Streamfunction-Vorticity Dynamics and High-Order Compact Schemes for the Time-Dependent Navier-Stokes Equations

We consider the Navier-Stokes equations in vorticity-streamfunction formulation. The main difficulty associated with this formulation is that there is no boundary condition for the vorticity. We show that every function in L₂ may be decomposed into a sum of a function in the image of the Laplacian operator, having appropriate boundary conditions, and a harmonic function. Thus, we project the vorticity onto the space of functions which reside in the image of the Laplacian operator. The projection is carried out by the biharmonic equation, which relates the streamfunction to the Laplacian of the vorticity. Numerical results for several test problems including the driven cavity are shown.

We also present a new fourth order compact scheme for the time-dependent Navier-Stokes equations. The idea is to extend existing compact schemes for the steady-state equations to the full time-dependent case. This is done by treating the time-derivative term as a source term, and then applying standard time marching schemes to the set of semi-discrete equations. The resulting fourth-order scheme has a nine-point (3x3) stencil.