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

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 Weissenberg number. In the wake, very large Weissenberg numbers are necessary to observe stresses that are proportional to W e 5 .