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An immersed interface method for simulating the interaction of a fluid with moving boundaries. (English) Zbl 1220.76058
Summary: In the immersed interface method, boundaries are represented as singular force in the Navier–Stokes equations, which enters a numerical scheme as jump conditions. Recently, we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3D with second-order spatial and temporal accuracy near the boundaries [SIAM J. Sci. Comput. 27, No. 6, 1948–1980 (2006; Zbl 1136.76346)]. In this paper we implement the immersed interface method to incorporate these jump conditions in a 2D numerical scheme. We study the accuracy, efficiency and robustness of our method by simulating Taylor–Couette flow, flow induced by a relaxing balloon, flow past single and multiple cylinders, and flow around a flapping wing. Our results show that: (1) our code has second-order accuracy in the infinity norm for both the velocity and the pressure; (2) the addition of an object introduces relatively insignificant computational cost; (3) the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76M20 Finite difference methods applied to problems in fluid mechanics
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PHYSALIS
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