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1. solid boundary force density Dirac  function circular Couette flow flow past moving cylinder(s) Efficiency Accuracy flow in a rotating sphere 3D relaxing balloon in a fluid swimming wing flexibility in flapping flight 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 flight[5] 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.