After verifying the accuracy of the 2D adaptive mesh refinement (AMR) method by the benchmarks of 2D lid-driven cavity flows and 2D backward facing step flows, this paper applied the AMR to simulate the flow over wall-mounted plate. The AMR method refines a mesh using the numerical solutions of the Navier-Stokes equations calculated on the mesh by an open source software Navier2D which implemented a vertex centered finite volume method (FVM) using the median dual mesh to form control volumes about each vertex. The application of the AMR produces the estimated coordinates of the vortex center after the plate after the AMR applied once to the initial mesh. The AMR method is proposed based on the qualitative theory of differential equations, and it can be applied to refine a mesh as many times as required and used to seek accurate numerical solutions of the mathematical models including the continuity equation for incompressible fluid or steady-state fluid flow with low computational cost.