The universal velocity profile in the turbulent core of smooth tubes was written as:

\[U^{+}=2.5 \ln y^{+}+5.5\]

This relationship holds when the friction Reynolds number is unity.

\[\operatorname{Re}_{f}=\frac{v^{*} \varepsilon}{v}=1 \quad \varepsilon-\text { roughness }(m)\]

Experiments have shown that for very rough tubes, the turbulent core velocity profile is better represented by:

\[U^{+}=2.5 \ln y^{+}+8.5\]

Following the example in the text, derive a Prandt1/von Karman universal resistance law for \(C_{f}\) for rough tubes.