A Langevin equation for turbulent velocity increments

Marcq, Philippe; Naert, Antoine

doi:10.1063/1.1386937

Cited by 30 publications

(28 citation statements)

References 11 publications

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“…(33) is again a Langevin equation for velocity increments in the scale space, with multiplicative and additive noises which is now expresses in an Eulerian form. Similar Langevin processes have been proposed before to explain the scale dependence of velocity increments [20,33,32] but without additive noise [19]. The noises in this Langevin equation are different from the noises appearing in the Lagrangian representation and they would have a complicated statistics if we assumed that σ and ξ were Gaussian in the Lagrangian representation.…”

Section: The Eulerian Descriptionmentioning

confidence: 99%

“…Other studies focused on the PDF of the modulus of one component. For example, Marcq and Naert [32] observe that the derivative has a highly non-Gaussian distribution, but with a correlation function which decays rapidly, and can be approximated by a delta function at scales large compared to the dissipative scale. In the present case, we observe different features.…”

Section: The Noisesmentioning

confidence: 99%

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Nonlocality and intermittency in three-dimensional turbulence

2001

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Numerical simulations are used to determine the influence of the non-local and local interactions on the intermittency corrections in the scaling properties of 3D turbulence. We show that neglect of local interactions leads to an enhanced small-scale energy spectrum and to a significantly larger number of very intense vortices ("tornadoes") and stronger intermittency (e.g. wider tails in the probability distribution functions of velocity increments and greater anomalous corrections). On the other hand, neglect of the non-local interactions results in even stronger small-scale spectrum but significantly weaker intermittency. Thus, the amount of intermittency is not determined just by the mean intensity of the small scales, but it is non-trivially shaped by the nature of the scale interactions. Namely, the role of the non-local interactions is to generate intense vortices responsible for intermittency and the role of the local interactions is to dissipate 1

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Section: The Eulerian Descriptionmentioning

confidence: 99%

Section: The Noisesmentioning

confidence: 99%

Nonlocality and intermittency in three-dimensional turbulence

2001

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“…The Markov property can be tested directly via its definition by using conditional probability densities [22] or by looking at the correlation of the noise of the Langevin equation [21]. For our case we have verified that the onedimensional processes of the longitudinal and transverse increments are Markovian [41], thus the two-dimensional processes should be Markovian, too [33].…”

mentioning

confidence: 99%

“…. , 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].…”

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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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“…With this knowledge of the Langevin equation (11) the noise can be reconstructed and analyzed with respect to its correlation [33,34]. …”

Section: Testing Proceduresmentioning

confidence: 99%

Stochastic analysis of different rough surfaces

Waechter

Riess

Schimmel³

et al. 2004

Eur. Phys. J. B

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Abstract. This paper shows in detail the application of a new stochastic approach for the characterization of surface height profiles, which is based on the theory of Markov processes. With this analysis we achieve a characterization of the scale dependent complexity of surface roughness by means of a Fokker-Planck or Langevin equation, providing the complete stochastic information of multiscale joint probabilities. The method is applied to several surfaces with different properties, for the purpose of showing the utility of this method in more detail. In particular we show evidence of the Markov properties, and we estimate the parameters of the Fokker-Planck equation by pure, parameter-free data analysis. The resulting FokkerPlanck equations are verified by numerical reconstruction of the conditional probability density functions. The results are compared with those from the analysis of multi-affine and extended multi-affine scaling properties which is often used for surface topographies. The different surface structures analysed here show in detail the advantages and disadvantages of these methods.

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A Langevin equation for turbulent velocity increments

Cited by 30 publications

References 11 publications

Nonlocality and intermittency in three-dimensional turbulence

Nonlocality and intermittency in three-dimensional turbulence

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

Stochastic analysis of different rough surfaces

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