EXPERIMENTAL STUDY OF NEAR WALL TURBULENCE USING PIV

Transcription

1 EUROMECH 411 Rouen, 9-31 May EXPERIMENTAL STUDY OF NEAR WALL TURBULENCE USING PIV J. Carler, J. M. Foucaut and M. Stanslas LML URA 1441, Bv Paul Langevn, Cté Scentfque, Vlleneuve d'ascq Cedex, France SHORT ABSTRACT Experments have been carred out by means of DC Partcle Image Velocmetry n a specfc turbulent boundary layer wnd tunnel whch allows to reach Reynolds numbers based on momentum thcness from 75 to 19. Double spatal correlaton, Proper Orthogonal Decomposton and Pattern Recognton were appled to the velocty maps n dfferent drectons to analyse coherent structures. Some quanttatve nformaton about hgh and low speed streas, large scale motons and eddy structures have been obtaned and are dscussed. INTRODUCTION What s the contrbuton of coherent structures and of ther nteractons n the generaton and self - sustanng of wall turbulence? To answer such a queston, spato - temporal data comng from numercal methods such as DNS and LES or from expermental methods such as PIV can be of great help. In ths paper, experments have been carred out by means of PIV n a turbulent boundary layer. The wnd tunnel s 1 x m n cross secton and m n length. The external velocty can been vared between 3 and 1 m/s, gvng a Reynolds number based on momentum thcness between 75 and 19. The boundary layer thcness s about.35 m. PIV measurements were performed wth a x 33 mj ND - YAG laser, a Koda DCS 46 (48 x 37 px ) and a Pulnx TM-971 (484 x 768 px ). The PIV records were analysed wth an n house software based on multgrd process wth zero paddng. Auto - correlaton was used wth the frst camera and cross - correlaton wth the second one. Two multgrd teratons were performed, respectvely wth wndow szes of 64 x 64 px and 4 x 4 px. The correlaton peas were nterpolated usng a gaussan pea fttng algorthm. The raw velocty felds were cleaned usng the procedure descrbed by Westerweel (1994). Sngle holes were flled by nterpolaton. DOUBLE SPATIAL CORRELATION One of the most mportant tools of the statstcal approach to random phenomenon s the correlaton operator. Correlaton ndcates possble determnstc lns between a pror random varables. In steady flows, one can defne the double spatal correlaton coeffcent of turbulent velocty fluctuatons n the followng form : x R x, x u' x, t u' x dx, t dx (1) u' x, t u' x dx, t dx s the dstance between a movng pont where s the poston vector of a fxed pont n space, and the fxed pont, u s the th turbulent velocty fluctuaton component and <> can be a temporal or an ensemble average operator. In PIV, due to the large tme separaton between each records, ensemble averagng s used. Fg 1 presents the double spatal correlaton coeffcent R 11 n a plane parallel to the wall at = yu / = 1 (where u s the frcton velocty and s the nematc vscosty) and R = 75. A homogenety hypothess was used to average R 11 n both drectons. The same double spatal correlaton coeffcent n a plane normal to the wall and parallel to the flow s presented n Fg also at R = 75. The fxed pont s at = 1. Ths coeffcent s averaged only n the x drecton. 1

2 EUROMECH 411 Rouen, 9-31 May dz d Fgure 1 - Double spatal correlaton coeffcent R 11 n a plane parallel to the wall at = d d Fgure - Double spatal correlaton coeffcent R 11 n a plane normal to the wall and parallel to the flow at = 1 for the fxed pont. These fgures gve a good dea of the three dmensonal shape of R 11 around the fxed pont. Fg shows a downstream angle to the wall of about 15. Ths angle s comparable to the value often mentoned n the lterature for the bacs of the large scale motons. However, the shape of R 11 n the plane parallel to the wall s clearly ndcatve of a streay structure of the flow. Ths streay structure extends far beyond the sze of the PIV wndow n the x drecton, whch s also n agreement wth the ltterature PROPER ORTHOGONAL DECOMPOSITION Lumley (1967) dd propose to defne the coherent structures n a turbulent flow as the structures havng the largest mean square proecton on the velocty feld. Ths problem of maxmzaton leads to a Fredholm ntegral problem : n n n u' x, t u' x', t x' dx' x () where s the spatal doman under study, u' x, t u' x', t s the double spatal correlaton (see eqn 1) and n and n are respectvely the th of the component egenvector of order n and the correspondng egenvalue. A POD analyss was performed wth the snapshots method ntroduced by Srovch (1987a, b, c), whch corresponds to solve the followng egenvalue problem :

3 EUROMECH 411 Rouen, 9-31 May 1 where C ux, tm ux, t, u x, t ml l M x C ml a n n n a l m (3) are the M snapshots of the velocty felds avalable, C ml s a MxM square proecton matrx and a n are the proecton coeffcents of the n th egenvector on the th nstantaneous velocty map. The egenvectors are thus obtaned by : M n n a u x, t (4) x 1 In the present study, the egenvalues and egenvectors were computed n a plane normal to the wall and parallel to the flow, wth snapshots for R = 75 and 135 and 1 maps for R = 15 and 19. Fg 3 presents these egenvalues dvded by the turbulence netc energy for the four values of the Reynolds number. For the modes above n = 1 whch correspond to turbulence (mode 1 s the mean flow), the egenvalues appear farly ndependent of the Reynolds number. It seems that velocty maps are enough to reach convergence untl the th mode. The contrbuton of the frst ten modes after n = 1 to the turbulence netc energy s of the order of 4%. Ths convergence poses the problem of the number of sgnfcant modes needed to descrbe turbulence wth a dynamcal system...15 R = 65 R = 1 R = 135 R = 19 /(E- 1 ) mode Fgure 3 Egenvalues dstrbuton n a plane normal to the wall and parallel to the flow. Fg 4 shows the second egenvector for R = 75 (the frst one s the mean velocty profle). Ths egenvector loos very much le a bac of a large scale moton. The angle of ths structure s qute comparable to the one observed n the spatal correlaton of Fg. On the man front gong through the map, several eddy structures can be observed. 3

4 EUROMECH 411 Rouen, 9-31 May Fgure 4 Second egenvector n a plane normal to the wall and parallel to the flow. PATTERN RECOGNITION It s qute dffcult to defne what s an eddy structure. Many defntons were revewed by Jeong and Hussan (1995) : maxmum value of vortcty magntude ; maxmum value of the second nvarant of the velocty gradent tensor ; mnmum negatve second egenvalue of S + wth S and beng respectvely the symmetrc and ant - symmetrc parts of the velocty gradent tensor. However, the spatal coherence concept s not consdered n these defntons. Ferré and Gralt (1989a, b) propose to use the Pattern Recognton analyss based on convoluton to dentfy part of a HWA velocty sgnal smlar to a model whch defne a reference coherent structure. In a smlar way to Scarano et al. (1999), ths method has been appled to detect eddy structures n the velocty maps. The model s, n polar coordnate, a tangental velocty component wth gaussan dampng : / ur, exp r e Measured velocty maps are correlated wth the model. The extremum values of ths product ndcate the presence of eddy structures. Fg 5 shows on the left, the corotatng mean eddy structure detected by ths procedure n velocty maps normal to the wall and parallel to the flow at = 15 and R = 75. On the rght, the counterrotatng mean eddy structure s presented. These mean values were obtaned over 15 samples n the frst case and 175 n the second case. The mean eddy structure parameters were determned by fttng an Oseen vortex (see eqn 6) to the spatal velocty dstrbuton. e r ur, 1 exp r / r r (5) (6) 4

5 EUROMECH 411 Rouen, 9-31 May a Fgure 5 Plane normal to the wall and parallel to the flow at = 15 and R = 75 : (a) corotatng mean eddy structure ; (b) counterrotatng mean eddy structure. For the corotatng mean eddy structure, the radus and the vortcty are n wall unts : r + = r u / = 16 ; + = /u =.15. At ths locaton, the local mean velocty gradent s : du + /d =. and s ncluded n +. For the counterrotatng mean eddy structure one obtans r + = 18 and + =.75. The mean eddy structure observed n Fg 5a s farly comparable to those of the mean front of Fg 4 and to the structures observed by several authors (Falco (1977), Head and Bandyopadhyay (1981), Zhou et al. (1999), etc). b CONCLUSION Three analyss methods based drectly or ndrectly on the correlaton operator were used to study the coherent structure n the turbulent boundary layer from PIV expermental data. The frst s the double spatal correlaton, the second s the Proper Orthogonal Decomposton and the thrd s the Pattern Recognton. These methods have allowed to dentfy eddy structures on slowly rasng fronts and to obtan some quanttatve nformaton about these coherent structures. REFERENCES Falco R. E. (1977), Coherent motons n the outer regon of turbulent boundary layers., Phys. Fluds, Vol., p Ferré J. A. & Gralt F. (1989a), Pattern-recognton analyss of the velocty feld n plane turbulent waes., J. Flud Mech., Vol. 198, p 7 64 Ferré J. A. & Gralt F. (1989b), Some topologcal features of the entranment process n a heated turbulent wae., J. Flud Mech., Vol. 198, p Head M. R. & Bandyopadhyay P. (1981), New aspects of turbulent boundary layer structure., J. Flud Mech., Vol. 17, p Jeong J. & Hussan F. (1995), On the dentfcaton of a vortex., J. Flud Mech., Vol. 85, p Lumley J. (1967), The structure of nhomogeneous turbulent flows., Atmospherc Turbulence and Rado Wave Propagaton, Yaglom A. M. & Tatars V. I., Mosow : Naua, p Scarano F., Benocc C. & Rethmuller M. L. (1999), Pattern-recognton analyss of the turbulent flow past a bacward facng step., Phys. Fluds, Vol. 11, No 1, p Srovch L. (1987a), Turbulence and the dynamcs of coherent structures. Part 1 : Coherent structures., Quarterly of Appled Mathematcs, Vol. XLV, No. 3, p Srovch L. (1987b), Turbulence and the dynamcs of coherent structures. Part : Symmetres and transformatons., Quarterly of Appled Mathematcs, Vol. XLV, No. 3, p Srovch L. (1987c), Turbulence and the dynamcs of coherent structures. Part 3 : Dynamcs and scalng., Quarterly of Appled Mathematcs, Vol. XLV, No. 3, p Westerweel J. (1994), Effcent detecton of spurous vectors n partcle mage velocmetry data., Experments n Fluds, Vol. 16, p Zhou J., Adran R. J., Balachandar S. & Kendall T. M. (1999), Mechansms for generatng coherent pacets of harpn vortces n channel flow., J. Flud Mech., Vol. 387, p