Solving the velocity field for a 2-D incompressible flow in the case of constant viscosity using both Navier-Stokes equation and stream function approaches

Two different 2-D incompressible flow problems with constant viscosities were investigated. The first problem was a fluid with constant density driven by the boundary condition. The time evolution of pressure and velocity was solved with the Navier-Stokes equation. Using different experimental materials, the result showed that reducing the viscosity affects the flow patterns and increases the time needed to reach the steady state. The second problem was a buoyancy driven flow in a vertical gravity field for a density structure with two vertical layers. The instantaneous velocity field as the flow starts was solved with the stream function – vorticity formulation. The maximum magnitude of the velocity field increases linearly with gravity and density contrast, but decreases inversely with viscosity. Finite element and finite difference approaches produced consistent solutions with only small differences. The accuracy of the two methods needs to be further investigated. The discrepancy did not reduce at all when the grid number was increased to twice of the default setting for both methods.