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Some simple immersed boundary techniques for simulating complex flows with rigid boundary

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 :.

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Some simple immersed boundary techniques for simulating complex flows with rigid boundary

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  1. 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

  2. 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

  3. 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 & McQueen) • Platelet aggregation during blood clotting (Fogelson, Fausi) • flow of particle suspensions (Fogelson & Peskin, Sulsky & Brackbill) • wave propagation in the cochlea (Beyer) • swimming organism (Fausi)

  4. • 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).

  5. 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.

  6. Mathematical formulation

  7. The force density is singular !

  8. Numerical algorithm

  9. 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

  10. IB formulation for the flow around a solid body

  11. 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

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