A lattice Boltzmann flux solver for numerical simulation of flows with the viscoelastic fluid

Abstract

In this paper, a viscoelastic lattice Boltzmann flux solver (VLBFS) is developed to simulate incompressible flows of viscoelastic fluids with linear and non-linear constitutive models. In this method, the macroscopic equations are solved by the finite volume method, where the fluxes at the cell interface are evaluated by local reconstruction of the solutions of lattice Boltzmann equations (LBE). Two sets of distribution functions are introduced to reconstruct the cell-interface fluxes, one used for mass and momentum fluxes and the other for the conformation tensor flux in the polymer constitutive equation. The elastic-viscous stress splitting (EVSS) and the solvent-polymer stress splitting (SPSS) techniques are incorporated into the present LBFS to improve the numerical stability. The standard lattice Boltzmann method (LBM) for solving the polymer constitutive equation contains redundant diffusion terms, but this problem is resolved in the current LBFS by setting the relaxation time corresponding to the true diffusion-free limit thus the correct polymer constitutive equation can be recovered. Furthermore, VLBFS eliminates other disadvantages of the standard LBM, such as the LBM on-grid advection coupling the time interval with grid spacing, complicated treatment of the mesoscopic boundary conditions, dependence on uniform grids, and the larger memory requirement due to solving the phase-space discrete distributions. Several flows of a viscoelastic fluid, namely, the two-dimensional plane Poiseuille flow, two-dimensional simplified four-roll mill flows, and three-dimensional Taylor–Green vortex flows, are considered to investigate the accuracy and stability of the present method. The results are found to be in good agreement with the analytical solutions and the previous numerical results. Numerical error analyses show that the present method owns a second-order accuracy in space. The developed VLBFS extends the application domain of LBFS and serves as a basis for simulating viscoelastic flows at high Weissenberg numbers.