Connection between attached eddies, friction factor, and mean-velocity profile

This work connects the attached eddies to the wall friction by relating the momentum transfer normal to the lower bound of the energetic range with the wall shear stress. Evidence is provided to show that this balance is the property of a specific layer, l approximate to 3 root Re-tau, which coincides with the smallest attached eddies. As a result, our model predicts successfully the transition of the friction factor after the Blasius regime to the extreme Reynolds numbers range, and the resulting friction equation shows the same accuracy as the well-known logarithmic laws. This transition also affects the mean velocity profile. From the developed friction equation, a mean velocity profile is derived which is in accordance with the well-known log laws. Finally, it is conjectured that for high enough Reynolds numbers, both Prandtl's 1/7 power law and the velocity profile approximate the mean velocity simultaneously, however at different distances from the wall.