NUMERICAL ANALYSIS OF TURBULENT FLOW WITH HEAT TRANSFER IN A SQUARE DUCT WITH 45 DEGREE RIBS

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1 Proceedngs of IONE19 19th Internatonal onference on Nuclear Engneerng May 16-19, 011, hba, Japan IONE NUMERIAL ANALYSIS OF TURBULENT FLOW WITH HEAT TRANSFER IN A SQUARE DUT WITH 45 DEGREE RIBS Yura Okagak Graduate School of Engneerng, Utsunomya Unversty 7-1- Yoto, Utsunomya, Tochg, Japan Phone: okagak.yura@jaea.go.jp Htosh Sugyama Graduate School of Engneerng, Utsunomya Unversty 7-1- Yoto, Utsunomya, Tochg, Japan Phone: sugyama@cc.utsunomya-u.ac.jp Naoto Kato Graduate School of Engneerng, Utsunomya Unversty 7-1- Yoto, Utsunomya, Tochg, Japan Phone: note@cc.utsunomya-u.ac.jp Atsuhko Terada Japan Atomc Energy Agency 400, Narta-cho, Oara-mach, Hgashbarak-gun, Ibarak, Japan Phone: terada.atsuhko@jaea.go.jp Ryutaro Hno Japan Atomc Energy Agency 400, Narta-cho, Oara-mach, Hgashbarak-gun, Ibarak, Japan Phone: hno.ryutaro@jaea.go.jp Keywords: omputatonal Flud Dynamcs, Heat Transfer, Algebrac Reynolds Stress Model, Turbulent Prandtl Number, Algebrac Turbulent Heat Flux Model, Heat Exchanger, HTGR ABSTRAT Turbulent flow and heat transfer characterstcs n a square duct of 100 mm heght wth 45 degree square rbs of 10 mm heght was analyzed numercally by usng algebrac Reynolds stress model ncludng the fxed turbulent Prandtl number and algebrac turbulent heat flux models. In ths research, analytcal results were compared wth the expermental and predcted data reported by Bonhoff et al. [1], whch were measured and analyzed turbulent flow felds at Reynolds number based on bulk velocty and duct heght by means of a PIV system and a Reynolds stress model. Analytcal results obtaned by algebrac Reynolds stress model showed a good agreement wth the expermental data of streamwse and spanwse mean velocty profles at the same Reynolds number of the experment. As for secondary flow n the duct, analytcal results showed relatvely good agreement wth the expermental data. Moreover, temperature dstrbutons were obtaned by usng the fxed turbulent Prandtl number and algebrac turbulent heat flux models. Temperature profles of algebrac turbulent heat flux model were not smlar to velocty profles. onsderng algebrac turbulent heat flux model has potentalty to reproduce ansotropc temperature, these results suggested that correlaton between momentum and heat transfer was not recognzed n ths turbulent feld. As a result of ths study, t was verfed that the presented method was able to predct turbulent flow n duct wth rbs through the comparson of calculated results wth the expermental data. 1. INTRODUTION hannels wth perodc rbs have been used n many engneerng felds due to ts hgh heat transfer performance, and are expected for applcaton of process heat exchange n nuclear heat applcaton systems such as for hydrogen producton usng a hgh-temperature gas-cooled reactor (HTGR). To develop a desgn code of heat exchangers wth rbs for HTGR heat applcaton systems, as a frst step, the one of effectve methods of heat transfer enhancement s to arrange perodcally rbs on duct wall []. Ths method produces actvely the ansotropc turbulent flow and complcated flow near the heat transfer surface. When the ansotropc turbulence s produced, heat transfer performance s strongly affected by pattern changes of secondary flows havng an effect on temperature felds. opyrght 011 by JSME

2 Accurate predcton of the heat transfer performance depends on turbulent models. To mprove predcton accuracy, many nvestgatons have reported analytcal studes on heat transfer enhancement by usng perodcally arranged rbs n ducts such as straght and U-shaped ducts, bend etc. Analytcal studes are categorzed as follows: heat transfer enhancement by the secondary flow of the second knd derved from ansotropc turbulent flow and that by the secondary flow of the frst knd due to the pressure gradent force. In the latter case, whle hghly heat transfer enhancement s expected, pressure drop ncreases wth heat transfer coeffcents. From the vew pont of compatble possblty wth hgh heat transfer coeffcents and rather low pressure drops, nclned rbs have been attracted. Han et al. [3], who determned the effect of rb angle of attack n a square duct wth two opposte rb-roughened walls, demonstrated that the best thermal performance s acheved at a 30 and 45 degree angle to the flow for both the rb ptch to heght ratos. He showed that the thermal performance was about 10-0 percent hgher than the rb wth a 90 degree angle to the flow, and the pumpng power requrement for the angled rb was about 0-50 percent lower than the transverse rb. It takes t as gven that a better performance than the rbs wth a 90 degree angle to the flow whch currently beng used n the modern gas turbne nternal coolng passages. Iacovdes et al. [4] reported results of the measurement and numercal analyss n a square U-bend wth nclned rbs of 45 degree. The measurement was performed by usng laser-doppler anemometry (LDA) n flow feld, and the lqud crystal technque n temperature feld. The numercal analyss was carred out by usng hgh and low-reynolds (Re) two equaton models. Murata and Mochzuk [5] have examned numercally the effect of the angled rb on the flow and heat transfer by performng the large eddy smulaton (LES) from the ponts of separaton and reattachment caused by the angled rbs. These effects were clarfed by makng a comparson between the lamnar and turbulent cases and also between angled 60 degree and transverse 90 degree rb cases. Jang et al. [6] presented calculated results for the turbulent flow and heat transfer for a two-pass square channel wth and wthout 60 degree angled parallel rbs. They used the fnte method, whch solves the Reynolds-averaged Naver-Stokes equaton n conjuncton wth a near-wall second-order Reynolds stress closure model, and a two-layer k- sotropc eddy vscosty model. omparng the second-moment and two-layer calculatons wth the expermental data demonstrated that angled rb turbulators and the 180 degree sharp turn of the channel produced strong non-sotropc turbulence, whch sgnfcantly affected the flow feld and heat transfer coeffcents. Bonhoff et al. [1] have performed expermental and numercal analyss of turbulent flow n a square duct wth 45 degree rbs, and compared analytcal results wth the expermental data. Standard k- turbulent model wth wall functon method, a dfferental Reynolds stress model and a standard k- turbulence model wth a two layer wall model has been adopted n predcton. Expermental data have measured by usng partcle mage velocmetry (PIV) method. To mprove predcton accuracy of heat transfer coeffcents dependng on turbulent models, avalablty of algebrac Reynolds stress model was verfed wth the expermental and analytcal results by Bonhoff et al. [1]. Ths paper presents analytcal results of flow and heat transfer characterstcs n a square duct of 100 mm heght wth 45 degree square rbs of 10 mm heght.. NOMENLATURE a : thermal dffusvty a t : turbulent thermal dffusvty D : sde length of square cross-secton h : rb heght P : wall statc pressure p : fluctuatng pressure T : mean temperature T' : temperature fluctuaton T w : wall temperature T n : mean temperature of nternal cross-secton u u j : Reynolds stresses u T' : turbulent heat flux correlaton U b : cross-secton mean velocty of square duct U : magntude of velocty vectors U : th mean velocty component X : th artesan coordnate : Kronecker delta j Plane A X X 3 Fg. 1 Dagram of square duct wth 45 degree rbs and defnton of coordnate system : knematcs vscosty : densty Plane B X 1 45 : ensemble averaged value S 0h 14.1h 3. NUMERIAL METHOD 3.1 Numercal Subject and Defnton of oordnate System The duct confguraton and coordnate system used n the calculaton are shown n Fg. 1. The operatng condtons correspond to those employed by Bonhoff et al. [1]. The rbs nclned 45 degree to streamwse velocty were located perodcally on the top and bottom wall of the square duct. The duct wth rbs cross-secton of 10 mm 10 mm had a rectangular cross-secton of 100 mm 100 mm. The Reynolds number was based on the hydraulc dameter and the bulk velocty. The measurng Plane A and Plane B are h D D opyrght 011 by JSME

3 Table 1 Modelng of the pressure-stran correlaton term j,1 j,1 1 u u j k j k 3 8 U U j ( Pj Pk j ) k 11 3 X j X j, j, 8 ( D j Pk j ) 11 3 [ ] j j w L L f f * ' * ' X w X w * ' L f X w U U U U P u u u u D u u u u j k k j k j k j k j k X X X X k k Table 3/4 3/ U k L Pk u ku l f X l x w X w onstants of the pressure-stran correlaton term * 1 * * 1 k ' Table 3 Modelng of the pressure-temperature gradent term u u j T,1 1T u T 1T j u jt k k k 3 U U m T, T u m T T u m T X m X T, w * L 1 T 1 T 1 1 T, w f X w * L 1 T 1 T 1 1 T, w f X w * L T T 1 T w f X w * L T T 1 T, w f X w Table 4 * 1T onstants of the pressure-temperature gradent term * 1 T * * T T 1 Tw, Tw, perpendcular to the streamwse drecton and the rbs, respectvely. Bonhoff et al. [1] had measured fully developed turbulent flow by use of a PIV system. The coordnate system s defned as shown n Fg. 1. Spanwse drecton s along the X 1 axs, the vertcal drecton s along the X and the streamwse drecton s along X 3. The orgn of the rectangular coordnate system s set at rght-hand bottom of channel. 3. Governng Transport Equaton 3..1 feld model The Reynolds averaged Naver-Stokes equaton (RANS) s expressed n the followng form: DU Dt = 1 P + ν U + U j u ρ X X j X j X u j (1) The Reynolds stresses that appear n the transport equaton must be solved n order to obtan the velocty felds completely. In ths calculaton, we have adopted the transport equaton of Reynolds stresses n order to accurately predct ansotropc turbulence. The transport equaton of Reynolds stresses s dsplayed exactly n the followng form: Du u j = u Dt u U j k +u X j u U k + p k X k ρ u + u j X j X u-ν X u j u u u j k + p k X k ρ δ jku +δ k u j ν u u j () X k X k It s mpossble to numercally solve the above equaton drectly, so t s necessary to rewrte several of the terms of the Reynolds stress equaton by ntroducng the concept of the turbulent model. Moreover, wth respect to the numercal analyss, the convecton term of the left-hand sde and the dffuson term of rght-hand sde of Eq. () make t dffcult to obtan a numercal soluton because these terms requre teratve calculatons n order to obtan stable results. In the present study, these terms are modeled by adoptng Rod s [7] approxmaton. As a result of ths approxmaton, these two terms are transformed nto algebrac form from dfferencng form, and transport equaton contrbutes to decrease n computatonal load. However, t s also true that nterdependence of physcal quantty dsappears compared wth dfferental equaton. Pressure-stan correlaton term plays an mportant role to dstrbute turbulent energy. The modelng process of pressure-stan correlaton term s descrbed n the other report [8] n detal. Tables 1 and show the modelng of pressure-stran term and model constants, respectvely. 3.. Temperature feld model It s ndspensable to predct turbulent heat fluxes n order to get precsely temperature profles. From ths pont of vew, the transport equaton of the turbulent heat flux s adopted to realze ansotropc temperature. The transport equaton of turbulent heat flux s expressed exactly as followng form: Du T' = u Dt u T k +u X k T' U k X k u+ X u k T' P k ρ T' δk ν u T ' T' aut X k X k + P T' T' u ν+a (3) ρ X X k X k where the terms on the rght hand sde represent the producton of turbulent heat flux from the mean flow, the dffuson of turbulent heat flux, the pressure-temperature gradent effect whch leads to nhomogenety among the ndvdual heat flux components, and ts dsspaton respectvely. The convecton and dffuson terms were modeled n a same manner to Rod s [7] approxmaton for opyrght 011 by JSME

4 Pre. by Bonhoff et al U 1/Ub (a) Streamwse velocty Pre. by Bonhoff et al. X -X 3 Plane X 1 -X 3 Plane X 1 -X XPlane 1 -X Plane Fg. Grds layouts for three dmensonal vew and cross sectonal planes U /Ub (b) Spanwse velocty Pre. by Bonhoff et al. Reynolds stress transport equaton. By ths approxmaton, the computatonal tme can be saved. The pressuretemperature gradent term are composed of T,1 and T,, whch are called as slow and rapd return, respectvely. The pressure-temperature gradent term s modeled as shown n Table 3. The model constants are lsted n Table 4. Pressure-temperature gradent term T,1 and T, have been made use of concept presented by Lumley s [9] model and Launder s [10] model, respectvely. Wall effect term T,w s modeled by ntroducng the functon f. The functon f has 1 near wall, and decreases as movng away from the wall. The dervaton and the valdty of the presented model have been descrbed n a separate report [11] n detal. In addton to algebrac turbulent heat flux model, the smplest method, assumng that the turbulent Prandtl number s fxed, s adopted n ths analyss. In ths smplest model, turbulent heat flux s expressed as follows: T =a u T' t (4) X In ths analyss, the turbulent Prandtl number s assumed to be. The calculated results of ths method are compared wth that of algebrac turbulent heat flux model to make model dfference clear. 3.3 Numercal Analyss In the presented numercal analyss, the perodc boundary condton s appled to straght duct wth two perodcal spans. Therefore, excepton for pressure, the values of nlet are set to be the same of outlet and, as for the pressure, the pressure gradent of nlet s equal to that of outlet. Fgure shows grd layouts. The number of computatonal grd ponts n the total - U 3/Ub (c) Vertcal velocty Pre. by Bonhoff et al U /Ub (d) Velocty magntude Fg. 3 omparson of velocty profles cross secton s 41 31, and 10 grd ponts are set along the streamwse drecton. Therefore, the total number of computatonal grds s 19,64. The grds located near the wall are fne because the physcal parameters changes rapdly near the wall. Snce the presented turbulent model can be classfed as a hgh-reynolds-number turbulent model, the wall functon method s used n the velocty feld and the wall temperature s set to be constant as a boundary condton. The opyrght 011 by JSME

5 buoyancy term s deleted from the momentum equaton assumng that the buoyancy effect to the velocty feld s neglgble small. The governng equatons were dscretzed by the dfferencng scheme, and QUIK (thrd-order upwnd dfferencng scheme) was used for the convecton term. Boundary-ftted coordnate system was used as a coordnate transformaton method. In unsteady calculaton wth the explct method, tme step s determned to satsfy ourant number s kept under RESULTS AND DISUSSION 4.1 Mean Velocty Feld Mean velocty profles are compared wth the expermental data measured by Bonhoff et al. [1] as shown n Fg. 3. It s noted that the calculated results of Reynolds stress model are dsplayed smultaneously wth the presented result. In Fg. 3, symbol (a), (b) and (c) show the velocty profles of streamwse, spanwse, and vertcal drecton, respectvely. Symbol (d) shows the compared results of the three dmensonal velocty vector magntude profles. The compared locaton s on lne along X 3 axs at X /D=0.5 n Plane A as shown n Fg. 1. In (a), the expermental dstrbuton of the streamwse velocty exhbts the maxmum value near the rb located the upper wall and small value n the central regon of the channel. Besdes, velocty profle ndcates a peak value wth approachng toward the lower wall. The calculated results can reproduce such characterstc phenomena. In (b), the expermental values show a postve value near the upper and lower walls and a negatve value n the central regon of the channel whch means that crculatng flow s generated. The expermental dstrbuton of spanwse velocty rapdly ncreases as approachng near to the upper rb whch also reproduces by algebrac Reynolds stress model, but mnmum value s predcted more accurately by Reynolds stress model compared wth algebrac Reynolds stress model. In (c), the expermental result shows that downstream flow s generated near the upper rb and the lower wall whle upstream rapd velocty s produced n the central regon of the channel. The two models can reproduce these characterstc profles. However, two models fal to predct precsely the expermental profles. In (d), ths profle mples total predcton accuracy of the three drectonal veloctes. omparng the calculated results of the presented model wth that of Bonhoff et al. [1], algebrac Reynolds stress model predcts the expermental value relatvely well. Fgures 4 (a) and (b) show the comparson of secondary flow vectors and spanwse velocty profle n Plane A, respectvely. From the top, the expermental and predcted results by Bonhoff et al. [1] and the presented results are ndcated n the condton of settng vewpont on downstream. In (a), the vector scale s not descrbed n Bonhoff s paper, but t can be estmated from the expermental value of the spanwse velocty n Fg. 3. Its estmated vector scale was shown n ths paper. However, the vector scale of calculated results by Bonhoff s unable to show because there s no descrpton about the vector scale. From secondary flow vectors as shown n (a), t s clear that crculatng flow near upper wall and counterclockwse crculatng flow near lower wall are generated. Addng to ths, X/D by Bonhoff et al. X/D X/D (a) Secondary flow vectors Ub Ub U/Ub - by Bonhoff et al. U/Ub - - t s also ponted out as characterstc feature from the experment that the secondary vector along X 3 /D=0.5 moves wavy n the mddle of cross secton. In (b), the relatvely large value formed near the upper and lower wall n the expermental result. Ths large velocty near upper and lower wall mpnges on the sde wall. As a result of ths mpngement, downward and upward flows are formed along sde wall. These two flows are merged n the mddle of sde wall and ts merged flow moves to the opposte sde wall. The presented result reproduces these features and predcts well the value of - measured n the mddle of cross secton. Fgure 5 shows the compared result of the velocty vectors projected on Plane B. The projected velocty vectors are dsplayed as a brd's-eye fgure whch s vertcally watchng Plane B from upstream. The expermental results show only the velocty vectors around the rb, but the calculated results of Bonhoff et al. [1] show the projected velocty vectors n entre Plane B as well as the present calculated results. However, the velocty vector scale s not ndcated n the fgure because there s no nformaton about vector scale n the paper presented by Bonhoff et al. [1]. In the expermental result, t s ponted out that the flow vectors near the rb located up stream show the nclne upward flow from the rb X/D X/D X/D U/Ub (b) Spanwse velocty dstrbutons Fg. 4 omparson of predcted and measured velocty profles n Plane A opyrght 011 by JSME

6 by Bonhoff et al. 0.5 S/D 0.5 S/D Fg. 5 omparson of predcted and measured vectors n Plane B corner to upper rght and the crculatng flow s recognzed large area behnd the rb. The calculated result by Bonhoff et al. [1] reproduces comparatvely well ths phenomenon. Although the presented calculaton reproduces the nclne upward flow, the sze of the crculatng flow behnd the rb s smaller than the experment. In addton, two models predct well reattachment pont of separaton behnd the rb. Judgng from the comparson results of mean velocty fled, the expermental characterstc features can be reproduced by algebrac Reynolds stress model as well as Reynolds stress model. The dfference between the two models s modelng of the convecton and dffuson terms that expressed n dfferental form or n algebrac form. If two models can reproduce the expermental results, model valdaton depends on the producton and pressure-stran correlaton terms. Snce t s not necessary to model the producton term, the dfference of the pressure-stran correlaton term affects to the numercal soluton. In case of the Reynolds stress model used by Bonhoff et al. [1] whch s commercal numercal code, the pressure-stan correlaton term sn t descrbed n detal n ther paper. In general, the secondary flow of the second knd depends greatly on the pressure-stan correlaton term modelng n ansotropc turbulent flow. It has been already confrmed that the algebrac Reynolds stress model used n the presented analyss can reproduce the secondary flow of the second knd [1]. Fgure 6 shows the comparson results of sx Reynolds stress components formed n cross-secton of Plane A. Reynolds stresses u 1, u, and u3 are normal stresses for streamwse, spanwse, and vertcal, respectvely. Reynolds stresses u, 1 u u, 1 u 3 and u u 3 are shear stress. In turbulence Ub u1 /Ub X/D u /Ub X/D u3 /Ub X/D of square duct wthout rb, the normal stress for streamwse dsplays the hghest value compared wth the other normal stresse. By contrast, calculated results show that all normal stresses ndcate hgh value near the rb and generate actvely turbulence, whch are consdered as a characterstc dstrbuton for turbulent flow n straght duct wth the nclned rbs. On the other hand, about the share stress, n the case of turbulent fled of square duct wthout rb, shear stress u u 3 s small absolute value, but shear stress u u 3 of ths case s large absolute value whch are the same value of other share stresses. These results suggest that momentum transfer was performed actvely n three drectons by settng nclned rb. 4. Mean Temperature Feld Fgures 7 and 8 show the streamwse velocty dstrbuton and the temperature dstrbuton n Plane A, respectvely. Symbol (a), (b) n Fg. 8 are calculated contour maps of the fxed turbulent Prandtl number model and algebrac turbulent heat flux model, respectvely. In the case of the fxed turbulent Prandtl number model, Fg. 8 (a) suggest that there s a correlaton between heat transfer and momentum transfer, judgng from that temperature dstrbuton s smlar to streamwse velocty dstrbuton whch s shown n Fg. 7. By contrast, the contour map n Fg. 8 (b) s not smlar to streamwse velocty dstrbuton. onsderng the algebrac turbulent heat flux opyrght 011 by JSME X/D X/D 0 uu3/ub X/D Fg. 6 omparson of predcted dstrbuton of normal and shear stresses n Plane A u1u/ub u1u3 /Ub 3

7 (T-Tn)/(Tw-Tn) X/D 0.50 (a) The fxed turbulent Prandtl number model model has potentalty to predct ansotropc temperature feld, t s concluded that the correlaton between heat transfer and momentum transfer s not realzed n ths turbulent flow wth nclned rbs. Fgure 9 shows the compared results of dstrbuton for Nusselt number over lower wall. Bonhoff et al. [1] predcted temperature feld by usng results of (a) Reynolds stress model and (b) k- model. The method for analyzng temperature feld sn t descrbed n detal n ther paper. On the other hand, the presented results was predcted temperature feld by (c) the fxed turbulent Prandtl number model and (d) algebrac turbulent heat flux model by usng the velocty feld solved by algebrac Reynolds stress model. In (a), the area of hgh heat transfer s formed n the mddle of bottom wall, whle, n (b), hgh Nusselt number s recognzed just near downstream of the rb. About the phenomenon of (b), Bonhoff et al. [1] ponted out that the locaton of hgh Nusselt number corresponds to the locaton of reattachment pont n Plane B whch feature s related wth the secondary flow. Both results of (c) and (d) show hgh Nusselt number s also formed n the same area of (b). Besdes, It s found n fgure (d) that the second hghest area s generated n the center of bottom wall whch means algebrac turbulent heat flux model predct hgh value of Nuseselt number not only near just downstream behnd rb but also center of bottom wall. 5. ONLUSIONS A numercal analyss has been performed for turbulent flow wth heat transfer n a square duct wth 45 degree rbs by usng the algebrac Reynolds stress, the fxed turbulent U1/Ub X/D Fg. 7 Predcted streamwse velocty dstrbuton n Plane A (T-Tn)/(Tw-Tn) X/D Fg. 8 omparson of predcted dstrbuton of temperature n Plane A (b) Algebrac turbulent heat flux model X3/h X3/h X3/h by Bonhoff et al X 1/h (a) Reynolds stress model by Bonhoff et al X 1/h (b) k- model X 1/h (c) The fxed turbulent Prandtl number model X3/h X 1/h (d) Algebrac turbulent heat flux model Fg. 9 omparson of predcted dstrbuton of Nusselt number Prandtl number, and the algebrac turbulent heat flux models. Analytcal results were compared wth the expermental and predcted data reported by Bonhoff et al. [1] n order to confrm the valdty of the presented models. As results of ths analyss, the followng conclusons are summarzed: (1) The presented method was able to predct three drectonal veloctes except for vertcal velocty, whch tendency was the same as that obtaned wth the Reynolds stress model. Although the agreement wth opyrght 011 by JSME

8 the expermental results were not perfect, characterstc features are realzed by the algebrac Reynolds stress model. () The algebrac Reynolds stress relatvely well reproduces the streamwse flow, the vector pattern and the reattachment pont n Plane B as well as those obtaned wth the Reynolds stress model. The algebrac Reynolds stress model can predct ansotropc turbulent flow n duct wth 45 degree rbs to present sx Reynolds stresses. (3) The temperature contour map obtaned by the algebrac turbulent heat flux model was not smlar to velocty contour map. Snce the algebrac turbulent heat flux model has potentalty to reproduce ansotropc temperature, correlaton between momentum and heat transfer was not organzed n ths knd of turbulent feld. (4) The fxed turbulent Prandtl number and algebrac turbulent heat flux models predcted smlar values of Nusselt number. REFERENES [1] Bonhoff, B., et al., 1999, Expermental Numercal Study of Developed and Heat Transfer n oolant hannels wth 45 degree Rbs, Int. J. Heat and Flud, 0(3), pp [] Sanokawa, K., et al., 1999, Thermal and Flud-dynamc Research and Development for Hgh Temperature Gas ooled Reactor n JAERI, JAERI-Revew, 98-04, pp [n Japanese]. [3] Han, J.., et al., 1985, Heat Transfer Enhancement n hannels wth Turbulence Promoters, the ASME, J. Eng. Gas Turbnes and Power, 107(3), pp [4] Iacovdes, H., et al., 003, and Heat Transfer n Straght oolng Passages wth Inclned Rbs on Opposte Walls: an Expermental and omputatonal Study, Exp. Therm. Flud Sc., 7(3), pp [5] Murata, A., and Mochzuk, S., 001, omparson between Lamnar and Turbulent Heat Transfer n a Statonary Square Duct wth Transverse or Angled Rb Turbulators, Int. J. Heat and Mass Trans., 44(14), pp [6] Jang, Y-J., et al., 001, omputaton of and Heat Transfer n Two-Pass hannels wth 60 deg Rbs, Trans. ASME J. Heat Transfer, 13(3), pp [7] Rod, W., 1976, A new algebrac relaton for calculaton the Reynolds stresses, Z. Angew. Math. Mech., 56, pp [8] Sugyama, H., and Htom, D., 005, Numercal analyss of developng turbulent flow n a 180 bend tube by algebrac Reynolds stress model, Int. J. Numer. Meth. Fluds, 47(1), pp [9] Lumley, J. L., 1975, Introducton n Methods for Turbulent s, Lecture Seres, 76, von Karman Inst., Belgum. [10] Launder, B. E., 1975, Progress n The Modelng of Turbulent Transport, Lecture Seres, 76, von Karman Inst., Belgum. [11] Sugyama, H., et al., 00, alculaton of Turbulent Heat Flux Dstrbutons n a Square Duct wth One Roughened Wall by Means of Algebrac Heat Flux Models, Int. J. Heat and Flud, 3(1), pp [1] Sugyama, H., et al., 006, Numercal Analyss of Developng Turbulent n a Rectangular Duct wth Abrupt hange from Rough to Smooth Wall, Trans. JSAE, 37(), pp [n Japanese]. opyrght 011 by JSME