Conditional statistics of temperature fluctuations in turbulent convection

10 Pages·0.12 MB·English

by  Emily S. C. Ching

Conditional Statistics of Temperature Fluctuations in Turbulent Convection
Emily S.C. Ching and K.L. Chau
Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong
(February 1, 2008)
1
0 Wefind that theconditional statistics of temperature differenceat fixedvaluesof thelocally av-
0 eraged temperature dissipation rate in turbulent convection become Gaussian in the regime where
2 themixingdynamicsisexpectedtobedrivenbybuoyancy. Hence,intermittencyofthetemperature

n fluctuations in this buoyancy-driven regime can be solely attributed to the variation of the locally
a averaged temperature dissipation rate. We further obtain the functional behavior of these condi-
J tional temperature structure functions. This functional form demonstrates explicitly the failure of

6 dimensional agruments and enhances theunderstandingof thetemperature structurefunctions.

]
h
c
In turbulent fluid flows, physical quantities such as velocity, temperature and pressure exhibit seemingly irregular
e
m fluctuations both in time and in space. A key issue in turbulence research is to make sense of these fluctuations.
The central result of the seminal work of Kolmogorov in 1941 (K41) [1] is that the fluctuating velocity field in high
-
t ReynoldsnumberNavier-Stokesturbulenceisself-similaratscaleswithinthe inertialrange,therangeoflengthscales
a
t that are smaller than those of energy input and larger than those affected directly by molecular dissipation. K41
.s predicted that the velocity structure functions h[u(x+r)−u(x)]pi scale as rξp with scaling exponents ξp equal to
at p/3 when r is within the inertial range. Experimental and numerical results, however, indicate that ξp is a nonlinear
m function of p andthat turbulent velocity fluctuations are scale-dependentinthat the shape of the probabilitydensity
function (PDF) of the velocity difference u(x+r)−u(x) changes with the scale r even when r is within the inertial
-
d range. This deviation from the K41 results is associated with intermittency or the uneven distribution of turbulent
n
activity of the velocity field in time and in space.
o
Extensive efforts have been devoted to the understanding of the problem of intermittency or anomalous scaling.
c
[ In his Refined Similarity Hypothesis (RSH) [2,3], Kolmogorov attributed this intermittent nature of the velocity

fluctuations to the spatial variations of the energy dissipation rate. Various models have been put forth for the
1
statistics of the locally averaged energy dissipation rate. The most recent model of She and Leveque [4] proposed a
v
1 hierarchicalstructureforthemoments,whichleadstopredictionsthatareingoodagreementwithexperiments. This
7 moment hierarchy was later shown to be naturally satisfied by log-Poissonstatistics [5,6].
0 High Rayleighnumber convectionhas been a well-studiedmodel systemfor investigatingturbulence. Fluid motion
1
isdrivenbyanappliedtemperaturedifferenceacrossthetopandthebottomplatesofaclosedexperimentalcellfilled
0
with fluid. The temperature field in convection is thus an so-called active scalar. The flow state is characterized by
1
0 the geometryof the cell andtwo dimensionless parameters: the Rayleighnumber Ra =αg∆L3/(νκ) and the Prandtl
/ number Pr = ν/κ, where L is the height of the cell, ∆ is the applied temperature difference, g the acceleration due
t
a to gravity,and α, ν, and κ are respectively the volume expansion coefficient, the kinematic viscosity and the thermal
m
diffusivity of the fluid. When Ra is large enough, the convection becomes turbulent.
- In turbulent convection, the temperature fluctuations are also intermittent [7]. As for velocity fluctuations in
d
high Reynolds number Navier-Stokes turbulence, it is of interest to understand the intermittency of temperature
n
o fluctuations in high Rayleigh number convection. Turbulent convection poses additional interesting questions of
c its own. There is the issue of whether and how the characteristics of turbulence are affected by the presence of
: buoyancy. One expects the mixing dynamics to be driven by buoyancy at scales larger than the Bolgiano scale,
v
Xi lB ≡¯ǫ5/4/[χ¯3/4(αg)3/2] [8], where ǫ¯and χ¯ are respectively the averageenergyand temperature (variance)dissipation
rates. On the other hand, for length scales smaller than l , the mixing dynamics is expected to be driven by the
B
r inertial force of the fluid motion and the temperature field is effectively passive. Recently, one of us (Ching) [9]
a
has analyzed the intermittency of temperature field in turbulent convection. The normalized temperature structure
functions haveindeedbeenfoundto havedifferentscalingexponentsinthe buoyancy-drivenandin the inertia-driven
regimes.
In our present project, we attempt to understand the intermittency problem of temperature by separating it into
two parts: the understanding of the conditional statistics of temperature fluctuations at fixed values of the locally
averaged temperature dissipation rate and the understanding of the statistics of the local temperature dissipation.
In this paper, we report our study of the first part. The second part of our study is reported elsewhere [10]. This
separationallowsustoespeciallyaddresswhetherRSHtypeideaswouldbefruitful. Weshallseethattheintermittent
1
nature of the temperature fluctuations in the buoyancy-driven regime can indeed be attributed to the variations of
the locally averaged temperature dissipation rate. Moreover, a change in the statistical features of the temperature
fluctuations is again observed when the Bolgiano scale l is crossed. This change manifests itself as a change in the
B
behavior of the conditional PDFs of the temperature difference at fixed value of the locally averaged temperature
dissipation rate.
We use temperature data obtained in the well-documented Chicago experiment of low-temperature helium gas
[11,12]forouranalyses. Theexperimentalcellheatedfrombelowis cylindricalwithadiameterof20cmandaheight
of 40 cm. A mean circulating flow is present for Ra ≥ 108. The temperature at the center of the cell, T(t), was
measuredasafunctionoftime t. Weevaluatethe temperaturedifference betweentwotimes: T (t)≡T(t+τ)−T(t).
τ
TheintermittencyofthetemperaturefluctuationsismanifestedasachangeintheshapeofthePDFofT asτ varies.
τ
Inourearlierstudyofthis τ-dependence [7],the dissipativeandthe circulationtime scales,τ andτ ,wereidentified.
d c
A time scale corresponding to l is naturally defined by τ =τ l /L. It was shown [13] that l can be written as
B B c B B
1
Nu2L
l = (1)
B 1
(RaPr)4
where the Nusselt number (Nu) is the heat flux normalized by that when there was only conduction. Thus, τ can
B
be easily evaluated using the measured values of Nu, Ra, and Pr.
The locally averaged temperature dissipation rate χ is the spatial average of κ|∇T|2 over a ball of radius r. We
r
estimate it by χ , which is defined as
τ
1 t+τ κ ∂T 2
′
χ (t)≡ dt (2)
τ τ hu2i ∂t′
Zt c (cid:18) (cid:19)
and can be calculated using the one-point temperature measurements. Here, hu2i is the mean square velocity fluctu-
c
ations at the center of the cell.
We start by investigating the conditional PDF of T , at fixed values of χ . We consider those T (t) whose cor-
τ τ τ
responding lnχ (t) assumes a certain value within a small range δ, and calculate the conditional PDFs P(Y |χ )
τ τ τ
where
T
τ
Y ≡ (3)
τ
hT2|χ i
τ τ
As the conditional mean hT |χ i is approximately zerop, P(Y |χ ) is standardized with zero mean and unit standard
τ τ τ τ
deviation. For a given τ, P(Y |χ ) is found to be independent of χ for a range of χ that contains most of the
τ τ τ τ
data. The conditional PDFs for different values of τ are plotted in Fig. 1. We measure the value of χ in units of
τ
χ≡κh(∂T/∂t)2i/hu2i. In the limit τ →0,χ ∼T2, therefore, the conditionalPDF is bimodal for small τ, as seen in
c τ τ
the figure. As τ increases, P(Y |χ ) changes from bimodal to a function with one maximum and varies with τ. But
τ τ
forlargerτ, itbecomesastandardizedGaussiandistributionandisthus independentofτ. Suchachangeinbehavior
occurs at τ ≈τ .
B
Hence, achangeinthe statisticalfeaturesofthe temperature fluctuations is againobservedasthe Bolgianoscaleis
crossed,demonstratingthatbuoyancydoeshaveaneffectonthecharacteristicsofturbulenceinconvection. Moreover,
the physical nature of the presently observed change is clear. We have the interesting result that the temperature
fluctuations at fixed values of χ become self-similar and thus non-intermittent in the regime where the mixing
τ
dynamics is expected to be drivenby buoyancy. In other words,intermittency of the temperature fluctuations in this
buoyancy-drivenregime can be solely attributed to the variations of χ .
τ
In the remaining of this paper, we shall obtain the functional dependence of the conditional temperature structure
functions h|T |p|χ i on p, τ, and χ .
τ τ τ
It is illuminating to first work out what functional form is predicted by simple phenomenology and dimensional
agruments. One expects T , the temperature difference across a scale r, depends on r, χ , and u , the velocity
r r r
differenceacrossthesamescaler. Intheinertia-drivenregime,u isrelatedtothelocallyaveragedenergydissipation
r
rate ǫ by u ∼(rǫ )1/3 while in the buoyancy-drivenregime, u is generated by buoyancy: u2/r ∼αgT . Hence, we
r r r r r r
have
r1/3ǫ−1/6χ1/2 r <l
T ∼ r r B (4)
r r1/5χ2/5(αg)−1/5 r >l
(cid:26) r B
Equation (4) implies that
2
hu2ip/6τp/3χp/2hǫ −p/6|χ i τ <τ
h|T |p|χ i∼ c τ τ τ B (5)
τ τ (hu2cip/10τp/5χ2τp/5(αg)−p/5 τ >τB
if T , χ , and ǫ have the same scaling behavior in τ as the corresponding quantities with subscript r in r with
τ τ τ
r =hu2i1/2τ.
c
If the variations of χ and ǫ are both ignored, (5) implies that the temperature frequency power spectrum has
τ τ
a scaling ω−75 for frequency ω < ωB and ω−5/3 for ω > ωB, where ωB = 2π/τB. The former scaling behavior was
reportedforthetemperaturefrequencypowerspectrameasuredinwater[13]andhelium[14]whilethe latteronewas
reported for that measured in low Pr mercury [15].
Now we proceed with the analyses. From the result that P(Y |χ ) is independent of χ , we get
τ τ τ
h|T |p|χ i=F (τ)σp(τ,χ ) (6)
τ τ p τ
where
σ(τ,χ )≡ hT2|χ i (7)
τ τ τ
p
By definition, F (τ)=1. For τ >τ , P(Y |χ ) becomes a standardized Gaussian, thus
2 B τ τ
2p p+1
F (τ >τ )= Γ( ) (8)
p B
π 2
r
is independent of τ. For τ <τ <τ , we find that F (τ) can be fitted by a power law (see Fig. 2), that is
d B p
F (τ)≈C ταp τ <τ <τ (9)
p p d B
This τ dependence of F echoes that of P(Y |χ ) for τ < τ . Using (5), such dependence can be attributed to the
p τ τ B
additionalvariationofthelocalenergydissipationrateǫ evenwhenthe localtemperaturedissipationrateχ isheld
τ τ
fixed. The scaling exponents α are plotted in Fig. 3. Since α = α = 0 by definition, α has to be a nonlinear
p 0 2 p
function of p, as is found.
Next, we analyze the functional dependence of σ. We fix τ and study its dependence on χ . When τ is not too
τ
large, σ(τ,χ ) indeed scales with χ for a range of χ that contains most of the data. The scaling exponent b(τ),
τ τ τ
however, varies with τ. When τ is large, the data scatter. Thus, we have
σ(τ,χ )=G(τ)χb(τ) (10)
τ τ
From the relation χ ∼ T2 in the limit of τ → 0, one gets b(τ) → 1/2 as τ → 0. Indeed, as shown in Fig. 4, b(τ)
τ τ
is about 1/2 for τ ≤ τ . It then crosses over to an approximately linear function of lnτ, and has a value of 2/5 at
d
τ ≈ τ . This is, therefore, in contrary to the behavior of σ(τ,χ ) ∼ τ1/3χ1/2 and σ(τ,χ ) ∼ τ1/5χ2/5 respectively
B τ τ τ τ
in the inertia-driven (τ < τ < τ ) and buoyancy-driven regimes (τ < τ < τ ) that simple phenomenology and
d B B c
dimensional agruments would predict [see (5)]. In Fig. 5, we plot σ(τ,χ )(χ /χ)−b(τ) for various values of χ . The
τ τ τ
linear fit of b(τ) in lnτ is used for τ > τ . The data for different values of χ collapse to one single curve, thus
d τ
confirming (10). We take the averageof the data to get an estimate of G(τ)χb(τ), which is shown in the inset. It can
be fitted by a power law for τ >τ with an exponent about 0.27.
B
The temperature structure functions h|T |pi are related to the conditional ones at fixed values of χ as follows:
τ τ
∞
h|T |pi= h|T |p|χ iP (χ )dχ (11)
τ τ τ τ τ τ
Z0
where P (χ ) is the PDF of χ . Using (6) and (10), we thus get
τ τ τ
h|T |pi=F (τ)Gp(τ)hχpb(τ)i (12)
τ p τ
Equation (12) implies that the change in the scaling exponent of the normalized structure functions h|T |pi/hT2ip/2
τ τ
pb(τ)
observed,when τ is crossed[9], is the combined effect of the change in the τ dependence of F (τ) and hχ i. The
B p τ
comparison of (12) with data will be presented elsewhere.
Insummary,wehavestudiedsystematicallytheconditionalstatisticsofthetemperaturefluctuationsatfixedvalues
of local temperature dissipation χ in turbulent convection. We have found that such conditional statistics become
τ
3
self-similar in the buoyancy-driven regime, demonstrating that the intermittency of the temperature field in this
regimecanbeattributedsolelytothevariationsofχ . Wehaveworkedoutthefunctionalbehavioroftheconditional
τ
structure functions h|T |p|χ i. There is indeed scaling behavior in χ but the scaling exponent b(τ) depends on
τ τ τ
τ, in contrary to what simple phenomenology and dimensional agruments might predict. We emphasize that this
τ dependence demonstrates explicitly the failure of dimensional arguments. Together with the knowledge of the
statistical properties of χ , this functional behavior would enable us to better understand the temperature structure
τ
functions.
ESCC wouldliketo acknowledgediscussions withT. Witten. This workis supportedby agrantfromthe Research
Grants Council of the Hong Kong Special Administrative Region, China (RGC Ref. No. CUHK 4119/98P).
[1] A.N. Kolmogorov, C.R. Acad. Sci. URSS30, 301 (1941).
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[4] Z.-S.Sheand E. Leveque,Phys.Rev.Lett. 72, 336 (1994).
[5] B. Dubrulle,Phys. Rev.Lett. 73, 959 (1994).
[6] Z.-S.Sheand E. C. Waymire, Phys.Rev. Lett. 74, 262 (1995).
[7] E.S.C. Ching, Phys. Rev.A 44, 3622 (1991).
[8] A.S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Massachusetts, 1975).
[9] E. S.C. Ching, Phys. Rev.E 61, R36 (2000).
[10] E. S.C. Ching and C. Y.Kwok, Phys. Rev.E. 62, R7587 (2000).
[11] F. Heslot, B. Castaing and A. Libchaber, Phys.Rev. A 36, 5870 (1987).
[12] M. Sano, X.-Z Wu and A. Libchaber, Phys.Rev A 40, 6421 (1989).
[13] F. Chili´a, S. Ciliberto, C. Innocenti,and E. Pampaloni, NuovoCimento D 15, 1229 (1993).
[14] X.-Z.Wu,L. Kadanoff, A. Libchaber, and M. Sano, Phys. Rev.Lett. 64, 2140 (1990).
[15] S.Cioni, S. Ciliberto and J. Sommeria, Europhys.Lett. 32, 413 (1995).
4
FIGURE CAPTIONS
FIG 1. The conditional PDFs P(Y |χ ) versus Y for Ra = 6.0 × 1011 and χ /χ = 0.18 for various values of
τ τ τ τ
τ. τ = 8 (dotted line), τ = 16 (dashed line), τ = 32 (dot-dashed line), τ = 64 (circles), τ = 128 (squares),
and τ = 256 (triangles). It can be seen that P(Y |χ ) becomes a standard Gaussian distribution (solid line) for
τ τ
τ > τ ≈ 70. All times are in units of the sampling time = 1/409.6 s. The conditional PDFs are found to be
B
independent of χ .
τ
FIG. 2. The logarithm of the normalized conditional temperature structure functions F (τ)≡h|T |p|χ i/hT2|χ ip/2
p τ τ τ τ
versus lnτ for Ra = 7.3×1010 and χ /χ = 0.43 for various values of p. The three time scales τ , τ and τ are
τ d B c
approximately6,60and1750respectively,andareindicatedbythedashedlines. Alltimesareinunitsofthesampling
time = 1/320 s. p=0.5 (circles), p=1.5 (diamonds), p=1.75 (triangles), p=2.25 (crosses), p=2.5 (squares), and
p=2.75(pluses). Forτd <τ <τB, Fp(τ) canbe fitted bya power-lawCpταp (solidlines) andforτ >τB, it becomes
2p/πΓ((p+1)/2) (dot-dashed lines) and is thus independent of τ.
p
FIG. 3. The scaling exponent α versus p for Ra = 4.0 × 109 (circles), Ra = 7.3 × 1010 (squares), and Ra =
p
6.0×1011 (diamonds).
FIG. 4. The scaling exponent b(τ) versus lnτ for Ra = 4.0×109. The time scales τ and τ are approximately 8
d B
and 50 respectively, and are indicated by the dashed lines. All times are in units of the sampling time = 1/160.8 s.
It can be seen that b(τ) is close to 1/2 for τ ≤τ and can be fitted by a linear function in lnτ (solid line) for τ >τ .
d d
Moreover,b(τ)≈2/5 at τ ≈τ .
B
FIG. 5. lnσ(τ,χ )(χ /χ)−b(τ) versus lnτ for Ra = 7.3×1010 for χ /χ = 0.13 (circles), χ /χ = 0.35 (squares), and
τ τ τ τ
χ /χ ≈ 0.96 (triangles). The three sets of data collapse into a single function of τ (= G(τ)χb(τ)) confirming (10).
τ
Thetimes areinunits ofthesamplingtime=1/320swhileσ isinunits ofthestandarddeviationofthe temperature
fluctuations. Shown in the inset is the average of the three sets of data (solid line), which can be fitted by a power
law (dot-dashed line) for τ >τ (indicated by dashed line).
B
5
0
10
−1
10
)
τ
χ
|
τ
Y
(
P
−2
10
−3
10
−4 −3 −2 −1 0 1 2 3 4
Y
τ
0.5
0.4
0.3
0.2
)
τ
(
p 0.1
F

n
l
0.0
−0.1
−0.2
−0.3
0 1 2 3 4 5 6 7 8
τ
ln
0.20
0.15
0.10
p 0.05
α
0.00
−0.05
−0.10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
p
0.6
0.5
)
τ
0.4
(
b
0.3
0.2
0 1 2 3 4 5 6
τ
ln
τ
σ τ χ χ χ −b( )
ln [ ( , )( / ) ]
τ τ
− − −
3 2 1 0 1
0
2
σ τ χ χ χ −b(τ)
ln [ ( , )( / ) ]
τ τ
4
l
n
− − −

3 2 1 0 1
τ
0
2
6
l
n
4

τ
6
8
8