Lagrangian velocity covariance in helical turbulence

Abstract

The effect of helicity on the Lagrangian velocity covariance U^L(t) in isotropic, normally distributed turbulence is examined by computer simulation and by a renormalized perturbation expansion for U^L(t). The first term of the latter represents Corrsin's (1959) conjecture (extrapolated to all t), which relates U^L(t) to the Eulerian covariance and the distribution G(x, t) of fluid-element displacement. Truncation of the expansion at the first term yields the direct-interaction approximation for G(x, t). The expansion suggests that with or without helicity Corrsin's conjecture is valid as t → ∞ and that in either case U^L(t) behaves asymptotically like $t^{-(r+\frac{3}{2})}$ if the spectrum of the Eulerian field varies like k^r+2 at small wavenumbers. Corrsin's conjecture breaks down at small and moderate t if there is strong helicity while remaining accurate at all t in the mirror-symmetric case. Computer simulations for a frozen Eulerian field with spectrum confined to a thin spherical shell in k space indicate that strong helicity induces an increase in the Lagrangian correlation time by a factor of approximately three. Direct-interaction equations are constructed for the Lagrangian space-time covariance and the resulting prediction for U^L(t) is compared with the simulations. The effect of helicity is well represented quantitatively by the direct-interaction equations for small and moderate t but not for large t. These frozen-field results imply good quantitative accuracy at all t in time-varying turbulence whose Eulerian correlation time is of the order of the eddy-circulation time. In turbulence with weak helicity, the directinteraction equations imply that the Lagrangian correlation of vorticity with initial velocity is more persistent than U^L(t), by a substantial factor.