Temporal Artificial Stress Diffusion for Numerical Simulations of Oldroyd-B Fluid Flow

Marília Pires,Tomáš Bodnár

doi:10.3390/math10030404

Abstract

This paper presents a numerical evaluation of two different artificial stress diffusion techniques for the stabilization of viscoelastic Oldroyd-B fluid flows at high Weissenberg numbers. The standard artificial diffusion in the form of a Laplacian of the extra stress tensor is compared with a newly proposed approach using a discrete time derivative of the Laplacian of the extra stress tensor. Both methods are implemented in a finite element code and demonstrated in the solution of a viscoelastic fluid flow in a two-dimensional corrugated channel for a range of Weissenberg numbers. The numerical simulations have shown that this new temporal stress diffusion not only efficiently stabilizes numerical simulations, but also vanishes when the solution reaches a steady state. It is demonstrated that in contrast to the standard tensorial diffusion, the temporal artificial stress diffusion does not affect the final solution.

Highlights

The mathematical modeling and numerical simulation of non-Newtonian fluid flows became one of the fastest developing and most challenging problems of contemporary computational fluid dynamics
The results presented are based on a comprehensive set of new numerical simulations performed to verify and support the conclusions of this paper
The viscoelastic fluid flow through the 2D corrugated channel was solved for a range of Weissenberg numbers at a single fixed Reynolds number

Summary

Introduction

The mathematical modeling and numerical simulation of non-Newtonian fluid flows became one of the fastest developing and most challenging problems of contemporary computational fluid dynamics. It has numerous physical and technical applications of high interest. The most distinct property of viscoelastic fluids is the ability to store and recover mechanical energy. Such fluids, including blood, can be mathematically described by the class of tensorial constitutive relations, from which the Oldroyd-B model (possibly generalized) is one of the simplest and most often used [1,2]

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