IMPLEMENTATION OF AN ALGEBRAIC BYPASS TRANSITION MODEL INTO TWO-EQUATION TURBULENCE MODEL FOR A FINITE VOLUME METHOD SOLVER

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1 Colloquium FLUID DYNAMICS 2007 Insiue of Thermomechanics AS CR, v. v. i., Prague, Ocober 24-26, 2007 p.1 IMPLEMENTATION OF AN ALGEBRAIC BYPASS TRANSITION MODEL INTO TWO-EQUATION TURBULENCE MODEL FOR A FINITE VOLUME METHOD SOLVER Jiří Dobeš 1), Jaroslav Foř 1), Jaromír Příhoda 2) 1 ) Deparmen of Technical Mahemaics, Faculy of Mechanical Engineering, Czech Technical Universiy in Prague, Karlovo náměsí 13, Praha 2, Czech Republic 2) Insiue of Thermomechanics AS CR, v.v.i., Dolejškova 5, Praha 8, Czech Republic 1 Inroducion The mahemaical model of urbulen flow, based on he finie volume mehod and wo-equaion urbulence model, was exended by an algebraic model of he bypass ransiion aking ino accoun he effec of free-sream urbulence and pressure gradien on he laminar/urbulen ransiion. The inermien feaure of flow in he ransiion region is described by he algebraic relaion for he inermiency parameer and empirical relaions for he onse and lengh of he ransiion region. 2 Formulaion of he problem We solve a sysem of averaged Navier-Sokes equaions in wo spaial dimensions w + f + g = r + s (1) x y x y wih w vecor of conserved variables, f and g are inviscid flux vecors, r and s are viscous flux vecors. We consider perfec gas (κ = 1.4) only. The sysem of governing equaion was closed by he urbulence model wih he urbulen viscosiy. The woequaion k-ω SST model proposed by Mener [1] was considered. Appropriae iniial and boundary condiions are prescribed. 3 Transiion model Transiion models are based on he algebraic and/or ranspor equaion for he inermiency coefficien γ. Neverheless, all he ransiion models are dependen in some exen on empirical relaion for he ransiion onse. Therefore, exising models of bypass ransiion have a limied range of applicabiliy. We deal wih he ransiion model based on he empirical relaion for he inermiency coefficien γ = 1 exp nˆ σ( Re ) 2 x Re x (2) proposed by Narasimha [2]. For simpliciy, i is supposed ha he inermiency coefficien is dependen on x coordinae only. The posiion of he ransiion onse is described by he Reynolds number Re x deermined by means of he momenum Reynolds number Re ϑ = f (Tu, λ ) where Tu (%) is he free-sream urbulence level and λ is he pressure gradien parameer. Boh parameers are considered a he locaion of he ransiion onse. Příhoda e al. [3] proposed he relaion

2 p.2 ( 40λ ) ( ) 40λ 1 exp Re ϑ = Reϑo exp( Tu) (3) exp based on experimens of Fasihfar and Johnson [4] and he relaion for he fla plae flow according o Mayle [5]. The lengh of he ransiion region is given by parameers describing he spo generaion rae ˆn and spo propagaion parameer σ. The simples correlaion uses he empirical parameer N = n$ σ Re 3 ϑ (4) proposed by Narasimha [2]. For fla plae flow, i can be expressed by he relaion N o = 0.86 x10 Tu (5) given by Goselow, Blunden and Walker [6]). The effec of he pressure gradien is correlaed by empirical relaions N = 0.86 x10 N = 0.86 x exp Tu ( 2.134λ ln Tu 59.23λ ln Tu) exp ( 10 λ ) for λ < 0 for λ > 0 (6) proposed by Solomon, Walker, Goselow [7]. To avoid he calculaion of he momenum Reynolds number in cases wih complicaed geomery and unsrucured meshes, he voriciy Reynolds number Re ν is used. According o Mener e al. [8], hese wo parameers can be correlaed for he Blasius boundary layer by he relaion Reν max Reϑ = (7) where Re νmax is he maximum of he voriciy Reynolds number 2 y Ω Reν max = max (8) y ν where y is he disance o he neares wall and Ω is he absolue value of voriciy Ω = 2Ω Ω (9) ij ij The relaion (7) does no change oo much for flows wih pressure gradien. 4 Numerical soluion The finie volume mehod of he cell cenered ype wih he modificaion of he approximaive Roe's Riemann solver, he linear leas square reconsrucion and he Barh's limier was developed. The viscous fluxes are discreized in he cenral manner on a mesh dual o he cell faces. Time inegraion is performed wih linearized Euler backward formula and local ime sepping is used. The sysem of equaions is solved wih GMRES mehod and ILU(0) precondiioning. The mehod works on general unsrucured meshes. The inermiency coefficien resuling from he ransiion model gives is value on he boundary. Inside he compuaional domain we ake he value γ a he poin on he wall boundary, which lies closes o he considered place. Using he Boussinesq s hypohesis, he inermiency coefficien influences in he ransiion region he effecive viscosiy given by he relaion µ = µ + γµ (10) ef

3 Colloquium FLUID DYNAMICS 2006 Insiue of Thermomechanics AS CR, Prague Ocober 25-27, 2006 p.3 where µ is he urbulen viscosiy. Following corresponding modificaions of he SST urbulence model were made in ranspor equaions for he urbulen energy k and for he specific dissipaion rae ω. The ransiion model is no valid on he leading edge. This region has o be excluded from he consideraion. 5 Numerical resuls 5.1 ERCOFTAC es cases We consider hree es cases relaed o fla-plae ransiional 2D boundary layers flows wih differen free-sream urbulence level. Tes cases T3A, T3B, and T3Aminus from he ERCOFTAC daabase [9, 10] were chosen. The domain of soluion consiss of recangle [-0.2, 1.58] [0, 0.3]. The wall boundary is locaed a y = 0, x 0. The symmery is imposed a y = 0, x < 0. Oupu is a x = A all he oher boundaries he free sream is prescribed. Consan value of free sream saic pressure p and densiy ρ is considered. As he free sream urbulence is isoropic, he decay of urbulence level can be approximaed as 2 u 5/ 7 Tu (%) = 100 = C( 1000x + 610) (11) U wih values of C given in Tab.1. The equaion (11) was used o deermine inle boundary condiions for k and ω. The oher condiions are given in Tab.1 as well. Case U e (m/s) Tu o (%) C T3A T3B T3Aminus Table 1: Free sream condiions for ERCOFTAC es cases T3A T3B T3Aminus Figure 1: Disribuion of skin fricion coefficien for ERCOFTAC es cases The compuaions for es cases T3A and T3B give a good agreemen wih heoreical invesigaion and experimenal daa. Bu for es case T3Aminus wih urbulence level Tu abou 1%, he prediced ransiion onse sars sooner in comparison wih experimenal daa. I can be caused by many facors. Besides free-sream urbulence and pressure gradien, he ransiion is influenced by surface roughness, noise and vibraions of he experimenal faciliy. Furher experimenal daa can be influenced by he used experimenal echnique and by he mehod of deermining of he

4 p.4 ransiion onse. Therefore a considerable scaer exis in experimenal daa used for empirical correlaions. 5.2 SE1050 urbine cascade The furher es case represens he ransonic flow hrough SE1050 blade cascade. The chosen blade cascade SE 1050 was designed for he las sage of an axial seam urbine of large oupu. The ransonic flow has a relaively complex srucure of he flow field especially in he exi par of he cascade [11]. Experimens on he SE 1050 blade cascade were carried ou in he High-Speed Wind Tunnel of he Insiue of Thermomechanics [12]. Sagnaion values of he pressure, densiy and inle angle are prescribed a he inle. Mean value of he pressure is prescribed a he oule. We also prescribe zero normal derivaives of he conserved variables a he inle and oule. No-slip boundary condiion was used on he adiabaic wall. Suiable boundary condiions are chosen for he SST urbulence model. The seleced es case is characerized by he oule isenropic Mach number Ma 2 = 1.198, he inle angle α 1 = 19.3 o and he Reynolds number Re = We have chosen wo differen urbulence levels Tu = 1% and Tu = 3%. The hybrid mesh consiss of nodes and elemens. Layers of quadrilaeral elemens are presen along he blade, in he wake and along he oule boundary (o preven reflecion of he ou-running shock waves). The compuaion was carried ou for hree modes: laminar (γ 0), wih he ransiion model included, and urbulen (γ 1). The flow-fields depiced by he Mach number isolines are compared in Fig.2. The skin-fricion disribuion is shown in Fig.3. The difference beween he laminar and urbulen shock wave/boundary layer ineracion is clearly visible, boh on Mach number isolines and he skin-fricion disribuion. The survey of loss coefficiens is given in Tab.2. The problem wih urbulence level of Tu = 3% didn' converge properly, due o limi-cycle ype feedback beween urbulence and ransiion model.

5 Colloquium FLUID DYNAMICS 2006 Insiue of Thermomechanics AS CR, Prague Ocober 25-27, 2006 p.5 Figure 2: Mach number isolines for he SE1050 blade cascade, lef o righ: laminar compuaion (γ 0), ransiion model included, urbulen compuaion (γ 1) a) Tu = 1% b) Tu = 3% Figure 3: Disribuion of he skin fricion coefficien for he SE1050 blade cascade Loss coefficien ξ(%) Tu = 1% Tu = 3% Laminar compuaion Compuaion wih ransiion model included Turbulen compuaion Experimen 4.5 Table 2: Loss coefficien for he SE1050 blade cascade

6 p.6 6 Conclusions The proposed algebraic bypass ransiion model was implemened ino he unsrucured finie volume mehod solver. The calculaion procedure was validaed for ransiional fla-plae boundary layers wih differen free-sream urbulence level and for ransonic flow hrough he SE1050 blade cascade. The ransiion in inernal flows is influenced by many various facors and so he predicion of he bypass ransiion is he mos problemaic par of he whole calculaion. Furher progress can be achieved by he applicaion of he ransiion model based on a ranspor equaion for he inermiency coefficien. Acknowledgemen The research was suppored by he research projec No.A funded by he Gran Agency of AS CR as well as by he projec No.101/07/1508 funded by he Czech Science Foundaion and by he Research Plan No.AV0Z References [1] Mener F.R. (1994): Two-equaion eddy-viscosiy urbulence models for engineering applicaions, AIAA J., 32, [2] Narasimha R. (1985): The laminar-urbulen ransiion zone in he boundary layer, Progress in Aerospace Science, 22, [3] Příhoda J., Hlava T., Kozel K. (1997): Tesing of he ransiion lengh using a wolayer urbulence model, Proc. Conf. Engineering Mechanics 97, Svraka, Vol. 4, (in Czech) [4] Fasihfar A., Johnson M.V. (1992): An improved boundary layer ransiion correlaion, ASME Paper No.92-GT-245 [5] Mayle R.E. (1991): The role of laminar-urbulen ransiion in gas urbine engines, Trans. ASME, J. Turbomachinery, 113, [6] Goselow J.P., Blunden A.R., Walker G.J. (1992): Effecs of free-sream urbulence and adverse pressure gradiens on boundary layer ransiion, ASME Paper No.92-GT-380 [7] Solomon W.J., Walker G.J., Goselow J.P. (1996): The laminar-urbulen ransiion zone in he boundary layer, Trans. ASME, J. Turbomachinery, 118, [8] Mener F., Esch T., Kubacki S. (2002): Transiion modelling based on local variables, Proc. 5 h In. Symposium on Engineering Turbulence Modelling and Experimens (Eds. Rodi W., Fueyo N.), Elsevier Science Ld., [9] Roach P.E., Brierley D.H. (1990): The influence of a urbulen free sream on zero pressure gradien ransiional boundary layer developmen, Par 1: escases T3A and T3B, In: Numerical simulaion of unseady flows and ransiion o urbulence, (Eds. Pironneau D., Rode W., Ryhming I.L.), Cambridge Universiy Press [10] Tescase C20, Classic ERCOFTAC Daabase, hp://cfd.mace.mancheser.ac.uk/ercofac/ [11] Kozel K., Příhoda J., Šafařík P. (2003): Transonic flow hrough plane urbine cascade: Experimenal and numercal resuls, QNET-CFD Nework Newsleer, 2, 3, [12] Šťasný M., Šafařík P. (1990): Experimenal analysis daa on he ransonic flow pas a plane urbine cascade, ASME Paper No.90-GT-313, New York