The Effect of Near-Wall Vortices on Wall Shear Stress in Turbulent Boundary Layers

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1 Engneerng, 00,, do:0.436/eng Publshed Onlne March 00 (htt:// The Effect of Near-Wall Vortces on Wall Shear Stress n Turbulent Boundary Layers Shuangx Guo, Wanng L School of Cvl Engneerng and Mechancs, Huazhong Unversty of Scence and Technology, Wuhan, Chna The Key Laboratory of Mechancs on Western Dsaster and Envronment of the Mnstry of Educaton of Chna, Lanzhou, Chna Emal: gsx_00@63.com Receved June 0, 009; revsed July 9, 009; acceted July, 009 Abstract The objectve of the resent study s to exlore the relaton between the near-wall vortces and the shear stress on the wall n two-dmensonal channel flows. A drect numercal smulaton of an ncoressble two-dmensonal turbulent channel flow s erformed wth sectral method and the results are used to examne the relaton between wall shear stress and near-wall vortces. The two-ont correlaton results ndcate that the wall shear stress s assocated wth the vortces near the wall and the maxmum correlaton-value locaton of the near-wall vortces s obtaned. The analyss of the nstantaneous dagrams of fluctuaton velocty vectors rovdes a further exresson for the above conclusons. The results of ths research rovde a useful sulement for the control of turbulent boundary layers. Keywords: Sectral Methods, Two-Dmensonal Turbulence, Wall Shear Stress, Two-Pont Correlaton. Introducton The flow henomenon of turbulent boundary layers s common n nature. It s closely related to aerosace, marne, envronmental energy, chemcal engneerng and other felds. In aeronautcal engneerng, the colex turbulent vortex structures n boundary layers not only affect the workng stablty and securty of the arcraft but also ncrease the skn frcton on the wall remarkably. So the research of control of turbulent boundary layers s sgnfcant. In recent years the related lteratures focus manly on two asects: control of near-wall turbulent structures and wall skn frcton. Essentally, the wall skn frcton s closely related wth the near-wall turbulent structures, so the researches on these two asects are n accordance. In flat wall flows, the wall shear stress consttutes wall skn frcton. Sheng, Malkel and Katz dd much n-deth and colete study on the relaton between wall shear stress (streamwse and sanwse) and near-wall flow structure (streamwse, sanwse and outsde structure) by exerment method []. Most researchers agreed that the near-wall streamwse vortces were the man effect factors of wall shear stress [-5]. The wall normal and sanwse veloctes boundary condtons were resented by the methods of wall blowng and sucton or sanwse-wall oscllaton, whch drectly changed the near-wall streamwse vortces and acheved the urose of control of the wall shear stress [6-8]. Though the detaled mechansm has not been coletely clear so far, the above-mentoned control methods have made good effectveness on wall shear stress reducton. Recently Y. S. Park et al also researched the control of wall shear stress by the method of wall blowng and sucton. Instead of vertcal to the wall, certan angles were resented between the blowng-sucton drecton and the streamwse drecton. Ths meant that the velocty boundary condtons brought by ther control method were normal and streamwse veloctes nstead of normal and sanwse veloctes. Ther exerments showed a better effectveness of wall shear stress reducton f the angle was roer [9]. In fact, wall normal and streamwse velocty boundary condtons changed the near-wall sanwse vortces drectly. As well as streamwse vortces, sanwse vortces are also the man characterstcs n turbulent boundary layers, whle are they also the crucal effect factors on wall shear stress as the streamwse vortces? Essentally, turbulent flow s absolutely three-dmensonal, but some certan turbulence moton, such as atmoshere or ocean flows, s behavng quas-two-dmensonally. The horzontal scales are hundreds of klometers n the ocean and thousands of klometers n the at-

2 9 moshere, whle ther vertcal scale s only a few klometers. So the turbulent motons n the vertcal drecton are suressed and can be gnored can be treated as twodmensonal turbulence [0-]. The saturaton states of two-dmensonal turbulence have the smlar characterstcs as the three-dmensonal turbulence, such as njecton, swee and other burstng henomenon [-3], whle they have lots of dfferences from the three-dmensonal turbulence, such as self-organzaton and nverse energy cascade [0-]. In addton, the smulaton of two-dmensonal turbulence requres less exensve coutatonal resources n coarson wth that of three-dmensonal turbulence. So t s also valuable for the study of two-dmensonal turbulence. Moreover, scalar vortcty n two-dmensonal turbulence s controlled by the normal and streamwse velocty, and has the same exresson as the sanwse vortcty n three-dmensonal v u turbulence,. So n the resent aer the x y two-dmensonal scalar vortcty s taken as the major subject of study nstead of the three-dmensonal sanwse vortcty. The objectve of the resent study s to exlore the relaton between the near-wall vortces and the shear stress on the wall n two-dmensonal channel flows. So far, the researches of two-dmensonal turbulence are manly lmted to the models wth unbounded condton, or wth the dentcal bounded condton such as square or crcular domans. The lteratures of two-dmensonal channel flow are rare. W. Kramer, H. J. H. Clercx and G. J. F. van Hejst have done some oneerng study on ths subject. They have researched the nfluence of the asect rato of the channel and the ntegral-scale Reynolds number on the large-scale self-organzaton of the flow n detal and obtaned lots of ortant consequence []. In the resent aer, the numercal rocess to smulate the two-dmensonal turbulent channel flows drectly wth sectral method s frstly ntroduced, and the accuracy and stablty of the roosed algorthm s verfed wth two exales. Secondly the relaton between the wall shear stress and near-wall vortces s exlored and the maxmum correlaton-value locaton of the near-wall vortces s obtaned. Fnally the nstantaneous dagrams of fluctuaton velocty vectors and the near-wall model are analyzed to rovde a further exresson for the conclusons obtaned..numercal Processes.. Numercal Method Wth the develoment of coutatonal technology and resources, the drect numercal smulaton (DNS) s more and more wdely used as the basc research aroach of turbulence. Sectral method s one of the most common methods for the DNS of turbulence, whch has many advantage, such as hgh degree of accuracy, quckly seed on convergence, and analytcally satal dervaton for flow varables [3]. Many researchers have done lots of oneerng and sgnfcant achevements for ths method, such as John Km, Mon & Moser [4], Kleser & Schumann [5], Hu, Morfey & Sandham [6]. So sectral method s aled to solve the Naver-Stokes equaton drectly n the resent study. The governng equatons for two-dmensonal ncoressble channel flow can be wrtten as the followng forms: u t f u x Re u x 0 Here, all varables are non-dmensonalzed by the channel half-wdth and lamnar Poseulle flow central velocty Uc ; Non-lnear term f ncludes the convectve terms and the mean ressure gradent [4]; and Re denotes the Reynolds number defned as Re U c /, where s knematc vscosty. Vortcty s defned as v u x y. Equaton () can be reduced to yeld a second-order equaton for the vortcty as follows: f f t Re z x 3 () () (3) The velocty coonent equatons can be deduced due to and contnuty Equaton (): u, v y x Fully develoed turbulent channel flow s homogeneous n the streamwse drecton, and erodc boundary condtons are used n ths drecton. All unknown quanttes are exanded wth Fourer seres n the streamwse drecton and Chebyshev olynomal n the normal drecton as follows: N / N qxyt (,, ) q ( t)ex( xt ) ( y) mn / P0 where m m. s amount of streamwse erod, defned as / xn, xn s the non-dmensonal wdth of coutatonal doman n the streamwse drecton; m s streamwse wave number; T ( y ) s -order Chebyshev olynomal. m (4)

3 9 Substtute the exansons of all unknown quanttes nto Equatons (3) and (4) resectvely, and the sectral coeffcent equatons for each Fourer wave number can be obtaned: N [ ( )] ˆ m D ( t) T 0 t Re ( y) N, m, 0 [ fˆ ( td ) fˆ ( t)] T( y) (5) ˆ ˆ m 0 0 ( D ) u () t T ( y) () t DT ( y) (6) ( ) ˆ ( ) ( ) ˆ m m ( ) 0 0 D v t T y t T ( y) (7) where dfferental oerator D defnes as D d. Equatons (5), (6) and (7) are the sectral coeffcent formulas dy for the vortcty, streamwse and normal velocty coonents. Boundary condtons of sectral coeffcents of vortcty can be derved from the vortcty defnton and wall non-sl condton as follows: ˆ ˆ () t u() t 0 0 ˆ 0 0 ( ) ( t) uˆ ( t) ( ) uˆ (8) ˆ Because of current tme ste s unknown, on the wall can t been solved from (8) drectly. In ths aer the teraton method s adoted to solve the boundary condtons of ˆ n every tme ste. The detaled rocess s as follows: ) Solve the ˆ on wall from (8) wth uˆ of revous tme ste. ) Solve the equaton (5) and (6) for ˆ and uˆ. 3) Coare current uˆ wth that of revous tme ste. If the dscreancy s greater than the gven crteron, return to ste ) wth current u. In the total numercal rocess the most coutng tme s sent on the FFT and IFFT for sectral method, so ths teraton rocess doesn t remarkably ncrease the coutng tme. The coutatonal results ndcated that method can rovde satsfactory accuracy. Boundary condtons of streamwse and normal veloctey-sectral coeffcents can be nduced smlarly: ˆ uˆ ( t) 0, ( ) uˆ ( t) vˆ ( t) 0, ( ) vˆ ( t) 0 (9) 0 0 Equatons of vortcty-sectral coeffcent and veloctey-sectral coeffcents wth corresondng boundary condtons coose to resectve closed equatons sets, whch can be solved and the corresondng sectral coeffcents can be obtaned. The tme advancement s carred out by sem-lct scheme: Crank-Ncolson for the vscous terms and Adams-Bashforth for the nonlnear terms. The detaled dscrete rocess can be consulted n Reference [4]... Coutatonal Model Two-dmensonal channel flow s chosen as the numercal model. The flow geometry and the coordnate system are shown n Fgure. The non-dmensonal sze of coutatonal doman s [0, ] [,]. Unform grds are aled n the streamwse drecton and non-unform grds n the normal drecton as follows: x x ( )/( N ) = N n y j cos( j), j ( j)/( N ) = N j where x n s the non-dmensonal wdth of coutatonal doman n the streamwse drecton. N and N are the grd numbers n the streamwse drecton and normal drecton resectvely..3. Small-Perturbaton Analyss The attenuaton of small erturbaton n lamnar Poseulle flow and lnear growth n the transton rocess of small erturbaton are resectvely smulated to rove the accuracy and stablty of the roosed algorthm. The coutaton s carred out wth 460 grd onts (64 65) for a Reynolds number 500, whch s lower than the transton crtcal Reynolds number. The tme ste s 0.00, whch satsfes CFL stablty condton. The coutaton lasts tll the soluton s steady. The ntal flow feld s lamnar Poseulle flow wth a small Fgure. Coordnate system of two-dmensonal channel.

4 93 erturbaton. Fgure (a) shows the devaton between ntal flow feld and Poseulle flow. Fgure (b) shows that the rofle of steady velocty soluton s consstent (a) (b) wth that of Poseulle flow. Fgure (c) shows the varaton of maxmal devaton wth tme between coutatonal flow feld and Poseulle flow. It can be seen that the maxmal devaton gradually reduces and aroxmately equals zero, even less than 0 8. Coutatonal results reflect the attenuaton of small erturbaton n lamnar flow and rove the accuracy and stablty of the roosed algorthm. The Reynolds number s ncreased to 7500, whch s hgher than the transton crtcal Reynolds number. The objectve s to calculate the lnear growth rate of small erturbaton wth the roosed algorthm n ths aer and coare that wth the results by solvng the Orr- Sommerfeld equaton. The ntal flow feld wth small erturbaton s set as follows: uxy (, ) y u, vxy (, ) v where u and v are n accordance wth the most nstable model of stablty theory wth the erturbaton wave number.0 and altude = Knetc ene- rgy of erturbaton s defned as: () ( ) 0 E t u v dxdy Accordng to the lnear theory of small erturbaton, the knetc energy ncreases exonentally wth tme, ct Et () E(0) e. The lnear growth rate c s by solvng the Orr-Sommerfeld equaton. Fgure 3 shows that the coutatonal results of knetc energy of erturbaton wth the roosed algorthm n resent aer are n good agreement wth the theoretcal ones by solvng the Orr-Sommerfeld equaton. 3. Wall Shear Stress Analyss Y. S. Park et al. nvestgated the effect of erodc blowng and sucton on a turbulent boundary layer at three dfferent blowng-sucton angles ( 60, 90 and 0 ) (c) Fgure. (a) Contrast between ntal velocty rofle and Poseulle rofle; (b) contrast between solved steady velocty rofle and Poseulle rofle; (c) varaton of maxmal devaton wth tme between coutatonal flow feld and Poseulle flow, the fgure nsde s artal magnfcaton ( u and u a are the coutatonal streamwse velocty and Poseulle velocty). Fgure 3. Lnear growth of small erturbaton.

5 94 wth PIV. They found that a better effectveness of wall shear stress reducton was obtaned f the blowng-sucton angle was 0 rather than 90 [9]. The blowng and sucton at ths angle changed the streamwse and normal veloctes. Whle the scalar vortcty s defned as v u, the vortcty s obvously changed by ths x y control method. So t cam be redcted that the near-wall vortces are also the crucal nfluence factors to wall shear stress. A smulaton of two-dmensonal channel for the Reynolds number 7500 wth 856 grd onts (64 9) s carred out to verfy ths redcton. The ntal flow feld s Poseulle flow wth the most unstable erturbaton whch s solved from lnear erturbaton theory. The turbulence statstcs almost no longer change after the non-dmensonal tme s 600. The coutaton s contnued for 00 tme-stes and the results are as the sale database. The turbulent statstcs (such as mean velocty, root-mean-square velocty fluctuatons and Reynolds shear stress normalzed by frcton velocty) are show n Fgure 4, whch agree well wth those n reference []. The wall shear stress s defned as w u/ y, w where the subscrt w denotes the wall. The relaton between the vortces above the wall and the wall shear stress can be analyzed wth two-ont correlaton: u R ( x,0) ( x, ry ) (0) y where ry s the satal dstance n y drecton and denotes an average over y and tme t. The two-ont correlaton functon s shown n Fgure 5. Note that the lace y = s the locaton of uer wall of the channel. It can be seen that the relaton between the near-wall vortces and the wall shear stress really exsts. Wth the ncrease of satal dstance n y drecton, the correlaton value radly ncreases to the maxmum eak, and then decreases reosefully wth the second eak occurrng. The satal dstances of the two eaks from the uer channel-wall are 0.03 and 0. resectvely (.e. y = 0.97 and y = 0.8). A further exresson for the relaton between near-wall vortces and the wall shear stress can be resented wth the nstantaneous dagrams of fluctuaton velocty vectors. Four successve nstantaneous dagrams of fluctuaton velocty vectors are shown n Fgure 6. It can be clearly seen that a ar of near-wall vortces moves downstream. The y-coordnate of the vortces center A s aroxmatvely 0.8, whch just corresonds to the second eak n Fgure 5. Ths s because the greater vortcty n the center causes the greater two-ont correlaton value. The streamwse velocty ncreases from the center to the outsde of the near-wall vortces, whle t s zero on the wall. So a velocty nflecton ont occurs n the nearwall regon. Ths can be clearly seen from Fgure 7, shear stress u y (a) (b) (c) Fgure 4. The turbulent statstcs of two-dmensonal turbulent channel flow: (a) the mean velocty; (b) the rootmean-square velocty fluctuatons (sold lne: u + rms ; dash lne: v + rms ); (c) Reynolds shear stress and total shear stress (sold lne: Reynolds shear stress; dash lne: total shear stress). Fgure 5. Two-ont correlaton between the near-wall vortcty and the wall shear rate (where y = s the locaton of uer wall of channel).

6 95 Fgure 7. A near-wall vortex model. (a) where the locaton B s just the nflecton ont. It s easy to see that the velocty of the nflecton ont affects the wall shear rate drectly. Therefore the nflecton ont B just corresonds to the maxmum eak n Fgure Conclusons (b) (c) A drect numercal smulaton of an ncoressble twodmensonal turbulent channel s erformed wth sectral method. The numercal results are used to examne the relaton between wall shear stress and near-wall vortces. It s found that the wall shear stress s assocated wth the near-wall vortces and the maxmum correlaton-value locaton of near-wall vortces s obtaned. It should be onted out that the three-dmensonal smulaton has not been carred out due to the shortage of coutatonal resources, but we beleve that the qualtatve concluson of near-wall sanwse vortces affectng wall shear stress exsts objectvely. Ths achevement s a good sulement to tradtonal understandng that the wall shear stress s affected by near-wall streamwse vortces. It rovdes a theoretcal gudance for the control of turbulent boundary layers. An otmal control method of turbulent boundary layer may be dscovered f the near-wall sanwse vortces are managed to be controlled as well as the streamwse vortces, whch s also our research focus for the next ste. 5. Acknowledgments We are grateful to Professor J. S. Luo and Dr H. L. Xao for ther hel n numercal rocess. We also thank the fnancal suort rovded for ths research by Natonal Natural Scence Foundaton of Chna (Grant No ) and Oen Foundaton of the Key Laboratory of Mechancs on Western Dsaster and Envronment of the Mnstry of Educaton of Chna. 6. References (d) Fgure 6. Instantaneous dagrams of fluctuaton velocty vectors. (a) t = 60; (b) t = 670; (c) t = 70; (d) t = 770. [] J. Shen, E. Malkel and J. Katz, Usng dgtal holograhc mcroscoy for smultaneous measurement of 3D

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