The dilemma of turbulence modelling

TL;DR

The paper explores the limitations of turbulence models, showing that simplified models lack uniqueness, which raises questions about their ability to accurately represent turbulent flows and the need for more complex force considerations.

Contribution

It introduces a new construction technique for multiple solutions of the Euler equation, linking them to Navier-Stokes solutions, and discusses the implications for turbulence modeling and solution uniqueness.

Findings

Multiple solutions of Euler equations can be constructed in strong spaces.

Simplified turbulence models often lack solution uniqueness.

Restrictions on external forces are necessary for solution uniqueness.

Abstract

A new construction technique of multiple solutions of the Euler equa- tion in strong spaces is introduced which reveals the relationship to multi- ple Navier Stokes equation solutions with special force terms while avoid- ing viscosity limit constructions. This shows that severe restrictions have to be imposed on time dependent external force terms in order to ob- tain uniqueness for the Navier Stokes equation Cauchy problem. Such restrictions are imposed in the statement of the so-called millenium prob- lem. Minimal turbulence models should arguably incorporate weaker force terms in order to account for boundary conditions and forces. However, we show that models of this type which have been proposed recently, do not have a unique solution. This lack of determinism of mimimal turbu- lence models indicates a dilemma: either models are too simple to capture turbulence but may have unique smooth solutions, or there is a modelling gap as the model does not determine a unique solution.