On the initial conditions for the problem of flow rising in front of an obstacle

Abstract

Flowing water encounters a fixed obstacle, which causes a subsequent unsteady process whereby the water level rises. This process is modeled by the Euler equations with natural initial conditions and boundary conditions. If the impermeability boundary conditions of the obstacle do not match the initial data, a curious and unexpected phenomenon occurs: the velocity and pressure almost instantly depart from the initial values, and their new values should be considered as the “forced” initial conditions. The forced initial conditions depend on the real conditions but differ from them in the case of the mismatch mentioned above. The problem analysis assumes a smooth and local deformation of the boundary conditions from the given initial values at the initial time to the real (impermeability) values after the short transition period (and in the limit as the transition period approaches zero). A similar phenomenon may occur for hyperbolic systems of equations in general.