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Properties of two equations describing the evolution of the probabilitydensity function (PDF) of the relative dispersion in turbulent flow arecompared by investigating their solutions: the Richardson diffusion equationwith the drift term and the self-similar telegraph equation derived byOgasawara and Toh [J. Phys. Soc. Jpn. 75, 083401 (2006)]. The solution of theself-similar telegraph equation vanishes at a finite point, which representspersistent separation of a particle pair, while that of the Richardson equationextends infinitely just after the initial time. Each equation has a similaritysolution, which is found to be an asymptotic solution of the initial valueproblem. The time lag has a dominant effect on the relaxation process into thesimilarity solution. The approaching time to the similarity solution can bereduced by advancing the time of the similarity solution appropriately.Batchelor scaling, a scaling law relevant to initial separation, is observedonly for the telegraph case. For both models, we estimate the Richardsonconstant, based on their similarity solutions.

Telegraph-type versus diffusion-type models of turbulent relative dispersion