No 3 March 1975, pp 368-374. 4. Milne-Thomson, L.M.: Theoretical Hydrodynamics, 5 th Edition, London Macmillan & Co. Ltd., 1968. 5. Brebbia, C.A.: The Boundary element Method for Engineers, Pentech Press (1978). 6. Brebbia, C.A. and Walker, S.: Boundary Element Techniques in Engineering, Newnes-Butterworths (1980). 7. Shah Nor Basri, et al: Incompressible Potential Flow Analysis Using Panel Method, Universiti Purta, Malaysia (2005). 8. Dambaru, D. Bhatta: Fluid Flow Around An Elliptic Cylinder Using Finite Element Method, Journal of Mathematical Sciences & Mathematics Education Volume 1, Number 1, February 2006, PP. 7 12. 9. N.A. Shah, Ideal Fluid Dynamics, A One Publishers, Lahore Pakistan (2008). 10. G. Muhammad,N. A.Shah & M. Mushtaq : Indirect Boundary Element Method for the flow past a circular cylinder with linear element approach, Int. J of Applied Engg. Research(Accepted for Publication) Journal of Mathematical Sciences & Mathematics Education, Vol. 5 No. 1 51

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