Different Kinds of Boundary Elements for Solving the Problem of the Compressible Fluid Flow around Bodies-a Comparison Study

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1 Proceedgs of the Word Cogress o Egeerg 8 Vo II WCE 8, Ju - 4, 8, Lodo, U.K. Dfferet Kds of Boudar Eemets for Sovg the Probem of the Compressbe Fud Fow aroud Bodes-a Comparso Stud Lumta Grecu, Gabrea Dema ad Mha Dema Abstract The paper presets a comparso stud betwee the umerca soutos obtaed whe usg dfferet kds of boudar eemets for sovg the probem of the bdmesoa compressbe fud fow aroud obstaces, b appg the boudar eemet method. The drect boudar eemet method wth sources dstrbuto apped to ths probem offers a sguar boudar tegra equato whch s soved usg costat, ear ad quadratc boudar eemets. For some partcuar cases eact soutos est for ths probem. Some computer codes are made for each of the cosdered boudar eemets ad umerca soutos are obtaed for the case of a crcuar obstace ad a eptca oe. The umerca soutos obtaed these cases are compared wth the eact oes ad the errors are aazed. Ver good resuts are obtaed, eve for sma umbers of boudar eemets, whe quadratc boudar eemets are used. Ide Terms boudar eemet method, compressbe fud fow, ear boudar eemet, quadratc boudar eemet. I. ITRODUCTIO The boudar tegra method (BEM) s a moder umerca techque used to sove boudar vaue probems for sstems of parta dffereta equatos. There est two pra varats of appg ths method: the drect method ad the drect oe. Both of them offer the pra advatage of the BEM over the other umerca method - the abt to reduce the probem dmeso b oe. Ths propert s advatageous as t reduces the sze of the sstem the probem s equvaet wth, ad so mproves computatoa effcec. To acheve ths reducto of dmeso t s ecessar to formuate the goverg equato as a boudar tegra equato, whch s usua a sguar oe (see [], []), ad for ths, both techques the drect techque ad the drect oe ca be used. Ths paper s focused o sovg the sguar boudar tegra equatos obtaed whe the frst varat s apped for the bdmesoa probem of a vscd, compressve subsoc fud fow aroud bodes, cosderg the case of a o-ftg obstace, b usg dfferet tpes of boudar eemets. A comparso stud betwee the umerca soutos obtaed these cases for the same umber of chose o the boudar s aso made. The probem of a uform, stead, poteta moto of a dea vscd fud of subsoc veoct U, pressure p ad dest ρ that s perturbed b the presece of a fed bod of a kow boudar, oted C, assumed to be smooth ad cosed s descrbed, usg dmesoess varabes, b the foowg mathematca mode: u v + = v u =, () wth the boudar codto: ( β + u) + β v = o C, ad m v =, () where u ad v are the compoets aog the aes of v, the dmesoess perturbato veoct, s the orma ut vector outward the fud, β has the usua sgfcato, β = M ad M the Mach umber for the uperturbed moto. We wat to fd out the perturbed moto, ad the fud acto o the bod. II. THE BOUDARY ITEGRAL EQUATIO Appg the drect method wth sources dstrbuto the sguar boudar tegra equato the probem s reduced at s obtaed (see [3]): Assmatg the boudar wth a dstrbuto of sources of ukow test, f, frst there are deduced the compoets of the perturbato veoct the fud doma ad the wth a mt process ther epressos o the boudar. Usg the boudar codto a sguar boudar tegra equato s obtaed. For gettg ths boudaru tegra equato the defto of the Cauch pra vaue of a tegra s used ad the ukow fucto f s assumed to satsf a höder codto-esseta for the estece of the boudar equato. The boudar tegra equato has the foowg form: ( ) ( ) ( ) ( ) + β + β f + f ds = C = β () 3 where, are the compoets of the orma ut vector outward the fud evauated at a pot stuated o C. The boudar tegra equato s a sguar oe. The sg " ' " deotes the Cauch pra vaue of the tegra. I order to sove the sguar boudar tegra equato we use three tpes of boudar eemets: costat boudar eemets, ear soparametrc boudar eemets ad quadratc oes ISB: WCE 8

2 Proceedgs of the Word Cogress o Egeerg 8 Vo II WCE 8, Ju - 4, 8, Lodo, U.K. A. Case of costat boudar eemets We appromate the boudar b a pogoa e{ L }, =, wth the o the rea boudar ad we cosder that the ukow s costat o each segmet. We cosder that o each L the ukow s equa wth the vaue take the mdpot of the segmet, oted =, {,,..., }, =. (4) I (3) we cosder tha = ad we deduce the dscrete form of the sguar boudar tegra equato: ( ) + ( ) + β β f ( ) + f ds = = L = β (5) Imposg reato (5) to be satsfed o ever mdpot, we get (see [4]) the foowg ear agebrac sstem whch ukows are the vaues of the sources test for the mdde pots of the segmets: a f + A f = A, =,, (6) = where a ( ) ( ) = + β A = ( ) U + β ( ) V A = ( ) β (7) The coeffcets deped o o the coordates of the chose for the boudar dscretzato. A the coeffcets (6) ca be aatca evauated ad o errors appear due to ther evauato. After sovg the sstem (6) the compoets of the veoct are foud ad the the oca pressure coeffcet. B. Case of ear boudar eemets I order to sove the sguar boudar equato we chose ow the case of ear soparametrc boudar eemets. We appromate the cotour C wth a pogoa e havg the segmets L, =, ad the etremes: (, ) ş (, ) a oca umberg sstem. We have reatos: (, ) = ( +, + ), ad (, ) = (, ), cotour C beg cosed. For descrbeg the geometr of a boudar eemet we use a oca sstem of coordates whch has the org the frst ode of a eemet, ad so we have the reatos: = ϕ + ϕ, t [,], (8) = ϕ + ϕ where ϕ,ϕ are the form fuctos gve b () t = t, ϕ () t = t ϕ. (9) Usg soparametrc boudar eemets we have, for the ukow f, the oca represetato: where f + = f ϕ f ϕ, () f f are the oda vaues of the ukow, t meas, the vaues of f at the etremes of the boudar eemet L, the oca umberg. These vaues satsf the reatos: f = f +,, ad f = f. For =,, (3), we get a agebrac = sstem of equatos each of them of the foowg form: + β = L ( ) ( ) ( ) + β f ϕ ϕ ds = β f + () For smpfg the wrtg we sha ot use the prm sg to specf that a tegra must be uderstad ts Cauch sese. Wth the otatos: ( ) + β ( ) a = ϕ ds b L ( ) + ( ) β = ϕ L we get the foowg equvaet form for ( ): = f a + f = b = β ds (), (3) where a = a for, ad a = ( + β ) + a,, =,. (4) After some cacuous we obta for the coeffcets of sstem (3) the foowg epressos: [ + t( )] a = ( t) dt + at + bt β [ + t( )] + ( t) dt = at + bt = [( ) I + ( + ) I ( ) I ] + β β ( ) I + ( + ) I I [ ] ( ) [ + t( )] b = t dt + at + bt β [ + t( )] + t dt = at + bt β ( ) I + ( ) I + [ ] ( ) I + = ISB: WCE 8

3 Proceedgs of the Word Cogress o Egeerg 8 Vo II WCE 8, Ju - 4, 8, Lodo, U.K. ( ) I β + where : k t I k = dt, k =,,. (5) at + bt For the compoets of the orma ut vector outward the fud we have the foowg formuas: =, =, =,. (6) The osguar tegras ca be computed aatca ad for the sguar oes the defto of the Cauch pra vaue ca be used (see [5]). I paper [6] ther epressos are gvve. Usg the same otatos as before, the compoets of the veoct o the boudar (for the ode, =, ) ca be evauated wth the formuas: u = f f = f = f f = = v = f f [( ) I + ( + ) I ( ) I ] [( ) I + ( ) I ] f [( ) I + ( + ) I ( ) I ] [( ) I + ( ) I ] ( 7) f So the oca pressure coeffcet ca be obtaed. C. Case of quadratc boudar eemets f I ths paragraph we use quadratc soparametrc boudar eemets of Lagragea tpe to sove to sove the sguar boudar tegra equato (3), so the ukow fucto s appromated b poomas of secod degree, ad the boudar b curved arcs. For obtag the dscret equato the boudar s dvded to udmesoa quadratc boudar eemets, each of them wth three : two etreme ad a teror oe. For gettg ths mesh we eed o the boudar. Cosderg that the dscrete equato s satsfed ever ode, we have for =, : + β + f = L f ( ) + + β ( ) ( ) ( ) ds = β (8) The quadratc soparametrc boudar eemet uses the same set of basc fuctos, oted,, 3, for descrbg the geometr ad the ukow fucto. Usg the trsc sstem of coordates, wth the org the teror ode, these fuctos have the epressos: ( ξ ) ( ξ + ) ξ ξ ( ξ ) =, ( ξ ) = ξ, 3( ξ ) =, ξ [, ] (9) Usg a matrca otato we obta the foowg equato: 3 + β f ( ) + a f = β, () = = where ([ ]{ } ) + β ([ ]{ } ) a = J ξ d [ ] ( ) =, { }, { }, 3 [ ]{} ( ) ξ are coum matrces made wth the goba coordates of the eemet L, ad f =,, =,,3 are the oda vaues of the ukow fucto for the three of the metoed eemet (the vaue of the ukow for the ode umber of the eemet umber ). Returg to the goba sstem of otato we obta the foowg ear agebrac sstem: [ A]{ f } = { B}, A M ( R), { f } R,{ B} R f = f ( ) B =, =,. () β For gettg the matr [ A ] we eed to evauate the tegras that appeare. Oe of them are usua tegras, but the other are sguar tegras. For the sguar tegras that appear there ca be used more techques, some of them beg preseted [7]. Oe of these methods are: the trucato method, the Cauch pra vaue method ad the reguarzato method. We have used ths paper the reguarzato method because the stud made [8] shows that ths method eads case of quadratc boudar eemets to the best resuts. After sovg the sstem (), so after we fd the vaues of f for the choose for the dscretzato of the boudar we ma aso compute the veoct for these. We deduce (see [9]): 3 u( ) = f ( f b b 3 b ), = 3 v( ) = f ( f c c 3 c ), () = The coeffcets from the above epressos deped o o the coordates chose for the boudar dscretzato ad the ca be foud [9]. III. UMERICAL RESULTS I some partcuar cases the cosdered probem has eact souto. I [] there s preseted the eact souto for the probem of the uform dea compressbe subsoc fud fow aroud a crcuar obstace. Some computer codes made MATHCAD, aow us to compare the umerca soutos obtaed whe usg ISB: WCE 8

4 Proceedgs of the Word Cogress o Egeerg 8 Vo II WCE 8, Ju - 4, 8, Lodo, U.K. costat, ear ad quadratc boudar eemets wth the aatca oe, ad to evauate the errors that appear each stuato. The comparso s made through the oca pressure coeffcet, oted c p, obtaed whe there are used ad for the boudar dscretzato. The foowg graphcs show good agreemets ad the demostrate the fact that usg quadratc boudar eemets ad a adequate method for evauatg the sguartes we get ver good resuts eve for a sma umber of boudar eemets. We cosder frst the case of o the boudar ad we perform the oca pressure coeffcet, c p evauated at these, o the oe had whe dfferet kds of boudar eemets are used ad o the other had for the eact souto.,5,5 -,5 - -,5 - -, costat eact Fg.. c p for the eact souto ad the umerca oe obtaed for costat boudar eemets.,5,5 -, ,5 - -,5-4 ear eact Fg.. c p for the eact souto ad the umerca oe obtaed for ear boudar eemets.,5,5 -, ,5 - -,5-4 costat eact ear Fg.3. c p for the eact souto, the umerca oes obtaed for costat ad ear boudar eemets. Evauatg the errors that appears we get the foowg graphc. errors,7,6,5,4,3,, errors Fg.4. The absoute error betwee the eact souto ad the umerca oe obtaed: for the case of costat boudar eemets (Error), ad ear boudar eemets (Error). As we otce the error s smaer for the case of ear boudar eemets case of 6 of the. Whe usg quadratc boudar eemets we get ver good resuts as we ca see from the et graphc.,5,5 -,5 - -,5 - -, error error quadratc eact Fg.5. c p for the eact souto ad the umerca oe obtaed for quadratc boudar eemets. We ca see that the vaues obtaed for the oca pressure coeffcet case of usg quadratc boudar eemets are amost equa wth the eact vaues. That s wh we ca see o oe e o the graphc. The absoute error that appears s performed the foowg graphc.,4,35,3,5,,5,,5 errors error3 Fg.6. The absoute error betwee the eact souto ad the oe obtaed for quadratc boudar eemets. The errors are so sma ot o for the reaso of usg quadratc boudar eemets but aso because the sguar ISB: WCE 8

5 Proceedgs of the Word Cogress o Egeerg 8 Vo II WCE 8, Ju - 4, 8, Lodo, U.K. tegras that appear have bee treated wth a speca atteto. Usg a good method for evauatg the sguar tegras that appear s a stage of great practca mportace because the coeffcets gve b these sguar tegras are domats ad stuated ear ad o the dagoa of the sstem matr, ad so the pa a mportat roe for a we behavor of the sstem. A the oda vaues obtaed for o the boudar are performed the et fgure.,5,5 -,5 - -,5 - -, costat eact ear quadratc Fg.7. c p for the eact souto, the umerca oe for costat, ear ad quadratc boudar eemets. Whe there are used for the boudar dscretzato the umerca resuts are performed the foowg graphc ad the comparso s aso made through the oca pressure coeffcet.,5,5 -,5 - -,5 - -, ear costat eact quadratc Fg.8. c p for the eact souto, the umerca oe for costat, ear ad quadratc boudar eemets, case of. As we ca otce the umerca resuts are ot as good as before especa whe costat ad ear boudar eemets are used. The best resuts, for o the boudar, are obtaed as before case of quadratc boudar eemets. The foowg graph shows the errors that appear ths case. error,4,35,3,5,,5,,5 Errors Fg.9. The errors case of quadratc boudar eemets ad. Comparg the errors from Fg.6 ad Fg.9 we deduce the fact that the umerca souto obtaed whe are used for the boudar dscretzato s we mproved. We ca ru the computer codes for dfferet umbers of to see whch s the optma umber of each case, a umber bg eough to ead to a sma eough error ad aso ot to bg for a ustfed computatoa effort. We ca deduce whch s the best umber of that must be chose for the boudar dscretzato for obtag the best rato computatoa effcec good resuts. The computer codes ca be used for obstaces wth dfferet geometres. I paper [] the eact souto of the metoed probem for the case of a eptca obstace ca be foud. I the foowg graphc there s made, for a eptca obstace, a comparso betwee the eact souto ad the umerca oes obtaed for the same cases of boudar eemets. There are used for the boudar dscretzato.,5,5 -,5 - -, Error ear eact quadratc costat Fg.. c p for the eact souto, the umerca oe for costat, ear ad quadratc boudar eemet for a eptca obstace. As we see from the above graphc the umerca souto obtaed whe usg quadratc boudar eemets s the best ad s earb the eact oe eve whe we choose o the boudar. The dstrbuto of errors that appear for the chose tpes of boudar eemets case of a eptca obstace s performed the foowg graph. ISB: WCE 8

6 Proceedgs of the Word Cogress o Egeerg 8 Vo II WCE 8, Ju - 4, 8, Lodo, U.K. eroors,6,5,4,3,, Errors Fg.. The errors betwee the eact souto ad the umerca oe obtaed: for the case of costat boudar eemets (Error), ear boudar eemets (Error), ad quadratc boudar eemets (Error3). For better seeg the errors case of quadratc boudar eemets we have the foowg graphc. Error Error Error3 [3] L. DRAGOŞ, Mathematca Methods aerodamcs, Ed. Academe Româe, Bucureşt. [4] L. GRECU, Ph.D. these: Boudar eemet method apped fud mechacs, Uverst of Bucharest, Facut of Mathematcs, 4. [5] I. K. LIFAOV, Sguar tegra equatos ad dscrete vortces, VSP, Utrecht, Theetherads, 996. [6] L. GRECU, A Souto of the Boudar Itegra Equato of the Theor of the Ifte Spa Arfo Subsoc Fow wth Lear Boudar Eemets", Aas of Bucharest Uverst, Mathematcs, Year LII, r. (3), pp [7] H. M. ATIA, umerca Methods for Scetsts ad Egeer", Brkhause, [8] L. GRECU, Aspects about the evauato of the sguartes whe appg the boudar eemet method to sove probems of fud fow aroud bodes., Buet of the Trasvaa Uverst of Brasov, seres B, Tom 3(48), 6 pag [9] L. GRECU, A Souto of the Boudar Itegra Equato of the D Fud Fow aroud Bodes usg Quadratc Isoparametrc Boudar Eemets, ROMAI Joura, vo, r., 6, pag [] L. DRAGOŞ, Fud Mechacs I (Mecaca fudeor I), Bucureşt, Edtura Academe Româe, 999. Errors,45,4,35,3 eroors,5, Error3,5,, Fg. The errors betwee the eact souto ad the umerca oe obtaed for the case of quadratc boudar eemets (Error3). As t s atura better resuts ca be obtaed b usg hgher order boudar eemets or more for the boudar dscretzato, but as we see the resuts are satsfactor whe choosg quadratc boudar eemets ad o o the boudar. From the above graphcs we ca observe that the aazed obstaces are o-ftg oes because of the smmetr of the oca pressure coeffcet: for correspodg o the upper ad the ower boudar t takes the same vaue. As we kow ths s a cosequece of the fact that the aazed profes have smooth boudares. Wth the same computer codes umerca soutos ca be obtaed for a kd of compressbe fud fows, for dfferet vaues of Mach umber, ot o for the dea case ad for other kds of obstaces wth smooth boudares too. For profes wth cusped trag edge usg a Kutta-Jukovsk codto ad makg adequate chages to the computer codes, umerca soutos of the metoed probem ca be foud too. REFERECES [] C. A. BREBBIA, J. C. F. TELLES, L. C. WOBEL, Boudar Eemet Theor ad Appcato Egeerg, Sprger-Verag, Ber,984. [] M. BOE, Boudar tegra equato methods for sods ad fuds, Joh We ad Sos, 995. ISB: WCE 8