Multiple light scattering in turbulent flow

ELSEVIER

Physica B 204 (1995) 20 26

Multiple light scattering in turbulent flow D.J. Bicout a'*, R. Maynard b a Laboratoire de Modblisation et de Simulation, Institut de Biologic Structurale, CEA-CNRS, 41 Avenue des Martyrs, F-38027 Grenoble Cedex 01, France b Laboratoire d'Expbrimentation Numbrique, Universitb Joseph Fourier, Maison des Magistkres-CNRS, B,P. 166, F-38042 Grenoble Cedex, France

Abstract The time autocorrelation function, C1 (t), of the multiply scattered light field can reflect the nature of the flow of a fluid containing scatterers as markers which follow the fluid motion. It is well-established that, for instance, in the thick medium C~(t) asymptotically behaves like C~(t) = 1 - t ~, with ~ = ½ for the Brownian motion and ~ = 1 for the homogeneous laminar flows. For more complex flows, even for laminar flows, the t~-dependence (with ~ ~ ½, 1) of C~ (t) is no more valid. Here we derive the general expression of C~(t) for a simplified model of nonhomogeneous turbulent flow consisting of a regular array of sheets of discontinuous shear flow embedded in a chaotic velocity field.

1. Introduction Multiple light scattering spectroscopy (MLS) turned out to be a valuable technique for studying particle dynamics in turbid media [1 3]. Recently, it has been demonstrated [4, 5] that when a dense fluid is dragged in a laminar and stationary flow, the accurate and noninvasive measurement [6] of velocity gradients can be performed using MLS. In this case, the time autocorrelation function, C1 (t), of the scattered intensity in sensitive to the velocity gradients at the scale of mean free path. It has been also shown [6] that velocity gradient measurements allow one to study hydrodynamic instabilities. Until now, all works studying multiple light scattering in flows have considered homogeneous and nonhomogeneous flows separately [4,5]. Within this framework, MLS could then be extended to study the fully developed homogeneous turbulence. But it is wellknown that appearance of hydrodynamic instabilities or the stage preceding full turbulence is characterized by a highly non-homogeneous and nonisotropic flow which * Corresponding author.

is constituted, for instance, of laminar regions embedded in turbulent flow or conversely [7]. The study of such nonhomogeneous flows would allow one, in principle, to characterize hydrodynamic instabilities and to gain informations on the nature of the flow preceding turbulent regime for a given scenario of the transition to turbulence. This short paper deals with the study of such nonhomogeneous flows by deriving the expression of Cl(t) and by extracting characteristic time and length scales of the flow which are accessible using the MLS. By using a heuristic and very simplified model of nonhomogeneous turbulence - sheets of discontinuous shear flow embedded in chaotic velocity field - we obtain an insight of different mechanisms contributing to the decorrelation of Cl(t) in such flows.

2. Time autocorrelation function Cx (t) In quasi-elastic multiple light scattering spectroscopy, the motion of particles is probed by monitoring the time autocorrelation function Cl(t) = (E(O)E*(t))/(IE(O)I2),

0921-4526/95/$09,50 © 1995 Elsevier Science B.V. All rights reserved SSDI 09 2 I -4 5 2 6 ( 9 4 / 0 0 2 30-I~

D.J. Bicout, R. Maynard/ Physica B 204 (1995) 20-26 where E(t) is the scattered electric field of light collected by the detector. In order to compute C~(t) we consider light which is multiply scattered by a random distribution of point-like particles, and hence performs a random walk through the sample. Within the weak localization regime (kl >> l; k and l being the wave number and the scattering mean free path of light, respectively) and in the strong multiple scattering regime (L >> I; L being the linear dimension of the sample cell), the intensity of light leaving the sample can be described in the diffusion approximation [8]. The time autocorrelation function is given by [1 3,5]

r(rv(t),Av)t according to (ii). The quantity of fundamental interest in the theory of turbulence is the velocity difference r(r(t), A), that a pair of points in the turbulent fluid separated by a distance A differ in velocity by an amount r(r(t), A) = V(r(t) + A) - V(r(t)), where V(r(t)) is the local velocity of the fluid. One can show that [91 for incompressible fluids the leading nonvanishing term in a cumulant expansion of Eq. (1) is the second cumulant which corresponds to the fluctuations of the phase difference. We have (exp{iA¢.(t)})=exp{-(Atk2(t))/2}, where the fluctuations of A~b.(t) are given by the relation: (Aq~.z(t)) = (kt)2n ((e. r(r, A))2)t,,,

C~(t) = I o ~ G(n)(exp{iAO,(t)}),

(1)

n=l

where G(n), the fraction of total intensity Io scattered in all nth-order scattering paths, describes the diffusive transport of light and is related to the sample geometry. A¢,(t) = ~ = 1q~" Ar~(t) is thus the phase difference of the scattered electric field between time t = 0 and t, associated to a given multiple scattering path of nth order, and contains the dynamical information about the scatterers. (-..) denotes both the configurational average of Art(t) = r~(t) - r~(0) (displacements during time t of vth scatterer) and the average over all possible scattering wave vector q~ = kv - k~_ 1. We consider now a flowing fluid, containing independent elastic scatterers which follow the fluid motion, and we neglect the additional Brownian motion of scatterers due to the finite temperature. In order to describe the decorrelation caused by the flow, we make the following assumptions: (i) the properties of the multiple scattering of light are not affected by the flow; there is no correlation between, for instance, the scattering mean free path and the velocity field and its derivatives. (ii) the characteristic time scale of the time autocorrelation function is much smaller than the shortest characteristic time scale of velocity changes allowing to consider the flow as stationary or quasi-stationary at the correlation time scale of Ca(t). The phase difference of electric field scattered by particles following a flowing fluid is given by Aq~.(t) =

-

ke~.

dt'[V(r~(t')) - V(r~+1(t'))]

kt ~ e~. r(r~(t), Av),

(2)

v=l

where the corrections due to the first and last scattering events have been ignored, e~ is the emergent unit vector of the vth scattering. Here v(r~(t), AO is the velocity difference between the two successive scatterers v and v + 1 with (IA~I) = I. The integral Eq. (2) is replaced by

21

(3)

where the double average is performed in a given order: the first average involves the photon variables on a sample of frozen velocity field. It is justified by the fact that the speed of the light, c, inside the scattering medium is much faster than the typical flow velocity, U: c >> U. Hence,

'fff

((...))~,,) = ~

(...) p,(r)P(A) d ~ dA d~r,

(4)

where p,(r), which depends upon the geometry of the sample, is the density distribution of end-to-end diffusion path of nth order [5, 9], P(A) = I- 1 exp ( - A/l) is the density distribution of the distance separating two successive scattering events, and the angle average (d¢2) stands for scattering angles. The second average is the configurational average of the velocity difference field:

(...--] = f(...) II(v) dr,

(5)

with H(v) being the density probability of the velocity difference. Nevertheless, since it is assumed that the set of multiple scattering paths of order n at time t coincides with those ones at time t = 0 (ergodic hypothesis), the order of performing these two averages can be reversed. When the velocity difference field is deterministic, the double average in Eq. (3) is reduced to the simple one of cloud of light paths. The average in Eq. (4) now weights the configurational mean velocity difference field by the cloud of light paths and describes the nonhomogeneous nature of the flow. Due to the spatial dependence of the velocity difference, the dephasing of light is not the same everywhere in the scattering medium: for each geometrically different nth-order scattering path, one expects a different contribution to the dephasing of light. This complex situation, which can occur for moderate Reynolds numbers, corresponds to the very nonhomogeneous and nonisotropic turbulent flows like for turbulent jets and wakes where the turbulent fluid is limited in certain regions of the space surrounded by nonturbulent fluid.

22

D.J. Bicout, R. Maynard/Physica B 204 (1995) 2~26

Here, we are interested to the case where the flowing fluid is constituted of mixed laminar and turbulent regions. Such situation appears, for example, in the transition to turbulence in the pipe flow where one observes the growth and transport of turbulent slugs embedded in laminar flow [7]. The flow is then intermittent. This occurs also in the Taylor-Couette instability [10] when the two cylinders are rotating in opposite senses, but the outer one faster. Under certain conditions, one observes that the stage preceding full turbulence is intermittent turbulence similar to that observed in pipe flow. The motion of a fluid particle is alternately laminar and turbulent. Such intermittency is also observed in fully developed homogeneous and isotropic turbulence for length scales very close to the vicinity of the Kolmogorov dissipation scale which separates scales of laminar behavior from turbulent one. This intermittency phenomenon is often observed in catastrophic transition to turbulence where an intermediate range of Reynolds numbers in which a stable mixed laminar-turbulent configuration can exist. In order to handle such situations, we assume that the velocity difference field can be approximated as the incoherent summation of laminar and turbulent flows:

v(r,A) = vt.rb(A)+ Vlam(r,A),

(6)

where the indices h and nh denote the homogeneous and nonhomogeneous components of the velocity difference, respectively. In what follows, we consider the heuristic model for the nonhomogeneous turbulence where the chaotic velocity field, described by Vh(A), surrounds a comb of sheets of discontinuous shear flow described by Vnh(r,A).

(8) where Mp are the cumulants of order p of the projection along the scattering direction ev of the velocity difference. Note that because of this regime the turbulence is isotropic, Mp are only function of I. On the other hand, the scattering directions ev are uniformly distributed over the unit sphere for isotropic scattering. So (ev) = 0 implies M1 = (evi vi(A~)) = 0 for all distributions of v since there is no correlation between the scattering of light and the velocity field as stated in (i). For the sake of simplicity, we consider only the case of the symmetric 1 probability density for the velocity difference centered at 6 = 0. The cumulant expansion in Eq. (8) gives (exp{iAq~,(t)}) = e x p { - 2(t/rh) 2 × n[1 -

2(t/Zh)2(M4/M z) + ...]},

(9)

where we have introduced the correlation time given by (10)

Av(I) = x/(v2(l)) - (v(l)) 2 is the fluctuation of the velocity difference and the double average ( 7=..) is an average over the velocity difference distribution ll(v) and over the exponential distribution P(A) of A. zh is the time required by a pair of scatterers initially separated by a distance I to move a relative distance k - 1 due to the fluctuation Av(l) of their velocity difference. Note that according to Eq. (6) the correlation time in turbulent flow is zturb = rh/~b. The knowledge of the velocity structure function of second order, S2(A), given by S2(A) = 1)2 __ /72

(11)

leads to derive the fluctuation Av(I) of the velocity difference obtained by averaging S2(A) over the density distribution P(A) to give

2.1. Homogeneous velocity difference: Fully homogeneous turbulence We consider here to the limit of large Reynolds numbers where the flow is in the regime of fully developed homogeneous and isotropic turbulence. Thus, the velocity difference is independent on the locations of scatterers in the flow but depends only on [A~I, says vh(A0 = v(IA~]), and the index h is dropped for simplicity. Thus, the average of Eq. (1) writes

(exp{iA4a,(t)})=(exp{-ikt~

fexp{-iktv~=lev'v(IAvl)}l = exp {np~__~l(- ikt)PMp},

x/~ Zh - k Av(1)'

where lam and turb stand for laminar and turbulent flows, respectively. Let q~ be the fraction of turbulent regions, we have then v(r, A) = ¢ v . ( A ) + (1 - 4,)r..(r, a ) ,

Then, for independent scatterers, the right-hand side of Eq. (7) becomes

(7,

Av2(/) = I ~ e-xS2(lx) Jo

dx = LT[S2(lx)],

(12)

1 In the first analysis, it is more easy for calculations to neglect the nonsymmetrical nature of the probability density function of the velocity difference. However, in the locally homogeneous and isotropic turbulence, the third-order moment of velocity difference in the inertial range is not zero and, in addition, experimental distributions of velocity increment exhibit the skewness [l 1].

D.J. Bicout, R. Maynard/ Physica B 204 (1995) 20-26

23

where LT[...] denotes the Laplace transform. Thus, the correlation time %, given by Eq. (10), provides a way for studying S2(r) at the scale of I. We restrict ourselves to small delay times t <~ rh and retain only the first term in the cumulant expansion. This impose to work at time scales such that

L/r1 "~ R 3/4, so that for three-dimensional turbulent flow,

rh

S2(r) = (Cv2(q)(r/~]) 2/3+u/9,

there are at least of the order of (L/r/) 3 "" R 9/4 dynamically active degrees of freedom per volume L 3. It is well-known that Kolmogorov-Obukhov hypotheses [ 12, 13] lead to the following form for the structure function of the velocity difference:

)eq2/15v(r/q) 2, ~/2M4 = ~/ 3 ~

1.

(13)

The time autocorrelation function due to the turbulent flow is therefore given by Cltturb)(t) = f ~ G(n)exp{-2(t/zturb)2n}dn"

(14)

Cltturb~(t) has a Gaussian decay exp{--2(t/r,,rb) 2} per scattering event, which will be characteristic of deterministic motion although v(l) is a stochastic variable. This directly stem from assumption (ii). It is worth noting that, under these above approximations, i.e. the homogeneous and isotropic turbulence, the mathematical expression of Cltt~b~(t) for turbulent flow is similar to that we obtain for laminar flows [9]. This allow us to define the effective velocity Ueff(/) at the scale l, which originates the loss of the temporal correlation of scattered intensity, and such that % = x//~/kUeff to obtain

uo~tt) = , ~ Av(t).

(15)

Note that U~ef"~ Feffl (F~ff being the effective velocity gradient) for the velocity difference field varying slowly at the scale l [4, 5, 9]. Let us consider now the expression of S2(r) for the fully turbulence. It is customary to classify turbulent flows in terms of the so-called Reynolds number, which is a nondimensional measure of the nonlinearity in the Navier-Stokes equation and is defined as R = UL/v, where U is a typical velocity, L is a typical length scale and v is the kinematic viscosity. For small Reynolds number, the flow is laminar and is characterized by a unique length scale L. As the Reynolds number increases (up to a critical Reynolds number) flows typically undergo a sequence of instabilities until, at some large enough value of R, they become fully turbulent. At large scales, the mechanism producing turbulence generally single out a scale L. For scale smaller than the Kolmogorov dissipation scale q ~- (v3/,s) TM (where e is the rate of viscous dissipation of energy per unit mass) the viscosity becomes important; it is at these scales that viscous dissipation occurs. This define hence a dissipation range, r ,~ q, and a range r/,~ r ,~ L, also called Kolmogorov inertial range, in which holds a fully turbulence which is universally homogeneous and isotropic. The extent of the inertial range in terms of R is

r ~ q, 11 ~ r ~ L,

(16)

where C is a constant,/~ is the Kolmogorov-Obukhov intermittency parameter [12] and Vk(tl) = (
et12/3v)l/2 l/tl"

l ~ q,

(_5CF[(6: 1~)/9])l/2vk(tl)(1/tl)1/3+u/ls, q ~ l ~ L,

(17) where F(...) is the Gamma function. It clearly appears that the behavior of S2(l/q) as a function of l/q is well reproduced by Ude(l/~l) illustrating the possibility of studying turbulence even under conditions of strong multiple scattering. It is also possible to study the transition between the dissipation range (l ,~ q) and the inertial Kolmogorov range ( t / , ~ l ~ L) since S2(r) could be probed at the adjustable length scale of the scattering mean free path. On the other hand, Eq. (14) is valid if condition (ii) is fulfilled, i.e. if the correlation time Zh is smaller tha shortest turbulent turnover time, ~ ~ q / ~ . In the large Reynolds number limit, this restriction writes: Vh <~ Z, <:~ kq >> ~geff(l

(18)

meaning that, at the extreme limit kq >> 1, the turbulent velocity difference field must vary slowly at the scale of the optical wavelength to consider the flow as quasistationary. This impose to work at Reynolds numbers such that: R/Rc ~ (kL) '*/3.

2.2. Nonhomogeneous velocity difference." Comb of sheets of discontinuous shear The nonhomogeneous velocities difference have been already treated in Ref. [9]. If V,h slowly varies at the scale of l, in this case the quantity of interest is its strain tensor oil(r) and the corresponding fluctuations, (A$2(t))~oh~,

D.J. Bicout, R. Maynard/ Physica B 204 (1995) 20 26

24

are given by Eqs. (40)-(41) of Ref. [9]. For instance, for a comb of continuous sheets of vorticity 2 parallel to the z-direction and separated by a distance ~ (which may be a random variable), the strain tensor is given by ra=oo

axe(Z) =

~

vm6(z- m~),

(19)

where the discontinuous shear Vmis either a deterministic function of the position r or a stochastic variable or both. For the sake of simplicity, we assume that v,. is a random variable for which the configurational averages are given by

v~ = O,

v.v.~, = ~ 6~.~,,

(20)

2.3. Total time autocorrelation.['unction Consider now the total velocity difference obtained from the linear combination of laminar and turbulent parts. The total phase fluctuations. (A~,2(t)), caused by the whole flowing fluid are obtained by summing incoherently the two contributions defined above as follows: (A~bn2(t)) = (Agb2(t))flam) + (A~bnz(t))lturb).

The total time autocorrelation function, C~(t), is then given by

Cl(t)=flG(t'l)exp{-2( t~12n-2(

,'Clam/

where v is a constant. F o r the diffusing paths starting on the sheet z = 0, one obtains

\Z.h} m= ~

1-/3~2\ 1/2

×.,=-~o y~ erfc[t~)ImlJ,

]

(2l)

where z,h = ( 3 0 , ~ / k 2 ] ) 1/2 is the characteristic time needed by a pair of scatterers initially separated by a distance 1 a part of a continuous sheet of vorticity to move a relative distance k-~ due to the fluctuation of their velocity difference v2. According to Eq. (6) the correlation time in laminar flow is Zlam = z.d(1 - ~b). The time autocorrelation function due to the laminar flow is therefore given by

×~/].= ~erfcLIm'[~) J~d..

]'1 x.,=~_~erfc[~ff~n) IrnlJ~dn, " =~

F/'3~2\ 1/2

'cturb

l -- (~ Th -

(25)

~'|am

(22)

where we have taken the continuum limit by replacing in Eq. (1) the summation over n by the integral. When the inter-sheet distance, ~, is small compared to I but larger than the instability thickness, the discontinuous nature of the velocity field disappears and the summation over m can be changed into an integral to give: (A4~,z(t))~.h~ = (tl/~Z,h)2n, which corresponds to the situation where the velocity gradient is the same everywhere in the whole scattering medium. This latter case has been already studied in Refs. [4, 5, 9],

~

Tnh

and the order, n, of multiple scattering paths. For instance, for m = 0, i.e. when ~ >) 1, corresponding to a continuous sheet of vorticity located at the starting point of light paths, the time autocorrclation function writes Cl(t)--

G(n)expl-2( l

t~2n[1 + \z,.,b/

dn. (26)

L

The final expression of Cl(t) is obtained by integration over G(n) for a given experimental geometry. F o r instance, for reflection of an extended plane wave source from a thick medium, we have Ct(t)=i

n-3/2exp 1

2 Tangential contact of two layers of superimposed nonmiscible fluids flowing with different constant velocities give rise in their contact plane to a flow corresponding to an infinitely thin and continuous sheet of vorticity, generally instable. This is the manifestation of the Kelvin-Helmholtz instability I-7].

(24)

It is remarkable to note that, in most of the cases, the time autocorrelation function for the nonhomogeneous turbulent flow is different from that we obtain for the laminar flow although the quasi-stationary (hypothesis (ii)) of flow is assumed for the two contributions. In addition, despite the separation of the total velocity difference field into two incoherent contributions, Cl(t) is equal neither to Cl(lam)(t)× Cl(turb)(t) nor t o Cl(lam)(t) + Cl.u,b)(t) but stems from the nontrivial combination of Cao.m)(t) and Cl,.,b~(t). The relative importance of CUl.m)(t) and Cl(turb)(t ) into Cl(t) is controlled by both the contrast factors, ~. -

Cl(lam,(t) = f ~ G ( n ) e x p { - 2(t/'C,am)2X~

(23)

-2

--

n 1+--

'cturb

dn. (27)

N~

The asymptotic expressions of C~ (t) can be written as

C1 (t) ~-

{: -(t/zt"'b)'(t/z'"rb)~l/~2' (t/Zl.m} 2, 1/e ~ (t/Zl.m) < 1.

(28)

D.J. Bicout, R, Maynard/ Physica B 204 (1995) 20-26 It clearly appears that in the real turbulence, two contributions, at least, act competitively to the decorrelation of Cl(t). As an example, consider a fluid flow in a Taylor-Couette cell with the inner cylinder (of radius R~ = 1.25 cm) rotating at speed Vand the outer cylinder (of radius R2 - 1.6 cm) fixed. The Taylor rolls appear when the Reynolds number R exceeds a critical value, R¢ ~ 84, with R defined as R = (R 2 - Rt)V/v, where v is the kinematic viscosity of fluid (here, water). For the transport mean free path l* ~ 60 lam and the wavelength 2 = 387 nm of light, we find that the characteristic time % of Cl(t) corresponding to R¢ is: r~---695 ~ts. For R ~> 22R, (i.e. for correlation times r ~< 32 ITS), for instance, where the velocity becomes turbulent [10], and if we assume that laminar and turbulent regions rotate with the same mean speed as in spiral turbulence [10], therefore, e only depends on 4, as e=

1-4,

(29)

Thus, Cx (t) exhibits a quadratic, t 2, and linear, t, behavior with time for ~b = 0 and 4, = 1, respectively. In general, the contrast factor e is a function of l, Kolmogorov dissipation scale q, characteristic scale ~ of the nonhomogeneous laminar flow and Reynolds number R. For a mixture of laminar and fully developed turbulence (without intermittency) fluids, for instance, can be expressed as

(30)

This indicates that 4, is a function of both R and I. One expects ~b to increase with R and decrease with length scale I. Hence, 4, can be regarded as the intermittency factor defined as the fraction of fluid participating in the turbulent motion. So, we have t l '¢ l, q .~ I < L ~ ¢ (homogeneous turbulence), • ~ 1, q < l < ¢ < L (nonhomogeneous turbulence), [ e >> 1, 1 < ~ < q

(nonhomogeneous flow).

(31) For example, if 4, = 1/2, R/R¢ -,~ 100 and the typical width L -~ 2 mm of the sample, the behavior of the velocity difference field probed by light paths at scale I is then:

25

Correspondingly, the window of accessible Reynolds numbers is: R/Rc ~ 106, for a typical value kl ~- 103 in multiple light scattering experiments.

3. Conclusion The main interest of this short paper is to show that the time dependence of Cl(t) is not straightforward for a turbulent flow as illustrates the heuristic and simplified model of turbulence considered above. For a regular array of sheets of discontinuous shear embedded in a distribution of a chaotic velocity field (at the scale of mean free path of light), we have obtained a general expression for C~(t) parametrized by the contrast factor ~ = f(I, ~1,4, R). The functional form of CI (t) depends both on the regime of the flow (laminar or turbulent flow) and on the nature and characteristics of the flow at the scale of mean free path of light. In general, C~ (t) is sensitive to the second-order structure function of the velocity difference at the scale of mean free path of light. Schematically, two mechanisms, at least, contribute to the decorrelation of Cl(t ) involving two characteristic times ~'lam and "Cturb. The first characteristic time corresponds to the laminar component of the velocity difference field, which is often nonhomogeneous except for shear flow, and the second one to the homogeneous turbulence. The model of a turbulent flow considered here is obviously oversimplified. Specially, the coupling between antisymmetric ~ij and symmetric au tensors of the velocity field has not been discussed although the multiple light scattering spectroscopy be only sensitive to aij [9]. This contribution must be considered as a first step towards this complicated problem where the sheets of discontinuous velocity gradient field are changed into discontinuous lines. It is certainly difficult to determine the nature of the turbulent flow from the sole analysis of Cl(t). However, such investigations lead to extract characteristic time and length scales of the problem like the typical width of structures of turbulent (or laminar) fluid. The study of contrast factor, e, versus the intermittency factor 4, would allow one, in principle, to follow how the flow is disorganized before it becomes turbulent and, therefore, to gain qualitative i~asight on the scenario of the transition to turbulence.

References f ! ,~ 60 p_m (laminar), 60 p.m (laminar-turbulent), > 60~tm

(turbulent).

(32)

[1] G. Maret and P.E. Wolf, Z. Phys. B 65 (1987) 409. [2] D.J. Pine, D.A. Weitz, P.M. Chaikin and E. Herbolzheimer, Phys. Rev. Lett. 60 (1988) 1134.

26

D.J. Bicout, R. Maynard/Physica B 204 (1995) 2 6 2 6

I-3] D.J. Pine, D.A. Weitz, P.E. Wolf, G. Maret, E. Herbolzheimer and P.M. Chaikin, in: Scattering and Localization of Classical Moves in Random Media, ed. P. Sheng (World Scientific, Singapore, 1990) p. 312. I-4] X-L. Wu, D.J. Pine, P.M. Chaikin, J.S. Huang and D.A. Weitz, J. Opt. Soc. Am. B 7 (1990) 15. [5] D. Bicout, E. AkkeI:mans and R. Maynard, J. Phys. I France 1 (1991) 471. [6] D. Bicout and G. Maret, Multiple Light Scattering in Taylor-Couette Flow, to be published in Physica A. 1-7] D.J. Tritton, Physical Fluid Dynamics, Sections 2.6, 17.5 and 18.3 (Clarendon Press, Oxford, 1992).

[8] E. Akkermans, P.E. Wolf, R. Maynard and G. Maret, J. Phys. France 49 (1988) 77. [9] D. Bicout and R. Maynard, Physica A 199 (1993) 387. [10] D. Coles, J. Fluid. Mech. 21 (1965) 385. [11] B. Castaing, Y. Gagne and E.J. Hopfinger, Physica D 46 (1990) 177. [12] A.N. Kolmogorov, J. Fluid Mech. 13 (1962) 82; A.M. Obukhov, J. Fluid Mech. 13 (1962) 77. For the last versions, see: A.N. Kolmogorov, Proc, R. Soc. Lond. A 434 (1991) 9-17. [13] L. Landau and E. Lifchitz, M6canique des Fluides (Librairie du Globe, MIR Edition, 1989; 2nd ed.) ch. lIl.