Velocity for fully turbulent flow in rough pipe

Criterion. Flow is fully turbulent (hydrodynamically rough) when the roughness Reynolds number (Nikuradse) is

$$\frac{u_* k}{\nu} \ge 100$$

Take $\nu = 1.0\times10^{-6}$ m²/s (water at about 20 °C), $k = 1$ mm $= 0.001$ m, $D = 1.2$ m.

Step 1: rough-pipe friction factor (von Karman-Prandtl, $D/k = 1200$):

$$\frac{1}{\sqrt f} = 2\log_{10}\frac{D}{k} + 1.14 \;\Rightarrow\; f = 0.01877, \qquad \sqrt{f/8} = 0.04844$$

Step 2: shear velocity at the limit

$$u_* = \frac{100\,\nu}{k} = \frac{100(1.0\times10^{-6})}{0.001} = 0.100\ \text{m/s}$$

Step 3: mean velocity

$$V = \frac{u_*}{\sqrt{f/8}} = \frac{0.100}{0.04844} = 2.06\ \text{m/s}$$

Answer: The flow becomes fully turbulent (rough) for $V \ge 2.06$ m/s, about 2 m/s. (If the limit $u_*k/\nu = 70$ is used, the velocity is 1.45 m/s.)