Lagrangian velocity autocorrelation in isotropic turbulence

An analysis is made of the Lagrangian and the Eulerian velocity autocorrelations RL(s,t) and RE(s,t) at times s and t for small time lag τ=s−t in isotropic decaying turbulence, and it is shown that under appropriate conditions RE≤RL for small τ in three dimensions, irrespective of the form of the energy spectrum. The behaviors of RE and RL not limited to small τ are studied by direct numerical simulations in two dimensions. The values of RL obtained by the Lagrangian renormalized approximation are found to be in good agreement with the simulations and with the existing data of three-dimensional simulations and experiments.