A filamentary cascade model of the inertial range

Abstract

This paper develops a simple model of the inertial range of turbulent flow, based on a cascade of vortical filaments. The filaments are taken to be helical, one turn of the helix playing the role of a turbulent eddy. A binary branching structure is proposed, involving the splitting of filaments at each step into pairs of daughter filaments with differing properties, in effect two distinct simultaneous cascades. Neither of the cascades of this bimodal structure, in isolation, has the Richardson exponent of 1/3. If cascades are assumed to be initiated continuously and throughout space we obtain a model of the inertial range of stationary turbulence. We impose the constraint associated with Kolmogorov’s four-fifths law and then adjust the splitting to achieve good agreement with the observed structure exponents ζp. The presence of two elements to the cascade is responsible for the nonlinear dependence of ζp upon p. The proposed binary branching cascade provides a model for the initial-value problem of the Navier–Stokes equations in the limit of vanishing viscosity. To simulate this limit we let the cascade continue indefinitely, energy removal occurring in the limit. We are thus able to compute the decay of energy in the model.