Some simple immersed boundary techniques for simulating complex
Some simple immersed boundary techniques for simulating complex flows with rigid boundary Ming-Chih Lai Department of Applied Mathematics National Chiao Tung University 1001, Ta Hsueh Road, Hsinchu 30050 Taiwan mclai@math. nctu. edu. tw
Outline of the talk: • Review of the Immersed Boundary Method (IB method) • Immersed boundary formulation for the flow around a solid body • Feedback forcing + IB method • Direct forcing approach • Volume-of-Fluid approach • Interpolating forcing approach • Numerical results
Review of the IB method: A general numerical method for simulations of biological systems interacting with fluids (fluid interacts with elastic fiber). Typical example: blood interacts with valve leaflet (Charles S. Peskin, 1972, flow patterns around heart valves) Applications: • computer-assisted design of prothetic valve (Peskin & Mc. Queen) • Platelet aggregation during blood clotting (Fogelson, Fausi) • flow of particle suspensions (Fogelson & Peskin, Sulsky & Brackbill) • wave propagation in the cochlea (Beyer) • swimming organism (Fausi)
• arteriolar flow (Arthurs, et. al. ) • cell and tissue deformation under shear flow (Bottino, Stockie & Green, Eggleton & Popel) • flow around a circular cylinder (Lai & Peskin) • valveless pumping (Jung & Peskin) • flapping filament in a flowing soap film (Zhu & Peskin) • falling papers, sails, parachutes, insect flight, …… Recent review : C. S. Peskin, Acta Numerica, pp 1 -39, (2002).
Idea: Mathematical formulation: • Treat the elastic material as a part of fluid. • The material acts force into the fluid. • The material moves along with the fluid. Numerical method: • • Finite difference discretization. Eulerian grid points for the fluid variables. Lagrangian markers for the immersed boundary. The fluid-boundary are linked by a smooth version of Dirac delta function.
Mathematical formulation
The force density is singular !
Numerical algorithm
Numerical issues of IB method: • • • simple and easy to implement first-order accurate numerical smearing near the immersed boundary high-order discrete delta function numerical stability, semi-implicit method
IB formulation for the flow around a solid body
Numerical Results • • • Decaying vortex problem Lid-driven cavity problem A cylinder in lid-driven cavity Flow around a circular cylinder The flow past an in-line oscillating cylinder
- Slides: 52