Different methods to estimate the Einstein-Markov coherence length in turbulence.

: We study the Markov property of experimental velocity data of different homogeneous isotropic turbulent flows. In particular, we examine the stochastic "cascade" process of nested velocity increments ξ(r):=u(x+r)-u(x) as a function of scale r for different nesting structures. It was found in previous work that, for a certain nesting structure, the stochastic process of ξ(r) has the Markov property for step sizes larger than the so-called Einstein-Markov coherence length l(EM), which is of the order of magnitude of the Taylor microscale λ [Phys. Lett. A 359, 335 (2006)]. We now show that, if a reasonable definition of the effective step size of the process is applied, this result holds independently of the nesting structure. Furthermore, we analyze the stochastic process of the velocity u as a function of the spatial position x. Although this process does not have the exact Markov property, a characteristic length scale l(u(x))≈l(EM) can be identified on the basis of a statistical test for the Markov property. Using a method based on the matrix of transition probabilities, we examine the significance of the non-Markovian character of the velocity u(x) for the statistical properties of turbulence.