One‐ and two‐step semi‐Lagrangian integrators for arbitrary Lagrangian–Eulerian‐finite element two‐phase flow simulations

Abstract

Semi‐Lagrangian (SL) schemes are of utmost relevance to simulate two‐phase flows where advection dominates. The combination of SL schemes with the finite element (FE) method and arbitrary Lagrangian–Eulerian (ALE) dynamic meshes yields a strong ingredient to deal with applications in Earth sciences, environmental engineering, and oil and gas industry whose numerical background is incipient. This article is intended to propose a second‐order time multistep SL/ALE/FE scheme to track particle trajectories traveling amidst single‐ or two‐phase incompressible flows both in fixed and moving mesh situations. The novel scheme is a BDF2‐like implementation that considers the mesh velocity as part of its backward‐in‐time integration. Trajectory departure points are searched through extrapolation and followed by a multistep interpolation corrector that works both for constant and varying time steps. A series of two‐phase benchmark flow simulations are carried out by using the cubic element to compare the performance of the new integrator against single‐step approximations as well as to analyze enhancements in representing the two‐phase flow dynamics. We use the 2D axisymmetric Navier–Stokes equations as underlying model. Error analyses, convergence tests, quantitative, and qualitative comparisons are presented and discussed to highlight the superior conservative feature of the novel scheme.

Numerical Investigation of Gas Bubble Interaction in a Circular Cross-Section Channel in Shear Flow

“…The Navier-Stokes equations described above are solved through the finite element and Galerkin methods. The non-linear convective term is treated explicitly using a firstorder semi-Lagrangian method [19], which is unconditionally stable and preserves the convection operator symmetry. The surface tension and gravity forces are also added explicitly to the resulting linear system of equations.…”