## 1 Introduction  
The measurement of irrigation water consumption in irrigation districts is significant in the implementation of a stringent water resources management system [1]. Rural canals can be approximated by rectangular open channels. The standard method for calculating discharge in an open channel is the velocity-area method, which requires velocity measurements to be made at many points throughout the area. Although this method is considered to be particularly reliable, a large number of measurements results in an associated time cost of measurement. Numerous researchers [2] have sought to define methods for estimating discharge on the basis of an extremely reduced number of velocity observations. One important approach in this respect is to estimate the discharge by measuring the velocity at a point which stands for the depth average velocity on a vertical measuring profile. However, there is no relevant research on how to obtain the location of this point. For practical reasons, it is helpful to separate the flow regions and to separate the bed or side-wall local friction velocity, which is crucial for velocity distribution. It is universally acknowledged that the flow partitioning theory is extremely important for velocity distribution. Nevertheless, the usual application of the flow partitioning theory is in calculating the shear stress in a practical channel. Thus, how to use the flow partitioning theory to find a point which stands for the average velocity on a vertical measuring profile has provoked researchers' interest [3]. The depth average velocity of natural rivers is the basis for studying sediment movement and riverbed evolution, and it has been a key issue in hydraulics for a long time, which is directly related to its function in the calculation of cross-sectional flow and the longitudinal dispersion coefficient [4]. The study of velocity distribution is critical to the calculation of the depth average velocity. There have been many studies on velocity distribution. In 1938, [PERSON] [5] proposed that a logarithmic expression can be used to describe the average velocity distribution of fully developed uniform turbulence in open channels:  
\[u^{+}=A\log y^{+}+B \tag{1}\]  
where \(u^{+}=u/u_{*}\), \(y^{+}=y/u_{*}\), \(A=2.3/k\), \(u_{*}\) is frictional velocity, \(v\) is the coefficient of kinematic viscosity, \(A\) and \(B\) are constants, and \(\kappa\) is the Karman constant. [PERSON] [6] proposed a wake function to explain the turbulent boundary layer flow development process:  
\[\omega\left(\frac{y}{H}\right)=2\frac{\Pi}{k}\sin^{2}\left(\frac{\pi}{2}\frac{ y}{H}\right) \tag{2}\]  
where \(\Pi\) is the wake function. Further more, [PERSON] proposed that the wake function can correct the deviation of logarithmic functions of the velocity distribution of a two-dimensional incompressible turbulent boundary layer flow near the water surface:  
\[\frac{u}{u_{*}}=A\lg\frac{u_{*}y}{v}+B+\frac{2\Pi}{k}\sin^{2}\left(\frac{\pi}{ 2}\frac{y}{H}\right) \tag{3}\]  
[PERSON] [7] made an artificial sand rough pipe by using pasting sand with a uniform particle size on the wall of a pipe. The analysis of the experimental data of the smooth sand tube showed that the values of parameters \(k\) and \(B\) in logarithmic law are 0.4 and 5.5, as shown in Equation (4):  
\[u=u_{*}\Big{(}5.75\lg\frac{u_{*}y}{v}+5.5\Big{)} \tag{4}\]  
However, with the deepening of understanding in this research area, researchers have developed different views on the scope of the applications of logarithmic law. As the description of logarithmic law near the water surface is not perfect, some scholars have begun to try and find a more consistent functional form to describe the velocity distribution in the whole water depth. [PERSON] [8] conducted experiments which showed that the velocity distribution on the same vertical line turns out to be different with an increasing water depth in different water depth ranges.