A polar turbulence invariant map with applicability to realisable machine learning turbulence models

What is it about?

Machine learning is a promising tool for developing improved turbulence models, but realizability, or the requirement that turbulent kinetic energy be non-negative, is traditionally difficult to enforce in a machine learning context. In this paper we introduce a novel invariant map that reformulates realizability in terms of an angle and a normalized anisotropy magnitude. Using the polar invariant map, realizability can be reframed from a set of eigenvalue constraints to a constraint on the anisotropy magnitude. We show that this new formulation provides a foundation for simpler, more efficient approaches to enforcing realizability.

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Photo by CasSa Paintings on Unsplash

Photo by CasSa Paintings on Unsplash

Why is it important?

Previous approaches to realizability in machine learning-based turbulence models have been restricted by the need to calculate the eigenvalues and eigenvectors at prediction time. This has limited models that don’t directly predict in terms of the eigenvalues and eigenvectors to enforce realizability as a post-processing step, or to learn realizability through training. By reformulating realizability as a scaling problem, the polar invariant map not only can make current methods more computationally efficient, but also provides a framework for modifying general machine learning-based turbulence models to implicitly enforce realizability.