## 2 Materials and Methods  
[PERSON] [17] demonstrated the existence of a dividing line in a smooth channel flow with a smooth flat or curved bed by analyzing the average velocity distribution, which can be used to obtain the velocity distribution in different flow areas. In order to understand how flow division theory affects the velocity distribution, [PERSON]'s theory needs to be revisited. [PERSON] [18] proposed the concept of mixing length to express eddy viscosity.  
\[v_{t}=l_{m}^{2}\bigg{|}\frac{du}{dy}\bigg{|} \tag{5}\]  
\[-\overline{\rho u^{\prime}v^{\prime}}=\rho l_{m}^{2}\bigg{|}\frac{du}{dy} \bigg{|}\frac{du}{dy}=\rho u_{\star}^{2} \tag{6}\]  
[PERSON] assumed that the mixing length \(l_{m}\) is directly proportional to the distance from the boundary, \(l_{m}=\kappa y\), where \(\kappa\) is the Karman constant. Generally, the value of \(\kappa\) is 0.4. Under different conditions, the value of \(\kappa\) may be different [16]. Substituting the boundary conditions \(u=0\) and \(y=y_{0}\) into Equation (6), the equation for velocity is obtained.  
\[\frac{u}{u_{\star 1}}=\frac{1}{\kappa}\ln\frac{y}{y_{0}} \tag{7}\]  
where \(u_{\star 1}\) is the frictional velocity, and \(y_{0}\) may be related to local velocity of the sticky lower layer, according to Equation (8).  
\[y_{0}=\frac{v}{cu_{\star s}} \tag{8}\]  
where \(c\) is the coefficient measured by experiments, and \(u_{\star s}\) is the local friction, which may be different from \(u_{\star 1}\).  
Therefore, if a flow region can be divided, then the local shear stress can be used, which can better reflect the actual side wall effect instead of the concept of average shear stress. Additionally, the local boundary shear stress may be determined using Equations (9) and (10).  
\[\tau=\rho u_{\star s}^{2} \tag{9}\]  
\[u_{\star s}=\sqrt{gL_{\star}s} \tag{10}\]  
where \(L_{\star}\) is the vertical distance from the boundary to the boundary line, as proposed by [PERSON] and [PERSON] [15].  
According to YLM, the division line in a rectangular open channel can be expressed using Equation (11):  
\[z=ky \tag{11}\]  
where \(z\) and \(y\) are normal to the boundary and \(k\) is the slope of division line. Furthermore, \(k\) is calculated using Equations (12) or (13).  
\[k^{3}+(2h/b)k-2=0;b/h\geq\alpha \tag{12}\]  
or  
\[k^{3}+(b/2h)k-2=0;b/h\leq\alpha \tag{13}\]  
where \(h\) is the depth of water, \(b\) is the width of the channel, and \(\alpha\) is the critical aspect ratio (width to depth ratio), which is equal to 2 in [PERSON] and [PERSON]'s theory.  
For a steady, uniform, and fully developed turbulent flow, the classical formulation which can describe the velocity distribution is a logarithmic function that increases from zero velocity at theboundary of the channel and reaches its maximum at the water surface. The velocity distribution can be expressed using Equations (14) and (15).  
\[\frac{u}{u_{*}}=2.5\ln(u_{*1}y/v)+B \tag{14}\]  
\[\frac{u}{u_{*}}=2.5\ln(u_{*1}z/v)+B \tag{15}\]  
where \(u\) is the flow velocity, \(u_{*}\) is the frictional velocity, which is given by the correlation: \(u_{*1}=\sqrt{gsL_{n}}\) and \(u_{*1}\) is the local frictional velocity, where \(g\) is the gravitational acceleration, \(s\) is the energy slope and \(L_{n}\) is the distance from the division line to the boundary. Furthermore, \(y/z\) is the distance to the boundary, \(v\) is the dynamic viscosity, and \(B\) is an empirical coefficient, which is usually equal to 5.5.  
According to [PERSON] and [PERSON]'s theory, the flow region can be divided according to the schematic shown in Figure 2 (aspect ratio \(>2\)). Furthermore, the three regions can be defined in half cross-section.  
There is a rectangular channel, whose depth and width are \(h\) and \(b\), respectively. According to the flow partition theory and the value of \(h/b\), the three partitions can be determined.