The Unified Coordinate System in Computational Fluid Dynamics (Communications in Computational Physics, 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 recentlydeveloped unified coordinate system (UC). For one-dimensional flow, UC uses a material coordinate and thus coincides with Lagrangian system. For two-dimensional flowit uses a material coordinate, with the other coordinate determined so as to preservemesh othorgonality (or the Jacobian), whereas for three-dimensional flow it uses twomaterial 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 shownin this paper. (a) For 1-D flow, Lagrangian system plus shock-adaptive Godunovscheme is superior to Eulerian system. (b) The governing equations in any movingmulti-dimensional coordinates can be written as a system of closed conservation partial differential equations (PDE) by appending the time evolution equations – calledgeometric conservation laws – of the coefficients of the transformation (from Cartesianto the moving coordinates) to the physical conservation laws; consequently, effectsof coordinate movement on the flow are fully accounted for. (c) The system of Lagrangian gas dynamics equations is written in conservation PDE form, thus providinga foundation for developing Lagrangian schemes as moving mesh schemes. (d) TheLagrangian system of gas dynamics equations in two- and three-dimension are shownto be only weakly hyperbolic, in direct contrast to the Eulerian system which is fullyhyperbolic; hence the two systems are not equivalent to each other. (e) The unifiedcoordinate system possesses the advantages of the Lagrangian system in that contactdiscontinuities (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. Numericalexamples are given to confirm these properties. Relations of the UC approach with theArbitrary-Lagrangian-Eulerian (ALE) approach and with various moving coordinatesapproaches are also clarified.