A projection method for incompressible viscous flow on a deformable domain

A second-order accurate finite difference method is presented for numerical solution of the incompressible Navier-Stokes equations on a deformable domain. The target problem is flow in a flexible tube. Fluid motion is forced by movement of the solid wall which requires special treatment of boundary conditions at inflow and outflow. The problem is formulated in terms of a moving, body-fitted coordinate system in (r, z). 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. The velocity field is split into vortical and potential components. The splitting allows the incompressibility constraint to be enforced in the vortical part and the time-dependent boundary conditions to be carried in the potential component.;The discretization method is based on a new predictor-corrector time discretization which generalizes the Bell-Colella-Glaz method for incompressible flow. Advection terms are obtained by an explicit, second-order Godunov scheme. A MAC projection ensures proper treatment of the pressure and enforcement of the constraint in the advection step. Crank-Nicholson discretization is used to obtain the viscous terms which contain higher-order boundary conditions. Pressure and divergence-free velocity are computed by the application of a Hodge projection. The algorithm is second-order accurate in space and time for a range of laminar flows and domain motions.