Commun. Comput. Phys., 6 (2009), pp. 997-1021.


An Indirect-Forcing Immersed Boundary Method for Incompressible Viscous Flows with Interfaces on Irregular Domains

Zhijun Tan 1*, K. M. Lim 2, B. C. Khoo 2, Desheng Wang 3

1 Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore.
2 Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore; and Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore.
3 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore.

Received 14 August 2008; Accepted (in revised version) 20 March 2009
Available online 8 May 2009

Abstract

An indirect-forcing immersed boundary method for solving the incompressible Navier-Stokes equations involving the interfaces and irregular domains is developed. The rigid boundaries and interfaces are represented by a number of Lagrangian control points. Stationary rigid boundaries are embedded in the Cartesian grid and singular forces at the rigid boundaries are applied to impose the prescribed velocity conditions. The singular forces at the interfaces and the rigid boundaries are then distributed to the nearby Cartesian grid points using the immersed boundary method. In the present work, the singular forces at the rigid boundaries are computed implicitly by solving a small system of equations at each time step to ensure that the prescribed velocity condition at the rigid boundary is satisfied exactly. For deformable interfaces, the forces that the interface exerts on the fluid are computed from the configuration of the elastic interface and are applied to the fluid. The Navier-Stokes equations are discretized using finite difference method on a staggered uniform Cartesian grid by a second order accurate projection method. The ability of the method to simulate viscous flows with interfaces on irregular domains is demonstrated by applying to the rotational flow problem, the relaxation of an elastic membrane and flow in a constriction with an immersed elastic membrane.

AMS subject classifications: 65N06, 76D05, 65M12

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Key words: Incompressible Navier-Stokes equation, fast Poisson solvers, immersed boundary method, projection method, Cartesian grid, irregular domain, finite difference methods.

*Corresponding author.
Email: smatz@nus.edu.sg (Z.-J. Tan), mpelimkm@nus.edu.sg (K. M. Lim), mpekbc@nus.edu.sg (B. C. Khoo), desheng@ntu.edu.sg (D. S. Wang)
 

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