A Newton algorithm for steady Johnson-Segalman viscoelastic fluids based on a new non-singular log-conformation formulation and an incompressible finite element method
Résumé
A new log-conformation formulation of viscoelastic fluid flows is presented in this paper. It is non-singular for vanishing Weissenberg numbers and allows a direct steady numerical resolution by a Newton method. Moreover, an exact computation of all the terms of the linearized problem is provided. The use an exact divergence free finite element method for velocity-pressure approximation and a discontinuous Galerkin upwinding treatment for stresses leads to a robust discretization. A demonstration is provided by the computation of steady solutions at high Weissenberg numbers for the difficult benchmark of the lid driven cavity flow. Numerical results are in good agreement, both qualitatively with experiment measurements on real viscoelastic flows, and quantitatively with computations performed by others authors. The numerical algorithm is both robust and very efficient, as it requires few and mesh-invariant number of linear systems resolution to reach solutions at high Weissenberg number. An adaptive mesh procedure is also presented: it permits to catch accurately both boundary layers and main and secondary vortex.
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