The interaction between flapping wings and unsteady flows is governed by the Navier-Stokes equation coupled to the dynamic boundaries. The typical Reynolds number for insect flight is in the range of 10-10000, where both the viscous and inertial forces are relevant. While some of the most interesting fluid dynamics originates near a moving sharp interface, computational schemes typically encounter great difficulty in resolving moving interfaces, a known difficulty in nearly all fluid-structure simulations.

We have improved a number of computational methods and developed new algorithms for solving the Navier-Stokes equation coupled to moving interfaces. The first set of codes are Navier-Stokes solvers for simulating a 2D rigid flapping wing, based on an essentially compact 4th order finite difference scheme in vorticity and stream function formulation. In these solvers we took advantage of coordinate transformations and conformal mapping to resolve sharp wing tips so as to avoid grid-regeneration. With these methods, we have investigated the unsteady aerodynamics of forward and hovering flight as well as of plates freely falling in a fluid. To simulate multiple bodies, we have further developed a Cartesian method coupled to an overset grid that are attached to the moving geometries.

To go beyond 2D simulations of rigid objects, we have recently developed a more general purpose code for simulating multiple flexible freely objects. The main advance in the new immersed interface method is to obtain a 2nd order accuracy of the sharp moving surface at an intermediate range of Reynolds flows. To avoid introducing ad-hoc boundary conditions at the moving interface, we derived systematically from the 3D Navier-Stokes equation the jump conditions on the fluid variables caused by the singular force. Our analysis showed that a 2nd order accuracy along the sharp interface requires jump conditions on derivatives of velocity that are of higher order than those appearing in the principal jump conditions. In addition, the temporal jump conditions must be included in order to have a correct scheme. To handle the spatial and temporal jump conditions in the finite difference scheme, we derived a generalized Taylor expansions for functions with discontinuity of an arbitrary order.

This code has been adapted to studies of passive wing pitching in insect flight. Most recently, the code has been further developed to simulate hydrodynamic interactions of multiple objects at intermediate Reynolds numbers.

Particle collisions in fluids are ubiquitous, but to compute the collision dynamics in a Navier-Stokes flow remains challenging. In addition to capturing the two-way coupling between the fluid and the particles, a key difficulty is to resolve the collision dynamics mediated by the flow. The gap between particles during collision is minuscule. This introduces a small length scale which needs to be resolved simultaneously with the flow at the large scale. Our goal is to develop a numerical scheme that is accurate and efficient in computing the Navier-Stokes flow around moving particles while taking into account the effect of the lubrication forces on the collisions. Our method integrates the immersed interface method with the lubrication theory in a way that directly couples all three parts, the bulk flow, the flow in the gap, and the dynamics of the freely moving particles. We present a general algorithm for computing the collision. To test the method, we study four fundamental cases involving normal and tangential collisions, so that we can compare numerics against analytic solutions in the lubrication layer. In addition, we provide the lubrication solution needed for computing collisions between surfaces of any shapes in arbitrary relative motions, so that the method can be applied to other cases.

In immersed interface methods, solids in a fluid are represented by singular forces in the Navier-Stokes equations, and flow jump conditions induced by the singular forces directly enter into numerical schemes. This paper focuses on the implementation of an immersed interface method for simulating fluid-solid interaction in 3D. The method employs the MAC scheme for the spatial discretization, the RK4 scheme for the time integration, and an FFT-based Poisson solver for the pressure Poisson equation. A fluid-solid interface is tracked by Lagrangian markers. Intersections of the interface with MAC grid lines identify finite difference stencils on which jump contributions to finite difference schemes are needed. To find the intersections and to interpolate jump conditions from the Lagrangian markers to the intersections, parametric triangulation of the interface is used. The velocity of the Lagrangian markers is interpolated directly from surrounding MAC grid nodes with interpolation schemes accounting for jump conditions. Numerical examples demonstrate that (1) the method has near second-order accuracy in the infinity norm for velocity, and the accuracy for pressure is between first and second order; (2) the method conserves the volume enclosed by a no-penetration boundary; and (3) the method can efficiently handle multiple moving solids with ease.

In this paper, we systematically derive jump conditions for the immersed interface method [SIAM J. Numer. Anal., 31 (1994), pp. 1019-1044; SIAM J. Sci. Comput., 18 (1997), pp.709-735] to simulate three-dimensional incompressible viscous flows subject to moving surfaces. The surfaces are represented as singular forces in the Navier-Stokes equations, which give rise to discontinuities of flow quantities. The principal jump conditions across a closed surface of the velocity, the pressure, and their normal derivatives have been derived by Lai and Li [Appl. Math. Lett., 14 (2001), pp. 149-154]. In this paper, we first extend their derivation to generalized surface parametrization. Starting from the principal jump conditions, we then derive the jump conditions of all first-, second-, and third-order spatial derivatives of the velocity and the pressure. We also derive the jump conditions of first- and second-order temporal derivatives of the velocity. Using these jump conditions, the immersed interface method is applicable to the simulation of three-dimensional incompressible viscous flows subject to moving surfaces, where near the surfaces the first- and second order spatial derivatives of the velocity and the pressure can be discretized with, respectively, third- and second-order accuracy, and the first-order temporal derivatives of the velocity can be discretized with second-order accuracy.

Solving flows numerically in open geometry requires specification of flows at some finite computational boundary. I find a simple scheme for implementing the far-field boundary condition exactly for the Poisson equation with source term of finite support. This numerical scheme avoids introducing mixed boundary conditions in the far field, thus it can be used efficiently with FFT.