EFFECTS OF SWIRL IMPARTED AT INLET ON THE RECIRCULATION BUBBLE GENERATED IN THE TURBULENT FLOW IN AN AXI-SYMMETRIC SUDDEN EXPANSION

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1 Poceedings of the Intenational Confeence on Mechanical Engineeing 011 (ICME011) 180 Decembe 011, Dhaka, Bangladesh ICME11FL004 EFFECTS OF SWIL IMPATED AT INLET ON THE ECICULATION BUBBLE GENEATED IN THE TUBULENT FLOW IN AN AXISYMMETIC SUDDEN EXPANSION Snehamoy Majumde, Debajit Saha, Aindam Mandal and Vikam oy Depatment of Mechanical Engineeing, Jadavpu Univesity, Kolkata70003, India. ABSTACT The tubulent fluid flow though an axisymmetic sudden expansion is a geneal phenomenon associated with the expansion flows in the pipe lines, dump combusto chambe etc., whee in geneal the eciculation bubbles ae geneated. In case of combustion chambes swil pomotes extensive mixing of the fuel and ai and theeby inceases the combustion iciency with the augmentations of the stong adial flow. In the pesent study the numeical analysis of the tubulent fluid flow though an axisymmetic sudden expansion with the swil pesent at the inlet has been caied out by using modified tubulence model. The steamline contou and vecto plots have been pesented to illustate the ects of the vaiation of the swil numbe on the size and stength of the eciculation bubble. A coelation has been obtained between the eattachment length and the inlet swil angle to analyze the ects swil. Keywods: Sudden Expansion, eciculation Bubble, eattachment Length, Swil. 1. INTODUCTION Tubulence is the most common state of fluid motion. The tubulent fluids flows though sudden expansion passage have both fundamental scientific inteest and numeous pactical applications. The engineeing applications such as the sepaated flow associated with pipelines, dump combustos, diffeent ducting devices etc. ae examples of the tubulent flow though the sudden expansion. To impove the pefomance of such applications, an accuate desciption of the flow pattens ae equied. The tubulent flow in such devices is vey complicated owing to geneation of eciculation bubble and theeby changing the downsteam flow stuctues. The swiling flow is of immense significance in the combustion chambes. The pupose of the swil is to assist in the ceation of the tooidal flow evesal to obtain the highe ate of mixing. The eciculation zone is an advese pessue gadient egion. This phenomenon of advese pessue gadient ceates high tubulent fluctuation. Theefoe the tubulent mixing and combustion ae benefited when the eciculation zones ae fomed in the combustos. The eciculation obtained due to the swil povides bette mixing. Because of the inheent impotance of swiling flow in many engineeing applications, thei investigations have been the subject of continued inteest among the fluid mechanics eseaches. In the pesent analysis the ect of the swil in the downsteam of the flow field has been obseved. Duing the last five decades the tubulent fluid flow though the sudden expansion passage fo both swil and without swil condition has been igoously investigated by the eseaches thoughout whole wold. Talbot [1] was fist to investigate the lamina swiling flow in the pipe. Chauvedi [] fist analyzed the flow chaacteistics in the axisymmetic expansion. The ects of the sudden expansions and compessions on the tubulent bounday laye wee pesented by White [3]. The tubulent flow in the downsteam of the eawad facing step was expeimentally analyzed by Etheidge et al. [4]. Thei tubulence measuements evealed the development of new shea laye, which splits at eattachment with about onesixth of the mass flow deflected in the upsteam. Thei esults indicated that nea the wall the length scales inceases moe apidly with distance fom the wall than in an attached bounday laye. Habib et a.l [6] numeically simulated the flow and heat tansfe of the swiling tubulent flow behind the sudden expansion pipe. They pedicted that with the incease in the maximum Nusselt numbe the swil numbe inceases. Lilley et al. [7] pesented the Fivehole pitot tube pobe timemean velocity measuements fo the confined swiling flows. Both the noneacting swiling and non swiling flows wee investigated in the axisymmetic test section with an expansion atio of. They obseved that the nonswiling confined jet possessed the cone eciculation zone with no cental eciculation zone. The ICME011 1 FL004

2 pesence of the swile shotens the cone eciculation zone and geneates a cental eciculation zone followed by a pecessing votex coe. The expeimental analysis of the flow with the swil in the cicula duct was pesented by Aoki et al. [8]. The velocity and the tubulence of the swiling flow in the cicula duct wee measued by using the Lase Dopple Velocimete (LDV). They illustated the chaacteistics of the tubulent intensity, eynolds stess, tubulent enegy and tubulent dissipation atio fo the diffeent swil numbes. Hallet [9] pesented the model fo estimating the minimum o citical swil intensity equied to poduce cental eciculation in a swiling sudden expansion flow. The esults fom thei poposed model showed that the swil equied fo the cental eciculation could be educed by poducing an inlet flow with the small solid body coe o a velocity deficit on the axis. A momentum integal model was developed by Hallet et al. [10] to pedict the size and the maximum eciculating mass flow ate fo the cental eciculation poduced by swiling flow in a sudden expansion. Thei esults showed that with all othe things being equal the eciculation ate inceases linealy with inlet swil intensity and expansion atio. The numeical simulation of the tubulent flow with inlet swil in the axisymmetic sudden expansion was pesented by Guo et al. [11]. They poposed that the pecessing votex coe was an inheent chaacteistic of the swiling flow inside the axisymmetic sudden expansion. The featues of the swiling flow in sudden expansion wee expeimentally descibed by Palm et al. [1]. The velocity pofile and the velocity gadient in the annula coss section wee measued by using the Lase Dopple technique. They poposed that the axial velocity pofile became inceasingly asymmetic with inceased swil intensity. Vanieschot et al. [13] theoetically studied the influence of swil on the eattachment length in an axisymmetic sudden expansion. An analytical expession was deived by them to pedict the influence of swil on the eattachment length in an axisymmetic sudden expansion. It was poposed by them that the eattachment length depends on the swil numbe and the expansion atio. In the pesent study the attention is mainly focused on detemining the ects of the swil impated at inlet on the eciculation bubble geneated due to the tubulent fluid flow though the sudden expansion passage. A coelation between the eciculation bubble length and inlet swiling angle has been developed. The adial distibution of axial velocity has also been analyzed. The contol volume fomulation of Patanka with SIMPLE algoithm and powelaw scheme has been employed fo the computational analysis.. GEOMETICAL DESCIPTION In figue.1 the flow geomety has been shown in the cylindical coodinate system with axes. The geomety of the consideed poblem is symmetic with the xaxis. Hee the swiling condition has been taken into consideation. The flow at the inlet of the passage is along the axial diection. In the above figue the inlet axial velocity has been indicated as. X A Fig 1. Schematic diagam of the flow though axisymmetic sudden expansion with swil. The axial inlet velocity is 15 m/s. The inlet velocity emains constant thoughout the analysis pocess. The swil is impated fom the inlet of the passage. Fo the analysis the swil numbe is vaied. The vaiation of the swil numbe is obtained by vaying the inlet swil angle. The inlet swil angle is vaied fom 0 to 75. The upsteam length of the duct is epesented as L i. The value of L i is taken as m. The downsteam length of the duct is epesented as L. The value of the downsteam length of the duct is 0.75 m. Both the upsteam and downsteam length emains constant thoughout the analysis. In the above figue the adius of the duct at the upsteam is epesented by i. The value of the upsteam adius emains constant and it is m. The adius of the swile is half of the adius of the passage at the upsteam. The adius of the swile taken fo the analysis is m. The downsteam adius of the cicula passage is denoted by and its value is taken as m. The value of the geometical paametes emains constant thoughout the analysis. The expansion atio fo this geometical model is. 3. GOVENING EQUATIONS In the pesent analysis the consideed steady, incompessible, tubulent flow is modeled by the momentum and continuity equations. 3.1 Continuity Equation ( u ) 1 ( v) 0 x 3. Momentum Equations Axial Component (xcomponent): u u p u 1 u v u x x x x x L i / u 1 x L WALL WA v x OUTLET ICME011 FL004

3 adial Component (component): v v p v 1 v v u x x x u 1 v v w x Whee, u and v ae the mean velocity components along x and diections espectively. The ective viscosity is, l t Whee, l and t ae molecula o lamina viscosity and eddy o tubulent viscosity espectively. Equation (1), () and (3) ae govening equations fo descibing the mean flow chaacteistics of a tubulent flow. The eddy viscosity is given by, t C k / Whee C is an empiical coicient. The modification of the empiical constant is given by, K K C 1 k U U U 1 8K S S S 1 n C c Hee, U u v s c is the adius of cuvatue of the concened steamline ( constant). The values of K 1 and K ae taken as 0.7and espectively. 3.3 Tubulence Model The equations ae given by, Equation: k k t k 1 k u v t G x x l l k x k Whee, G is the poduction tem and given by: v u v u v G t x x Equation: t 1 u v t l l C G C x x x 1 k k Hee, C 1, C, k and ae the empiical tubulence constants, and some typical values of these constants in the standad kε model ae ecommended by Launde and Spalding [14] which ae given below C 1 = 1.44 C = 1.9 k = 1.0 = 1.3 The final discetized equations have been solved by applying the unifom gid system with points. It was obseved that the esults convege at lesse gid points than this gid system, howeve fo the sake of any unwanted instability the pesent wok has been done with a gid system of above mentioned value. It has been obseved that the gid independent study has shown 0.001% change in the steam wise velocity. The esults of the pesent analysis calculated by employing the unifom gid system ae compaed with the benchmak numeical esults obtained by applying the nonunifom gid system and it is found to be satisfactoy. 3. The Swil Numbe The degee of the influence of swiling flow is usually chaacteized by the swiling numbe, which is a nondimensional numbe. The swil numbe is the atio between the axial fluxes of the swil momentum to the axial flux of the axial momentum. S 0 UW 0 U d d Whee, is the adius of the pipe at diffeent axial positions. and ae the mean axial and tangential velocity components, espectively. 4. BOUNDAY CONDITIONS The govening equations by themselves do not yield solution to the given poblem. Theefoe additional bounday infomation is equied at the inlet, outlet, the axis and the solid wall. The bounday conditions ae as follows, At the Inlet: (a) The inlet axial velocity is unifom. (U in = Constant) (b) The tangential velocity W in = U in tan. (c) Fully developed velocity pofile has been taken at the inlet. (d) The tubulence enegy and dissipation ate ae taken as: 3/ c kin kin 0.003u in and in At the Axis: Zeo shea stess condition has been taken fo the axis. ICME011 3 FL004

4 At the Wall: (a) No slip wall bounday condition has been taken fo the solid wall (u = v = w = 0). (b) Both k and ε ae handled by the wall function. At the Outlet: Fully developed flow condition has been taken at the x outlet. (i.e. 0 ) 5. VALIDATION OF THE PESENT NUMEICAL METHOD In figue the vaiation of the eattachment length with the swiling angle has been plotted. The benchmak expeimental esults of Lilley [7] and numeical esults of Chuang et al [15] ae plotted along with esults obtained fom the pesent calculations. By measuing the timemean velocity in the confined swiling flow using Fivehole pitot pobe, Lilley [7] pesented the expeimental esults fo the inlet swiling angle of 0, 15 and 45 degees espectively. Chuang et al [15] got the numeical esult by applying the nonunifom gid system with gid points. The esults of Chuang et al [15] wee fo the inlet swiling angle of 0, 15 and 45 degees espectively. The inlet swiling angles fo the pesent calculation ae 0, 15, 45, 60 and 75 degees espectively. In the pesent analysis the unifom gid system with 51 X151 points has been applied. angle is moe fo the expeimental esult than that of the numeical esults. The eattachment length obtained fom the expeimental esult is less than the othe two numeical esults. At the swiling angle of 45 thee is small deviation between the expeimental analysis esults and the numeical analyses esults. The ate of decease of the eattachment length in between 15 and 45 is moe linea fo the esults of Lilley [7] and Chuang et al [15]. In pesent analysis fo the inlet swiling angle of 45 the eattachment length (x/d) is The eattachment length obtained fom the esults of Lilley [7] and Chuang et al [15] ae 0.65 and 0.78 espectively. So fo the swiling angle of 45 the eattachment length fom the pesent analysis is geate than the othe two esults. In the pesent analysis the inlet swiling angle upto 75 is consideed. In between 45 to 60 the ate of decease of the eattachment length is steady. In the pesent calculation the eattachment length (x/d) at the inlet swiling angle 60 is The eattachment length in between 60 and 75 deceases moe apidly than the pevious obseved intevals of the inlet swiling angles. The eattachment length (x/d) at the inlet swiling angle of 75 is The esults obtained fom the benchmak expeimental and numeical analyses ae found to be in good ageement with the pesent calculations. A coelation has been obtained between the eattachment length and the inlet swil angle which is given below. Fig. Vaiation of eattachment length with espect to swiling angles Fom the above figue it is obseved that fo the nonswiling condition i.e., when the inlet swiling angle and the swil numbe is zeo, the eattachment length calculated fom the pesent analysis is almost same as that of the expeimental analysis of Lilley [7] and the numeical analysis of Chuang et al [15]. In between the inlet swiling angle 0 to 15 the eattachment length obtained fom the thee esults shows almost the same ate of decement of the eattachment length with the incease in inlet swiling angle. Fo the swiling angle of 15 the eattachment length fom the expeimental esults is moe than the othe numeical esults. The esult of Chuang et al [15] fo the swiling angle of 15 is appoximately equal to the esults of the pesent calculations. In between the inlet swiling angle 15 to 45 the ate of decease of the eattachment length with the swiling eattachment Length (L ) = Fom the coelation it is possible to estimate oughly the eciculation size changes with the swiling angle. It can be concluded fom the above that fo the same eynolds numbe and the expansion atio the eattachment length deceases with the swiling angle o the swil numbe. 6. ESULTS AND DISCUSSIONS The ects of the swil on the eciculation bubble ae analyzed by vaying the inlet swil angle. The inlet swil angle is vaied keeping the geomety of the cicula passage constant. The diamete of the cicula passage at the upsteam is m. The downsteam diamete of the cicula passage is 0.15 m. The length of the passage at the upsteam is 0.04 m. The downsteam length of the cicula passage is 0.75 m. The expansion atio i.e. the atio between the diamete of the cicula passage at the downsteam and the diamete of the cicula passage at the upsteam fo the pesent analysis is taken as. The axial inlet velocity consideed fo the analysis is 15 m/s. The eynolds numbe fo the analysis is , it is based on the axial bulk inlet velocity at the upsteam. The geometic paametes, axial inlet velocity, expansion atio and eynolds numbe emains constant thoughout the pesent analysis. The inlet swiling angles consideed fo the analysis ae 0, 15, 45, 60 and 75 degees espectively. The coesponding swil numbes ae 0, 0.18, 0.67, 1.16 and.49 espectively. The tangential velocity depends on the inlet swiling angle so with the change of the inlet swiling angle the tangential velocity also changes. The tangential velocities fo the inlet swiling angles 0, 15, 45, 60 and 75 degees ae 0 m/s, 4 ICME011 4 FL004

5 m/s, 15m/s, 6 m/s and 56 m/s espectively. The steamline contou, flooded contou and vecto plots ae utilized to illustate the ects of the eciculation bubbles on the main flow. The eattachment length o the eciculation bubble lengths fo diffeent swiling angles o swil numbes ae obtained. In the fig. 3 the ects of the swil numbe on the eciculation bubble have been shown in steamline and flooded contou plots and in fig.4 the ects of the same paamete on the eciculation bubble have been shown in vecto diagam. In both the figues the fist steamline and contou plot and vecto diagam ae obtained espectively fo the inlet swiling angle of 0. The coesponding swil numbe fo the fist plot is zeo. The zeo swil numbe epesents the nonswiling flow. The next fou plots of both the above espective figues ae fo the inlet swiling angle geate than zeo, so that plots ae obtained fo the swiling condition. The eciculation bubble is geneated in the cone of the passage. Fo the nonswiling flow condition fom the fist plot of the above figues it is obseved that the size and stength of the eciculation bubble is moe than the swiling condition. Fom the vecto diagam of the fig.4 it is obseved that the flow in the eciculation egion is in the anticlockwise diection. Fo the second plots of both the figues the inlet swiling angle is inceased to 15 keeping the inlet velocity and geometical paametes constant. The coesponding swil numbe also inceases to In this case also the eciculation bubble is geneated at the cone of the passage. Fom the steamline plots of the figue it is obseved that fo the swiling condition the dividing steamline is moe towads the cone potion of the passage than the non swiling condition. This implies that the size of the eciculation bubble geneated fo the swiling condition is smalle than the eciculation bubble geneated fo the non swiling condition. The stength of the eciculation bubble obtained fom the swiling condition is less than the nonswiling condition. The flow diection emains same fo both the swiling and nonswiling condition. In the thid plots of both the above figues the swiling angle is taken as 45 and the coesponding swil numbe is The inlet velocity and the geometical paametes emain same as that of the pevious two descibed cases. With the incease in the swil numbe the dividing steamline moves towad the cone potion of the passage. Hee in this case size and the stength of the geneated eciculation bubble is futhe smalle than the pevious two cases. The diection of the flow in the eciculation egion emains same as that of the pevious cases. Fo the foth plot of both the above espective figues the consideed inlet swil angle is 60 keeping the axial inlet velocity and geometic paametes as same as that of the pevious cases. The coesponding swiling numbe is With incease in the swil numbe to 1.16 the eattachment point futhe moves towads the cone of the passage. The eciculation bubble geneated in the passage fo the above swil numbe is much smalle than the pevious thee cases. Fig 3. Effect on eciculation size with the vaiation of swiling numbe In the fifth plots of both the above figues the inlet swiling angle is inceased to 75. The othe paametes ae consideed same as that of descibed in the pevious cases. The swil numbe fo this swiling angle is.49. Hee in this case also the dividing steamline and the eattachment point shifts towads the cone of the passage with incease in the swil numbe. This shows that the eciculation bubble obtained in this case is of smalle size and stength than the othe fou pevious cases. Fom the above vecto diagam it is found that the flow diection in the eciculation egion emains same in all the cases. Fom the steamline plots of the fig. 3 it is obseved the dividing steamline and the eattachment point moves moe towads the cone of the passage with the incease in the swil numbe. Hee the shifting of the eattachment point towads the cone of the passage implies that the size and shape of the eciculation zone deceases with the incease in the swiling numbe o the swiling angle. Fo the nonswiling flow condition the size of the eciculation zone geneated at the cone of the passage is much geate than the eciculation bubble geneated fo the swiling condition. It is obseved that with the incease in the swiling numbe the eciculation bubble size deceases. The swil numbe epesents the swiling stength. Fom steamline and flooded contou plots of the above figue it is found that the swiling stength has a stong ect on the flow field. With incease in the swil numbe the flow field becomes moe stabilized. Theefoe we can conclude that the swil has the stabilizing ect on the eciculation bubble. ICME011 5 FL004

6 Tubulent Enegy Fig 4. Effect on eciculation size with the vaiation of swiling numbe The tubulent kinetic enegy is inceased by means of the swiling ect. The swiling flow inceases the tubulent mixing. The tubulent mixing enhances the unifom distibution of the tempeatue in the combustion chambe. Thus the combustion iciency is inceased with the lage swil numbe. the expansion atio of 1.6. In the fist case the vaiation of the tubulent enegy has been obtained at the axial position of 8 m fom the upsteam side of the passage. In this case it is obseved that befoe the eciculation egion the vaiation of the tubulent enegy is steady with espect to the adius. The maximum tubulent kinetic enegy is in the eciculation egion as it is physically expected to be. Afte that the tubulent enegy is in deceasing tend because towads the solid wall thee exist the lamina sub laye. We know in a lamina sub laye the flow is no longe tubulent and the tubulent intensity has to be zeo at the vicinity of the wall. Hence this dooping natue of the tubulent enegy is physically possible. In the second case the distibution of the tubulent enegy at the axial position of 16 m fom the left side of the passage has been obtained. In this case it is obseved that the tubulent enegy fluctuates with espect to the adius and the highest tubulent enegy is aound the eciculation zone. Fom this we can say that the eciculation zone is the egion of highest tubulent in the flow field. In fig. 6 and 7 the adial distibutions of the axial velocity at diffeent axial positions fo the expansion atio of 1.6, have been plotted. In the fig. 6 the axial velocity distibution is obtained at the axial position of 8 m and in fig. 7 the axial velocity distibution fo the axial position of 16 m. The existence of the eciculation egion is obseved in both the cases by the flow evesal. At the axial position of 8 m the negative axial velocity is found to be moe than the negative axial velocity distibution fo axial position of 16 m X= 8m Fig 6. adial distibution of axial velocity at x=8 m / Fig 5. adial distibutions of tubulent enegy at x=8 m and x=16 m In the fig. 5 the adial distibutions of the tubulent enegy fo diffeent axial position have been plotted. The above adial distibutions of the tubulent enegy ae fo ICME011 6 FL004

7 U / X=16m Fig. 7. adial distibution of axial velocity at x=16 m Fom the above figues it is obseved that the eciculation zone at the axial position of 16 m is less than the eciculation zone at the axial position of 8 m. The axial position of 8 m is nea to the expanded potion of the passage and the axial position of 16 m is futhe towads the downsteam fom the expanded potion of the passage. The coe of the eciculation zone fo the expansion atio of 1.6 is appoximately aound the axial position of 8 m. Fom the axial velocity pofiles it can be said that on moving fom coe of the eciculation zone towads the downsteam of the flow field axially the size and ect of eciculation bubble deceases. 7. CONCLUSION The numeical analysis of the tubulent fluid flow though the sudden expansion passage has been caied out by applying modified kε model. The ects of inlet swil on the eciculation bubble have been investigated. The size and the stength of the eciculation bubble deceases with the incease in the inlet swil angle o the swil numbe. It can be concluded fom the above analyses that the swil has the stabilizing ect on the tubulent flow though the sudden expansion passage. These flow paametes ae needed to be contolled fo the geneation of the eciculation bubble as equied fo combustion o any othe puposes like the chemical pocesses etc. A coelation has been developed between the eattachment length and the swiling angle. 6. EFEENCES 1. Talbot, L., 1954, Lamina Swiling Pipe Flow, ASME J. Appl. Mech., pp Chatuvedi, M. C., 1963, Flow Chaacteistics of AxiSymmetic Expansion, Jounal. Hydaulic div.poc., ASCE, 89: pp White,. A., 1966, Effect of Sudden Expansions o Compessions on the Tubulent Bounday Laye, AIAA Jounal, 4(1): pp Etheidge, D. W., and Kemp, P.H., 1978, Measuements of Tubulent Flow Downsteam of a eawad Facing Step, Jounal of Fluid Mechanics, 86(): pp Patanka, S. V., 1981, Numeical Heat Tansfe and Fluid Flow, McGawHill, New Yok. 6. Habib, M. A., and McEligot, D.M., 198, Tubulent Heat Tansfe in a Swil Flow Downsteam of an Abupt Pipe Expansion, Intenational J. Heat tansfe, Poceedings of the Seventh Intenational Confeence, Munich, West Gemany, Pape Fc9, Hemisphee Publishing Cop., pp Lilley, D. G., and Yoon, H. K., 1983, Five hole Pitot Pobe Timemean Velocity Measuements in Confined Swiling Flows, AIAA Pape, pp Aoki, K., Shibata, M., and Nakayama, Y., 1986, Study on the Flow with a Swil Flow in a Cylindical Combusto, Bulletin of JSME, 9(58): pp Hallet, W. L. H., 1988, A Simple Model fo the Citical Swil in a swiling Sudden Expansion Flow, Jounal of Fluid Engineeing, 110: pp Hallet, W. L. H., and Ding, C. Y., 1995, A Momentum Integal Model fo Cental eciculation in Swiling Flow in a Sudden Expansion, The Canadian Jounal of Chemical Engineeing, 73: pp Aloui, F., and Souha, M., 000, Expeimental Study of Tubulent Asymmetic Flow in a Flat Duct Symmetic Sudden Expansion, Jounal of Fluids Engineeing, 1: pp Palm,., Gundmann, S., Weismülle, M., Šaic, S., Jakilic, S. and Topea, C., 006, Expeimental Chaacteization and Modelling of Inflow Conditions fo a Gas Tubine Swil Combusto, Intenational Jounal of Heat and Fluid Flow, 7( 5): pp Vanieschot, M., and Van den Bulck, E., 008, The Influence of Swil on the eattachment Length in an Abupt Axisymmetic Expansion, Intenational Jounal of Heat and Fluid Flow, 9: pp Launde, B.E., and Spalding, D.B., 1974, the Numeical Computation of Tubulent Flows, Compute Methods in Applied Mechanics and Engineeing, 3: pp Chuang, SH., Lin, HC., Tai, FM and Sung HM., 199, Hot flow analysis of swiling suddenexpansion dump combusto, Intenational Jounal of numeical Methods in Fluids, 14: NOMENCLATUE Symbol Meaning Unit Empiical Constant Empiical Constant Empiical Constant Time mean velocity (m/s) along Z axis. Time mean velocity (m/s) along axis Aveage inlet velocity (m/s) Swil Numbe ICME011 7 FL004

8 S G k L L i L i e Z ε ate of Poduction Tubulent kinetic enegy adial coodinate acoss the duct Length at the down steam Length at the upsteam eattachment length adius at inlet eynolds numbe Axial coodinate along the duct Molecula o lamina viscosity Tubulent viscosity Effective viscosity Tubulence kinetic enegy dissipation ate Pandtl numbe of the tubulent kinetic enegy Dissipation Enegy Density (m /s ) (m) (m) (m) (m) (Ns/m ) (Ns/m ) (Ns/m ) (m /s 3 ) (Kg/m 3 ) 8. MAILING ADDESS D. Snehamoy Majumde Associate pofesso Depatment of Mechanical Engineeing Jadavpu Univesity, Kolkata West Bengal, INDIA. Phone: (mob) ID: sg_maj@yahoo.com ICME011 8 FL004