An arbitrary Lagrangian–Eulerian‐based finite element strategy for modeling incompressible two‐phase flows

Abstract

In this work, we develop a numerical strategy for solving two‐phase immiscible incompressible fluid flows in a general arbitrary Lagrangian–Eulerian framework. As we use a conforming mesh moving with one of the fluids, there is no need to track or reconstruct the interface explicitly. A sharp interface, discontinuity in the fluid properties, and the jump in the pressure field are all accurately modeled using the dummy‐node technique. One of the challenges that this work addresses is to obtain a C0‐continuous approximation for the surface tension force term on the reference configuration, and to obtain its consistent linearization within the context of a Newton–Raphson strategy. It is shown by means of various numerical examples that the strategy is computationally efficient and robust, and circumvents the need for remeshing upto significantly large deformations.

An ALE‐based finite element strategy for modeling compressible two‐phase flows

“…The detailed derivation of the surface tension force term is presented in Reference 35. The final expression is given by …”

“…Assuming that the solution at time $$$ {t}_n $$$ is known, we find the solution at time $$$ {t}_{n+1} $$$. In order to develop a Newton–Raphson strategy, we linearize the variational formulation given by Equations (17)–(19) in a manner similar to that presented in References 33 and 35. The mappings used for the linearization are described in the Appendix.…”

“…In this work, we do not deal with two‐phase flow problems involving topological changes since that requires re‐meshing of the domain; however, one can handle such problem with this algorithm by using an appropriate re‐meshing technique. The strategy developed in this work is a generalization of the earlier works in References 33‐35 to include compressibility effects in the two‐phase formulation. The salient features of this work are The use of a Lagrangian setup to model Fluid 1 (for example, the bubble) ensures conservation of mass in each fluid with very high fidelity, and also circumvents the need to track or construct the interface explicitly by solving an additional diffusion equation as in the case of an Eulerian approach; simply adding the displacement field to the original position vector yields the new position vector for points on the interface. The proposed strategy is monolithic, with all the field variables in Fluid 1 and in Fluid 2 (for example, the fluid surrounding the bubble) solved for simultaneously to ensure faster convergence of the overall scheme. The nodes on the interface define the interface accurately, which leads to an accurate computation of the surface tension forces.…”

Prediction of multiphase flows with sharp interfaces using anisotropic mesh optimisation

“…Falling into this category are such methods as interface tracking [10]; Volume-Of-Fluid (VOF) methods [7,11,12], which use an indicator field to model the interface(s); Arbitrary Lagrangian Eulerian methods [13,14], in which elements are deformed to follow the interface and its evolution; the immersed boundary method [15], in which the domain boundaries can be placed arbitrarily inside the domain; and compressive advection techniques [2,16], which keep the indicator field sharp whilst being advected through the domain. A relatively new approach known as CutFEM aims to make the discretisation independent of the geometry [17] and allows any interfaces to cut through elements in a similar manner to XFEM [18].…”