This study investigates the interaction of the vertical velocity v and the streamwise velocity u in a gradually accelerating flow. The analytical result shows that the momentum of uv driven by the mean velocities in a non-uniform flow is not negligible. This additional momentum directly results in the concave profiles of Reynolds shear stress in gradually accelerating flows, a departure from the expected linear profile. Consequently, this momentum causes the maximum velocity to be located below the free surface, i.e., the dip-phenomenon. This paper investigated the interactions of the Reynolds shear stress, non-zero vertical velocity and dip-phenomenon, it is found that the non-zero vertical velocity causes the dip-phenomenon. The approach is tested using the experimental data of Song and others, and good agreements between the predicted and measured velocity profiles have been achieved.
