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

Abstract

The log conformation representation proposed in [1] has been implemented in a fem context using the DEVSS/DG formulation for viscoelastic fluid flow. We present a stability analysis in 1D and attribute the high Weissenberg problem to the failure of the numerical scheme to balance exponential growth. A slightly different derivation of the log based evolution equation than in [1] 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 far beyond the limiting Weissenberg numbers in the standard scheme. 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 for this are: artefacts of the model (Oldroyd-B) or unresolved small scales (Giesekus). 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.

Free-energy-dissipative schemes for the Oldroyd-B model

“…However, these results have not led yet to a complete understanding of the numerical instabilities [28], despite some progress [15,23]. Roughly speaking, we can distinguish between three possible causes of the HWNP:…”

“…Moreover, any solution to the logformulation satisfies the free energy estimate (which is not the case for the usual formulation in terms of τ . This may be related to the fact that the log-formulation has been reported to be more stable than the formulation in terms of τ (see [23]). …”

“…is discontinuous, we have discretized the advection term for σ h with a sum of jumps similar to the usual upwind technique, where φ + = 1 2 φ + {φ} (see [13,23]). Remark 3.2.…”

“…The advection terms u · ∇σ and u · ∇ψ will be discretized either through a characteristic method in the spirit of [3,40,48], or with the discontinuous Galerkin (DG) method in the spirit of [23]. Notice already that the characteristic method requires the velocity field to be more regular than the discontinuous Galerkin method in order to define the flow associated with the vector field u h .…”

Modeling Flow-Induced Crystallization

“…The governing equations are solved sequentially per time step with semi-implicit Euler time stepping and semi-implicit formulation of the stress in the momentum balance [96]. A number of stabilization techniques were employed: discrete elastic-viscoelastic stress split (DEVSS) [97,98], streamline-upwind Petrov-Galerkin (SUPG) [99], and log-conformation representation (LCR) [100]. …”