A modified vorticity-velocity formulation for the two-dimensional Navier-Stokes equations is proposed with the goal of imposing an arbitrary unsteady vortex stretching into the two–dimensional fluid flows. To this end, the velocity field is (Helmholtz) decomposed, allowing for presence of an arbitrary dilatation, which results in some modifications in both the continuity equation, and the vorticity transport equation. To show the applicability of the method, two entirely different classes of turbulent flows are analyzed; that is, the isotropic turbulence (solved via pseudospectral method), and the near-wall turbulent flow (simulated via the finite difference method). The results show ability of the method in mimicking some essential features of three-dimensional flows.