In natural rivers, the dispersion is mainly caused by the non-uniform distribution of longitudinal flow velocity in the cross-section. The inhomogeneity of the velocity distribution is more obvious when there is partial emergent vegetation in the river. In order to study the longitudinal dispersion in the flow with partially emergent vegetation, assuming that the variables are constant in the vertical direction, a two-dimensional domain consisting of flow direction and lateral direction was simulated, and the Random Displacement Model (RDM) based on Lagrangian method was introduced. This model directly uses discrete particles with independent motion to describe the transport of matter, and it is no longer necessary to solve the advection-diffusion equation. The RDM simulates the distribution of pollutants mainly based on the transverse distribution of the vertically average flow velocity and transverse turbulence diffusion coefficient. Among them, the lateral velocity distribution formula adopts a simplified empirical formula based on shear vortex. The advantage of this formula is that it does not require rating the secondary flow coefficient, and the physical meaning is clear; with the dimensional relationship and the genetic algorithm, the explicit empirical formula of the outer layer width of the shear vortex was proposed, which is mainly determined by the water depth and the bottom friction. The diameter and density of the vegetation have little effect on the outer structure of the vortex. Besides, the model of the lateral distribution of the turbulent diffusion coefficient was given based on previous measured data, revealing the dynamic characteristics that the lateral turbulence diffusion at the junction is much greater than that of the stable zone. In order to verify the accuracy of the random displacement model, an indoor experiment was conducted in a long and straight partial vegetated water tank. The measured values of the concentration process lines at different points in the flow field agree well with the simulated values, indicating that the RDM can effectively simulate longitudinal dispersion of flow with partial vegetation, and provide a new method for the assessment of the mixing and transport of pollutants in ecological rivers.
