Fully Developed Turbulent Channel Flow

[narbij]

Homework Statement

Hey everyone I was wondering if you could help me - is there any known derivation for the volumetric flow rate through a fully developed turbulent channel flow, I've been looking for ages and can't find anything only for laminar flow.

thanks in advance

nar

Homework Equations

The Attempt at a Solution

 

[narbij]

Is there an equation that relates the friction factor to the change in pressure or head loss for a closed channel flow - I am aware there exists equations that exits for open channel flow