In this talk, we justify the low Mach number limit of the steady irrotational Euler flows for the airfoil problem, which is the first result for the low Mach number limit of the steady Euler flows in an exterior domain. The uniform estimates on the compressibility parameter, which is singular for the flows, are established via a variational approach based on the compressible-incompressible difference functions. The limit is on the Holder space and is unique. It is noticeable that, due to the feature of the airfoil problem, the extra force dominates the asymptotic decay rate of the compressible flow to the infinity. And the effect of extra force vanishes in the limiting process from compressible flows to the incompressible ones, as the Mach number goes to zero.Also, we will mention the work on the bounded nozzle. This is the joint work with Mingjie Li and Wei Xiang.