Enhanced formulation of the probability principle based on maximum entropy to design the bank profile of channels in geomorphic threshold

Abstract

It is essential to consider the geometric shape and dimensions of a stable channel bank profile in the design and implementation of open channels. A threshold channel refers to a state in which particles do not move along the channel (especially the banks) even after the channel has reached stability. This type of channel is a simple model of rivers and natural channels. The way sediment particles form at the channel banks is significant. The transverse bank slope (St) varies monotonically from zero at the channel bed to a maximum value at the channel centerline. Accordingly, studying the transverse bank slope based on maximizing its entropy value can offer acceptable results in predicting the shape and dimensions of stable channels. The mean and maximum channel transverse slope values at any discharge (Q) are useful for predicting channel dimensions. In different Q ranges, a channel adjusts its dimensions and geometric shape to attain a constant range of entropy parameter values. In the present study, the entropy concept is employed to study the bank profile shape after a threshold channel with a gravel bed reaches equilibrium. Moreover, the ratio (β) between the maximum (SX) and mean (SN) transverse slope values of a channel bank profile (β = SN/SX) and the relation with the entropy parameter (K) are assessed based on a wide range of observations and field data in various hydraulic conditions. According to the results, the ratio of SN to SX (β) at the banks of a stable channel at each discharge tends towards a constant value and does not change with changes in discharge and time. Subsequently, the mean value of the entropy parameter (K) per discharge is constant at approximately 1–3. Therefore, four common threshold channel types are defined in this study based on the variations in Q and mean sediment size (d50). New relationships between K, β and other parameters in open channels are developed to facilitate the calculation and design of stable channels. The proposed method produces the transverse bank slope values and vertical alignment of a stable channel bank’s cross sections with high accuracy. Thus, it is possible to calculate the shape and dimensions of a stable channel in different hydraulic conditions (varying Q and d50).