Research Journal of Appled Scences, Engneerng and Technolog 4(): 53-535, 0 ISSN: 040-7467 Mawell Scentfc Organzaton, 0 Submtted: December 0, 0 Accepted: Januar, 0 Publshed: June 0, 0 Grd Generaton around a Clnder b Comple Potental Functons Hassan Davar, Mahd Chekan and Al Asghar Davar Department of Mechancal Engneerng, Islamc Azad Unverst, Roudan Branch, Roudan, Iran Abstract: In ths stud, orthogonal and structured grd generaton around a clnder s descrbed b usng potental functons. In ths method, the orthogonalt of and n functons are used for grd generaton. Frst, coordnates of ponts are gven b usng the algebrac method on clnder boundares and then, accordng to known potental functons n terms of and values n the eternal flow around a clnder, coordnates of other network's ponts are calculated through solvng a sstem of two nonlnear equatons wth two unknowns. Iteraton methods are used for solvng ths sstem of equatons. The generated grd besdes orthogonal propert has small dstances on the surface of clnder and graduall as t goes farther awa from the clnder, the dstance betweeodes rses. Ths knd of grd can be useful n solvng the flow feld around a clnder. Ke words: Comple potental functon, grd generaton, orthogonal grd INTRODUCTION Computatonal flud dnamcs s a branch of Mechancal Engneerng n whch numercal method s used to solve equatons governng flud dnamcs and heat transfer. Equatons governng flud moton are usuall of partal dfferental tpe that partal dervatves should be appromated to solve these knds of equatons. Wth these appromatons, the equatons of partal dervatves are converted to fnte dfference epressons that dfferental equatons are converted n to algebrac equatons. Obtaned algebrac equatons are called fnte dfference equatons that should be solved n the specal grd and network. So wthn the related doman and on ts borders, some grd ponts are determned. Furthermore, the effect of grd's qualt on accurac and convergence of numercal methods has caused partcular mportance of computatonal grd generaton. Generated grds are classfed nto two groups of structured and unstructured. Structured grd s generated n a wa that grd ponts can be easl dentfed on a regular bass compared to grd lnes that are defned regularl. In other words, each pont of the grd can be defned wth a certan raton of and j. There are dfferent methods to generate grds. The smplest method s algebrac method generated b usng smple algebrac equatons wth respect to geometr of the approprate grd doman. Another method s Comple Varable method on two-dmensonal spaces developed b Churchll (948), Morett (979), Davs (979) and Ives (98). The thrd common method of generatng structured grds s the method of partal dfferental equatons. The dea of usng dfferental equatons s based on research works of Crowle (96) and Wnslow (966) and also based on dea of changng phscal domato a computatonal doman. For solvng dfferental equatons on some specfc spaces wth respect to phscs of the problem, orthogonal grds can have ver benefcal and reduce calculaton's sze. For eample, when we nvestgate flow around an arfol, we know that pressure gradent n the drecton perpendcular to the surface s zero;.e. pressure n the vertcal drecton on the surface wll not change. Accordng to orthogonal grd generaton b methods of dfferental equatons, ths method has been used etensvel. In ths stud b choosng a dfferent method we are lookng for generatng an orthogonal grd around a clnder n order to help us be able to analze the flow around t. In ths stud, orthogonal propert of R and n functons s used for grd generaton. In fact, we wll generate an orthogonal grd b help of potental functons around a clnder and ts relaton wth and values. It should be noted that ths method can be used etensvel for all domans that ther R and n functons are known. MATERIALS AND METHODS Phscal doman: It s assumed that we want to generate an orthogonal grd around a clnder b Comple Potental Functons method. We assume clnder's the radus as m and defne phscal space as show Fg.. As we know, ths method s based on ths assumpton that R and n functons are orthogonal. Therefore, t s enough just to know the functons propert based on and, then ou can easl generate an orthogonal grd: Snce shape has smmetr, frst t s possble to consder one fourth of t as t s show Fg.. Net, we can use mrrorng aganst the horzontal and orthogonal as of the grd n order to make the whole shape. Mesh Correspondng Author: Hassan Davar, Department of Mechancal Engneerng, Islamc Azad Unverst, Roudan Branch, Iran 53
Res. J. Appl. Sc. Eng. Technol., 4(): 53-535, 0 Start Fg. : The Phscal doman Insert N, N, R N = / N- = ( +) = r cos = r sn prnt, No Yes End N No -RN- -R = 0R = 0 X + = X +r - prnt, Fg. : One forth of the phscal doman should be generated n a wa that n areas close to the clnder s there wll be less space betweeodes and graduall as t goes farther awa from the clnder, the dstance betweeodes rses. Grd generaton method: Frst assume grd for the j =,.e. ABC lne. In ths area, the shape wll be dvded nto two parts: AB arc: We dvde ths arc nto N parts. Therefore, t gves: N ( ) r cos rsn () () (3) (4) BC lne: As t can be observed, the value on ths lne s equal wth zero and space betweeodes should be n a wa that as t goes farther awa from the clnder, the dstance betweeodes rses. Therefore, we should calculate the grd rato n ths stage. Node spaces on the AB arc wll be presented wth * and the frst node on the BC lne wll be consdered wth the same amount. Then graduall we ncrease the space betweeodes accordng to the rato of R n order to get the C pont. We wll show number of nodes on ths lne wth N. N R R BC (5) =+ =+ Fg. 3: Grd algorthm generated on j: R or grd raton for ever j =, = N+N! s known. Therefore, the onl unknow ths equaton s R. Ths equaton s a knd of frst-degree non-lnear equaton that can be solved b teraton method of Newton- Raphson, and R value can be calculated. Wth gven values of and t s possble to use and n values through relatons of comple potental functons for the flow around the clnder: A r sn A r A r cos A r (6) (7) Parameter A s a fed number. It s assumed as. Therefore, t wll be possble to calculate values of and on the j = lne. You can observe grd algorthm generated on j = lne n the Fg. 3. In the net step, we consder the AE lne. Space betweeodes and on the AB arc equals wth * that s a known value. Frst node on AE lne wll be assumed wth the same space and then space betweeodes wll be ncreases graduall wth R rato. We want to generate N3 nodes on ths lne. R rato n ths case can be calculated b followng relaton: N 3 R AE R (8) It s obvous that coordnates of the ponts located on AE lne segment equals wth zero. Therefore, and values can be calculated on ths lne too. So (, j) and (, j) for ever ( =, j = N3-) s known. 53
Res. J. Appl. Sc. Eng. Technol., 4(): 53-535, 0 To solve the sstem's equatons, we assume that (", $ ) s the desred answer of the sstem and ( 0, 0 ) s an appromaton of (, ). So, we can wrte: " = 0 +h 0 (7) Fg. 4: The fed R lne Now we assume the j = lne. Because and s known for (I =, j = ) pont, then t wll be possble to calculate R and n based on gven relaton. In Fg. 4 ou can see the fed lne: Now we move on the fed R lne or j =. It s obvous that: R(, ) = R(, ) (9) Now we move on the fed R lne R or move perpendcular to the drecton: n(, ) = n(, ) (0) Wth known values of and for ths pont, t wll be possble to calculate and values for these ponts through solvng two equatons and two unknown sstem (equatons). Method of solvng nonlnear two equatons and two unknown sstem s gve the net secton. We can contnue n the same wa: n(3, ) = n(3, ) () R(3, ) = R(, ) () Therefore, on j=constant lnes we wll have: n(, j) = n(, j-) (3) R(, j) = R(-, j) (4) Therefore, b the above-mentoned relaton, and n values can be easl used for varous ponts. Furthermore, and values can be calculated through solvng two equatons and two unknown sstem. Method of solvng two equatons and two unknown sstem: Our equaton sstem s as follows n whch wth known values of and, the value of and or nodes coordnates wll be calculated (A and B parameters are assumed as ): f (, ) 0 (5) $ = 0 +k 0 (8) we use Talor Epanson method to calculate h 0 and k 0 parameters n Eq. (3) and (4). In the performed epanson, the epressons above second order are gnored: ƒ(", $) = ƒ( 0 + h 0, 0 + k 0 ) (9) f ( 0, 0) f ( 0, 0) f ( 0, 0) k0 Snce (", $) s an appromaton for answers of the problem, then we wll have: 0 f ( 0, 0) f (, ) f ( 0, 0) k0 It wll be calculated n the same wa for g functon: g ( 0, 0) g ( 0, 0) k0 0 g ( 0, 0) (0) () B substtuton of ƒ and g values n Eq. (6) and (7), we wll have: 0 0 0 k 0 0 0 0 0 0 0 0 ( ) ( ) 0 0 k0 ( 0 0 ) ( 0 0 ) 0 0 0 0 () (3) The ntal assumpton of 0 = 0 = s consdered. Therefore, Eq. (8) and (9) can be smplfed as follows: h 3 0 k0 k0 (4) (5) The above mentoned sstem s a sstem of lnear equatons based on h 0 and k 0 that wll be solved b Kramer Order: g (, ) 0 (6) h 0 3 7 (6) 533
Res. J. Appl. Sc. Eng. Technol., 4(): 53-535, 0 k 0 3 5 (7) B gettng h 0 and k 0 values, then 0 + h 0 value wll be a better appromaton of 0 for " and also 0 + k 0 wll be a better appromaton of 0 for $. Therefore, = 0 + h 0 (8) 0 = 0 + k 0 (9) Fg. 5: The generated grd for the one-fourth of the shape related to the N: 50; N: 50; N3: 50 case Now we repeat the same operaton for and and calculate and appromatons. Generall, f n and n to be calculated, then b solvng the followng sstem: n n hn k 0 n n n ( ) ( ) n n n hn kn n ( n) ( ) n (30) (3) Fg. 6:The generated grd for the one-half of the shape related to the N:50; N:50; N3:50 case The h n and k n values wll be calculated and we wll have: n+ = n + h n (3) n+ = n +k n (33) We repeat the teraton operaton to the pont that: n+! n < 0.000 (34) Fg. 7: Magnfcaton of some parts of Fg. 6 n+! n < 0.000 (35) RESULTS AND DISCUSSION In ths stud b usng comple potental functon method we are lookng for generatng an orthogonal grd around a clnder n order to help us be able to analze the flow around t. The generated mesh s defned n a wa that n areas close to the clnder s there wll be less space betweeodes and graduall as t goes farther awa from the clnder, the dstance betweeodes ncreases. Accordng to smmetr of the phscal space, frst grd s produced for one fourth of t and then s generalzed to the whole problem accordng to the smmetr. In Fg. 5, ou can see the generated grd for the one fourth and one-half of the shape related to the N = 50, N = 50, N3 = 50 case. To represent orthogonal grds more clearl, ou can observe magnfcaton of some parts of Fg. 6 n the Fg. 7. As ou can observe n these fgures, as we go Fg. 8: The generated grd for the whole shape related to the N = 50, N = 50, N3 = 50 case farther awa from the clnder surface, the dstance betweeodes ncreases because our calculaton's sze decreases. On the other hand, the generated mesh s orthogonal that can be used effcentl to solve the flow feld n the eternal flows. In the Fg. 8, ou can see generated grd for the whole shape related to the N = 50, N = 50, N3 = 50 case. 534
Res. J. Appl. Sc. Eng. Technol., 4(): 53-535, 0 CONCLUSION In ths stud, an orthogonal mesh s generated around a clnder b usng orthogonal propert of R and n functons. The generated grd besdes orthogonal propert has small dstances on the surface of clnder and graduall as t goes farther awa from the clnder, the dstance betweeodes ncreases. Ths knd of grd can be useful n solvng the flow feld around a clnder. Ths knd of mesh besdes ts orthogonal propert s generated n a wa that there are man lttle dstances on the clnder surface, and as we go farther awa from the clnder surface, the dstances betweecreases, because the closer dstances est betweeodes, the more accurate answers wll generate for solvng mesh equatons. Snce responses on the surface of the clnder and n the closer ponts are of more mportance, then fewer dstances are consdered betweeodes. Most mportant advantage of ths method s orthogonal mesh generaton wth a low amount of calculatons n comparson wth other methods of grd generaton lke partal dervatves method. However, dsadvantage of ths method s that n specal spaces we consder R and n functons based on r and or and, or n other words t s possble to calculate the Functon Comple Potental. RECOMMENDATIONS In ths stud onl the grd generated around a clnder s descrbed b usng potental functons method. It s suggested that ths ssue to be consdered n other spaces wth known functons lke wedge and obtaned results to be analzed. On the other hand, t s suggested to solve a flow equaton around a clnder wth generated mesh for nvestgaton of valdt of the method. Net, obtaned results wll be compared wth results of other methods. REFERENCES Churchll, R.V., 948. Introducton to Comple Varables. McGraw-Hll, New York. Crowle, W.P., 96. Internal Memorandum, Lawrence Radaton Laborator, Lvermore, Calforna. Davs, R.T., 979. Numercal Methods for Coordnate Generaton Based on Schwarz-Chrstoffel Transformatons. AIAA Paper 79-463, Wllamsburg, Vrgna. Ives, D.C., 98. Conformal Grd Generaton, Numercal Grd Generaton, Proceedngs of a Smposum on the Numercal Generaton of Curvlnear Coordnate Sstems and Ther Use n the Numercal Soluton of Partal Dfferental Equatons. Thompson, J.F., (Ed.), Elsever, New York, pp: 07-30. Morett, G., 979. Conformal Mappngs for the Computaton of Stead Three-Dmensonal Supersonc Flows, Numercal j Laborotor Computer Methods n Flud Mechancs. Pourng, A.A. and V.I. Shah, (Eds.), ASME, New York, pp: 3-8. Wnslow, A., 966. Numercal Soluton of the quaslnear posson equaton. J. Comput. Phs., : 49-7. 535