# The mean conformation tensor in viscoelastic turbulence

## TL;DR

This paper introduces geometric and log-Euclidean means as better alternatives to the arithmetic mean for conformation tensors in turbulent viscoelastic flows, aligning with the tensor manifold's structure and better representing ensemble behavior.

## Contribution

It proposes and validates new mean definitions based on Riemannian geometry that are more physically meaningful for analyzing turbulent viscoelastic flows.

## Key findings

- Geometric and log-Euclidean means better represent the ensemble of tensors.
- These means preserve scalar geometric means of volume and stretches.
- They are physically interpretable and useful for turbulence modeling.

## Abstract

This work demonstrates that the popular arithmetic mean conformation tensor frequently used in the analysis of turbulent viscoelastic flows is not a good representative of the ensemble. Alternative means based on recent developments in the literature are proposed, namely, the geometric and log-Euclidean means. These means are mathematically consistent with the Riemannian structure of the manifold of positive-definite tensors, on which the conformation tensor lives, and have useful properties that make them attractive alternatives to the arithmetic mean. Using a turbulent FENE-P channel flow dataset, it is shown that these two alternatives are physically representative of the ensemble. By definition these means minimize the geodesic distance to realizations and exactly preserve the scalar geometric mean of the volume and of the principal stretches. The proposed geometric and log-Euclidean means have a clear physical interpretation and provide an attractive quantity for turbulence modelling.