The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

Abstract

Peskin's Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be first-order accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier-Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a second-order projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully second-order accuracy for the velocity and nearly second-order accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper.

A velocity decomposition approach for moving interfaces in viscous fluids

“…The fluid velocities u and u r are known at time levels t n and t n−1 , but only at grid points, which likely do not coincide with the upstream positions x n and x n−1 that are needed in (19), (23), and (24). Thus, cubic Lagrange interpolation in the spatial variable is used to estimate u and u r at other locations.…”

“…In (23) u n+ 1 2 , the value at location x 0 , is approximated using the time extrapolation 3 2 u n − 1 2 u n−1 , and for u ·, t n+ 1 2 at other locations, the same extrapolation is used in time, as well as spatial interpolation. Once x n and x n−1 are known, they are used to findũ n r andũ n−1 r in (19), and the forcing term F n+1 b is treated analogously.…”

“…One approach for achieving better accuracy is the immersed interface method [16,18] (or similarly, Mayo's method [20]), which first expresses the jumps in the solution and its derivatives in terms of the boundary force and its derivatives, and then incorporates these jumps into a finite difference scheme as correction terms [16,19,32]. This method was first applied to the Stokes equations [16], the simplified model at zero Reynolds number in which acceleration and advection are neglected.…”

“…The method can also be applied to the full Navier-Stokes equations (1) and (2). This was first done in [19] and [15]. In principle this method can be quite accurate, but it becomes rather involved because of the large number of cases and corrections needed.…”

“…For the regular part, it appears that if we were to solve (13) and (14) by means of the immersed interface method, we should not have to impose corrections for jumps; see [19]. However, the computed solution will still be sensitive to the choice of numerical method.…”