The immersed boundary approach for the modeling of complex geometries in incompressible flows is examined critically from the perspective of satisfying boundary conditions and mass conservation. The system of discretized equations for mass and momentum can be inconsistent if the real velocities are used in defining the forcing terms used to satisfy the boundary conditions. As a result, the velocity is generally not divergence free and the pressure at locations in the vicinity of the immersed boundary is not physical. However, the use of the pseudo velocities in defining the forcing (as frequently done when the governing equations are solved using a fractional step or projection method) combined with the use of the specified velocity on the immersed boundary is shown to result in a consistent set of equations which allows a divergence free velocity but, depending on the time step used to obtain a steady state solution, is shown to have an undesirable effect of allowing significant permeability of the immersed boundary. An improvement is shown if the pressure gradient is integrated in time using the Crank-Nicholson scheme instead of the backward Euler scheme. However, even with this improvement a significant reduction in the time step and hence increase in computational expense is still required for sufficient satisfaction of the boundary conditions.
Mass Conservation in the Immersed Boundary Method
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Muldoon, F, & Acharya, S. "Mass Conservation in the Immersed Boundary Method." Proceedings of the ASME 2005 Fluids Engineering Division Summer Meeting. Volume 1: Symposia, Parts A and B. Houston, Texas, USA. June 19–23, 2005. pp. 411-419. ASME. https://doi.org/10.1115/FEDSM2005-77301
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