This talk is concerned with the development of numerical research for the 2D vorticity-stream function formulation and its application in vortex merging at high Reynolds numbers. A novel numerical method for solving the vorticity-stream function formulation of the Navier-Stokes equations at high Reynolds number is presented. We implement the second-order linear scheme by combining the finite difference method and finite volume method with the help of careful treatment of nonlinear terms and splitting techniques, on a staggered-mesh grid system, which typically consists of two steps: prediction and correction. We show in a rigorous fashion that the scheme is uniquely solvable at each time step. A verification algorithm that has the analytical solution is designed to demonstrate the feasibility and effectiveness of the proposed scheme.