数理解析研究所講究録922 巻 1995 年 70-81

70

乱流における速度ゆらぎの間欠性 :4.2K での陽ガスの実験

INTERMITTENCY OF VELOCITY FLUCTUATIONS IN: AN EXPERIMENT AT 4. .
$2\mathrm{K}$$\mathrm{T}\mathrm{U}\mathrm{R}^{-}\mathrm{B}\mathrm{u}\mathrm{L}\mathrm{E}\mathrm{N}\mathrm{C}\mathrm{E}$A.

$\mathrm{N}\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{t}2$

B.

),

(1)
(2)
(3)
(4)

) ,
$n$

$\mathrm{C}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{g}\{1$

B. Chabaud(1), F.

),$\mathrm{C}\mathrm{h}\mathrm{i}||\grave{\mathrm{a}}\mathrm{t}3$

B. H\’ebralll), J.

$\mathrm{P}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{k}\mathrm{e}\{4)$

CRTBT-CNRS, BP 166, 38042 Grenoble Cedex 9, France
RIEC, Tohoku University, 2-1-1 Katahira Aobaku Sendai 980-77, Japan
Lyon, France
Ecole Normale Sup\’erieure de Lyon, 46 All\’ee
Experimental Physik II, Universitat Bayreuth, D-95440 Bayreuth, Germany$\mathrm{d}^{\mathrm{t}}1\mathrm{t}\mathrm{a}|\mathrm{i}\mathrm{e}69364$

Also at lnstitut Universitaire de France

Among the most important questions about turbulence, is the origin ofintermittency in a flow. Another, closely related, is whether there is anultimate, high Reynolds number regime, and if so, how it ari ses and how tocharacterize it.
The main difficulty in studying these questions is that an experimentusually gives access only to a limited range of Reynolds numbers. Physicists
have to compare results from different experimental configurations, such as
grid, jet and tunnel flow wi th the hope that the Taylor based Reynolds
is the unique and universal parameter for turbulence [1].
numberit becomes possible
With cryogenic experiments using gaseouswith the same geometry. Close to its criticalto cover a large range ofpoint (2.2 bar, 5.2 K), gaseous Helium has the smallest known kinematic, but can be an appreciably viscous fluid at low
viscositypressure. We have built the first open flow working undersuch conditions, an axisymetric jet. We have developed the corresponding
anemometry techniques [4]. In the following we present briefly theexperimental set-up and the anemometer. Then we discuss the velocity
intermittency results at inertial scales in developed turbulence, and a new
method of analysis using conditional histograms.
$\mathrm{R}_{\Theta}$$4\mathrm{H}\mathrm{e}[2]$$\mathrm{R}_{\Theta}$$(\mathrm{V}=210- 8\mathrm{m}2/\mathrm{s})$$\mathrm{P}(\mathrm{v}\mathrm{P}=\mathrm{c}\mathrm{o}\cap \mathrm{S}\iota.)[3]$

71

THE EXPERIMENTAL SET UP
The experimental system consists of a chamber (fig.1) immersed in aliquidbath which also acts as a ballast for the working gas. Afterlaminarization, this gas enters the chamber through a contracting nozzle ofsection ratio 100 and diameter. The
develops downward in thechamber, and is pumped outward to ensure the circulation. The return flow
is isolated from the jet by a jacket.
$4\mathrm{H}\mathrm{e}$$2\mathrm{m}\mathrm{m}$$\mathrm{i}\mathrm{e}\mathrm{t}$Fig. 1: Experimental set-up: (1) gas
injection, (2) laminarization honeycomb, (3)nozzle, (4)(5) stabilization grid,
exit,
(6)
(7)
pressure
pumpingmeasurement, (8) symbol for thermometer,(9) jacket.
$\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{c}\iota \mathrm{o}\mathrm{r}_{\mathrm{I}}$An exact scale reproduction of the set-up has been made out of glass tovisualize the flow with dyed water. It showed that a grid is necessary
downstream to stabilize the. Without it, the jet randomly attaches tothe walls, producing strong bursts. For more details on this point, seeref. [5].
$\dot{|}\mathrm{e}\mathrm{t}$72

Our anemometer is a cryogenic adaptation of the hot-wire techniqueusing superconducting materials [4]. Its typical space resolution is about 10. The wire is operated at(constant resistance)
through a home-made 10 MHz lock-in amplifier. The total response time to a
perturbation can be less than 1 ps depending on the loop gain [4]. The voltage
signal is recorded through a 12 bit HP 5183
converter driven by aMacintosh Quadra 700 computer. The average signal is about 3.5 to 5.5 V, andthe instrumental noise is about. The calibration of the wire wasmade in the potential cone of the jet as described in ref. [6].
The probe is placed on the axis of the jet at a distance of 50 times the),nozzle diameter
upstream rom the grid. At this distance, thediameter of the jet is about 3 cm and the diameter of the jacket is 13 cm.
The mean velocity , the turbulent ratio (25%) and the dissipation rate arein agreement with what is commonly observed [7].
$\mu \mathrm{m}$$|\mathrm{l}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{I}|$$\mathrm{A}/\mathrm{D}$$10^{-7}2/\mathrm{H}\mathrm{z}$$\mathrm{t}^{]\mathrm{o}_{\mathrm{C}}\mathrm{m}}$$7\mathrm{c}\mathrm{m}$$\mathrm{f}$$\cup$

It is traditional [1] to study intermittency by investigating thestatistics of the velocity differences
between two times at the sameprobe. This time lag is converted to a distanceusing the Taylor frozen
turbulence hypothesis.
For each Reynolds number, a sample ofdata points is recorded. A
typical sampling frequency atisand the whole recording timeis less than 10 minutes.
$6\mathrm{v}_{\mathrm{r}}$$\mathrm{r}=\mathrm{U}\tau$$\tau$$10^{7}$$\mathrm{R}_{\Theta}=700$$60\mathrm{k}\mathrm{H}\mathrm{z}$EXPERIMENTAL RESULTS
We investigate the shape of the probability density function (hereafterPDF) ofand its evolution with the scale, , for different Reynolds
numbers. We discuss it in terms of the parameter.
It is well known that the shape of these PDFs depends on the scale, . It
goes from nearly Gaussian at large scales, , to nearly exponential at small(Kolmogorov) scales , where dissipation occurs [1]. The PDF at scalecanbe written in a normalized way :$6\mathrm{v}_{\mathrm{r}}$$\mathrm{r}$$\lambda^{2}(\mathrm{r})$$\mathrm{r}$$\mathrm{L}$$\eta$$\mathrm{r}$$\frac{1}{\sigma_{\mathrm{r}}}\mathrm{P}_{\mathrm{r}}(_{\frac{6\mathrm{v}_{\mathrm{r}}}{\sigma_{\mathrm{r}}}})$

(1)

73

(
which defines the function). We represent thisis the variance of
PDF at scale by a superposition of large scale PDFs [8] :$\mathrm{P}_{\mathrm{r}}$$\sigma_{\mathrm{r}^{2}}$$6\mathrm{v}_{\mathrm{r}}$$\mathrm{r}$(2)$\frac{1}{\sigma_{\mathrm{r}}}\mathrm{P}_{\mathrm{r}(\frac{6\mathrm{v}_{r}}{\mathrm{o}_{\mathrm{r}}})=}\int_{\infty}^{\mathrm{R}}\mathrm{G}_{r}(|\cap 0)_{\frac{1}{\sigma}\mathrm{p}(_{\frac{6\mathrm{v}}{\sigma})}}\llcorner \mathrm{d}|\cap\sigma$

The physical basis of this superposition is given by the analysis ofconditional histograms. In the Kolmogorov-Obukhov i theory 11,10]
simply Gaussian :$\mathrm{G}_{\mathrm{r}}\mathrm{i}\mathrm{s}$$\mathrm{G}_{\mathrm{r}}(\ln\sigma)=arrow \mathrm{X}\lambda\sqrt{2\pi}]\mathrm{t}^{-_{\frac{1\cap^{2}(\frac{\sigma}{\mathrm{o}_{\mathrm{m}}})}{2\lambda^{2}}}}]$(3)

It has been shown in previous papers $[9,11]$ that the shape of
weaklydepends on the actual shape of.
is mainly determined by the variance. We then fi the hi stogram of the increments
wi th the formul a(2) usi ng the simple gaussian distribution (3) as an ansatz. The val ue ofch gi ves the best fi of the hi stogram will be to a good approximation
equal to.
We can then characterize the change of shape of the histograms withscale, simply by investigating thedependence of N2. Fig. 2 illustrates theassertion that the shape of
mainly depends on. PDFs corresponding tomeasurements at several Reynolds numbers and various scales , but having
the same measured
have been plotted versus. The difference
between them is small. The function
corresponding to this value ofisshown by a solid line.
Within the energy cascade scheme, the measured
can be interpreted
as proportional to the number of cascade steps [9]. We named this important
physical quantity.
$\mathrm{P}_{\mathrm{r}}$$\mathrm{G}_{\mathrm{r}}$$<(\ln\sigma)2>$$\mathrm{P}_{\mathrm{r}}$$\mathrm{t}$$6\mathrm{v}_{\mathrm{r}}$$\lambda 2$$\mathrm{w}\mathrm{h}\mathrm{i}$$\mathrm{t}$$<(1\mathrm{n}\mathrm{o})2>$$\mathrm{r}$$\lambda^{2}$$\mathrm{P}_{\mathrm{r}}$$\mathrm{r}$$\lambda^{2}$$6\mathrm{v}_{\mathrm{r}}/\sigma_{\mathrm{r}}$$\lambda^{2}$$\mathrm{P}_{\mathrm{r}}$$\lambda 2$$\mathrm{I}\iota_{\mathrm{C}\mathrm{a}}\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}^{1}1$74$10^{0}$$!’$.

$10-2$$\vee \mathrm{h}\wedge\infty>$$\mathrm{A}14^{\wedge}$$10-4$$\mathrm{o}^{\mathrm{A}}B$$\mathrm{a}_{\mathrm{A}}$$10-6$$- 10$

$- 5$

5

$0$

$\mathrm{I}0$$6\mathrm{v}_{\mathrm{r}}/_{0}$Fig. 2: Histograms of(the variance is normalized to 1) for different Reynolds numbers anddifferent scales but the same value of . The open symbols correspond to the following parameters(is the dissipative Kolmogorov scale) : triangle,; circle,,,diamond,; square,,. The solid line represents the PDF calculated
with.
$6\mathrm{v}_{\mathrm{r}}$$\lambda^{2}$$\mathrm{R}_{9}=90$$\eta$$\mathrm{R}_{6}=328,$$\mathrm{r}=24\eta$$\mathrm{R}_{\Theta^{=}}598$$\mathrm{r}=20\eta$$\mathrm{R}_{\mathfrak{g}}=161$$\mathrm{r}=24\eta|$$\mathrm{r}=62\eta$$\lambda^{2}=0.09$

There are two theoretical prediction for the behaviour ofversus .
One is the Kolmogorov-Obukhov logarithmic lawln(r/L), the other is a
power lawpredicted by themodel [11]. It has been
observed in several flows $[11,12]$ thatbe closer to the power law
than to the logarithmic law, even though we can observe in fig.
theexistence of the latter over a narrow range at large scale.
order to firmly
establish on a purely experimental basis the power law dependence, we) versus In(r). This is possible thanks to the quality of ourdifferentiate
determination of N2 by systematically minimizing aquantity [9]. We
observe in fig.
a plateau region which is a clear affirmation of the powerlaw dependance ofin this range.
$\lambda^{2}$$\mathrm{r}$$\lambda 2\alpha$$\lambda^{2}\alpha \mathrm{r}-\beta$$\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\iota \mathrm{l}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$$\lambda 2\mathrm{m}\mathrm{l}\mathrm{g}\mathrm{h}\mathrm{t}$$3\mathrm{a}$$\ln$$\mathrm{l}\mathrm{n}1\lambda 2$$\mathrm{X}^{2}$$3\mathrm{b}$$\lambda^{2}$

75

0.20

0. $15$

0. $10$

0.05

0.00
10

1

103

$10^{2}$$\mathrm{r}/\eta$

0.5
$0$

$0.0$

$*$

$0$

$. \mathrm{r}_{\mathrm{O}}\vee\subseteq\wedge\sim\frac{\vee=}{\varpi}\wedge\prec.- 1^{\cdot}0- 0.5$$\nwarrow_{\mathrm{a}_{\mathrm{s}_{u_{4}}}^{\mathrm{t}\mathrm{b}_{\mathrm{n}_{\mathrm{A}}\prod_{\wedge}\mathrm{D}}}}0*_{\mathrm{b}\triangleleft m\triangleleft 4_{4\tilde{\Leftrightarrow}\alpha*\mathrm{n}}^{\mathrm{o}}}\infty \mathrm{R}\Pi \mathrm{n}_{\mathrm{H}\mathrm{n}},\mathrm{f}\mathrm{f}\mathrm{l}_{\mathrm{q}_{\mathrm{D}}}\mathfrak{o}_{1}\mathrm{A}\Phi\Phi \mathrm{A}\circ\triangleleft 4\mathrm{A}y\wedge \mathrm{o}_{\Phi_{\circ}}\mathrm{o}\prime \mathrm{O}\phi_{\wedge}$$\triangleright$$\circ \mathrm{u}_{\mathrm{Q}}$$\mathrm{A}$$- 1.5$102

10$\mathrm{r}/\eta$: Dependence of. The solid diamond corresponds to a logarithmicon for
Fig. 3open
diamond corresponds to a double logarithmic plot.
linear plot, whereas the( : dln
as a function of for fferent
values (the different symbols
represent the same parameters as in fig. 2).
$\lambda^{2}$$\{\mathrm{a}\}$$\mathrm{r}$$\mathrm{b}\}$$(\lambda^{2})/\mathrm{d}\ln(\mathrm{r})$$\mathrm{R}_{\Theta^{=}}328$$\mathrm{r}$$\mathrm{d}i$$\mathrm{R}_{6}$in the plateau
which is the derivative
The exponent
region, has been measured in the whole range of accessible Reynolds$-\beta$$\mathrm{d}\ln(\lambda^{2})/\mathrm{d}\ln(\mathrm{r})$76

numbers and compared to previous experiments. Following the proposed[11], we plotted
scaling of) in fig. 4. Good agreement isversusfound with other jet experiments. It is thus possible to characterize with
only one exponent
the whole statistics of the velocity fluctuations within
the scaling range.
$1/\beta$$\beta$$\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{R}_{\Theta}$$\beta$5.0
4.0$\blacksquare$$\blacksquare$

3.0$\Phi_{-}$$\underline{\sim}$

2.0.

$\bullet$$0^{\mathrm{q}}$$\circ$$\blacksquare$O$\blacksquare$$\bullet$oo$\epsilon^{l}0$

1.0

o.o

103

$10^{2}$$\mathrm{h}$Fig. 4: Inverse of the scaling exponent as a function of . Open circles are the results of ourcryogenic jet experiment. The solid squares correspond to the same analysis for different
experiments (see ref. [12]). The four middle ones correspond to air jets, the two smallest ones into grid flow and the two highest ones to wind tunnel measurements.
$\beta$$\mathrm{R}_{6}$$\mathrm{R}_{6}$Coming back to fig.
, we note that the plateau di ssappears for thelowest Reynolds number. This is a signature of the transition by which theflow goes from 1lhard turbulencell [13] in which thertial range can be
assimilated to the plateau region, to llsoft turbulence
which no inertial
range is evident. This transition occurs at, which explains why
we report no value of
below this threshold, even though we have done
measurements down to.
Curiously enough, the grid experiments referred to in fig. 4 give valuesoffor
below the transition. It is possible that the turbulence producedin the wake of a grid is qualitatively different from that produced by jets orwind tunnels.
$3\mathrm{b}$$\mathrm{I}\mathrm{I}\mathrm{i}\mathrm{n}.\mathrm{e}$$|\cap$$\mathrm{R}_{\Theta}=160\pm 40$$\beta$$\mathrm{R}_{\Theta}=90$$\beta$$\mathrm{R}_{\Theta}$

77, we note the analogy with the isotherms of a
Looking again at fig.
thermodynamic liquid-gas system. The transition between soft and hard
turbulence is then analogous to a critical point. This thermodynamic analogy
has been developed in ref. [14].
As often, the nature of the transition gives a hint to characterize theturbulent states above and below. At low Reynolds numbers, the soft
turbulence could be said to be dissipative in nature, as the scales in which, inenergy is injected and those at which it is dissipated overlap. At large$3\mathrm{b}$$\mathrm{R}_{\Theta}$

hard turbulence, the plateau introduces a clear inertial separation between
these scales.

CONDITIONAL HISTOGRAMS
It has been found recently that a condition applied to velocity
increments selects Gaussian histograms [15]. The conditional statistics are
Gaussian from the inertial range down to the dissipative one. The authors
being defined as :constant, the functionused as condition$\mathrm{e}_{\mathrm{r}}(\mathrm{x})=$$\mathrm{e}_{\mathrm{r}}$$\mathrm{e}_{\mathrm{f}}(\mathrm{x})=\frac{15\mathrm{v}}{\mathrm{r}}\int_{\mathrm{X}}^{\mathrm{X}+}r\mathrm{x}(\frac{\partial \mathrm{v}(\mathrm{x})1}{\partial \mathrm{x}^{1}})^{2}\mathrm{d}1-15\mathrm{V}(\frac{6\mathrm{v}(\mathrm{x})}{\mathrm{r}})^{2}$(4)

The first term of the sum is the dissipation averaged over a volume of size
, in the case of locally isotropic turbulence. The second term can be seen as
thus reflects the energy
the dissipation at scale . The differencetransfer rate at the considered scale .
We worked out this calculation with our velocity signal, and found thefollowing features [5] :$\mathrm{r}$$\mathrm{r}$$\mathrm{e}_{\mathrm{r}}$$\mathrm{r}$fig. 5 for a few
The conditional PDFs are Gaussian as we can see
values of the condition. This is valid not only in the inertial but also in thedissipative range.
1)$\mathrm{i}\mathrm{n}$78$-1\angle$$-‘\pm$$-\circ$OVr$\cup$$\perp\angle$$\overline{\mathrm{o}_{\mathrm{r}}}$Fig. 5: Conditional histograms of the increments
condition. Here,.

$6\mathrm{v}_{\mathrm{r}}$at scale$\mathrm{r}=100\eta$for a few values of the$\mathrm{R}_{6}=328$$\mathrm{e}_{\mathrm{r}}(\mathrm{m}\mathrm{W}/\mathrm{k}\mathrm{g})$The larger the energy transfer rate,, the wider the Gaussianconditional histogram : the variance of the conditional PDF is related to thecondition
by a power law as can be seen
fig. 6 for several scales.
2)$\mathrm{e}_{\mathrm{r}}$$\mathrm{i}\mathrm{n}$$\mathrm{e}_{\mathrm{r}}$$\perp\cup$$\perp \mathrm{U}^{-}$$\cup.\cup\cup 1$$\cup.\cup 1$$\cup.\perp$$\perp$$\mathrm{e}_{\mathrm{r}}$Fig. 6 : Standard deviationof the conditional histograms as a function of the condition
several scales. The vertical line gives the limit of resolution of . Here,.
$\sigma$$\mathrm{e}_{\mathrm{r}}$$\mathrm{R}_{\Theta}=328$$\mathrm{e}_{\mathrm{r}}$

at

79

.
This relation gives a physical basis to the measurement method ofIndeed, the mixture of pure Gaussian regimes supposed by equation (2) now
appears natural.
$\lambda^{2}$

We can easily compute the PDF of the quantity
by integrating theoi nt. The result for various
over all possible increments
scales can be seen in fig. 7.
3)$\mathrm{j}$$\mathrm{e}_{\mathrm{r}}$$6\mathrm{v}_{\mathrm{r}}$$\mathrm{h}\mathrm{I}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{m}$In

Fig. 7 : Histograms of the condition.
by a factor 10.

$\mathrm{q}$$\mathrm{t}_{r\mathrm{r}}^{\mathrm{e}_{1^{-}<}}\mathrm{I}\mathrm{n}(\mathrm{e})>$at several scales. Each of these has been shifted verticaly$\mathrm{R}_{\{\}}=328$

As
is related to , this gives a way to check the validity of the Iognormal ansatz. But as we have already mentioned, the statistics of velocity
increments are weakely sensitive to the shape of the PDF of . As aconsequense, a very large statistical sample of velocity is necessary topoints is slightly above
make any statement on this PDF. Our sample ofthe limit of statistical noise as evaluated in [9]. At this point we cannot(or ) isdetermine wether the distribution of-normal or not.
$\mathrm{e}_{\mathrm{r}}$$\sigma$$\mathit{0}$$10^{7}$$\mathrm{e}_{\mathrm{r}}$$\mathit{0}$$\log$

80

CONCLUSION
We have built an original cryogenic experiment allowing broad variationof Reynolds number in the same geometry. Measuring the physical quantity
which characterizes the intermittency, we have shown for the first timethat the inertial range appears in a turbulent flow through a well definedllsecond orderll transition (analogous to a critical point).
$\lambda^{2}$

The Reynolds number dependence of the exponent shows that viscosity
has some influence ar beyond the dissipative (Kolmogorov) scale, up to thevicinity of the integral scale. This result is of prime mportance forcharacterizing turbulent flows when the small scale resolution cannot bereached, for instance in large eddy simulations.
$\beta$$\mathrm{f}$$i$

The conditional statistics of the velocity fluctuations give new insight
into the structure of the cascade process. We have shown that the nonGaussian statistics of the velocity fluctuations is due to a mixing in theflow, of
(Gaussian) regimes. Behind the non-Gaussian behaviour of thevelocity fluctuations, there is a hidden simple dynamics, though it is not yet
understood.
$\mathrm{l}\mathrm{l}\mathrm{P}^{\mathrm{u}\mathrm{r}}\mathrm{e}^{\mathrm{l}\mathrm{I}}$REFERENCES:[1]

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (The MIT Press, Cambridge,1975)[2]

Closed flows have already been studied in such conditions. See for instance, D. C. Threlfall, J.
Fluid. Mech. 67, 17 (1975); M. Sano, X. Z. Wu, and A. Libchaber, Phys. Rev. A 40, 6421
(1989); P. Tabeling, G. Zocchi, and J. Maurer (to be published).

[3]

R. D.

[4]

B. Castaing, B. H\’ebral, and B. Chabaud, Rev. Sci. lnstrum. 63, 4167 (1992).

[5]

A. Naert, Thesis, Universit\’e Joseph Fourier (Grenoble, France), 1995 (unpublished)[6]

B. Chabaud, Thesis, Universit\’e Joseph Fourier (Grenoble, France), 1992 (unpublished)[7]

R. A. Antonia, B. R. Satyaprakash, and A. K. M. F. Hussain, Phys. Fluids 23, 695$\mathrm{M}\mathrm{C}\mathrm{c}\mathrm{a}\mathfrak{n}\mathrm{h}\mathrm{y}$, Nal. Bur. Stand. (US) Tech. Note No. 631.

(1980)

81[8]

The well known asymetry of the PDF (skewness) is here entirely contained in
As in ref. [11], we use the following ansatz :$\mathrm{P}_{\mathrm{L}}(\mathrm{x})=$

where
[9]

$\mathrm{a}_{\mathrm{s}}=0.18$

A$\mathrm{e}\mathrm{x}q^{\mathrm{x}_{2}}-\dashv^{2}$]$\mathrm{P}_{\mathrm{L}}$.

$+ \mathrm{a}\mathrm{s}\frac{\mathrm{x}}{\{1+\mathrm{X}2)2})]$and A is a normalization constant.

A. Naert, L. Puech, B. Chabaud, J. Peinke, B. Castaing, B. H\’ebral, J. Phys. II (France), 4,

215 (1994).
[10] A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962), A. M. Obukhov, J. Fluid Mech. 13,77$(^{]9}62)$[11] B. Castaing, Y. Gagne, E. Hopfinger, Physica (Amsterdam) 46D, 177, (1990).
[12] B. Castaing, Y. Gagne, M. Marchand, Physica (Amsterdam) 68D, 387, (1993).

[13] While the mechanisms are different in convective flows, it is worth noticing that, in thiscase, the transition fromtoturbulence has been also associated with thedevelopment of an inertial range. See F. Heslot, B. Castaing, and A. Libchaber, Phys. Rev. A
36, 5870 (1987).
$1\iota_{\mathrm{S}\mathrm{O}}\mathrm{f}\mathrm{t}\iota 1$$1|\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{l}t$[14] F. Chill\‘a, B. Chabaud, A. Naert, J. Peinke, B. Castaing, and B. H\’ebral, inthe 5th European Turbulence$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{C}\mathrm{e}||$$||\mathrm{p}_{\mathrm{r}\mathrm{o}}\mathrm{c}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{S}$, Sienna, Italy, 1994, Edited by R. Benzi (to bpublished).
[15] Y. Gagne, M. Marchand, B. Castaing, J. Phys. II (France), 4, 1-8 (1994).

of$\mathrm{e}$