Commun. Comput. Phys., 2 (2007), pp. 577-610.

The Unified Coordinate System in Computational Fluid Dynamics

W. H. Hui 1*

1 Division of Mechanics, Research Centre for Applied Sciences, Academia Sinica, Nankang, Taipei 115, Taiwan.

Received 17 May 2006; Accepted (in revised version) 10 November 2006
Communicated by Chi-Wang Shu
Available online 8 January 2007


A fundamental issue in CFD is the role of coordinates and, in particular, the search for ``optimal" coordinates. This paper reviews and generalizes the recently developed unified coordinate system (UC). For one-dimensional flow, UC uses a material coordinate and thus coincides with Lagrangian system. For two-dimensional flow it uses a material coordinate, with the other coordinate determined so as to preserve mesh othorgonality (or the Jacobian), whereas for three-dimensional flow it uses two material coordinates, with the third one determined so as to preserve mesh skewness (or the Jacobian). The unified coordinate system combines the advantages of both Eulerian and the Lagrangian system and beyond. Specifically, the followings are shown in this paper. (a) For 1-D flow, Lagrangian system plus shock-adaptive Godunov scheme is superior to Eulerian system. (b) The governing equations in any moving multi-dimensional coordinates can be written as a system of closed conservation partial differential equations (PDE) by appending the time evolution equations -- called geometric conservation laws -- of the coefficients of the transformation (from Cartesian to the moving coordinates) to the physical conservation laws; consequently, effects of coordinate movement on the flow are fully accounted for. (c) The system of Lagrangian gas dynamics equations is written in conservation PDE form, thus providing a foundation for developing Lagrangian schemes as moving mesh schemes. (d) The Lagrangian system of gas dynamics equations in two- and three-dimension are shown to be only weakly hyperbolic, in direct contrast to the Eulerian system which is fully hyperbolic; hence the two systems are not equivalent to each other. (e) The unified coordinate system possesses the advantages of the Lagrangian system in that contact discontinuities (including material interfaces and free surfaces) are resolved sharply. (f) In using the UC, there is no need to generate a body-fitted mesh prior to computing flow past a body; the mesh is automatically generated by the flow. Numerical examples are given to confirm these properties. Relations of the UC approach with the Arbitrary-Lagrangian-Eulerian (ALE) approach and with various moving coordinates approaches are also clarified.

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PACS: 47.11.-j, 47.40.-x
Key words: Unified coordinates, Eulerian coordinates, Lagrangian coordinates, contact discontinuities, automatic mesh generation, moving mesh, conservation form.

*Corresponding author.
Email: (W. H. Hui)

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