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A Cartesian grid method for computing flows with complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. A mixed Eulerian–Lagrangian framework is employed, which allows us to treat the immersed moving boundary as a sharp interface. The incompressible Navier–Stokes equations are discretized using a second-order-accurate finite-volume technique, and a second-order- accurate fractional-step scheme is employed for time advancement. The fractional-step method and associated boundary conditions are formulated in a manner that properly accounts for the boundary motion. A unique problem with sharp interface methods is the temporal discretization of what are termed “freshly cleared” cells, i.e., cells that are inside the solid at one time step and emerge into the fluid at the next time step. A …