Fractal Potential Flows: An Idealized Model for Fully Developed Turbulence

TL;DR

This paper introduces a mathematical model called Fractal Potential Flows to represent Fully Developed Turbulence, capturing its self-similar fractal features and providing a new theoretical framework for understanding this complex phenomenon.

Contribution

It presents a novel idealized model based on recursive fluid dynamics and proves the existence of a unique attractor, linking turbulence characteristics to fractal structures and measure theory.

Findings

Existence of a unique invariant flow attractor.

Fractal sink singularities form an IFS fractal, confirming Mandelbrot's Conjecture.

Establishes an isometric isomorphism between flows and probability measures.

Abstract

Fully Developed Turbulence (FDT) is a theoretical asymptotic phenomenon which can only be approximated experimentally or computationally, so its defining characteristics are hypothetical. It is considered to be a chaotic stationary flow field, with self-similar fractalline features. A number of approximate models exist, often exploiting this self-similarity. The idealized mathematical model of Fractal Potential Flows is hereby presented, and linked philosophically to the phenomenon of FDT on a free surface, based on its experimental characteristics. The model hinges on the recursive iteration of a fluid dynamical transfer operator. The existence of its unique attractor - called the invariant flow - is shown in an appropriate function space, which will serve as our suggested model for the FDT flow field. Its sink singularities are shown to form an IFS fractal, explicitly resolving Mandelbrot's Conjecture. Meanwhile an isometric isomorphism is defined between flows and probability measures, hinting at a wealth of future research. The inverse problem of representing turbulent flow fields with this model is discussed in closing, along with explicit practical considerations for experimental verification and visualization.