Numerical Simulations of 3D Free-Surface Flows with the Generalized Total Linearization Method

The Total Linearization Method (TLM), initially introduced by Kruyt et al. (1988), was derived for the numerical solution of the 2D die-swell problem by the finite element method (FEM); it was thus limited to two dimensions and did not accommodate contact line dynamics. The TLM is predicated on a linearization of the FEM weak form and therefore boasts quadratic convergence rates.

In this talk, we introduce an extension of the TLM to general, capillary, free-surface flows together with a preconditioner that permits the use of Krylov-space methods in simulations. Herein, the calculus is extended to three dimensions in a nontrivial step involving a generalization of calculus on 2D curves to one over 3D surfaces. It is furthermore generalized so as to accommodate contact line problems. Contrary to standard, 3D, Newton techniques, the present extension yields a problem formulation containing domain degrees of freedom only on the free surface. This nearly halves the problem size when compared to traditional methods, but preserves quadratic convergence rates nevertheless. In addition, our preconditioner enables scalable iterative solution of large, 3D, free-surface flows. Numerical results of the extended TLM formulation are presented for a 2D magnetohydrodynamics slot-coating flow, a 2D thermocapillary problem, and the 3D die-swell problem, verifying preconditioner adequacy, the hypothesized rate of convergence, earlier simulations and previous experimental measurements.