# Two-Equation Thermal Model for Heat Transfer Predictions

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1 (SYB) 3-1 Two-Equaton Thermal Model for Heat Transfer Predctons Pascale Kulsa, Nathale Marcnak Laboratore de Mécanque des Fludes et d'acoustque-lmfa-umr5509 Ecole Centrale de Lyon BP ECULLY Cedex France Notatons k turbulent knetc energy U, u mean and fluctuatng velocty k temperature varance uv Reynolds stress Prt turbulent Prandlt number v normal turbulent heat flux P pressure y + dmensonless coordnate q heat flux α t turbulent thermal dffusvty R tme scales rato ε dsspaton rate of k R t turbulent Reynolds number ε dsspaton rate of k T, mean and fluctuatng temperature τ knematc tme scale t tme τ thermal tme scale Introducton The very hgh temperature level reached n actual turbne components requres accurate smulaton tools to predct the heat transfers. The turbne flow s very complex. It s domnated by hgh pressure gradents, three-dmensonal vscous effects, and hgh heat fluxes. Moreover, the turbulence plays a major role n the heat transfer level. Very effcent turbulence models have been developed for the predcton of dynamc flows. Modelsaton of thermal feld s relatvely nferor to the modellng of dynamc feld. The current practce s to use assumpton of constant turbulent Prandtl number. The objectve of ths work s to develop and assess thermal flux models for turbne aerothermal feld predcton [1]. Physcal model The physcal model s based on the three-dmensonnal compressble Favre averaged Naver- Stokes equatons. The equatons are wrtten n a conservatve form for cartesan coordnates. Contnuty equaton : ρ ρu + = 0 Momentum equatons : ρu ρuu j P σ s j ρuu j + = + j j j Energy equaton : ρe ρeu τ ju j q ρue + = where U, ρ, P and µ are respectvely the velocty, the densty, the pressure and the vscosty. Paper presented at the RTO AVT Symposum on Advanced Flow Management: Part B Heat Transfer and Coolng n Propulson and Power Systems, held n Loen, Norway, 7-11 May 001, and publshed n RTO-MP-069(I).

3 (SYB) 3- These equatons are mplemented n the Naver-Stokes code CANARI, developed by ONERA []. The equatons are dscretsed n space wth a central fnte volume scheme assocated wth a four step Runge-Kutta tme scheme. Second and fourth order dsspatve terms are added to ensure numercal stablty. An mplct smoothng resdual could be used to accelerate the convergence. Turbulence modellng To close the momentum equatons, the Reynolds stresses u u j have to be determned. The classcal Boussnesq hypothess s postulated whch lnks the unknown Reynolds stresses to the known velocty feld wth the turbulent eddy vscosty µ t. Based on a dmensonal analyss, the modelsaton of µ t s bult wth a velocty scale and a tme scale. The turbulent knetc energy k and ts dsspaton rate ε are used to express these two scales and µ t. The two transport equatons of k and ε are solved. Dampng functons of Hattor-Nagano-Tagawa [3] are ntroduced to solved the equatons up to the wall. Thermal flux models The turbulent thermal model s based on a two-equaton model. Smlar to the modellng of the Reynolds stresses, the thermal fluxes are lnked to the mean temperature gradent by a scalar : the turbulent dffusvty α t. A dmensonal analyss s carred out to express α t. Ths quantty has the same dmenson as µ t. The most representatve velocty scale s that of the turbulent phenomenon, so the root of the knetc energy s chosen. The tme scale s expressed as a product of the velocty tme scale τ=k/ε and the thermal tme scale τ =k /ε. k s the temperature varance and ε ts dsspaton rate. αt = C λ f ρk λ τ τ Ths knd of model sgnfes that the knematc and the thermal phenomena are coupled and the k-ε-k -ε equatons are solved smultaneously [4], [5], [6]. The k -ε equatons are developed for ncompressble flows but on the Morkovn hypothess [7] they could be adapted to compressble flows f the Mach number fluctuatons are small compared to the mean Mach number value. In the k -equaton, the only term to be modeled s the turbulent dffuson and a gradent type representaton (lke for the k-equaton) s used. Varous models have been proposed for the ε equaton. The model we mplement has two producton terms and two destructon terms. For the turbulent dffuson term, a gradent approach s adopted. The low-reynolds number model of Hattor-Nagano-Tagawa [3] has been chosen because t solves the pseudo-dsspaton ε ' transport equaton, whch allows zero-value at the wall. The use of ths varable mples the ntroducton of addtonal terms : ( k ) and E. The modellng equaton of k s : ρk ρu k T αt k k + = ρu + α + ρε σ convecton producton dffuson dsspaton The modellng equaton of ε ' s : ρε ρε ρε ε α t ε ρε + U = C p1 f p1 P + C p f p P + E + α + C f ρε d1 d1 Cd f t x ρ k x σ φ x ρ convecton producton dffuson destructon d ε ρε k

4 (SYB) 3-3 where : f p1, f p, f d1 =1, σ φ, σ =1, P u u U = ρ j, j + 0 [ exp( y 30) ] 1+ exp[ ( R 10) ] P T = ρu α T f µ = t ; E = αt ( 1 fε ) Rt ρ y y P = + r 7. 9 Rt f λ 1 exp 1 exp 1 exp R t 10 + = ( 1. 9 fε 1) 0 9 ; f 1 exp( y P 3 ) 1 ε = r 30 f d. ( ) Channel test case Ths test case s a wall-bounded shear flow wth thermal gradents. The results of the calculaton are compared to the Drect Numercal Smulatons of Kasag [8]. A constant wall heat flux s appled, and the Reynolds number based on the frcton velocty and the half heght of the channel s Re=150. The buoyancy effects are neglected. The grd contans 00 ponts n the axal drecton and 60 ponts n the normal to the wall drecton. The frst node s located at y + =0.05. The mean velocty and temperature dstrbutons are presented fgures 1 and. For these two quanttes, the model shows a good behavour. In partcular at the wall, the results are very satsfactory. Fgure 3, the calculated Reynolds stress shows very good agreement wth the DNS data. The same remark may be made for the turbulent heat flux, fgure 4. The rato of the thermal and dynamc tme scales, R, s presented fgure 5. Its evoluton s qualtatvely reproduced by the twoequaton model. We can remark that R s not constant as mpled by a constant Prt model. Turbne blade test case A turbne stator blade s now tested. The expermental faclty s a lnear transonc cascade nvestgated at the Von Karman Insttute [9], [10]. The Mach number s at the nlet and 0.85 at the outlet. The stagnaton temperature of the flow equals 416.8K. The wall s heated at Tw=96.5K. The turbulence ntensty at the nlet s 4%. A mult-doman H-O-H grd s used. The upstream H grd contans 9x49x ponts. A O-type grd s defned around the blade to ensured a good descrpton of the wall boundary layers. The nodes dstrbuton s 301x65x, and the frst pont s located at y + =1. The downstream H-type grd contans 189x49x nodes. The heat transfer coeffcent H along the blade s avalable and compared to the expermental data, fgure 6. On the sucton sde (s/s0>0), the heat transfer coeffcent s hgh near the stagnaton pont. Up to 0% of the curvlnear abscssa, H decreases, due to the effect of the acceleraton on the lamnar boundary layer. The calculaton results are n good agreement wth the expermental data. At s/s0 0.5, a change n the velocty gradent nduced the begnnng of the lamnar-turbulent transton of the calculated flow. The transton produced the strong ncrease of the heat transfer coeffcent. In the experment, the lamnar-turbulent transton appends more downstream and the ncrease of H s delayed. The k-ε model s unable to smulate precsely the transton poston. When the transton s completed, H decreases slowly up to the tralng edge. Along ths regon, the heat transfer level s well predcted by the computaton. On the pressure sde (s/s0<0), the acceleraton of the lamnar boundary layer nduced the decrease of H. At s/s0= -0.1, a small separaton bubble occurs and ntates the lamnar-turbulent transton. These two phenomena produce the ncrease of the heat transfer level. Qualtatvely, the calculaton predcts evolutons n agreement wth the expermental ones. But the transton s nduced too late by the smulaton. Downstream, the heat transfer coeffcent ncreases up to the tralng edge. Ths ncrease s lmted by the effect of the acceleraton n ths regon. The computaton seems to be unable to account ths effect and overestmates the H level.

5 (SYB) 3-4 Concludng remarks A two-equaton model for the turbulent thermal feld has been mplemented, n addton of a k-ε model for the dynamc feld. These four-equaton model has been developed n a Naver-Stokes code. Ths approach s more unversal and avods to ntroduce the assumpton of a constant turbulent Prandtl number whch s not vald when strong temperature gradents exst. The model s valdated for a channel flow. The results are very satsfactory. The heat fluxes are well predcted. A turbne cascade flow s smulated. The physcal phenomena are reproduced by the model. Although, the poston of the lamnar-turbulent transton s not correctly predcted by the code and nduced some dscrepances n the heat transfer level. However, the capablty of solvng ths sophstcated closure model n a full Naver-Stokes code, for complex turbne flows, s demonstrated. References [1] N. Marcnak, Modélsaton aérothermque à bas nombre de Reynolds des écoulements en turbnes, Thèse de Doctorat, Ecole Centrale de Lyon, [] A.M. Vullot, V.Coualler, N. Lams, 3d turbomachnery Euler and Naver-Stokes calculaton wth multdoman cell-centered approach, AIAA paper (1993). [3] H. Hattor, Y. Nagano, M. Tagawa, Analyss of turbulent heat transfer under varous thermal condtons wth two-equaton models, n : Engneerng Turbulence Modellng and Experments, Elveser publshers, 1993, pp [4] Y. Nagano, C. Km, A two-equaton model for heat transport n wall turbulent shear flows, Journal of Heat Transfer 110 (1988). [5] T.P. Sommer, R.M.C. So, Y.G. La, A near-wall two-equaton model for turbulent heat fluxes, Int. Journal of Heat and Mass Transfer 35 (199). [6] M.S. Youssef, Y. Nagano, M. Tagawa, A two-equaton model for predctng turbulent thermal felds under arbtrary wall thermal condtons, Int. Journal of Heat and Mass Transfer 35 (199). [7] M.W. Morkovn, Effects of compressblty on turbulent flow, The mechancs of turbulence, CNRS, Pars, 196. [8] N. Kasag, Y. Tomta, A. Kuroda, Drect numercal smulaton of passve scalar feld n a turbulent channel flow, Journal of Heat Transfer 114 (199). [9] T. Arts, A. Bourgugnon, Thermal effects of a coolant flm along the sucton sde of a hgh pressure nozzle gude vane, AGARD CP 57, Heat transfer and coolng n gas turbne, 199. [10] T. Arts, I. Lapdus, Behavour of a coolant flm wth two rows of holes along the pressure sde of a hgh-pressure nozzle gude vane, Journal of Turbomachnery, vol. 11, 51-51, 1990.

6 (SYB) 3-5 Fgures Fg. 1 : Dmensonless velocty profle U + Fg. : Dmensonless temperature profle T + Fg.3 : Dmensonless Reynolds stress uv + Fg.4 : Dmensonless turbulent heat flux v + H [W/m.K] 1800 Data 1600 Model s/s0 Fg. 5 : Tme scales rato R=τ /τ Fg. 6 : Heat transfer coeffcent H

7 Ths page has been delberately left blank Page ntentonnellement blanche

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