APPLICATION OF THE SCHWARZ-CHRISTOFFEL TRANSFORMATION IN SOLVING TWO-DIMENSIONAL TURBULENT FLOWS IN COMPLEX GEOMETRIES

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1 APPLICATION OF THE SCHWARZ-CHRISTOFFEL TRANSFORMATION IN SOLVING TWO-DIMENSIONAL TURBULENT FLOWS IN COMPLEX GEOMETRIES R. Moosav Deparmen of Mechancal Engneerng, Yasou Unversy P.O. Box 75914, Yasou, Iran S.A. Gandalhan Nassab* Deparmen of Mechancal Engneerng, Shahd Bahonar Unversy P.O. Box , Kerman, Iran *Correspondng Auhor (Receved: March 1, 008 Acceped n Revsed Form: Sepember 5, 008 Absrac In hs paper, wo-dmensonal urbulen flows n dfferen and complex geomeres are smulaed by usng an accurae grd generaon mehod. In order o analyze he flud flow, numercal soluon of he connuy and Naver-Soes equaons are solved usng CFD echnques. Consderng he complexy of he physcal geomery, conformal mappng s used o generae an orhogonal grd by means of he Schwarz-Chrsoffel ransformaon. The sandard - urbulence model s employed o smulae he mean urbulen flow feld, usng a lnear low-re - model for near wall regon. The governng equaons are ransformed n he compuaonal doman and he dscrezed forms of hese equaons are obaned by he conrol volume mehod. Fne dfference forms of he governng equaons are solved n he compuaonal plane and he SIMPLE algorhm s used for he pressurevelocy couplng. The mporan par of he presen wor s based on he numercal negraon of Schwraz-Chrsoffel ransformaon n grd generaon for smulang flud flow n dfferen complex geomeres. To valdae he compuaonal resuls, he heoarecl daa s compared wh ha of heorecal resuls acheved by oher nvesgaors, whch are n reasonable agreemen. Keywords Schwarz-Chrsoffel Transformaon, Grd Generaon, Turbulen Flow چكيده در اين مقاله نمونه هايی از جريان های دو بعدی ا شفته و غير قابل تراکم با هندسه های پيچيده و متفاوت با بکارگيری تکنيک ديناميک سيالات محاسباتی (CFD حل شده اند. يکی از مهمترين قسمت ها در روش حل عددی جريان سيال توليد شبکه محاسباتی است که در کار حاضر از طريق نگاشت همديس و استفاده از تابع تبديل شوارتز-کريستوفل به انجام رسيده است. روش نگاشت همديس استفاده شده مبتني بر انتگرال گيري عددي از تبديل شوارتز-كريستوفل ميباشد. از مزاياي اين روش ميتوان به سادگي و دقت زياد ا ن در عين توانايی قابل ملاحظه در توليد شبکه با هندسه پيجيده اشاره كرد. در اين مطالعه به منظور شبيه سازی جريان سيال حل عددي معادلات پيوستگی و ناوير-استوكس بصورت هم زمان انجام شده و از مدل توربولانس - استاندارد برای محاسبه تنش های ناشی از نوسانات سرعت به همراه يک مدل خطی برای نواحی نزديک ديواره استفاده شده است. به اين ترتيب که معادلات حاکم به صفحه محاسباتی منتقل گرديده و با روش حجم محدود مجزا سازی شده اند و نهايتا فرم اختلاف محدود اين معادلات توسط ا لگوريتم شناخته شده سيمپل بصورت عددی حل شده است. مطابقت نزديك بين پيش بيني هاي روش حاضر و نتايج تي وريک و تجربي ديگران مويدصحت روش بکار گرفته شده در توليد شبکه و حل عددی معادلات حاکم میباشد. 1. INTRODUCTION The numercal soluon of many problems such as flud flow consss n dscrezng he doman upon whch he governng equaons mus be solved. These equaons are lnearzed nsde small conrol IJE Transacons A: Bascs Vol. 1, No. 4, November

2 volumes where he conservaon equaons are appled. The fas ncrease n compuer echnology and daa sorage have led o he applcaon of Compuaonal Flud Dynamc (CFD n smulaon of lamnar and urbulen flows wh dfferen geomeres. The applcaon of CFD mehods need o have a dscrezed regon and s of grea mporance o be able o generae grd. The mehods of generang mesh n he domans wh arbrary boundares are dvded o, algebrac and dfferenal mehods and boh orhogonal and nonorhogonal grds can be generaed. Normally, for all numercal mehods, s beer ha he generaed grd s srucured and orhogonal. In orhogonal grd, he coordnae lnes are muually perpendcular o each oher. In hs ype of grd, s an easer applcaon of boundary condons nvolvng he normal dervaves o he boundares. Besdes, he ransformed forms of governng equaons n he compuaonal doman wh orhogonal grd have less number of erms n comparson o he non-orhogonal case. One of he accurae mehod n orhogonal grd generaon whou any resrcon on he ype of flow, s he conformal mappng echnque [1-3]. In pracce, he generaed grd lnes whch are perpendcular o each oher may be chosen o concde wh he sreamlnes and equpoenal lnes of an equvalen poenal flow problem. Several effcen mehods have been developed usng he conformal mappng echnque o oban wo-dmensonal meshes. Tradonally many mes, hs echnque has been used o carry ou he soluon of poenal flow abou complcaed geomercal shapes [4]. Mansour, e al [5] used smple mappng n orhogonal grd generaon for exernal flows over bodes wh a varey of shapes. In anoher wor by he same nvesgaors, some poenal flows over bodes wh dfferen geomeres were smulaed n whch he mesh generaon was done usng he Schwarz-Chrsoffel ransformaon [6]. In he presen wor, wo-dmensonal urbulen flows over some bodes wh dfferen and complex geomeres are smulaed usng orhogonal grd generaed by conformal mappng echnque. The mappng of physcal plane no compuaonal one aes place by he Schwarz-Chrsoffel ransformaon. The governng equaons conssng of he connuy and Naver-Soes equaons wh he equaons governng he nec energy of urbulence and he dsspaon rae are ransformed n he compuaonal plane and solved by CFD echnques. To calculae he values of Reynolds sresses, combnaon of he sandard - urbulence model and a lnear low-re - one for near wall regon s employed.. THEORY.1. Mean Flow Equaons The conservaons of mass and momenum for a seady ncompressble wo-dmensonal urbulen flow may be wren as: Connuy U = 0 Momenum (U U 1 P U = + ( u u ρ (1 ( n whch U are he mean velocy componens and u u are he Reynolds sresses... Turbulence Model Equaons The urbulence model employed n he smulaon of urbulen flow s he sandard - model [7]. In hs urbulence model, he Reynolds sresses are calculaed va he eddy-vscosy approxmaons as follows: U u U u = ( + δ + (3 3 and he urbulen vscosy,, s gven by = c μ ( Vol. 1, No. 4, November 008 IJE Transacons A: Bascs

3 In whch he urbulen nec energy and urbulen dsspaon rae are obaned based on he sandard - model from he followng ranspor equaons: (U (U = = [( [( σ σ ] + P ] + c 1 TABLE 1. Emprcal Consans for he - Model. c μ c 1 c σ P c σ σ θ (5 (6 Where P whch s he generaon rae of urbulen nec energy ha can be compued from Equaon 7 P U = u u (7 The coeffcens n Equaons 4 o 6 are gven n Table Near Wall Regon In he presen wor, for modelng he near-wall regon, a lnear low-re - model s employed. In hs urbulence model, he urbulen vscosy,, s obaned from = c μ f μ (8 and he ranspor equaons for he urbulence nec energy,, and homogenous dsspaon rae, are as follows: (U = [( + σ ] + P ( x (9 (U [( + = σ ] + c 1 P c f + E + S (10 The homogeneous dsspaon rae and dampng facors are calculaed from = ( (11 f (R / (R / 400 μ = 1 exp[ ] f (1 = 1 0.3exp( R (13 Where R = / s he local urbulen Reynolds number, P = u u ( U / he generaon rae of urbulen nec energy and he erm E s defned as U E = ( (14 x x and he exra source erm S n Equaon 10 sands for he sandard Yap correcon erm whch was nroduced frs me by Yap, e al [8]. The deals of hs model wh he model coeffcens are gven by Rasee, e al [9]..4. Boundary Condons In he numercal soluon of governng equaons, he followng boundary condons n he physcal plane are consdered: A he nle secon or n he upsream regon far from body, s assumed ha he flud flow has a unform velocy dsrbuon equal o U wh urbulen nec energy of = 0.001U and dsspaon rae of = ρc μ / v such ha / = 10. IJE Transacons A: Bascs Vol. 1, No. 4, November

4 A he oule secon or n he downsream regon far from body, a zero graden n sreamwse drecon s consdered for all dependen varables. On he sold walls, no slp condon s employed, he urbulen nec energy s equaed o zero and dsspaon rae s compued by [9]: = ( / n (15 n whch n sands for normal drecon o he sold surface z 1 3. GRID GENERATION As noed before, he grd generaon n he presen wor s based on he numercal negraon of he Schwarz-Chrsoffel ransformaon. By hs ransformaon, a polygon n he z(x,y-plane, s mapped ono he upper half of γ(ξ,η-plane as shown n Fgure 1. The relaon beween he z-plane as physcal doman o he γ -plane as compuaonal one s as follows: α dz N n = A ( γ ξ π dγ n n = 1 α1 y z n (az-plane η ξ 1 ξ ξ n ξ Ν α n (b γ -plane Fgure 1. Mappng of a polygon n z(x,y-plane ono he upper half of γ(ξ,η-plane. z N ( x ξ In he above equaon, α n s he angle of counerclocwse roaon a each apex and N s he number of polygon apces. The pons ξ n are posons on he real axs n γ -plane, where each of hem corresponds o an apex of he polygon n z -plane. The values of parameers ξ are n unnown whch wll be deermned eravely durng he numercal procedure. Also, n Equaon 16, A s a complex consan whch depends on he geomery of physcal doman. Accordng o Remann heorem, e al [4], he posons of hree pons of ξ n a he real axs of compuaonal plane are arbrary. The ransformaon funcon z( γ denong he relaon beween physcal and compuaonal axes can be obaned by negraon of Equaon 16 as follows: γ α N n z ( γ = A (γ ξ π n dγ + B (17 γ n = 1 0 In Equaon 17, B s a complex consan and γ 0 s a pon on he upper half of compuaonal plane. As noed before, he correc selecon of pons ξ n nvolves an erave procedure. The deals of hs ransformaon and he relaed numercal procedure are gven compleely n Reference [6]. I mus be menoned n ha reference, he Schwarz- Chrsoffel ransformaon was used o generae orhogonal grds whch also solves some nernal and exernal poenal flows n dfferen geomeres. Bu, n he presen wor, hs grd generaon echnque s employed o smulae vscous urbulen flows n a wde varey of complex geomeres. By hs echnque, he relaon beween physcal and compuaonal planes s deermned, from whch he values of merc coeffcens, whch are needed o ransform he governng equaons no compuaonal doman can be obaned. The ransformed form of he governng equaons n he compuaonal plane for any dependen varable Ψ, can be wren n he followng common form: Ψ Ψ [JAΨ Γ ] + [JBΨ Γ ] = S ξ Ψ ξ η Ψ η Ψ ( Vol. 1, No. 4, November 008 IJE Transacons A: Bascs

5 In whch, J s he Jacoban of ransformaon and he values of A, B, Ψ, Γ Ψ and S Ψ are dfferen for each governng equaon. 4. OUTLINE OF SOLUTION STRATEGY The governng equaons whch were ransformed n he compuaonal plane have a common form as was presened n Equaon 18. These paral dfferenal equaons were made dscree by negrang over an elemenal cell volume usng he fne volume mehodology. Such ha, he saggered ype of conrol volumes for he ξ-and η- velocy componens were used when oher varables of neres were compued a he grd nodes. The dscrezed forms of all ranspor equaons were obaned by employng hybrd dfferencng scheme for approxmaon of he convecve erms n hese equaons and were solved by he SIMPLE pressure correcon algorhm of Paanar, e al [10]. Numercal soluons were obaned eravely hrough lne-bylne mehod unl he convergence condon was acheved. Ieraons were ermnaed when sum of he absolue resduals was less han 10-4 for each equaon. Numercal calculaons were coded no a compuer program n FORTRAN. Based on he grd-ndependen sudy, he opmum grds wh 350 o 450 nervals n he ζ-drecon and 10 o 180 nervals n he η-drecon dependen on he flow geomery were employed for he numercal analyss along wh cluserng near he sold boundares and n he regons wh sharp gradens. Calculaons were run wh a Penum 5 personal compuer and he smulaon mes ranged from 3000 o 4000 secs dependng on he es case condons. wo cubes, wo rangular sharp bumps and car profle were analyzed here. An neresng ype of flud flow s he flow over forward or bacward seps. The exsence of flow separaon and reaachmen due o sudden expanson and conracon n hese ypes of geomeres, plays an mporan role n he desgn of many engneerng applcaons. As he frs es case, he flud flow over double forward facng sep (DFFS was smulaed. The basc flow confguraon s shown n Fgure. The channel has wo forward facng seps wh heghs of h 1 and h, respecvely. H and L are he hegh and lengh of he channel, where b, c and a are he lengh of wo seps and he lengh of boom wall, respecvely. The sream lnes along wh he values of sream funcon for flud flow over DFFS are shown n Fgure 3a. In hs fgure, he values of geomercal parameers h 1 /H, h /H, a/l, b/l, and c/l are 0., 0.4, 0.65, 0.15 and 0.5, respecvely. These daa are he same as used n Reference [11] o mae a comparson beween resuls. As s seen, several crculaon zones are generaed on he facng seps. The varaon of pressure coeffcen s shown n Fgure 3b. In he vcny of nle secon, he pressure conours are vercal o he boom and op walls unl he frs sep, whch ndcaes ha he flud flow becomes hydrodynamcally developed n ha regon. The value of pressure coeffcen decreases n he flow drecon, such ha he pressure coeffcen a he second sep s less han ha of he frs one. Besdes, due o he complexy of flow geomery near he seps, he pressure coeffcen has a very complex varaon n ha regon. The conours of urbulen nec energy (TKE are ploed n Fgure 3c. TKE has small values near he sold x 5. RESULTS To verfy he performance and accuracy of he presen mehod and o smulae wo-dmensonal urbulen flows n dfferen geomeres, many es cases of flud flows over dfferen bodes ncludng forward/bacward seps, one and wo cylnders, h 1 U H h a b c Fgure. Sech of he problem geomery. IJE Transacons A: Bascs Vol. 1, No. 4, November

6 (a (b (c (d Fgure 3. Sreamlnes, conours of pressure coeffcen and urbulen nec energy for DFFS flow a Re = (a Sreamlnes, (b Pressure coeffcen, (c Turbulen nec energy and (d Sreamlnes, Reference [11]. boundares and he maxmum values occur a he vcny of he wo seps along wh a complex varaon. The flud flow over DFFS was also suded numercally by Ylmaz, e al [11], usng commercal FLUENT code. The grd was 40 - Vol. 1, No. 4, November 008 IJE Transacons A: Bascs

7 generaed by Gamb pre-processor whch was good enough o oban a grd-ndependen soluon. In order o valdae he presen numercal procedure, one can compare he compued sreamlnes n Fgure 3a, wh hose obaned n Reference [11] shown n Fgure 3d. Concernng he sreamlnes and he values of sream funcon, he presen numercal resuls show good agreemen wh he heorecal resuls obaned by he FLUENT code. I should be noed ha he x-and y- coordnae axes n Fgure 3d have no he same scale facor whch maes an apparen dfference wh Fgure 3a. However, can be concluded ha he appled grd generaon mehod based on he Schwarz-Chrsoffel ransformaon wh he presen CFD echnques n solvng he governng equaons are very powerful and effcen n smulang many urbulen flows wh a wde varey of flow geomery. For he above es case, he nfluence of grd refnemen on he numercal resuls s also suded. The dsrbuon of x-componen of he mean velocy n he duc a axal secon before he seps wh x = 0.3 m s shown n Fgure 4. I s seen ha he dfferences beween he predced velocy dsrbuons on he fne ( and fner ( meshes are farly small ndcang ha he numercal resuls are reasonably grdndependen. A furher grd refnemen has been underaen wh addng 50 nodes n y-drecon bu no maor change was shown n he compued resuls. Thereby, resuls obaned on he ( mesh are regarded as grd-ndependen. I should be noed ha n he compuaon of all es cases, he frs 0 grd nodes were consdered whn he regon near he wall such ha, a he nerface beween he near wall and fully urbulen regon, he value of y + = [y τ / ρ]/ w was around 50. In Fgure 5, a comparson s made beween he presen resuls wh expermen and also wh he resuls by DNS mehod. In hs es case, he flud flow over a rgh angle, bacward facng sep n a duc s smulaed. The flow condon and geomercal parameers of hs es case are gven n Reference [1]. Fgure 5 shows he varaon of sn frcon facor, c f, on he boom wall afer he sep surface a Re = Besdes, o show he effec of usng lnear low-re urbulence model Fgure 4. Dsrbuon of x-componen of mean velocy n he duc a he axal secon x = 0.3 m usng four dfferen grd szes. Lnear low-re model, Sandard - model Expermen [1], DNS [1] Fgure 5. varaon of sn frcon facor on he surface of boom wall afer he sep. Re = near he wall regon, he resuls of sandard - model whou any wall funcon are also compared wh hose obaned by lnear model for he regon near he sold wall. I should be noed ha n IJE Transacons A: Bascs Vol. 1, No. 4, November

8 Fgure 5, he x-axs sars afer he sep surface. Fgure 5 shows a good conssency beween heorecal resuls wh expermen nocng ha he resuls by lnear urbulence model are very close o hose obaned by DNS mehod and also wh he expermen. In anoher es case he urbulen flow over a bacward facng sep s also smulaed and he mean velocy dsrbuon a x = 10.1 cm s shown n Fgure 6 and compared wh expermen [13]. In he expermenal sudy of Km, e al [13], he sep hegh was H = cm, H/H = 1.5 and he value of Reynolds number based on he duc hegh H, was equal o Fgure 6 ndcaes ha he axal secon x = 10.1 cm s locaed n he recrculaon zone such ha he flud velocy s negave n he regon near he wall. However, he model predcons for he mean velocy appear o be n good agreemen wh he expermen. As was menoned before, he flow smulaon for several oher es cases were done n he presen sudy and he resuls are shown n he followng Fgures, 7 o 17. Besdes, for space savng, he pressure conours and TKE dsrbuons are shown for some of hese es cases. Such as he ones for flow over a cylnder, wo cubes and car profle, he pressure and TKE conours are ploed n Fgures, 10, 13, 14 and 17. These fgures ndcae small value of pressure and large value for urbulen Fgure 7. Sreamlnes for urbulen flow over wo bacward seps a Re = Fgure 8. Sreamlnes for urbulen flow over a forward sep wh nclnaon angle of θ = 80 a Re = U n H H 0.6 y y/h 0.5 x Presen wor Exp. ( Km e al. [13] x-componen of mean velocy (m / s Fgure 6. Comparson of mean velocy profle wh expermen. Fgure 9. Sreamlnes for urbulen flow over a cylnder a Re = nec energy n he recrculaon zones. The compued sreamlnes and conours of pressure and TKE are n agreemen wh leraures, hose used dfferen mehods n mesh generaon and CFD echnques [11-13] Vol. 1, No. 4, November 008 IJE Transacons A: Bascs

9 Fgure 10. Dsrbuon of TKE for urbulen flow over a cylnder a Re = Fgure 14. Dsrbuon of TKE for urbulen flow over wo cubes a Re = Fgure 11. Sreamlnes for urbulen flow over wo cylnders a Re = Fgure 15. Sreamlnes for urbulen flow over wo sharp bumps a Re = Fgure 1. Sreamlnes for urbulen flow over wo cubes a Re = Fgure 16. Sreamlnes for urbulen flow over a car profle a Re = Fgure 13. Pressure conours for urbulen flow over wo cubes a Re = Fgure 17. Pressure conours for urbulen flow over a car profle a Re = IJE Transacons A: Bascs Vol. 1, No. 4, November

10 6. CONCLUSION Ths paper has presened an effcen mehod based on he Schwarz-Chrsoffel ransformaon n generang orhogonal grds for smulaon of wodmensonal, ncompressble urbulen flows. The governng equaons are solved by CFD echnques usng he sandard - urbulence model n he fully urbulence regon and a lnear low-re - model for he regon near he sold surface. The ransformed forms of he governng equaons n he compuaonal doman were dscrezed wh fne volume mehod and solved by he SIMPLE Algorhm. The model predcons are compared wh he avalable expermenal daa and a good conssency s found. Thereby, can be concluded ha by he presen mehod, many urbulen flows n dfferen and complex geomeres can be smulaed accuraely. 7. ACKNOWLEDGEMENT The auhors graefully acnowledge he nd help of Dr. S. M. Hossen Sarvar n he numercal negraon of he Schwarz-Chrsoffel ransformaon and orhogonal grd generaon. 8. REFERENCES 1. Thompson, J. F., Wars, Z. U. A. and Masn, C. W., Boundary-Fed Coordnae Sysem for Numercal Soluon of Paral Dfferenal Equaons, A Revew, Journal of Compuaonal Physcs, Vol. 47, (198, Srdhar, K. P. and Davs, R. T., A Schwarz-Chrsoffel Mehod of Generang Two-Dmensonal flow Grds, Journal of Flud Engneerng, Vol. 197, (1985, Moayer, M. S. and Taghdr, M. A., Boundary Conformng Orhogonal Grds for Inernal flow Problems, Iranan Journal of Scence and Technology, Vol. 17, No. 3, (1993, Mlne-Thomson, L. M., Theorecal Hydrodynamcs, 4 h Edon, Macmllan, New Yor, U.S.A., ( Mansour, S. H., Hossen Sarvar, S. M., Keshavarz, A. and Rahnama, M., An Analycal Numercal Mehod for Grd Generaon by Mahemaca, Proc. of 6 h Annual Iranan Mahemacs Conference, Shahd Bahonar Unversy of Kerman, Kerman, Iran, Vol. 1, (1995, Mansour, S. H., Mehraban, M. A. and Hossen Sarvar, S. M., Smulaon of Ideal Exernal and Inernal flows wh Arbrary Boundares usng Schwarz-Chrsoffel Transformaon, In. Journal of Eng., Trans. A, Vol. 17, No. 4, (004, Chowdhury, S. J. and Ahmad, G., A Thermohydrodynamcally Conssen Rae Dependen Model for Turbulence-Par II. Compuaonal Resuls, In. Journal of Non-Lnear Mechancs, Vol. 7, No. 4, (199, Yap, C. R., Turbulen Hea and Momenum Transfer n Recrculaon and Impngng Flows, Ph.D. Thess, Faculy of Technology, Unversy of Mancheser, Mancheser, U.K., ( Rasee, M. and Heaz, S. H., Applcaon of Lnear and Non-Lnear Low-Re - Models n Two Dmensonal Predcons of Convecve Hea Transfer n Passages wh Sudden Conracons, In. Journal of Hea and Flud Flow, Vol. 8, (007, Paanar, S. V. and Spaldng, B. D., A Calculaon Procedure for Hea, Mass and Momenum Transfer n Three-Dmensonal Parabolc Flows, In. Journal of Hea and Mass Transfer, Vol. 15, (197, Ylmaz, I. and Ozop, H. F., Turbulence Forced Convecon Hea Transfer Over Double Forward Facng Sep Flow, In. Communcaon n Hea and Mass Transfer, Vol. 33, (006, Le, H., Mon, P. and Km, J., Drec Numercal Smulaon of Turbulen flow Over a Bacward-Facng Sep, J. Flud Mech., Vol. 330, (1997, Km, J., Klne, S. J. and Johnson, J. P., Invesgaon of Separaon and Reaachmen of a Turbulen Shear Layer: Flow Over a Bacward Facng Sep, Repor MD-37, Thermo-Scence Dvson, Dep. of Mech. Engng, Sanford Unversy, Sanford, CA, U.S.A., ( Vol. 1, No. 4, November 008 IJE Transacons A: Bascs