An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries.

*(English)*Zbl 0957.76043Summary: We develop a Cartesian grid method for simulating two-dimensional unsteady viscous incompressible flows with complex immersed boundaries. A finite volume method based on a second-order accurate central-difference scheme is used in conjunction with a two-step fractional-step procedure. The key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cell that are cut by these boundaries. To this end, we present a new interpolation procedure which allows systematic development of a spatial discretization scheme that preserves the second-order spatial accuracy of the underlying solver. The presence of immersed boundaries alters the conditioning of the linear operators, and this can slow down the iterative solution of these equations. The convergence is accelerated by using a preconditioned conjugate gradient method, where the preconditioner takes advantage of the structured nature of the underlying mesh. The accuracy of the solver is validated by simulating a number of canonical flows, and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated.

##### MSC:

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

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

##### Keywords:

Cartesian grid method; two-dimensional unsteady viscous incompressible flows; complex immersed boundaries; finite volume method; second-order accurate central-difference scheme; two-step fractional-step procedure; boundary conditions; interpolation procedure; spatial discretization; iterative solution; convergence; preconditioned conjugate gradient method
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