Unified coordinates and resolution of discontinuities for Euler and Shallow water equations
by Sergey Kudriakov
THESIS
2000
Ph.D. Mathematics
xiv, 229 leaves : ill. (some col.) ; 30 cm
Abstract
Two general coordinate systems have been used extensively in computational fluid dynamics: the Eulerian and the Lagrangian. The Eulerian coordinates cause excessive numerical diffusion across flow discontinuities, slip lines in particular. The Lagrangian coordinates, on the other hand, can resolve slip lines sharply but cause severe grid deformation, resulting in large errors and even breakdown of the computation. Recently, Hui et. al. (W.H. Hui, P.Y. Li and Z.W. Li, A Unified Coordinate System for Solving the Two-Dimensional Euler Equations, Journal of computational Physics, Vol. 153 (1999), pp. 596-637) have introduced a unified coordinate system which moves with velocity hq, q being velocity of the fluid particle. It includes the Eulerian system as a special case when h = 0 and the L...[ Read more ]
Two general coordinate systems have been used extensively in computational fluid dynamics: the Eulerian and the Lagrangian. The Eulerian coordinates cause excessive numerical diffusion across flow discontinuities, slip lines in particular. The Lagrangian coordinates, on the other hand, can resolve slip lines sharply but cause severe grid deformation, resulting in large errors and even breakdown of the computation. Recently, Hui et. al. (W.H. Hui, P.Y. Li and Z.W. Li, A Unified Coordinate System for Solving the Two-Dimensional Euler Equations, Journal of computational Physics, Vol. 153 (1999), pp. 596-637) have introduced a unified coordinate system which moves with velocity hq, q being velocity of the fluid particle. It includes the Eulerian system as a special case when h = 0 and the Lagrangian when h = 1, and was shown for the two-dimensional Euler equations of gas dynamics to be superior to both Eulerian and Lagrangian systems in resolution of flow discontinuities.
The main purpose of this thesis is to study the effect the unified coordinate system makes on resolution of discontinuities, especially slip lines, in the 1 - D and 3 - D flow modeled by the Euler equations of gas dynamics and 1 - D and 2 - D flow modeled by shallow water equations by utilizing the new degree of freedom in choosing h through employing Godunov shock-capturing scheme.
Systematic study on eight test problems as described by the one-dimensional Euler equations of gas dynamics, shows that while shock resolution is independent of h, the Lagrangian coordinates give the best accuracy and resolution of slip lines. Furthermore, with the use of shock-adaptive Godunov scheme based on Lagrangian coordinates, infinite shock resolution can also be achieved.
The unified coordinate system is adopted to solve the shallow water equations. It is shown that computational results using the unified system with h chosen to preserve grid angles are superior to existing results based on either Eulerian system or Lagrangian system in that it (a) resolves slip lines sharply, especially for steady flow, (b) avoids grid deformation and computation breakdown in Lagrangian coordinates, and (c) avoids spurious flow produced by Lagrangian coordinates.
In the other application of the unified coordinate system - to the three-dimensional Euler equations of gas dynamics, the free function h is chosen to preserve grid skewness. It has been tested on three problems and found that with the free function h so chosen, the unified coordinate system is superior to both Eulerian and Lagrangian systems in that: (a) it resolves slip lines as sharply as the Lagrangian system, especially for steady flows, (b) it avoids the severe grid deformation of the Lagrangian system which causes inaccuracy and breakdown of computation.
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