Analysis of the two-point velocity correlations. in turbulent boundary layer ows. By M. Oberlack

Transcription

1 Center for Turbulence Research Annual Research Brefs Analyss of the two-pont velocty correlatons n turbulent boundary layer ows By M. Oberlack. Motvaton obectves Two-pont Rapd Dstorton Theory (RDT) has become an mportant tool n the theory of homogeneous turbulence. Modelers try to mplement approprate results from RDT n ther statstcal turbulence models, for example n the structure based model developed by Kassnos Reynolds (994). On the other h, n non-homogeneous equlbrum ows the logarthmc law s one of the cornerstones n statstcal turbulence theory. Expermentalsts have found the log-law n a broad varety of derent turbulent wall shear ows, statstcal models have been made to be consstent wth the log-law. The logarthmc law was rst derved by von Karman (930a, 930b) usng dmensonal arguments. Later Mllkan (939) derved the law-of-the-wall more formally usng the so called \velocty defect law", also rst ntroduced by von Karman (930b). Even though the dervaton was much more comprehensve from a physcal pont of vew, the velocty defect law s essentally an emprcal observaton. A rst dervaton of the law-of-the-wall usng asymptotc methods n the Naver- Stokes equatons was gven by Mellor (972). Mellor needed the vscous sub-layer to obtan the log-regon, hs scalng of the nertal range n the log-regon s n error because t does not gve the one-pont lmt of producton equals dsspaton. The general obectve of the present work s to explore the use of RDT n analyss of the two-pont statstcs of the log-layer. RDT s applcable only to unsteady ows where the non-lnear turbulence-turbulence nteracton can be neglected n comparson to lnear turbulence-mean nteractons. Here we propose to use RDT to examne the structure of the large energy-contanng scales ther nteracton wth the mean ow n the log-regon. The contents of the work are twofold: Frst, two-pont analyss methods wll be used to derve the law-of-the-wall for the specal case of zero mean pressure gradent. The basc assumptons needed are one-dmensonalty n the mean ow homogenety of the uctuatons. It wll be shown that a formal soluton of the two-pont correlaton equaton can be obtaned as a power seres n the von Karman constant, known to be on the order of 0:4. In the second part, a detaled analyss of the two-pont correlaton functon n the log-layer wll be gven. The fundamental set of equatons a functonal relaton for the two-pont correlaton functon wll be derved. An asymptotc expanson procedure wll be used n the log-layer to match Kolmogorov's unversal range the one-pont correlatons to the nvscd outer regon vald for large correlaton dstances.

2 20 M. Oberlack 2. Governng equatons of the two-pont velocty correlaton functon Usng the stard Reynolds decomposton U =u + u P =p + p, the Reynolds averaged Naver Stokes (RANS) k = p u u k k 2 k the uctuaton equaton, later on referred to as N -equaton, s N @u k + u k u k The correspondng contnuty equatons are u 2 k =0 : k k =0 : (3) The ve two-pont correlaton tensor functons that appear n the two-pont correlaton equaton (5), further below, are dened as R (x r t) =u (x t) u (x () t) pu (x r t) =p(x t) u (x () t) u p(x r t) =u (x t) p(x () t) R (k) (x r t) =u (x t) u k (x t) u (x () t) R (k) (x r t) =u (x t) u (x () t) u k (x () t) : (4) All tensors n (4) are functons of the physcal the correlaton space coordnates x r = x () ; x respectvely. The double two-pont correlaton R, later on smply referred to as two-pont correlaton, converges to the Reynolds stress tensor u u n the lmt of zero separaton r. The well known two-pont correlaton equaton (Rotta (972)) can be wrtten as DR (x (x t) = ;R k ; R k (k) R(k) ; R k x+r ; [u k (x + r t) ; u k k 2 k ; k R (5) For both two-pont velocty-pressure correlatons, u p pu aposson equaton can be derved. The of equaton (5) leads to a Posson equaton for pu,

3 Two-pont velocty correlatons n turbulent boundary layers 2 x 2 ū (x 2 ) x x 3 Fgure. Sketch of the coordnate system the mean velocty pu ; pu k 2 R l ; R l R l = k k (6) the leads to the correspondng Posson equaton for u 2 u p = k(x l k R l (7) where the vertcal lne means that the dervatve s taken wth respect to x but wll be evaluated at x + r. All of the dependent varables n (5){(7) have to satsfy the =0 p p =0 @r For the analyss of the self-smlar, two-pont correlaton equaton further below, two denttes are mportant. They can easly be derved from a geometrcal consderaton by nterchangng the two ponts x x () = x + r R (x r) =R (x + r ;r) u p(x r) =pu (x + r ;r) : (0) The latter denttes are the key elements for the dervaton of some boundary condtons for the deeper understng of the self-smlar two-pont correlatons. There exsts a smlar dentty for the trple correlaton whch wll not be used here.

4 22 M. Oberlack 3. The log-law { a self-smlar form of the two-pont correlaton equaton Asketch of the coordnate system the mean velocty eld adopted n the proceedng paper s gven n Fg.. Wthn ths subsecton t wll be shown that the logarthmc part of the law-of-the-wall mean velocty prole can be derved from the two-pont correlaton equaton hence from the Naver-Stokes equaton f there exsts a regme where the followng assumptons hold: the mean velocty s parallel to the wall the statstcs n that doman are ndependent of vscosty tme the Reynolds number s hgh no mean pressure acts on the ow eld. The last assumpton can be elmnated, but n ths approach t wll be focused on the zero pressure gradent case. Besde the above assumptons no other condtons are needed n order to determne the log-law mean velocty prole the selfsmlarty of the correlaton functons. Inferrng the above assumpton n the Reynolds averaged Naver-Stokes equatons (), t s easy to conrm that the gradent of the Reynolds stress tensor on the rght h sde s the only remanng term. Integrated one tme we obtan that u u s ndependent ofx. However, ths s not necessarly true for the two-pont correlaton tensor R. It could depend on x f the dependence vanshes n the zero separaton lmt. Ths can only be acheved by havng the followng dependence on a new varable ~r = rg(x) () where g(x) has to be determned later no other hdden dependence on x can be n the correlaton functons. Of course, the latter denton of ~r can be generalzed to derent unknown scalng functons for every component of r, but from equaton (5) t can be vered that only a sngle scalng functon exsts. Wth the above gven assumptons, denng x = ~x usng the @~x + g the R -equaton (5) = k 0=;R 2 du (x 2 ) dx 2 ; R 2 du (x 2 ) dx 2 x2 +r 2 ; [u (x 2 + r 2 ) ; u (x 2 @pu ~r k ; + @~r k ~r l k R(k) ; R (k) : (3)

5 Two-pont velocty correlatons n turbulent boundary layers 23 As mentoned above, there s no hdden x dependence n the correlaton functon therefore all ~x dervatves comng from (2a) have been omtted. Obvously, equaton (3) can only have a non-trval soluton, thus be ndependent of x, f all the coecents have the same functonal dependence on x. Hence, the followng set of derental equatons determne the u g dependence on g for = 2 3 du (x 2 ) dx 2 g (4) g du (x 2 ) = f (~r) 6= f 2 (x) u (x 2 + r 2 ) ; u (x 2 )=f 3 (~r) 6= f 4 (x) : (5) dx 2 x+r are addtonal consstency condtons for u g. The last equaton n (4) determnes g to depend only on x 2. Hence, the equatons (4) have two ndependent sets of solutons gven by g(x 2 )= c () 2 (x 2 ; c () ) u (x 2 )= c () 3 c () 2 ln(x 2 ; c () )+c() 4 (6) g(x 2 )=c (2) u (x 2 )=c (2) 2 c(2) x 2 + c (2) 3 (7) where the c (p) q 's are ntegraton constants or proportonalty factors. Obvously, only the rst set of equatons correspond to a boundary layer ow because the solutons (7) dene homogeneous shear turbulence whch contradcts the assumpton to be ndependent of tme. Both equatons (5) requre c () = 0 c () 2 can be absorbed n the correlaton functons. In common notaton we nally obtan u = u ln(x 2)+C (8) ~r = r x 2 (9) where u s dened as u = s du dx 2 x2 =0 : (20) Insertng (8) (9) nto equaton (3) multplyng by the von Karman constant the nal form of the R -equaton results:

6 24 M. Oberlack 0=;R 2 ; R 2 ; ln( + ~r 2 +~r 2 ~r k (2) + ~r l R ; R (k) k (2) where R = R u 2 R (k) = R (k) u 3 R (k) = R (k) u 3 pu = pu u 3 u p = u p (22) The procedure descrbed above can be extended to the three-pont trple-correlaton equaton any hgher order correlaton equaton f an addtonal spatal pont sntroduced for each addtonal tensor order. As a result t s easy to verfy that the whole set of equatons dene an nnte set of lnear tensor equatons but whch are far too complex to be solved n general. Nevertheless, t s worthwhle to analyze some features of the soluton. In prncple ths nnte set of equatons could be solved by the followng procedure. Begnnng wth the two-pont correlaton equaton, the trple correlaton can be consdered as an nhomogeneous part of the R equaton. Once the homogeneous soluton s obtaned, the nhomogeneous soluton can be computed by stard methods. In the next step, the trple-correlaton equaton has to be tackled ts soluton wll be substtuted n the soluton for R, so forth for hgher correlatons. In each equaton the von Karman constant only appears as a factor of the hghest order tensor hence the nal soluton for R s a power seres n a (0) R = X m=0 u 3 a (m) m : (23) represents the soluton of the two-pont correlaton equaton after neglectng the trple-correlatons all hgher order terms. The structure of the formal soluton n equaton (23) admts the hope that a truncated seres may provde some nsght n the log-law statstcs. Hence, n the followng the trple-correlatons wll be neglected. Usng the smlarty varable n the posson the contnuty equatons, the pu equaton (6) becomes : ~r k ~r 2 k ~r l k ~r 2 pu +2~r k ~r +2~r 2 = (24) the u p equaton (7) becomes

7 Two-pont velocty correlatons n turbulent boundary layers 2 u k ~r k = ; 2 the contnuty equatons (8) (9) yeld ~r @u 2 p =0 k +~r @~r =0 (26) =0 : (27) The denttes (0) can also be transformed n a smlar manner. Introducng the transformaton (9) nto the equaton (0a), we obtan the relaton R (x 2 x 2 ~r) = R (x 2 ( + ~r 2 ) ;x 2 ~r). Because t was prevously assumed that all two-pont correlaton functons are solely functons of ~r, only the rato of the rst the second parameter can appear n R. Ths argumentaton can be extended to the pressure velocty correlaton. Thus, we nally obtan R (~r) ;~r =R (28) +~r 2 u p (~r) =pu ;~r +~r 2 : (29) The latter dentty also holds f u p pu are nterchanged. These two relatons gve valuable nsght nto the structure of the soluton. Relaton (28) connects derent ~r domans to each other provdes boundary condtons n the ~r 2 drecton. One nterestng feature of (28) s that t can be consdered as a functonal equaton for each trace element. It s easy to verfy that one soluton, but probably not the most general soluton to equaton (28), s gven by the followng form R [] (~r) =F ln( + ~r 2 ) 2 ~r ~r 2 ~r 3 ~r 2 where R s one of the three trace elements of [] R. In addton f the soluton for any o-dagonal R element ( 6= ) sknown, (28) provdes the soluton for the R. A smlar feature for u p pu s gven by relaton (29). If boundary condtons have to be satsed n nnty, all correlaton functons decay to zero. Therefore, any soluton of equatons (2) (24){(27) have toobey the boundary condtons (30) R (~r k!)=pu (~r k!)=u p (~r k!)=0 for k = 3 (3)

8 26 M. Oberlack x 2 = r 2 x 2 r 2 x 2 =0 r 2 Fgure 2. Sketch of the boundary condton n the x 2 -r 2 plane. R (~r 2!)=pu (~r 2!)=u p (~r 2!)=0 : (32) To better underst the boundary condtons n the wall-normal drecton, a sketch ofthex 2 -r 2 plane s gven n Fg. 2. Pckng any value for x 2, the negatve part of r 2 can not be smaller than x 2 hence one bound s on the lne x 2 = ;r 2. The bound for the physcal coordnate s at x 2 =0. Usng the denton of the scaled non-dmensonal coordnate (9), t s clear from Fg. 2 that ~r 2 represents the nverse of the slope gven by any straght lne through the orgn rangng between the two latter bounds. Hence, the doman for ~r 2 s restrcted to ; ~r 2 <. Usng (28) (29) together wth (32) one obtans R (~r 2 = ;) = 0 (33) pu (~r 2 = ;) = u p (~r 2 = ;) = 0 : (34) Obvously, the boundary condtons are all homogeneous one may expect the soluton to be zero. In secton (5) t wll be dscussed why the equatons mght have a non-trval soluton, but a rgorous proof s stll outstng. In the next secton an ntegral relaton comng from the one-pont equatons wll be derved, whch closes the mssng nformaton regardng the scalng of the two-pont correlatons. 4. Kolmogorov's unversal range one-pont correlatons The self-smlarty of the correlaton functons ntroduced n secton 3 s only vald n the lmt of large Reynolds number, based on the wall dstance the frcton velocty Re = u x 2 (35)

9 Two-pont velocty correlatons n turbulent boundary layers 27 Ths s also the denton of y +.From experments t s known that the log regon starts at about y + = 40 extends to y + =0:2U=. The analyss n the prevous chapters s nvscd, hence s not a regular expanson n Re. It s not applcable for small correlaton dstances, as wll be explaned n some detal now. An nner vscous layer n correlaton space has to be ntroduced n order to meet the requrement that vscosty s mportant for the dsspaton tensor " n the one-pont lmt. Comparng the two-pont correlaton equaton (5) n ts most general form to the nvscd verson n the log-layer (2), no vscous term has been retaned. In contrast to that, the Reynolds stress transport equaton n the log-layer ; [u u 2 + u 2 u ] u x 2 + ; " =0 (36) contans the vscosty n the dsspaton tensor, dened by @2 R =2 lm k k the pressure-stran tensor s dened by = + (37) : (38) The contracton of equaton (36) together wth u u 2 = ;u 2 determnes the scalar dsspaton " = " kk 2 = u3 x 2 : (39) As mentoned above we nd from equaton (36) that the asymptotc arguments we have used so far are not vald for correlaton dstances on the order of the Kolmogorov length scale l. The Kolmogorov length velocty scale are gven by 3 4 l = = x2 Re ; u =(") 4 = u Re ; 4 ; 4 : (40) " The only scalng of the ndependent varables wth whch the correct balance can be acheved n the two-pont correlaton equaton s gven by = r l = Re 3 4 ; 4 r x 2 : (4) In lne of Kolmogorov's arguments, the scalng of the dependent varables must be R = u 2 u u ; u 2 R (k) = u 3 h B (0) h D (0) (k) ()+O ()+O Re ; 4 Re ; 4

10 28 M. Oberlack h R (k) = u 3 D (0) ()+O Re ; 4 (k) pu = u 3 u p = u 3 h h M (0) N (0) ()+O ()+O Re ; 4 Re ; 4 : (42) Puttng (4) (42) nto (5), (6) (7) the leadng order terms n each equaton are gven by ; u 2 u ; u u @ ; (0) B (0) k (0) (k) 2 M k = (0) D(kl) 2 N (0) = k (0) l : (45) In order to obtan a unform soluton there must an overlappng regon that matches the nner the outer soluton together. From (42a) we see that the lmt!n the nner layer of the two-pont correlaton converges to the Reynolds stress tensor the same must be vald for a soluton of the equatons (2) (24){(29) n the outer layer for the lmt ~r! 0. Usng the same lmts for both regons n the trple- the pressure-velocty correlatons, they both drop to zero as they should do. As a result, the matchng between the nertal subrange obvously speces the outer soluton R at r = 0 to be u u, but the actual numercal value of Reynolds stress tensor s stll unknown. Note, that the equaton correspondng to (43) n Mellor's paper (972) (hs equaton (59)) has a serous error. It does not have the producton terms whch, of course, are responsble for the energy transfer rate. As mentoned above the nner layer does not determne the absolute value of the Reynolds stress tensor because the trple correlatons can not be neglected n (43){(45). Thus an addtonal assumpton s needed to determne the values of u u. In Kolmogorov's orgnal hypotheses t was suggested that n the lmt of large Reynolds number the dsspaton wll be sotropc. Saddough's (994) very hgh Reynolds number experment of a turbulent boundary layer n a wnd tunnel supports ths dea of sotropy. Hence, we take " = 2 3 " : (46) Usng ths, the three trace elements of can be obtaned from the Reynolds Stress tensor equaton whch n non-dmensonal form can be wrtten as

11 Two-pont velocty correlatons n turbulent boundary layers 29 ; [u u 2 + u 2 u ]+ ; 2 3 =0 wth = x 2 u 3 or n component notaton (47) = ; = = 2 3 : (48) Note, that the latter result for the pressure-stran correlaton holds no matter what s assumed for the trple-correlatons. As a result, all hgh Reynolds number secondmoment closure models should be consstent wth ths result. In most second moment models ths could only be ensured by addng wall reecton terms to the pressure-stran model. Because the system (2) (24){(29) has homogeneous boundary condtons on all boundares, there s nothng that speces the ampltude of R or the value of u u as mentoned above. In fact, ths would also be true f hgher correlaton functons would have been taken nto account. The denton (38) together wth the result (48) can be used to calculate the values for the Reynolds stress tensor. The term on the rght-h sde of (38) can be rewrtten as an ntegral of the two-pont correlaton some boundary ntegrals. Ths was necessary because the lmt r! 0 has to be evaluated wthn the dsspaton range where not enough s known about the two-pont velocty-pressure correlaton. It can be found that the dsspaton range, whch s of the order of l,makes a hgher order contrbuton to the above mentoned ntegral n the lmt of large Reynolds number thus can be neglected. After neglectng the trple-correlatons we nd Z = ; R l +~r l R 2 dv (~r) +( $ ) ~r where ( $ ) abbrevates the addton of the prevous term wth ndces nterchanged. No boundary ntegral has to be kept due to the homogeneous boundary condtons for all varables. Once a soluton to the equatons (2) (24){(27) are computed the scalng of the two-pont correlatons can be calculated by equatng (48) (49). Usng ths, the value for u u can be taken from R at r =0as has been proven by the matchng between the Kolmogorov unversal range the outer nvscd soluton. 5. Future plans There are bascally two outstng problems wthn the whole approach of RDT n the log-layer. The rst one s the fact that t has to be proven that the system (2), (24){(29) has a non-zero soluton even though all boundary condtons are homogeneous. A strong hnt towards ths character of the equaton s ganed by the analyss of the dscretsed set of equatons whch, of course, s lnear. To see why the equatons may have a non-zero soluton, a result from lnear algebra may

12 220 M. Oberlack be recalled. If n a lnear system of the form Ax = 0 the matrx A has the rank <nwhere n s the number of equatons, then the system has nontrval solutons. In ths partcular case consderng the dscretsed equatons (2), (24){ (27), A s a quadratc matrx ts rank can only be smaller than n f there s some redundancy n the equatons. In fact, ths redundancy s due to the denttes (28) (29). Even though the structure of the dscretsed system provdes some nformaton, the proof of a correspondng feature n the derental equatons s stll outstng. Once the prevous problem s solved, a numercal algorthm has to be coded to solve the dscretsed equatons (2) (24){(29) because t s very unlkely that an analytcal soluton can be found. In the next step of post-processng the numercal results, the ablty of the asymptotc lmts used n the RDT of the loglayer has to be revsed f necessary enhanced by ncludng hgher correlatons n the analyss. Fnally, the results of the theory wll be compared wth DNS data from the turbulent channel ow (Km et al. 987). Acknowledgment The author s sncerely grateful to Wllam C. Reynolds Paul A. Durbn for many valuable dscussons crtcal comments durng the development of the work. The work was n part supported by the Deutsche Forschungsgemenschaft. REFERENCES von Karman, Th. 930 Mechansche Ahnlchket und Turbulenz. Nachr. Ges. Wss. Goettngen. 68. von Karman, Th. 930 Mechansche Ahnlchket und Turbulenz. Proc. 3rd Int. Congr. Appl. Mech., Stockholm.. Kassnos, S. C. & Reynolds, W. C. 994 A structure-based model for rapd dstorton of homogeneous turbulence. Thermoscences Report No. TF-6. Km, J., Mon, P. & Moser, R. D. 987 Turbulence statstcs n fully developed channel ow at low Reynolds number. J. Flud Mech. 77, Mellor, G. L. 972 The large Reynolds number asymptotc theory of turbulent boundary layers. Int. J. Engng Sc. 0, Mllkan, C. B. 939 A crtcal dscusson of turbulent ows n channels crcular tubes. Proc. 5rd Int. Congr. Appl. Mech., Cambrdge. Rotta, J. C. 972 Turbulente Stromungen. Teubner Verlag Stuttgart. Saddough, S. G. & Veeravall, S. V. 994 Local sotropy n turbulent boundary layers at hgh Reynolds number. J. Flud Mech. 268,