In this paper, a scaling law relating the mesh size to the Reynolds number was proposed to ensure consistent results for large eddy simulation (LES) as the Reynolds number was varied. The grid size scaling law was developed by analyzing the length scale of the turbulent motion by using DNS data from the literature. The wall-resolving LES was then applied to a plane channel flow to validate the scaling law. The scaling law was tested at different Reynolds numbers $(Re_τ$ = 395, 590 and 1000), and showed good results compared to direct numerical simulation (DNS) in terms of mean flow and various turbulent statistics. The velocity spectra analysis shows the evidence of the Kolmogorov –5/3 inertial subrange and verifies that the current LES can resolve the bulk of the turbulent kinetic energy by satisfying the grid scaling law. Meanwhile, the near-wall turbulent flow structures can also be well captured. Reasonably accurate predictions can thus be obtained for flows at even higher Reynolds numbers with significantly lower computational costs compared to DNS by applying the mesh scaling law.