For a fully-developed pipe flow, which of the following options is/are correct?

The correct options are:
1. For the same maximum velocity, the average velocity is higher in the turbulent regime than that of the laminar regime
2. For laminar flow, the friction factor is independent of surface roughness

Here is a detailed, step-by-step breakdown of why these options are correct:

1. Analysis of velocity profiles (Laminar vs. Turbulent):
For a fully developed pipe flow, the shape of the velocity profile determines the relationship between the average velocity and the maximum (centerline) velocity.

For laminar flow, the velocity profile is parabolic and is mathematically expressed as:
$$u\left(r\right)=u_{\max }$$
By integrating this profile over the cross-section of the pipe, we find that the average velocity ($V_{\mathrm{avg}}$) is exactly half of the maximum velocity ($u_{\max }$):
$$V_{\mathrm{avg},\mathrm{laminar}}=0.5u_{\max }$$

For turbulent flow, rapid mixing across the pipe causes the velocity profile to be much flatter and fuller across the core area, dropping sharply only near the pipe wall. Using the turbulent power-law velocity profile, the ratio of average velocity to maximum velocity typically ranges from 0.75 to 0.85:
$$V_{\mathrm{avg},\mathrm{turbulent}}\approx 0.8u_{\max }$$
Comparing the two, for the same maximum velocity ($u_{\max }$), we get:
$$V_{\mathrm{avg},\mathrm{turbulent}}>V_{\mathrm{avg},\mathrm{laminar}}$$
Thus, the average velocity is higher in the turbulent regime for the same maximum velocity.

2. Analysis of the friction factor in laminar flow:
For fully developed laminar flow in a circular pipe, the Darcy friction factor ($f$) is derived analytically from the Hagen-Poiseuille equation:
$$f=\frac{64}{\mathrm{Re}}$$
where $\mathrm{Re}$ is the Reynolds number. This relation demonstrates that the friction factor depends solely on the Reynolds number and is completely independent of the pipe surface roughness. Thus, the third option is correct.

Why the remaining options are incorrect:
• "Compressibility effects are important if Mach number is less than 0.3": This is incorrect because compressibility effects only become significant when the Mach number exceeds 0.3. For Mach numbers less than 0.3, the flow is treated as incompressible.
• "For laminar flow, friction factor decreases with decrease in Reynolds number": Since the friction factor is inversely proportional to the Reynolds number ($f=64/\mathrm{Re}$), decreasing the Reynolds number causes the friction factor to increase, not decrease.