Generalized Finite Difference Method for Analyzing Multi-Dimensional Natural Convection Problems

• Main Authors: Chen, Chun-Yu, 陳俊宇
• Other Authors: Fan, Chia-Ming
• Format: Others
• Language: zh-TW
• Published: 2016
• Online Access: http://ndltd.ncl.edu.tw/handle/47596205849128024653

碩士 === 國立臺灣海洋大學 === 河海工程學系 === 104 === In this thesis, we used the generalized finite difference method (GFDM) and the projection method to accurately and efficiently analyze multi-dimensional natural convection problems. The Navier-Stokes equations are adopted as the governing equation since the Navier-Stokes equations are well-known to describe the fluid dynamics such as air and liquid. Thus, we adopted the primitive-variables formulation of the Navier-Stokes equations as the governing equation to simulate multi-dimensional natural convection problems. Natural convection appeared in our daily life, such as the convection in the atmosphere, ocean current and mantle convection, even if you boil the water or use the computer will cause natural convection. It’s very important to accurately simulate natural convection problems, so we can understand some physics of natural convection, and apply it to various engineering problems such as greenhouse and thermal design. In this study, we used the GFDM and the projection method to analyze the partial differential equations of natural convection. The GFDM is a localized domain-type meshless method, which can avoid the time-consuming task of mesh generation. The GFDM can yield sparse matrix rather than full matrix and ill-conditioning matrix. On the basis of these advantages, the GFDM can be adopted to solve problems accurately and efficiently, especially for large-scale problems. On the other hand, we can separate the velocity and pressure fields of the Navier-Stokes equations into three steps by using the projection method. The projection method can enhance the efficiency and avoid the complex calculations of coupled equations. Thus, we adopted these two methods and Matlab programming to build the meshless numerical model and to analyze multi-dimensional problems of natural convection. In this thesis, we analyzed two-dimensional and three-dimensional natural convection problems. The accuracy and efficiency of the numerical model can be verified by numerical comparisons in these examples. For the two-dimensional natural convection problems, we provided three examples. The first two examples are used to verify that this proposed numerical model can accurately solve the problem by comparing with results in the past study. The computational domain of the third example is designed by our own to test the stability and consistency of this numerical model. We also adopted different Rayleigh numbers in each example to validate that the proposed numerical model is suitable for various kinds of flow fields. After the numerical simulation of two-dimensional problems, we extended the proposed numerical model to accurately analyze three-dimensional problems. By testing different parameters in the proposed meshless numerical model, we can assure that this proposed meshless numerical model in this thesis is very stable and accurate as well as has great potential to be extended to realistic engineering problems of natural convection.