Entropy-Based Velocity and Shear Stress Distributions for Trapezoidal Channel

A multitude of studies have derived one-dimensional velocity distributions, assuming open channels to be rectangular, and have extended their applications to rivers, canals, and other open channels. A cumulative distribution function (CDF) of velocity distribution is proposed based on a channel’s cross-sectional geometry. This paper derives a one-dimensional velocity distribution for wide rigid trapezoidal channels using the maximization of Shannon entropy and proposed CDF. Two new parameters, geometric parameter ($a$), and linear weightage factor ($k$), are introduced and the dependency of CDF on these two parameters is discussed. These two parameters take into account the cross-sectional geometry of the section and the location of flow velocity measured along the depth, respectively, and, hence, their contributions to velocity distribution, which were not considered when formulating Chiu velocity distribution. The proposed velocity distribution has been found to work significantly well, especially in the middle zone of channel depth, when compared to Chiu’s velocity distribution. This has been verified using experimental as well as river field data (assuming the river’s cross-section channel to be trapezoidal). The paper also explains the limitation of Chiu’s velocity distribution when applied to obtain the average flow velocity in trapezoidal channels. Further, the shear stress distribution (in vertical direction along the depth of channel perpendicular to flow direction) is also derived by entropy maximization based on channel geometry, and the relation of shear stress at the channel bed with bed width and cross-sectional area is established. Finally, the CDF formulation is also proposed for velocity distribution in exponential channels based on channel geometry.