(244j) The Log-Conformation Reformulation (LCR) for Viscoelastic Flow Simulations in Openfoam®

The log-conformation reformulation (lcr)
for viscoelastic flow simulations in OpenFOAM®

Florian Habla¹, Tan Ming Wei¹, Johannes Hasslberger¹, and Olaf
Hinrichsen¹

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

Introduction

Simulation
of complex viscoelastic flows at high Weissenberg numbers is an outstanding
challenge. Fortunately, the last years provided significant progress in
developing stable and accurate numerical algorithms. Fattal and Kupferman^1,2
proposed the so-called log-conformation reformulation (LCR), in which a
logarithmized evolution equation for the conformation tensor is solved instead
of solving the constitutive equation itself. This removes the exponential
variation of the stress (and also the conformation tensor) at stagnation
points. The new variable (the logarithm of the conformation tensor) can better
be approximated by polynomial-based interpolation schemes than the
exponentially behaving conformation tensor (or stress) itself and thereby help
to extenuate the High Weissenberg Number Problem (HWNP). We implemented the
log-conformation reformulation in the collocated finite-volume based
open-source software OpenFOAM®.

Numerical method

The implementation is done such
as to be most flexible in terms of meshing in order to allow for tetrahedral
and polyhedral cell types. It is also very efficient by including a
computationally inexpensive eigenvalue and eigenvector routine, which is
necessary to compute the logarithm of the conformation tensor.  

Our solver is first validated
with the analytical solution for a startup Poiseuille flow of a viscoelastic
fluid. The result for an elasticity number of E = 10 is shown in Fig. 1. The
perfect agreement with the analytical solution proves the correct
implementation.

Fig.
1: Centerline velocity U₀ as a function of time T
during the start-up of a Poiseuille flow of an Oldroyd-B fluid at a retardation
ratio of β = 0.01.

The
algorithm finally is second-order accurate both in time and space as proved by
Fig. 2.

 

(a)     
                                                                         (b)

Fig.
2: Error of the calculated flow rate Q_num as a function of the time-step size Δt (a) and
the cell size Δy (b) for the start-up of a Poiseuille flow problem.

Simulation of cavity flows

We applied the solver to the three-dimensional and transient
simulation of a lid-driven cavity flow (cf. Fig. 3a), in which the viscoelastic
fluid is modeled by the Oldroyd-B constitutive equation. The simulations were
performed on various hexahedral meshes of different size (cf. Fig. 3b) in order
to check for mesh convergence of the results and a tetrahedral mesh (cf. Fig.
3c) to show the applicability of our numerical algorithm to unstructured
meshes.

(a)                                                            (b)
                                                       (c)        

 

Fig.
3: Domain of the three-dimensional cavity (a), a typical hexahedral (b) and the
tetrahedral (c) mesh we used in our study.

Results are obtained for various values of the Weissenberg number and
presented and discussed with respect to the location of the primary vortex
center, streamline patterns and velocity and stress profiles besides others. A
distinct effect of viscoelasticity is prevailing, see Figs. 4a and 4b, which
show the difference of the primary vortex pattern between an inelastic fluid
and a viscoelastic fluid at We = 2. We are able to obtain sufficiently mesh
converged results for Weissenberg numbers, which would have been impossible to
obtain without use of the log-conformation reformulation. Moreover, no upper
limit in the Weissenberg number could be found in terms of stability and we
present results for a Weissenberg numbers of We = 160, see Fig. 4c. However,
questions of accuracy at such large Weissenberg numbers remain and generally
finer meshes and time-steps would be necessary, which, however, is out of
question due to the overhead in computation time.

    

(a)                                                            (b)
                                                       (c)        

 

Fig.
4: Primary vortex for an inelastic fluid (a), a viscoelastic fluid at We = 2
(b) and a chaotic pattern at We = 160 (c).

References

[1]      
Fattal, R., Kupferman, R.: Constitutive laws for the
matrix-logarithm of the conformation tensor. J. Non-Newton. Fluid Mech. 123
(2004) 281-285.

[2]      
Fattal, R., Kupferman, R.: Time-dependent simulation of
viscoelastic flows at high Weissenberg number using the log-conformation
representation, J. Non-Newton. Fluid Mech. 126 (2005) 23-27.