A filamentary cascade model of the inertial range

TL;DR

This paper introduces a novel filamentary cascade model for the inertial range in turbulence, featuring a bimodal splitting process that aligns with observed structure exponents and extends to energy decay in the Navier-Stokes limit.

Contribution

It proposes a new bimodal cascade model with two simultaneous splitting processes, differing from traditional models, and demonstrates its consistency with turbulence scaling laws.

Findings

The model reproduces observed structure exponents $z_p$.

It explains the nonlinear dependence of $z_p$ on $p$.

The cascade continues indefinitely, modeling energy decay in vanishing viscosity limit.

Abstract

This paper develops a simple model of the inertial range of turbulent flow, based on a cascade of vortical filaments. 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 these cascades has the Richardson-Kolmogorov exponent of 1/3. This bimodal structure is also different from bifractal models as vorticity volume is conserved. 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 $\zeta_p$. The presence of two elements to the cascade is responsible for the nonlinear dependence of $\zeta_p$ upon $p$. A single 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.