Assessment of polymer feedback coupling approaches in simulation of viscoelastic fluids using the lattice Boltzmann method

The lattice Boltzmann method (LBM) has emerged as a popular approach to simulate complex non-Newtonian fluid mechanics problems, such as those involving viscoelastic fluids. In these cases, the LBM solver often requires modelling additional non-linear and non-ideal hydrodynamic contributions to the flow resulting from the addition of polymer molecules. These additional contributions, commonly referred to as the polymer feedback, are either incorporated directly in the Boltzmann stress (pressure method) or alternatively in the momentum field as a forcing contribution (forcing method). The choice of coupling approach has in previous works been strongly speculated to impact the numerical accuracy and stability of LBM solvers for viscoelastic fluids, however no formal comparison has been conducted. In this work, we compare the two popular polymer feedback coupling approaches, as well as the various schemes commonly used in the forcing method, namely, the explicit forcing (EF) scheme (He et al., 1998), the Guo et al. scheme (Guo et al., 2002), and the exact-difference-method (EDM) (Kupershtokh et al., 2009). First, through a theoretical comparison in recovering the macroscopic equations, we show that both the EF and Guo et al. forcing schemes retain the exact hydrodynamic representation up to second-order, whereas the EDM scheme includes additional lattice error terms. Similarly, the pressure method retains additional unphysical terms, which are known to violate Galilean invariance. Through numerical tests, involving the steady and time-dependent Poiseuille flow cases, as well as the four-roll mill problem under both steady and transient regimes, it is shown that the forcing method has enhanced numerical accuracy and stability at varying elastic effects. Alternatively, owing to the violation of Galilean invariance, the pressure method proved to be unstable even for moderate elastic effects. The obtained results are useful in determining the optimal polymer feedback coupling approach when simulating viscoelastic fluids using LBM, as well as further testing the generality of previous LBM findings on non-ideal coupling approaches for viscoelastic fluids.