Efficient Symmetric Boundary Condition for Galerkin Finite Volume Solution of 3D Temperature Field on Tetrahedral Meshes

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1 Proceegs of the 5th IASME/WSEAS It. Coferece o Het rsfer, herl Egeerg Evroet, Athes, Greece, August 5-7, 7 8 Effcet Syetrc Boury Coto for Gler te Volue Soluto of D eperture el o etrherl Meshes SAEED-REA SABBAGH-ADI KN oos Uversty of echology, Cvl Egeerg Deprtet, No.6 Vlsr Street, ehr, IRAN Sz@tu.c.r NIKOS E. MASORAKIS Mltry Istutes of Uversty Eucto (ASEI) Hellec Nvl Acey er Chtzyrou 859, Prues, GREECE stor@wses.org Astrct: I soe of the egeerg proles, t s ecessry to lyze the three-esol teperture profles. I orer to solve typcl prole uerclly, the three-esol teperture ffuso equto s chose s the thetcl oel. he fte volue forulto s erve usg Gler pproch for the esh of tetrherl eleets, whch fclttes solvg teperture proles wth coplcte geoetres. I ths pproch, the Posso equto s ultple y the pece wse ler shpe fucto of tetrherl eleet tegrte over the cotrol volues whch re fore y gtherg ll the eleets eetg every coputtol oe. he ler shpe fuctos of the eleets vsh y soe thetcl pultos the resulte forulto c e solve explctly for ech coputtol oe. he lgorth ot oly s le to hle the essetl oury cotos ut lso the turl oury cotos usg ovel techque. Accurcy effcecy of the lgorth s ssesse y coprso of the uercl results for ech r prole of het geerto trsfer loc wth ts lytcl soluto. he, trouce techque for posg turl oury cotos o ustructure tetrherl esh s exe for cses wth cle syetrc oures. Key-Wors: Nuercl Sulto of Het rsfer, Gler te Volue Metho, Syetrc Boury Cotos Itroucto Avllty of the fst powerful persol coputers otvtes the use of uercl ethos for solvg teperture fels of egeerg pplctos. I orer to prect the therl ehvor of the sol sttes wth terl source of het geerto rte severl uercl solvers re evelope usg vrous ethos such s te Dfferece Methos, te Eleet Methos te Volue Methos. he te Dfferece Methos [] covert fferetl for of the goverg equtos to sple forultos the expese of soe errors whch egres the ccurcy of the uercl solutos. But the prole of the te Dfferece Methos s serous ffcultes ther pplcto to solve rel worl proles ue to ecessty of the use of structure grs for geoetrc cretzto. he te Eleet Metho [] Boury Eleet Metho [] overcoe the foreetoe prole y pplcto of sophstcte thetcl pultos o the tegrl for of the goverg equtos forultos whch e up wth coplcte soluto proceures. Cosequetly, the te Eleet Methos ot oly c hle coplex geoetres ut lso prove ccurte uercl solutos for the oury vlue proles. However, ther hevy coputtol wor los, te-cosug coplcte trx coputtos plct solutos of rel worl pplctos wth geoetrcl coplextes soe tes re eyo the vlle hr wre effceces. he trtol te Volue Methos [] covert the tegrl for of the goverg equtos for sptl proles to sple lgerc forultos. hese ethos y hve soe vtges over the te Dfferece Methos ut the requre structure eshes rg up jor restrctos errors wth oelg of os wth coplex geoetres rregulr oures. he te Volue Methos sutle for the ustructure eshes [5] c hle the geoetrcl coplextes usg reltvely sple forultos coputtol proceures. herefore, f the evelope lgorth of these types of te Volue Metho c stsfy the ccurcy requreets of the esre prole, t woul e effcet es of coputer sultos of the

2 Proceegs of the 5th IASME/WSEAS It. Coferece o Het rsfer, herl Egeerg Evroet, Athes, Greece, August 5-7, 7 9 egeerg pplctos o orry hr wre systes. I ths pper, three esol fte volue het trsfer solver oule of NASIR (Nuercl Alyzer for Scetfc Iustrl Requreets) softwre s trouce. I ths pper, the uercl soluto lgorth of ths softwre for the teperture fel uer the effects of terl het geerto rte s well s essetl turl oury cotos s escre. Here, the ustructure fte volue soluto lgorth trouce for the vscous flow coputtos [6] s opte for teporl soluto of ffusve equto of het geerto trsfer. he goverg equto s ultple y the pece wse ler shpe fucto of tetrherl eleets of ustructure esh the t s tegrte over ll cotrol volues fore y the eleets eetg every coputtol oe (vertces of the eleets). he lgorth tes vtge fro the fct tht the frst ervtves of the ler terpolto fucto for the teperture re costt se ech eleet. By pplcto of Guss vergece theore usg the property of the ler shpe fucto, whch stsfes hoogeeous oury coto o the epeet vrle, the oury tegrl ters c e otte for every cotrol volues usg surroug ol vlues. After soe pultos, the resultg forultos c e solve explctly wth rther lght coputtol efforts [7]. Usg ovel uercl techque for posg turl oury cotos syetrc oures re use for reucg coputtol efforts. Hece, effcet solver s evelope for the soluto of three-esol teperture fels wth coplex oures whch geoetrclly c e oele y the use of ustructure esh of tetrherl eleets. I orer to ssess the perforce of the evelope solver, the uercl soluto results of teperture typcl loc re copre wth ts lytcl soluto. Goverg equto Assug sotropc therl propertes for the sol terls, the flr equto efg het geerto trsfer s of the for, α α ( + ) + Q κ () Where the preters re ρ ( Kg ) esty, o C ( K J Kg c) specfc het, ( o c) teperture, o κ ( W c) het coucto coeffcet Q ( K J h) rte of het geerto per ut volue the therl ffuso s efe s α κ / ρc. Nuercl Sulto Although for the soluto of reltvely sple pheoeo lytcl proceures c e cosere, for the jorty of the cses lytcl solutos re possle or very restrctve. As ltertve, uercl etho c e effcet powerful es of solutos for such cses. Severl ttepts hve ee e to solve trset het trsfer equtos usg well-ow ethos such s the fte fferece, fte volues fte eleets ethos. Ech of the hs ts ow shortcogs ffcultes. Here, lgorth s escre for the uercl soluto o ustructure esh of tetrherl. he ervto of the screte forultos strts usg Gle pproch. he pultos e up wth screte equto wthout the ler shpe fucto of eleet t c e solve explctly for every coputtol oe surroue y the eleets. hs uercl techque c ccurtely solve the three-esol teperture fel wth coplex oures wth coserle effcecy cheveet. Coser the goverg equto for het geerto trsfer hoogeous o s, + ( α ) (,,) S () Where (teperture) s the uow preter S s the het source. If teperture gret flux recto (secory vrle) s efe s, (,,) α () A hece, the equto tes the for: (,, ) + ( ) S () By pplcto of the Vrtol Metho, fter ultplyg the resul of the ove equto y the

3 Proceegs of the 5th IASME/WSEAS It. Coferece o Het rsfer, herl Egeerg Evroet, Athes, Greece, August 5-7, 7 5 test fucto φ tegrtg over suo, we hve, φ + φ Sφ (,, ) (5) he ters cotg sptl ervtves c e tegrte y prt over the su-o the equto (5) y e wrtte s, tes the vlue of (the vlue of the epeet vrle cetrl oe ). By ths property of the shpe fucto ϕ ( ϕ o oury Γ of the suo ), the oury tegrl ter equto (7) tes zero vlue for cotrol volue whch the vlues of ssue ow t oury oes. φ + φ φ S φ (,,) (6) Usg guss vergece theore the equto tes the for: φ + αφ (. Γ ) (,,) Γ φ S φ (7) Where Γ s the oury of o. ollowg the cocept of weghte resul ethos, y coserg the test fucto equl to the weghtg fucto, the epeet vrle se the o c e pproxte y pplcto of ler coto, such s N oes ϕ [8]. Accorg to the Gler etho, the weghtg fucto φ c e chose equl to the terpolto fucto ϕ. I fte eleet ethos ths fucto s systetclly copute for esre eleet type clle the shpe fucto. or tetrherl type eleet (wth four oes), the ler shpe fuctos, ϕ, tes the vlue of uty t esre oe, zero t other eghorg oes of ech trgulr eleet ( ) [9]. Exteg the cocept to su-o to the cotrol volue fore y the eleets eetg oe (gure ), the terpolto fucto ϕ tes the vlue of uty t the ceter oe of cotrol volue zero t other eghorg oes (t the oury of the cotrol volue Γ ). Noteworthy tht, ths s essetl property of weght fucto, ϕ, whch shoul stsfy hoogeeous oury coto o t oury of su-o []. ht s why the tegrto of the N oes ler coto ϕ (s pproxto) over eleets of su-o gure - Su-o ssocte wth oe of the coputtol fel After ottg zero ter, the equto (7) tes the for, ϕ ϕ Sϕ (,,) (8) I orer to rve the lgerc forulto, every sgle ter of the ove equto frst s pulte for ech eleet the the tegrto over the cotrol volue s perfore. he resultg forulto s vl for the cetrl oe of the cotrol volue. or the ters cotg o ervtves of the shpe fucto ϕ, exct tegrto forul s use s Λ c ϕϕϕϕ 6Λ (!! c!!) ( + + c+ + ) Λ / (for c), where Λ s the volue of the tetrherl eleet [6]. hs volue c e copute y the tegrto forul s, Λ l Λ x ( Λ) [ x l ] δ where x δ re the verge recto coortes projecte re (orl to recto) for every se fce opposte to oe of the eleet. herefore, the trset ter φ for ech t tetrherl eleet Λ (se the su-o) c e wrtte s, Λ ( ) φ Λ. Λ t Cosequetly, the trset ter of equto (8) for

4 Proceegs of the 5th IASME/WSEAS It. Coferece o Het rsfer, herl Egeerg Evroet, Athes, Greece, August 5-7, 7 5 the su-o (wth cetrl oe ) tes the for, ϕ t (9.) Slrly, the source ter of equto (8), Sϕ, for ech eleet Λ (se the suo), wll e wrtte s, ϕ S Λ ( Λ ) S. he the source ter Λ of the equto (8), for the cotrol volue (wth cetrl oe ) tes the for, ϕ S S (9.) Now we try to screte the ters cotg sptl ervtve, ( φ ) equto (8). Sce the oly uow epeet vrle s ϕ the shpe fuctos, ϕ, re chose pece wse ler every tetrherl eleet, the teperture gret flux ( s fore y frst ervtve) s costt over ech eleet c e te out of the tegrto. O the other h, the tegrto of the shpe fucto sptl ervto over tetrherl eleet c e coverte to oury tegrl usg Guss vergece theore [9], hece, ϕ Λ (. ) x ϕ. Here s copoet of the Λ se fce eleet orl to the recto. he screte for of the le tegrl c e wrtte s, ϕ.( ) [ ϕδl ] Λ, where [ ϕ δl ] s fore y coserg the se of the eleet opposte to the oe, the, ultplcto of ts copoet perpeculr to the recto y ϕ the verge shpe fucto vlue of ts three e oes. Hece, the ter N ( φ for cotrol ) [ Λ ( ϕδl x ) ] volue (cotg N eleets shrg ts cetrl oe). Sce the shpe fucto ϕ tes the vlue of uty oly t cetrl oe of cotrol volue s zero t the oes locte t the oury of cotrol volue, ϕ / for the fces coecte to the cetrl oe of cotrol volue ϕ for the oury fces of the cotrol volue. O the other h the su of the projecte re (orl to recto) of three se fces of every tetrherl eleet equtes to the projecte re of the fourth se fce, hece the ter cotg sptl ervtves recto of the equto (8), c e wrtte s, ϕ M [ δl ] (9.c) Where [ δ l ] s the copoet of the oury fce (opposte to the cetrl oe of the cotrol volue ) perpeculr to recto. Note tht, s copute t the ceter of tetrherl eleet of the cotrol volue, whch s ssocte wth se. he teperture gret flux recto, α, t ech tetrherl eleet c e clculte usg Guss vergece theore, α Λ α Λ ( ), where ( ) s the projecto of se fces of the eleet perpeculr to recto. By expressg the oury tegrl screte for s, ( ) ( δ l ), for ech eleet se the cotrol volue. herefore, we hve, [ ] Where, Λ ( l ) δ () δ l s the copoet of th fce of tetrherl eleet (perpeculr to the recto) s the verge teperture of tht fce Λ s the volue of the eleet. Note worthy tht for cotrol volues t the oury of the coputtol o, cetrl oe of the cotrol volue loctes t ts ow oury. or the oury ses coecte to the to the oe there re o eghorg eleet to ccel the cotruto. Hece, ther cotrutos re they ct s the oury ses of the su-o. herefore, there s o chge to the escre proceure for coputto of the sptl ervtve ters ( ϕ ). lly, usg expressos (9.), (9.) (9,c), the equto (8) c e wrtte for cotrol volue (wth ceter oe ) s:

5 Proceegs of the 5th IASME/WSEAS It. Coferece o Het rsfer, herl Egeerg Evroet, Athes, Greece, August 5-7, 7 5 t S M [ δ l ] (,,) () he volue of cotrol volue, c e copute y suto of the volue of the eleets ssocte wth oe. Reeer, the het source for ech oe cocrete oy s efe y S α Q ( te) κ. he resulte uercl oel, whch s slr to No-Overlppg Schee of the Cell-Vertex te Volue Metho o ustructure eshes, c explctly e solve for every oe (the ceter of the su-o whch s fore y gtherg eleets shrg oe ). he explct soluto of teperture t every oe of the o of terest c e oele s, N t+ t t + t S ( l ) δ (,) () Now we ee to efe lt for the explct te step, δ t. Coserg therl ffusvty s α κ ρc wth the ut ( s ), the crtero for esurg the lty of terl for teperture chge. Hece the rte of teperture chge c e expresse s, δ t α. herefore, the pproprte sze for locl te steppg c e cosere s, δ t β ( ) α β () β s cosere s proportolty costt coeffcet, whch ts gtue s less th uty. or the stey stte proles ths lt c e vewe s the lt of locl coputtol step towr stey stte. However, there re fferet szes of cotrol volues ustructure eshes. hs fct ples tht the u gtue of the ove relto e cosere. Hece, to t the stlty of the explct te steppg the glol u te step of the coputtol fel shoul e cosere, so, δ t β ( ) ( β ) () α Noteworthy tht for the soluto of stey stte proles o sutle fe ustructure eshes, the use of locl coputtol step ste of glol u te step y coserly reuce the coputtol efforts. Boury Cotos wo types of oury cotos re usully pple ths uercl oelg. he essetl turl oury cotos re use for teperture teperture gret flux (grets) t oures, respectvely [9]. or those oury oes where ol tepertures re to e pose (essetl oury cotos), there s o ee to copute the teperture. Hece, copute teperture t those oe hve to e replce y the gve cert vlues t the e of ech coputtol step. Cotrrly, there s o ee to chge the copute teperture t the oury oes where the turl oury coto s to e pose. I orer to pose gve teperture gret orl to the oury fces, G (the rte of het exchge per ut volue of the surfce), the orl vector of the oury fces,, ) c e utlze ( ( G, G, G to copute G ) t the esre oures. Although sple techques for posg gret t oury c e pple for the cses tht the oury orl s prllel to oe of the rectos of coorte syste, coputtol ffcultes rse for the cle or curve oures. or overcog the prole, the copute gret flux vector,,, ), t the cetre of jcet ( eleet y e ofe t the e of ech coputtol step. rst, the vector of teperture gret tget to the esre oury fce s ecopose fro the copute gret t the cetre of jcet eleet, gtl G (. ) (5.) he, the orl vector of teperture gret c e pose s, Norl G (5.) lly, the teperture gret vector t the cetre of eleet jcet to the esre oury fce s cosere s, + (6) Mofe gtl Norl Usg ove eto techque the ffcultes ssocte wth cle or curve oures re

6 Proceegs of the 5th IASME/WSEAS It. Coferece o Het rsfer, herl Egeerg Evroet, Athes, Greece, August 5-7, 7 5 overcoe. herefore, the propose techque sutes the preset lgorth whch s opte for the os wth coplex oures scretze usg ustructure eshes. 5 Structure etrherl Meshes Structure tetrherl esh c e geerte y coserg 5 or 6 tetrhero etwee eght oe cuc esh spcg (fgure). A geerl vew of two typcl eshes whch re fore y coserg 5 or 6 tetrhero etwee eght oe cuc esh spcg re presete fgure. 6 Verfcto of the oel he ccurcy of the soluto of sptl ervtve ters s vestgte y coprso of the results of the uercl solver wth the lytcl soluto of the followg stey stte ffuso equto (oury vlue prole) wth costt source ter s [,9], x Q (,) (7) the sptl fel of { < (, y) < } x. Coserg the costts of the ove equto s Q s well s the oury cotos t x, y s (syetrc oury cotos) t x, x. he lytcl soluto s gve y [7], ( x, y) (8) {( y ) + π ( ) cos[ ( ) πy / ] cosh[ ( ) πx / ] ( ) cosh[ ( ) π / ] I orer to ot teperture fel slr to the two esol soluto of ths prole o secto of the cue, the requreet of posg turl oury coto s relxe y oulg the eso, hece, essetl oury coto ( ) s pose over four the oures {( x, y), } turl oury coto ( z,. he ) t {( ) } tetrherl esh whch s geerte y coserg 6 tetrherl etwee cuc esh spcg wth eght oes s presete fgure.. hs esh s fore y gr pots. he result of the uercl soluto of equto (7) s show fgure the for of teperture cotour ps. he ccurcy of uercl soluto c e ssesse fgures 5 y coprso etwee the coputtol the lytcl soluto two rectos log the esh. 7 Icle Syetrc Cotos I orer to ssess the perforce of trouce techque for posg turl oury cotos (.e. syetrc coto ), severl reuctos o coputtol fel s oe y vg the orgl fel to sller prts (fgure 6). As c e see soe of the syetrc surfces (6.c 6.) re cle. he copute teperture fels o the eshes presete t fgure 6 re show fgure 7 the for of color coe ps of teperture. As c e see the pple techque for posg turl oury coto preserves the ccurcy of teperture grets, eve o cle surfces. he use of proper techque for syetry coto proves coserle svg coputtol efforts. 8 Cocluso he equto of het geerto trsfer s solve o trgulr eleet esh utlzg ler shpe fucto s ltertve test fucto. he resulte lgorth proves lght explct coputto of te epeet proles. he splcty of the lgorth es t esy to progr exteso for further evelopets. he uercl oel ws verfe two stges. rstly, y usg oury vlue prole ts lytcl soluto, the ccurcy of the soluto of the sptl ters ws ssesse. he results of the evelope oel preset resole greeets to the lytcl Secoly, the trouce techque for posg turl (syetry) oury cotos o ustructure tetrherl esh s exe for cses wth cle oury surfces. he pple techque for posg turl oury coto ot oly preserves the ccurcy of teperture grets o cle surfces, ut lso proves coserle svg coputtol efforts y pvg the wy for pplcto of syetrc cle oury surfces.

7 Proceegs of the 5th IASME/WSEAS It. Coferece o Het rsfer, herl Egeerg Evroet, Athes, Greece, August 5-7, 7 5 Refereces: [] Aerso D.A., ehll J.C., Pletcher R.H., Coputtol lu Mechc Het rsfer, Corge Hespher Press, 98. [] Rey J.N. Grtlg D.K., he te Eleet Metho Het rsfer lu Dycs, CRC Press,. [] Bere C.A., elles J.C.., Wroel L.C., Boury Eleet echque heory Appcto Egeerg, Sprger Verlg, Berl, 98 [] Ptr S.V., Nuercl Het rsfer lu low, McGrw Hll, 98. [5] hopso J.., So B.K. Wetherll N.P., H oo of gr geerto, CRC Press, New or, 999. [6] Syes L.A., Developet of wo- Desol Nver-Stoes Algorth for Ustructure rgulr Grs, ARA Report 8, Aprl 99. [7] Sgh-z S.R., Bger A.R. Coputer Sulto of Ceet Het Geerto eperture Profles Mss Cocrete Structures, IUS Itertol Jourl of Egeerg Scece, No., Vol.5, sprg. [8] eewcz O.C. ylor R.L., "he te Eleet Metho, Volue Bsc orulto Ler Proles", McGrw-Hll, 988 [9] Rey J.N., A Itroucto to the te Eleet Metho, McGrw-Hll,, 99 gures g. A geerl vew of two typcl eshes whch re fore y coserg 5 or 6 tetrhero etwee eght oe cuc esh spcg g. Copute teperture fel cuc prs ANALICAL COMPUAIONAL(INIE VOLUME)..6.5 ANALICAL COMPUAIONAL(INIE VOLUME) (cc).5..5 (oc) () ().8.6. g. 5 Coprso etwee the coputtol the lytcl soluto, : log the le y (<x<) : log the le yx (<x<) g. wo cuc esh spcg fore y eght oes whch re flle y ) 5 ) 6 tetrherl.8.6.

8 Proceegs of the 5th IASME/WSEAS It. Coferece o Het rsfer, herl Egeerg Evroet, Athes, Greece, August 5-7, c c g. 6 Usg syetrc cotos for vg the orgl cuc fel to sller prts s; ) Dvg the orgl cuc y ) Dvg the orgl cuc y 8 c) Dvg the orgl cuc y 6 ) Dvg the orgl cuc y 6 g. 7 Usg syetrc cotos for teperture fel wth sller esh s; ) Dvg the orgl cuc y ) Dvg the orgl cuc y 8 c) Dvg the orgl cuc y 6 ) Dvg the orgl cuc y 6