A Parallel Boundary Conforming Method on Adaptive Moving Meshes for Multiphase Flows

Abstract

This thesis presents a novel parallel boundary conforming method for simulating two-dimensional incompressible flows on adaptive moving unstructured meshes. The proposed method is especially helpful for studying immiscible multiphase flow problems as the interfaces are represented explicitly by mesh lines, and the grids are able to conform the movements of the interfaces. The fluid flow equations are studied based on the Arbitrary Lagrangian-Eulerian formulation and the Finite Element Method (Li, 2013); parallel linear solvers provided by PETSc are used to solve the discretised incompressible Navier-Stokes equations. There are several important advantages of the boundary conforming method on unstructured meshes. The method is accurate in computing curvature and in resolving the dynamic boundary condition at the interface. It is also able to handle large interface deformation. Due to the complexities of the mesh generator/optimiser, the method only exists in sequential version in the literature, and its application range is severely limited by the computing time and memory of a single processor. The motivation of this work is to take the advantage of the parallel computing platform to extend the sequential method to a parallel method. This will enormously enlarge the application range of the method so that more realistic multiphase flow problems can be investigated. The two major difficulties for the parallel adaptive moving mesh method are: (1) the consistent local remeshing at the inter-processor boundaries between processors; and (2) the dynamic load balancing due to mesh adaptation. To address these issues, a parallel fast marching method is developed such that the characteristic mesh length can be computed globally and this length also determines the condition for local remeshing. The synchronised local remeshing algorithm on inter-process boundaries is then developed in order to preserve consistent partition boundaries. Repartition of the adapted mesh is achieved by ParMETIS and further mesh migration is performed to enforce contiguous sub-meshes. The parallel Laplacian smoothing method is performed to relocate each interior vertex to the arithmetic mean of its neighbouring incident vertices. The parallel adaptive moving mesh is validated by moving bodies with prescribed velocity imposed on the dynamic interfaces. The accuracy of our method increases quadratically as the increase in the initial number of vertices on the interfaces. The discretisations of the incompressible Navier-Stokes equations for multiphase flows are studied; the parallel vectors and matrices resulted from FEM are formed via PETSc linear algebra implementations. Arange of computational fluid dynamics problems involving flow past a circular cylinder, an oscillating drop/bubble and a rising bubble are used to examine the accuracy of the parallel flow solver on both fixed and moving meshes. The parallel performance of the adaptive moving mesh method, flow solver and flow solver on adaptive moving meshes is investigated according to strong and weak scaling analyses. It is found that the adaptive moving mesh method does not scale well but it only takes little computation effort (approximately 2% of the total running time) when studying multiphase flow problems. Most of the computation effort is spent on assembly and solving the linear system. A favourable speedup is observed for multiphase flow problems and a super-linear speedup is achieved up to 32 processors. Generally, for a fixed size problem the best performance of the parallel multiphase flow solver occurs when there are approximately 13000 elements per processor. The code is finally used to simulate 64 rising bubbles. This thesis concludes with a discussion of further improvements and future work to be undertaken.