Prediction of mean turbulent flow velocity in a permeable-walled pipe

Determining the characteristics of pipe flow and the effect of friction resistance is relevant to a variety of applications in hydraulic and hydrological engineering such as flood control, river restoration and ground runoff prediction. In the Nikuradse's experiments, the relative roughness and the bulk Reynolds number are used to scale the Darcy friction factor thereby determining the mean velocity. Recent observations using permeable-walled pipes however indicate that the friction factor can significantly deviate from the results obtained in Nikuradse's approach. Here, we propose a concise model to estimate the mean velocity of the pipe flow over permeable walls. The model is formulated using a data-driven approach based on the analytical solution of the mean velocity and the similarity analysis (dimension conservation) of the two extreme cases: impermeable and porous pipe flow. We show that the proposed approach can derive a physically meaningful formula for the mean velocity. Furthermore, we propose a simple momentum transfer model which confirms the findings from the data-driven approach. The momentum transfer model indicates that the friction factor can be scaled with two different relative hydraulics radius scales, which are intrinsically determined by two different eddy sizes. Lastly, the present approach is examined by comparing it with another approximate formula and the limited experimental data in the literature.