Flow around a circular cylinder is a classic scenario which invariably draws the attention of the fluid mechanics circle, because its relevant studies are of both theoretical and practical significances. However, most experiments are conducted below transcritical Reynolds number（Re) regime（Re3.5106) due to the limitations of the wind tunnel modeling technique, which makes the obtained results inapplicable to some full-scale conditions. To this end, the field measurements for wind-induced pressures on a 167-meter high large cooling tower are conducted at Re=6.59107 to enrich the experimental results of flow past a circular cylinder in transcritical Re regime. Besides, the wind effects at low Re（Re=2.1105-4.19105) are also obtained by tests on a 1:200 rigid cooling tower model in a wind tunnel with considering 4 types of wind speeds, 8 types of surface roughness, and 2 flow fields. Employing the data obtained from both field measurements and wind tunnel model tests, the variations of static/dynamic flow characteristics with Re increasing are studied. It is found that 1) with the increase of Re, the drag coefficient for the smooth-walled tower in the uniform flow field decreases dramatically in the critical Re regime and increases slowly in the supercritical regime, which accord with Roshko's and Achenbach's results; 2) for smooth-walled tower, both the base pressure coefficient and pressure coefficient increase significantly with the increase of Re in critical and supercritical regimes, which qualitatively accord with Shih's results; and 3) the finding of the Strouhal number is supportive to Shih's result（i.e., shedding from the rough cylinder persists throughout the Re range tested). More importantly, special attention is paid to the Re-independence phenomenon of fluid flow, which is a typical phenomenon occurring in transcritical Re regime. Results indicate that the Re-independence exists in an Re range from 2105 to 1108 for a circular cylinder with a relative roughness greater than 0.01, and the increased free-stream turbulence can also induce Re-independence which probably exists in a narrow low Re range. Considering the flow mechanism, a reasonable explanation can be found for the Re-independence phenomenon, i.e., the critical and supercritical regimes narrow and move to lower Re range with the increase of surface roughness or the increase of free-stream turbulence, so Re independence can occur at a very low Re.