Analytical solutions of flow velocity profile based on the morphological response of flexible vegetation.

• New power law of velocity distribution is established on vegetated riverbed. • New formula of Darcy–Weisbach coefficient of flexible vegetated flow is derived. • New relationship between Darcy–Weisbach coefficient with power law index is built. • New expression for morphological response of vegetation is given. Hydrodynamics affected by vegetation in channel is one of the hot topics in ecological hydraulics. Flexible vegetation plays an important role in natural flow region, which changes flow structure and affects the transport of pollutants. Therefore, it is important to illustrate the physical process of the interaction between water flow and morphological response of flexible vegetation. Due to the complex resistance characteristics of flexible vegetation, the simulation of velocity distribution is more complicated. In this paper, velocity distribution in the form of Bessel function is derived by solving the governing equation based on linear resistance-velocity expression of flexible vegetation. Meanwhile, a new power law of velocity distribution is established by adopting the concept of vegetated riverbed. Moreover, a new expression of Darcy–Weisbach friction is derived for flow with flexible vegetation, and the relationship between vegetation bending angle and hydraulic parameters is given. Quantitative relationship between the power law index and the Darcy–Weisbach friction coefficient in the free surface layer is derived for flexible vegetated flow. This research reveals the velocity distribution characteristics under the influence of flexible vegetation and the resistance features on vegetated riverbed. [ABSTRACT FROM AUTHOR]