(98be) Simulation Of Viscoelastic Flows In Complex Geometries At High Elasticities

Simulation of viscoelastic flows in
complex geometries at high elasticities

Florian Habla¹, Johannes Hasslberger¹, Laura J. Dietsche²
and Olaf Hinrichsen¹

¹Catalysis Research Center and Chemistry
Department, Lichtenbergstraße 4, D-85748 Garching b. München, Germany

²The Dow Chemical Company, Midland, MI
48674, USA

Introduction

Viscoelastic
flows can be found in a wide range of industrial applications such as food
processing, injection molding or polymer blending. It is of major importance to
understand and predict such type of flows.

Flows
of viscoelastic fluids are very complex in nature and often result in complex
secondary flows and transient flow patterns even in very simple geometries.
Thus, a numerical method for viscoelastic flows must necessarily be transient
and three-dimensional. Due to that, finite volume methods are becoming more and
more popular in the field of numerical rheology due to their efficiency in time
and memory savings.

Our
research focuses on developing a comprehensive finite-volume CFD method for the
open-source software OpenFOAM® to simulate complex viscoelastic flows including
multi-phase flows^1-2 and non-isothermal flows³. In this
work, we will provide an overview of our developments and show simulation
results for various complex problems.

Fig. 1: Three-dimensional simulation of the flow of
a simplified Phan-Thien-Tanner fluid through a 4:1 square-square contraction at
Wi = 500. Streamlines are colored with the axial normal stress.

Simulation of single-phase flows

The high Weissenberg number problem (HWNP) is probably the central
problem in the numerical simulation of viscoelastic flows for over three
decades now. The Weissenberg number is a measure of elasticity of the flow. In
critical regions of the domain stresses show exponential profiles. Due to the
deficiency of approximating these profiles with polynomial-based discretization
schemes, a recent contribution by Fattal and Kupferman⁴ suggests to
logarithmize the constitutive equation to overcome this issue. Use of this
technique perceivably extenuates the HWNP, however, the problem still remains
at hand.

Additionally to using that method, we developed a reformulation,
which allows to handle the constitutive equation semi-implicitly, which
strengthens the diagonal-dominance of the matrix and thereby strongly increases
the stability of our numerical method⁵. In Fig. 1 the flow of a
viscoelastic fluid through a three-dimensional contraction at a very large Weissenberg
number is shown proofing the robustness of our method.

Simulation of multi-phase flows

In many applications multi-phase flows of viscoelastic fluids are
present such as polymer blending or extrusion processing. In Fig. 2 a fully
three-dimensional simulation of a droplet deforming in a viscoelastic matrix
due to an external shear flow is shown using a highly accurate two-phase model
derived from conditional volume averaging². Validation of our method
with experimental data shows generally good agreement.

Fig. 2: Three-dimensional simulation of a droplet
deforming due to a shear flow in a viscoelastic matrix with adaptive mesh
refinement of the interfacial region.

Besides the conditionally volume-averaged two-phase model, another
method to simulate multi-phase flows of viscoelastic fluids based on the
well-known volume-of-fluid (VoF) approach was implemented. The method was
validated with a number of different flows such as the die-swell effect, the
Weissenberg effect and droplet deformations¹. In Fig. 3 the VoF
model is used in to simulate the flow in a partially filled twin-screw
extruder. The complex movement of the screws is coped with a transient Immersed
Boundary Method (IBM).

Fig. 3: Two-dimensional simulation of a partially
filled twin-screw extruder using the Immersed Boundary Method (IBM). The blue
phase is air, the red phase is the extruded viscoelastic fluid governed by the
Oldroyd-B equation.

References

[1]      
Habla, F., Marschall, M., Hinrichsen, O., Dietsche, L.: Numerical
simulation of viscoelastic two-phase flows using OpenFOAM®. Chem. Eng. Sci. 66
(2011) 5487-5496.

[2]      
Habla, F., Dietsche, L., Hinrichsen, O.: Modeling and simulation
of conditionally volume averaged viscoelastic two-phase flows. AIChE Journal.
DOI: 10.1002/aic.14095.

[3]      
Habla, F., Woitalka, A., Neuner, S., Hinrichsen, O.: Development
of a methodology for numerical simulation of non-isothermal viscoelastic fluid
flows with application to axisymetric 4:1 contraction flows. Chem. Eng. J.
207-208 (2012) 772-784.

[4]      
Fattal, R., Kupferman, R.: Constitutive laws for the
matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123
(2004) 281-285.

[5]      
Habla, F., Obermeier, A., Hinrichsen, O.: Semi-implicit
formulation for viscoelastic models: Application to three-dimensional
contraction flows. Submitted to Journal of Non-Newtonian Fluid Mechanics.