Analysis of accelerations in turbulence based on generalizaed statistics

An analytical expression of probability density function (PDF) of accelerations in turbulence is derived with the help of the multifractal analysis based on generalized entropy, i.e., the Tsallis or the R\'{e}nyi entropy. It is shown that the derived PDF explains quite well the one obtained by Bodenschatz and coworkers in the measurement of fluid particle accelerations in the Lagrangian frame at $R_\lambda = 970$.

The multifractal analysis of turbulence by the statistics based on the generalized entropy of Rényi's or of Tsallis' has been developed by the present authors [ 1,2,3,4,5,6,7,8,9]. The Rényi entropy [ 10] has the extensive character as the usual thermodynamical entropy does, whereas the Tsallis entropy [ 11,12,13] is non-extensive. The multifractal analysis belongs to the line of study based on a kind of ensemble theoretical approaches that, starting from the log-normal model [ 14,15,16], 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 [ 2,3,4,5], and to determine the probability density function (PDF) of velocity fluctuations [ 5,6,7,8] 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 R λ = 970 (the Taylor microscale Reynolds number) obtained in the Lagrangian measurement 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 of an incompressible fluid, consists of a cascade of eddies with different sizes ℓ n = δ n ℓ 0 where δ n = 2 −n (n = 0, 1, 2, · · ·). The quantities ρ, p and ν represent, respectively, 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 ǫ n that represents the rate of transfer of energy per unit mass from eddies of size ℓ n to those of size ℓ n+1 (the energy cascade model). The Reynolds number Re of the system is given by Re is the energy input rate to the largest eddies with size ℓ 0 . 1 Introducing the pressure (divided by the mass density) difference δp n at two points separated by the distance ℓ n , i.e., δp n = |p/ρ(•+ℓ n )−p/ρ(•)|, and the acceleration a n = δp n /ℓ n belonging to the nth step in the energy cascade, one can estimate accelerations by | a| = lim n→∞ a n . For high Reynolds number Re ≫ 1, or for the situation where 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 transformation [ 24,19]: r → λ r, u → λ α/3 u, t → λ 1−α/3 t and (p/ρ) → λ 2α/3 (p/ρ). The exponent α is an arbitrary real quantity which specifies the degree of singularity in the acceleration for α < 1.5, i.e., lim n→∞ a n = lim ℓn→0 δp n /ℓ n ∼ lim ℓn→0 ℓ (2α/3)−1 n → ∞ which can be seen with the relation δp n /δp 0 = (ℓ n /ℓ 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 ℓ n having a value of the degree of singularity in the range α ∼ α + dα is given by [ 2,3,4,5] 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] that is a manifestation of scale inveriance and reveals how densely each singularity, labeled by α, fills physical space. Within the present model, the multifractal spectrum f (α) is given by [ 2,3,4,5] To make the paper self-contained, we put here its brief derivation. The distribution function (2) is derived by taking an extremum of the generalized entropy, (1) (α) q or the Tsallis entropy [ 11,12,13] under the two constraints, i.e., the normalization of distribution function: dαP (1) (α) = const. and the q-variance being kept constant as a known quantity: In spite of the different characteristics of these entropies, the distribution function giving their extremum has the common structure (2). The dependence of the parameters α 0 , X and q on the intermittency exponent µ is determined, self-consistently, with the help of the three independent equations, i.e., the energy conservation: ǫ n = ǫ, the definition of the intermittency exponent µ: ǫ 2 n = ǫ 2 δ −µ n , and the scaling relation: 2 The average · · · is taken with P (n) (α). For the region where the value of µ is usually observed, i.e., 0.13 ≤ µ ≤ 0.40, the three self-consistent equations are solved to give the approximate equations [ 8]: α 0 = 0.9989 + 0.5814µ, X = −2.848 × 10 −3 + 1.198µ and q = −1.507 + 20.58µ − 97.11µ 2 + 260.4µ 3 − 365.4µ 4 + 208.3µ 5 .
Since there are two mechanisms in turbulent flow to rule its evolution, i.e., the one controlled by the kinematic viscosity that takes care 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) (y n )dy n to find the scaled pressure fluctuations |y n | = δp n /δp 0 in the range y n ∼ y n + dy n has two independent origins: The singular part Λ (n) S (|y n |) of the PDF stemmed from multifractal distribution of the singularities, and the correction part ∆Λ (n) (y n ) from thermal dissipation and/or measurement error. 3 The former is derived through Λ We used the normalization: 1 = 1. The quantity is the so-called scaling exponent of velocity structure function, whose expression was derived first by the present authors [ 2,3,4,5]. The formula explains quite well experimental data [ 2,3,4,5,6,8]. Note that the formula is independent of the length ℓ n , and, therefore, independent of n. Let us derive the PDFΛ (n) (ω n ) of the accelerations by dividing it into two parts with respect to ω n , i.e., where the scaled variable ω n is defined by |ω n | = a n / a 2 Assuming that, for smaller accelerations |ω n | ≤ ω † n , the contribution to the PDF comes, mainly, from thermal fluctuations related to the kinematic viscosity or from measurement error, we take for the PDFΛ (n) < † (ω n )ω n = Λ (n) S (y n )dy n + ∆Λ (n) (y n )dy n a Gaussian function, i.e.,Λ (n) with |ω n,0 | =ω n δ 2α 0 /3−ζ 4 /2 n . The point ω † n was defined by ω † n =ω n δ 2α † /3−ζ 4 /2 n with α † being the smaller solution of ζ 4 /2 − 2α/3 + 1 − f (α) = 0, at whichΛ (n) (ω † n ) has the least n-dependence for large n. Here,Λ (n) < † (ω n ) andΛ (n) †< (ω n ) 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.
The comparison between the present PDF of accelerations and that measured in the experiment [ 21,22] at R λ = 970 is plotted in Fig. 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 scatter largely in the log scale. Substituting the extracted value of µ into the selfconsistent equations, we have the values of parameters: q = 0.431, α 0 = 1.149 and X = 0.307. With these values, other quantities are determined, e.g., ∆α = 1.248, α + − α 0 = α 0 − 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 ω n , 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 ζ m , 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 ζ 6 by the relation µ = 2 − ζ 6 within the present analysis. The flatness F (n) a ≡ a 4 n / a 2 n 2 = ω 4 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 = ℓ 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].
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 analyzed successfully the beautiful experiment conducted by Bodenschatz and coworkers [ 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 ζ m , the PDF's of the velocity 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 of accelerations is different from the one for the PDF's of velocity fluctuations or of velocity derivatives. The empirical PDF,Λ emp (ω) = C exp {−ω 2 / [(1 + |ωβ/σ| γ ) σ 2 ]} with β = 0.539, γ = 1.588, σ = 0.508 and C = 0.786 proposed in [ 21,22] for the data at R λ = 970, gives a line very close to the one provided by the present PDF Λ (n) (ω) of (8) for the region |ω| < 30 where the data of PDF exist. They deviates, however, for |ω| > 30, i.e.,Λ emp (ω) <Λ (n) (ω). The extraction of the PDF of accelerations out of the DNS data obtained by Gotoh et al. [ 27] is one of the attractive investigations in order to check the validity of the present formula (8) derived by the unified approach providing various PDF's [ 5,6,7,8,9], since the accuracy of their DNS data for PDF's is very high up to the order of 10 −10 [ 28].
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