June 2016

EPL, 114 (2016) 50011 www.epljournal.orgdoi: 10.1209/0295-5075/114/50011

Entropy production and Fluctuation Relation in turbulent

thermal convection

Francesco Zonta1,2 and Sergio Chibbaro3

1 Department of Electrical, Management and Mechanical Engineering, University of Udine - 33100, Udine, Italy2 Institute of Fluid Mechanics and Heat Transfer, Technische Universitat Wien - 1060 Wien, Austria3 Sorbonne Universites, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d’AlembertF-75005 Paris, France

received 9 February 2016; accepted in final form 15 June 2016published online 8 July 2016

PACS 05.20.Jj – Statistical mechanics of classical fluidsPACS 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motionPACS 47.27.-i – Turbulent flows

Abstract – We report on a numerical experiment performed to analyze fluctuations of the entropyproduction in turbulent thermal convection, a physical configuration taken here as a prototypeof an out-of-equilibrium dissipative system. We estimate the entropy production from instan-taneous measurements of the local temperature and velocity fields sampled along the trajectoryof a large number of pointwise Lagrangian tracers. The entropy production is characterized bylarge fluctuations and becomes often negative. This represents a sort of “finite-time” violation ofthe second principle of thermodynamics, since the direction of the energy flux is opposite to thatprescribed by the external gradient. We clearly show that the entropy production normalized bya suitable small-scale energy verifies the Fluctuation Relation (FR), even though the system istime-irreversible.

Copyright c⃝ EPLA, 2016

Introduction. – Fluctuations of physical systemsclose to equilibrium are well described by the classicallinear-response theory [1,2] that leads to the fluctuation-dissipation relation. The current knowledge of the dy-namics of systems far away from equilibrium is insteadmuch more limited, because of the lack of unifying prin-ciples. The introduction of the so-called Fluctuation Re-lation (FR) [3–6] has therefore represented a remarkableresult in this area of physics. The FR for nonequilibriumfluctuations reduces to the Green-Kubo and Onsager rela-tions close to equilibrium [7–10] and represents one of thefew exact results for systems kept far from equilibrium.

We recall that the FR concerns the symmetry of a rep-resentative observable, which is typically linked to thework done on the system, and, through dissipation, toirreversibility. For a Markov process, whose dynamics isdescribed by x(t) = u(x(t), t), the representative observ-able can be written as [11]

βWt =1

tlog

Π({x(s)}t0)

Π({Ix(s)}t0)

, (1)

where {x(s)}t0 and {Ix(s)}t

0 are the direct and the time-reversed trajectories in the time interval [0, t], respectively,

while Π indicates probability and β is a suitable energyscale of the system. When FR applies,

logΠ(βWt = p)

Π(βWt = −p)= tp, (2)

and Wt is usually called entropy production rate. Despitethese results, a general response theory for nonequilib-rium systems is still to be produced. This suggests thatnew analyses are required to investigate the behavior ofnonequilibrium fluctuations, in particular for macroscopicchaotic systems [12,13].

Turbulence represents the archetype of a macroscopicdynamical system characterized by a large number ofdegrees of freedom and by strong fluctuations. More-over, for its intrinsic chaotic nature, turbulence appearsas a paradigmatic case to which FR applies, cum granosalis [14]. If positively verified, this would strengthenthe link between turbulence and nonequilibrium statisticalmechanics. In particular, this would justify the hypothe-sis that a general response theory can be applied also todeterministic time-irreversible systems, at least with thepurpose of computing their statistical properties. Giventhe theoretical and practical importance of these issues,

50011-p1

Francesco Zonta and Sergio Chibbaro

FR in turbulent flows has been already investigated in thepast [15–22]. However, a satisfying statistical descriptionof entropy fluctuations in turbulence is still to be obtained,essentially because of the technical problems associated tothe experimental measurement of fluctuations in chaoticsystems [23], and also for the difficulty in performing accu-rate numerical simulations. Note that the choice of a rep-resentative observable is the central issue while discussingFR in macroscopic systems [13].

In this work we focus on turbulent Rayleigh-Benard con-vection to address the following issues: i) the choice ofa representative observable to compute entropy fluctua-tions; ii) the presence of large deviations of this quantitybeyond the linear regime; iii) the applicability of the FRto turbulent thermal convection. To do this, we run Di-rect Numerical Simulations (DNS) of turbulent Rayleigh-Benard convection inside a vertically confined fluid layerand we track the dynamics of pointwise tracers, which weuse here as probes to measure the local thermodynamicquantities of the system. The fundamental idea of ourapproach is that turbulence shares similarities with themicroscopic nature of heat flows, and turbulence fluctua-tions correspond to thermal fluctuations [24]. With this inmind, we have chosen a configuration similar to that stud-ied in stochastic thermodynamics [25], which consists of asystem kept in contact with two thermostats at differenttemperatures and characterized by a fluctuating energyflow. A key ingredient in our study is the use of a La-grangian point of view, which is specifically suited to studythe global transport properties of the flow [26]. We willshow that entropy production can be evaluated by lookingat the work done by buoyancy on moving fluid particles.Provided that the correlations of the measured quantitiesdecay fast enough, we show that the finite-time entropyproduction exhibits large fluctuations (being often nega-tive) and fulfills FR. Our results complement recent workson granular matter, a simpler but complete model used todescribe macroscopic irreversible systems. For granularmatter, theory and numerical simulations agree in verify-ing FR, but only if the correct fluctuating entropy produc-tion and temperature are defined [27–29].

Physical problem and modeling. – We consider aturbulent Rayleigh-Benard convection, in which a hori-zontal fluid layer is heated from below. Horizontal andwall-normal coordinates are indicated by x1, x2 and x3, re-spectively. Using the Boussinesq approximation, the sys-tem is described by the following dimensionless balanceequations:

∂ui

∂xi= 0, (3)

∂ui

∂t+ uj

∂ui

∂xj= −

∂P

∂xi+ 4

!

Pr

Ra

∂2ui

∂x2j

− δi,3θ, (4)

∂θ

∂t+ uj

∂θ

∂xj= +

4√PrRa

∂2θ

∂x2j

, (5)

x1

x3

x2

x1

x3

x2 a)

b)

Fig. 1: (Colour online) (a) Two-dimensional contour maps ofthe temperature distribution computed near the bottom walland on two vertical slices for Rayleigh-Benard convection atRa = 109. The domain has dimensions Lx1

× Lx2× Lx3

=2πh × 2πh × 2h and is discretized using 512 × 512 × 513 gridnodes. (b) Example of tracers trajectories for Ra = 109 coloredby the local velocity magnitude.

where ui is the i-th component of the velocity vector, Pis the pressure, θ = (T − T0)/∆T is the dimensionlesstemperature, ∆T = TH − TC is the imposed tempera-ture difference between the hot bottom wall (TH) and topcold wall (TC), whereas δ1,3θ is the driving buoyancy force(acting in the vertical direction x3 only). The referencevelocity is the free-fall velocity uref = (gα0h∆T/2)1/2,with h = 0.15m the half domain height and g the ac-celeration due to gravity. The fluid kinematic viscosityν0, thermal diffusivity κ0 and thermal expansion coeffi-cient α0 are evaluated at the reference fluid temperatureT0 = (TH +TC)/2 ≃ 29 ◦C. The Prandtl and the Rayleighnumbers in eqs. (4), (5) are defined as Pr = ν0/k0 andRa = (gα0∆T (2h)3)/(ν0k0). The size of the domain isLx1

×Lx2×Lx3

= 2πh×2πh×2h. Periodicity is imposedon velocity and temperature along the horizontal direc-tions x1 and x2, whereas no slip conditions are enforcedfor velocity at the top and bottom walls. In our simula-tions, we keep the Prandtl number Pr = 4 and we vary theRayleigh number between Ra = 107 and Ra = 109. Anexample of the temperature distribution inside our con-vection cell is given in fig. 1(a). To measure the localvalues of the field variables, we make use of a Lagrangianapproach. The dynamics of Np = 1.28 · 105 Lagrangian

50011-p2

Entropy production and Fluctuation Relation in turbulent convection

Table 1: Summary of the simulations performed with corre-sponding details of the computation grids. Nx1

, Nx2and Nx3

correspond to the number of nodes along x1, x2 and x3, whereas∆x3,max

h and∆x3,min

h correspond to the maximum and mini-mum grid spacings of the (nonuniform) grid along x3. Thevalue of the Kolmogorov space/time scales (ηK , τK) is alsogiven.

Simulations

S1 S2 S3

Ra 107 108 109

Nx1128 256 512

Nx2128 256 512

Nx3129 257 513

∆x1

h 4.9 · 10−2 2.4 · 10−2 1.2 · 10−2

∆x2

h 4.9 · 10−2 2.4 · 10−2 1.2 · 10−2

∆x3,max

h 2.4 · 10−2 1.2 · 10−2 6.1 · 10−3

∆x3,min

h 3 · 10−4 7.5 · 10−5 1.8 · 10−5

ηK 1.1 · 10−3 5.3 · 10−4 2.5 · 10−4

τK 11 2.4 0.5

tracers is computed as

xp = u (xp(t), t) θp = θ (xp(t), t) , (6)

with xp = (xp,1, xp,2, xp,3) the tracers position, xp =(up,1, up,2, up,3) their velocity and θp their temperature.A visualization of different particle trajectories is shownin fig. 1(b), highlighting also the chaotic nature of the flow.

Equations (4), (5) are discretized using a pseudo-spectral method based on transforming the field variablesinto wave number space, through Fourier representationsfor the periodic (homogeneous) directions x1 and x2,and Chebychev representation for the wall-normal (non-homogeneous) direction x3. Time advancement is done bya combined Crank-Nicolson scheme for the viscous terms,and an explicit Adams-Bashforth scheme for the nonlin-ear convective terms. Further details on the numericalmethod can be found in [30–33]. For the Lagrangian track-ing, we have employed 6th-order Lagrangian polynomialsto interpolate the fluid velocity and temperature at thetracers position. A 4th-order Runge-Kutta scheme is usedfor time advancement of the tracers equations (6) [34]. Abrief validation of the method is provided in the appendix.An overview of the simulations performed with the corre-sponding details of the computational grid and the valueof the Kolmogorov space/time scales is given in table 1.

Results. – Entropy production for a Markov processis usually defined based on the dynamical probability, aquantity that is generally not accessible in complex sys-tems. To find a representative observable for the system,we start from the balance equation for the turbulent ki-netic energy Et = 1/2⟨u′

iu′i⟩, where brackets ⟨ ⟩ indicate

statistical average and u′i = ui−⟨ui⟩ velocity fluctuations.

For the present flow configuration, ⟨ui⟩ = 0. Since the

system is homogeneous along the x1 and x2 directions, weobtain

∂Et

∂t+

∂

∂x3(⟨Etu

′3⟩ + ⟨u′

3p′⟩) = ⟨u′

3θ′⟩ − ⟨ϵ⟩, (7)

where ⟨ϵ⟩ = 4"

PrRa

#

∂2⟨Et⟩∂x2

3

+$

∂u′

i

∂xj

∂u′

i

∂xj

%&

is the turbulent

dissipation and θ′ = θ − ⟨θ⟩ is the temperature fluctu-ation. The volume-averaged steady-state solution gives⟨u′

3θ′⟩ = ⟨ϵ⟩. This provides also an estimate of the entropy

production, which from thermodynamics is ⟨ϵ⟩/⟨T ⟩ (with⟨T ⟩ the average absolute temperature of the system). Wespecifically focus on the term W = ρ0gα0θ′pu

′3,p measured

along the path of the Lagrangian tracers (hence we haveu′

3,p and θ′p), which represents the power spent by buoy-ancy to displace a fluid parcel. Note that W quantifies thevertical flux of energy and is therefore also linked to thevertical Nusselt number [33,35]. Warm fluctuations θ′p > 0produce a positive energy flux when associated to posi-tive vertical velocities up,3 > 0, whereas cold fluctuationsθ′p < 0 produce a positive energy flux when associated tonegative vertical velocity up,3 < 0.

We now consider the time-averaged (but fluctuating)expression of the work (per unit volume and time) doneby buoyancy on the system,

Wτ =1

τ

' τ

0Fext(t) · u(t)dt =

1

τ

' τ

0ρ0gα0θ

′pu

′p,3dt, (8)

where Fext is the external force field due to gravity. Thisobservable is similar to that proposed to analyze the localFR [15]. By contrast, previous experimental studies onmacroscopic chaotic systems focused on the behavior ofthe injected power [16,18,19], which is a measurable quan-tity that however was found to depart from the predictionsgiven by the FR [27,36]. Although work and heat fluctu-ations are generally different in stochastic systems [37],we believe that Wτ can give a good estimate of entropyproduction in turbulent convection.

In the following, we assume that time averages areequivalent to ensemble averages (ergodicity). This as-sumption is justified if the auto-correlation of W , RW =⟨W (t)W (t + τ)⟩/⟨W 2⟩, computed after a statisticallysteady state is reached, exhibits a fast decrease intime [38]. To verify this, we explicitly compute RW (τ)for each Ra as a function of the time lag τ . Results areshown in fig. 2. We observe that RW is characterized byan exponential decay exp(−t/Γ), whose decay rate 1/Γincreases with increasing Ra. As a consequence, velocityand temperature fluctuations decorrelate faster for largeRa, due to the larger fluctuations observed for increas-ing Ra. From the behavior of the correlation functionRW , we are able to compute the integral correlation timeτC =

( ∞0 RW (τ)dτ . Upon rescaling of the time lag τ by

τC , the correlation functions RW computed at different Racollapse (inset of fig. 2). For τ/τC > 1 the value of the cor-relation is already RW < 0.2, indicating that from τ ≃ τC

50011-p3

Francesco Zonta and Sergio Chibbaro

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300

RW

(τ)

τ

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

RW

(τ)

τ/τCRa

Fig. 2: Lagrangian correlation RW as a function of the timelag τ for the three different Rayleigh numbers. The collapseof the correlation function upon rescaling of the time lag τ bythe integral time scale τC is explicitly shown in the inset. Notethat τC = 100 s for Ra = 107, τC = 25 s for Ra = 108 andτC = 7 s for Ra = 109.

1e-07

1e-06

1e-05

1e-04

0.01 0.1

E (

kη

)

kR η

1

k η

Ra=107

Ra=108

Ra=109

Fig. 3: (Colour online) Time-averaged energy spectra of theturbulent kinetic energy, computed at the center of the channel(on the central plane explicitly shown in the inset) to avoidanisotropy effects due to the nonhomogeneity along the verticaldirection. The representative energy scale of the system β−1 =(kBT )turb = 1/2ρ

R

k>kRE(k)dk, computed assuming kRη = 1,

gives β = 0.49 for Ra = 107, β = 0.33 for Ra = 108 andβ = 0.3 for Ra = 109.

the signal is only barely reminiscent of the initial condi-tions. This means that τC is a representative time scaleof the system and suggests that FR can be convenientlytested for τ/τC > 1.

As already discussed, one of the crucial aspects to com-pute the FR is the choice of the representative energy scale(β) of the system. This energy scale cannot be the thermalenergy kBT [22], with kB the Boltzmann constant and Tthe absolute temperature. The main reason is that en-tropy fluctuations in turbulence are determined by smallscale mixing rather than by molecular agitation. Thereforean effective temperature must be introduced, as success-fully shown for other systems [27,39]. In turbulent flows,following the Kolmogorov cascade picture, we study the

0.001

0.01

0.1

1

-5 0 5 10 15

Π(p

)

p

τ/τC=0τ/τC=1

τ/τC=10τ/τC=40

Fig. 4: (Colour online) Probability density function Π of thenormalized energy flux p = Wτ/⟨Wτ ⟩ for Ra = 109 and for dif-ferent values of the averaging time window τ/τC . The behaviorof a Gaussian distribution is explicitly indicated by the solidline. Results from simulations at Ra = 107 and Ra = 108 arenot shown here because they are qualitatively similar to thoseat Ra = 109 and do not add much to the discussion.

0

0.5

1

1.5

2

0 0.5 1 1.5 2

(β⟨W

τ⟩)-1

τ-1 l

og

(Π(p

)/Π

(-p))

p

-2

-1.5

-1

-0.5

0

-1 -0.5 0 0.5 1 1.5 2 2.5

Ra

Fig. 5: Behavior of (β⟨Wτ ⟩)−1τ−1 log (Π(p)/Π(−p)) as a func-

tion of p for 1 < τ/τC < 30, i.e. when the probability densityfunction is not Gaussian. Results are shown for each value ofthe Rayleigh number Ra (as indicated by the arrow). The solidline represents the theoretical prediction given by the FR, a lin-ear function with slope equal to unity. In the inset, the collapseof the rescaled probability distributions of Wτ at large timesonto the large-deviation function is shown (for 5 < τ < 25).

nature of entropy fluctuations assuming that β−1 is theenergy (per unit volume) of the dissipative scales [5,40,41]β−1 = (kBT )turb = 1/2ρ

(

k>kRE(k)dk, with kR the wave

number characterizing dissipation (i.e. kRη ≃ 1, with ηthe Kolmogorov length scale). Following this approach,the value of β is obtained directly from the turbulentkinetic-energy spectrum E(k) (with dimensions m2/s2),as shown in fig. 3.

Starting from the Lagrangian measurements of W , wecompute the probability density function Π of the normal-ized flux p = Wτ

⟨Wτ ⟩, for different values of τ , as shown in

fig. 4. For τ/τC = 0, Π(p) is highly asymmetric, with themost probable value occurring for p = 0 and with pos-itive fluctuations being larger than negative ones. The

50011-p4

Entropy production and Fluctuation Relation in turbulent convection

asymmetry of Π(p) persists also for increasing τ/τC anddisappears only when the average is done on a large timewindow (τ/τC ≥ 40). In particular, for τ/τC ≃ 40 thedistribution peaks around p ≃ 1 and recovers an almostGaussian distribution (explicitely shown by the solid linein fig. 4). Note that at this stage (τ/τC ≥ 40), theprobability of negative events becomes essentially zero.Altogether these observations suggest that, although theimposed mean temperature difference between the wallsinduces a net positive vertical energy flux (i.e. ⟨W ⟩ > 0),W can often be negative. The occurrence of countegra-dient fluxes of global transport properties (such as theNusselt number) is an extremely important phenomenonthat has been also observed in other situations [42]. Froma physical point of view, small positive and negative val-ues of W are produced by turbulence, which is uncorre-lated with the temperature field. These small positiveand negative values of W balance each others and do notcontribute to the average heat transport [22]. Only largevelocity and temperature fluctuations produced by ther-mal plumes (rising hot plumes and falling cold plumes) arecorrelated and contribute to the positive mean heat flux.

Based on these observations, it is reasonable to expectthat the fluctuations of p are governed by a large deviationlaw Π(p) ∼ eτζ(p), with ζ concave. Then, from the behav-ior of Π(p), we measure the quantity

ζ(p) − ζ(−p) = σ(p) =1

τlog

Π(p)

Π(−p)(9)

for different averaging time τ/τC taken in the range 1 <τ/τC < 30. The resulting behavior, given in fig. 5, nicelyshows that σ(p) is a linear function of p,

σ(p) = γp. (10)

In particular, we observe that the slope γ of the curveincreases with increasing Ra and tends (for Ra = 109)to γ = β⟨Wτ ⟩. This indicates that the FR introduced ineq. (2) is verified at large times, within numerical and sta-tistical errors. We finally show (inset of fig. 5) the collapseof ζ(p) ∼ log Π(Wτ )/τ onto the large-deviation functionfor 5 < τ < 25. The Cramer function ζ(p), rescaled usingits standard deviation στ and its slope Cτ as suggestedby [43], collapses for the different values of the averagingtime τ shown here (for 5 < τ < 25).

Conclusion. – In this letter, we have analyzed thebehavior of entropy fluctuations in turbulent thermal con-vection, taken here as a paradigmatic case of a complexout-of-equilibrium system. We have performed Direct Nu-merical Simulations of a turbulent Rayleigh-Benard flowinside a vertically confined fluid layer and we have fol-lowed the dynamics of pointwise Lagrangian tracers tomeasure the local quantities of the flow. We have shownthat entropy production can be evaluated by looking at thework done by buoyancy on fluid particles, Wτ ∝ θ′pu

′3,p.

We have found that Wτ is often negative, and is char-acterized by fluctuations that follow the FR beyond the

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

⟨ θ

' 2 ⟩

1/2

x3/h

∆1

∆2

∆3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.5 1 1.5 2

⟨ θ

' 3 ⟩

/⟨ θ

' 2 ⟩

3/2

x3/h

∆1

∆2

∆3

(a)

(b)

Fig. 6: Grid convergence analysis: wall-normal behavior of theroot mean square ⟨θ′2⟩1/2 (panel (a)) and of the skewness factor⟨θ′3⟩/⟨θ′2⟩1/2 (panel (b)) of temperature fluctuations. Resultsare obtained from simulations at Rayleigh number Ra = 107

and using three different grids : grid ∆1 has 64×64×65 nodes;grid ∆2 has 128×128×129 nodes; grid ∆3 has 256×256×257nodes in the streamwise (x1), spanwise (x2) and wall-normal(x3) direction, respectively.

linear regime, provided that a representative effectivetemperature is employed. Here we defined the effectivetemperature as the kinetic energy of the small scales,which can be taken as a sort of temperature. However,this is a crucial point that deserves further investigation,since other physical quantities (for instance the energy dis-sipation rate ϵ) can be used for this scope as well [44].

Present results shed new light on turbulence, allowingan a priori estimate of the behavior of fluctuations of en-ergy flux or entropy production and giving access to theCramer function. New simulations in different configura-tions and at higher Reynolds/Rayleigh numbers are fore-seen to assess the robustness of present results.

∗ ∗ ∗

We thank Massimo Cencini, Andrea Crisanti,Andrea Puglisi, Lamberto Rondoni, Dario Villa-maina and Angelo Vulpiani for fruitful discussions.

Appendix

For validation purposes, in fig. 6 we show the wall-normal behavior (as a function of x3/h) of the rootmean square ⟨θ′2⟩1/2 (fig. 6(a)) and of the skewness fac-tor ⟨θ′3⟩/⟨θ′2⟩1/2 (fig. 6(b)) of temperature fluctuationsfor Ra = 107. Results are compared using three differ-ent grid resolutions ∆1 (64 × 64 × 65 nodes), ∆2 (128 ×128 × 129 nodes) and ∆3 (256 × 256 × 257 nodes). Note

50011-p5

Francesco Zonta and Sergio Chibbaro

that the computational grids have a nonuniform spacing(with near-wall refinement) along the wall-normal coordi-nate x3, due to the adoption of Chebychev polynomials(Tn3

(x3) = cos(n3 cos−1(x3/h)) is the Chebychev polyno-mial of order n3 along x3). Results in fig. 6(a), (b) indi-cate that the computational grid ∆2 is accurate enough toproperly resolve all the flow scales at Ra = 107, and fur-ther grid refinement is not required (the difference with afiner grid, ∆3, is always below 3% for both second- andthird-order moments). Increasing Ra, temperature andvelocity flow structures become smaller and the computa-tional grid must be refined accordingly (see table 1). Thegrid resolutions employed here are consistent with thosefound in the literature [45].

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