On velocity structure functions and the spherical vortex model for isotropic turbulence

Abstract

We investigate a stochastic model for homogeneous, isotropic turbulence based on Hill's spherical vortex. This is an extension of the method of Synge and Lin [Trans. R. Soc. Can. 37, 45 (1943)], to the calculation of higher even-order velocity structure functions. Isotropic turbulence is represented by a homogeneous distribution of eddies, each modeled by a spherical vortex. The cascade process of eddy breakdown is incorporated into the statistical model through an average over an assumed log-normal distribution of vortex radii. We calculate the statistical properties of the model, in particular order-n velocity structure functions defined by rank-n tensors for the ensemble average of a set of incremental differences in velocity components. We define Di[centered ellipsis]s = <(ui(x + xi )–ui(x))[centered ellipsis](us(x + xi )–us(x))>, where <[centered ellipsis]> denotes the ensemble average. Specifically Dij, Dijkl, and the longitudinal component of Dijklmn are calculated directly from the spherical vortex ensemble. Matching the longitudinal components of Dij and Dijkl with experimental results fixes two independent model parameters. The lateral and mixed components of Dijkl and the longitudinal component of Dijklmn are then model predictions.