Water flow on steep slopes is not uniform because of the effects of the gravity, surface friction, viscous force, and turbulent stress of the fluid. Instabilities appear and gradually develop over the flow distance on free surfaces
affected by disturbances, which finally expand to form a series of roll waves. The instability phenomenon has negative consequences for water conservancy structures. For example, pressure changes and periodic impacts of water flow
cause structural damage to the channel, rapid rise of water depth in the channel and overflow, and even cause water
splash to introduce soil erosion around the channel. Therefore, this study explores the instability phenomenon both theoretically and empirically. The theoretical derivation uses the Vedernikov number to distinguish instability phenomena.In addition, the derivation processing of critical conditions considers the influence of the slope on the Froude number, and the critical conditions for the occurrence of instability phenomena in different channels were determined.The rectangular and trapezoidal shape, slope, Manning roughness coefficient, water depth, and channel width were the influencing factors. For the parabolic cross-section and semicircular cross-sections, the influencing factors were the slope, Manning roughness coefficient, water surface width, and water depth. It was the easiest to maintain the stability
of flow in different sections of the rectangular channel. In the experiments, the rectangular channel was selected for
testing. On the basis of the comparison between the test and judgment results of the theory in this study, when the water
flow conditions are close to the critical value, the conditions could successfully be determined.
Key word: Steep slopes, Instability, Vedernikov number