Viscoelastic Flow through an Axisymmetric Contraction Using the Grid‐by‐Grid Inversion Method

H M Park

doi:10.1155/2012/420189

Abstract

The newly developed algorithm called the grid‐by‐grid inversion method is a very convenient method for converting an existing computer code for Newtonian flow simulations to that for viscoelastic flow simulations. In this method, the hyperbolic constitutive equation is split such that the term for the convective transport of stress tensor is treated as a source which is updated iteratively. This allows the stress tensors at each grid point to be expressed in terms of velocity gradient tensor at the same location, and the set of stress tensor components is found after inverting a small matrix at each grid point. To corroborate the robustness and accuracy of the grid‐by‐grid inversion method, we apply it to the 4 : 1 axisymmetric contraction problem. This algorithm is found to be robust and yields accurate results as compared with other finite volume methods. Any commercial CFD packages for Newtonian flow simulations can be easily converted to those for viscoelastic fluids exploiting the grid‐by‐grid inversion method.

Highlights

Contrary to the techniques of computational fluid dynamics for Newtonian fluids, the numerical algorithms for viscoelastic flows are not so matured
Discretizing 2.4 in time implicitly and representing D in terms of the velocity gradient ∇v, we find the following local matrix equations defined at each grid point
Viscoelastic flows through the contraction generate complex flows exhibiting strong shear and uniaxial extension, which is a good test bed for the robustness and accuracy of a new numerical algorithms

Summary

Introduction

Contrary to the techniques of computational fluid dynamics for Newtonian fluids, the numerical algorithms for viscoelastic flows are not so matured. Since the momentum balance equation is elliptic in steady state and parabolic in unsteady state, the complete set for viscoelastic flows is a mixed type, hyperbolic-elliptic, or hyperbolic-parabolic This situation is difficult to treat numerically since it is difficult to devise a numerical algorithm that works for mixed systems. The six stress tensor components for the cases of a three-dimensional flow are found after inverting a six by six matrix at each grid point and are substituted into the Navier-Stokes equation as a source term In this way, the numerical solution of viscoelastic flows becomes as straightforward as that of Newtonian fluids. The numerical solution of viscoelastic flows becomes as straightforward as that of Newtonian fluids We call this algorithm the grid-by-grid inversion method since the viscoelastic stress tensor is obtained by the grid-by-grid inversion of a matrix equation at each grid point. When applied to the 4 : 1 axisymmetric contraction problem, it is found that the grid-by-grid inversion method yields accurate results efficiently in comparison with numerical results of traditional algorithms

Governing Equations and the Grid-by-Grid Inversion Method

Viscoelastic Flow through an Axisymmetric Contraction

Results

Conclusion