The lattice Boltzmann method (LBM) is a method for modeling fluid dynamics which is based on discrete kinetic theory. It discretizes the continuous Boltzmann equation in velocity, spatial and temporal space and solves a transport equation for particle populations. To enable an efficient time-space discretization with discrete velocity sets, common LBM solvers employ uniform grids making them prohibitively expensive for the simulation of high Reynolds number flows. This paper describes the implementation of a vertex-centered grid refinement method with the BGK operator in the Open-source Lattice Boltzmann code, OpenLB. The method is validated for flow around a 3D circular cylinder with Reynolds number varying from 100 to 300 and the 3D lid-driven cavity with Reynolds number equals 3200. The results are in good agreement with experimental measurements and the reference solutions with uniform grid spacing. The simulation with grid refinement allows for remarkable computational cost savings for a given accuracy.