THE TOPOLOGY OF SURFACE SKIN FRICTION AND VORTICITY FIELDS IN WALL-BOUNDED FLOWS

THE TOPOLOGY OF SURFACE SKIN FRICTION AND VORTICITY FIELDS IN WALL-BOUNDED FLOWS M.S. Chng Department f Mechanical Engineering The University f Melburne Victria 3010 AUSTRALIA min@unimelb.edu.au J.P. Mnty and I. Marusic Department f Mechanical Engineering The University f Melburne Victria 3010 AUSTRALIA ABSTRACT In previus studies, the three invariants f the velcity gradient tensr have been used t study turbulent flw structures. Fr incmpressible flw the first invariant P is zer and the tplgy f the flw structures can be investigated in terms f the secnd and third invariants, Q and R respectively. Hwever, these invariants are zer at a n slip wall and can n lnger be used t identify and study structures at the surface in a wall-bunded flw. At the wall, the flw field can be described by a n slip Taylr-series expansin. Like the velcity gradient tensr, it is pssible t define the invariants P, Q and R f the n slip tensr. In this paper it will be shwn hw the tplgy f the flw field n a n slip wall can be studied in terms f these invariants. INTRODUCTION A critical pint is a pint in a flw field where the velcity u 1 = u 2 = u 3 = 0 and the streamline slpe is indeterminate. Clse t the critical pint the velcity field can be described by the linear terms f a Taylr-series expansin. Tw types f critical pints can be described: ne is the free slip critical pint which is lcated away frm a n slip wall and the ther is the n slip critical pint which ccurs n a n slip wall. Free-slip critical pints Fr free slip critical pints, the velcity u i in x i space is given by. u 1 x 1. dx 1 /dt A 11 A 12 A 13 x 1 u 2 = x 2. = dx 2 /dt = A 21 A 22 A 23 x 2 u 3 x 3 dx 3 /dt A 31 A 32 A 33 x 3 where A i j = u i / x j is the velcity gradient tensr. The invariants f the velcity gradient tensr Fr an bserver mving in a nn-rtating frame f reference with any particle in a flw field, the flw surrunding the particle is described in terms f the nine cmpnents f the velcity gradient tensr A i j. The characteristic equatin f A i j is λ 3 i + Pλ 2 i + Qλ i + R = 0, (2) where λ i are the eigenvalues and P, Q and R are the tensr invariants which are defined as P = (A 11 + A 22 + A 33 ) Q = A 11 A 22 A 12 A 21 + A 11 A 33 A 13 A 31 +A 22 A 33 A 23 A 32 and R = A 11 A 23 A 32 A 11 A 22 A 33 + A 12 A 21 A 33 A 12 A 23 A 31 A 13 A 21 A 32 + A 13 A 22 A 31 (3) The characteristic equatin (2) can have (i) all real rts which are distinct, (ii) all real rts where at least tw rts are equal, r (iii) ne real rt and a cnjugate pair f cmplex rts. In the P Q R space, the surface which divides the real slutins frm the cmplex slutins is given by 27R 2 + (4P 3 18PQ)R + (4Q 3 P 2 Q 2 ) = 0 (4) r. x i = dx i = A i j x j (1) dt Fr incmpressible flw, the first invariant P is zer and hence all free-slip critical pints can be described by the secnd and third invariants, i.e. Q and R respectively. The invariants 1

f the velcity gradient tensr have been used t study turbulent flw structures in rder t extract infrmatin regarding the scales, kinematics and dynamics f these structures (see Davidsn, 2004, pg. 268 and Elsinga & Marusic, 2010). These invariants can als be used t study turbulent structures using data frm Direct Numerical Simulatins (DNS), fr example in the DNS f hmgeneus istrpic turbulence by Oi et al. (1999) and in wall-bunded shear flws by Chng et al. (1998). N slip critical pints In wall-bunded flws, there is n slip at the wall, i.e. u 1 = u 2 = u 3 = 0 when x 3 = 0, where x 3 is the wall-nrmal directin. If the rigin f a Taylr-series expansin is lcated n the n slip wall, i.e. at x 3 = 0, the linear terms f the expansin are given by u 1 0 0 A 13 x 1 u 2 = 0 0 A 23 x 2 u 3 0 0 0 x 3 where A 13 = u 1 / x 3 and A 23 = u 2 / x 3 ; x 1 is the streamwise directin and x 2 is the spanwise directin. All the invariants f the abve tensr are zer at the wall and hence they cannt be used t map the tplgy f the flw pattern at a n slip surface. T satisfy the n slip cnditin, the Taylr-series expansin at the wall (i.e. at x 3 = 0) can be re-written t include higher rder terms and at the same time satisfy the n slip bundary cnditin. The expansin is given by r x i = dx i dτ = A i jx j. (6) In this paper, the A i j will be referred t as the n slip tensr. The abve vectr field can be integrated with τ t generate surface streamlines (limiting streamlines r skin frictin lines). The invariant f the n slip tensr T satisfy bundary cnditins n a n slip surface, the n slip tensr is given by A 11 A 12 A 13 A i j = A 21 A 22 A 23 (7) 0 0 1 2 (A 11 + A 22 ) By substitutin f the abve expansin int the Navier-Stkes equatin, it can be shwn that A 11, A 12, A 21 and A 22 are related t the vrticity gradients and A 13 and A 23 are related t pressure gradients. Like the velcity gradient tensr, the n slip tensr has three invariants P, Q and R. Fr incmpressible flw, the first invariant P is n lnger zer and is given by P = 1 2 (A 11 + A 22 ). (8) u 1 = A 13 x 3 + (A 11 x 1 + A 12 x 2 + A 13 x 3 )x 3 u 2 = A 23 x 3 + (A 21 x 1 + A 22 x 2 + A 23 x 3 )x 3 u 3 = (A 31 x 1 + A 32 x 2 + A 33 x 3 )x 3 Since u 1 = u 2 = u 3 = 0 at the wall (at x 3 = 0), we cannt integrate the velcity field at the wall t btain the surface flw pattern r skin frictin lines. Hwever, by defining x i as x= dx i dτ where dτ = x 3 dt. the n slip velcity field can be expressed as u 1 = x 1 = dx 1 x 3 dτ = A 13 + (A 11 x 1 + A 12 x 2 + A 13 x 3 ) u 2 = x 2 = dx 2 x 3 dτ = A 23 + (A 21 x 1 + A 22 x 2 + A 23 x 3 ) u 3 = x 3 = dx 3 x 3 dτ = (A 31x 1 + A 32 x 2 + A 33 x 3 ) A critical pint ccurs n the wall when A 13 = A 23 = 0 and at this critical pint x 1 dx 1 /dτ A 11 A 12 A 13 x 1 x 2 = dx 2 /dτ = A 21 A 22 A 23 x 2 x dx 3 3 /dτ A 31 A 32 A 33 x 3 (5) It can als be shwn that the relatinship between the three invariants f the n slip tensr is given by 2P 3 + PQ + R = 0. (9) SURFACE SKIN FRICTION AND SURFACE VORTICITY VECTOR FIELDS The main aim f this study is t investigate the tplgy f skin frictin fields and the surface vrticity field at the wall in wall-bunded flws. Wall shear stress r skin frictin is defined as the nrmal derivative f the velcity vectr at the wall. Hence the directin f the near-wall velcity vectr when prjected nrmal t the wall is in the same directin as the wall shear stress vectr, i.e. surface streamlines are knwn as skin frictin lines. Figure 1 shws skin frictin lines in a streamwise-spanwise plane at the n slip wall using data frm the Direct Numerical Simulatin (DNS) f channel flw by del Alam et al. (2004). The simulatin is fr Re τ = hu τ /ν = 934 where h is the channel half height, ν is the kinematic viscsity and u τ is the frictin velcity. Skin frictin lines (limiting streamlines at the n slip wall) are btained frm integrating the streamwise velcity u 1 and spanwise velcity u 2 in the first plane frm the wall, and it generally assumed that this first plane is very clse t the wall. Hw clse this plane has t be t prduce sufficiently accurate skin frictin lines, is an pen debatable issue. In mst cmputatins f wall bunded flws it is generally 2

Figure 1. Typical surface skin frictin lines at the wall using data frm DNS f channel flw at Re τ = 934 (see del Alam et al., 2004 fr details f cmputatins). The regin shwn is fr a randm area f the entire channel wall in the cmputatinal dmain. Figure 3. Surface vrtex lines (red) superimpsed n the surface skin frictin field shwn in figure 2. Surface vrtex lines are rthgnal t the surface skin frictin lines. Nte that critical pints in the surface skin frictin field are als critical pints in the surface vrticity field. A saddle in the skin frictin field is als a saddle in the vrticity field. A nde in the skin frictin field is a fcus in the vrticity field. Figure 2. Surface skin frictin field shwing critical pints (a saddle and a nde) at the wall frm the same data as that in figure1. agreed that the first plane is such that x 3 + = x 3u τ /ν 1. In the cmputatin by del Alam et al. (2004), the first plane ff the wall is fr x 3 + = 0.0313. The surface skin frictin lines shwn in figure 1 are fr a small regin f the flw, i.e. fr x 1 + x+ 2 plane 300 925, the size f the entire streamwisespanwise channel wall, in viscus units, is 3072 2304. The figure is fairly typical f surface skin frictin lines which cnsist f bifurcatin lines which are assciated with surface streak lines and streamwise vrtical mtins (r streamwise eddying mtins) at the wall. A questin this study seeks t answer is: Are there critical pints in the surface skin frictin field? This questin is best addressed by cnsidering the surface vrticity lines which are rthgnal t the surface skin frictin lines (see Chng et al., 1998). It is cnjectured in the DNS f wall bunded flws, that since the flw is peridic in the spanwise directin and the utflw is re-cycled and fed back as inflw, the surface skin frictin field at the n slip wall can be mapped n t the surface f a trid. The Pincare-Hpf index therem r Pincare Bendixsn therem (see Hunt, et al., 1978, Lighthill, 1963 r Flegg, 1974) states that the tplgy f a smth vectr field n a tridal surface is such that the number f ndes minus the number f saddles is equal t zer, a saddle having a Pincare index f -1 and a nde (r a fcus) having a Pincare index f +1. Hence, if the skin frictin field fr the entire n slip wall is similar t that shwn in figure 1, where n critical pints can be seen, then the Pincare-Hpf index therem is satisfied. This led t the initial cnclusin that there are prbably n critical pints n the surface f a n slip wall since a single critical pint (a single saddle r a single nde) will vilate the Pincare-Hpf therem fr the tplgy f the flw field n the surface f a tridal surface. Hwever, cntrary t expectatins, there are critical pints in the surface skin frictin field at the wall. Figure 2 shws a clse-up (x 1 + x+ 2 plane 100 225) frm anther area f a surface skin frictin field using data frm the same cmputatin as that used t prduce figure 1. This figure shws an example f critical pints which are fund t exist n the n slip wall; these critical pints cme as a saddle-nde pair s that the Pincare index arund the pair f critical pints is such that it is zer arund a clsed circuit enclsing the tw critical pints. Critical pints in the surface skin frictin field are als critical pints in the surface vrticity field. In fact, all critical pints n the surface cme as a saddle-nde pair s that fr the entire surface in the cmputatinal dmain, the Pincare-Hpf index therem is satisfied. The tplgy f the surface vrticity field als satisfies the cnditin that the surface vrtex lines must frm clsed lps and can nly crss at critical pints in vrticity. Critical pints can als be fund n the surface using data frm the DNS f a pipe flw by Chin et al. (2010) and als in the recent spatial cmputatin f a zer pressure gradient turbulent bundary layer by Wu & 3

Min (2009). Taylr-series expansin slutins f the Navier-Stkes equatins T investigate the three-dimensinal structure f the flw n a n slip surface, the technique described in Perry & Chng (1986) will be used t generate smth three-dimensinal skin frictin vectr fields which are lcal slutins f the Navier- Stkes equatins. The velcity field fr a cmplex flw can be represented by Taylr-series expansins f arbitrary rder N as N u i = S(a i,b i,c i )x a i 1 xb i 2 xc i 3 (10) n=0 where i = 1,2,3, and x i are the rthgnal spatial crdinates. (a i,b i,c i ) uniquely specify the terms in the expansin and the pwers f x 1, x 2 and x 3 respectively. The factr 1 S is given by and S = (a i + b i + c i )! a i!b i!c i! (11) n = a i + b i + c i (12) The abve representatin f the expansin fr the velcity field is ideal fr generating algrithms s that cmputer prgrams can be written t btain Taylr-series expansin fr the velcity, t arbitrary rders. Once generated, the expansins in the abve ntatin can be translated t a mre cnventinal ntatin. Fr example, the third rder Taylr-series expansin fr the streamwise velcity u 1, in a cnventinal ntatin, is given by u 1 = P 1 + P 2 x 1 + P 3 x 2 + P 4 x 3 + P 5 x 2 1 + P 6x 2 2 + P 7x 2 3 + 2P 8x 1 x 2 + 2P 9 x 1 x 3 + 2P 10 x 2 x 3 + P 11 x 3 1 + P 12x 3 2 + P 13x 3 3 + 3P 14x 2 1 x 2 + 3P 15 x 2 1 x 3 + 3P 16 x 1 x 2 2 + 3P 17x 1 x 2 3 + 3P 18x 2 2 x 3 + 3P 19 x 2 x 2 3 + 6P 20x 1 x 2 x 3 (13) where the P s are the cefficients f the Taylr expansin. The expansin fr the spanwise velcity u 2 and the wall nrmal velcity u 3 are similar t the abve expansin, except that the cefficients are dented by Q s and R s respectively 2. It can be shwn that the number f unknwn cefficients fr a N th -rder expansin is given by N K+1 N c = 3 J (14) K=0 J=0 1 S was intrduced in the riginal technique fr generating the Taylr-series expansins and is retained in the analysis given in this paper - fr details, see Perry & Chng (1986). 2 Nte that P s, Q s and R s are cefficients f the expansin fr the velcity field and nt t be cnfused with the invariants f the velcity gradient tensr. 4 The Navier-Stkes equatin and cntinuity equatin in tensr ntatin 3 is given by u i t u i + u j = p + ν x j x i 2 u i (15) x j x j u i x i = 0 (16) where p is kinematic pressure = p/ρ and ν is the kinematic viscsity. Differentiating the velcity expansins (13), substitutin int the cntinuity equatin (16), and equating like pwers f x 1, x 2 and x 3, relatinships between the cefficients can be generated. Examples f cntinuity relatinships generated frm equatin (16) are: P 2 + Q 3 + R 4 = 0 P 5 + Q 8 + R 9 = 0 P 8 + Q 6 + R 10 = 0... etc (17) Fr an N th -rder expansin, the number f these relatinships is given by N n E c = J (18) n=0 J=0 By equating crss-derivatives f pressure in the Navier-Stkes equatins, and by cllecting terms fr like pwers f x 1, x 2 and x 3, the Navier-Stkes relatinships can be generated. A typical example f a Navier-Stkes relatinship fr a fifth rder expansin is given belw: R 8 P 10 +R 2 P 8 + R 3 Q 8 + R 4 R 8 + R 5 P 3 + R 6 Q 2 +R 8 P 2 + R 8 Q 3 + R 9 R 3 + R 10 R 2 + 3R 14 P 1 +3R 16 Q 1 + 3R 20 R 1 P 2 P 10 P 3 Q 10 P 4 R 10 P 6 Q 4 P 7 R 3 P 8 P 4 P 9 P 3 P 10 Q 3 P 10 R 4 3P 18 Q 1 3P 19 R 1 3P 20 P 1 12ν(R 24 + R 26 + R 35 P 33 P 28 P 29 ) = 0 (19) Fr an N th -rder expansin, the number f these relatinships is given by N n 1 E NS = (2J 1) (20) n=3 J=2 Nte that fr time-dependent flw, these relatinships are rdinary differential equatins (hence determining the dynamics f the flw). Fr steady flw prblems these relatinships, 3 Using Einstein s ntatin where repeated indices implies summatin ver indices 1,2 and 3.

u 3 = R 7 x 2 3 + R 13x 3 3 + 3R 17x 1 x 2 3 + 3R 19x 2 x 2 3 (21) T further simplify the prblem, the flw is assumed t be symmetrical abut the x 1 x 3 plane, i.e. u 1 is even in x 2, hence P 10 = P 19 = P 20 = 0 and u 2 is dd in x 2, hence Q 4 = Q 7 = Q 9 = Q 13 = Q 15 = Q 17 = Q 18 = 0. Frm cntinuity R 19 = (P 20 + Q 18 ) = 0 and the expansins simplify t u 1 = P 4 x 3 + P 7 x 2 3 + 2P 9x 1 x 3 + P 13 x 3 3 Figure 4. Lcal three-dimensinal separatin pattern. Flw is in the x 1 directin. x 1 x 2 is the n slip surface and x 1 x 3 is a plane f symmetry. like the cntinuity relatinships, are algebriac (kinematic) relatinships. The number f unknwn cefficients fr different rder expansins and the number f cntinuity and Navier-Stkes relatinships generated is given in the fllwing table. N N c E c E NS 0 3 0 0 1 12 1 0 2 30 4 0 3 60 10 3 4 105 20 11 5 168 35 26........ 15 2448 680 1001 In all cases the ttal number f relatinships generated exceeds the ttal number f unknwn cefficients f a given rder expansin N, and hence further equatins are needed fr clsure. Three-dimensinal separated flw n a n slip wall A third rder slutin fr separated flw abve a n-slip surface will be used t illustrate hw a slutin can be generated. Applying the n-slip cnditin (all cefficients withut x 3 are zer) and cntinuity, the third-rder Taylr-series expansins fr the velcity field given in (13) expansin are reduced t u 1 = P 4 x 3 + P 7 x 2 3 + 2P 9x 1 x 3 + 2P 10 x 2 x 3 + P 13 x 3 3 + 3P 15 x 2 1 x 3 + 3P 17 x 1 x 2 3 + 3P 18x 2 2 x 3 + 3P 19 x 2 x 2 3 + 3P 15 x 2 1 x 3 + 3P 17 x 1 x 2 3 + 3P 18x 2 2 x 3 u 2 = 2Q 10 x 2 x 3 + 3Q 19 x 2 x 2 3 + 6Q 20x 1 x 2 x 3 u 3 = R 7 x 2 3 + R 13x 3 3 + 3R 17x 1 x 2 3 (22) The abve expansin has t satisfy cntinuity, i.e. P 9 + Q 10 + R 7 = 0 P 15 + Q 20 + R 17 = 0 P 17 + Q 19 + R 13 = 0 (23) Fr a steady slutin f the Navier-Stkes equatin, R 17 P 15 P 18 P 13 = 0 (24) The n-slip velcity field is given by x 1 = u 1 /x 3 = P 4 + P 7 x 3 + 2P 9 x 1 + P 13 x 2 3 + 3P 15 x1 2 + 3P 17x 1 x 3 + 3P 18 x2 2 x 2 = u 2 /x 3 = 2Q 10 x 2 + 3Q 19 x 2 x 3 + 6Q 20 x 1 x 2 x 3 = u 3 /x 3 = R 7 x 3 + R 13 x3 2 + 3R 17x 1 x 3 (25) Further equatins can be generated by specifying the lcatin and prperties f critical pints in the flw field. Fr example, in generating a separated flw pattern, the prperties f separatin/reattachment pints can be expressed as x 1 a 0 B x 1 x 2 = 0 na 0 x 2 (26) x 3 0 0 1 2 a(n + 1) x 3 Fr an angle f separatin/separatin θ, B is given by + 6P 20 x 1 x 2 x 3 u 2 = Q 4 x 3 + Q 7 x3 2 + 2Q 9x 1 x 3 + 2Q 10 x 2 x 3 + Q 13 x3 3 + 3Q 15 x1 2 x 3 + 3Q 17 x 1 x3 2 + 3Q 18x2 2 x 3 + 3Q 19 x 2 x3 2 + 6Q 20 x 1 x 2 x 3 B = 2 1 a(n + 3) tanθ (27) 5

Using a few simple parameters, varius threedimensinal flw patterns can be generated. Figure 4 shws the lcal surface skin frictin field, cnsisting f a saddle-nde pair, which is similar t the tplgy f the skin frictin field arund critical pints fund n the wall in the DNS f wall bunded flws. Figure 4 als shws the three-dimensinal structure abve the surface (in the x 1 x 3 plane) indicating the saddle-nde pairs are the ftprints f lcal three-dimensinal separated flw at the surface, caused by lcal vrticity and pressure gradients. Althugh ne wuld expect that the flw structures t be tw dimensinal n the surface, the abve study shws that the structures at the surface are three dimensinal. Hwever, these instantaneus lcal vrtical structures are exceedingly small and the significance f these structures and effects n turbulent structures further frm the wall has yet t be investigated. In an incmpressible flw, vrticity is nly generated at the n-slip wall. Many questins remained unanswered, fr example: Is lcal instantaneus flw separatin necessary fr the creatin f three-dimensinal vrtical structures at the wall? The invariants f the n slip tensr at the wall may als be used t help lead t a further understanding f flw structures at the wall. Figure 5 is a scatter plt shwing the distributin f the three n slip invariants fr the channel flw simulatin f del Alam et al. and figure 6 shws that the secnd and third invariants can be nrmalized as indicated by equatin (9). These tw nrmalized invariants can be used t describe structures n the surface f a n slip wall in a wall bunded flw simulatin. Hw these invariants are related t the tplgical structures (critical pints) in the surface skin frictin and surface vrticity fields is the subject f future wrk in this study f near wall structures in wall bunded flws. Figure 5. The invariants f the n slip tensr fr the DNS f channel flw. CONCLUSION A initial study f the tplgical structure f the surface skin frictin field and surface vrticity field is described in this paper. The imprtance f these lcal structures n much larger mean flw structures fund in turbulent wall bunded flws has yet t be fully investigated. ACKNOWLEDGEMENTS The authrs will like t acknwledge the use f data frm varius DNS f wall bunded flws. This wrk is funded by ARC Grant DP1093585. Figure 6. The invariants f the n slip tensr nrmalised (see equatin (9) ). REFERENCES [1] Chin, C., Oi, A.S.H., Marusic, I., and Blackburn, H., 2010, The influence f pipe length n turbulence statistics cmputed frm direct numerical simulatin data. Phys. Fluids 22, 115107, 1-10. [2] Chng, M.S., Sria, J., Perry, A.E., Chacin, J., Cantwell, B.J. and Na, Y., 1998, Turbulence structures f wallbunded shear flws fund using DNS data. J. Fluid Mech., 357:225-247. [3] Davidsn P.A. Turbulence Oxfrd University Press. 2004. [4] Elsinga, G.E. and Marusic, I., 2010, Evlutin and lifetimes f flw tplgy in a turbulent bundary layer. Physics f Fluids, 22, 2010, 015102. [5] Hunt, J.C.R., Abell, C.J., Peterka, J.A., W, H., 1987, Kinematic studies f the flws arund free r surfacemunted bstacles; applying tplgy t flw visualizatin. J. Fluid Mech., 86:179-200. [6] Lighthill, M.J., 1963, Attachment and separatin in threedimensinal flw. In Laminar Bundary Layers, ed. L. Rsenhead, pp. 72-82. Oxfrd: Oxfrd University Press. [7] Flegg, G.C., 1974, Frm Gemetry t Tplgy. English Universities Press. [8] Oi, A., Chng, M.S. and Sria, J., 1999, A study f the invariants assciated with the velcity gradient tensr in hmgeneus flws. J. Fluid Mech., 381:141-174. [9] Perry, A.E. and Chng, M.S., 1987, A descriptin f eddying mtins and flw patterns using critical-pint cncepts. Ann. Rev. Fluid Mech., 19:125-155. [10] Perry, A.E. and Chng, M.S., 1986, A series-expansin study f the Navier-Stkes equatins with applicatin t three-dimensinal separatin patterns. J. Fluid Mech., 173:207-223. [11] Wu, Xu. and Min, P., 2009, Direct numerical simulatin f turbulence in a nminally zer-pressure-gradient flat-plate bundary layer. J. f Fluid Mech., 630:5-41. 6