Local flow modification near engineering structures such as – local velocity, flow separation, circulation, vortex structures, and increased turbulence, leads to local scour near foundations. Local flow velocity is forming the scour hole at the structures, but not approach flow velocity as it is accepted now by different authors. Based on test results, comparison of local velocity with approach flow velocity is presented. At steady flow conditions, when scour is developing in time, the local velocity is reducing and the critical velocity is increasing, but when the local velocity becomes equal to a critical one, the scour stops. According to test results and formulas, it presented that the value of the local and the critical velocities depend on contraction rate, depth, grain size, Froude number of flow and the river bed stratification. The value of the relative scour depth is dependent on local to critical velocities ratio at the initial stage, when there is no scour. Formulas for calculation of local and critical velocities are presented and confirmed by test results.