An accurate numerical method for DNS of turbulent pipe flow

In contrast with channel flow, there are only few numerical methods for DNS of turbulent pipe flow [1]. The reason for this is the singularity at the pipe axis, which results from transformation of the governing equations to cylindrical coordinates. Most numerical methods which are presently being used can be divided in two classes. In the first class [2, 3, 4] use is made of a finite volume approach on a staggered grid. In such an approach it is not so obvious how to find suitable boundary conditions at the pipe axis. In the second class [5, 6] a spectral approach is chosen with a Fourier expansion in the two periodic (streamwise and azimuthal) directions and a Chebyshev collocation method in the radial direction. In order to avoid the clustering of collocation points near the pipe axis, where it is undesired, the radial direction is divided into several elements, which are coupled using continuity conditions for the velocity components and their radial derivatives. In such a method it is hardly possible to satisfy the continuity equation within machine accuracy. Moreover, the division of the radial direction into elements is somewhat arbitrary, reduces the global accuracy of the spectral method and makes it almost impossible to use the method for large-eddy simulation. Therefore, in the present work a novel spectral method for DNS of turbulent pipe flow is developed, which circumvents the disadvantages of the spectral element method mentioned above. The method yields a fully divergence free velocity field and satisfies all regularity conditions at the axis of the pipe