Very high-order accurate sharp immersed interface method: Application to direct numerical simulations of incompressible flows

Development of robust and very high-order accurate Immersed Interface Method (IIM) for solving the incompressible Navier-Stokes equations in the vorticity-velocity formulation on non-equidistant grids is presented. For computation of spatial derivatives on regular grid points, a seventh-order upwind combined compact difference (CCD) scheme for firstderivative and sixth-order central CCD scheme for second derivative are employed. The coefficients of the CCD schemes are constructed for a non-equidistant grid instead of using a coordinate transformation. Corrections to the finite difference schemes are used for irregular grid points near the interface of the immersed boundary to maintain high formal accuracy. For the interface points, the CCD schemes are tuned and adjusted accordingly to obtain numerically stable schemes and no jump correction will therefore be required. To demonstrate the numerical stability of the new IIM, both semi- and fully-discrete eigenvalue problems are employed for the one-dimensional pure advection (inviscid) and the pure diffusion, and advection-diffusion equations. The new IIM satisfies both necessary and sufficient conditions for numerical stability. The new IIM was first applied to two-dimensional linear advection equation to demonstrate its stability. Then the development of a new, efficient and high-order sharp-interface method for the solution of the Poisson equation in irregular domains on non-equidistant grids is presented. The underlying approach for this is based on the combination of the fourth-order compact finite difference scheme and the multiscale multigrid (MSMG) method. The computational efficiency of the new solution strategy for the Poisson equation is demonstrated with regard to convergence rate and required computer time, which shows that the MSMG method is equally efficient for domains with immersed boundaries and for simple domains. To validate the application of IIM for incompressible flows, the results from the new method is compared with the benchmark solutions for the flow past a circular cylinder and the propagation of Tollmien-Schlichting wave in a boundary layer.