Numerical Simulations of the Two-phase flow and Fluid-Structure Interaction Problems with Adaptive Mesh Refinement

Abstract

Numerical simulations of two-phase flow and fluid structure interaction problems are of great interest in many environmental problems and engineering applications. To capture the complex physical processes involved in these problems, a high grid resolution is usually needed. However, one does not need or maybe cannot afford a fine grid of uniformly high resolution across the whole domain. The need to resolve local fine features can be addressed by the adaptive mesh refinement (AMR) method, which increases the grid resolution in regions of interest as needed during the simulation while leaving general estimates in other regions. In this work, we propose a block-structured adaptive mesh refinement (BSAMR) framework to simulate two-phase flows using the level set (LS) function with both the subcycling and non-subcycling methods on a collocated grid. To the best of our knowledge, this is the first framework that unifies the subcycling and non-subcycling methods to simulate two-phase flows. The use of the collocated grid is also the first among the two-phase BSAMR framework, which significantly simplifies the implementation of multi-level differential operators and interpolation schemes. We design the synchronization operations, including the averaging, refluxing, and synchronization projection, which ensures that the flow field is divergence-free on the multi-level grid. It is shown that the present multi-level scheme can accurately resolve the interfaces of the two-phase flows with gravitational and surface tension effects while having good momentum and energy conservation. We then develop another consistent scheme, in which the conservative momentum equations and the mass equation are solved in the aforementioned BSAMR framework. This consistent mass and momentum transport treatment greatly improves the accuracy and robustness for simulating two-phase flows with a high density ratio and high Reynolds number. We demonstrate that the consistent scheme results in a numerically stable solution in flows with high density ratios~(up to $10^6$) and high Reynolds numbers~(up to $10^6$), while the inconsistent scheme exhibits nonphysical fluid behaviors in these tests. For solving single- and multiphase fluid-structure interaction (FSI) problems, we present an adaptive implementation of the distributed Lagrange multiplier (DLM) immersed boundary (IB) method on multilevel collocated grids. We also developed a force-averaging algorithm to maintain the consistency of Eulerian immersed boundary (IB) forces across multiple levels. The efficacy of the force averaging algorithm is validated using the lid-driven cavity with a submerged cylinder problem. We demonstrate the versatility of the present multilevel framework by simulating problems with various types of kinematic constraints imposed by structures on fluids, such as imposing a prescribed motion, free motion, and time-evolving shape of a solid body. The accuracy and robustness of the codes are validated using several canonical test problems.