Comparative study of 1D entropy-based and conventional deterministic velocity distribution equations for open channel flows

Abstract

Velocity distributions for open channel flows have been investigated using deterministic and probabilistic approaches. It is well known that the vertical velocity profiles in wide open channels (i.e. aspect ratio width/depth > 5) can be approximated by logarithmic velocity laws and power laws. Recently the entropy concept in the forms of Shannon entropy and Tsallis entropy has been employed to estimate velocity distributions in open channels with different aspect ratios. The accuracy of conventional velocity equations is highly dependent on their parameters that can only be estimated by empirical or semi-empirical analytical relations which requires either a good knowledge of velocity field and/or physical properties of the channel, such as topographic conditions, sedimentation conditions and boundary roughness. In contrast, the entropy based velocity distributions derived based on the least-biased probability density function (PDF) by treating time-averaged velocities as random variables are resilient regardless of the flow and channel conditions. However, a comparison of the velocity profiles computed using deterministic approaches and probabilistic approaches has not been rigorously conducted. Furthermore, the accuracy and reliability of associated velocity distribution equations have not been tested thoroughly using data sets collected using advanced techniques. This paper presents a comprehensive and comparative study to analyze the distinctions and linkages between four commonly used velocity laws and two entropy-based velocity distributions theoretically and quantitatively using selective laboratory and field measurements available in the literature, considering typical sedimentation and channel hydraulic conditions. Amongst all, Tsallis entropy based velocity distribution developed from a generalized form of informational entropy exhibits universal validity to sediment-laden flows in wide alluvial open channels, and is found to be superior to others to predict velocity profiles in large waterways with unmanageable rough beds.