Numerical solution of three-dimensional stream function vector components of vorticity transport equations

The advantage of solving for the new three-dimensional stream function vector components is explored for the three-dimensional flow. To this end, a single scalar stream function for two dimensions is expanded into three stream function vector components for three dimensions. As a result, the fourth-order three component partial differential equations of vorticity transport in terms of three-dimensional stream function vector components are formulated. The generalized Galerkin finite element solutions of the governing equations for incompressible flow is then carried out, and the results are shown to be very accurate for the lid-driven three-dimensional cavity problems examined herein. This numerical accuracy is attributed to the correct definition of three-dimensional stream function vector components, appropriate finite element interpolation functions, associated physically simple boundary conditions, and the rigorous Newton-Raphson scheme of solution process, as well as the governing equations being free from pressure oscillations. To the best of our knowledge, the present study marks the first attempt to solve the three-dimensional fourth-order partial differential equations of vorticity transport for the three-dimensional stream function vector components. This will make it possible to obtain the three-dimensional version of the Orr-Sommerfeld type eigenvalue solutions for hydrodynamic instability in the future.