Direct transversal integration of the Navier-Stokes equations

Abstract

This work concerns the numerical integration of the vorticity-streamfunction formulation of the Navier-Stokes equations in their Fourier Space form. The expansion is done with respect to the direction along which the flow is periodic (the expansion direction), and the direct integration is performed along the direction normal to that (transversal direction). In this paper, the Navier-Stokes equations are expanded in Fourier series along the axial direction for plane Couette flow, along the azimuthal coordinate θ for the case of plane flow around a circular cylinder 1, and along the longitudinal coordinate z for the case of axisymmetric flow between concentric rotating cylinders 2. The resulting vorticity transport equations contain convolution terms which can be efficiently calculated by the use of the Fast Fourier Transform (FFT) algorithm 3. The streamfunction-vorticity equation becomes an ordinary differential equation in Fourier space. Its numerical integration is performed by means of a scheme introduced here for the first time 4. Another important feature of the scheme concerns the determination of the vorticity at the boundaries through integral relations which involve the values of the vorticity within the flow domain and the values of the streamfunction at the boundaries 5.