| We investigate a stochastic model for homogeneous, isotropic turbulence based on Hill's spherical vortex. This is an extension of the original work done by Synge and Lin in the early 1940's. The spherical vortex is an inviscid, steady, rotational solution to Euler's equation and its structure makes it a natural candidate for a model of a turbulent eddy. The model assumes that a field of locally isotropic turbulence is generated by a homogeneous distribution of Hill's vortices. The cascade process of eddy breakdown is incorporated into the statistical model through an average over vortex size. Dissipation field characteristics are assumed for the vortex size distribution. We are interested in the statistical properties of the model, in particular order-n structure functions defined by rank-n tensors for the ensemble average of a set of incremental differences in a given field property. We are primarily concerned with the second order pressure structure function, , and the velocity structure functions, orders 2–6, where and ⟨…⟩ denotes the ensemble average.; Specifically Dp, Dij, Dijkl, and the longitudinal component of Dijklmn are calculated. Comparing Dij and Dijkl with experimental curves fixes the two independent model parameters while the sixth-order longitudinal velocity structure function is a predicted result. The pressure structure function is calculated both from first principles and directly from its integral relation with the fourth-order velocity structure function and compared with direct numerical simulation. |
