Simulation of vorticity dominated flows using a hybrid approach. II: Numerical method

A family of numerical methods for the solution of the incompressible Navier–Stokes equations in cylindrical coordinates is developed. The formulation for hybrid spectral-finite difference discretizations in domains with cylindrical boundaries presented in Kollman, in press [Kollmann W, Simulation of vorticity dominated flows using a hybrid approach: I formulation, in press] forms the basis of the solvers. The solution methods use the Fourier-spectral approach for the azimuthal and a set of finite-difference operators for the radial and axial directions. The convective terms are discretized with a linear combination of upwind-biased and central difference operators applied to the non-conservative and conservative formulations, respectively. All other spatial derivatives are discretized with central operators. The time integration is specified as a minimum storage, state space, fourth order Runge–Kutta method. The convergence of the solvers as the formal accuracy of the finite-difference operators varies is tested for an axi-symmetric flow for fixed discretization and time integrator. The results show satisfactory convergence with respect to order of accuracy and the convective operators.