Kármán–Howarth theorem for the Lagrangian-averaged Navier–Stokes–alpha model of turbulence

Abstract

The Lagrangian averaged Navier–Stokes–alpha (LANS-α) model of turbulence is found to possess a Kármán–Howarth (KH) theorem for the dynamics of its second-order autocorrelation functions in homogeneous isotropic turbulence. This KH result implies that alpha-filtering in the LANS-α model of turbulence does not affect the exact Navier–Stokes relation between second and third moments at separation distances large compared to the model's length scale α. Moreover, at separations r that are smaller than α, the KH scaling between energy dissipation rate and longitudinal third-order autocorrelation changes to match the scaling found in two-dimensional incompressible flow. This is consistent with the corresponding change in scaling of the kinetic energy spectrum from k−5/3 for larger scales with kα < 1, which switches to k−3 for smaller scales with kα > 1, as discovered in Foias, Holm & Titi (2001).