Generalized Fractal Dimension for a Dissipative Multi-fractal Cascade Model for Fully Developed Turbulence

In this paper (Shivamoggi et al.), we explore a variant for the simple model based on a binomial multiplicative process of Meneveau and Sreenivasan that mimics the multi-fractal nature of the energy dissipation field in the inertial range of fully developed turbulence (FDT), and uses the generalized fractal dimension (GFD) prescription of Hentschel and Proccacia, Halsey et al. However, the presence of an even infinitesimal dissipation in the inertial range is shown to lead to a singularity in the GFD $D_q$ (at $q=1$) of the energy dissipation field and leads to a breakdown of the Meneveau-Sreenivasan binomial multiplicative formulation for a dissipative inertial cascade. The purpose of this paper is to demonstrate that this can be resolved by introducing a new appropriate ansatz for the definition of the GFD $D_q$ to incorporate the effect of a scale-invariant dissipation via a phenomenological dissipative parameter $K$ $( 0 < K < 1)$. This dissipation parameter is also shown to cause a steeper energy spectrum in the inertial range, as to be expected. This ansatz is then generalized to incorporate a more symmetric dissipation via two dissipative parameters $K_1,$ and $K_2$ $( 0 < K_1$ and $K_2 < 1)$