Vorticity-velocity formulation of the 3D Navier-Stokes equations in cylindrical co-ordinates

A finite difference method is presented for solving the 3D Navier-Stokes equations in vorticity-velocity form. The method involves solving the vorticity transport equations in ʹcurl-formʹ along with a set of Cauchy-Riemann type equations for the velocity. The equations are formulated in cylindrical co-ordinates and discretized using a staggered grid arrangement. The discretized Cauchy-Riemann type equations are overdetermined and their solution is accomplished by employing a conjugate gradient method on the normal equations. The vorticity transport equations are solved in time using a semi-implicit CrankNicolson/Adams-Bashforth scheme combined with a second-order accurate spatial discretization scheme. Special emphasis is put on the treatment of the polar singularity. Numerical results of axisymmetric as well as non-axisymmetric flows in a pipe and in a closed cylinder are presented. Comparison with measurements are carried out for the axisymmetric flow cases.