On a mathematical definition of laminar and turbulent fluid flow
F. Javier Garcia Garcia, Pablo Fariñas Alvariño
TL;DR
This paper introduces a rigorous mathematical framework to distinguish laminar and turbulent fluid flows by analyzing ensembles of nearly-identical realizations and their ensemble-averaged (mean) fields. Laminar flow is characterized by a mean field that is a quasi-solution to the Navier–Stokes equations, arising when nonlinearity is restricted, so realizations are repeatable; turbulence results when general nonlinearity drives non-repeatable realizations and the mean diverges from any single realization. The authors formalize proximity notions, construct physical function classes of range $\epsilon$, define ensembles, Reynolds decomposition, and Reynolds-averaged equations, and discuss the local geometry of the flow manifold and steady-state behavior. A practical finite-N method based on a Cauchy criterion is provided to estimate the required ensemble size. The framework offers a constructive, non-probabilistic route to classify flows and has potential to underpin rigorous analytical results and transitions, with extensions to compressible flows and additional appends like solved examples.
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 Kàrmàn vortex street and a laminar flow with imposed white-noise perturbation.