A second-order cut-cell method for the numerical simulation of 2D flows past obstacles.

*(English)*Zbl 1365.76186Summary: We present a new second-order method, based on the MAC scheme on cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. In this approach, the solid boundary is embedded in the cartesian computational mesh. Discretizations of the viscous and convective terms are formulated in the context of finite volume methods ensuring local conservation properties of the scheme. Classical second-order centered schemes are applied in mesh cells which are sufficiently far from the obstacle. In the mesh cells cut by the obstacle, first-order approximations are proposed. The resulting linear system is nonsymmetric but the stencil remains local as in the classical MAC scheme on cartesian grids. The linear systems are solved by a fast direct method based on the capacitance matrix method. The time integration is achieved with a second-order projection scheme. While in cut-cells the scheme is locally first-order, a global second-order accuracy is recovered. This property is assessed by computing analytical solutions for a Taylor-Couette problem. The efficiency and robustness of the method is supported by numerical simulations of 2D flows past a circular cylinder at Reynolds number up to 9500. Good agreement with experimental and published numerical results are obtained.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

76M12 | Finite volume methods applied to problems in fluid mechanics |

76M15 | Boundary element methods applied to problems in fluid mechanics |

##### Keywords:

immersed boundary method; Cartesian grid method; cut-cell method; incompressible viscous flows; complex geometry
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