Cell-Centered Finite Volume Solution of The Two-Dimensional Navier-Stokes Equations. pressure. Pr and. where W r, F r, and . (3) where.

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1 JOURNA OF AERONAUTICS AND SPACE TECHNOOGIES JUY 5 VOUME NUMBER (7-36) CE-CENTERED FINITE VOUME SOUTION OF THE TWO-DIMENSIONA NAVIER-STOKES EQUATIONS Muat UYGUN Hava Hap Okulu Yeşilyut-İSTANBU m.uygun@hho.eu.t Kai KIRKKÖPRÜ İTÜ Makina Fakültesi Gümüşsuyu- İSTANBU kikkopuk@itu.eu.t ABSTRACT Cell-centee finite volume metho with multistage time-stepping is successfully applie to two-imensional mass-weighte, time-aveage Navie-Stokes equations fo the computation of viscous flows. In the cellcentee scheme, flow quantities ae associate with the cente of a cell. Convective flues at the cell faces ae evaluate by means of upwin Roe Flu Diffeencing Scheme (Roe FDS) with Monotone Upwin Schemes fo Scala Consevation aws (MUSC) appoach. Geen s theoem is employe fo evaluation of gaients in computation of viscous flues. Five stage hybi time-stepping scheme is implemente fo integation to steay state. Convegence is acceleate by utilizing local time stepping an esiual smoothing. The accuacy of the pesent Navie-Stokes solve is veifie by compaing flat-plate lamina bounay-laye solutions with theoetical solutions of Blasius an by compaing lamina aifoil solutions with those available in liteatue. Convegence own to machine zeo attaine in the computations inicates a goo sign fo the efficiency of the pesent solve. Tubulence closue fo Reynols stesses is obtaine using two-laye algebaic ey viscosity moel of Balwin an oma. Compute esults fo tubulent flows ae valiate with available epeimental esults. Keywos: Viscous Flows, Roe Flu Diffeencing Scheme, Geen s Theoem, Balwin-oma Tubulence Moel.. INTRODUCTION: Eule solves ae matue enough to be use in a vaiety of engineeing applications fo flows with comple geometies. Howeve, eclusion of iffusive effects limits thei applicability foeal flows, although the ominating convective chaacte is successfully involve. Hence, Navie-Stokes solves ae essential fo moe accuate engineeing computations [, ]. In a pevious stuy [3], Uygun an Kıkköpü stuie two imensional Eule solve implementing Roe s Flu Diffeencing Scheme []. They conclue that Roe FDS is woking quite efficient in esolving gaients pesent in the flow omain at the cost of CPU time an computational memoy. In the cuent wok, two-imensional Navie-Stokes solve base on cell-centee finite volume iscetization technique is veifie an valiate. Finite volume metho is popula ue to its fleibility in teatment of abitay geometies an applicability of consevation laws in integal fom. While employing Roe FDS, the flu vectos at the mipoint of a cell face ae compute by utilizing two flow vaiables name the left an the ight states, which ae intepolate fom left an ight 7 sie of the cell face using MUSC appoach [5]. Nonphysical oscillations in the solutions nea the egions of stong flow gaients ae eliminate by means of Van Albaa flu limite [6]. The vaiables, which ae equie fo the computation of viscous tems, ae aveage at a cell face. Gaients at the mipoints of a cell face ae compute by means of Geen s theoem. Five stage hybi time-stepping scheme is applie to avance the solution in time. Convegence of the eplicit scheme is acceleate by applying local timestepping an implicit esiual smoothing techniques. Tubulence closue fo Reynols stesses is ealize by using two-laye algebaic ey viscosity moel of Balwin an oma [7], which eliminates the ifficulty in etemining the bounay laye ege eisting in the moel of Cebeci an Smith [8]. It is a computationally cheap an easy to implement moel, which yiels accuate esults fo fully attache o milly sepaate flows, whee histoy effects ae not impotant. The solutions pesente in this wok consist of investigation of lamina an tubulent flow cases fo which theoetical an numeical solutions ae available in efeences 9-5.

2 . GOVERNING EQUATIONS: Application of Reynols (o time) aveaging (enote by oveba) to ensity an pessue an of Fave (o mass) aveaging (enote by tile) to the emaining flow vaiables in the compessible Navie-Stokes equations yiels the so calle Fave Aveage Navie- Stokes (FANS) equations []: ρ + ( ρv~ i ) = t i % % % () t j j j ( ρe% ) ρhv + j viτij τijui τiju i t %% = % + + j j T% uuu F i i j k + ρcpu jt + ρv% iτij + ρ j j with v% v% i j v% k τ ij = µ + δij. () j i 3 k ( ρvi) + ( ρvv i j) = ( pδij) + ( τij ρvv i j) The vaiables i, t, p, ρ, T %, v% i ae catesian cooinates, time, pessue, ensity, tempeatue an velocity components espectively. E % is total enegy, H % is total enthalpy, µ is lamina (molecula) F viscosity an τ = ρ vv is Reynols-stess tenso. ij i j FANS equations ehibit a tem by tem coesponence with thei lamina flow countepats, ecept that the stess tenso is augmente by the Reynols stesses an the heat flu vecto is augmente by aitional tubulence heat flu tems an mean enegy issipation tems. The components of the Reynols stesses ae moele though the Boussinesq ey-viscosity hypothesis: F v% v% i j v% k τ ij = µ T + δij ρkδ. (3) ij j i 3 k 3 It assumes that shea stess is elate linealy to mean ate of stain, as in the case of a lamina flow. µ is T calle the tubulent (o ey) viscosity. ρk = ρ uu i i accounts fo the effect of Faveaveage tubulent kinetic enegy an it is neglecte in this stuy. Afte applying Boussinesq ey-viscosity appoach (eqn. 3) to the aveage fom of the govening equations (eqn. ), the effective viscosity µ = µ + µ T () eplaces the lamina viscosity. Similaly, the effective themal conuctivity coefficient k T k = k + kt = C µ µ p + (5) P PT eplaces the lamina themal conuctivity coefficient. C enotes the specific heat coefficient at constant p pessue. P an P T ae lamina an tubulent Pantl numbes, which ae taken as.7 an.9 fo ai. Coefficient of lamina viscosity, µ is compute using the Suthelan fomula: 3/.5T% 6 µ =. (6) T% + Tubulent viscosity µ T is compute by means of two-laye algebaic ey viscosity moel of Balwin an oma [7]. Assuming ai as an ieal gas, pessue an tempeatue ea ~ + ~ ~ u v p p = ( γ ) ρ E, T% =. (7) ρ R Integal fom of Navie-Stokes equations can be witten as W + ( Fc Fv) S = t v, (8) whee W, F, an C F ae the vectos of V consevative vaiables, convective an viscous flues: ρ ρv% ρu% W = uv n p F ρ% % + n % τ + n % yτ y C = F ρv% ρvv % % + ny p v = (9) n % τ y + n % yτ yy ρe % ρhv %% n Θ+ % nyθ% y with the Fave-aveage contavaiant velocity, V% = v% n = nu% + nyv%. The tems escibing the wok of viscous stesses an the heat conuction ea T% Θ % = u%% τ + v%% τy + ( k + kt) () T% Θ % y = u%% τ y + v%% τ yy + ( k + kt) y whee u% v% % τ = ( µ + µ T ) 3 y v% u% % τ yy = ( µ + µ T ) 3 y u% v% % τy = % τ y = ( µ + µ T ) 3 y 3. SPATIA DISCRETIZATION: () Afte witing the equation (8) fo all contol volumes, a system of ODE of fist oe is obtaine. Fo a paticula D contol volume, equation (8) becomes W IJ, = ( FC FV) m Sm t () IJ, m= 8

3 I an J locate the paticula contol volume an m ientifies contol volume faces. S enotes the aea m of the face m. Steay solution can be eache iteatively: n n t + W I, J = WI, J R (3) I, J I, J R = F F S whee, ( ) R I, J I J c V m m m=. () is calle the esiual. The finite volume iscetization equies an evaluation of the convective an viscous flues at each cell face. In this wok, convective flues wee evaluate using upwin Roe FDS, which is popula ue to its goo esolution of shocks an bounay layes. It is base on the ecomposition of the flu iffeence ove a face of the contol volume into a sum of wave contibutions. Total convective flu at the face of a contol volume eas [, 6] Fc( WR) + Fc( W) F = + ( c ) I /, J A W W ( ) Roe I+ /, J R 5 Roe R = k k A W W F with ( ) whee p c V u~ cn ~ ~ F = V c~ ~~ ρ, c ~ v ~ cn ~ y ~ H cv ~ ~ ~ ~ ρ u F,3, = V p + ~ c v~ ~ q ~ u Vn, ~ρ V v Vn y u ~ u + v~ v + w~ ~ w V V p c V u~ + cn ~ ~ F = V + c~ ~~ ρ. 5 c ~ v ~ + cn ~ y ~ H + cv ~ ~ (5) (6) The jump conition is efine as () = () R ( ). Roe-aveage vaiables (enote by tile) ae compute fom the left an left state by the specific fomulae given in efeences an 6. The sonic point in Roe FDS is intouce by moifying the moulus of the eigenvalues using Haten s entopy coection [7]. eft an ight states wee evaluate using MUSC intepolation [5] with Van Albaa imite [6]: a( b + ε ) + b( a + ε ) U R I + a + b + ε (7) a( b + ε ) + b( a + ε ) U I + a + b + ε ar I + U I + br I + U I whee, an. a I + U I b I U I The paamete ε pevents the activation of the limite in smooth flow egions [8]. The shifte contol volume fo the evaluation of viscous flues is selecte. The vaiables, which ae equie fo the computation of viscous tems, ae aveage at a cell face. Gaients at the mipoints of a cell face ae compute by employing Geen s theoem with the ai of an auiliay contol volume. The eivative of the flow vaiable U in the -iection is efine as U S UmS, m (8) m= whee is the auiliay contol volume.. INTEGRATION TO STEADY STATE: Fo steay poblems with cell-centee iscetization, hybi eplicit scheme eas ( ) n WI, J = WI, J () ( ) tij, ( ) ( ) WI, J WI, J α = Rc R I, J I, J ( ) ( ) tij, () ( ) α = Rc R I, J IJ, ( 3) ( ) tij, ( ) (,) α = 3 Rc R, IJ, ( ) ( ) tij, ( 3) (,) α = Rc R, IJ, ( n+ ) ( ) t I, J ( ) (,) WI, J = WI, J α 5 Rc R I, J, R whee = β3r + β3 R, R = β R + β R I J I J ( ) ( ) ( ) ( ) I, J ( ) ( ) ( ) (,) an ( Rc) =Σ Fc( WI+ ) S, = ( R) =Σ Fv S + D I, J I, J k = 5 5 I J k k k (9) () () 9

4 D IJ, accounts fo the issipation tem (eqn.6). Table pesents stage coefficients, α m an blening coefficients, β use in all computations. m Table. Hybi Multistage scheme Upwin Scheme Stage α β Convegence ate of the eplicit time-stepping scheme is futhe acceleate by local time stepping an implicit esiual smoothing techniques. In local time stepping, the solution at each cell is avance at the maimum t allowe by the stability. Stability limitations ue to iffusive in aition to convective effects ae consiee. In implicit esiual smoothing technique, the eplicit scheme is given an implicit chaacte [9]. The oiginal esiual vecto, R is eplace by the smoothe esiuals, R by solving implicit equations R I = RI + ε R () I at each cell. is the aplacian opeato an ε is the smoothing coefficient. 5. TURBUENCE MODE: Closue fo the system of FANS equations is ealize by using two-laye algebaic ey viscosity moel of Balwin an oma [7]. Fo wall boune flows, ( µ T) fo y y inne c µ T = (3) ( µ T) fo y > y oute c whee y is the nomal istance fom the wall an y c is the smallest value of y at which ( µ T) = ( µ T). inne oute Fo fee tubulent flows, µ T = ( µ T). () oute The inneegion tubulent viscosity is moele using Pantl-Van Diest fomulation: ( µ ) ρ T inne l = (5) whee ρ is the aveage ensity an l is the miing length efine as + + y A l = κ y( e ), (6) whee κ, an A + ae Von Kaman s an Van Diest s amping constants, taken as., an y A ( e ) + + is Van Diest s Damping facto. Magnitue of voticity vecto is efine as u ~ v ~ =. (7) y The outeegion tubulent viscosity is moele using ( µ T) = kc oute cpρfklebf (8) wake whee k is the Clause constant, taken as.68 an Ccp is a constant, taken as.6. y F Fwk = min (9) C % ma ma yma wk V iff Fma whee C wk is a constant taken as.5. V% = V% V% (3) iff ma min whee V % is the contavaiant velocity. F is the maimum value of ma + y e y + wall boune flows ( ) = A (3) F y y fee tubulent flows This function yiels the length an velocity scales fo the oute laye. y ma is the value of y coesponing to F ma. The tubulence length scales ae thus etemine by l in the inne laye, an by y ma in the oute laye. The cooinate + y is efine as + τ wρ w y = y (3) µ w whee τ w is the shea stess at the wall. Klebanoff intemittency facto is efine as 6 C kleb y F = (33) kleb ma y whee C is taken as.5. This facto accounts fo kleb the fact that as fee steam is appoache, the tubulence intensity eceases. 6. BOUNDARY CONDITIONS: Bounay conitions ae teate by using ghost cells, which ae obtaine by etening the iscetization stencil beyon the computational bounaies. At the boy suface, the velocity components ae zeo (no slip), an the nomal eivative of tempeatue is zeo (aiabatic wall). Fo the flat-plate case, a symmety conition is applie in font of the flat-plate on the symmety line. Fo the aifoil cases, a continuity conition is enfoce along the wake cut in the C-type mesh. The fafiel bounay conitions, which ae

5 base on Riemann invaiants fo -D flow, ae applie to the est of the bounaies fo all cases. 7. COMPUTATIONA RESUTS: The numeical esults given hee emonstate the accuacy an computational efficiency of the pesent Navie-Stokes solve fo lamina an tubulent flows. Compute esults agee well with those available in efeences 9-5. Figue - pesent the computational gis fo flat-plate an aifoil cases. The computational omain of the flat-plate is a ectangle incluing 66 gi points. With espect to the leaing ege of the flat-plate, the omain etens one plate length upsteam an two plate lengths ownsteam. The uppe bounay is two plate lengths above the flat-plate. Fo the aifoil case, C-type computational mesh incluing 36 mesh points with 9 points on the aifoil, an 6 points in the wake is use. The oute bounay is locate 5 chos away fom the aifoil. Fo all computations, the flow was initially unifom having the fa upsteam popeties. The solution was assume to each the steay state when the euction of ensity esiuals is less than oes of magnitue in lamina flow cases an oes of magnitue in tubulent flow cases. Couant-Fieichs-evy (CF) numbe of 5 is utilize in all computations. All computations wee pefome on a Compaq Wokstation incluing Gb memoy an 3 GHz ouble-cpu unning Winows XP. 7.. Flat-Plate Bounay-aye Accuacy of the pesent Navie-Sokes solve is veifie by epoucing the eact compessible bounay-laye solution ove a flat-plate. The gi spacing net to the wall is.5 plate lengths. The computations wee pefome fo a Mach numbe of.5 an a Reynols numbe base on the plate length of 5. Compute solutions ae compae to eact solutions of Blasius [9]. Figue 3 pesents the compute an eact solutions of steamwise velocity in the bounay-laye. The cooinate η is efine as η = y * sqt(re / ) (3) The skin fiction along the flat-plate is given in figue. Compute an eact solutions of velocity pofile an skin fiction agee well with each othe. Figue 5 pesents the convegence histoy fo the pesent case. 7.. Symmetic Flow ove NACA Secon test case is NACA aifoil at a Mach numbe of.5 with egee incience an a Reynols numbe of 5. The nomal gi spacing at the wall is. chos. Fo this case, as pesente in Figue 6, sepaation occus nea the tailing ege, an small eciculation bubbles ae fome in the wake egion. Figue 7-8 inicate the compute suface pessue istibution an velocity vecto plot. Figue 9 pesents the convegence histoy fo the pesent case. Compute solutions agee well with those available in liteatue [,, an 3] ow Reynols Numbe Flow ove NACA These thee test cases inclue vey low Reynols numbe flow ove NACA aifoil []. They povie a goo eample fo valiation of computation of viscous tems in Navie-Stokes equations, since viscous effects ominate in the flow fiel. The nomal gi spacing at the wall is. chos. In test case A, Mach numbe is.8, incience is egees, an Reynols numbe is 73. Figue pesents the Mach numbe contous of the compute solution. A api gowth of the bounay laye along the uppe suface is obseve. A small supesonic flow egion appeas on the uppe suface outsie the viscous laye. The compute solution agees well with that of efeence. In test case B, Reynols numbe is incease to 5, while Mach numbe an incience angle is kept same as in test case A. Figue pesents the Mach numbe contous of the compute solution. As a esult of inceasing Reynols numbe, slowe bounay laye gowth an lage supesonic flow egion is obtaine. A lage egion of eciculating flow occus ownsteam of the aifoil stating at 35% cho length on the uppe suface. In test case C, Mach numbe is, the incience is egees, an Reynols numbe is 6. Figue pesents the ensity contous, which inicates a stong bow shock. This solution agees well with that available in efeence. Figue 3 pesents the convegence histoies of test cases A, B, an C. 7.. Tubulent Flows fo Flat-Plate Two-laye algebaic ey viscosity moel of Balwin an oma in the pesent Navie-Stokes solve is valiate by computing the compessible bounaylaye solution fo a flat-plate (figue ) at fee steam Mach numbe of.5 an fee steam Reynols numbe of. 7. The gi spacing net to the wall is 6-6 plate lengths. Tansition fom lamina to tubulent flow is intouce by fiing the tansition at a cetain point. In figue, compute skin fiction along the plate is compae to the analytical ones obtaine using powe law an eact theoy: ( ) 5 ( ) C f =.59 Re (35) C =.55 ln.6re (36) f Compute solutions agee with analytical ones as well as numeical ones available in Ref.. Figue 5 inicates the convegence histoy fo the pesent case. 3

6 7.5. Tubulent Flows past NACA Tubulent flow past a NACA aifoil is consiee at fee steam Mach numbe of.5, incience angle of.77 eg., an a fee steam Reynols numbe of.9 6. The gi spacing net to the wall is 6-6 cho lengths, such that 5- cells ae accommoate in the bounay laye. No significant iffeence in the pessue istibutions between compute an epeimental one [] is obseve (figue 6). Figue 7 an 8 inicate the velocity vecto plot an convegence histoy. og(c F ) Eact Roe.5.5 X/ Figue. Skin fiction istibutions. Figue. Computational mesh employe fo the flatplate bounay-laye calculation og( ρ) Figue. Computational mesh employe fo the computations ove NACA aifoil Iteation Figue 5. Convegence histoy. 6 5 Eact Roe η u/u inf Figue 3. Steamwise velocity pofiles. Figue 6. Paticle pathlines. 3

7 C P X/C Figue 7. Suface pessue istibution. Figue. Mach numbe contous in Case A Figue 8. Velocity vectos at tailing ege Figue. Mach numbe contous in Case B og( ρ) Iteation Figue 9. Convegence histoy Figue. Density contous in Case C 33

8 og ( ρ) Case B Case A Iteation Figue 3. Convegence histoy Case C C P Ep. Roe Figue 6. Pessue istibutions og(c f ).3. BASIUS POWER AW EXACT THEORY ROE og(x/) Figue. Skin fiction istibutions Figue 7. Velocity vectos at tailing ege - - og( ρ) - og( ρ) Iteation Figue 5. Convegence histoy Iteation Figue 8. Convegence histoy. 3

9 8. CONCUSION: The capability of the pesent two-imensional Navie- Stokes solve base on cell-centee finite volume iscetization metho employing upwin Roe flu iffeencing scheme an two-laye algebaic tubulence moel of Balwin an oma to accuately calculate viscous flows has been investigate. Geen s theoem implemente fo evaluation of gaients in computation of iffusive flues yiele eliable solutions. Convegence own to machine zeo attaine in the computations inicates a goo sign fo the efficiency of the pesent solve. Numeical esults agee well with those available in the liteatue an they inicate that pesent solve is accuate, an eliable at low as well as high Reynols numbe flows. Futue wok will concentate on the impovement of the pesent solve to accuately calculate low Mach numbe flows in the incompessible limits an to incease the computational efficiency by implementing multigi technique. 9. ACKNOWEDGEMENT: Suppot povie by Aeonautics an Space Technologies Institute of Tukish Ai Foce Acaemy, Yesilyut-İstanbul is gatefully acknowlege.. REFERENCES: [] Uygun, M., Tunce, I.H., 3, A Computational Stuy of Subsonic Flows ove A Meium Range Cago Aicaft, AIAA pape [] Uygun, M., Tunce, I.H.,, Viscous Flow Solutions ove CN-35 Cago Aicaft, AIAA Jounal of Aicaft, Vol., No., pp [3] Uygun, M., Kıkköpü, K., Numeical Solution of the Eule Equations by Finite Volume Methos: Cental vesus Upwin Schemes, Jounal of Aeonautics an Space Technologies, Vol., No., pp.7-55, Januay 5. [] Roe, P.., 98, Appoimate Riemann Solves, Paamete Vectos, an Diffeence Schemes, Jounal of Computational Physics, 3: [5] ee, B. V., 977, Towas the Ultimate Consevation Diffeence Scheme IV; A New Appoach to Numeical Convection, Jounal of Computational Physics, 3: [6] Albaa, G.D., ee, B. V., an Robets, W.W., 98, A Compaative Stuy of Computational Methos in Cosmic Gas Dynamics, Aston. Astophysics, 8:76-8. [7] Balwin, B.S., an oma, H., 978, Thin aye Appoimation an Algebaic Moel fo Sepaate Tubulent Flows, AIAA pape [8] Cebeci, T., an Smith, A.M.O., Analysis of Tubulent Bounay ayes, Acaemic Pess, New Yok, 97. [9] White, F.M., Viscous Flui Flow, McGaw Hill Inc., New Yok, 97. [] Poceeings of the GAMM-Wokshop on Numeical Simulation of Compessible Navie- Stokes Flows, INRIA, Sophia-Antipolis; Notes on Numeical Flui Mechanics, Vol.8, Vieweg-Velag, 986. [] Chakabatty, S.K., 989, Numeical Solution of Navie-Stokes Equations fo Two- Dimensional Viscous Compessible Flows, AIAA Jounal, Vol. 7, No. 7, pp [] Chakabatty, S.K., 99, Vete Base Finite- Volume Solution of the Two-imensional Navie-Stokes Equations, AIAA Jounal, Vol. 8, No. 9, pp [3] Swanson, R.C., Tukel, E., 987, Atificial Dissipation an Cental Diffeence Schemes fo The Eule an Navie-Stokes Equations, AIAA pape [] Thibet, J.J., Ganjacques, M., an Ohman,. H., 979, NACA Aifoil, Epeimental Data Base fo Compute Pogam Assessment, AGARD-AR-38. [5] Calson, J.R., 996, High Reynols Numbe Analysis of Flat-Plate an Sepaate Afteboy Flow Using Non-linea Tubulence Moels, AIAA pape [6] Roe, P..; Pike, J., Efficient Constuction an Utilization of Appoimate Riemann Solutions, Computing Methos in Applie Sciences an Engineeing, R. Glowinski, J.. ions (es.), Noth Hollan Publishing, The Nethelans, 98. [7] Haten, A., a, P.D., Van ee, B., 983, On Upsteam Diffeencing an Gounov-Type Schemes fo Hypebolic Consevation aws, Soc. Inust. an Appl. Math. Rev., 5, No.. [8] Venkatakishnan, V., 993, On the Accuacy of imites an Convegence to Steay State Solutions, AIAA Pape [9] Jameson, A., Bake, T.J., 983, Solution of the Eule Equations fo Comple Configuations, AIAA pape

10 VITA Muat UYGUN He was gauate fom Aeonautical Engineeing Depatment at Tukish Ai Foce Acaemy, Istanbul in August 995. He eceive his M.Sc. egee in Aeospace Engineeing fom Mile East Technical Univesity, Ankaa in Septembe. ate, he joine Aeonautical Engineeing Depatment at Tukish Ai Foce Acaemy, Istanbul. His thesis an follow-on wok as a PhD stuent in Mechanical Engineeing at Istanbul Technical Univesity, involves numeical simulation of intenal an etenal viscous flow fiels. Duing -3, he conucte a stuy as a NATO Fellow at Defence Reseach Development Canaa-Atlantic (DRDC-Atlantic), Halifa, Canaa. His cuent eseach inteests ae Computational Flui Dynamics, Upwin-Finite Volume schemes, compessible flows, low Mach numbe peconitioning, convegence acceleation techniques, an tubulence moeling. He is a membe of Ameican Institute of Aeonautics an Astonautics (AIAA). Kai KIRKKÖPRÜ He obtaine his BS an MSc egees fom Mechanical Engineeing Depatment at Istanbul Technical Univesity (ITU) in 979 an 98, espectively. He eceive his PhD egee in Mechanical Engineeing fom the Univesity of Coloao (CU) at Boule in 988. He was Senio Reseach Associate in Mathematics an Physics Depatment at the Univesity of East Anglia in Englan between 988 an 99. He joine Mechanical Engineeing Depatment at ITU in 99. ate, he woke in the Applie Mathematics Depatment at CU Boule fo two yeas. He has been pofesso in Mechanical Engineeing Depatment at ITU since 998. His main eseach inteests ae Analytical an Computational Flui Dynamics, Flui Machiney, Asymptotic Techniques an Theoetical Combustion. 36