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The accurate and stable simulation of viscoelastic flows remains asignificant computational challenge, exacerbated for flows in non-trivial andpractical geometries. Here we present a new high-order meshless approach withvariable resolution for the solution of viscoelastic flows across a range ofWeissenberg number (up to Wi=128, depending on test case). Based on the LABFMapproach of King et al. J. Comput. Phys. 415 (2020):109549, highly accurateviscoelastic flow solutions are found using Oldroyd B and PPT models for arange of two dimensional problems - including Kolmogorov flow, planar Poiseulleflow, and flow in a representative porous media geometry. Convergence rates upto 9th order are shown. Three treatments for the conformation tensor evolutionare investigated (direct integration, Cholesky decomposition, andlog-conformation), with log-conformation providing consistently stablesolutions across test cases, and direct integration yielding better accuracyfor simpler unidirectional flows. The final test considers symmetry breaking inthe porous media flow at moderate Weissenberg number, as a precursor to afuture study of fully 3D high-fidelity simulations of elastic flowinstabilities in complex geometries. The results herein demonstrate thepotential of a viscoelastic flow solver that is both high-order (for accuracy)and meshless (for straightforward discretisation of non-trivial geometriesincluding variable resolution). In the near-term, extension of this approach tothree dimensional solutions promises to yield important insights into a rangeof viscoelastic flow problems, and especially the fundamental challenge ofunderstanding elastic instabilities in practical settings.

A mesh-free framework for high-order simulations of viscoelastic flows in complex geometries