Vorticity-Streamfunction Equations

Abstract

The vorticity-streamfunction equations present some advantages over the velocity-pressure equations in the case of two-dimensional flows in simply connected domains. These advantages are well known: (1) the velocity field is automatically divergence-free, (2) the mathematical properties of the equations permit the construction of simple and robust solution methods, (3) computing time is saved because of the smaller number of equations. In the present chapter, Fourier and Chebyshev methods for the solution of the vorticity-streamfunction equations are discussed. The classical difficulty associated with this formulation is the lack of boundary conditions for the vorticity at a no-slip wall, whereas the streamfunction and its normal derivative are prescribed. This difficulty is surmounted thanks to the influence matrix method which will be described in the following. The stability of the Chebyshev approximation and the stability of the timediscretization schemes will be discussed. Finally, examples of applications (Rayleigh-Bénard convection and flow in a rotating annulus) will be presented.