Derivation of Power Law and Logarithmic Velocity Distributions Using the Shannon Entropy

Abstract

Employing the Shannon entropy, this note derives the well-known power law and the Prandt–von Karman universal (or logarithmic) velocity distribution equations for open channel flow. The Shannon entropy yields probability distributions underlying these velocity equations. With the use of this entropy, one obtains an expression for the power law exponent in physically measurable quantities (surface flow velocity and average logarithmic velocity) and an expression for shear flow depth in flow depth, surface flow velocity and shear velocity, thus obviating the need for fitting.