Investigation of the FENE-L and FENE-LS constitutive models for the computation of viscoelastic turbulent flows

The viscoelastic nature of dilute polymer solutions can induce peculiar macroscopic effects, such as polymer drag reduction (PDR) in turbulent flows, elastic turbulence (ET) or elasto-inertial turbulence (EIT). Numerical simulations of such phenomena typically rely on the FENE-P model to represent the polymer dynamics. Fundamentally, the FENE-P model derives from the FENE model, that represents a polymer chain as a nonlinear dumbbell, i.e., as two beads connected by a Warner spring. While the nonlinearity of the spring ensures a finite extensibility of the polymer chain, in contrast to the Oldroyd-B model, it prevents the derivation of a constitutive equation in closed form. The FENE-P model thus relies on Peterlin’s approximation (pre-averaging) for closure. Although it can qualitatively represent most of the FENE dynamics, quantitative differences can be observed between the two models (e.g., stress overshoot in inception of shear flow). Another shortcoming of the FENE-P model is that it cannot predict the hysteresis phenomenon observed between the polymer stress and the conformation tensor. To address this latter limitation, more elaborate models have been proposed; this work focuses on the FENE-L and FENE-LS models. These two models rely on a simplified (decoupling of orientation and extension) and parametrized representation of the underlying probability distribution of the polymer end-to-end vector. The average stress can then be expressed in terms of higher moments of the distribution, for which transport equations can be derived. The use of more complex probability distributions (the FENE-P corresponding to a simple Dirac delta function) allows predicting the hysteresis of the FENE model. The first question that the present work aims to address is thus whether this hysteresis plays an important role in the PDR or EIT phenomena. Although the center-of-mass diffusion is often neglected in fundamental polymer models owing to the very low diffusivity of polymer chains, a certain level of (artificial) diffusion is usually still needed to stabilize Eulerian simulations of polymeric flows. In practice, an additional diffusion term is typically added in the transport equation for the conformation tensor. While this is straightforward for the FENE-P model, it is shown in this work that the same approach can lead the transported moments to leave the space of admissible solutions in the case of the FENE-L(S). The second question that this work addresses is thus how to best include diffusion in the FENE-L(S) models. In particular, an alternative approach in which diffusion acts on the parameters of the probability distribution and not on the transported moments is investigated. These two questions are discussed based on i) comparisons between FENE (Brownian dynamics), FENE-P and FENE-L(S) semi-coupled (passive polymers) Lagrangian simulations along trajectories in a Newtonian turbulent channel flow, and ii) comparison between FENE-P and FENE-L(S) coupled (active polymers) Eulerian simulations of canonical flows.