On the continuity of stochastic models for the Lagrangian velocity in turbulence

Abstract

Inertial sub-range statistics of stochastic models for the Lagrangian velocity in turbulent flow have been examined. For Markovian models it is shown that consistency with Kolmogorov's theory of local isotropy requires that the Lagrangian velocity be a continuous function of time. This limits suitable Markov models to those which can be represented by a stochastic differential equation. Markov models in which the velocity is discontinuous (and a class of non-Markovian jump models) are not consistent with Kolmogorov's theory. Modifications to (Kolmogorov's theory to account for the effects of intermittency are shown to be non-Markovian, but still correspond to a Lagrangian velocity which is continuous. In Gaussian homogeneous turbulence only continuous Markov models predict that the particle displacement is Gaussian. For a Markovian jump model, the particle displacement distribution is leptokurtic with a maximum excess of about 0.67, which is inconsistent with wind tunnel data in grid turbulence.