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.