Hong Kong University of Science and Technology
Hong Kong, China
There exist two different descriptions of fluid motion. The Eulerian one considers flow variables to be functions of time and of fixed space coordinates; consequently, it produces excessive numerical diffusion, across sliplines in particular. By contrast, the Lagrangian one considers flow variables to be functions of time and of some permanent identification of fluid particles; consequently, computational grids can be severely distorted, rendering the description invalid.
A unified description is given in which the flow variables are considered to be functions of time and of some permanent identification of pseudo-particles which move with velocity $hq$, $q$ being fluid velocity. It includes the Eulerian description as a special case when $h=0$ and the Lagrangian one when $h=1$. The free function $h$ is chosen to avoid excessive numerical diffusion and severe grid distortion. Numerical computations of $2-D$ flow have shown the unified coordinate system to be superior to the Eulerian and the Lagrangian one. As a by-product the Lagrangian description is found to have an additional deficiency, namely, the $2-D$ and $3-D$ unsteady Euler equations of gasdynamics based on Lagrangian coordinates are only weakly hyperbolic, with all its consequences on computation