An unstructured geometrical un-split VOF method for viscoelastic two-phase flows

TL;DR

This work introduces an unstructured geometrical VOF method for viscoelastic two-phase flows by embedding a conditional volume-averaged one-field formulation into DeboRheo, enabling robust interface tracking and flexible rheology on general meshes. A unified stabilization framework based on a square-root (root) conformation representation keeps simulations stable at high Weissenberg numbers up to De ≈ 16, while a PLIC-like interface reconstruction ensures sharp interfaces and compact stencils. The method is implemented in a segregated solver and validated via 2D and 3D drop deformation in simple shear for NN, NV, and VN configurations, with RDF curvature outperforming height-function in curvature accuracy. Key findings include intricate dependencies of deformation D and orientation θ on Capillary Ca and Deborah De numbers, the emergence of damped oscillations at higher De, and notable differences between Oldroyd-B and Giesekus models, all consistent with prior literature yet extended to higher De and unstructured meshes. The framework’s open-source integration and data availability support broader adoption and further validation across constitutive models and complex multiphase rheology.

Abstract

Since viscoelastic two-phase flows arise in various industrial and natural processes, developing accurate and efficient software for their detailed numerical simulation is a highly relevant and challenging research task. We present a geometrical unstructured Volume-of-Fluid (VOF) method for handling two-phase flows with viscoelastic liquid phase, where the latter is modeled via generic rate-type constitutive equations and a one-field description is derived by conditional volume averaging of the local instantaneous bulk equations and interface jump conditions. The method builds on the plicRDF-isoAdvector geometrical VOF solver that is extended and combined with the modular framework DeboRheo for viscoelastic computational fluid dynamics (CFD). A piecewise-linear geometrical interface reconstruction technique on general unstructured meshes is employed for discretizing the viscoelastic stresses across the fluid interface. DeboRheo facilitates a flexible combination of different rheological models with appropriate stabilization methods to address the high Weissenberg number problem.

Figures (26)

• Figure 1: Schematic diagram of a two-phase system in a domain $\Omega$.
• Figure 1: Drop deformation parameter $D$ and orientation angle $\theta$ as a function of time for the NN, VN, and NV systems at $\operatorname{Ca}=0.24$ and $\operatorname{Ca}=0.6$ and varying Deborah numbers.
• Figure 2: Different ways to compute $A_f^{n+1}$ for $\alpha_f$ weighting of the polymer stress $\tau_p$ in the pressure Poisson equation (\ref{['eq:peqn']}).
• Figure 2: Drop shapes at $t=10\,\dot{\gamma}^{-1}$.
• Figure 3: Schematic diagram of the two-dimensional drop deformation in shear flow.