Fokker-Planck equation for the energy cascade in turbulence

Naert, Antoine; Friedrich, Rudolf; Peinke, Joachim

doi:10.1103/physreve.56.6719

Cited by 46 publications

(85 citation statements)

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“…In a series of recent papers, Friedrich, Peinke and collaborators [40,41,42] have experimentally shown that, for a fixed position, the velocity or dissipation statistics in turbulence are Markov processes. Their Fokker-Planck approach can be shown to be linked (in case of lognormal cascades) to Castaing's semi-group decomposition approach [38,39,36].…”

Section: Semigroup Composition Propertiesmentioning

confidence: 99%

“…Our results can be compared to previous studies, some of them done in purely experimental grounds. Let us first note that in some cases (see [41]) the study is done considering a Fokker-Planck equation with drift and diffusion coefficients D 1 and D 2 . The associated stochastic differential equation (Langevin form) is:…”

Section: Differential Properties and A Langevin Equationmentioning

confidence: 99%

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A causal multifractal stochastic equation and its statistical properties

Schmitt

2003

The European Physical Journal B - Condensed Matter

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Abstract. Multiplicative cascades have been introduced in turbulence to generate random or deterministic fields having intermittent values and long-range power-law correlations. Generally this is done using discrete construction rules leading to discrete cascades. Here a causal log-normal stochastic process is introduced; its multifractal properties are demonstrated together with other properties such as the composition rule for scale dependence and stochastic differential equations for time and scale evolutions. This multifractal stochastic process is continuous in scale ratio and in time. It has a simple generating equation and can be used to generate sequentially time series of any length.

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Section: Semigroup Composition Propertiesmentioning

confidence: 99%

Section: Differential Properties and A Langevin Equationmentioning

confidence: 99%

A causal multifractal stochastic equation and its statistical properties

Schmitt

2003

The European Physical Journal B - Condensed Matter

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“…This method is definitely more general than the above mentioned analysis by structure functions, which characterize only the simple scale statistics p(u(r)) or p(v(r)). The Fokker-Planck method has attracted interest and was applied to different problems of the complexity of turbulence like energy dissipation [23,24,25], universality turbulence [26] and others [11,27,28,29,30,31]. The Fokker-Planck equation (here written for vector quantities) reads as…”

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confidence: 99%

Different cascade speeds for longitudinal and transverse velocity increments of small-scale turbulence

Siefert

Peinke

2004

Phys. Rev. E

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We address the problem of differences between longitudinal and transverse velocity increments in isotropic small scale turbulence. The relationship of these two quantities is analyzed experimentally by means of stochastic Markovian processes leading to a phenomenological Fokker-Planck equation from which a generalization of the Kármán equation is derived. From these results, a simple relationship between longitudinal and transverse structure functions is found which explains the difference in the scaling properties of these two structure functions.PACS numbers: 47.27. Gs, 47.27.Jv, 05.10.Gg Substantial details of the complex statistical behaviour of fully developed turbulent flows are still unknown, cf. [1,2,3,4]. One important task is to understand intermittency, i.e. finding unexpected frequent occurences of large fluctuations of the local velocity on small length scales. In the last years, the differences of velocity fluctuations in different spatial directions have attracted considerable attention as a main issue of the problem of small scale turbulence, see for example [5,6,7,8,9,10,11,12,13]. For local isotropic turbulence, the statistics of velocity increments [v(x + r) − v(x)] e as a function of the length scale r is of interest. Here, e denotes a unit vector. We denote with u(r) the longitudinal increments (e is parallel to r) and with v(r) transverse increments (e is orthogonal to r) [40].In a first step, this statistics is commonly investigated by means of its moments u n (r) or v n (r) , the so-called velocity structure functions. Different theories and models try to explain the shape of the structure functions cf. [2]. Most of the works examine the scaling of the structure function, u n ∝ r ξ n l , and try to explain intermittency, expressed by ξ n l − n/3 the deviation from Kolmogorov theory of 1941 [14,15]. For the corresponding transverse quantity we write v n ∝ r ξ n t . There is strong evidence that there are fundamental differences in the statistics of the longitudinal increments u(r) and transverse increments v(r). Whereas there were some contradictions initially, there is evidence now that the transverse scaling shows stronger intermittency even for high Reynolds numbers [9,16].A basic equation which relates both quantities is derived by Kármán and Howarth [17]. Assuming incompressibilty and isotropy, the so called first Kármán equation is obtained:Relations between structure functions become more and * Electronic address: peinke@uni-oldenburg.de; URL: http://www.uni-oldenburg.de/hydro more complicated with higher order, including also pressure terms [13,18].In this paper, we focus on a different approach to characterize spatial multipoint correlations via multi-scale statistics. Recently it has been shown that it is possible to get access to the joint probability distribution p(u(r 1 ), u(r 2 ), . . . , u(r n )) via a Fokker-Planck equation, which can be estimated directly from measured data [19,20,21]. For a detailed presentation see [22]. This method is definitely more general than the...

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“…While this scheme is simple, it has previously played a decisive role in explaining scale-correlations for observed multiplier distributions [3,4] and observed Markov properties [8,9]. For two-point statistics with constant bin-bin distance d < L/η, the appropriate averaging is given by…”

mentioning

confidence: 99%

“…The second complication arises because experimental cumulants are derived from measured moments rather than the other way round [6], requiring translational averaging over two-point moments rather than two-point cumulants for theory also. The proper procedure is hence to convert theoretical cumulants (8) to moments, average these to restore translational invariance, and then convert them back to translationally invariant cumulants for experimental comparison.…”

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confidence: 99%

Translationally invariant cumulants in energy cascade models of turbulence

2001

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In the context of random multiplicative energy cascade processes, we derive analytical expressions for translationally invariant one-and two-point cumulants in logarithmic field amplitudes. Such cumulants make it possible to distinguish between hitherto equally successful cascade generator models and hence supplement lowest-order multifractal scaling exponents and multiplier distributions.Although the underlying hydrodynamic equations are deterministic, the statistical description of fully developed turbulence has by now a long tradition [1]. Random multiplicative energy cascade models form a particularly simple and robust class of such statistical models. While different theoretical models can reproduce experimentally observed multifractal scaling exponents rather easily [2], observed multiplier distributions [3,4] eliminate many candidate cascade generators. Nevertheless, a number of competing generators remain, equally successful in reproducing both scaling exponents and multipliers. To make further progress in ferreting out the best cascade generator within this approximation, new observables are clearly called for.While most experiments have concentrated on measuring statistics in the energy dissipation density ε, we recently found a complete and analytical solution working in ln ε rather than ε itself [5]. In this letter, we show that cumulants in ln ε are analytically calculable even when restoring translational invariance to the solutions to emulate the spatial homogeneity of experimental turbulence statistics. Cumulants turn out to be powerful tools which for third and fourth

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Fokker-Planck equation for the energy cascade in turbulence

Cited by 46 publications

References 11 publications

A causal multifractal stochastic equation and its statistical properties

A causal multifractal stochastic equation and its statistical properties

Different cascade speeds for longitudinal and transverse velocity increments of small-scale turbulence

Translationally invariant cumulants in energy cascade models of turbulence

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