Entropy Errors in the Numerical Approximation of the Euler Flow along a Kinked Wall

Abstract

Numerical entropy generation is studied in the case of steady, subsonic Euler flow along a kinked wall. For a standard upwind finite volume discretization, the numerical entropy error appears to be zeroth-order in mesh size. Two possible causes of the zeroth-order entropy error are studied. First an investigation is made of the local discretization error on a kinked grid. In the neighbourhood of the kink, this error appears to be zeroth-order as well. For supersonic flow it can be easily removed. However, it seems that the zeroth-order entropy error is yet not caused by the zeroth-order discretization error. Next a study is made of the existence of a singularity in the exact solution. Probably, the Euler flow solution is singular at the kink in the wall. The form of the likely singularity is unknown. Therefore, the construction of a computational method which uses a priori knowledge of the singularity is not possible. Finally it is shown by numerical experiment that the subsonic Euler flow along a kinked wall still can be computed with vanishing entropy errors, by using a continuously curved wall which converges to the kinked wall in the limit of zero mesh width.