A theory of turbulence. Part 1:Towards solutions of the Navier-stokes equations

TL;DR

This paper introduces a visualization method for turbulent flows that isolates flow structures, simplifies Navier-Stokes solutions, and challenges traditional turbulence statistics, aiming to improve theoretical understanding and computational modeling.

Contribution

It proposes a new visualization approach, identifies flow domain-specific subsets of Navier-Stokes equations, and introduces a partial derivative to simplify turbulence analysis.

Findings

Reynolds stresses and turbulence statistics can be derived from laminar flow subsets.

The Kolmogorov scale does not necessarily represent eddy size or require isotropy.

A new partial derivative decouples diffusion and convection effects.

Abstract

In this visualisation the instantaneous local velocity is expressed in terms of four components to capture the development of and interactions between coherent structures in turbulent flows. It is then possible to isolate the terms linked with each major type of structure and identify the corresponding subsets of the Navier-Stokes equations that are easier to solve than the full version. Each of these subsets applies to a domain in the flow field not a flow regime. The traditional statistics of turbulence are shown to be not specific to turbulent flow. In particular the Reynolds stresses, the probability density function, the dissipation, the production and the energy spectrum are obtained from one subset associated with unsteady "laminar" flow. The evidence also indicates that the Kolmogorov scale does not represent an eddy size and more importantly that it does not require the assumption of local isotropy. Applications of this visualisation include a proposal for a theoretical closure of the Reynolds equations, the collapse of velocity and skin friction profiles for Newtonian and non-Newtonian fluid flows in all geometries into unique master curves. A new partial derivative is introduced that decouples the effect of diffusion and convection simplifying the mathematical solution of subsets of the NS equations with implications for a more efficient generation of computational fluid dynamics packages. Finally the consequences to turbulent heat and mass transfer are examined.