  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.
