A light scattering study of turbulence

ph.,,~,,,.a D 38 i I c~Sq i 134-140 North-Holland. Amsterdam

A L I G H T SCATTERING STI.ID~. O F T U R B U L E N C E

W.I. G O L D B U R G . P. T O N G ~ and H.K. PAK Dt pa, t,m'nr of Pht s~c.~and A ffronom.v Untver~t.l"ql Pm~buL~h. Ptttsburgh, P.4 15260. US4

B.,, scattering he.h~ from a turbule'tl flutd seeded ~,ith small parl,cles, one obtains mformation about turbulent ~,eloeity fluctuat~,)ns o~er vat3 mg spatial scales. /~'. The measured mtensit~ autocorrelat,on function, g(t). is related to the probabiht~, densil) P( I'( R I I of finding ~elocit~, fluctuations of magnitude I'( R ' ~,soc,ated ~,,th eddies of size R. The measurements described here s~rongl.~ suggest that the energy-containing eddies occupy, a fractal region whose dimension (or spectrum ofdimensions) increases ,,~th Ihe Re~ holds number Re ~.hen Re exceeds some threshold ~.alue.

I. Introduelion

In his seminal book. The Fractal G e o m e t ~ ' of Nature. Benolt Mandeibrot [ I ] makes clear his deep interest in the geometrical nature of turbulence. As he pnmts o u t the description of the ~.isual appearance of a turbulent fluid, such as smoke curling up from a ~:~garette. taxes our po,-ers of description. It seems that present-da,, speech JS not ~sell suited Io e~okmg 1he ,mage of s,.'It'-slmdar structures. After all. it takes a ser,es of ,mages. one magnified *',ith respect to the uther. ,o talent,f) fractal structures. And turbulence is. b) all e,, idence, a fractal thing at its roots] 2 i. There are man) ~,.a)s of re,.eahng the fractal or :.pott.~ nature of a turbulent fluid. One technique ~s to n~ea~ure the time ",ariat~on oflhe square of the ~eloc,t.~ at a point in the flu,d [3]. Another is to add a small amount of long-chain molecules to the fluid and ,:)bser~ c it througt, crossed polaroids [4]. The molecules are locally aligned by turbulent shear forces. These molecules, being amsotropic scatterers, depoIJfl_~e the ~ght Irl re~.[ons v.here tile' shea, ~s large. n~..a~ ng the local structure of the s l r o n g ',Orl!Clt.', d~re,. f[, , tsible.

Hereto ~e describe experiments, earned out at the L',r,., ecs.~t~ of Ph'sbur~h. ~h~ch pro~ ~de a ne,.~, anP,-~,mr .Jddf,:,~ Ex'..L, rl Rc~,:'arch .,~d l-,Jnda]~: [',ij. OSSill (I%_'A

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proach to the study o f the small-scale structure ofturbulence. The method involves a measurement of the autocorrelation function of the light intensity scattered by small particles suspended in the lurbulent fluid. For this technique there is no need to invoke the "'frozen lurbulence assumption" to translate temporal information to spatial information. According to this assumption, small-scale eddies (the ones of interest), are transported past a ~.elocily measuring device ~tlh the mean veloclt.~ U o f t h e flow. If these small-scale eddies remain intact for a long enough time, a time record of the velocity z'( 1 ) at a point v, ill reveal spatial features of the flow through the equation z'(t ) = l,(.¥/U). The frozen turbulence assumption fails unless the velocity fluctuations 1'( R ) associated with eddies of size R are uncoupled from the larger-scale eddies. The technique of photon correlation homodyne spectroscop.~ ( H S ) [5]. which we have used in our experiments, is that of recording the beating of scattered li~hi ~aves !ha~ have been Doppler shif~e0 b~ pairs of panicles seeded in the turbulent fluid. The teciln[que was ~mroduced man) )ears ago by Bourl,e eL al. [6], but seems largely to have been ignored. Being an optical technique, it permits non-in~.asi~e observation of velocity flucluations at ve~' small SCa [e.g

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I!'1. Goldburg e/al ,'.4 hght scatten ng study gf turl,ulent e

fig. I, which shows two moving particles at a particular instant of time when their separation is R and their velocities are t,~ and ~',. The seed particles are small enough that they scatter light isotroplcally. A photodetector ( P M T ) , located at an angle 0 with respect to the incident beam, receives the light from both particles. The scattered light from each particle is Doppler shifted by an amount k-t,t and k. L,_,respectively, where k is the scattering vector, o f magnitude k=(4nn/).) sin(½0). Here 3. is the vacuum wavelength of the light ( 2 = 4 8 8 nm in our experiments), and n is the refractive index o f the turbulent fluid: in our case the fluid was water. The photomultiplier, which receives the light from the particle pair, is a square-law detector, so that its output current, l ( t ) , contains a beating term proportional to cos[kl'~(R)t], where I'~ is the projection o f t h e velocity difference ~',-='., along the direction of k. Henceforth the subscript on 1'~(R) will be dropped, but its R dependence will be retained. The essential aspecl of turbulence is that the ~,elocity difference between two points in the fluid depends on the separation R of these two points. According to the theory of Kolmogorov [6]. the moments of the velocity fluctuations I"(R) obey a scaling law

(I'(R)")-,- u( R)"--, R':",

P( I ' ( R ) ) x {I-I- [ l ' ( R ) / u ( R ) ] ' }

-'

12)

for relatively small values of i'( R ). We also find that the scaling velocity u( R)-. R;, where ~ is a function of Reynolds number, Re. The measurements were made at very modest values ofthe Reynolds number. In fact the turbulence was so weak that one might not ha~e thought the flow would exhibit the self-similarity which was indeed obsetn.ed. Throughout this paper the Reynolds number is defined as Re=/..'/,,/e. where U is the mean veiocit.~ ofthe flow. I,, ts the outer scale of the turbulence, and v is thc kinemal~,.", ~'3co~ity o f t h e fired.

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Fig I ~. schematic d i a g r a m sho,~mg ~cauermg geometn.. The Scattering seclor ,~ = k x~ase sectors respecux el~

(1!

with ~.,, ~ - ~n. The homodyne technique is ~eii suuetl to measure Lhe lower moments of I ' R ), but not the higher moments. On the other hand the method yields information about the functional form of the probabilit~ oensity P( 1'( R ) ), that two points in the fluid, separated by a distance R, have velocity difference lying within I'(R ) and !'( R ) + d I "(R ). Our central finding is that P( I'(R ) ) is well represented by a Lorentzian function,

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II I Goldburg et al... 4 Itg_ltt s~attermg stud.r olturbuh'nce

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2. Experimental

The detailed experimental setup can be found m tel: [8]. The fluid flow was generated in a closed ,,.ater tunnel :omprised of a cylindrical pipe and a pum-~ of variable speed. The turbulence is generated by a grid within the pipe. The grid can be remo~,.ed to permit study of v,all-generated turbulence (pipe flow ). A baffle section placed in the high-pressure side of the grid. suppresses the turbulence generated by the pump and b~ those sections of pipe on the highpressure side of the grid. With this arrangement all of the turbulence is generated by the grid only. In most of the experiments discussed here the diameter ofthe pipe v,as 4.4 cm. and the aperture size o f t h e grid ,,,,'as 3. I mm. These parameters are taken to be I,, in calculating the Reynolds number. The measurements ~ere made 28 cm downstream from the grid. The water which flowed through the pipe was seeded with polystyrene spheres 60 nm in diameter. These panicles ~ere small enough to scatter light isotropically and in sufficient concentration that their mean separauon ~as much less than the Kolmogorov dissipation length/,i. ~hich ~as estimated to be a fraction of a millimeter. On the do~ nstream side ofthe grid there is an opt~call.x transparent section of pip,rig to admit the in,.~dcnt laser beam and obse~e the scattering. Because the 11o~. is seeded, a thin column of the scattered light ~s produced in Ihe ~ater and that light is imaged with a lens. on a slit of variable width. L. By var).mg L. the homodyne scheme perra,,s the probing ofveIocit.~ fluctuations i'( R I from the smallest scale Ij to that ofthe ~ldth of the sht. L. Using a standard light scattering apparatus and a digfial co,'relator. ~,.e measure the intensity autocorrelauon function, g(' t ) = ,.' I( t ) I( t' + t ) ) / ( !( t' ) i. :. "~.~1,:r,2 i ~ t t

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3~ sca~tcrl ng angle _9. and the angle brackets represent a .'.me a~cra~,.e c,~er t . One can sho~, [8] tha,. the ~:orrelatton funo~on ~_,ltI has the Ibllow~ng form: e;~l= I+l{

4~ G ( I ) .

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Th,.2 geometrical factor ;tA) ts of order un.,t.~ ~fthc

photodetector recei-~es light from only one coherence area [51. All of the interesting physics is contained in G( t ). which is proportional to a sum of the timeaveraged phase factors cos(ktl') coming from the Doppler shift of all particle pairs in the scattering volume. The function G( t } can be written as [ 8 ]

G(t)= d d R h l R )

dI'IR)Pll'{R))

xcos[kl'(R)t] .

(4)

~here hIR) is the probability of finding a particle pair. separated by R. m the columnar region o f length L. If the image on the slit is taken to be quasi-onedimensional, w h i c h is valid w h e n the slit w i d t h rem a i n s large compared to the diameter o f the laser beam, h ( R ) = 2( I - R / L )/L. Note that the inner integral in eq. (4) is the Fourier cosine transform of P( I'(R) ) and the G(t) may be thought o f a s a transform of the characteristic function. If the probability, density P ( I ' ( R ) I has the scaling form F ( I ( R ) t=Q[ I ( R ) / u ( R ) ]/ulR).eq. ~4) becomes I

G(t)=

t! d R h ( R ) F ( k t u ( R ) ) .

(51

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where F is the Fourier cosine transform of Q[ I ' ( R ) / u( R ) ]. It is easy to show that the scaling law in eq. ( I ) follows if the probability aensity function P¢ I ( R ) ) has the above mentioned scaling form. The above equations for g(tl have quite general validity. They hold. for exampl.~. even if the fired is stattonar3, and the seed panicles are undergoing Brownlan molion only. In that case. J'¢R) is independent of R and P( I'(RI I is a Gaussian function. Ti,en the function GI t ) is an exponentially decaying function [51. G l : ) = e x p ( -2D~Ztl. v, here D is the diffusi,, it.~ of the Brov.nian particles, and ~s given b.~ Stokes" la;v. This contribution to the deca3 of Glt) ~ill be presem, excn when Ihe fluid is turbulent. Ho~.e~er. in a turbulent fluid, the deca3 time T of Gtt) is much shorter than the diffusive deca~ time. 7,~ = I _"Ch-. s.a lha; d:,e I.-..:ie~ c~3ntribulion can be

I I ' l Goldburg fl al. / -I It,_hi ~calterotg _~tt4dt'ol turl,zdcncc

safely ignored. From eq. ( 5 ), it follows that the turbulent decay time should be of the order of T= I/ku( L ). because the fastest deca~, rate is associated with the largest eddies ofsize L.

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3. Results

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Over a wide range of slit widths and Reynolds numbers, we find that the function G( t ) exhibits the scaling form.

G(kz. L. Re) = G ( x ) .

(6)

~,ith x - k"L;t. This scaling behavior o f G(t) is observed only. when the Reynolds number exceeded a certain value Re,. In the case of the grid flow described above. Re~ was roughly 500, which corresponds to much weaker turbulence than that one normally associates with scaling behavior. It is quite possible that the scaling behavior is seen at such small x.alues of Re because the simultaneous velocity difference I "1R) is measured and no frozen turbulence assumption is needed in the data analysis. The exponent ,u in eq. (6) was measured as follows. For a fixed slit width L and a fi:.ed Re. G(t) ~,~as measured at several scattering angles and hence se~.erai values of k. All of the plots o f log [ G( t ) ] versus iog(tl could be superimposed by' horizontal translation of one graph ~,'ith respect to another. The amount of translation. 6(k 1. is found to be roughly proportional to k, i.e. ,u= I. ~,hen Re exceeded the critical value Re,. However. in the absence of flo~'~.. It=2. as expected for Brownian motion o f the seed particles. Similar measurements were made in which k and Re were held fixed, and L ~.as ,..aried. Again all the plots of log[ G( t ) ] ~ersus log( t ) could be superimposed. 5ielcing the result h',,.I,L-t, a s long as Re exceeded Re~. In these experiments. ,,."is found to be Re dependent. We return to this impo.r,~ar~ poser, alion below. Fig.. 2. a iog-log plot of G I ~ ) versus h'. shows the scaling beh.,vior of G( t ) discussed above. The measurement~ correspond to several ~ alues of scauering angle, or k-~alue, several slit ~.idths and at various

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~(e) Llmm) Re Flow t 60..rl I.O 1364 grid x 90 0.6 t's'~O0 pmpe • , 90 0.3 ;:'6000 o~pe

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Fig. 2. The scahng function G ( x ) ~ersus ~,= qztl/. )~ mn p~pe Ilo~. a n d gr,d flow.

Reynolds numbers. In one set of measurements (closed circles ). the grid v,.-s present: in the other two sets (crosses and triangles), th.* grid was removed (pipe flow). The correlation functigns G(t) ha~e been horizontally (and vertically) transla,ed so that they coincide. In the pipe flow measurements, the Reynolds number :. based on the pipe diameler making it an order of magnitude larger than that for the grid flo~, e~en when the mean Ilow velocH,es ,~' are comparable m both cases. An allernalt~ e ~.a) to delen~ine the exponent ~ ~as to plot. on a double logarithmic scale, the siit-~i0111 dependence of the decay time. 7. of G(t J. keeping Re a n d / , fixed. As is shown in fig. 3. linear variation of log(T) with Iogl L ) was seen at inlermedlate values o f L. The data in fig. 3 were obtained m the grid flow at three different Reynolds numbers R e = 4 6 0 . 1400. and 2200. Since 7-'~ I / k t l ( L ) and , ( L I - - L : . the slope of this line yields the exponenl ~. ~.hich ~s [/3 in the Kolmogorov theo~'. X,~,.'eha,.e ~enficd that ,'he power la~. beha,, iorat large L ",~as !:mi:e6 o 1.:~e 7u¢.:r scale./,,, of the turbulence. -".,t smali ~a~ue-~ o! L ~Jlc ,: ou~t~ a
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design to that used in the studies reported above, in this experiment, the opucally transparent pipe, where ~( t ) ~.as measured, was square in cross section, rather than c~lindrical, so that the laser beam was undistorted in passing through it. In this square pipe. the beam diameter was less than 0.1 mm, which is smaller than the smallest value of L at ~,hich g(/) ~as measured. Using laser Doppler velocimet~ and invoking the frozen turbulence assumption one can determine the smallest eddy size I.~. At R e = 8 5 0 , we obtained /,,=0.4 ram. At values of L between 0.4 mm (=1,4) and 0. I ram, the decay time of G ( t ) became independent of L, i.e. ¢ ~ 0 when L
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Re

Sl,lwidth L tram) F~g 3. The decay time T( L ) ~ersus slit w i d t h L m grid flo~. The number below a line ~s the slope of that hne.

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F:g. 4. The exponent ~ as a function of Re in p=pe flow. The solid line i ~.dra,,'n by e~e through the data points, and the dashed curve shows the oscillatory b e h a v i o r ore. The inset :hov, s ¢ versus Re m grid flow.

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the notion that the turbulence becomes increasingly three-dimensional as Re is increased above Re~ and that in tP.- vicinity of Re.., the turbulence is lwo-dime,asio,a. v 1.

- ' : ~ '~ . . . . . . . . . . . . .

~: ~s seen to increase from () to ~ 1/3 ( the Kolmogoro~. '~alue) as Re ~s increased. When the Reynolds number ~s below Re~, G(~ ) fails to exhibit scaling beha, ~our. The measured Re.. ~ 300-400 in the grid flow and Re, -,- 3000-4000 when the grid :,,as remo,,ed. Meas, rements in the ~mpro~ed water tunnel g.~e s,mflar results. These obsen afions are consistent with

t~rm r,.c ; ~r ). Fig. 5 is a semilog plot of G( t ~ versus t bn ptp~ fl(,x; at the indicated values of L, 0, and Re. The straight line is a linear fit to the data points at small t. it is seen that onl: at large time does the cur~e slam to deviate from :he linear behavior. If we assum,:- chat !he characlenslL funewion F(ku(R)t) in eq. i 5 ) has the Ibrm F - - e x p [ - k u ( R It], G(I) then

I1' I. Goldburg et a L / A Itght scattering study of turbulem'e

becomes an incomplete gamma function with ku(L )t as its argument [8]. This equation is well fitted to our measurement of G(t). An example of this good fit is shown in fig. 6. Note that the assumption o f F ( x ) being a single exponential decaying function implies that P( I.'(R) ) is o f Lorentzian form as shown in eq. (2). This function has a diverging second moment, to which the energy density in the fluid is proportional. Therefore G ( t ) cannot have this form for large values of 1"(R ). We indeed obsen.ed departures from this Lorentzian form for P ( r ' ( R ) ) ~ith very large values of I'(R) Ccorresponding to very small t for Glt) ) [ 10,1 ! ]. However, these observations will not be discussed further here. Most theories of turbulence concentrate on the scaling behavior of the moments of V(R), rather than in PC I'(R ) ) itself. Quite often, PC V(R) ) is assumed to be of Gaussian form, P ( I ' ( R ) ) -,- exp{ - [ I"(R)/u(R)]-'}, but this form of P( |'C R ) ) is clearly contrary to our findings. How can one understand that the exponent t~ increases from 0 to ~ I/3 as the fluid becomes increasingly turbulent? A fundamental understanding of lhis result is lacking, but it can be said that the obsera.alion is consistent with the notion that the turbulent active region is a fractal [12]. Let the fractal dimension o f t h e turbulent region be D. Since the turbulent energy is confined to active regions of dimension D < 3, the concentration ofthe turbulent energy is increased to smaller regions, relative to the case of ,~olume-fil!ing turbulence. Modifying the Kolmogorov

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theory Io lake lhis effecl inlo account I 13 ], one has ul R ) -,. R , with ~= ~( I + D - 3 ). According to this model, the increase of~ from 0 to ~ [ corresponds ~.o an increase of D from 2 to 3. It should be stressed that our measurements o f g ( t ) described above, do not directly give information about the fractal dimension of the energy-containing eddies: it can only be said :hat the data invite such an interpretation. The above imerpretation o f the data in fig. 4 is supported by the recent work of Shreenivasan ,,:t al. [3]. They measured the fractal dimension oftne interface of two counter-flowing fluids, one o f which has been dyed. Such measurements, made in the vicinity of Rec, support the conclusion that increasing Re above Re~- increases the dimensionality o f the turbulent active region. With one adjustable parameter, Rec, the data ofShreenivasan et al. can be directly superimposed on the measurements in fig. 4

[31. E~,en if the energy-containing eddies in a turbulent field occupy regions with dimensionality less than 3. it is not necessary that the entire lurbulent region be characterized by a homogeneous fractal. Benzi et al. [ 14] have proposed a model that the turbulem region is a rnultifractal object. In their model there ts a probability, x, that the turbulent region ts space fill,rig I D = 3 ) and a probabilit.~. I - . v that D = 2 . Our measurements are consistenl with this model, pro,,.ided one makes an additional assumption that .~."is a function of the Reynolds number. The details ofthe model have been worked out for a general function o f , v ( R e ) and filled to experiment 19]. At present. howe~.er, our measurements are not precise enough to confirm thai a mulufractal model is reclu~red to explain the observat,,ons.

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F~g b. A I~,plCd[ autocorrelal~on funcHon k'(i ) ~ersus t m grid flo~. The sohd hne ~s a fit to the i n c o m p l e t e g a m m a function.

Vv'h~A started out as a stud) ~:arried out in ~he l~rr,~ domain ( the measurement of g( t ~~. ~ s ended up b~ yielding spatial information about turbuient f~o'~. The homodyne experiments pro~tde further conl=,rma~ion of 1he notion of Mande!bro~ that 1.he en,~_rg~•

1411

ii .I. Goldburg el al ,' 4 h.ght ~:catterlng stz~dl" of n¢rbtd,mce

containing eddies are fractal in their geometrical structure. This find:ng is not new. W h a t seems to ha~e gone unnoticed before, is that the fractal dimension o f the turbulence changes with changing Reynolds number, when s o m e critical value o f this p a r a m e t e r is exceeded. T h e interpretation o f ~hese experiments makes no appeal to the frozen t u r b u l e n c e assumption. By using the technique o f p h o t o n correlation h o m o d y n e spectroscopy, we have been able to observe the self-similar behavior o f t u r b u l e n t flox~s al moderate R e y n o l d s n u m b e r that were heretofore regarded as too weak to exhibit universal features.

Acknowledgements

We have benefited from illuminating discussions and c o r r e s p o n d e n c e with K.R. Shreenivasan, M. Nelkin, and J. Stavans and have e n j o y e d a c,~ntinu ing fruilfill interaction ~ith A. Onuki. We are grateful for the collaboration of our colleague. A. Sirivat and to C.K. Chan for h~s essentl,,! c o n t r i b q l l o n s in the earl~ stages o f this ~ork. This research ~a.~ sup-

p o r t e d by the National Science F o u n d a t i o n under G r a n t No. DMR-861 i 666.

References [I I B.B. Mandelbrot. The Fractal GeomeLry of Nature ( Freeman. San Francisco. 1982 ). [2] B. Mandelbrot, Ph~,5. Flmds Suppl. l0 (1967) 5302. [31 K.R. Srecnivasan. R. Ramshankar and C. Mene~,eau. Proc. Ro!,..Soc. (London) A 421 (1989) 7q. [4] E.R. Lindgren, Arch. F,~,s. (Iq59) 97. [ 5 ] B.J. Berne and R. Pecora. Dynamic Light Scattering (Wde~, New York. 1976i. [6] P.J. Bourke et al.. J. Ph~,s. A 3 (1970) 216 [ 71A.N. Kolmogorov. C.R. ( Dokl. I Acad. Scl. URSS 30 ( 1941 ) 301:31 (1941) 538. [ 81 P. Tong et al.. Phys. Rex. A 37 ( 19881 2125: P. Tong and W.I. Goldburg, Ph~,s. Fluids 31 (1988) 2841. [ 9 ] P. Tong and W.i. Goldburg, Phys. Fluid 31 ( 1988 ) 3253. [ IO] A. Onuki. Phys. Lelt. A 127 (1988l 143. [ I I I P. Tongand W.I. Goldburg. Phys. Lett. A 127 (1988) 147. [ 121 B.B. Mandelbrot. J. Fluid Mech 62 (1974) 331: m: Lecture Notes in Ph~,,slcs.Val. 12. S|atislieal Models and Turbulence. M RosenbL~lauand C.~. Van Alia. eds. (Sprmger. Berhn. 1972 I, p. 333 [ 13[ LI Fnseh. P. ',ulem and M. Nelkm.J. I=lutdMech 87 ( 19781 71,~. [14]R.B,:nztel.d..I Ph~,s.A 17 f1%4 ) 3521.