A Lattice Boltzmann Method and Asynchronous Model Coupling for Viscoelastic Fluids

Abstract

The numerical algorithms of viscoelastic flows can appear a tremendous challenge as the Weissenberg number (Wi) enlarged sufficiently. In this study, we present a generalized technique of time-stably advancing based on the coupled lattice Boltzmann method, in order to improve the numerical stability of simulations at a high Wi number. The mathematical models of viscoelastic fluids include both the equation of the solvent and the Oldroyd-B constitutive equation of the polymer. In the two-dimensional (2D) channel flow, the coupled method shows good agreements between the corresponding exact results and the numerical results obtained by our method. In addition, as the Wi number increased, for the viscoelastic flows through contractions, we show that the prediction of our presented method can reproduce the same numerical results that were reported by previous studies. The main advantage of current method is that it can be applied to simulate the complex phenomena of the viscoelastic fluids.lattice Boltzmann method have been devised to solve the equations of an Oldroyd-B viscoelastic fluid in the reference [21], which did not need to adding extra distribution functions for the stress tensor. The coupled systems of the viscoelastic fluid include the macro scale constitutive equation and the mesoscopic lattice Boltzmann equation. There is a more disparity in timescales between the macro solver and meso solver. To distinguish this different time scale and enhance the computational efficiency, the asynchronously coupled method [22,23] was applied for the coupling time in reference [21]. That means, the data between macro constitutive equation and the lattice Boltzmann equation exchange at every time step. The constitutive solver may run at a slower pace than the meso-scale dynamics.The asynchronous coupling method allows for the coupled solvers to exchange the data at every loop, although the meso and macro time steps are unequal. This special treatment is actually a numerical approximation to the true solution. For viscoelastic fluids at low Wi number, asynchronously coupled method may get steady result in simulations if the macro time step can guarantee the adequate resolutions of each solvers because of the small scale-separation. However, for viscoelastic fluids at high Wi number, the computational stability of asynchronously coupling is greatly affected by the polymer stress as the elasticity interaction becomes relatively large. Therefore, the asynchronously coupling method has the limitations, such as poor numerical stability, which were limited to solve the low Wi number problem.The aim of the current paper is to develop a generalized technique for time advancing of the viscoelastic coupled lattice Boltzmann method, seeking to be immune to high Wi number problem. The basic idea of our new method is as follows. (i) Run the lattice Boltzmann solver using its own time step δt, while allowing for a flexible number of lattice Boltzmann solver steps N before the exchange of coupling variables; (ii) Run the constitutive ...