In a recent preprint (cond-mat/0601398), D. Funfschilling and G. Ahlers
describe a new effect, that they interpret as non-Boussinesq, in a convection
cell working with ethane, near its critical point. They argue that such an
effect could have spoiled the Chavanne {\it et al.} (Phys. Rev. Lett. {\bf 79}
3648, 1997) results, and not the Niemela {\it et al.} (Nature, {\bf 404}, 837,
2000) ones, which would explain the differences between these two experiments.
We show that:-i)Restricting the Chavanne's data to situations as far from the
critical point than the Niemela's one, the same discrepancy remains.-ii)The
helium data of Chavanne show no indication of the effect observed by D.
Funfschilling and G. Ahlers.Comment: comment on cond-mat/060139

Castaing, Bernard

Chabaud, Benoît

Chavanne, Xavier

Chillà, Francesca

Hébral, Bernard

Roche, Philippe-Emmanuel P. -E.

English

arXiv

In a recent preprint ( cond-mat/0601398 ), D. Funfschilling and G. Ahlers describe a new effect, that they interpret as non-Boussinesq, in a convection cell working with ethane, near its critical point. They argue that such an effect could have spoiled the Chavanne {\it et al.} (Phys. Rev. Lett. {\bf 79} 3648, 1997) results, and not the Niemela {\it et al.} (Nature, {\bf 404}, 837, 2000) ones, which would explain the differences between these two experiments. We show that:-i)Restricting the Chavanne's data to situations as far from the critical point than the Niemela's one, the same discrepancy remains.-ii)The helium data of Chavanne show no indication of the effect observed by D. Funfschilling and G. Ahlers

Roche, Philippe-Emmanuel

HAL-ENS-LYON

Comment on ”Turbulent heat transport near criticalpoints: Non-Boussinesq effects” ( cond-mat/0601398 )Xavier Chavanne, Philippe-Emmanuel Roche, Benoˆıt Chabaud, BernardHe´bral, Francesca Chilla`, Bernard CastaingTo cite this version:Xavier Chavanne, Philippe-Emmanuel Roche, Benoˆıt Chabaud, Bernard He´bral, FrancescaChilla`, et al.. Comment on ”Turbulent heat transport near critical points: Non-Boussinesqeffects” ( cond-mat/0601398 ). 2006. <hal-00020417>HAL Id: hal-00020417https://hal.archives-ouvertes.fr/hal-00020417Submitted on 9 Mar 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00020417, version 1 - 9 Mar 2006APS/123-QEDComment on “Turbulent heat transport near critical points: Non-Boussinesq effects”.X. ChavanneInstitut de Physique du Globe de Paris, Ge´omate´riaux et Environnement,4, place Jussieu - Case 89 - 75252 Paris Cedex 05P.-E. Roche, B. Chabaud, and B. He´bralCentre de Recherche sur les Tre`s Basses Tempe´raturesCNRS, Associe´ a` l’Universite´ Joseph Fourier25, Av. des Martyrs, BP 166, 38042 Grenoble Cedex 9F. Chilla` and B. CastaingEcole Normale Supe´rieure de Lyon,46, Alle´e d’Italie, 69364 Lyon Cedex 7, France(Dated: 9/march/2006)In a recent preprint ( cond-mat/0601398 ), D. Funfschilling and G. Ahlers describe a new effect,that they interpret as non-Boussinesq, in a convection cell working with ethane, near its criticalpoint. They argue that such an effect could have spoiled the Chavanne et al. (Phys. Rev. Lett. 793648, 1997) results, and not the Niemela et al. (Nature, 404, 837, 2000) ones, which would explainthe differences between these two experiments. We show that:-i)Restricting the Chavanne’s data to situations as far from the critical point than the Niemela’sone, the same discrepancy remains.-ii)The helium data of Chavanne show no indication of the effect observed by D. Funfschilling andG. Ahlers.PACS numbers: 47.27The question of whether or not the fully turbulentKraichnan regime [1, 2] has been observed in Rayleigh-Be´nard (RB) convection is a subject of vigorous contro-versy in the past decade. Chavanne et al.[3, 4] observeda rapid increase of the Nusselt number Nu versus theRayleigh number Ra, and interpreted it as the transitiontoward this ultimate regime. The difference between theobserved logarithmic slope of this dependence and thepredicted 1/2 exponent was attributed to the logarithmiccorrection proposed by Kraichnan himself [1, 2]. Rocheet al. [5] showed that rough boundaries fix the correctionas expected and induce a 1/2 exponent behaviour. Later,Niemela and Sreenivasan [6] reported the same kind ofbehaviour.HoweverWu et al. [7, 8] and Niemela et al. [9] exploredthe same range of Ra, with the same working fluid (he-lium) without observing the same increase in Nu. Sev-eral propositions have been made, but there exist no clearconsensus about the differences between the cells whichcould explain such a discrepancy.In a recent work, D. Funfschilling and G. Ahlers (FA)[10] observe a new effect, that they interpret as spurious(non-Boussinesq), in their convection cell working withethane, near its critical point. They remark that most ofthe working points of the works observing the Kraichnanlike behaviour [3, 4, 5, 6] are nearer the critical pointand/or the gas-liquid coexistence curve of helium thanthose not observing it [7, 8, 9]. They thus propose thatthis spurious effect explains the difference quoted above,which would mean that the Kraichnan regime has not yetbeen experimentally observed.In their figure 4, they indicate the position of the dif-0.80.911.10 0.5 1T / Tc! / !cFIG. 1: Phase diagram of 4He in reduced units (Tc = 5.1953K, ρc = 69.641 kgm−3) showing the location of various mea-surements of Nu versus Ra. Solid line: coexistence curve.Asterisc: critical point. Open diamonds: Niemela et al. [9].Open squares: Wu et al. [7, 8]. Triangles up and down: Cha-vanne et al. [4]; triangles up: β∆T < 0.1, triangles down:β∆T > 0.1, open: near the critical isochore, solid: far fromthe critical isochore.ferent working points in the phase space. It can be seenthat some of the points of Chavanne et al. are withinthe points of Niemela et al.. In our figure 1 we reproduceonly these points (triangles up and down). The temper-ature T is Tm = (Tb + Tt)/2, where Tt (resp. Tb) is thetop (resp. bottom) plate temperature. ρ is the fluid av-erage density, Tc and ρc are the critical temperature and2density. We also show the Niemela et al. [9] points (dia-monds), recently provided to us by J.J. Niemela, and theWu et al. [7, 8] points (squares). We note that some ofthe points of Niemela et al., not shown on the FA figure4, are very near the critical point, however not showingthe Kraichnan like increase in Nu [9].0.100.150.2010101011101210131014Nu / Ra0.306RaFIG. 2: The compensated Nusselt versus the Rayleigh num-ber. Circles: FA results as presented in their figure 5 (open:isothermal, solid: isobaric with ∆T = 1 K, P/Pc = 0.920).Other symbols are as in figure 1.In figure 1, and in the following one 2 we do several dis-tinctions between the Chavanne’s points we kept. Pointswith β∆T < 0.1, where β is the isobaric expansion co-efficient and ∆T = Tb − Tt, are plotted as triangles up.We can consider for these points that physical proper-ties are reasonably uniform within the cell. Points withβ∆T > 0.1 are plotted as triangles down. To eliminateany doubt, we also distinguish densities far from the criti-cal one (solid symbols) from those near it (open symbols).Following FA ([10] figure 3), we plot in our figure 2Nucomp = Nu/Ra0.306versus Ra for the Chavanne’spoints we kept. The symbols are the same than in figure1. Also shown are the FA results (circles) as presentedin their figure 5 (open: isothermal, solid: isobaric with∆T = 1 K), and the Niemela’s points [9]. The discrep-ancy remains for these points which have the same loca-tion in the phase space. The Kraichnan like increase forthe Chavanne’s points is obvious. We can easily concludethat the discrepancy between references [3, 4] and [9] hasnothing to do with the location in the phase space.Another question is to know if whether or not thecharacteristics of the effect observed by Funfschilling andAhlers can be found in the data of Chavanne et al.. Fun-fschilling and Ahlers have shown that, in their observa-tions, the compensated Nusselt Nucomp = Nu/Ra0.306only depends on the difference between Tt (not Tm) andthe equilibrium temperature Te at the same pressure (thiscorresponding to Tm and ρ).There are no constant pressure series of measurementsin Chavanne’s data [3, 4]. The reason is that ensuringa constant pressure ask for letting the filling capillary0.150.200 0.025 0.05Nu / Ra0.306Tt-Te (K)FIG. 3: Compensated Nusselt versus the difference betweenthe top plate temperature and the condensation temperatureat the considered pressure. All the points correspond to thesame density: 37.9 kgm−3. The circles with cross correspondto the same pressure within ±0.2% (P = 1.955 bar, P/Pc =0.86)0.150.201010101110121013Nu / Ra0.306RaFIG. 4: Same points as figure 3 plotted versus Ra.open up to a pressure gauge or ballast. On some pointsof the capillary, the temperature can go down to the con-densation temperature Te. Together with the importantcompressibility near the critical point, it can cause largetransfers of mass and heat in this capillary. However sev-eral series of measurements in [3, 4] have been made atconstant density ρ. On the figure 3 we plot the compen-sated Nusselt versus (Tt − Te) for the Chavanne’s pointscorresponding to ρ = 37.9 kgm−3. The strongly differentvalues do not merge on a curve. On the opposite, theynicely merge if plotted versus Ra (figure 4). We draw theattention of the reader on the points labeled with a crossin the circle. Their corresponding pressure is the samewithin ±0.2%, which is not as precise as the ±3 10−5 ofFunfschilling and Ahlers, but rather constant anyway.To conclude, the simple examination of the publisheddata of Chavanne et al. shows that the hypothesis raised3by Funfschilling and Ahlers [10], that the influence ofthe critical point could explain the apparent discrepancybetween references [3, 4] and [9], this hypothesis cannothold. Moreover, as far as the effect observed by them [10]is well characterized, we found no trace of it in the heliumdata of [3, 4]. We see here no reason to reconsider theoriginal interpretation for the increase of Nu observedby [3, 4, 5] as a transition toward the Kraichnan regime,even if the exact conditions for such a transition have yetto be elucidated.AcknowledgmentsWe are grateful to J.J. Niemela for providing us thenumerical values corresponding to the data in Ref. [9].We also thank G. Ahlers for supplementary details onthe cell and on ethane properties.[1] R. H. Kraichnan, “Turbulent Thermal Convection atArbitrary Prandtl Number”, Phys. Fluids, 5, 1374–1389,(1962).[2] E. Siggia, “High Rayleigh Number Convection” AnnualReview of Fluid Mechanics, 26, 137–168, (1994)[3] X. Chavanne, F. Chilla`, B. Castaing, B. He´bral, B.Chabaud, and J. Chaussy, “ Observation of the UltimateRegime in Rayleigh-Be´nard Convection” Phys. Rev. Lett.79, 3648–3651 (1997).[4] X. Chavanne, F. Chilla`, B. Chabaud, B. Castaing,B. He´bral, “Turbulent Rayleigh-Be´nard convection ingaseous and liquid He” Phys. Fluids 13, 1300–1320,(2001).[5] P.-E. Roche, B. Castaing, B. Chabaud, and B. He´bral,“Observation of the 1/2 power law in Rayleigh-Be´nardconvection” Phys. Rev. E 63, 045303–1–4 (2001)[6] J. J. Niemela, K. R. Sreenivasan, “Confined turbulentconvection” J. Fluid Mech., 481, 355–384 (2003)[7] X. Z. Wu,“Along a road to developped turbulence: freethermal convection in low temperature helium gas”, PhDThesis, Chicago University, (1991).[8] X. Z. Wu, A. Libchaber, “Scaling relations in thermalturbulence: The aspect-ratio dependence” Phys. Rev. A45, 842–845 (1992); M. Sano, X. Z. Wu, A. Libchaber,“Turbulence in helium-gas free convection” Phys. Rev. A40, 6421–6430 (1989)[9] J. J. Niemela, L. Skrbek, K. R. Sreenivasan, R. J. Don-nelly, “Turbulent convection at very high Rayleigh num-bers” Nature, 404, 837–841, (2000).[10] D. Funfschilling and G. Ahlers, “Turbulent heat trans-port near critical points: Non-Boussinesq effects” cond-mat/0601398 (2006)

Comment on "Turbulent heat transport near critical points: Non-Boussinesq effects" ( cond-mat/0601398 )

Hal - Université Grenoble Alpes

HAL-INSU

Hal-Diderot

Comment on ”Turbulent heat transport near critical
points: Non-Boussinesq effects” ( cond-mat/0601398 )
Xavier Chavanne, Philippe-Emmanuel Roche, Benoˆıt Chabaud, Bernard
He´bral, Francesca Chilla`, Bernard Castaing
To cite this version:
Xavier Chavanne, Philippe-Emmanuel Roche, Benoˆıt Chabaud, Bernard He´bral, Francesca
Chilla`, et al.. Comment on ”Turbulent heat transport near critical points: Non-Boussinesq
effects” ( cond-mat/0601398 ). 2006. <hal-00020417>
HAL Id: hal-00020417
https://hal.archives-ouvertes.fr/hal-00020417
Submitted on 9 Mar 2006
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destine´e au de´poˆt et a` la diffusion de documents
scientifiques de niveau recherche, publie´s ou non,
e´manant des e´tablissements d’enseignement et de
recherche franc¸ais ou e´trangers, des laboratoires
publics ou prive´s.
cc
sd
-0
00
20
41
7,
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er
sio
n 
1 
- 9
 M
ar
 2
00
6
APS/123-QED
Comment on “Turbulent heat transport near critical points: Non-Boussinesq effects”.
X. Chavanne
Institut de Physique du Globe de Paris, Ge´omate´riaux et Environnement,
4, place Jussieu - Case 89 - 75252 Paris Cedex 05
P.-E. Roche, B. Chabaud, and B. He´bral
Centre de Recherche sur les Tre`s Basses Tempe´ratures
CNRS, Associe´ a` l’Universite´ Joseph Fourier
25, Av. des Martyrs, BP 166, 38042 Grenoble Cedex 9
F. Chilla` and B. Castaing
Ecole Normale Supe´rieure de Lyon,
46, Alle´e d’Italie, 69364 Lyon Cedex 7, France
(Dated: 9/march/2006)
In a recent preprint ( cond-mat/0601398 ), D. Funfschilling and G. Ahlers describe a new effect,
that they interpret as non-Boussinesq, in a convection cell working with ethane, near its critical
point. They argue that such an effect could have spoiled the Chavanne et al. (Phys. Rev. Lett. 79
3648, 1997) results, and not the Niemela et al. (Nature, 404, 837, 2000) ones, which would explain
the differences between these two experiments. We show that:
-i)Restricting the Chavanne’s data to situations as far from the critical point than the Niemela’s
one, the same discrepancy remains.
-ii)The helium data of Chavanne show no indication of the effect observed by D. Funfschilling and
G. Ahlers.
PACS numbers: 47.27
The question of whether or not the fully turbulent
Kraichnan regime [1, 2] has been observed in Rayleigh-
Be´nard (RB) convection is a subject of vigorous contro-
versy in the past decade. Chavanne et al.[3, 4] observed
a rapid increase of the Nusselt number Nu versus the
Rayleigh number Ra, and interpreted it as the transition
toward this ultimate regime. The difference between the
observed logarithmic slope of this dependence and the
predicted 1/2 exponent was attributed to the logarithmic
correction proposed by Kraichnan himself [1, 2]. Roche
et al. [5] showed that rough boundaries fix the correction
as expected and induce a 1/2 exponent behaviour. Later,
Niemela and Sreenivasan [6] reported the same kind of
behaviour.
HoweverWu et al. [7, 8] and Niemela et al. [9] explored
the same range of Ra, with the same working fluid (he-
lium) without observing the same increase in Nu. Sev-
eral propositions have been made, but there exist no clear
consensus about the differences between the cells which
could explain such a discrepancy.
In a recent work, D. Funfschilling and G. Ahlers (FA)
[10] observe a new effect, that they interpret as spurious
(non-Boussinesq), in their convection cell working with
ethane, near its critical point. They remark that most of
the working points of the works observing the Kraichnan
like behaviour [3, 4, 5, 6] are nearer the critical point
and/or the gas-liquid coexistence curve of helium than
those not observing it [7, 8, 9]. They thus propose that
this spurious effect explains the difference quoted above,
which would mean that the Kraichnan regime has not yet
been experimentally observed.
In their figure 4, they indicate the position of the dif-
0.8
0.9
1
1.1
0 0.5 1
T
 /
 T
c
! / !
c
FIG. 1: Phase diagram of 4He in reduced units (Tc = 5.1953
K, ρc = 69.641 kgm
−3) showing the location of various mea-
surements of Nu versus Ra. Solid line: coexistence curve.
Asterisc: critical point. Open diamonds: Niemela et al. [9].
Open squares: Wu et al. [7, 8]. Triangles up and down: Cha-
vanne et al. [4]; triangles up: β∆T < 0.1, triangles down:
β∆T > 0.1, open: near the critical isochore, solid: far from
the critical isochore.
ferent working points in the phase space. It can be seen
that some of the points of Chavanne et al. are within
the points of Niemela et al.. In our figure 1 we reproduce
only these points (triangles up and down). The temper-
ature T is Tm = (Tb + Tt)/2, where Tt (resp. Tb) is the
top (resp. bottom) plate temperature. ρ is the fluid av-
erage density, Tc and ρc are the critical temperature and
2density. We also show the Niemela et al. [9] points (dia-
monds), recently provided to us by J.J. Niemela, and the
Wu et al. [7, 8] points (squares). We note that some of
the points of Niemela et al., not shown on the FA figure
4, are very near the critical point, however not showing
the Kraichnan like increase in Nu [9].
0.10
0.15
0.20
10
10
10
11
10
12
10
13
10
14
N
u
 /
 R
a
0
.3
0
6
Ra
FIG. 2: The compensated Nusselt versus the Rayleigh num-
ber. Circles: FA results as presented in their figure 5 (open:
isothermal, solid: isobaric with ∆T = 1 K, P/Pc = 0.920).
Other symbols are as in figure 1.
In figure 1, and in the following one 2 we do several dis-
tinctions between the Chavanne’s points we kept. Points
with β∆T < 0.1, where β is the isobaric expansion co-
efficient and ∆T = Tb − Tt, are plotted as triangles up.
We can consider for these points that physical proper-
ties are reasonably uniform within the cell. Points with
β∆T > 0.1 are plotted as triangles down. To eliminate
any doubt, we also distinguish densities far from the criti-
cal one (solid symbols) from those near it (open symbols).
Following FA ([10] figure 3), we plot in our figure 2
Nucomp = Nu/Ra
0.306
versus Ra for the Chavanne’s
points we kept. The symbols are the same than in figure
1. Also shown are the FA results (circles) as presented
in their figure 5 (open: isothermal, solid: isobaric with
∆T = 1 K), and the Niemela’s points [9]. The discrep-
ancy remains for these points which have the same loca-
tion in the phase space. The Kraichnan like increase for
the Chavanne’s points is obvious. We can easily conclude
that the discrepancy between references [3, 4] and [9] has
nothing to do with the location in the phase space.
Another question is to know if whether or not the
characteristics of the effect observed by Funfschilling and
Ahlers can be found in the data of Chavanne et al.. Fun-
fschilling and Ahlers have shown that, in their observa-
tions, the compensated Nusselt Nucomp = Nu/Ra
0.306
only depends on the difference between Tt (not Tm) and
the equilibrium temperature Te at the same pressure (this
corresponding to Tm and ρ).
There are no constant pressure series of measurements
in Chavanne’s data [3, 4]. The reason is that ensuring
a constant pressure ask for letting the filling capillary
0.15
0.20
0 0.025 0.05
N
u
 /
 R
a
0
.3
0
6
T
t
-T
e
 (K)
FIG. 3: Compensated Nusselt versus the difference between
the top plate temperature and the condensation temperature
at the considered pressure. All the points correspond to the
same density: 37.9 kgm−3. The circles with cross correspond
to the same pressure within ±0.2% (P = 1.955 bar, P/Pc =
0.86)
0.15
0.20
10
10
10
11
10
12
10
13
N
u
 /
 R
a
0
.3
0
6
Ra
FIG. 4: Same points as figure 3 plotted versus Ra.
open up to a pressure gauge or ballast. On some points
of the capillary, the temperature can go down to the con-
densation temperature Te. Together with the important
compressibility near the critical point, it can cause large
transfers of mass and heat in this capillary. However sev-
eral series of measurements in [3, 4] have been made at
constant density ρ. On the figure 3 we plot the compen-
sated Nusselt versus (Tt − Te) for the Chavanne’s points
corresponding to ρ = 37.9 kgm−3. The strongly different
values do not merge on a curve. On the opposite, they
nicely merge if plotted versus Ra (figure 4). We draw the
attention of the reader on the points labeled with a cross
in the circle. Their corresponding pressure is the same
within ±0.2%, which is not as precise as the ±3 10−5 of
Funfschilling and Ahlers, but rather constant anyway.
To conclude, the simple examination of the published
data of Chavanne et al. shows that the hypothesis raised
3by Funfschilling and Ahlers [10], that the influence of
the critical point could explain the apparent discrepancy
between references [3, 4] and [9], this hypothesis cannot
hold. Moreover, as far as the effect observed by them [10]
is well characterized, we found no trace of it in the helium
data of [3, 4]. We see here no reason to reconsider the
original interpretation for the increase of Nu observed
by [3, 4, 5] as a transition toward the Kraichnan regime,
even if the exact conditions for such a transition have yet
to be elucidated.
Acknowledgments
We are grateful to J.J. Niemela for providing us the
numerical values corresponding to the data in Ref. [9].
We also thank G. Ahlers for supplementary details on
the cell and on ethane properties.
[1] R. H. Kraichnan, “Turbulent Thermal Convection at
Arbitrary Prandtl Number”, Phys. Fluids, 5, 1374–1389,
(1962).
[2] E. Siggia, “High Rayleigh Number Convection” Annual
Review of Fluid Mechanics, 26, 137–168, (1994)
[3] X. Chavanne, F. Chilla`, B. Castaing, B. He´bral, B.
Chabaud, and J. Chaussy, “ Observation of the Ultimate
Regime in Rayleigh-Be´nard Convection” Phys. Rev. Lett.
79, 3648–3651 (1997).
[4] X. Chavanne, F. Chilla`, B. Chabaud, B. Castaing,
B. He´bral, “Turbulent Rayleigh-Be´nard convection in
gaseous and liquid He” Phys. Fluids 13, 1300–1320,
(2001).
[5] P.-E. Roche, B. Castaing, B. Chabaud, and B. He´bral,
“Observation of the 1/2 power law in Rayleigh-Be´nard
convection” Phys. Rev. E 63, 045303–1–4 (2001)
[6] J. J. Niemela, K. R. Sreenivasan, “Confined turbulent
convection” J. Fluid Mech., 481, 355–384 (2003)
[7] X. Z. Wu,“Along a road to developped turbulence: free
thermal convection in low temperature helium gas”, PhD
Thesis, Chicago University, (1991).
[8] X. Z. Wu, A. Libchaber, “Scaling relations in thermal
turbulence: The aspect-ratio dependence” Phys. Rev. A
45, 842–845 (1992); M. Sano, X. Z. Wu, A. Libchaber,
“Turbulence in helium-gas free convection” Phys. Rev. A
40, 6421–6430 (1989)
[9] J. J. Niemela, L. Skrbek, K. R. Sreenivasan, R. J. Don-
nelly, “Turbulent convection at very high Rayleigh num-
bers” Nature, 404, 837–841, (2000).
[10] D. Funfschilling and G. Ahlers, “Turbulent heat trans-
port near critical points: Non-Boussinesq effects” cond-
mat/0601398 (2006)


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