Derivation of Velocity, Eddy Viscosity, and Mixing Length Models in Pipe and Symmetric Channel Flow Using Incomplete Information–Based Entropy

A single hypothesis of the cumulative distribution function was proposed to derive the incomplete information–based entropy model for the mean velocity distribution in full pipe flow (fully filled with fluid) and symmetric channel flow. The velocity distribution model was then utilized to derive the informational entropy-based eddy viscosity and mixing length models. The previous entropy-based pipe velocity model employs two different cumulative distribution functions, and as a result the derived eddy viscosity model shows a singularity near the pipe centerline. The novelty of the study lies in the proposal of the modified smooth velocity model and the first smooth entropy-based models for the eddy viscosity as well as the mixing length for turbulent flows. The proposed velocity, eddy viscosity, and mixing length models were validated using different experimental datasets for pipe flow and symmetric channel flow with a wide range of the flow Reynolds numbers. These models showed satisfactory agreement with observed data. The velocity and the eddy viscosity models were also compared with existing analytical models for various experimental datasets of pipe flow and symmetric channel flow. For several cases, the proposed velocity and the eddy viscosity models performed better than the existing deterministic models. Statistical error analysis quantifies that, on average, the mean and standard deviation of the relative error for the velocity model have been reduced by 58.96% and 59.77% when compared with previous deterministic models and reduced by 41.24% and 39.36% when compared with existing entropy models, respectively.