On a mathematical definition of laminar and turbulent fluid flow

TL;DR

This paper offers a rigorous mathematical definition of laminar and turbulent flows based on their repeatability and nonlinearity, providing a foundation for formal theorems and clarifying the nature of turbulence.

Contribution

It introduces a novel mathematical framework for defining turbulent flows through experimental repeatability and nonlinearity distinctions, enabling formal analysis.

Findings

Turbulent flows are inherently non-repeatable and involve general nonlinearity.

Laminar flows only exist under conditions of restricted nonlinearity.

The paper presents examples illustrating the definitions and methods.

Abstract

As stated in the title, the present research proposes a mathematical definition of laminar and turbulent flows, i.e., a definition that may be used to conceive and prove mathematical theorems about such flows. The definition is based on an experimental truth long known to humans: Whenever one repeats a given flow, the results will not be the same if the flow is turbulent. Turbulent flows are not strictly repeatable. From this basic fact follows a more elaborate truth about turbulent flows: The mean flow obtained by averaging the results of a large number of repetitions is not a natural flow, that is, it is a flow that cannot occur naturally in any experiment. The proposed definition requires some preliminary mathematical notions, which are also introduced in the text: Proximity between functions, the ensemble of realisations, the method of averaging the flows, and the distinct properties of realisations (physical flows) and averages (mean flows). The notion of restricted nonlinearity is introduced and it is demonstrated that laminar flows can only exist in conditions of restricted nonlinearity, whereas turbulent flows are a consequence of general nonlinearity. The particular case of steady-state turbulent flow is studied, and an uncertainty is raised about the equality of ensemble average and time average. Two solved examples are also offered to illustrate the meaning and methods implied by the definitions: The von K\`arm\`an vortex street and a laminar flow with imposed white-noise perturbation.