## 1 Introduction
Additionally, according to the distribution characteristics of velocity in different water depth ranges, the whole water depth should be divided into three parts: the inner area, the outer area and the surface area. [PERSON] and [PERSON] [9] integrated the N-S equation along the flow direction to take the average value along the water depth and proposed the Shiono and Knight Model (SKM), which is most widely used in the calculation of the average flow velocity in the cross section of a compound channel. The research conducted by different researchers [10] shows that logarithmic law could be used on velocity distributions of open channel flows. However, under various conditions, there is no agreement on whether the vertical velocity correlation of the cross-section can be expressed using a unified correlation or not. The Division of flow region for 3-D channel flows has been shown in a number of textbooks on hydraulics [11]. [PERSON] [5] is the first to mention the need to divide the flow cross-sectional area into different sections. For turbulence in a prismatic channel, [PERSON] [12] suggested that the shear force applied to the bed should be separated from the shear force applied to the lateral boundary. [PERSON] and [PERSON] [13] briefly explained the physical meaning of [PERSON]'s thoughts, and reported that the residual energy in the mainstream should be transferred and eventually dissipated as heat at the boundary. Additionally, the energy in any unit flow is transported to the nearest boundary. In fact, all of these provide an excellent approach to calculating the shear stress in a cross-section. However, the most important issue is whether it is possible to use flow partitioning theory to obtain local frictional velocity separately in different regions. Hydraulic engineers use the concept of flow region division to assume that there is no shear force on the dividing line and have proposed various methods for dividing the cross-section of the channel. They have different views on the division of flow area, and various models have been proposed to express the division lines:  
Keulegan Method (KM): [PERSON] [5] proposed that the flow in a polygonal channel could be separated into three separate areas, and this is achieved by division lines bisecting the base angles, as shown in Figure 1. Unfortunately, no theoretical explanation for this treatment was provided by the authors. Daido Method (DM): [PERSON] [14] performed extensive research to derive an equation for the division lines of a rectangular open channel. The author applied the Karman-Prandtl velocity equation from the sidewall and bed for each dashed line shown in Figure 1. However, the validation of [PERSON]'s model in-flow has not been verified by experimental data. Guo and Julien Method (GJM): [PERSON] and [PERSON] [15] determined the bed and sidewall shear stresses in a rectangular open channel flow by solving the continuity and momentum equations. Conformal mapping was used to partition the flow and the equation for division lines was derived. The GJM method considers the control volume partitioned by the curved division lines by their first approximation, as illustrated in Figure 1.  
[PERSON] and Lim Method (YLM): Their work has been widely quoted in many textbooks and has had a great impact on fluid mechanics. [PERSON] and [PERSON] [16] proposed that there is a slanted straight line, which can be used to divide the cross-section into two different flow regions, as shown in Figure 1.  
There have been many ways to obtain the average velocity in a cross-section of an open channel. However, the flow division theory has almost never been applied to obtain the average velocity in open rectangular channels. It is very important to find out where the characteristic point is, which stands for the average velocity along the normal line of boundary.