Finite line method and its application in coupled heat transfer between fluid-solid domains

Abstract

In this paper, a completely new numerical method, called finite line method, is proposed and is used to solve fluid-solid coupled heat transfer problems. The extensively used finite element method is a method based on volume discretization; the finite volume method is a method operated on the surface of the control volume; the boundary element method is the one based on boundary surface discretization; the meshless method is the one constructing the computational algorithm using surrounding scatter points at a collocation point. The method proposed in the work is based on the use of finite number of lines, in which an arbitrarily high-order computational scheme can be established by using only two or three straight or curved lines at each point. The creative idea of the method is that by using a directional derivative technique along a line, high-order two- and three-dimensional spatial partial derivatives with respective to the global coordinates can be derived from the Lagrange polynomial interpolation formulation, based on which the discretized system of equations can be directly formed by the problem-governing partial differential equation and relevant boundary conditions. The proposed finite line method is very simple in theory and robust in universality, by using which the boundary value problems of partial differential equations in solid and fluid mechanics problems can be solved in a unified way. In solving fluid mechanics problems, the diffusion term is simulated by using the central line set to maintain a high efficiency, and the convection term is computed by using an upwind line set to embody its directional characteristic. A few of numerical examples will be given in this paper for fluid-solid coupled heat transfer problems for verifying the correctness and efficiency of the proposed method.