An immersed boundary method by the lattice Boltzmann approach in three dimensions with application

The immersed boundary (IB) method originated by Peskin has been popular in modeling and simulating problems involving the interaction of a flexible structure and a viscous incompressible fluid. The Navier-Stokes (N-S) equations in the IB method are usually solved using numerical methods such as FFT and projection methods. Here in our work, the N-S equations are solved by an alternative approach, the lattice Boltzmann method (LBM). Compared to many conventional N-S solvers, the LBM can be easier to implement and more convenient for modeling additional physics in a problem. This alternative approach adds extra versatility to the immersed boundary method. In this paper we discuss the use of a 3D lattice Boltzmann model (D3Q19) in the IB method. We use this hybrid approach to simulate a viscous flow past a flexible sheet tethered at its middle-line in a 3D channel and determine a drag scaling law for the sheet. Our main conclusions are 1)the hybrid method is convergent with first-order accuracy which is consistent with the immersed boundary method in general; 2)the drag of the flexible sheet appears to scale with the inflow speed which is in sharp contrast with the square law for a rigid body in a viscous flow. Preprint submitted to Computers & Mathematics with Applications January 29, 2010