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

Abstract

We analyse the relationship of longitudinal and transversal increment statistics measured in isotropic small-scale turbulence. This is done by means of the theory of Markov processes leading to a phenomenological Fokker-Planck equation for the two increments from which a generalized Kármán equation is derived. We discuss in detail the analysis and show that the estimated equation can describe the statistics of the turbulent cascade. A remarkable result is that the main differences between longitudinal and transversal increments can be explained by a simple rescaling symmetry, namely the cascade speed of the transverse increments is 1.5 times faster than that of the longitudinal increments. Small differences can be found in the skewness and in a higher order intermittency term. The rescaling symmetry is compatible with the Kolmogorov constants and the Kármán equation and gives new insight into the use of extended self-similarity (ESS) for transverse increments. Based on the results we propose an extended self-similarity for the transverse increments (ESST).

Improved estimation of Fokker-Planck equations through optimization

“…It has been shown that for a higher dimensional analysis a corresponding non-vanishing q (2) 1 term violates the second von Kármán equation (see Eqs. (31) and (32) in [25]). In Fig.…”

Atmospheric turbulence and its influence on the alternating loads on wind turbines

“…Note that based on Taylor's hypothesis of frozen turbulence, these temporal correlations correspond to spatial ones. The directional dependency of these correlations may be expressed by a simple rescaling known for local isotropic small‐scale turbulence 7…”

Identification of Markov process within a wind turbine array boundary layer

“…2,6,8 The test begins with computing the total number of inversions, Q, after samples from a ¼ ½a 1 ; a 2 ; …; a m and b ¼ ½b 1 ; b 2 ; …; b n are merged as well as sorted in ascending order. An inversion occurs when adjacent ordered values in the combined matrix change from values provided from a to values provided by b or vice versa.…”

“…One approach to characterize the statistics of small scale turbulence searches for the existence of Markov process. Markov processes have been discovered in several turbulent flows, such as high Reynolds number axisymmetric jet, 7 the wake behind a cylinder, 8 and fractal-generated grid turbulence. 9 In each of these studies, the flows demonstrated Markov process behavior for scales larger than or equal to the Taylor microscale, k, given by k 2 ¼ huðxÞ 2 i=hð @u @x Þ 2 i, where hÁi represents ensemble averaging.…”