Immersed boundary method

The immersed boundary method is an approach by which to model and simulate mechanical systems in which elastic structures interact with a fluid flow. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations. In the immersed boundary method approach the fluid is represented in an Eulerian coordinate frame and the structures in a Lagrangian coordinate frame. For Newtonian fluids governed by the Navier-Stokes the equations the immersed boundary method fluid equations are $\rho ({\frac {\partial {u}({x},t)}{\partial {t}}}+{u}\cdot \nabla {u})=\mu \Delta u(x,t)-\nabla p+f(x,t)$ with incompressibility condition $\nabla \cdot u=0.$ The immersed structures are typically represented by a collection of interacting particles $Z_{j}$ with a prescribed force law, where $F_{j}$ is the force acting on the j^th particle. The forces are accounted for in the fluid equations by the force density $f(x,t)=\sum _{j=0}^{N}\delta _{a}(x-Z_{j})F_{j}$ where $\delta _{a}$ is an approximation of the Dirac $\delta$-function smoothed out over a length scale $a$. The immersed structures are then updated using the equation ${\frac {dZ_{j}}{dt}}=\int \delta _{a}(x-Z_{j})u(x,t)dx.$ Variants of this basic approach has been applied to simulate a wide variety of mechanical systems involving elastic structures which interact with a fluid flow, see the references for more details.

References:

(1) C. S. Peskin, The immersed boundary method, Acta Numerica, 11, pp. 1– 39, 2002.

(2) R. Mittal, G. Iaccarino, Immersed Boundary Methods, Annual Review of Fluid Mechanics, Vol. 37: pp. 239-261, 2005.

(3) Y. Mori and C.S. Peskin. Implicit Second Order Immersed Boundary Methods with Boundary Mass Computational Methods in Applied Mechanics and Engineering, 2007.

(4) L. Zhua, and C. S. Peskin, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, JCP, Volume 179, Issue 2, Pages 452-468, 2002.

(5) P. J. Atzberger, P.R. Kramer, and C.S. Peskin, A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Journal of Computational Physics, Vol. 224, Iss. 2, 2007.

(6) A. M. Roma , C. S. Peskin , M. J. Berger, An adaptive version of the immersed boundary method, Journal of Computational Physics, v.153 n.2, p.509-534, 1999.