Implicit Central Moment Lattice Boltzmann Method for Viscoelastic Flows for a Wide Range of Weissenberg Numbers

Viscoelastic flows are characterized by nonlinear interactions between polymeric viscoelastic stress and the fluid motions. The solution of the viscoelastic stress (VES) tensor, whose behavior is prototypically modeled using the Oldroyd-B model, can encounter numerical stability issues under large disparities in the relaxation time scale of VES and the flow time scales, or the Weissenberg numbers (Wi). We present lattice Boltzmann (LB) schemes that use central moments and multiple relaxation times in their collision steps for the solution of both the VES and the fluid motions. We propose a novel approach to address the stability issue in the computation of VES by introducing a locally implicit formulation to account for the associated stiff source terms based on the velocity gradient tensor in the model using a L-stable and second order method based on the trapezoidal rule in conjunction with the backward difference formula (TR-BDF2) via a Strang splitting around the collision step. Furthermore, we propose time-dependent conditions on the VES at the boundaries that are fully consistent with the underlying model. We demonstrate the validity and robustness of our new formulation for different benchmarks including shear-driven viscoelastic flows for a wide range of Wi.