Average velocity from midway velocity reading

Use the velocity defect law (valid for smooth and rough pipes): $\dfrac{u_{max} - u}{u_*} = 5.75\log_{10}\dfrac{R}{y}$, and the mean velocity relation $\dfrac{u_{max} - V}{u_*} = 3.75$.

Step 1: use the midway reading

Midway between the wall and the centre line, $y = R/2$ and $u = 0.9\,u_{max}$:

$$\frac{u_{max} - 0.9u_{max}}{u_*} = 5.75\log_{10}\frac{R}{R/2} = 5.75\log_{10}2 = 1.7309$$ $$\frac{0.1\,u_{max}}{u_*} = 1.7309 \;\Rightarrow\; u_{max} = 17.309\,u_*$$

Step 2: average velocity

$$V = u_{max} - 3.75u_* = 17.309u_* - 3.75u_* = 13.559\,u_*$$ $$\frac{V}{u_{max}} = \frac{13.559}{17.309} = 0.7834$$ $$\boxed{V = 0.783\,u_{max}}$$

Step 3: relative roughness (rough pipe)

The velocity at the axis of a rough pipe: $\dfrac{u_{max}}{u_*} = 5.75\log_{10}\dfrac{R}{k} + 8.5$ (from $y = R$).

$$17.309 = 5.75\log_{10}\frac{R}{k} + 8.5 \;\Rightarrow\; \log_{10}\frac{R}{k} = \frac{17.309 - 8.5}{5.75} = 1.532$$ $$\frac{R}{k} = 34.04 \;\Rightarrow\; \frac{k}{D} = \frac{1}{2(34.04)} = 0.0147$$

Answer: $V = 0.783\,u_{max}$; relative roughness $k/D \approx 0.0147$ (about 1.5%).