Sit International Smposim on Trblence and Sear Flow Penomena Seol, Korea, -4 Jne 9 INVESTIGATION OF TURBULENT BOUNDARY LAYER OVER FORWARD-FACING STEP BY MEANS OF DNS Hirofmi Hattori Department of Mecanical Engineering, Nagoa Institte of Tecnolog Gokiso-co, Sowa-k, Nagoa, 466-8555, Japan attori@eat.mec.nitec.ac.jp Yastaka Nagano Department of Mecanical Engineering, Nagoa Institte of Tecnolog nagano@eat.mec.nitec.ac.jp ABSTRACT Tis paper presents observations and investigations of te detailed strctre of a trblent bondar laer over a forward-facing step. A trblent bondar laer wit reattacment and separation wic cases an increase in friction of flow and distrbance in transportation of scalar often occrs in flid ciner, te atmospere, etc. Ts, it is important for friction redction and environmental protection to know te transport penomena of a bondar laer wit reattacment and separation. However, little data on a bondar laer wit reattacment and separation are available in comparison wit an internal trblent flow sc as a cannel flow. Data of trblent eat and ss transfer in te bondar laer over a block are especiall difficlt to obtain. Terefore, in order to investigate te detailed trblent qantities of a bondar laer over an obstacle wic cases reattacment and separation, a DNS of te trblent bondar laer over a forward-facing step mst first be carried ot. Te present DNSs are condcted nder conditions wit varios Renolds nmbers based on step eigt, or based on momentm tickness so as to investigate te effects of step eigt and inlet bondar laer tickness. DNS reslts sow te qantitative trblent statistics and strctres of bondar laers over a forward-facing step, in wic te caracteristic transport penomena in te recirclation regions wic occr in front of and on te step are demonstrated. INTRODUCTION Te ators ave been condcting DNSs of a bondar laer (Hattori et al., 7, in wic nderstanding of te trblence penomena in te terll stratified bondar laer as been improved, and a flow database as been constrcted. Te trblent caracteristic of te terll stratified bondar laer is investigated in detail, and te prediction perfornce of trblence models is evalated sing DNS in order to improve te trblence model for prediction of trblent flows. Additionall, we need to condct DNS concerning more complicated trblence sc as a trblent flow over an obstacle wic is modeled as te tpical flow in a comple flow field and cases flow reattacment of separation, to compreend transport penomena tere. A trblent bondar laer wit reattacment and separation as been observed in drodnamic penomena, wic wit trblence become more comple. Since te reattacment and separation of flow cases an increase in friction of flow and distrbance in transport of scalar, te drodnamic penomena in a trblent flow wit reattacment and separation sold be investigated in detail so as to decrease friction and enance transport of scalar. A tpical flow wit reattacment and separation is a back-ward facing step flow, and n stdies as been reported b bot eperiment and calclation (e.g., Kasagi and Matsnaga, 995; Le et al., 997; Vogel and Eaton, 985. Ts, te most notable separated flow is a back-ward facing step flow. On te oter and, a flow wic cases reattacment and separation incldes a forward-facing step flow, bt sefl detailed data inclding trblence are scarce (e.g., Moss and Baker, 98; Sakoci et al., 999. Since a forward-facing step flow occrs in varios flow sitations, we sold know te detailed trblent penomena, flow properties and strctres. Terefore te objective of tis std is to investigate trblent bondar laers over a forward-facing step. In sc flows, owever, it is difficlt to predict and measre distribtions of trblence, since varios tpes of ciner, bilding strctres and terrain complicate transport penomena of te kind. Hence, direct nmerical simlation (DNS is a ver sefl and powerfl tecniqe in order to determine sc detailed trblence penomena as well as eperimental tecniqes. Especiall, te near-wall trblent strctre can be clarified b DNS. Ts, DNS of te trblent bondar laer over a forward-facing step is carried ot to obtain detailed knowledge of te effect of te forward-facing step in varios Renolds nmbers. In particlar, transport penomena near te step are selectivel eplored. NUMERICAL PROCEDURE Te governing eqations sed in DNS are te Navier- Stokes eqation witot boanc, te continit eqation for te velocit field, and te energ eqation for te terl field, in wic incompressibilit is assmed as follows: i i i t + i j j ( p i + Re δ,in i j j were te Einstein smtion convention applies to re- ( 7
Sit International Smposim on Trblence and Sear Flow Penomena Seol, Korea, -4 Jne 9 Table : Comptational metod and conditions Grid Staggered Grid Copling Algoritm Fractional Step Metod Time Advancement Adams-Basfort Metod Spatial scemes nd-order central difference Doin size Driver δ,in δ,in 4δ,in ( z Main δ,in δ,in 4δ,in Re 9, 8 Driver 9 8 8 Grid nmbers ( z Main 88 8 8 Re Driver 9 56 56 Grid nmbers ( z Main 576 56 56 - f C - -5 Re (é,in 9 8 Step -5 5 Table : Separation and reattacment points Re Sepa. point (/ Reat. point (/ 9 δ,in.9.4 8 δ,in.8.8 8 6δ,in.9.86 δ,in.7. - f C - (a δ,in Re 9 (é,in 8 (é,in 8 (6é,in Step peated indices, and a com followed b an inde indicates differentiation wit respect to te indeed spatial coordinate. i is te dimensionless velocit component in i direction, p is te dimensionless pressre, t is te dimensionless time, and i is te dimensionless spatial coordinate in te i direction, respectivel. All eqations are non-dimensionalized b te free stream velocit, Ū, and te momentm tickness, δ,in, at te inlet of te driver part. Figre sows a scetic of te forward-facing step and te coordinate sstem. Te present DNS based on te ig-accrac finite-difference metod (Hattori et al., 7 is carried ot nder conditions of Renolds nmbers based Figre : Coordinate sstem -5-5 5 (b δ,in, 6δ,in Figre : Distribtions of friction coefficients on te free stream velocit and te momentm tickness at te inlet of te driver part, Re δ,in, 6 and, in order to eplore te inflence of te Renolds nmber in te present condition. Te step eigt,, is fndamentall set to δ,in, bt is also set to 6δ,in in te case of Re δ,in in order to observe te effect of te step eigt on trblence. Tese eigts are abot / / of te bondar laer tickness at te inlet of te driver. Ts, te Renolds nmbers based on te step eigt become Re 9. Note tat te Renolds nmber based on te momentm tickness is sed in te calclation, bt te Renolds nmber based on te step eigt is emploed in te reslt, becase te proper Renolds nmber wic represent a forwardfacing step flow is Re. Te detailed comptational metod and conditions inclde te algoritm, doin size and grid nmbers of DNS. Te bondar conditions for te velocit field are te non-slip conditions on te walls, and /, w/, v/ ( / + w/ z on te pper bondar (free stream. At te otlet of bot parts, convective bondar conditions are applied, and periodic bondar conditions are sed in te spanwise direction. Step p Re (é,in C 9 8 - Re (6é,in 8 - -5 5 Figre : Distribtions of pressre coefficients RESULTS AND DISCUSSION Fndamental Flow Properties First, te fndamental flow properties of present bondar laer over a forward-facing step are sown. Figre sows te pressre coefficients arond te step in all cases. Obviosl, te case of step eigt 6δ,in indicates a dissimilar distribtion in comparison wit te step eigt δ,in of te step. On te oter and, te distribtions of friction coefficients of te case of te step eigt δ,in are sown in Fig. (a and te case of te step 74
Sit International Smposim on Trblence and Sear Flow Penomena Seol, Korea, -4 Jne 9 eigt 6δ,in, in wic te cases of Re and 6 wit te step eigt δ,in are inclded for comparison, are sown in Fig. (b. Also, te separation points in front of te step and te reattacment points on te step are listed in Tab.. Te nearest separation point from te step is given in te case of te igest Renolds nmber, Re, and te longest reattacment point on te step is obtained in te case of te lowest Renolds nmber, Re 9. Te eperimental reslt (Moss and Baker, 98; Re 5, indicated.5. for te detacment point and 4.7 for te reattacment point. Te obvios reason for tis disagreement is te Renolds nmber difference. In particlar, te step eigt in comparison wit te bondar laer tickness of te eperiment is iger tan tat of te DNS, i.e., te step eigt eceeds te bondar laer tickness in te eperiment, bt te step eigt of all cases of present DNSs are witin te bondar laer tickness. As for te distribtion of friction coefficient, te case of Re 9 gives a dissimilar distribtion in comparison wit te oter cases on te step near te corner. However, no clear difference in distribtions of friction coefficients in te oter cases can be fond. In order to investigate te flow sitation, te streamlines arond te step of all cases are sown in Fig. 4. It is well-known tat te separation regions occr in front of and on te step in te forwardfacing step flow. Te present DNSs obviosl represent te. UU.4 vu Re 9 (é,in - - - (a Re 9 ( δ,in.4. vu UU - Re 9 (é,in (a Re 9( δ,in. (b Re 8 ( δ,in UU.4 vu Re 8 (é,in - - - (c Re ( δ,in Re 8 ( 6δ,in Figre 4: Streamlines arond te step (d Re 8 (é,in.4. UU vu - (b Re 8 ( δ,in Figre 5: Near-wall profiles of streamwise mean velocit and Renolds sear stress arond te step 75
Sit International Smposim on Trblence and Sear Flow Penomena Seol, Korea, -4 Jne 9 caracteristic flow configration of forward-facing step flow. Te sllest separation region in front of te step can be observed in te case of Re. Ts, te nearest separation point from te step is given in tis case. On te oter and, te separation region on te step in te case of Re 9 reveals a can be observed sligt difference. Ts, dissimilar distribtions of friction coefficients of te case are. UU.4 vu Re (é,in - - -.4. vu UU - Re (é,in (c Re ( δ,in..4 UU vu Re 8 (é,in - - - Re 8 (6é,in.4. vu UU - (d Re 8 ( 6δ,in Figre 5: (contined given in comparison wit te oter cases. Trblent Statistics Te near-wall profiles of streamwise mean velocities and Renolds sear stresses in front of and on te step are sown in Fig. 5. Te mean velocities of te case of Re 8 wit te step eigt 6δ,in rerkabl accelerate on te step..5..5..5..5 v S v S v S v S Re 8 (6é,in. Re 9 (é,in..5.5 v S.5 Re 9 (é,in (a Re 9 ( δ,in Re 8 (é,in..5.5 v S.5 Re 8 (é,in (b Re 8 ( δ,in Re (é,in..5.5 v S.5 Re (é,in (c Re ( δ,in.5.5 v S.5 Re 8 (6é,in (d Re 8 ( 6δ,in Figre 6: Relations between Renolds sear stress and mean velocit gradient in vicinit of te wall 76
Sit International Smposim on Trblence and Sear Flow Penomena Seol, Korea, -4 Jne 9 (a Re 9 ( δ,in v.6 Q Q Q Re 9 (é,in - (b Re 8 ( δ,in v.4 Re 9 (é,in Q Q Q -.5 (c Re ( δ,in.55 (d Re 8 ( 6δ,in Figre 7: Vorte strctres arond te step v Q Q Re Q 9 (é,in -.5.98 v 9. Q Q Q Re 9 (é,in - Figre 8: Fractional contribtions of different qadrant motions at inlet bondar laer de to iger step eigt tan tat of te oter cases. Te Renolds sear stress increases near te wall on te step de to te increase in te sear of te mean velocit. Also, te Renolds sear stress increases in front of te step, were te separation region casing te increase in te sear of te mean velocit eists. Since te different distribtion v Q Q Re Q 9 (é,in -.5 Figre 9: Fractional contribtions of different qadrant motions arond te step of Renolds sear stresses is fond in te case arond te step, it can be considered tat te dissimilar distribtion of friction coefficients is given. It can be fond tat te conter gradient diffsion penomenon (CGDP between te streamwise mean velocit and Renolds sear stress appears near te corner of te step (/. Especiall, te CGDP rerkabl ap- 77
Sit International Smposim on Trblence and Sear Flow Penomena Seol, Korea, -4 Jne 9 pears in te case of te lowest Renolds nmber. Note tat we define te CGDP as te opposite sign between te Renolds sear stress, v, and te mean velocit gradient, S Ū/ + V/, simltaneosl occr. Altog it can be seen tat te profiles of bot Renolds sear stress and streamwise mean velocit are almost similar on te step in all cases as indicated in Fig. 5, S of te case of te igest Renolds nmber gives te negative sign as indicated in Fig. 6(c. Ts, te CGDP on te step is less apt to occr as te Renolds nmber increases. In te downstream of te corner /.5, te CGDP appears in te vicinit of te wall in all cases, bt te CGDP vanises at te region awa from te wall in te case of iger Renolds nmber tan te case of Re 9 as sown in Fig. 6. Trblent Strctres and Qadrant Analsis From investigating te trblent strctre of te bondar laer over te forward-facing step in detail, te vorte strctres arond te step be sown as in Fig. 7, were te vorte strctre is determined b te second invariant of te velocit gradient. It can be seen tat te vorte strctre develops de to te effect of te step toward downstream direction on te step in all cases, were te continos vorte strctre can be observed clearl. Wit an increase in Renolds nmber, te vorte strctre obviosl becomes fine, and te step cases prodction of a finer vorte tan tat in front of te step. In te case of te lowest Renolds nmber, owever, a discontinos vorte strctre appears over te recirclation region de to te bondar laer redevelopment. Wit an increase in step eigt, te vorte strctre is promoted more tan te case of Re 9 wose inlet condition of bondar laer is identical wit te case of te igest step. Te effect of te step obviosl inflences te redevelopment of te bondar laer in te downstream region. Finall, a qadrant analsis (Nagano and Tagawa, 988 is condcted to investigate in detail trblence motion arond te step. Sc analsis is especiall sed to eplore te case of te lowest Renolds nmber, in wic te CGDP rerkabl occrs. In general, te Q motion (ejection and te motion (sweep dominate te prodction of Renolds sear stress in te wall trblent sear flow as indicated in Fig. 8. Note tat te motions are norlized b te imm Renolds sear stress at te point, and te total means te Renolds sear stress itself. In particlar, te motion dominates near te wall, and te Q motion affects te region awa from te wall for te Renolds sear stress. Figres 9 sow te fractional contribtions of different qadrant motions arond te step. At /.6, wic is te recirclation region in front of te step, te Renolds sear stress as tree inflection points, in wic te Q motion is similar in te profile of Renolds sear stress. At sc points, te motion increases near te wall, bt te Q (otward interaction and Q (wallward interaction motions inflence te Renolds sear stress. Ts, te Renolds sear stress decreases near te wall, wereas te motion increases. On te step near te corner (/.4 in wic te CGDP appears, te Q and Q motions rerkabl affect te negative Renolds sear stress. Te Q motion especiall contribtes to te region were te Renolds sear stress indicates a negative sign. In te region awa from te wall, te Q motion dominates as te sal wall trblent sear flow. At te downstream point in te recirclation region on te step (/.55, te profiles of Renolds sear stress significantl var. In te vicinit of te wall at /.55, te motion is observed as te dominative motion. Te Q motion becomes te dominative motion in te region were te Renolds sear stress indicates te negative sign. At te point (/.98 were te Renolds sear stress completel indicates te positive sign, te dominative motion is similar in te sal wall trblent sear flow as indicated in Fig. 8, bt te dominative region of becomes wide near te wall. CONCLUSIONS DNSs of trblent bondar laers over forward-facing step are carried ot to investigate in detail te transport penomena of sc flows. Te present DNS indicates te fndamental and detailed caracteristics of trblent bondar laers over a forward-facing step nder varios step and flow conditions. In particlar, prononced conter-gradient diffsion penomena (CGDP are observed. Terefore, it is conclded tat te obvios trblent bondar laer over te forward-facing step can be investigated in detail b means of DNS. Also, DNS of a iger Renolds nmber is carried ot in order to investigate te Renolds nmber dependence of sc flows. Wit an increase in Renolds nmber, te CGDP is less apt to occr. Ts, te CGDP appears rerkabl in te case of a sll Renolds nmber. On te oter and, in order to investigate trblence motion arond te step in detail, qadrant analsis is condcted. Te qadrant analsis clearl sows te dominative motion for te prodction of Renolds sear stress arond te step. Terefore, we can observe and investigate te detailed strctre of a trblent bondar laer over a forward-facing step. ACKNOWLEDGEMENT Tis researc was spported b a Grant-in-Aid for Scientific Researc (S, 76, from te Japan Societ for te Promotion of Science (JSPS. REFERENCES Hattori, H., Hora, T. and Nagano, Y., 7, Direct nmerical simlation of stable and nstable trblent terl bondar laers. Int. J.Heat and Flid Flow, 8, pp.6 7. Kasagi, N., and Matsnaga, A., 995, Tree-dimensional particle-tracking velocimetr measrement of trblence statistics and energ bdget in a backward-facing step flow. Int. J. Heat Flid Flow, 6 pp. 477 485. Le, H., Moin, P., and Kim, J., 997, Direct nmerical simlation of trblent flow over a backward-facing step. J. Flid Mecanics,, pp.49 74. Moss, W. D. and Baker, S., 98, Re-circlating flow associated wit two-dimensional steps. Aeronatical Qarterl, Agst, pp.5-7. Nagano, Y. and Tagawa, M., 988, Statistical caracteristics of wall trblence wit passive scalar. J. Flid Mecanics, 96, pp.57 85. Sakoci, T. Ando, T., and Nakano K., 999, Flow caracteristics over forward facing step and control of separated flow. Trans.JSME, Ser. B, 65, pp.8 4. Vogel, J. C., and Eaton, J. K., 985, Combined eat transfer and flid dnamic measrements downstream of a backward-facing step. Trans. ASME, J. Heat Transfer, 7, pp. 9 99. 78