Taylor expansions in powers of time of Lagrangian and Eulerian two-point two-time velocity correlations in turbulence

Abstract

A method is developed for generating the Taylor expansions in powers of the time difference of the Lagrangian and Eulerian two-point two-time velocity correlations in turbulence. The expansions are based on the Taylor series of the Eulerian and Lagrangian velocity fields subject to given dynamics along with initial and boundary conditions. The lowest few coefficients in the expansions enable us to construct approximations to the correlations. An application of the method to turbulence obeying the Navier-Stokes dynamics yields approximations, particularly Padé approximations that agree well with direct numerical simulations of homogeneous isotropic turbulence at moderate Reynolds numbers. The ratios of the second-order to the zeroth-order coefficients of the Taylor series of the Lagrangian and Eulerian correlations give, respectively, the estimates for the Lagrangian and Eulerian micro time scales τ L and τ E . An analysis of a high resolution ( 512 3 grid points) direct numerical simulation database at large Reynolds number suggests the scalings τ L ∝k −2/3 and τ E ∝k −1 for wave numbers k in the inertial subrange. The role of flow structures in turbulence in determining the time scales is also discussed.