The wire mesh liner usually possesses a better acoustic linearity to flow than perforated liners, but it still lacks an impedance model. So, the grazing flow effect of the mesh liner is studied experimentally herein. It reveals that, relative to perforated liners, the mesh liner has qualitatively similar but quantitatively weakened trends in the grazing flow resistance Rg and reactance Xg. As a mesh occurs or as the mesh becomes finer on the flow-saturated perforated surface, Rg is suppressed gradually especially at high frequencies, by lowering its sinking segment, delaying its increasing segment, and decreasing its increasing slope. Furthermore, the effect of a fine mesh can be superimposed with that of a larger orifice diameter dp on lowering the sinking and delaying the increasing, but the former can weaken the effect of the latter on increasing the slope. On another aspect, Xg can also be suppressed evidently by a fine mesh. By contrast, a tested liner with a fine mesh embedded in the middle of perforated plates rather than on the surface exhibits no such a suppressing effect on Rg. Accordingly, a more complete impedance model of the mesh-on-perforated-surface liner considering the grazing flow, mesh type and frequency is established.