A Projection Method for Incompressible Viscous Flow on Moving QuadrilateralGrids

We present a second-order accurate projection method for numerical solution of the incompressible Navier–Stokes equations on moving quadrilateral grids. Our approach is a generalization of the Bell–Colella–Glaz (BCG) predictor–corrector method for incompressible flow. Irregular geometry is represented in terms of a moving, body-fitted cylindrical coordinate system. Mapped coordinates are used to smoothly transform in both time and space the moving domain onto a logically rectangular domain which is fixed in time. To treat the time dependence and inhomogeneities in the incompressibility constraint introduced by the presence of deforming boundaries, we introduce a nontrivial splitting of the velocity field into vortical and potential components to eliminate the inhomogeneous terms in the constraint and a generalization of the BCG algorithm to treat time-dependent constraints. The method is second-order accurate in space and time, has a time step constraint determined by the advective CFL condition, and requires the solution of well-behaved linear systems amenable to the use of fast iterative methods. We demonstrate the method on the specific example of viscous incompressible flow in an axisymmetric deforming tube.