An Immersed Interface Method for Fluid-Solid Interaction

1
An Immersed Interface Method for Fluid-Solid
Interaction
Sheng Xu, Department of Mathematics, Southern
Methodist University
Introduction
The immersed interface method (IIM) 1 is a
variant of the immersed boundary (IB) method 2.
In both methods, solids in a fluid are modeled as
forces in the Navier-Stokes equations and tracked
with Lagrangian markers or level sets. The IB
method spreads the forces with the use of
discrete Dirac ? functions. The IIM incorporates
force-induced flow jump conditions into finite
difference schemes, which better resolves
fluid-solid boundaries and their effects. This
poster presents the recent work to derive the
necessary jump conditions 3 and to implement
them in 2D 4 and 3D. The results indicate that
the IIM (1) achieves near 2nd order accuracy in
the infinity norm, (2) introduces relatively
insignificant cost with the addition of a solid,
and (3) conserves mass enclosed by
non-penetration boundaries.
For example, a 2nd order accurate central finite
difference scheme can be modified as the
following.
Numerical Implementation
Governing Equations
The MAC scheme, the classical 4th order RK
integration, and an FFT Poisson solver are used
to implement jump conditions in both 2D and 3D.
To find solid-boundary grid-line intersection
points and to interpolate jump conditions from
Lagrangian markers to them, cubic splines are
used in 2D, and parametric triangulation is used
in 3D. The velocity of a Lagrangian marker is
interpolated from the intersection points in 2D
with cubic splines. In 3D, it is interpolated
directly from surrounding grid nodes with
interpolation schemes accounting for jump
conditions.
A solid in a fluid is modeled as a force
distribution in the Navier-Stokes equations.
Results and Applications
Jump Conditions
Because of the force singularity in the form of
the Dirac ? function, the flow field is
generally not smooth across the boundary. Derived
from the governing equations, the principal
spatial jump conditions of the velocity, the
pressure, and their normal derivatives are
accounting for temporal jumps, otherwise wrong
here, but fine in viscous flow
where
Conservation
vorticity and velocity fields induced by a 2D
relaxing balloon
Temporal jumps
Using the facts that a jump condition is a
function of time and Lagrangian parameters, and a
jump operation commutes with differentiation, the
following spatial jump conditions have been
derived.
A flow quantity ? at a fixed point in space can
have a jump with respect to time when the
boundary passes the point at time ts . The
temporal jump condition is related to the
corresponding spatial jump condition.
Applications
wing rotation in dragonfly flight5
others
References
1 Randall J. LeVeque Zhilin Li, The immersed
interface method for elliptic equations with
discontinuous coefficients and singular sources,
SIAM J. Numer. Anal. 31 (4) (1994) 1019-1044 2
Charles S. Peskin, Flow patterns around heart
valves a numerical method, J. Comput. Phys. 10
(1972) 252-271 3 Sheng Xu Z. Jane Wang,
Systematic derivation of jump conditions for the
immersed interface method in three-dimensional
flow simulation, SIAM J. Sci. Comput. 27 (6)
(2006) 1948-1980 4 Sheng Xu Z. Jane Wang, An
immersed interface method for simulating the
interaction of a fluid with moving boundaries, J.
Comput. Phys. 216(2) (2006) 454-493 5 Attila J.
Bergou, Sheng Xu Z. Jane Wang, Passive wing
rotation in dragonfly flight, J. Fluid Mech.
(submitted)
Finite Differences
Without losing accuracy, a usual finite
difference scheme has to be modified to take into
account jump conditions when its stencil is
crossed by the boundary. The modification is
based on the following generalized Taylor
expansion for a non-smooth function.