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

Abstract

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...

Joint multi-scale statistics of longitudinal and transversal increments in small-scale wake turbulence

“…These two aspects were already presented in [47], but here we go in more detail and add some new aspects.…”

Longitudinal and transverse structure functions in high-Reynolds-number turbulence

“…In [18], evidence from experimental data Taylor-based Reynolds numbers between 180 and 550 was presented to support this view. The success of the approach introduced by Siefert and Peinke [18] motivated us to extend their re-interpretation of differential relations to structure functions of higher orders, making use of the exact relationships derived by Hill and Boratav [2], Hill [1] and Yakhot [3]. Hill derived these relations directly by inventing a clever matrix algorithm which allowed him to efficiently simplify the derivation and calculations.…”

“…which is exact and contains no contribution from the pressure-it is a statement of incompressibility. Siefert and Peinke [18] observed that the structure function is a smooth function of r and that if the scale r is chosen in the inertial range, i.e. 'small' compared to the integral scale L, equation (1) can be seen as a Taylor expansion:…”

Internal variability and pattern identification in cirrus cloud structure: The Fokker‐Planck equation approach

“…We present a method for deriving an underlying mathematical or model‐free equation, the Fokker‐Planck equation, that governs the time‐dependent probability distributions of the fluctuations at different delay times starting from observations of the backscattering cross section. Such an approach provides a stochastic method for describing turbulent flows, and its use has been of interest in the community in recent years [ Friedrich and Peinke , 1997; Siefert and Peinke , 2004]. In particular, the time‐dependent tails of the probability distribution functions are a signature of intermittency in turbulent flows [ Monin and Yaglom , 1975; Frisch , 1995].…”