Abstract
We present a generalized picture of intermittency in turbulence that is based on the theory of stochastic processes. To this end, we rely on the experimentally and numerically verified finding by R. Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that allows for an interpretation of the turbulent energy cascade as a Markov process of velocity increments in scale. It is explicitly shown that phenomenological models of turbulence, which are characterized by scaling exponents ζn of velocity increment structure functions, can be reproduced by the Kramers–Moyal expansion of the velocity increment probability density function that is associated with a Markov process. We compare the different sets of Kramers–Moyal coefficients of each phenomenology and deduce that an accurate description of intermittency should take into account an infinite number of coefficients. This is demonstrated in more detail for the case of Burgers turbulence that exhibits pronounced intermittency effects. Moreover, the influence of nonlocality on Kramers–Moyal coefficients is investigated by direct numerical simulations of a generalized Burgers equation. Depending on the balance between nonlinearity and nonlocality, we encounter different intermittency behavior that ranges from self-similarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot’s mean field theory [Phys. Rev. E 63 026307 (2001)]) to shock-like behavior (purely nonlinear Burgers case).
Highlights
The phenomenon of homogeneous and isotropic turbulence can still be considered as one of the main unsolved problems in classical physics [1,2]
Considerable efforts have been devoted to the development of phenomenological models of turbulence that all try to account for the intermittent character of the local energy dissipation rate such as the log-normal model [4,5] or the popular model by She and Leveque [6]
The present paper underlines the importance of the multi-scale approach devised by Friedrich and Peinke [8], which is capable of capturing the general effects of anomalous scaling in turbulence embodied in Equation (17)
Summary
The phenomenon of homogeneous and isotropic turbulence can still be considered as one of the main unsolved problems in classical physics [1,2]. It predicts the scaling of the structure functions according to h(δr v)n i = Cn hεi 3 r 3 Lr where L is the integral length scale and μ is the so-called intermittency coefficient which is of the order μ ≈ 0.227 (recent experiments [26], suggest a value of μ = 0.17 ± 0.01) As it has been discussed by Friedrich and Peinke [8], this reduces the Kramers–Moyal expansion to a Fokker–Planck equation with drift and diffusion coefficient. According to a theorem due to Pawula [34] (see [25]), the vanishing of the fourth-order Kramers–Moyal coefficient implies that all higher coefficients are zero as well and the Kramers–Moyal expansion (7) reduces to an ordinary Fokker–Planck equation The latter is suitable for modeling approaches via its corresponding Langevin equation as well as the undemanding determination of statistical quantities via the exact short-scale propagator of the Fokker–Planck equation [25]. Note that the She–Leveque model possesses a nearly linear slope in the semi-logarithmic representation
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