An immersed interface method for incompressible flow with moving boundaries and high order time integration

Abstract

In this work we present a novel Immersed Interface Method (IIM) for simulating two dimensional incompressible flows involving moving rigid bodies immersed in an unbounded fluid domain. To do so, we solve the Navier-Stokes equations in vorticity-stream function form, using a second order IIM spatial discretization that allows for the use of high order explicit Runge-Kutta time integration. We begin by reviewing existing work on the immersed interface method, and developing novel algorithms for stencil calculation, geometry processing, and integration over irregular domains. We then introduce a stable IIM discretization of the advection-diffusion equation, and describe an improved version of the IIM Poisson solver developed by Gillis [9]. We review vorticity-based formulas for calculating the local tractions and global forces acting on an immersed body, and present a novel extension of the control-volume force calculation methods developed by Noca [16].

This first section culminates in the presentation of an IIM Navier Stokes solver for problems on stationary domains, which is shown to have second-order spatial accuracy and third-order temporal accuracy. The second portion of this work develops a general IIM framework for discretizing PDEs on moving domains. We focus on schemes that are compatible with explicit high-order Runge-Kutta methods, and demonstrate that our method introduces a mixed spatial-temporal error term not seen in stationary IIM discretizations. We also consider CFL-like restrictions that limit the maximum time step used in problems with moving domains, and develop geometric criteria to ensure that these restrictions are met. Using these new methods, we extend our existing IIM Navier Stokes solver to allow for moving boundaries, and verify that the method retains its second-order spatial and third order temporal accuracy.

Finally, we demonstrate the applicability of the algorithm to complex two-dimensional flow problems by calculating the time-dependent lift, thrust, and moment coefficients of a flapping airfoil.