Determining topological charge (TC) of optical vortex (OV) beams is important for many applications, such as optical measurement and information transmission through long-distance propagation. In this application, the OV beams usually have an annular intensity profile at the receiving end and are inevitably distorted by aberrations during propagation. In this paper, we propose a simple, direct, and aberration-resistible method to determine the TC value of this annular-shaped OV beam with a Shack-Hartmann wavefront sensor (SH-WFS). Our approach involves finding a closed-path along the annular intensity distribution ridge, and then determining the TC value by a discrete circulation calculation with only the phase-slope vectors along the closed-path. Verification experiments were performed using a phase-only spatial light modulator to generate OV beams and aberrations, while SH-WFS to measure the intensity profile and phase-slopes. The results show that our method can determine the TC value of up to ± 20. The robustness against wavefront aberration and simulated atmospheric turbulence was evaluated. We also verified the proposed closed-path's superiority to other circular closed-paths for TC determination.