Q1: For turbulent flow in smooth circular pipe, the velocity distribution is a function of the distance 'd' measured from the wall of the pipe and the friction velocity 'v', and it follows a __________ relationship.

ANS:A - logarithmic

For turbulent flow in a smooth circular pipe, the velocity distribution profile across the pipe diameter is typically characterized by a logarithmic relationship.

Explanation:

1. Turbulent Flow in Pipes:
   • In turbulent flow within a smooth circular pipe, the velocity profile across the pipe diameter is not perfectly uniform but varies from zero at the pipe wall to a maximum at the centerline.
2. Logarithmic Velocity Profile:
   • The velocity distribution in turbulent flow is often described by the logarithmic law (also known as the logarithmic velocity profile). This law states that the velocity uuu at a distance yyy from the wall of the pipe can be expressed as: u(y)=vκln⁡(yy0)u(y) = \frac{v}{\kappa} \ln \left( \frac{y}{y_0} \right)u(y)=κv​ln(y0​y​) where:
     • vvv is the friction velocity,
     • κ\kappaκ is the von Kármán constant (approximately 0.41),
     • yyy is the distance from the wall,
     • y0y_0y0​ is the roughness height (usually very small for smooth pipes).
3. Characteristics:
   • The logarithmic velocity profile indicates that the velocity increases logarithmically as you move away from the wall towards the center of the pipe.
   • This profile is valid across the entire cross-section of the pipe under fully developed turbulent flow conditions.

Conclusion: