An Immersed Interface Method for the Incompressible Navier–Stokes Equations with Discontinuous Viscosity Across the Interface

Abstract

We present an immersed interface method for solving the incompressible Navier–Stokes equations with discontinuous viscosity across the interface and singular forces. The method is based on the augmented strategy proposed by Li, Ito, and Lai [Comput. Fluids, 36 (2007), pp. 622–635] to decouple the jump conditions of the fluid variables through the introduction of two augmented variables. In the proposed method, the immersed interface is represented by a number of Lagrangian control points, and the augmented interface variables applicable along the interface are solved by the LU method or the GMRES iterative method. In addition, we also derive a new second order accurate bilinear interpolating scheme for the discontinuous velocity and a new jump condition for the normal derivative of the pressure. The jumps in both pressure and velocity and the jumps in their derivatives are related to the augmented interface variables and/or the forces which are interpolated using cubic splines. For a flexible interface, the forces that the interface exerts on the fluid are computed from the constitutive relation of the flexible interface and are applied to the fluid through the jump conditions. The position of the flexible interface is updated implicitly within each time step. The Navier–Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method. The numerical results show that the overall scheme is second order accurate.