A piecewise linear contour to avoid critical points in inviscid flow stability analyses

In inviscid flow stability analyses critical points are encountered which spoil the convergence of spectral methods for the computation of unstable modes as they become neutral. One way to alleviate this problem is to make a detour in the complex plane, which is often done by using a parabolic or cubic mapping. A piecewise linear profile has also been proposed in the literature for shooting methods. Its use with a spectral collocation method is investigated in the present paper. The method is applied to solve the linearized Euler equations for the computation of a stable surface mode in a channel flow with a lined wall, modelled as a mechanical oscillator.