A Langevin equation for turbulent velocity increments
Résumé
Recently, Friedrich and Peinke demonstrated empirically that a Fokker–Planck equation describes
the scale dependence of probability distribution functions of longitudinal velocity increments vr in
fully developed turbulent flows. Thanks to the analysis of an experimental velocity signal, the
stochastic process vr is further investigated by examining the related Langevin equation. This
process is found to be Markovian in scale because the turbulent velocity field is correlated over
distances much larger than the correlation length r of its spatial derivative. A Gaussian
approximation for the random force yields evolution equations for the structure functions ^vr
n&.
Analytic solutions are obtained, in agreement with experimental data for even-order moments when
the scale r is larger than a few times r. The third-order moment ^vr
3& is found linear in r, as predicted
by Kolmogorov’s four-fifths law.
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