Benchmarking diffuse interface approaches for fluid–solid interaction problems

  • Reder, Martin (Karlsruhe University of Applied Sciences)
  • Weichel, Marcel (Karlsruhe Institute of Technology)
  • Nestler, Britta (Karlsruhe Institute of Technology)
  • Schneider, Daniel (Karlsruhe Institute of Technology)

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Diffuse-interface approaches, and in particular the phase-field method, are widely employed to simulate fluid flow in complex geometries involving solid–fluid interactions. In these frameworks, the sharp fluid–solid boundary is replaced by a diffuse interface, which necessitates the consistent formulation of boundary conditions for the flow field across the interfacial region. While several diffuse formulations for imposing such conditions have been proposed in the literature (see, e.g., [1, 2]), a rigorous and quantitative comparison of their performance is still missing. In the present study, we therefore introduce a set of benchmark problems for diffuse fluid–solid interactions, encompassing configurations with stationary boundaries as well as cases with moving walls. Several established model formulations from the literature are revisited and systematically compared using well-defined reference solutions. By applying these models to flows with varying interface thicknesses and different phase-field profile types, we evaluate their accuracy and convergence properties. Our results demonstrate that the performance of a given diffuse-interface formulation depends strongly on the specific problem setup. For configurations without wall motion, the approach proposed by Beckermann [1] yields the highest accuracy. In contrast, when non-zero wall velocities are present, locally formulated models exhibit significantly improved performance. For more complex geometries, the discrepancies between the different formulations become less pronounced. Moreover, we observe that a thinner diffuse interface leads to more accurate results than a wider but better resolved interface. Overall, this work provides the first systematic and quantitative comparison of diffuse boundary-condition models for fluid flow, offering practical guidance for the selection of suitable formulations depending on the physical scenario and numerical resolution. REFERENCES [1] C. Beckermann, H.-J. Diepers, I. Steinbach, A. Karma, and X. Tong., Modeling melt convection in phase-field simulations of solidification, Journal of Computational Physics, 154(2):468–496, 1999. [2] X. Li, J. Lowengrub, A. R¨atz, and A. Voigt., Solving pdes in complex geometries: a diffuse domain approach, Communications in mathematical sciences, 7(1):81–107, 2009.