Computational Fluid Dynamics Based on
The Unified Coordinates
Professor W H Hui（许为厚）
Emeritus Professor of Applied Mathematics
University of Waterloo, Canada and
Hong Kong University of Science & Technology
This lecture highlights the monograph of the above title by W H Hui and K Xu (Springer and Science Press, 2012). It is well known that the numerical solution to a given flow depends on the coordinates (mesh) used in the computation. The two well-known coordinate systems, Eulerian and Lagrangian, have advantages as well as drawback. Eulerian methods is relatively simple, but it smears contact discontinuities badly and requires generation of a body-fitted mesh prior to computing flow past a body. In contrast, Lagrangian method resolves contact discontinuities sharply, but the gas dynamics equations in 2D and 3D could not be written in conservation partial differential equation (PDE) form, rendering numerical computation complicated. It also breaks down due to mesh tangling.
A unified coordinate system (UC) is introduced via Hui's transformation with three degrees of freedom (the mesh velocity). Its main contributions to the theory of CFD are: (1) the governing equations in any moving coordinates can be written as a system of closed conservation PDEs; (2) the system of Lagrangian system of gas dynamics equations is written in conservation PDE form; and (3) the 2D or 3D lagrangian system of gas dynamics equations is shown to be weakly hyperbolic and is , therefore, not equivalent to the Eulerian system. Computationally, its advantages are :(1) it is superior to both Eulerian and Lagrangian systems in that contact discontinuities are resolved sharply without mesh tangling; (2) it avoids the tedious and time-consuming task of mesh generation for flow past a body (the UC mesh is automatically generated by the flow); and (3) it provides a new dynamic moving mesh method. Numerous examples are given to illustrate these advantages.