A novel sharp interface method is considered for solving the Helmholtz equation with interface in cylindrical coordinates, and is applied to solve incompressible axisymmetric flows. In order to overcome the singularity at \(r=0\) and the reduction in precision of numerical solutions near the interface, a modified second-order finite difference scheme is constructed using the immersed interface method and Taylor series expansion, combined with interface relationships on a staggered grid. The projection method is utilized to address the incompressible axisymmetric flow problem with an interface, then the original problem is transformed into several Helmholtz equations with an interface. The sharp interface method is then repeatedly used to numerically determine the physical quantities of fluid velocity and pressure. Numerical experiments verify the effectiveness of the proposed method.
