Circular-crested weirs are overflow structures with a reasonably high discharge coefficient, compared with that of sharp- and broad-crested weirs. In this study, combining the potential flow around a circular cylinder and the free vortex flow, to determine the discharge coefficient and velocity distribution simulated flow over circular-crested weirs. Both the potential flow around a fixed-circular cylinder and the free vortex flow independently consider a symmetrical flow field. However, in realized conditions, the inertia force induces deviation of the flow streamlines over the circular cylinder. Therefore, a more comprehensive theory is essential to simulate the asymmetry of the flow streamlines properly. For this purpose, in the present study, both the mentioned theories were combined to simulate the flow around the circular-crested weirs using the potential flow past a circular cylinder with circulation. The analytical results were calibrated and compared with the experimental data of former investigations. Results indicate that the suggested semi-analytical model can predict the discharge coefficient and velocity distribution satisfactorily for the relative crest curvature (H1/RbH1/Rb) up to 7. In addition, results of the sensitivity analysis demonstrated that H1/RbH1/Rb is the most prominent parameter influencing the discharge coefficient of circular-crested weirs.