A numerical method for the computation of topological and geometric properties of iso-surfaces with and without boundaries developed by the authors is applied to randomly generated surfaces and surfaces embedded in numerical solutions of the Navier-Stokes equations. The Euler number of level surfaces corresponding to site percolation is shown to change drastically at the critical value for percolation. Numerical solutions of the Navier-Stokes equations are used to investigate the topological and geometric effects of vortex stretching and viscosity on iso-surfaces of enstrophy, dissipation rate and kinetic energy.