SIMULATION OF TURBULENT FLOW AND HEAT TRANSFER OVER A BACKWARD-FACING STEP WITH RIBS TURBULATORS

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1 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp SIMULATION OF TURBULENT FLOW AND HEAT TRANSFER OVER A BACKWARD-FACING STEP WITH RIBS TURBULATORS b Khdheer S. MUSHATET Mechanical Engineering Department, College of Engineering, Thi-qar Uniersit, Nassiria, Iraq Original scientific paper UDC: :536.4 DOI: 10.98/TSCI M Simlation is presented for a backward-facing step flow and heat transfer inside a channel with ribs trblators. The problem was inestigated for Renolds nmbers p to The effect of a step height, the nmber of ribs and the rib thickness on the flow and thermal field were inestigated. The compted reslts are presented as streamlines conters, elocit ectors, and graphs of Nsselt nmber and trblent kinetic energ ariation. A control olme method emploing a staggered grid techniqes was imposed to discretize the goerning continit, fll Naier-Stockes and energ eqations. A compter program sing a SIMPLE algorithm was deeloped to handle the considered problem. The effect of trblence was modeled b sing a k-є model with its wall fnction formlas. The obtained reslts show that the strength and size of the recirclation zones behind the step are increased with the increase of contraction ratio (i. e. with the increase of a step height). The size of recirclation regions and the reattachment length after the ribs are decreased with increasing of the contraction ratio. Also the reslts show that the Renolds nmber and contraction ratio hae a significant effect on the ariation of trblent kinetic energ and Nsselt nmber. Kewords: backward facing, ribs, trblent dct flow Introdction A backward-facing step channel flow is considered an interesting topic for man researchers since it incldes the phenomena of separation and reattachment. Ribs trblators are widel sed in some engineering applications sch as serpentine cooling air channel for the internal cooling of the gas trbine, heat echanges and cooling of electronic deices. In some applications, the control on the size and strength of the separation zone is needed to get the desired heat transfer. Howeer the geometr is simple bt it is still one of the problems that hae a comple flow field. Ths etra work inoling std of some changes on the considered geometr is needed to get a better nderstanding. In relating channels with a backward facing, man inestigations has been reiewed. Lio et al. [1] performed a nmerical inestigation oer a backward-facing. The emphasis in this std is gien to the effect of abrpt epansion entrance on the local heat transfer characteristics. The obtained reslts show that the separation jst after the step has a significant effect on heat transfer especiall in the *nathors; khdheer004@ahoo.com

2 46 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp entrance region. Wang et al. [] sed a large edd method along with Lagrangian approach to analze the trblent flow oer a backward-facing step. The std showed that the particles follow a specified path when the orticit of a gas phase is small. The effect of a step height on the separated flow for a conectie flow adjacent to a backward-facing step was inestigated nmericall b Nie et al. [3], Thangam et al. [4]. Chen et al. [5] simlated the three dimensional laminar forced conection flow adjacent to inclined backward-facing step in a rectanglar dct. The effect of a step inclination angle on the flow and heat transfer distribtion was stdied. An eperimental std to isalize the trblent separated flow and to measre the wall pressre oer a backward-facing step was performed b Feng et al. [6]. The std show that the separation bbble and reattachment zone and the negatie peak of the time-aring wall pressre was in phase with passage of the local large scale ertical strctre. Concerning the channel flow with ribs, Han [7], Ra et al. [8], Web et al. [9], Lio et al. [10], and Hane et al. [11] inestigated the trblent flow in channels roghened with ribs. The aim of inestigations was to predict the thermal field and friction factor. Fie low trblence models was sed b Tsai et al. [1] to simlate the fll deeloped trblent flow in a smmetric-ribbed channel. The reslts was alidated with the work of Lio et al. [10]. Lacoides et al. [13] stdied the flow and heat transfer in a rotating U-bend with inclined ribs. The std was performed for Renolds nmber p to while the ribs angle of inclination was fied b 45. In this paper, an attempt is made to incorporate the ribs trblators (aliened in normal direction to the bottom wall of the channel) to the backward-facing flow. As shown in fig. 1, the bottom wall of the channel is hot (T h = 50 C) while the inlet channel flow was cold (T c = 5 C). The combined problem was inestigated for different parameters sch as the epansion ratio, the nmber of ribs, the rib thickness, and Renolds nmber. To the knowledge of the researcher, there is no std docmented on this flow sitation p to date. So the objectie of the present work is to show how the ribs trblators can affect the conentional featres of the trblent flow and heat transfer of the backward-facing step in confined flows. Figre 1. Schematic diagram of the considered problem, H = 0.05 m, L = 0.4 m, 1 = m, H/w = 11, P = 0.1

3 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp Model description The trblent flow and heat transfer throgh a channel backward-facing step with rib trblators are described b fll Naier-Stockes, energ and continit eqations. The working flid is an air with constant properties. The Bossinesq approimation is sed. Ths, the mentioned goerning eqations are listed as: T p p 0 (1) eff eff eff () T eff eff eff (3) T T eff eff (4) m eff = m + m t (5) t eff, T (6) Pr where m eff is the combined laminar and trblent iscosit and G eff is an effectie echange coefficient. Trblence model The standard trblence k-e model proposed b Lander et al. [14] is adopted here to handle the effect of trblence in the flow. This model incldes two transport eqations, one for trblent kinetic energ and the other for the rate of dissipation of trblent kinetic energ: k k k k eff, k eff, G Pr (7) where eff, eff, C1 G C k k t G (9) (8) eff, k e k, eff, e (10) the edd iscosit is obtained b the formla: k C (11) t

4 48 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp and the model coefficients are: σ k, σ e, C 1e, C e, and C µ is 1.0, 1.3, 1.44, 1.9, and 0.09, respectiel. The flow parameters at inlet are described as: in. 05 in k 0 U, T in = T c = 5 C k h 1.5 in in, Re in l = (1) U in h where k in, U in, and T in, are the trblent kinetic energ, elocit, and temperatre at a channel inlet, respectiel. At the walls, no slip conditions are imposed; = = 0, k = 0, e/ = 0, T w = T h = 50 C. To treat the large steep gradient near the walls of the channel and step, wall fnction laws sed b Versteege [15] is adopted. The local N along the bottom wall is epressed as N = q/, at = 0. Zero gradients are imposed on the channel eit for considered ariables. Nmerical method In this work, the nmerical comptations are performed on non-niform staggered grid sstem. A finite olme method described b Versteege [15] is considered to integrate the eqations from (1) to (5): ( )d di( grad )d S d (13) c c c This gies a sstem of discretization eqations which means that the sstem of fll elliptic partial differential eqations is transformed in a sstem of algebraic eqations. Then the soltion of these transformed eqations is done b semi implicit line b line Gass elimination scheme. An elliptic finite olme compter code is deeloped to obtain the reslts of the nmerical procedre throgh sing pressre-elocit copling (SIMPLE algorithm) [15]. This code is based on hbrid scheme. Becase of this strong copling and non-linearit inherent in these eqations, relaation factors are needed to ensre conergence. The relaation factors sed for elocit components, pressre, temperatre, and trblence qantities are 0.5, 0.8, 0.7, and 0.7, respectiel. Howeer these relaation factors hae been adjsted for each case stdied to accelerate the conergence criterion defined as the relatie deference of eer dependent ariable between iteration steps, ma f k (i,j) f k 1 (i,j) To ensre that the trblent flid flow soltions are not significantl effected b the mesh, the nmerical simlations are eamined nder different grid sizes ranging from 6 8 ntil 8 5 control olmes. An additional increase in grid points on 6 8 does not significantl effect on the reslts. Reslts and discssion Simlation of the backward-facing step flow with rib trblators has been performed. The problem is inestigated for different ales of a step height (contraction ratio SR) besides to the nmber of ribs for Renolds nmber p to So the present reslts are smmarized as follows.

5 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp Figres and 3 show the flow field distribtion for different ales of contraction ratios and mltiple rib trblators. It can be seen that the bondar laer is separated jst after the step forming a recirclating zone. The etension of this zone and reattachment are inhibited b the presence of the rib. So it can be emphasized that this zone is distorted de to the presence of the rib. The size and strength of this recirclation zone is increased with the increase of the step height as shown in (a) for the both mentioned figres and ice ersa as shown in (b) and (c). After that, the flow is accelerated oer the first rib along with the incoming main flow forming another recirclating zone. Also the reattachment is inhibited b the second rib and so on for the third rib. As a reslt the reattachment length is effected. It is clear that the size of the recirclation zone behind the first rib is larger compared with the other ribs. This will enhance the rate of heat transfer. Figre. Compted elocit ectors for ribs and different ales of contraction ratios, Re = 16000, H/w = 11 Figre 3. Compted elocit ectors for 3 ribs and different ales of contraction ratios, Re = 16000, H/w = 11

6 50 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp Figre 4. Stream lines distribtion for ribs and different ales of contraction ratios, Re = 16000, H/w = 11 Figre 5. Stream lines distribtion for 3 ribs and different ales of contraction ratios, Re = 16000, H/w = 11 The stream lines distribtion for the stdied parameters is depicted in figs. 4 and 5. As can be mentioned in preios figres, the flow separates down stream the step and ribs forming a recirclation zone. The zone behind each rib works as a sdden epansion. The bondar laer attaches the wall of the channel at some distance down stream of the ribs. The presence of ribs will accelerate the flow and the recirclation zone seems to be larger at the first rib. The presence of ribs will promote miing and enhance the rate of heat transfer as shown in fig. 9.

7 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp The distribtion of aial elocit for different ales of Re is depicted in fig. 6. As the Figre shows, the maimm ales of the aial elocities are significantl increase with the increase of Re. This indicate that the strength of recirclation regions are increased with the increase of Re. The increase in Re, will increase the inertia force in the icinit of the ribs creating a large re-circlating region downstream of the ribs. Conseqentl the rate of heat transfer will be enhanced as shown in fig. 10. Figre 6. Effect of Re on distribtion of aial elocit for 3 ribs, SR = 0.5, H/w = 11 (color image see on or web site) Figre 7 shows the effect of Re on the ariation of trblent kinetic energ for three ribs and SR = 0.5. For the bottom wall at (a), the trblent kinetic energ is significantl increased with the increase of Re and this increase is along the length of the channel. The maimm and minimm ales of the trblent kinetic energ (semi wa ariation) are de to the presence of rib trblators. This distribtion is changed when Re At the pper wall, (b), in which there is no ribs, the distribtion of trblent kinetic energ is some different where the maimm elocities oer the ribs and the minimm elocities after the ribs affects the mentioned distribtion. Howeer the trblent kinetic energ is increased with the increase of Re. At the middle of the channel, the trblent kinetic energ is also increased with the increase of Re. Howeer at Re = 8000, the distribtion is similar to the bottom wall. The maimm ales are larger than that of the bottom and pper wall becase at the center of the channel, the maimm ales of aial elocities are fond and the stress is high, conseqentl the kinetic energ is increased. The effect of SR on the distribtion of trblent kinetic energ at the lower and pper wall of the channel is demonstrated at fig. 8. As the figre shows, the ales of the trblent kinetic energ are increased with the decrease of contraction. This ma be de to

8 5 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp Figre 7. Effect of Re on ariation of trblent kinetic energ [m s ] for 3 ribs, SR = 0.5, Re = 16000, H/w = 11, P = 0.1 Figre 8. Effect of Re on ariation of trblent kinetic energ [m s ] for 3 ribs, SR = 0.5, Re = 16000, H/w = 11, P = 0.1 In fig. 9, the ariation of local Nsselt nmber is seen. The local Nsselt nmber is increased with the increase of SR for and 3 ribs at the region near the step. Also the Nsselt nmber is enhanced with the decrease of SR for the ribbed region as shown in (a). The presence of ribs will promote the conection heat transfer de to increase of the trblence miing. Howeer the Nsselt nmber is decreased at the second rib. This will change when the nmber of ribs are increased to three as shown in (b). The trblent kinetic energ is enhanced with increasing of Re as shown in fig. 10.

9 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp Figre 9. Effect of contraction ratio on ariation of N for Re = 16000, H/w = 11, P = 0.1 Figre 10. Effect of Re on ariation of local N for 3 ribs, SR = 0.5, H/w = 11 The thermal performance of a backward- -facing withot ribs is compared with or problem as in fig. 11. The ales of N are enhanced at the recirclation jst after the edge of the step after that N is linearl decreased. While with ribs, the rate of heat transfer is significantl enhanced and the ales of N are shifted and there is no difference between two and three ribs. Howeer at the third rib, the ale of N is increased. The ariation of N for different ales of SR is fond in fig. 1 for ribs. The ales of N are increased with the decrease Figre 11. Comparison of ariation of local N for the stdied cases, SR = 0.5, Re = 16000, H/w = 11 Figre 1. Effect of contraction ratio on ariation of N for ribs, H/w = 11, Re = 16000

10 54 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp of SR for the region containing the ribs and ice ersa. The case arises to the increase of trblence and that leads to incrase the rate of heat transfer. The effect of increasing of the rib width on heat transfer is shown in fig. 13 for different ales of SR. It is eident that the rate of heat transfer is enhanced after the second rib and ice ersa occrs at the step region. The alidation of the present code is eamined throgh the comparison of the present reslts with aailable pblished eperimental data as depicted in fig. 14. The comparison indicated an acceptable agreement. Conclsions Figre 13. Effect of the rib width on N ariation for Re Figre 14. Comparison between the present simlation and pblished eperimental data (pblished reslts of Lio et al. [10], H/B = 1, Re = 6000) The present std was performed for different ales of Re, ribs trblators, and step height. A finite olme based on the staggered method is sed to simlate the backward- -facing step flow with ribs trblators. Ths, the following conclding remarks can be reported. Adding of the ribs trblators to the backward-facing step flow is significantl enhanced the rate of heat transfer for the all stdied Renolds nmbers. The strength and size of recirclation zones at the step and after the ribs is increased with the increase of Re. Howeer this increase is larger after the first rib. The strength and size of recirclation zones are increased with the increase of SR (0.5 SR 0.5). The trblent kinetic energ near the walls is decreased with the increase of SR. The nmerical analsis shows that the heat transfer is increased with the increase of Re and SR and when sing more ribs. The local Nsselt nmber decreases as the rib width increases.

11 THERMAL SCIENCE, Year 011, Vol. 15, No. 1, pp Nomenclatre G generation term, [kgms 3 ] H height of the channel, [m] k trblent kinetic energ,[ m s ] L length of the channel, [m] N local Nsselt nmber, [ ] P pitch, [m] p pressre, [Nm ] Pr Prandtl nmber, [ ] Re Renolds nmber, [ ] s step height, [m] SR contraction ratio (= S/H), [ ] S f sorce term, [ ] T c cold wall temperatre, [ C] hot wall temperatre, [ C] T h, aial and normal elocit, [ms ], Cartesian co-ordinates, [m] Greak letters G eff effectie echange coefficient,[kgms] e trblence dissipation rate, [m s 3 ] m dnamic iscosit, [Nsm ] m t trblent iscosit, [Nsm ] r air densit, [Kgm 3 ] s k, s e trblent Schmidt nmbers, [ ] q dimensionless temperatre [=(T T c )/(T h T c ), [ ] f constant propert, [ ] References [1] Lio, T., Hwang, J., Deeloping Heat Transfer and Friction in a Ribbed Rectanglar Dct with Flow Separation at Inlet, ASME J. Heat Transfer, 114 (199), 3, pp [] Wang, B., Zhang, H. Q., Wang, X. L., Large Edd Simlation of Particle Response to Trblence Along its Trajector in a Backward-Facing Step Trblent Flow, Int. J. Heat and Mass Transfer, 49 (006), 1-, pp [3] Nie, J. H., Armal, B. F., Three Dimensional Conectie Flow Adjacent to a Backward Facing Step- Effects of Step Height, Int. J. Heat Mass Transfer, 45 (00), 1, pp [4] Thangam, S., Knight, D., Effect of Step Height on the Separated Flow Past a Backward Facing Step, Phs. Flids, 1 (1989), 3, pp [5] Chen, Y. T., et al., Three Dimensional Conectie Flow Adjacent to Inclined Backward Facing Step, Int. J. Heat and Mass Transfer, 49 (006), 5-6, pp [6] Feng, K. E., Li, Y.-Z., Chen, H.-P., Simltaneos Flow Visalization and Wall Pressre Measrement of the Trblent Separated and Reattachment Flow oer a Backward Facing Step, Jornal of Hdrodnamics, 19 (007),, pp [7] Han, J. C, Heat Transfer and Friction Characteristics in Rectanglar Channels with Rib Trblators, ASME Jornal of Heat Transfer, 110 (1988),, pp [8] Ra, G., et al.,the Effect of Periodic Ribs on the Local Aerodnamics and Heat Transfer Performance of a Straight Cooling Channel, ASME Jornal of Trbo machiner, 10 (1988),, pp [9] Webb, R. L, Eckert, E. R. G., Goldsten, R. J., Heat Transfer and Friction in Tbes with Repeated Rib Roghness, International Jornal of Heat and Mass Transfer, 14 (1984), 4, pp [10] Lio, T. M., Hwang, G. G., Chen, S. H., Simlation and Measrements of Enhanced Trblent Heat Transfer in Channels with Periodic Ribs on One Principal Wall, International Jornal of Heat Mass Transfer, 36 (1997),, pp [11] Han, J. C., Park, J. S., Deeloping Heat Transfer throgh Rectanglar Channels with Rib Trblators, International Jornal of Heat mass Transfer, 31 (1988), 1, pp [1] Tsai, W. B, Lin, W. W, Chieng, C. C., Comptation of Enhanced Trblent Heat Transfer in a Channel with Periodic Ribs, Int. J. of Nmerical Methods for Heat and Flid Flow, 10 (000), 1, pp [13] Lacoides, H., et al., Flow and Heat Transfer in a Rotating U-Bend with 45 Ribs, Int. J. Heat and Flid Flow, (001), 3, pp [14] Jones, W. P., Lnder, B. E., The Prediction of Laminarization with a Two Eqation Model of Trblence, International Jornal on Heat and Mass Transfer, 15 (197),, pp [15] Versteege, H. K., Malalasekera, W., An Introdction of Comptational Flid Dnamics, Hemisphere Pblishing Corporation, Washington, D. C., USA, 1995 Paper sbmitted: September 6, 009 Paper reised: Ma 4, 010 Paper accepted: Febrar 5, 010