A hierarchical Cartesian method for conjugate heat transfer involving moving bodies

Abstract

Electrical Discharge Machining (EDM) is a manufacturing process in which controlled electrical discharges are used to accurately remove material. It allows realizing complex contours and cavities on pre-hardened steel without heat treatment. The machining tool is insulated from the workpiece by a dielectricum such that the sparks are only produced when the voltage between them is high enough to break its resistivity. The heat release as well as the debris deposition during machining play an essential role in the efficiency and stability of the manufacturing process.1 Improper flushing affects the process efficiency,2 surface finish,3 and can result in significant tool wear.4 The underlying physics of the flushing process is still not well understood1 which is why further investigations are performed. The motivation of this work is to develop a numerical framework for studying the energy transfer mechanisms during the flushing step of the EDM process. The fluid flow around the moving particles as well as the heat conduction within the particles themselves are fully-resolved with a cut-cell finite-volume method5 while the particle interface is described by the level-set method.6 The heat conduction and the fluid flow problems are loosely coupled. Octrees and related data-structures (like e.g.7–9) allow for unstructured grids that retain enough structure to enable automatic mesh generation, adaptive mesh refinement, domain decomposition via e.g. space filling curves as well as dynamic load balancing. In our approach the spatial domain is partitioned as a single entity independently of wether a region belongs to the fluid or the solid phase by means of an octree, and a mapping between the octree nodes and the cells of the different meshes is established and maintained during mesh adaptation. In this study we show that such a strategy facilitates the coupling of different meshes under adaptive mesh refinement. This work is structured as follows. In section II the physical models considered are introduced. In section III the numerical methods for the fluid and the solid phase are detailed. The coupling conditions at the fluid-solid interface as well as the coupling scheme used are provided. Last, the usage of hierarchical Cartesian grids for exchanging boundary conditions between the fluid and the solid domains is detailed. In section IV results are presented for a configuration involving four spherical particles undergoing forced motion and heating, and for configurations of two oblate and two prolate ellipsoidal particles undergoing forced heating.