A Jacobian-free-based IIM for incompressible flows involving moving interfaces with Dirichlet boundary conditions

Abstract

In this paper, a finite difference marker-and-cell (MAC) scheme is presented for the steady Stokes equations with moving interfaces and Dirichlet boundary condition. The moving interfaces are represented by Lagrangian control points and their position is updated implicitly using a Jacobian-free approach within each time step. The forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions. The proposed Jacobian-free Newton-generalized minimum residual (GMRES) method avoids the need to form and store the matrix explicitly in the computation of the inverse of the Jacobian and betters numerical stability. The Stokes equations are discretized on a MAC grid via a second-order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate very well the accuracy and effectiveness of the proposed method. The present algorithm has been applied to solve incompressible Navier-Stokes flows with moving interfaces. © 2010 John Wiley & Sons, Ltd.