To systemically study the effect of varying slope coefficients in the upstream side on the discharge coefficient at free overflow over a broken-line practical weir with vertical downstream face and without lateral contraction, different slope coefficients of upstream side, including 0, 0. 3, 0. 5, 0. 8, 1. 0, 1. 5 and 2. 0, are simulated by the volume of fraction(VOF)method and the RNG k-ε turbulence model. Numerical results show that the rising of total head H₀ reduces the curvature of free surface profile and increases the discharge coefficient. Meanwhile, the influence of slopes in upstream side on the discharge coefficient decreases with H₀. Furthermore, a discharge coefficient formula including the variable slope coefficient of upstream side is derived based on the method of non-linear regression, in which as the slope coefficient of upstream side increases, the discharge coefficient increases firstly and then decreases for a given H₀.