Analysis of accelerations in turbulence based on generalized statistics
An analytical expression of probability density function (PDF) of accelerations in turbulence is derived with the help of the statistics based on generalized entropy (the Tsallis entropy or the Ren´ yi entropy). It is revealed that the derived PDF explains the one obtained by Bodenschatz et al. i...
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Інститут фізики конденсованих систем НАН України
2003
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| Цитувати: | Analysis of accelerations in turbulence based on generalized statistics / T. Arimitsu, N. Arimitsu // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 85-92. — Бібліогр.: 28 назв. — англ. |
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irk-123456789-1206962017-06-13T03:06:05Z Analysis of accelerations in turbulence based on generalized statistics Arimitsu, T. Arimitsu, N. An analytical expression of probability density function (PDF) of accelerations in turbulence is derived with the help of the statistics based on generalized entropy (the Tsallis entropy or the Ren´ yi entropy). It is revealed that the derived PDF explains the one obtained by Bodenschatz et al. in the measurement of fluid particle accelerations in fully developed turbulence at Rλ = 970 . За допомогою статистики, яка базується на узагальненій ентропії (ентропії Цалліса чи ентропії Рені) виведено аналітичний вираз для функції густини ймовірності прискорень при турбулентності. Виявлено, що дана функція пояснює результат, отриманий Боденшацом та ін. при вимірюванні прискорень частинок флюїду за умов повністю розвинутої турбулентності при Rλ = 970 . 2003 Article Analysis of accelerations in turbulence based on generalized statistics / T. Arimitsu, N. Arimitsu // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 85-92. — Бібліогр.: 28 назв. — англ. 1607-324X PACS: 47.27.-i, 47.53.+n, 47.52.+j, 05.90.+m DOI:10.5488/CMP.6.1.85 http://dspace.nbuv.gov.ua/handle/123456789/120696 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
An analytical expression of probability density function (PDF) of accelerations
in turbulence is derived with the help of the statistics based on generalized
entropy (the Tsallis entropy or the Ren´ yi entropy). It is revealed that
the derived PDF explains the one obtained by Bodenschatz et al. in the
measurement of fluid particle accelerations in fully developed turbulence
at Rλ = 970 . |
| format |
Article |
| author |
Arimitsu, T. Arimitsu, N. |
| spellingShingle |
Arimitsu, T. Arimitsu, N. Analysis of accelerations in turbulence based on generalized statistics Condensed Matter Physics |
| author_facet |
Arimitsu, T. Arimitsu, N. |
| author_sort |
Arimitsu, T. |
| title |
Analysis of accelerations in turbulence based on generalized statistics |
| title_short |
Analysis of accelerations in turbulence based on generalized statistics |
| title_full |
Analysis of accelerations in turbulence based on generalized statistics |
| title_fullStr |
Analysis of accelerations in turbulence based on generalized statistics |
| title_full_unstemmed |
Analysis of accelerations in turbulence based on generalized statistics |
| title_sort |
analysis of accelerations in turbulence based on generalized statistics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120696 |
| citation_txt |
Analysis of accelerations in turbulence
based on generalized statistics / T. Arimitsu, N. Arimitsu // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 85-92. — Бібліогр.: 28 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT arimitsut analysisofaccelerationsinturbulencebasedongeneralizedstatistics AT arimitsun analysisofaccelerationsinturbulencebasedongeneralizedstatistics |
| first_indexed |
2025-07-08T18:25:25Z |
| last_indexed |
2025-07-08T18:25:25Z |
| _version_ |
1837104238262484992 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 85–92
Analysis of accelerations in turbulence
based on generalized statistics
T.Arimitsu 1 ∗, N.Arimitsu 2 †
1 Institute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan
2 Graduate School of EIS, Yokohama Nat’l. University,
Kanagawa 240-8501, Japan
Received October 14, 2002
An analytical expression of probability density function (PDF) of accelera-
tions in turbulence is derived with the help of the statistics based on gener-
alized entropy (the Tsallis entropy or the Rényi entropy). It is revealed that
the derived PDF explains the one obtained by Bodenschatz et al. in the
measurement of fluid particle accelerations in fully developed turbulence
at Rλ = 970 .
Key words: multifractal analysis, fully developed turbulence, PDF of fluid
particle accelerations, Rényi entropy, Tsallis entropy
PACS: 47.27.-i, 47.53.+n, 47.52.+j, 05.90.+m
The multifractal analysis of turbulence by the statistics based on the general-
ized entropy of Rényi’s or of Tsallis’ has been developed by the present authors
[1–9]. Similarly to the usual thermodynamic entropy, the Rényi entropy [10] is of
an extensive character, whereas the Tsallis entropy [11–13] is non-extensive. The
multifractal analysis belongs to the line of study based on a kind of ensemble theo-
retical approaches that, starting from the log-normal model [14–16], continues with
the β-model [17], the p-model [18,19], the 3D binomial Cantor set model [20] and so
on. After a rather preliminary investigation of the p-model [1], we developed further
to derive the analytical expression for the scaling exponents of velocity structure
function [2–5], and to determine the probability density function (PDF) of veloc-
ity fluctuations [5–8] and of velocity derivative [9] by a self-consistent statistical
mechanical approach.
In this paper, we will derive the formula for the PDF of the accelerations of a
fluid particle in fully developed turbulence by means of the multifractal analysis.
With the theoretical PDF, we will analyze the PDF of accelerations at Rλ = 970
(the Taylor microscale Reynolds number) obtained in the Lagrangian measurement
∗E-mail: arimitsu@cm.ph.tsukuba.ac.jp
†E-mail: arimitsu@ynu.ac.jp
c© T.Arimitsu, N.Arimitsu 85
T.Arimitsu, N.Arimitsu
of particle accelerations that was realized by Bodenschatz and co-workers [21,22]
by raising dramatically the spatial and temporal measurement resolutions with the
help of the silicon strip detectors.
We assume that the turbulent flow, satisfying the Navier-Stokes equation
∂~u/∂t + (~u · ~∇)~u = −~∇ (p/ρ) + ν∇2~u (1)
of an incompressible fluid, consists of a cascade of eddies with different sizes `n =
δn`0 , where δn = 2−n (n = 0, 1, 2, · · ·). The quantities ρ, p and ν represent, respec-
tively, the mass density, the pressure and the kinematic viscosity. The acceleration
~a of a fluid particle is given by the substantive time derivative of the velocity:
~a = ∂~u/∂t + (~u · ~∇)~u. At each step of the cascade, say at the nth step, eddies break
up into two pieces producing an energy cascade with the energy-transfer rate εn that
represents the rate of transfer of energy per unit mass from eddies of size `n to those
of size `n+1 (the energy cascade model). The Reynolds number Re of the system is
given by Re = δu0`0/ν = (`0/η)4/3 with the Kolmogorov scale [23] η = (ν3/ε)1/4,
where ε (= ε0) is the energy input rate to the largest eddies with size `0 .1 Introducing
the pressure (divided by the mass density) difference δpn at two points separated by
the distance `n, i.e., δpn = |p/ρ(• + `n) − p/ρ(•)|, and the acceleration an = δpn/`n
belonging to the nth step in the energy cascade, one can estimate accelerations by
|~a| = limn→∞ an. For a high Reynolds number Re � 1, or for the situation where
the effects of the kinematic viscosity ν can be neglected compared with those of the
turbulent viscosity, the Navier-Stokes equation (1) is invariant under the scale trans-
formation [24,19]: ~r → λ~r, ~u → λα/3~u, t → λ1−α/3t and (p/ρ) → λ2α/3 (p/ρ). The ex-
ponent α is an arbitrary real quantity which specifies the degree of singularity in the
acceleration for α < 1.5, i.e., limn→∞ an = lim`n→0 δpn/`n ∼ lim`n→0 `
(2α/3)−1
n → ∞
which can be seen with the relation δpn/δp0 = (`n/`0)
2α/3.
The multifractal analysis rests on the assumption that the distribution of the
exponent α is multifractal, and that the probability P (n)(α)dα to find, at a point in
physical space, an eddy of size `n having a value of the degree of singularity in the
range α ∼ α + dα is given by [2–5]
P (n)(α) ∝
[
1 − (α − α0)
2/(∆α)2
]n/(1−q)
(2)
with (∆α)2 = 2X/[(1 − q) ln 2]. The range of α is αmin 6 α 6 αmax with αmin =
α0−∆α and αmax = α0 +∆α. Here, we assume that the distribution function at the
nth multifractal depth has the structure P (n)(α) ∝ [P (1)(α)]n. This is consistent with
the relation [19,5] P (n)(α) ∝ δ
1−f(α)
n that is a manifestation of scale inveriance and
reveals how densely each singularity, labeled by α, fills the physical space. Within
the present model, the multifractal spectrum f(α) is given by [2–5]
f(α) = 1 + (1 − q)−1 log2
[
1 − (α − α0)
2 / (∆α)2] . (3)
1The velocity fluctuation δun is defined by δun = |u(• + `n) − u(•)| where u represents a
component of the velocity field ~u.
86
Analysis of accelerations in turbulence
To make the paper self-contained, we put here its brief derivation. The dis-
tribution function (2) is derived by taking an extremum of the generalized en-
tropy, the Rényi entropy [10] SR
q [P (1)(α)] = (1 − q)−1 ln
∫
dαP (1)(α)q or the Tsallis
entropy [11–13] ST
q [P (1)(α)] = (1 − q)−1 (∫
dα P (1)(α)q − 1
)
, under the two con-
straints, i.e., the normalization of distribution function:
∫
dαP (1)(α) = const and
the q-variance being kept constant as a known quantity: σ2
q = (
∫
dαP (1)(α)q(α −
α0)
2)/
∫
dαP (1)(α)q. In spite of the different characteristics of these entropies, the
distribution function giving their extremum has the common structure (2). The de-
pendence of the parameters α0, X and q on the intermittency exponent µ is deter-
mined, self-consistently, with the help of three independent equations, i.e., the energy
conservation: 〈εn〉 = ε, the definition of the intermittency exponent µ: 〈ε2
n〉 = ε2δ−µ
n ,
and the scaling relation:2 1/(1 − q) = 1/α− − 1/α+ with α± satisfying f(α±) = 0.
The average 〈· · ·〉 is taken with P (n)(α). For the region where the value of µ is usually
observed, i.e., 0.13 6 µ 6 0.40, the three self-consistent equations are solved to give
the approximate equations [8]: α0 = 0.9989 + 0.5814µ, X = −2.848 · 10−3 + 1.198µ
and q = −1.507 + 20.58µ − 97.11µ2 + 260.4µ3 − 365.4µ4 + 208.3µ5.
Since there are two mechanisms in a turbulent flow to rule its evolution, i.e., the
one controlled by the kinematic viscosity that takes care of thermal fluctuations, and
the other by the turbulent viscosity that is responsible for intermittent fluctuations
related to the singularities in acceleration, it may be reasonable to assume that the
probability Λ(n)(yn)dyn to find the scaled pressure fluctuations |yn| = δpn/δp0 in the
range yn ∼ yn + dyn has two independent origins:
Λ(n)(yn)dyn = Λ
(n)
S (|yn|)dyn + ∆Λ(n)(yn)dyn. (4)
The singular part Λ
(n)
S (|yn|) of the PDF stemmed from multifractal distribution
of the singularities, and the correction part ∆Λ(n)(yn) from the thermal dissipa-
tion and/or the measurement error.3 The former is derived through Λ
(n)
S (|yn|)dyn ∝
P (n)(α)dα with the transformation of the variables: |yn| = δ
2α/3
n . The mth moments
of the pressure fluctuations are given by
〈〈|yn|
m〉〉 ≡
∫ ∞
−∞
dyn|yn|
mΛ(n)(yn) = 2γ̃(n)
m + (1 − 2γ̃
(n)
0 ) a2m δζ2m
n (5)
with a3q̄ = {2/[
√
Cq̄(1 +
√
Cq̄)]}
1/2, Cq̄ = 1 + 2q̄2(1 − q)X ln 2 and
2γ̃(n)
m =
∫ ∞
−∞
dyn |yn|
m∆Λ(n)(yn). (6)
2The scaling relation is a generalization of the one derived first in [25,26] to the case where the
multifractal spectrum has negative values [1].
3Needless to say that each term in (4) is a multiple of two PDF’s, i.e., the PDF for one of the two
independent origins to realize and the conditional PDF for a value yn in the range yn ∼ yn + dyn
to come out. This is, of course, in a generalized sense in which the second correction term may
weaken the first singular contribution.
87
T.Arimitsu, N.Arimitsu
0 10 20 30
10–6
10–4
10–2
100
0 2 4 6
0
0.2
0.4
0.6
0.8
ω
Λ
<
=0.260, =16.2µω
n
n
ω
=0.260, =16.2µ
n
(a)
(b)
(
)
n(
)
Λ<
ω
n
(
)
(
) n
Figure 1. PDF of accelerations plotted on (a) log and (b) linear scale. Compari-
son between the experimentally measured PDF of fluid particle accelerations by
Bodenschatz et al. at Rλ = 970 (Re = 62 700) and the present theoretical PDF
Λ̂(n)(ωn). Open squares are the experimental data points on the left hand side
of the PDF, whereas open circles are those on the right hand side. Solid lines
represent the curves given by the present theory (8) with µ = 0.259 (q = 0.431)
and n = 16.4.
We used the normalization: 〈〈1〉〉 = 1. The quantity
ζm =
α0m
3
−
2Xm2
9
(
1 +
√
Cm/3
) −
1
1 − q
[
1 − log2
(
1 +
√
Cm/3
)]
(7)
is the so-called scaling exponent of velocity structure function, whose expression was
originally derived by the present authors [2–5]. The formula quite well explains the
experimental data [2–6,8]. Note that the formula is independent of the length `n,
and, therefore, independent of n.
Let us derive the PDF Λ̂(n)(ωn) of the accelerations by dividing it into two parts
with respect to ωn, i.e.,
Λ̂(n)(ωn) =
{
Λ̂
(n)
<† (ωn) for |ωn| 6 ω†
n
Λ̂
(n)
†< (ωn) for ω†
n 6 |ωn| 6 ωmax
n
, (8)
where the scaled variable ωn is defined by |ωn| = an/〈〈a2
n〉〉
1/2 = ω̄nδ
2α/3−ζ4/2
n with
ω̄n = [2γ̃
(n)
2 δ−ζ4
n + (1 − 2γ̃
(n)
0 )a4]
−1/2, and ωmax
n = ω̄nδ
2αmin/3−ζ4/2
n . Assuming that, for
88
Analysis of accelerations in turbulence
smaller accelerations |ωn| 6 ω†
n, the contribution to the PDF comes, mainly, from
thermal fluctuations related to the kinematic viscosity or from the measurement
error, we take for the PDF Λ̂
(n)
<† (ωn)ωn = Λ
(n)
S (yn)dyn + ∆Λ(n)(yn)dyn a Gaussian
function, i.e.,
Λ̂
(n)
<† (ωn) = Λ̃
(n)
S exp
{
−
1
2
[
1 +
3
2
f ′(α†)
]
[
(
ωn/ω
†
n
)2
− 1
]
}
(9)
with Λ̃
(n)
S = 3(1 − 2γ̃
(n)
0 )/(4ω̄n
√
2πX| ln δn|). On the other hand, we assume that
the main contribution to Λ̂
(n)
†< (ωn) may come from the multifractal distribution of
singularities related to the turbulent viscosity, i.e., Λ̂
(n)
†< (ωn)dωn = Λ
(n)
S (|yn|)dyn:
Λ̂
(n)
†< (ωn) = Λ̃
(n)
S
ω̄n
|ωn|
[
1 −
1 − q
n
(3 ln |ωn/ωn,0|)
2
8X| ln δn|
]n/(1−q)
(10)
with |ωn,0| = ω̄nδ
2α0/3−ζ4/2
n . The point ω†
n was defined by ω†
n = ω̄nδ
2α†/3−ζ4/2
n with α†
being the smaller solution of ζ4/2− 2α/3 + 1− f(α) = 0, at which Λ̂(n)(ω†
n) has the
least n-dependence for large n. Here, Λ̂
(n)
<† (ωn) and Λ̂
(n)
†< (ωn) were connected at ω†
n
under the condition that they should have the same value and the same slope there.
The specific form of Gaussian function (9) comes out through the connection.
With the expression (9), we can obtain ∆Λ(n)(yn), and have the formula to
evaluate γ̃
(n)
m in the form
2γ̃(n)
m =
(
K(n)
m − L(n)
m
)
/(
1 + K
(n)
0 − L
(n)
0
)
, (11)
where
K(n)
m =
3 δ
2(m+1)α†/3−ζ4/2
n
2
√
2πX| ln δn|
∫ 1
0
dz zm exp
{
−
1
2
[
1 +
3
2
f ′(α†)
]
(
z2 − 1
)
}
, (12)
L(n)
m =
3 δ
2mα†/3
n
2
√
2πX| ln δn|
∫ 1
zmin
dz zm−1
1 −
1 − q
n
(
3 ln |z/z†0|
)2
8X| ln δn|
n/(1−q)
(13)
with zmin = ωmin/ω
†
n = δ
2(αmax−α†)/3
n and z†0 = ωn,0/ω
†
n = δ
2(α0−α†)/3
n . Now, the PDF
of fluid particle accelerations (8) is determined by the intermittency exponent µ and
the number n of steps in the energy cascade which gives the eddy size `n.
The comparison between the present PDF of accelerations and that measured in
the experiment [21,22] at Rλ = 970 is plotted in figure 1 on log and linear scale. The
intermittency exponent µ = 0.259 and the number n = 16.4 of steps in the cascade
are extracted by the method of least squares with respect to the logarithm of PDF’s
as the best fit of our theoretical formulae (8) to the observed values of the PDF
[21,22] by discarding those points whose values are less than ∼ 2 · 10−6 since they
89
T.Arimitsu, N.Arimitsu
Table 1. The values of the scaling exponents derived by the present theory for
the acceleration is also listed with the corresponding values for q, α0 and X.
m ζm m ζm m ζm m ζm
1 0.3661 6 1.741 11 2.599 16 3.175
2 0.6989 7 1.945 12 2.731 17 3.268
3 1.000 8 2.130 13 2.854 18 3.356
4 1.272 9 2.299 14 2.969 19 3.438
5 1.519 10 2.455 15 3.075 20 3.515
scatter largely in the log scale. Substituting the extracted value of µ into the self-
consistent equations, we have the values of parameters: q = 0.431, α0 = 1.149 and
X = 0.307. With these values, other quantities are determined, e.g., ∆α = 1.248,
α+ − α0 = α0 − α− = 0.7211 and ω†
n = 0.557 (α† = 1.005). We see an excellent
agreement between the measured PDF of accelerations and the analytical formula
of PDF derived by the present self-consistent multifractal analysis. The value of ω†
n
tells us that the contribution from thermal fluctuation and/or measurement error
is restricted to smaller values of ωn, i.e., less than about one half of its standard
deviation. Then, the values of α responsible for the intermittency due to the scale
invariance turn out to be smaller than α† ≈ 1. This is the range within the condition
α < 1.5 in which the singularity appears in fluid particle accelerations.
The values of the scaling exponents ζm, given by (7), for m = 1, . . . , 20 are listed
in table 1 for future convenience in comparison with experiments or other theories.
Note that µ is related to ζ6 by the relation µ = 2 − ζ6 within the present analysis.
The flatness F
(n)
a ≡ 〈〈a4
n〉〉/〈〈a
2
n〉〉
2 = 〈〈ω4
n〉〉 of the PDF of accelerations has the value
F
(n)
a = 57.8 which is compatible with the value of the flatness ∼ 60 reported in [21,22]
as it should be. It is quite attractive to see that the distance r = `n corresponding
to the extracted value n = 16.4 reduces to r = 0.821µm (r/η = 0.0456), which is
close to the value of the spatial resolution 0.5µm (1/40 of the Kolmogorov distance)
of the measurement in [21,22]. Here, we used `0 = 0.071 m reported in [21,22].
We derived, in this letter, the formula of the PDF of accelerations (8) within
the approach of the multifractal analysis constructed by the present authors, and
successfully analyzed the beautiful experiment conducted by Bodenschatz and co-
workers [21,22] in the Lagrangian frame. We expect that other data for the same
system, in addition to the PDF of accelerations, will be provided in the near future,
such as the intermittency exponent µ, the scaling exponents ζm, the PDF’s of the ve-
locity fluctuations or of the velocity derivatives. Then, we can cross-check the validity
of the present analysis based on the formula (8).4 Note that the formula for the PDF
4We have checked, with the help of the DNS data reported by Gotoh et al. in [27], the accuracy
of the value of the intermittency exponent extracted out of the measured PDF’s. With the formula
(7), we determined in [8] the value µ = 0.240 for the longitudinal velocity fluctuations by fitting,
with the method of least squares, the ten data of the scaling exponents ζm (m = 1, 2, · · · , 10) at
Rλ = 381. On the other hand, we have extracted the value µ = 0.237 by the method of least
90
Analysis of accelerations in turbulence
of accelerations is different from the one for the PDF’s of velocity fluctuations or of
velocity derivatives. The empirical PDF, Λ̂emp(ω) = C exp {−ω2/ [(1 + |ωβ/σ|γ) σ2]}
with β = 0.539, γ = 1.588, σ = 0.508 and C = 0.786 proposed in [21,22] for the
data at Rλ = 970, gives a line very close to the one provided by the present PDF
Λ̂(n)(ω) of (8) for the region |ω| < 30 where the data of PDF exist. They deviate,
however, for |ω| > 30, i.e., Λ̂emp(ω) < Λ̂(n)(ω). The extraction of the PDF of the
accelerations out of the DNS data obtained by Gotoh et al. [27] is one of the attrac-
tive investigations in order to check the validity of the present formula (8) derived
by the unified approach providing various PDF’s [5–9], since the accuracy of their
DNS data for PDF’s is very high up to the order of 10−10 [28].
Acknowledgement
The authors would like to thank Prof. C.Tsallis for his fruitful comments with
encouragement. The authors are grateful to Prof. E.Bodenschatz for his kindness to
show his data prior to publication.
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