Viscoelastic flows of a lid-driven cavity using spectral element methods

The performance of a spectral element method in the DEVSS-G formulation for the solution of non-Newtonian flows is assessed by means of a systematic analysis of the benchmark lid-driven cavity problem. It is first validated by comparison with the creeping Newtonian and Oldroyd-B flows, where in the latter case a lid velocity regularisation scheme must be employed to remove the singularity at the lid-wall interfaces. In both instances, excellent agreement is found with the literature for stable, time-independent flows, and in fact it is shown that higher Weissenberg numbers can be obtained using the present methodology for these types of flow. Some physical aspects of the solutions are also presented and discussed, however at increasing Weissenberg numbers, the methodology breaks down due to a lack of convergence in the BDF/FPI time advancement scheme. By systematically assessing the effects of the levels of $hp$-refinement and temporal refinement on the flow fields, as well as the introduction of the extension-limiting Giesekus mobility parameter in the constitutive equations, it is demonstrated that in each instance the inability to accurately resolve the stress gradients leads to a compounding of errors in the BDF/FPI regime, ultimately causing it to diverge.