INVISCID INCOMPRESSIBLE AND COMPRESSIBLE FLOW EQUATIONS UNDER SPACE-TIME TRANSFORMATION

INTRODUCTIONCoordinate transformation is frequently usedin fluid mechanics such as in computational fluiddynamics with curvilnear geometries.Tradition-ally,the physical time is directly used in thetransformed frame.In this paper we consider amore general transformation in which the time inthe new frame,called pseudo time for conve-nience,depends not only on the original physicaltime,but also on the space.First of all,thestudy of such a general transformation has obvi-ous academic signific…

INVISCID INCOMPRESSIBLE AND COMPRESSIBLE FLOW EQUATIONS UNDER SPACE-TIME TRANSFORMATION

The equations governing incompressible and compressible inviscid flows and written in the physical frame (t ,x,y,z) are known to be linearly well-posed and exhibit elliptic or hyperbolic nature. The linear well-posedness is considered here for these equations under a space-time transformation (t,x,y,z)→(τ,ξ,η,ζ), where the pseudo-time τ and the new space coordinate (ξ,η,ζ) all depend on (t,x,y,z). Such a transformation could be useful for uniformly treating problems in which the flow is fast unsteady somewhere and slow unsteady or steady elsewhere. It is found that the transformation may alter the ellipticity, the hyperbolicty, and even the well-posedness of the original equations. In one dimension, the transformed incompressible flow equations become weakly hyperbolic and the compressible ones could degenerate to elliptical equations. In high dimensions there are conditions such that the transformed equations become ill-posed.``