Solving the Problem of the Compressible Fluid Flow around Obstacles by an Indirect Approach with Vortex Distribution and Linear Boundary Elements

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1 Lumta Grecu Sov the Probem of the ompressbe Fud Fow aroud Obstaces b a Idrect Approach wth Vorte Dstrbuto ad Lear Boudar Eemets LUMINITA GREU Departmet of Apped Sceces, Navato ad Evromet Protecto Facut of Eeer ad Maaemet of Techooca Sstems, Dr Tr Sever, Uverst of raova st auare street, Dr Tr Sever, 37 ROMANIA umrecu@hotma.com Abstract: - I the preset paper there s preseted a souto wth ear boudar eemets of araea tpe for the suar boudar tera equato obtaed b a drect techque wth vorte dstrbuto for the bdmesoa compressbe fud fow aroud bodes. The suar boudar tera equato the probem s reduced at s formuated terms of prmar varabes-the compoets of the veoct o the boudar. Numerca soutos for the compoets of the veoct ad the oca pressure coeffcet are obtaed, for dfferet tpes of obstaces, wth some computer codes made MATHAD, based o the method eposed. For some partcuar cases, whe aatca soutos est a comparso stud betwee the umerca soutos ad the eact oes s aso doe. It ca be see, from the raphcs obtaed, that the umerca soutos are ood areemet wth the eact soutos of the probem. The paper s aso focused o a comparso stud betwee the umerca soutos obtaed whe the drect method wth sources dstrbuto s used ad the umerca souto preseted ths paper whe boudar eemets of same tpe are used for sov both suar boudar tera equatos. Ke-Words: - ompressbe fud fow, boudar eemet method, vorte dstrbuto, ear boudar eemets. INTRODUTION For sov boudar vaues probems for sstems of parta dffereta equatos dfferet umerca methods ca be used. Most of them are abe to fd the soutos b us the dffereta equatos as the are ve, wthout a further mathematca mapuato. The appromate the dffereta operators the equatos b smper oes vad at a seres of odes wth the reo, ke the fte dfferece method, or the represet the reo tsef b fte eemets whch are assembed to provde a appromato of the sstem voved, ke the fte eemet methods. The Boudar Eemet Method (BEM), aso kow as the Boudar Itera Method, s a moder umerca techque whch ca be cuded, toether wth the Fte Eemet Method, the are cass of Gaerk methods. These are a cass of methods for covert a cotuous operator probem to a dscrete probem. I pre, ths s doe b covert the equato to a weak formuato. There est two pra techques of app BEM method: the drect BEM method; the drect BEM method. Both of these methods offer the pra advatae of the BEM over other umerca methods - the abt to reduce the probem dmeso b oe. Ths propert s advataeous as t reduces the sze of the sstem the probem s equvaet wth, ad so mproves computatoa effcec. Whe sov a probem wth ths method two mportat steps have to be made: frst, we must obta a equvaet boudar formuato for the probem voved, fact a boudar tera equato or a sstem of boudar tera equatos, ad the, ths boudar tera equato whch ISSN: Issue 7, Voume 7, Ju 8

2 Lumta Grecu usua s a suar oe must be soved. For sov the boudar tera equato ma tpes of boudar eemets ca be used: costat, ear quadratc or hher order boudar eemets. The Boudar Eemet Method reduces the probem to a sstem of ear equatos (see [], [], [3]), ad further the probem ca be soved wth a computer. The am of the paper s to sove the probem of the compressbe fud fow aroud a obstace us a boudar eemet approach based o the drect method wth a vorte dstrbuto, ad to sove the suar boudar equato that resuts wth ear soparametrc boudar eemets of Laraea tpe.. Advataes brouht b app BEM wth vorte dstrbuto The probem has bee studed b ma authors, wth dfferet kds of techques. There have bee made dfferet assumptos for smpf the mathematca mode of the probem. Some ear techques dea wth the case of the compressbe fud fow ad use ear equatos, ear boudar codtos ad sometmes the boudar codto was satsfed ot o the boudar but o the chord of the profe. B app the BEM to sove ths probem o the frst assumpto s st use. So the BEM uses the oear boudar codto whch s satsfed o the obstace s boudar, ot o ts chord. The BEM was frst apped o for the compressbe case ad the boudar tera formuato was obtaed terms of poteta fucto or stream fucto. The measures of terest for the probem, ke the veoct for eampe, were obtaed after evauat the dervatves of the ukows of the probem, ad so, ew errors were troduced at ths stae. The BEM wth vorte dstrbuto, preseted ths paper, besdes the advataes brouht b the BEM, offers aso the advatae that deas wth the compressbe case ad eads to a boudar formuato of the probem terms of prmar varabes-the compoets of the veoct fed emat so the errors that coud appear b evauat the dervatves, ad br so more accurac to the umerca souto. We frst preset the probem to sove: a uform, stead, poteta moto of a dea vscd fud of subsoc veoctu, pressure p ad dest ρ s perturbed b the presece of a fed bod of a kow boudar, oted, assumed to be smooth ad cosed. We wat to fd out the perturbed moto, ad the fud acto over the bod. Deot b v the perturbato veoct (u, v ts compoets ao the aes) ad us dmesoess varabes we have the foow mathematca mode: u v + v u = = wth the boudar codto: ( β u) + β v = +, () o, () where s the orma ut vector outward the fud, β has the usua sfcato, β = M, ad M the Mach umber for the uperturbed moto. It s aso requred that the perturbato veoct vashes at ft: m v =. 3. The boudar tera equato - vorte dstrbuto The fudameta souto of vorte tpe s the souto of the foow sstem (see [6] ): * * u v + = * * v u = δ ( ξ, η ) Its ame comes from the fact that the perturbato produced b the presece of δ appears the secod equato of (), equato whch epresses the fact that the perturbed moto s rrotatoa, ad t has the foow epresso (see[6]): ISSN: Issue 7, Voume 7, Ju 8

3 Lumta Grecu u v * * = = η ( ξ ) + ( η) ξ ( ξ ) + ( η) Appromat the boudar wth a cotuous dstrbuto of such fudameta soutos, hav the ukow test ( ), the compoets of the perturbato veoct for a pot stuated the fud doma are frst foud. The are ve b the formuas: u v ( ξ ) = ( ) ( ξ ) = ( ) η ds ξ ξ ξ For obta the compoets of the perturbato veoct we must take the mt of the above teras for ξ, a reuar pot o the boudar. As t ca be observed, the above teras are suar for such pots. It s ecessar to use the cocept of the auch Pra Vaue of a tera for dea wth the suar teras see for eampe []. Ths cocept s defed ma books ad ts defto s ver smpe ad atura. F.. For evauate the mt of a tera that has a suart we have to soate ths pot wth a crce of a ver sma radus, oted ε, that tersect ds the cosdered boudar ao the arc, oted c. So we have: = +. c If, forε, the tera c teds to a fte c mt, the the mt s caed the PV of the tera. Not wth the prm s the PV of a tera, we have the reato: ' = m ε c. Assum that s a H o.. der fucto o, [6] s obtaed a tera formuato for the probem. the compoets of the perturbato veoct for a reuar pot o the boudar are foud. The are ve b the foow epressos: u v ( ) = ( ) + ( ) ( ) = ( ) ( ) where, ' ' ds (3) ds are the compoets of the orma ut vector outward the fud evauated at. Us the boudar codto the suar boudar equato s deduced ad has the foow form: M = β ( ) + ( ) ' β ( ) ( ) ds = wth the same otatos as before. The oa of ths paper s to sove the suar boudar tera (4) us boudar eemets that offer a oba cotut for the ukow of the probem, so for the ukow test. For sov tera equatos method of successve appromato, orthooa poomas, or Krov subspaces ca be used for eampe. I case of sov suar boudar tera equatos or more eera, suar boudar terodffereta equatos, appromate soutos ca be ( 4) ISSN: Issue 7, Voume 7, Ju 8

4 Lumta Grecu obtaed b us the coocato method as [4]ad [5]. For the suar boudar tera equato (4) [8] a coocato method s used ad ood umerca resuts are obtaed. 4. Lear boudar eemets for sov the suar boudar era equato I ths paper, order to sove the suar boudar tera equato (4) we use ear soparametrc boudar eemets of Laraea tpe (see [], [], [3]). We choose N odes o the boudar, so o, ad we appromate the boudar wth a pooa e hav the semets, =,N ad the etremes: (, ) ad (, ) a oca umber sstem. We have reatos: (, ) = ( +, + ), N ad (, N ) N = (, ), cotour be cosed. L = ϕ + ϕ = ϕ + ϕ, t [, ], (5) where ϕ,ϕ are the shape fuctos ve b: ( t ) = t, ϕ ( t = t ϕ ). (6) Us soparametrc boudar eemets we have, for the ukow, the oca represetato: where = ϕ + ϕ, (7), are the oda vaues of the ukow, t meas the vaues of at the etremes of the boudar eemet, the oca umber. L These vaues satsf the reatos: = +, N, ad N =. For smpf the wrt we sha ot use the prm s to specf that a tera must be uderstad ts auch sese. For =, =, N equato (4) we obta a aebrac sstem of N equatos each of them of the foow form: M ` N + = L ( ϕ + ϕ ) + β ( ) ( ) (8) ds = β F.. A soparametrc boudar eemet uses the same shape fuctos for oca descrbe theukow ad the eometr of the eemet. For descrbe the eometr of a boudar eemet we use a oca sstem of coordates whch has the or the frst ode of a eemet, ad so we have the reatos: 5. oeffcets evauato a b Wth the otatos: β = ϕ L β = ϕ L ( ) ( ) ( ) ( ) ds, we et the foow equvaet form for (8): ds (9) ISSN: Issue 7, Voume 7, Ju 8

5 Lumta Grecu N = N a + b = = β, () c =. where: a = a for ( M ) a = a +, ad,, =, N. () After do some cacuous we et the foow reatos for the above coeffcets: a [ + t( )] β = ( t) dt at + bt ( t) [ + t( )] dt = at + bt [( ) I + ( + ) I ( ) I ] β = β b [( ) I + ( + ) I ] ( ) I β = t t β = Wth teras: I [ + t( )] at + bt [ + t( )] at + bt [( ) I + ( ) I ] [( ) ( I + ) I ] I k, k =,, k t = dt, k =,, at + bt dt = dt () we have oted the foow k, where a =, ( )( ) + ( )( ) b =, For the compoets of the the orma ut vector we use the reatos: =, =, = N. (3), A computer code ca be use to evaute these teras but, for mtt the errors that appare because of the umerca approach, the osuar teras are computed aatca ad for the suar oes the defto of the auch Pra Vaue s used. a) The osuar case For whe =, N ad N whe =, we et the foow epressos: I I I = ac b arct a + b = a c a ac b c + b b ac b, arct ac b c + b b a + b b ac = + I. (4) a a c a b) The suar case For = ( =, N ) ad for = N whe =, so for the suar teras that appear we et: I = I =, ad I =. (5) Achev ths stae we ca observe a mportat aspect: a the coeffcets () ca be aatca evauated ad the deped o o the coordates of the odes chose for the boudar dscretzato. Retur to the oba sstem of otato, so cosdr that: = + = + for =, N, N = =, ad ot:, ISSN: Issue 7, Voume 7, Ju 8

6 Lumta Grecu A A + = a + b for = N,, = a bn ad T β, = =, N (6) we deduce the foow equvaet epresso for sstem (): N A = T, =, N. (7) = For evauat the suar teras that appeare whe sov secod order eptc equato of Posso tpe, for the three dmesoa case, wth Boudar Eemet Method, a appromate techque based o the auto sod ae evauato ca be used as []. 6. Evauat the oda vaues of the veoct s compoets ad of the oca pressure coeffcet o the boudar After sov ths sstem ad fd the oda vaues for the ukow fucto, fact the oda vaues of the vorte testes, oted, =, N, the compoets of the veoct o the boudar (for the ode, =, N ) ca be evauated start from formuas (3). Wth the same otatos as before we et the foow epressos: u = N + = N + v = = N N = = + [( ) I + ( + ) I ( ) I ] [( ) I + ( ) I ], ( 8) + [( ) I + ( + ) I ( ) I ] [( ) I + ( ) I ]. ( 9) We ca compute the fud veoct for dfferet + po.ts of the fud doma too us the dscretzated epressos of (3). Reard the fud acto over the bod, we ca evauate the oca pressure coeffcet, oted c p, us the reato: c p = u v u. () Ths coeffcet s oe of reat mportace for the probem because t s used to obta the ft force. It s kow that for profes wth smooth boudar the ft force does t appear because of the same vaues of the oca pressure coeffcet o the tra ad etrados of the profe. It s mportat to specf that a the coeffcets sstem (7) have aatca epressos ad therefore o errors are troduced for ther evauatos. A these coeffcets deped o o the coordates of the odes used for the boudar dscretzato, ad so, t ca be use a computer code to sove the probem. 7. Numerca resuts ad cocusos For sov sstem (7) ad for evauat the fud veoct ad the oca pressure coeffcets there s deveoped a computer code MATHAD that uses reatos (8), (9), (). These umerca soutos are compared wth the eact soutos that est for the partcuar case of a crcuar obstace ad a compressbe fud (M=). I [7] the bdmesoa probem of the compressbe fud fow aroud a crcuar obstace + s eact soved. The epressos of the compoets of the perturbed fud veoct are obtaed ad the are ve b the foow reatos: u = U cos θ, v = U s θ For the dmesoess compoets we et: u = cosθ, v = s θ, () ad further, for the oca pressure coeffcet, the foow epresso: = os θ. () Aother computer code ves us the souto for ths case. Both prorams ca be ru for dfferet umber of odes used for the boudar dscretzato. ISSN: Issue 7, Voume 7, Ju 8

7 Lumta Grecu For the case whe we use odes for the dscretzato the soutos obtaed are represeted the foow raphcs. I F. 3. there are represeted the vaues obtaed for the veoct compoet ao the O as. I F. 4. there are represeted the vaues obtaed for the veoct compoet ao the O as. The pressure coeffcet s represeted F. 5. The umerca souto s ood areemet wth the eact oe. We ca verf wth ths raphc a we kow resut too: the crcuar obstace s a o-ft profe because of the oca pressure coeffcet smmetr. For correspod odes 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 profe has a smooth boudar. v,5,5 -, v v eact v ear,5,5 -,5 - -,5 - -,5, odes F. 5. The oca pressure coeffcet for the case of vorte dstrbuto wth ear boudar eemets, ad the eact souto. As we ca see from the raphcs the umerca soutos are ood areemet wth the eact souto, ad a sma umber of eemets () s suffcet for obta satsfactor resuts. As t s atura the umerca souto s fueced b the umber of odes chose for the boudar dscretzato. We ca observe ths from the foow raphcs where the oda vaues of the oca pressure coeffcet are performed for dfferet umber of odes o the boudar. There were cosdered, 5, 5, ad 3 odes for the boudar dscretzato. eact ear -,5 odes F. 3. The veoct ao the O as: case of vorte dstrbuto wth ear boudar eemets ad the eact souto ,5 v - - vorte eact,5-4 v -, v eact v ear odes F. 6. The oca pressure coeffcet for the case of odes: umerca souto ad eact souto. - -,5 odes F. 4. The veoct ao the O as: case of vorte dstrbuto wth ear boudar eemets ad the eact souto. ISSN: Issue 7, Voume 7, Ju 8

8 Lumta Grecu,5,5 the errors that appear each of the above cases. We otce that these vaues decrease wth the rowth of the odes umber. -, errors - -,5 - -,5,5 odes vorte eact ma error,6,4,,8,6 errors F. 7. The oca pressure coeffcet for the case of 5 odes: umerca souto ad eact souto.,4, umber of odes,5,5 -,5 - -,5 - -,5, odes vorte eact F. 8. The oca pressure coeffcet for the case of 5 odes: umerca souto ad eact souto.,5,5 -,5 - -,5 - -,5, odes vorte eact F. 7. The oca pressure coeffcet for the case of 3 odes: umerca souto ad eact souto. As epected, better resuts are obtaed whe us more odes o the boudar, but the resuts are ver ood whe us, 5 ad 3 odes. For better observ the umerca souto mprovemet brouht b the rowth umber of odes we cosder F. 8 the mamum vaues for F. 8. The mamum errors for, 5,, 5 ad 3 odes o the boudar. We ca aso see from the raphc that a umber of odes ber tha for the boudar dscretzato does ot ead to a substata mprovemet so much that to ustf the computatoa effort. It appears reasoabe to epect better resuts b us hher order boudar eemets for sov the suar boudar tera equato because the aow a better appromato of the eometr, I paper [9] the boudar tera equato, obtaed as a equvaet form for the voved probem, b app the drect method wth sources dstrbuto of ukow testes, s soved wth ear soparametrc boudar eemets of Laraea tpe. I the same paper the umerca souto s compared to the eact oe for the same partcuar case: the crcuar obstace ad the compressbe fud, ad there were obtaed ood resuts. For the boudar dscretzato there were aso used odes. I the foow pararaphs the umerca souto obtaed ths paper s compared to the eact oe, ad to the oe obtaed case of sources dstrbuto, for the metoed partcuar case-the crcuar obstace. The comparso stud s made throuh the oca pressure coeffcet,. I the foow raphcs we perform the eact oda vaues of ad the umerca oes obtaed wth sources dstrbuto ad vorte dstrbuto ad the errors that appear each case order to see whch of the two umerca soutos offers a better resut. ISSN: Issue 7, Voume 7, Ju 8

9 Lumta Grecu,5,5 -,5 - -,5 - -,5, eact sorces vorte F. 9. The oca pressure coeffcet: eact souto, sources dstrbuto ad vorte dstrbuto. The errors that appear are represeted the foow raph. Because of the smmetr of the profe the umerca souto s aso smmetrca ad so the errors are. error vaue,6,5,4,3, errors error v error s,6,5,4,3,, error v ma. errors error s errors F.. The mamum errors for the case of vorte ad sources dstrbuto. The umerca resuts preseted the above pararaphs show that the drect boudar eemet method wth vorte dstrbuto ad ear boudar eemets offers for the probem of the compressbe fud fow aroud a obstace a better souto tha the oe that uses a sources dstrbuto ad ear boudar eemets, ad ver ood resuts for a qute sma umber of dscretzato odes. Wth the same computer code based o the method preseted ths paper, umerca soutos ca be obtaed for a kd of compressbe fud fows, for dfferet vaues of Mach umber, ot o for the compressbe case, ad for other kds of obstaces wth smooth boudares too.,, odes F.. The errors betwee the eact oda vaues of the oca pressure coeffcet ad the umerca oes obtaed: wth vorte dstrbuto (error v) ad sources dstrbuto (error s). As we ca see the errors obtaed whe the obstace s boudar s assmated wth a vorte dstrbuto are smaer tha the errors obtaed case of the sources dstrbuto for ma odes ( from odes) ad aso there s a b dfferece betwee the two mamum errors vaues obtaed these cases. Ths ca be better otce from the foow fure. Refereces: [] Brebba. A., Tees J.. F., Wobe L.., Boudar Eemet Theor ad Appcato Eeer, Sprer-Vera, Ber,984. [] Brebba. A., Waker S., Boudar Eemet Techques Eeer Butterworths, Lodo 98 [3] Boe M., Boudar tera equato methods for sods ad fuds, Joh We ad Sos, 995. [4] araus I., Mastoraks N. E., The Numerca Souto for Suar Itero- Dffereta Equatos Geerazed Hoder Spaces, Wseas Trasacto o Mathematcs, Issue 5, vo 5, Ma 6, pa [5] araus I., Mastoraks N. E., overece of the coocato methods for suar terodffereta equatos Lebesue spaces, Wseas Trasacto o Mathematcs, Issue, vo 6, November 7, pa ISSN: Issue 7, Voume 7, Ju 8

10 Lumta Grecu [6] Draoş L., Mathematca Methods Aerodamcs, Ed. Academe Româe, Bucureşt. [7] Draoş L., Fud Mechacs Vo.. Geera Theor. The Idea Icompressbe Fud (Mecaca Fudeor Vo. Teora Geeraă Fudu Idea Icompresb) Edtura Academe Româe, Bucureşt, 999. [8] Grecu L., Ph.D. these: Boudar eemet method apped fud mechacs, Uverst of Bucharest, Facut of Mathematcs, 4. [9] Grecu L., A Souto of the Boudar Itera Equato of the Theor of the Ifte Spa Arfo Subsoc Fow wth Lear Boudar Eemets, Aas of Bucharest Uverst, Mathematcs, Year LII, Nr. (3), pp [] Lfaov I. K., Suar tera equatos ad dscrete vortces, VSP, Utrecht, TheNetherads, 996. [] Rubo D, Troparevsk M.I., O the appromato of the auto sod ae for sov tera eqautos", Wseas Trasacto o Mathematcs, Issue, vo 3, Jauar 4, pa 3 ISSN: Issue 7, Voume 7, Ju 8