A Direct Effective Viscosity Approach for Modeling and Simulating Bingham Fluids with the Cumulant Lattice Boltzmann Method

Martin Geier,Manfred Krafczyk,Konstantin Kutscher

doi:10.4236/ojfd.2021.111003

Abstract

Modeling of fluids with complex rheology in the lattice Boltzmann method (LBM) is typically realized through the introduction of an effective viscosity. For fluids with a yield stress behavior, such as so-called Bingham fluids, the effective viscosity has a singularity for low shear rates and may become negative. This is typically avoided by regularization such as Papanastasiou’s method. Here we argue that the effective viscosity model can be re-interpreted as a generalized equilibrium in which no violation of the stability constraint is observed. We implement a Bingham fluid model in a three-dimensional cumulant lattice Boltzmann framework and compare the direct analytic effective viscosity/generalized equilibrium method to the iterative approach first introduced by Vikhansky which avoids the singularity in viscosity that can arise in the analytic method. We find that both methods obtain similar results at coarse resolutions. However, at higher resolutions the accuracy of the regularized method levels off while the accuracy of the direct method continuously improves. We find that the accuracy of the proposed direct method is not limited by the singularity in viscosity indicating that a regularization is not strictly necessary.

Highlights

A popular way to model the stress of non-Newtonian fluids is by imposing an effective local viscosity
We implement a Bingham fluid model in a three-dimensional cumulant lattice Boltzmann framework and compare the direct analytic effective viscosity/generalized equilibrium method to the iterative approach first introduced by Vikhansky which avoids the singularity in viscosity that can arise in the analytic method
In this paper we present an implementation of the Bingham fluid in the context of the cumulant lattice Boltzmann method (LBM)

Summary

Introduction

A popular way to model the stress of non-Newtonian fluids is by imposing an effective local viscosity. Implementing non-Newtonian fluids in Navier-Stokes solvers either requires storing the stress tensor explicitly or applying the chain rule to the effective viscosity field which can result in rather complicated differential operators. Implementing non-Newtonian behavior through an effective viscosity comes naturally in the LBM with all differential operators applied in a consistent order. Many substances of industrial interest are subject to non-Newtonian stress-strain relationships. One such example is fresh concrete [1]. Among the many properties of freshly mixed concrete, the yield stress behavior is of particular importance in additive manufacturing processes which highly depends on a controlled transition from the fluid to the solid phase

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