### Peter Wapperom and Mike F. Webster

### Simulation for Viscoelastic Flow by a Finite Volume/Element Method

* Computer Methods in Applied Mechanics and
Engineering* **180** (1999) 281-304

#### Abstract

Stability of a second-order finite element/finite volume hybrid scheme
is investigated on the basis of flows with increasing Weissenberg number.
Finite elements are used to discretise the balances of mass and
momentum. For the stress equation a finite volume method is used, based
on the recent development with fluctuation distribution schemes for pure
convection problems.
Examples considered include a start-up channel flow, flow past a
cylinder and the non-smooth 4:1 contraction flow for an Oldroyd-B fluid.
A considerable gain in efficiency per time step can be obtained compared
to an alternative pure finite element implementation.
A distribution based on the flux terms is unstable for higher
Weissenberg numbers, and this is also true for a distribution based on
source terms alone. The instability is identified as being caused by the
interaction of the balance equations and stress equation. A combination
of distribution schemes based on flux and source terms, however, gives a
considerable improvement to the hybrid FE/FV implementation. With respect
to limiting Weissenberg number attenuation, the hybrid scheme is more
stable than the pure finite element alternative for the smooth flow past a
cylinder, but less so for the non-smooth contraction flow.
The influence of additional strain-rate stabilisation techniques is also
analysed and found to be beneficial.
#### Keywords

Hybrid finite element/finite volume; stability; flux and
source distribution; Oldroyd-B