The Immersed Interface Method for Navier-Stokes Equations with Interfaces in Cylindrical Coordinates

Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates. In this paper, the immersed interface method (IIM) tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed. Note that, while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature, there is a gap in dealing with Navier-Stokes equations, where the non-linear term includes an additional component involving the coordinate \(\varphi \) $\phi$ , even if the geometry and force term are axis-symmetric. Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity, in addition to the pole singularity at \(r=0\) $r=0$ . In the newly developed algorithm, we have derived the jump conditions under the cylindrical coordinates. The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates. The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface. As expected, the method is shown to be second-order accurate for the velocity. The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.