Markovian properties of passive scalar increments in grid-generated turbulence

Abstract

Recent research (Renner, Peinke and Friedrich 2001 J. Fluid Mech. 433 383) has shown that the statistics of velocity increments in a turbulent jet exhibit Markovian properties for scales of size greater than the Taylor microscale, λ. In addition, it was shown that the probability density functions (PDFs) of the velocity increments, v (r), were governed by a Fokker–Planck equation. Such properties for passive scalar increments have never been tested. The present work studies the (velocity and) temperature field in grid-generated wind tunnel turbulence for Taylor-microscale-based Reynolds numbers in the range 140⩽Rλ⩽582. Increments of longitudinal velocity were found to (i) exhibit Markovian properties for separations r⪆λ and (ii) be describable by a Fokker–Planck equation because terms in the Kramers–Moyal expansion of order >2 were small. Although the passive scalar increments, δ(r), also exhibited Markovian properties for a similar range of scales as the velocity field, the higher-order terms in the Kramers–Moyal expansion were found to be non-negligible at all Reynolds numbers, thus precluding the PDFs of δ(r) from being described by a Fokker–Planck equation. Such a result indicates that the scalar field is less Markovian than the velocity field—an attribute presumably related to the higher level of internal intermittency associated with passive scalars.