KS input spectrum, some fundamental works on the vibration spectrum of a self-similar linear chain

Abstract

The turbulence energy spectrum is a significant input parameter of KS modeling. In parallel to KS, fractal approaches have been developed in fluid mechanics (often by the same team of researchers doing KS) either experimentally or numerically to interfere with spectral law. We add another direction of research to this interesting problem by looking at what analytical mechanics can teach us about the vibration spectrum of a self similar chain hoping that one day that knowledge will help our understanding of spectral laws and fractal forcing in fluid mechanics. We consider some general aspects of the construction of self-similar functions and linear operators and deduce a self-similar variant of the Laplacian operator and of the d’Alembertian wave operator. The derived self-similar wave operator describes the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We also show that the self-similar wave equation in a certain approximation corresponds to (nonlocal) fractional derivatives.