High order vorticity–velocity method for the simulation of pipe flow transition Original Research Article

In this paper a numerical method for the direct simulation of pipe flow transition is described. The algorithm is based on a vorticity transport formulation of the Navier–Stokes equations for incompressible flow. The basic equations are discretized in space using compact finite differences with an accuracy of at least 7th order for the axial and radial coordinates, and using a fully de-aliased Fourier pseudo-spectral method for the azimuthal coordinate of the cylindrical coordinate system. Time integration is done using a modified 4th order Runge–Kutta scheme that integrates some of the linear terms exactly to alleviate time-step restrictions of our explicit scheme. The incompressibility constraint and vorticity definition are enforced using an influence matrix technique. Comparisons with predictions from stability theory for the limiting case of small disturbances as well as results from fully nonlinear simulations are presented.