A 3D Eulerian meshless conservative level set method for two-phase flows with complex geometries

In this paper, we develop a 3D Eulerian meshless method based on the conservative level set method targeted to solve two-phase flows with complex geometries. The method combines the advantages of Eulerian methods and meshless methods. Being an Eulerian method, it does not require neighbourhood estimation every time step. At the same time, being a meshless method, it does not require mesh connectivity between points in the domain and consequently, alleviates the difficulty of mesh-generation, makes point cloud adaptation and simulation with complex geometries relatively straight forward. In this method, we use a point cloud generating algorithm as a part of the fluid-solver, which can be used to generate or change the point cloud whenever there are any geometric changes in the domain. The meshless method is based on the Generalized Finite Difference Method (GFDM), which uses differential operators that are derived from a least-squares error minimization procedure. The advection equation of the volume fraction $\alpha$ uses the Directional Flux Error Minimization (DFEM) scheme. The 5th order WENO scheme is used for the flux-reconstruction in the advection equation. The interface sharpening step is performed at regular intervals to ensure that the sharpness of the interface is retained, thus, reducing the mass losses associated with the dissipative errors in the advection step. To further improve the accuracy, we propose the adaptation of the point cloud in the vicinity of the interface using the convolution of the volume fraction function ($\alpha$). The method is validated using benchmark test cases. Additionally, some flow problems involving complex geometries are presented — flow through a porous cavity with uniform and randomly distributed obstacles and the flow of molten metal in the casting of a helical bevel gear.