Application of relative entropy theory to streamwise velocity profile in open-channel flow: effect of prior probability distributions

Abstract

Applying the concept of relative or cross-entropy and the principle of minimum cross-entropy, this study derives the velocity distribution in a wide open channel. Previous studies have employed Shannon entropy and the principle of maximum entropy to derive distributions of various flow variables, including velocity. Relative entropy is a generalized form of entropy and can specialize into Shannon entropy if the prior probability distribution is taken to be a uniform distribution. The prior distribution is often formulated, based on intuition, experience, or thought experiment. When deriving the velocity distribution in wide open channels, this study assumes four prior probability distributions and analyzes the effect of these assumed priors. It is found that a normal-type and a gamma-type prior can significantly influence the velocity profile, especially near the channel bed, and their prediction accuracies are superior to the previously obtained velocity distribution based on Shannon entropy. Furthermore, closed-form explicit analytical solutions are obtained for the nonlinear differential equations that arise while incorporating these priors. Experimental and field data are used to verify the derived velocity distributions, and an error analysis is carried out to evaluate their performance.