Table Of Content“Clusterization” and intermittency of temperature fluctuations in turbulent
convection
A. Bershadskii1,2, J.J. Niemela1, A. Praskovsky3 and K.R. Sreenivasan1
1International Center for Theoretical Physics, Strada Costiera 11, I-34100 Trieste, Italy
2ICAR, P.O. Box 31155, Jerusalem 91000, Israel
3National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA
TemperaturetimetracesareobtainedinturbulentthermalconvectionathighRayleighnumbers.
Measurementsaremadeinthemidplaneoftheapparatus,nearthesidewallbutoutsidetheboundary
layer. A telegraph approximation for temperature traces is generated by setting the fluctuation
amplitude to 1 or 0 depending on whether or not it exceeds the mean value. Unlike the standard
4 diagnostics of intermittency, the telegraph approximation allows one to distinguish the tendency
0 of events to cluster (“clusterization”) from their large-scale variability in amplitude. A qualitative
0 conclusion is that amplitude intermittency might mitigate clusterization effects.
2
PACSnumbers: 47.27.Te;47.27.Jv
n
a
J
I. INTRODUCTION seconds. NotingfromRef. [3]thatthemeanspeedofthe
9
large-scalecirculation(“meanwind”)fortheseconditions
2
is about 6 cm/s, a typical length scale of these events is
We consider turbulent convection in a confined con-
] of the order of 50 cm, which is the characteristic dimen-
D tainer of circular crosssection and 50 cm diameter. The
sionoftheapparatus. Thatis,iftheselarge-scaleplumes
aspectratio(diameter/height)isunity. Thesidewallsare
C originate from the boundary layer, it is as if the entire
insulatedandthebottomwallismaintainedataconstant
. boundarylayeronthebottomwallparticipatesonceina
n temperature,whichishigherbyasmallamount∆T than
while in this activity that we have called the large-scale
i that of the top wall. The working fluid is cryogenic he-
l plume. The maximum temperature in these large-scale
n lium gas. By controlling the temperature difference be-
[ tween the bottom and top walls, as well as the thermo- plumesisafractionoftheexcesstemperatureofthebot-
1 dynamic operating point on the phase plane of the gas, tomplate(namely ∆T/2),so,presumably,the fluidthat
is participating in the formation of a typical large-scale
v the Rayleigh number (Ra) of the flow could be varied
plume comes from the top parts of the boundary layer;
4 between107 and1015. We measuretemperature fluctua-
or, if a plume does indeed come from the very bottom
4 tionsatvariousRayleighnumberstowardstheupperend
0 partsof the boundarylayer,it is alreadypartiallymixed
ofthisrange,inwhichtheconvectivemotionisturbulent.
1 by the time it reaches the probe midway between the
Time traces of fluctuations are obtained at a distance of
0 topandbottomwalls. Theyarecertainlynotsmall-scale
4.4 cm from the sidewall on the center plane of the ap-
4 events that scale on the thickness of the boundary layer.
0 paratus. This position is outside of the boundary layer
Thisdescriptiondoesnotapplytosmall-scaleplumesa-i,
/ region for the Rayleigh numbers considered here. More
n though, presumably,they too belong to the same family.
details of the experimental conditions and measurement
i InFig.1,weshowthecaseofhotplumes,thatis,thecase
l procedure can be found in Ref. [1].
n whenthewindatthemeasurementpointarrivesfromthe
: A significant part of convection, even at the high
v hotter bottom plate. One can imagine that the wind di-
Rayleighnumbersthatconcernushere,is due toplumes
i rection could be just the opposite, leading to the arrival
X [2]. We use the term here merely to denote an orga-
at the probe of cold plumes coming from the colder top
r nizedactivityofconvectionwithoutimplyingmuchabout plate. We haveanalyzedsuchinstancesas well. Further,
a
their three-dimensional shapes and sizes, or the param-
beyond a certain Rayleigh number, as described in Ref.
eters on which they scale, though a few comments will
[3], the mean wind reverses itself randomly, so that a
be made momentarily. The primary goal of the paper
probe permanently held at one position sees hot plumes
is to learn about the tendency of the plumes to cluster
for some period of time and cold plumes for some other
together (“clusterization”).
period of time. We have analyzed the two parts sepa-
TheupperpartofFig.1showsashortsegmentoftem- rately by stringing together only hot parts or the cold
perature fluctuations at Ra=1.5×1011. There are four partsof ameasuredtemperature trace. The data for the
large-scale events within this segment (marked by the two cases have been examined separately. In each case,
letters A-D), and we imagine them to be the manifesta- thetelegraphapproximationisrelatedtospecificproper-
tion of large-scaleplumes. Eachevent consists of several ties of the underlying physical processes associated with
subevents, and there also exist a number of small events hot or cold plumes, as they encounter the probe during
marked a-i. While it may well be that the subevents their motion.
deserve to be considered separately,we regardthem col-
lectivelyhere. Underthesecircumstances,itisclearthat
a typical life-time of the large events is of the order of 8
2
100
0.2
A B C D 10-1
10-2
(t) 0.1 (f) 10-3
q a bcd e f g h i E 10-4
0 10-5 hot
full signal
-0.1 10-6 telegraph approximation
0 20 40 60 80 100 120 10-7
0.01 0.1 1 10 100
1
100
10-1
10-2
)
T(t E(f) 1100--43
10-5 cold
10-6 full signal
0 telegraph approximation
0 20 40 60 80 100 120 10-7
0.01 0.1 1 10 100
t [s] f [s-1]
FIG. 1: Anexample of themeasured temperaturetime trace FIG. 2: Spectra of the telegraph approximation of two real-
(upperpart) and its random telegraph approximation (lower izations ofthetemperaturetracecompared with thoseofthe
part). In addition to the small plumes marked a-i, there are temperature trace itself. The ordinate has been shifted so
several others such structures in the signals. They have not that thefirst points in both spectra coincide.
been marked merely because theyoccur below thezero line.
properties examined here are reasonably independent of
II. RANDOM TELEGRAPH APPROXIMATION the threshold; this comment will be made also at other
specific places in the paper. It may be useful to know
A more detailed discussion of the plumes will be pre- how the conventionalstatistics for the randomtelegraph
sented elsewhere but we limit ourselves here to a discus- approximation compare with those of the temperature
sion of their tendency to cluster together occasionally. signal. Figure 2 compares the spectral densities, E(f),
This is not obvious from the piece of the temperature of the telegraph approximations with those of the full
trace shown in Fig. 1, and a longer trace crowds the signal. Thecomparisonsaremadeseparatelyforhotand
plumes too much. It may be surmised that the clus- cold cases. It is clear that the spectra of the telegraph
terizationis indeed responsible for the mean wind in the approximationareclosetothoseoftheoriginalsignalina
apparatus. Inthe usualmethods ofanalysisofturbulent significantly large interval of scales. The main difference
signals [4], it is difficult to separate the clusterizationef- is that the telegraph approximation is richer in spectral
fectfromtheusualintermittencyeffectsarisingfromam- contentaboveacertainfrequency. Thisisnotdifficultto
plitudevariability. Toseparatethetwoeffects,weignore understand from a visual inspection of Fig. 1.
thevariationoftheamplitudefromoneplumetoanother Of particular interest is the power-lawbehavior of the
and replace the temperature trace of the type shown in spectral densities of the telegraph approximation. For
the upper partofFig.1 byits randomtelegraphapprox- both hot and cold cases, they follow a power-law of the
imation, shown in the lower part. This approximationis form
generatedfromthemeasuredtemperatureby settingthe
fluctuation magnitudes to 1 or 0 depending on whether E(f)∼f−β. (2)
theactualmagnitudeexceedsthemeanvalue(markedas
Though this power-law behavior is clear from Fig. 2, we
zero and shown by the dashed line in the upper part of
reproduce in Fig. 3 the spectra of the telegraph signal,
Fig. 1). Formally, for the temperature fluctuation Θ(t)
computedfromseveralrecordstoattainbetterstatistical
(with zero mean), the telegraph approximation T(t) is
convergence,inordertoemphasizethepower-lawscaling.
constructed as
The exponent β =1.38±0.02. A reasonable shift of the
1 Θ(t) thresholddoesnotchangethespectralexponentβ. From
T(t)= +1 . (1)
2(cid:18)|Θ(t)| (cid:19) theclosenessofthespectraofthetemperaturetracewith
its telegraphapproximation,it is inferredeasily that the
By definition, T can assume either 1 and 0. The tele- formerhasaspectralexponentof1.38aswell,albeitover
graph approximation can be generated by setting differ- asmallerrangeofscales. Observationofthetemperature
ent “thresholds” than the mean. It turns out that most trace spectra with such power law was first made in [5],
3
103 104
102 103
) -1.37
(f) 101 -1.38 tp( 102
E
100 101
hot
10-1 hot 100
0.1 1 10
10-2
0.01 0.1 1 10 100
103 104
102 103
) -1.37
)
(f 101 -1.38 tp( 102
E
100 101
cold
10-1 100
cold 0.1 1 10
10-2 t
0.01 0.1 1 10 100 [s]
f [s-1]
FIG. 3: Spectrum in the telegraph approximation computed FIG.4: Theprobabilitydensityfunctionofthedurationτ for
usingtwentyrealizationsofthetemperaturetrace,forthehot the telegraph approximation of the temperature signal (for
case (upper plot) and for the cold case (bottom part). The hot and cold cases). The straight lines (the best fits) are
straight lines (thebest fit) show that thepower-law approxi- drawn to indicate the scaling law (3).
mation (2) holds well for both hot and cold cases.
of the temperature signal and its telegraph approxima-
and was explored theoretically in [6], and is now a well- tionarecloserinconfinedconvectionthaninatmospheric
known result. turbulence.
The probability density function of the duration be-
tweenevents,τ,forthetelegraphapproximationisshown
in Fig. 4. Data for the hot and cold cases are given sep- III. QUANTIFYING CLUSTERIZATION
arately. The log-log scale has been chosen to emphasize
the power-law structure Thedifferencebetweentheobservedtelegraphspectral
exponent (β ≃ 1.38) and its value given by Eq. (4) (≃
p(τ)∼τ−α. (3)
1.63) is a quantitative measure of clusterization [7] of
plume-like objects observed in temperature traces. This
Forbothhotandcoldcases,weobserveα=1.37±0.03.
is a part of intermittency.
A reasonablevariation of the threshold in the vicinity of
Intermittency of the so-called temperature dissipation
the average value does not change the exponent α.
rate [4],[8] is characterized in turbulence by
It is known that, for non-intermittent cases (Ref. [7]
and Sec. 4), the relation between the exponents α and β dT2
is given by χ=| |. (5)
dt
β =3−α. (4) Following Obukhov [4], the local average
If we substitute in (4)the value of α≃1.37(as observed 1 t+τ
χ = χ(t)dt
inFig.4)weobtainβ ≃1.63. Thisisconsiderablylarger τ τ Z
t
than that actual value measured in Fig. 3, namely 1.38.
canbe usedtodescribetheintermittency ofχ. Thescal-
Thisdiscrepancyistheobjectofinteresttoushere;asal-
ing of the moments,
ready remarked, since there are no amplitudes involved,
itmustbe relatedtoclusterizationentirely(see also[7]). hχqi
It is of interest to note here that, for the telegraph ap- τ ∼τ−µq, (6)
hχ iq
proximation of the temperature fluctuation in the tur- τ
bulent atmospheric boundary layer, we have about the assuming that scaling exists, is a common tool for the
same values of α and β as the present, while the spec- description of the intermittency [4],[8]. Intermittent sig-
tral density of the temperature trace has a roll-off rate nalspossessanon-zerovalueoftheexponentµ . Ofpar-
q
ofabout1.66(consistentwiththeKolmogorov-Obukhov- ticular interest is the exponent µ2 for the second-order
Onsager-Corrsin theory [4]). It is clear that the spectra moment.
4
This definitionhas aprobleminthe telegraphapprox-
imation because dT/dt is composed of delta functions.
Fortunately, one is interested in the scaling of the dis-
crete representation of the dissipation field, given by
10
9
8 τ
2 67 χτ ∼ (Θn+1−Θn)2/τ,
> 5 nX=1
c<t 4 -0.47
/ where n specifies an interval of space. This discrete defi-
> 3
2 nitionofχ avoidstheproblemwithdeltafunctions. Ob-
t τ
c< 2 viously,forthe telegraphsignal,scalingexponentscalcu-
lated for the pulse-defined dissipation, Eqs. (5)-(6), and
hot for the discrete spike-like process are identical; this ob-
cold
servation is not true for the original temperature Θ(t).
1
0.1 1 10
t
[s]
IV. CLUSTERIZATION AND THE SPECTRUM
FIG. 5: Normalized second moment of the local dissipation
rate for the telegraph approximation plotted against τ, for The spectrum of the telegraph approximation can be
coldandhotcases. Thestraightlines(thebestfits)aredrawn related to the probability distribution p(τ) through
to indicate thescaling law (6).
E(f)= W(t)G(ft)p(t)dt, (7)
Z
The telegraph approximation is a composite of Heavi-
side step functions, so the dissipation rate (5) is a com- whereG(ft)isatransformfunctionandtheweightfunc-
positeofpulses(i.e.delta-functions)locatedattheedges tion W(t) is supposed to have a scaling form
of the boxes of the telegraph signal. For the uniform
W(t)∼tδ. (8)
random distribution of the pulses along the time axis,
µ =0. Non-zero values of µ mean that there is a clus-
q q Since the quantities G(ft) and p(t)dt are dimensionless,
terization of pulses. Figure 5 shows dependence of the
one can use dimensional considerationsto find the expo-
normalized dissipation rate hχ2i on τ for the telegraph
τ nent δ (cf. Ref. [4]) through
approximationofhotandcoldsignals. Thestraightlines
(the best fits) are drawn to indicate the scaling law (6), W(t)∼hχi·t2, (9)
forq =2. Itshouldbe notedthatscalingintervalforthe
dissipation rate is the same as for the PDF and for the and the use of (2),(3) and (7); this yields relation (4).
spectrum(cf. Figs.3 and4). Values ofthe intermittency To estimate the clusterization correction on the rela-
exponent µ2, calculated as slopes of the straight lines in tion between scaling spectrum and p(t), we should take
Fig. 5 is µ2 ≃ 0.47±0.03. The relatively large value of into account two-point correlations in the telegraph sig-
the exponent µ2 suggests that the clusterization of the nal. This can be characterized by the two-point corre-
pulses is quite strong. The temperature dissipation can lation function ξ(t). In a situation where the two-point
be also characterizedby the “gradient” measure [4],[8] correlationfunction exhibits the scaling behavior
(▽T)2dv ξ(t)∼t−γ, (10)
χ = vr ,
r R v
r the correlation exponent γ is the same as the exponent
where v is a subvolume with space-scale r (for detailed µ2. This is easily seen by the well-known result that the
r
justification of this measure, see Ref. [4], p. 381 and correlationdimension [9] D2 is related to γ through
later). The scaling law of the moments of this mea-
D2 =1−γ, (11)
sure are importantcharacteristicsof the dissipationfield
[8]. By Taylor’s hypothesis [4], we can replace dT/dx by and to µ2 Ref. [8] via
dT/huidt (where hui is the mean wind and x is the co-
ordinate alongthe directionof the wind), andcandefine D2 =1−µ2, (12)
the dissipation rate as
thus yielding γ = µ2. To estimate the weight function
τ(dT)2dt W(t)withthesamedimensionalconsiderationsasabove,
χ ∼ 0 dt ,
τ R τ andtotakeintoaccountthetwo-pointcorrelation(which
characterizes clusterization), we replace hχi by
where τ ≃ r/hui. This, too, should follow the scaling
relation (6). hχ2i1/2 ∼t−µ2/2. (13)
t
5
Replacing (9) by discussed in the paper are generally valid for turbulent
temperature fluctuations at all Rayleigh numbers cov-
W(t)∼hχ2i1/2·t2, (14) ered in the measurements. The main conclusion is that
t
the telegraph approximation captures the main statis-
we have tical features of the temperature time trace obtained
in convection. This approximation, which gives a clear
W(t)∼t2−µ2/2. (15)
separation between clusterization and magnitude inter-
mittency, has been useful in demonstrating that there is
The corresponding correction of Eq. (4) is
a significanttendency for the plumes to cluster together.
The telegraphapproximationturns outto be useful here
β =(3−µ2/2)−α. (16)
because of the specific process of heat transport, which
is determined in large measure by the random motion
Using the value of µ2 ≃ 0.47 from Fig. 5 and the value
of temperature plumes. However, one can expect that
α≃1.37 from Fig. 4, we obtain
this approximation (or its modifications) may be also
E(f)∼f−1.40, (17) useful in the description of other turbulent signals. The
exponentµ2 forthetelegraphapproximation,completely
which compares well with the behavior found in Fig. 3 determined by clusterization, is about 0.47. This should
(see also [5],[6]). be comparedwith the intermittency exponent computed
for the χ-χ correlation of the full temperature signal,
which is about 0.36—consistent with similar estimates
V. SUMMARY AND DISCUSSION available for passive scalars (see, e.g., Ref. [10]). The
clusterization exponent is thus larger than the classical
The results discussed so far are for a fixed Rayleigh intermittency exponent. From this, one can infer that
number. The same situation seems to occur at other the magnitude intermittency plays a smoothing role on
Rayleigh numbers. For those Rayleigh numbers where the clusterization effects within the scaling interval.
there is a reversal of the wind, the concatenated data
corresponding to one given direction of the wind follow We thank L.J. Biven, J. Davoudi and V. Yakhot for
the same statistics as well. Thus, the characteristics useful discussions.
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[3] K.R. Sreenivasan, A. Bershadskii, and J.J. Niemela, [8] K.R. Sreenivasan and R.A. Antonia, Annu. Rev. Fluid
Phys. Rev. E 65, 056306 (2002); J.J. Niemela and K.R. Mech, 29, 435 (1997).
Sreenivasan, Europhys.Lett. 62, 829 (2003). [9] P.GrassbergerandI.Procaccia,Phys.Rev.Lett.50,346
[4] A.S.MoninandA.M.Yaglom,StatisticalFluidMechan- (1983).
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