Flow of viscoelastic fluids past a cylinder at high Weissenberg number : stabilized simulations using matrix logarithms

The log conformation representation proposed in Fattal et al. has been implemented in a FEM context using the DEVSS/DG formulation for viscoelastic fluid flow. We present a stability analysis in 1D and identify the failure of the numerical scheme to balance exponential growth as a possible source for numerical instabilities at high Weissenberg numbers. A different derivation of the log based evolution equation than in Fattal at al. is also presented. We show numerical results for the flow around a cylinder for an Oldroyd-B and a Giesekus model. We provide evidence that the numerical instability identified in the 1D problem is also the actual reason for the failure of the standard FEM implementation of the problem. With the log conformation representation we are able to obtain solutions beyond the limiting Weissenberg numbers in the standard scheme. In particular, for the Giesekus model the improvement is rather dramatic: there does not seem to be a limit for the chosen model parameter(alpha=0.01). However, it turns out that although in large parts of the flow the solution converges, we have not been able to obtain convergence in localized regions of the flow. Possible reasons include artefacts of the model and unresolved small scales. However, more work is necessary, including the use of more refined meshes and/or higher-order schemes, before any conclusion can be made on the local convergence problems