Experimental indications for Markov properties of small-scale turbulence

Abstract

We present a stochastic analysis of a data set consisting of 1.25 × 10⁷ samples of the local velocity measured in the turbulent region of a round free jet. We find evidence that the statistics of the longitudinal velocity increment v(r) can be described as a Markov process. This new approach to characterize small-scale turbulence leads to a Fokker–Planck equation for the r-evolution of the probability density function (p.d.f.) of v(r). This equation for p(v, r) is completely determined by two coefficients D₁(v, r) and D₂(v, r) (drift and diffusion coefficient, respectively). It is shown how these coefficients can be estimated directly from the experimental data without using any assumptions or models for the underlying stochastic process. The solutions of the resulting Fokker–Planck equation are compared with experimentally determined probability density functions. It is shown that the Fokker–Planck equation describes the measured p.d.f.(s) correctly, including intermittency effects. Furthermore, knowledge of the Fokker–Planck equation also allows the joint probability density of N increments on N different scales p(v₁, r₁, …, v_N, r_N) to be determined.