Geometric decomposition of the conformation tensor in viscoelastic turbulence

TL;DR

This paper presents a novel geometric decomposition method for the conformation tensor in viscoelastic turbulence, enabling physically meaningful analysis of polymer fluctuations and dynamics.

Contribution

It introduces a geometric decomposition that ensures positive-definite tensor fields and provides new scalar measures based on non-Euclidean geometry, improving analysis of polymer behavior.

Findings

The new decomposition yields physically interpretable polymer fluctuation measures.

Application to turbulent channel flow demonstrates the method's effectiveness.

Analysis reveals detailed polymer deformation dynamics in turbulence.

Abstract

This work introduces a mathematical approach to analysing the polymer dynamics in turbulent viscoelastic flows that uses a new geometric decomposition of the conformation tensor, along with associated scalar measures of the polymer fluctuations. The approach circumvents an inherent difficulty in traditional Reynolds decompositions of the conformation tensor: the fluctuating tensor fields are not positive-definite and so do not retain the physical meaning of the tensor. The geometric decomposition of the conformation tensor yields both mean and fluctuating tensor fields that are positive-definite. The fluctuating tensor in the present decomposition has a clear physical interpretation as a polymer deformation relative to the mean configuration. Scalar measures of this fluctuating conformation tensor are developed based on the non-Euclidean geometry of the set of positive-definite tensors. Drag-reduced viscoelastic turbulent channel flow is then used an example case study. The conformation tensor field, obtained using direct numerical simulations, is analysed using the proposed framework.