@Article{CiCP-2-577,
author = {W. H. Hui},
title = {The Unified Coordinate System in Computational Fluid Dynamics},
journal = {Communications in Computational Physics},
year = {2007},
volume = {2},
number = {4},
pages = {577--610},
abstract = {<p style="text-align: justify;">A fundamental issue in CFD is the role of coordinates and, in particular,
the search for &quot;optimal&quot; 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/cicp/7919.html}
}