### Peter Wapperom and Michael Renardy

### Numerical prediction of the boundary layers in the flow
around a cylinder using a fixed velocity field

* Journal of Non-Newtonian Fluid Mechanics* **125** (2005) 35-48

#### Abstract

For the flow around a cylinder of an upper convected Maxwell or
Oldroyd-B fluid, thin stress boundary layers develop close to the
cylinder wall and in the wake. Numerical simulations of this
flow problem already fail to converge at a Weissenberg number
of order unity.
For the boundary layer and in the wake,
high-Weissenberg number stress scalings, using a given, Newtonian
velocity field have been predicted by Renardy (J. Non-Newtonian Fluid
Mech. 90 (2000) 13--23). We develop a purely Lagrangian technique
that is able to resolve thin stress boundary layers in an accurate and
very efficient manner up to arbitrarily large Weissenberg numbers.
This is in sharp contrast with a traditional method which has severe
difficulties in predicting the correct solution at relatively low
Weissenberg numbers and suffers from long computational times.
With the purely Lagrangian technique, we observe numerically the
existence of thin regions
with large stresses, just outside the boundary layer along the
cylinder and birefringent strand in the wake, just as predicted by the
asymptotic analysis.
All theoretical scalings are observed at larger values of
the Weissenberg number than can be reached in the benchmark
flow around a cylinder with non-fixed kinematics. Around the cylinder,
the asymptotic
results already appear to be valid at moderate values of the Weisenberg
number. In the wake very large Weissenberg numbers are
necessary to observe stresses that are proportional to We^{5}
#### Keywords

Boundary layers; Cylinder; High Weissenberg asymptotics;
Upper-Convected Maxwell model; Numerical simulation