Fluctuation-dissipation relation on a Melde string in a turbulent flow,
  considerations on a "dynamical temperature"
8 Fluctuation-dissipation relation on a Melde string in
a turbulent flow, considerations on a “dynamical
temperature”.
V Grenard, N B Garnier and A Naert.
Université de Lyon, Laboratoire de Physique, École Normale Supérieure de Lyon,
46 Allée d’Italie, 69364 Lyon Cedex 07, France.
E-mail: Antoine.Naert@ens-lyon.fr
PACS numbers: 05.70.Ln
PACS numbers: 05.40.-a
PACS numbers: 05.20.Jj
Abstract. We report on measurements of the transverse fluctuations of a string
in a turbulent air jet flow. Harmonic modes are excited by the fluctuating
drag force, at different wave-numbers. This simple mechanical probe makes
it possible to measure excitations of the flow at specific scales, averaged over
space and time: it is a scale-resolved, global measurement. We also measure
the dissipation associated to the string motion, and we consider the ratio of the
fluctuations over dissipation (FDR). In an exploratory approach, we investigate
the concept of effective temperature defined through the FDR. We compare our
observations with other definitions of temperature in turbulence. From the theory
of Kolmogorov (1941), we derive the exponent −11/3 expected for the spectrum
of the fluctuations. This simple model and our experimental results are in good
agreement, over the range of wave-numbers, and Reynolds number accessible
(74000 ≤ Re ≤ 170000).
1. Introduction
Turbulent flows exhibit a notoriously complex and unpredictable dynamics: they
present a huge number of degrees of freedom, and their dynamics are both far from
equilibrium and dissipative [1, 2, 3]. The kinetic energy injected at large scale by shear
instability mecanisms is dissipated into heat by the molecular viscosity at small scales.
That is, dissipation and injection scales are distinct. Therefore, a transport process
through scales is necessary for a flow to be stationary. It is suspected that instability
mechanisms associated with non-linearities generate harmonics, therefore transfering
energy to smaller scales almost without dissipation. An equivalent picture would
consist in vortices stretching each other in such a way that a non-zero energy transfer
occurs toward smaller scales. This picture of cascade process was first proposed by
Richardson [4]. The cascade stops approximately in the range of scales where the
viscosity becomes efficient to damp velocity gradients. In the late thirties, Kolmogorov
derived from this idea a phenomenological theory accounting for the fluctuations of
various observables in fully developed turbulence [5]. In the present work, we are
http://arxiv.org/abs/0704.0325v3
Measurements of a dynamical temperature in turbulence. 2
neither concerned by the large (energy injection) scales, nor by the small (dissipation)
scales, but by the intermediate range. In this intermediate inertial range, we study
the transport process through scales, expected to be universal. Instead of scale l, one
often refers to the wave-number k = 2π/l.
The control parameter of the flow is the Reynolds number: Re = V L
, where L is the
macroscopic scale of the flow (integral scale, or correlation length), V is a characteristic
shear velocity at large scale, and ν is the kinematic viscosity of the fluid. It is also
the mean ratio of the inertial by the dissipative contribution of the forcing over a fluid
particle. Interesting predictions were derived by Kolmogorov (1941), that we use in the
following. Especially, the range of scales over which fluctuations occur scales as Re3/4.
The prediction for the exponent of the power spectral density as 〈|ṽ|2〉 ∝ k−5/3 is
among the most famous successes of this theory [1, 2, 3].
Our experimental system is discribed in detail in the next section. It is a thin string
held by its ends at constant tension across a turbulent flow. To formalize briefly, it
is an oscillator with multiple resonances, coupled to a particular ’thermostat’: the
turbulent flow. This string is used to probe the inertial range of a flow of high enough
Reynolds numbers. The device is ’calibrated’ by measuring the average (complex)
response to an external perturbation, and then used to measure the free fluctuations
caused by turbulence alone. Measurement of the displacement r(t) caused by the
turbulent forcing f(t) is performed with small piezoelectric transducers. We measure
the average response, i.e. the displacement on one end caused by a known broad band
forcing on the other end. Then, measurements of the displacement on one end alone
give information on the forcing fluctuations. Our study goes a step forward, in an
exploratory way. Knowing the average response function of the string and measuring
r(t), we invoque a version of the Fluctuation-Dissipation Theorem extended out of
equilibrium, to define an effective temperature of the turbulent flow. This effective
temperature happends to be scale-dependant.
In this work, fully developped turbulence is addressed from the point of view of
statistical mechanics. We first recall one important break-through: the statement of
the Fluctuation-Dissipation Theorem (FDT). Consider a pair of conjugate variables
(displacement r and force f) of a small system in thermal contact with a large
heat reservoir. In the present case the small system is the string, coupled to the
turbulent flow which is the reservoir. Displacement r and force f are conjugate in
the sense that their product is the work exerted by the flow on the string. The
theorem originates from the idea that spontaneous fluctuations r(t) should have the
same statistical properties as the relaxation of r(t) after the removal of an external
forcing perturbation. The main hypothesis needed to derive this theorem are: –
linear response between f and r, – thermal equilibrium between the system under
consideration and the thermostat, – thermal equilibrium of the thermostat itself. The
response function Hr,f is such that: r(t) =
Hx,f (t − t
′)f(t′)dt′. Equivalently it
can be written in the Fourier space as: r̃(ω) = H̃r,f f̃(ω). Under some hypothesis, the
fluctuations of r (its 2-times correlation function) are linked by a very simple relation
with the dissipative response of the system to a perturbation of the conjugate variable
f (imaginary part of the average response function). It is simply proportional, and
the coefficient is nothing but the temperature multiplied by the Boltzman constant:
kBT [6]. The validity of the hypothesis has to be discussed in each case. If they are
satisfied, the correlation function of the spontaneous fluctuations is proportional to
the response function, i.e. the factor is unique and constant. Moreover, this factor
Measurements of a dynamical temperature in turbulence. 3
is the same for all couples of conjugate variables, and this factor is kBT , where T is
the temperature of the system. The Boltzman constant kB ≃ 1.38 10
−23JK−1 is an
universal constant. This relation can be expressed in spectral variables:
〈|r̃(ω)|2〉 =
2 kBT
Im[H̃r,f (ω)]. (1)
In this expression of the FDT, 〈|r̃(ω)|2〉 is the power spectral density of the fluctuations
of the displacement r, as H̃r,f(ω) is the response function on r to the conjugate
variable f . Because the string is very thin, the drag is purely viscous. It is therefore
proportional to the velocity, which is in quadrature with the displacement. The
dissipation is therefore proportional to the imaginary part of the average response
function: Im[H̃].
In the perspective of constructing a non-equilibrium thermodynamics, the FDT has
been reconsidered by L. Cugliandolo and J. Kurchan, while investigating amorphous
materials relaxing after a thermal quench through the glass transition [7, 8].
We present in the following an exploratory approach of the question of turbulent
fluctuations using their extended formalism. The Fluctuation-Dissipation Ratio
(FDR) can be rewritten:
ω 〈r̃(ω)2〉
Im[H̃r,f(ω)]
= 2 kBTeff.(ω), (2)
where the temperature is replaced by an ’effective’ temperature Teff., function of
frequency ω. The frequency dependence of Teff. expresses the fact that different degrees
of freedom are not at equilibrium with each other, resulting in internal energy fluxes.
In other words, in our system, each (independent) mode of the string couples to
(non-independent) scale of the flow. As the flow is stationary, we average our
measurements on time, and finally obtain the frequency dependance of Teff. as defined
by equation 2. Measurements of the fluctuations of the string give Fourier components
of the excitation of the flow. We measure independently the fluctuations, and the
complex average response function to a specified excitation, in a way discussed below.
We propose to analyse these measurements with the criteria discussed above.
The paper is organised as follows. The next section describes the experimental setup,
turbulent flow properties, and the setting of the string. General properties of a
vibrating Melde string are also discussed. The measurements are shown in section 3:
response, fluctuations, and the Fluctuation Dissipation Ratio of this system. In section
4, we derive from Kolmogorov’s theory a simple scaling model for the fluctuations of
the drag, and therefore the FDR, which accounts for the exponent observed in the
whole range of accessible Re. The section 5 is devoted to a discussion of our results,
especially in comparison to several definitions of temperature in turbulence proposed
in the literature.
2. The Melde string and the experimental setup
The experimental setup is sketched in Fig. 1. A turbulent air jet originates from
a nozzle of diameter 5 cm. The flow facility we used is thoroughly described in
[9]. A thin stainless steel string of length 60 cm is located 2 m downstream the
nozzle, perpendicular to the axis of the flow. At this distance, the length of the
string is about the diameter of the turbulent jet. The displacement of the string is
measured using piezoelectric multi-layer ceramics at each end of the string. A piezo
Measurements of a dynamical temperature in turbulence. 4
is deformed by a voltage. Reciprocally, if the ceramic in compressed, a voltage is
generated. The relation between voltage and deformation is linear, and the frequency
response is almost flat in the frequency range we consider here. It can be used as
actuator or sensor. We have two piezos, one on each end of the string. The two
different measurements we perform are the following. 1) complex response function:
one (input) piezo is feeded with a white noise voltage through a power amplifier. The
source is that of a HP3562A signal analyser. Standing transverse waves appear in
the string, weakly perturbed by the turbulent fluctuations. Mecanical displacement
on the other end is transformed into a voltage by the other (output) piezo. It must
be amplified, and both input and output voltages are recorded synchronously with a
24 bits A/D converter. The acquisition frequency is 50 kHz. We call response the
time averaged ratio of the voltage amplitudes on input and output piezos, recorded
simultaneously. Voltages in and out are proportional respectively to the displacement
and the constraint (on the piezos). The dimension of the actual response is the inverse
of a stiffness, as what we measure is the ratio of voltages. Dimentional prefactors are
omited for simplicity, as they are constant for the same setup (string and transducers).
The diameter of the string is 100 µm, less than the viscous scale of the flow which
is about η ≃ 170 µm at the largest Re accessible. The equation of motion of the
PIEZOS
STAND
Figure 1. Eperimental setup: the thin steel wire is pulled across a turbulent air
jet by a 4 Kg weight on a rigid stand. Piezoelectric transducers are in mecanical
contact with the wire at each end.
undamped and unforced string is a linear wave equation. Its solutions with fixed ends
are standing waves r(x, t) = A cos(ωn t − knx), where A is the amplitude, t is time
and x is position along the wire. The discrete wave numbers are kn = n
, where L
Measurements of a dynamical temperature in turbulence. 5
is the length of the string and n is a positive integer. In a first approximation, the
waves are not dispersive: ωn = c kn, where c is the phase velocity. T is the tension of
the string and µ its mass per unit length, c =
T/µ ≃ 300 m/s. With a 4 kg weight
on one end, the string’s fundamental frequency is f0 = 344 Hz.
Dissipation is mainly due to friction on air, and causes little dispersion. More precise
treatment would require terms of dissipation in the wire itself and in the piezoelectric
transducers that fix the ends. We neglect this, as the amplitude remains small (a
few tens of micrometers) if compared to the length of the ceramic pile (3mm), or
even the wire diameter (100µm). The possible coupling with compression wave is not
relevant, as the range of frequency is distinct. (Compression wave speed in steel is a
few thousands of m/s, larger than what we consider here: c ≃ 300 m/s.) When this
wire is immersed into the turbulent flow, the resonant modes are excited by the drag
forcing. The quantities measured are averaged along the wire. They are therefore
global in space but local in scale, or more precisely in Fourier-space. The vortices
at scale l are expected to excite modes of wave-number k = 2π/l. In that sense,
the string is acting like a mechanical spectrometer, almost exactly like a Fabry-Perot
interferometer.
3. Measurements
Modulus of the response function is plotted in Fig. 2. It shows that the resonance
peaks are indeed very narrow, ensuring a very precise selection of wave-numbers:
the quality factor is approximately Q ≃ 4000. The imaginary part of the response
function is giving the dissipation. The width of the peaks in the modulus is also
Figure 2. Modulus of the response function versus the harmonic number, at
Re = 154000. The abscissa is given in non-dimensional coordinates, normalised
by the fundamental frequency.
linked to the dissipation, as well as the damping time after a perturbation. We used
in the following the measurement of the imaginary part of the response, but checked
that these different methods coincide. Only the resonant frequencies are considered
in this study, as they are much more sensitive to the velocity fluctuations. This is
Measurements of a dynamical temperature in turbulence. 6
especially important at large k, as the kinetic energy of the flow is small. Spectrum
of the fluctuation excited by the turbulent drag is shown in Fig. 3. Fluctuations
resonance peaks are clearly identified. Spurious vibrations are visible, mainly caused
by the vibrations of the stand. Because the peaks are very thin, long acquisitions
are necessary, as well as large windows for the FFT calculations (150000 points), in
order to achieve a sufficient resolution (0.33Hz). The protocol we used to find the
resonance frequencies, the value of the amplitude of fluctuations, and imaginary part
of the response, is the following. Resonance frequency is obtained by spline smoothing
each peak around the maximum amplitude of the response. Then, imaginary part is
measured after being also smoothed. The amplitude of the fluctuations peaks are
collected on the spectrum, after local smoothing around the maxima. One can see the
Figure 3. Spectrum of the resonance modes of the string excited by turbulent
drag fluctuations, at Re = 154000.
FDR in Fig. 4, called kBTeff., for several values of Re. Uncertainties on this ratio have
multiple origins. Errors indicated by the size of the symbols are those coming from
the determination of the resonance frequencies. Spurious vibrations of the stand are
difficult to handle: we perform measurements of response and fluctuations in the same
conditions, to reduce its influence on the ratio. We believe the scattering of the points
in Fig. 4 comes mainly from the weakening of signal/noise ratio for large frequencies,
simply because there is less energy in the flow at large k, especially at small Re.
The only possible escape on this point is to improve the coupling between the string
and the sensors. The wave-number has been rescaled with the internal viscous scale
η ∝ Re−3/4. The ordinates have been rescaled by an estimated number of degrees
of freedom: (L/η)3 ∝ Re9/4. These Re scalings are both usual consequences from
Kolmogorov’s theory. In other words, the “thermal energy” kBTeff. that the FDR is
representing in the framework of Cugliandolo et al ’s theory, is given per degree of
freedom. Assuming the number of degrees of freedom is the total number of particles
of size η in the total volume is usual, but crude. A more realistic description should
involve correlations between them, reducing this number. However, all the curves
collapse to a single power-law with this scaling. The exponent is discussed in the
Measurements of a dynamical temperature in turbulence. 7
Figure 4. Spectrum of the FDR, labelled as thermal agitation per degree of
freedom. Axis are rescaled with proper Reynolds number dependence, between
74000 and 170000. The size of the symbols represents the uncertainty in the
determination of the maxima of the peaks. The solid line is a k−11/3 power-law
given as an eye guide.
following section.
Please note that the equipartition of energy at equilibrium would require this spectrum
to be constant. There is no equilibrium between the Fourier modes, because of
the energy flux through scales. Moreover, they are not independent, and probably
not Gaussian. There is no reason to expect equipartition. Considering a kinematik
temperature as poportional to the kinetic energy, like in the kinetic theory of gases, it
would be: T ∝ 〈ṽ2〉. And, because of Kolmogorov’s theory it would scale as k−5/3.
The dependance we observe with our definition is much steeper.
4. Scaling law
Because the susceptibility of the string is very high at resonance, the half-wave-length
modes nλ/2 match with velocity structures of scale l (n is an integer). Therefore, the
wave number of the standing wave in the string k = n 2π/λ is the same as k = 2π/l.
The necessary condition for this matching is resonance. It also ensures that velocities
of the string and fluid equalise, which is crucial for the following argument.
Displacement is proportional to the drag forcing, itself proportional to velocity, as
drag is viscous: the string diameter-based Reynolds number is small (about 10).
The Melde string is not dispersive: ω = 2πf = ck, c being the wave velocity.
Therefore, the displacement is r = v/ω = v/(ck), and its power spectrum is:
〈r̃(ω)2〉 = 〈ṽ(ω)2〉(ck)−2 ∝ k−11/3. Because the viscous dissipation at each
resonance is proportional to frequency, the FDR of Eq. 2 is simply proportional to
c k 〈r̃(ω)2〉 ∝ k−11/3. Following Eq. 2, an effective “thermal agitation” defined by the
FDR would be: kBTeff. ∝ k
−11/3, in the inertial range of fully developed turbulence.
This exponent is compatible with the spectrum we measured, as can be seen in Fig.
Measurements of a dynamical temperature in turbulence. 8
5. Discussion
Theoretical characterisation of turbulence in terms of temperature were proposed in
the past by several authors. The temperatures as defined by T. M. Brown [10] and
B. Castaing [11] do not depend on k throughout the inertial range. The qualitative
idea is that the cascade transport process is efficient enough to equalise a quantity
they call temperature. In another model invoking an extremum principle, B. Castaing
proposed a definition of temperature, which might depend on scale [12]. In any case,
none of these theories invoke the FDR. On different basis, R. Robert and J. Sommeria
proposed a definition of temperature [13], only valid for 2D turbulence. It is not
expected to apply in a 3D flow.
Now, let’s consider our experimental results from the perspective of the three points of
reflexion we proposed in the first section, in relation with the FDT. 1- Linear response:
as we mentioned, the coupling between the string and the flow is purely viscous.
Therefore, drag force is proportional to velocity: f(t) = γ v(t), γ being a friction
coefficient. It is also the time-derivative of the position f(t) = γ ω r(t). Response is
linear in r, but the coefficient depends on frequency. 2- Are fluctuations and dissipation
proportional ? As we have seen, the measurements of the FDR are consistent with a
k−11/3 scaling, it is definitely not constant with respect to k. As our system is out
of equilibrium but stationary, there is no time evolution like the relaxation of glasses.
3- Setting a string in a turbulent flow allows to perform measurements on a couple
of conjugate force-displacement variables. We have no other set of observables to
compare with, for now.
We may ask whether what we measure is actually a temperature, in a dynamical
sense. If one assumes that each mode of the string is a harmonic oscillator, and that
a harmonic oscillator at equilibrium with a bath gives the temperature of this bath
through the FDR, then equilibrium between modes of the string and modes of the flow
means the temperature is equal: measurements give the temperature of the flow at this
corresponding scale. Such interpretation still rely on the assumption that FDR on the
oscillator gives the temperature of the oscilaror: this is our working hypothesis. By
equilibrium between modes of the string and the flow, we mean a ’no-flux’ condition
on energy. This is ensured by the high susceptibility of the string at resonance. In
other words, the probe and the reservoir are in equilibrium with each other for each
k, but equilibium is obviously not expected between one scale and another.
We have performed measurements on a turbulent flow, coupling to it a set of harmonic
oscillators: a Melde string. At equilibrium with the flow, in the sense that each mode
of the string couples with the fluid at scale l = πc/ω. It gives informations much
like a spectrometer, even though the flow itself is strongly out of equilibrium. This
is true, of course, as long as the response of the string is fast enough compared to
the frequencies of the velocity fluctuations. The displacement spectra are recorded at
different values of Re, as well as the complex response of the string over an excitation
(contributions of all the standing waves).
The matching of the string’s modes and hydrodynamic structures, what we call
equilibrium between the string and the flow, is still a questionable working hypothesis.
However, drawing inspiration from Cugliandolo et al ’s theory of non-equilibrium
temperature based on the FDR, we measured the Fluctuation over Dissipation Ratio
of our string in a turbulent flow, for different values of Re. The FDR, multiplied
by an appropriate power of the Reynolds number exhibits a unique power law, when
Reynolds number is between 74000 and 170000. The exponent is consistent with a
Measurements of a dynamical temperature in turbulence. 9
value −11/3 given by a very simple model derived from Kolmogorov 1941 theory.
Acknowledgments
We acknowledge B. Castaing, E. Leveque, P. Borgnat, F. Delduc, S. Ciliberto,
E. Bertin, and K. Gawedzki for many discussions. We also thank V. Bergeron,
T. Divoux, and V. Vidal for corrections on the manuscript and for many discussions.
Thanks to F. Dumas for his help in the construction of positioning devices. As this
system became a teaching experiment, several students contributed to this study as
part of their graduate lab-course. They are gratefully acknowledged: A. Louvet,
G. Bordes, I. Dossmann, J. Perret, C. Cohen, and M. Mathieu. We also thank the
guitar maker D. Teyssot, from Lyon, who gently gave us his thinnest E strings.
[1] L.D. Landau and E.M. Lifshitz. Course of Theoretical Physics: Fluid mechanics. Mir, 1971.
[2] A.S. Monin and A.M. Yaglom. Statistical fluid mechanics. MIT Press, Cambridge, 1975.
[3] U. Frisch. Turbulence: the legacy of A.N. Kolmogorov. Cambridge Univ. Press., 1995.
[4] L.F. Richardson. Weather prediction by numerical process. Cambridge Univ. Press, 1922.
[5] A.N. Kolmogorov. C. R. Acad. Sci. U.S.S.R., 30, 1941.
[6] M. Toda R. Kubo and N. Hashitsume. Statistical Physics II: Nonequilibrium Statistical
Mechanics, volume II. Springer, 1985.
[7] L. Cugliandolo and J. Kurchan. Phys. Rev. Lett., 71, 1993.
[8] J. Kurchan L. Cugliandolo and L. Peliti. Phys. Rev. E, 55, 1997.
[9] P. Marcq and A. Naert. Phys. of Fluids, 13, 2001.
[10] T.M. Brown. J. Phys. I, 15, 1982.
[11] B. Castaing. J. Phys. II, 6, 1996.
[12] B. Castaing. J. Phys. II, 50, 1989.
[13] J. Sommeria and R. Robert. J. Fluid Mech., 229, 1991.
	Introduction
	The Melde string and the experimental setup
	Measurements
	Scaling law
	Discussion
