Lagrangian velocity covariance in helical turbulence

Abstract

The effect of helicity on the Lagrangian velocity covarianceUL(t) in isotropic, normally distributed turbulence is examined by computer simulation and by a renormalized perturbation expansion forUL(t). The first term of the latter represents Corrsin's (1959) conjecture (extrapolated to allt), which relatesUL(t) to the Eulerian covariance and the distributionG(x, t) of fluid-element displacement. Truncation of the expansion at the first term yields the direct-interaction approximation forG(x, t). The expansion suggests that with or without helicity Corrsin's conjecture is valid ast→ ∞ and that in either caseUL(t) behaves asymptotically like$t^{-(r+\frac{3}{2})}$if the spectrum of the Eulerian field varies likekr+2at small wavenumbers. Corrsin's conjecture breaks down at small and moderatetif there is strong helicity while remaining accurate at alltin the mirror-symmetric case. Computer simulations for a frozen Eulerian field with spectrum confined to a thin spherical shell inkspace 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 forUL(t) is compared with the simulations. The effect of helicity is well represented quantitatively by the direct-interaction equations for small and moderatetbut not for larget. These frozen-field results imply good quantitative accuracy at alltin 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 thanUL(t), by a substantial factor.