Numerical Solution of the Incompressible Navier-Stokes Equations in Doubly-Connected Regions

A transient computational procedure for the numerical solution of the incompressible Navier-Stokes equations in doubly-connected domains is described. The tangential velocity boundary conditions are satisfied by introducing shear layers at the boundaries at each time-step. The requirement that the pressure be single-valued is explicitly included in the formulation. The computational procedure is applied to the viscous flow in the annulus between two eccentric cylinders when either of them is rotating. Transient and steady-state results are obtained for both Stokes flow (inertia neglected) and Navier-Stokes flow. Rotational speeds of the outer cylinder are explored for the entire extent of laminar flow and of the inner cylinder up to the point of incidence of Taylor vortices. The agreement between the steady-state numerical results and the exact analytical results for the Stokes flow is very good. It is believed that inclusion of inertia does not appreciably affect the solution accuracy.