(142bj) CFD Simulation of Non-Isothermal Flows of Viscoelastic Fluids in 4:1 Contractions

Introduction

Non-isothermal viscoelastic fluid flows can be found
in a wide range of industrial applications such as injection molding, polymer
blending and extrusion processing. It is of course of major importance to understand
and predict such type of flows. Fortunately, numerical simulation evolved as an
important tool to guide the engineer in the task of developing and improving
such processes. However, there is still a gap between the engineer's needs and
the current state-of-the-art, especially in complex flows such as is the case
for non-isothermal viscoelastic fluids flows.

We therefore focus our research on developing a
comprehensive CFD method to simulate such type of flows. In this work, focus is
set on the thermo-rheological modeling and the numerical method. Furthermore,
we apply the developed model to simulation of non-isothermal viscoelastic
contraction flows.

Theory

In viscous fluids, mechanical energy is
fully dissipated as heat, whereas in solids the mechanical energy is stored and
can be released into mechanical energy again [1]. The behavior is known as pure
entropy elasticity and energy elasticity, respectively. Viscoelastic fluids
behave in between those of fluids and solids. As a result, viscoelastic fluids
do show both of the aforementioned behaviors, which severely complicates the
thermo-rheological modeling. One of the present approaches uses an a-priori
defined split factor [2], which is being adopted in this work.

Fig. 1: Streamlines for different Weissenberg numbers
in an axisymmetric contraction.

The Oldroyd-B model is used to model
viscoelasticity [3]. Temperature dependence of the viscosity and the relaxation
time is modeled with the Williams-Landel-Ferry model (WLF) [4].

The finite volume method is used for
discretization. Special attention has to be paid to the convection scheme,
which has a major influence on the accuracy and stability of the numerical
method. We therefore evaluated several state-of-the-art differencing schemes in
a stagnation point flow. The DEVSS technique is used to explicitly stabilize
the solution method by introducing an elliptic operator into the momentum
equation [5].

Results and discussion

Contraction flows are extensively studied in
literature. Especially for viscoelastic fluid flows, this is an important test
case, aiming at the stability of numerical algorithms due to the singularity at
the reentrant corner. The failure in obtaining converged solutions at high
Weissenberg numbers is probably the central problem in literature, which is
known as the High-Weissenberg-Number-Problem (HWNP).

In this work, we were able to obtain
converged solutions up to Weissenberg numbers of We = 10 for an axisymmetric
contraction, see Fig. 1. As expected, the recirculating flow length increases
with increasing Weissenberg numbers and the vortex area becoming more bulgy,
see also Fig. 1. Initially the vortex is solely at the salient corner, but from
We = 1 the vortex connects with the re-entrant corner.

When imposing a temperature jump at the
walls in the area of the contraction, we find the vortex length to increase
with increasing temperature jump, see Fig. 2. Herein, pure entropy elasticity
was assumed.

Fig. 2: Dimensionless vortex length as a function of
the temperature jump at the contraction for We = 1.

Conclusions

In this work we developed a general finite volume model capable of
simulating non-isothermal viscoelastic fluid flows. The model was used to
simulate axisymmetric 4:1 contraction flows and results were obtained up to a
Weissenberg number of We = 10. The recirculating flow length was found to
increase both with increasing Weissenberg number and temperature jump.

References

[1]       G. W. M. Peters, F. P. T. Baaijens, Modelling of non-isothermal viscoelastic
flows, J. Non-Newtonian Fluid Mech. 68 (1997) 205-224.

[2]       A. Wachs, J.-R. Clermont, Non-isothermal viscoelastic flow computations
in an axisymmetric contraction at high Weissenberg numbers by a finite volume
method, J. Non-Newtonian Fluid Mech. 95 (2000) 147-184.

[3]       R. B. Bird, O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, second
ed., Wiley Interscience, 1987.

[4]       J. D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1980.

[5]       R. Guénette, M. Fortin, A new mixed finite element method for computing
viscoelastic flows, J. Non-Newtonian Fluid Mech. 60 (1995) 27-52.