A Langevin equation for turbulent velocity increments

International audienceRecently, Friedrich and Peinke demonstrated empirically that a Fokker–Planck equation describesthe scale dependence of probability distribution functions of longitudinal velocity increments vr infully developed turbulent flows. Thanks to the analysis of an experimental velocity signal, thestochastic process vr is further investigated by examining the related Langevin equation. Thisprocess is found to be Markovian in scale because the turbulent velocity field is correlated overdistances much larger than the correlation length r of its spatial derivative. A Gaussianapproximation for the random force yields evolution equations for the structure functions ^vrn&.Analytic solutions are obtained, in agreement with experimental data for even-order moments whenthe scale r is larger than a few times r. The third-order moment ^vr3& is found linear in r, as predictedby Kolmogorov’s four-fifths law