Internal flows — shape of velocity profile in turbulence For fully developed turbulent flow through a closed conduit (e.g., a smooth pipe), which functional form best describes the mean velocity profile across the radius?

Introduction / Context:
Velocity profiles reveal the physics of flow regimes. Laminar pipe flow exhibits a parabolic profile, whereas turbulent flow shows a much flatter core and a characteristic near-wall behavior described by empirical laws.

Given Data / Assumptions:

• Flow is fully developed and turbulent in a circular conduit.
• No strong roughness effects altering the classic wall law.
• Mean velocity profile is considered (time-averaged).

Concept / Approach:
The law of the wall for turbulent boundary layers and internal flows gives a logarithmic relation between mean velocity and distance from the wall in inner variables. Hence, the overall shape is commonly termed “logarithmic,” not parabolic (laminar) or linear.

Step-by-Step Solution:
Recall laminar Hagen–Poiseuille profile → parabolic.Recall turbulent wall law → logarithmic mean velocity in the inner region and flattened core.Select “logarithmic.”

Verification / Alternative check:
Experimental data and turbulence models (e.g., log-law with von Kármán constant) support the logarithmic near-wall form.

Why Other Options Are Wrong:

• Parabolic: laminar regime only.
• Linear: typical only in special Couette flows.
• Hyperbolic: not a standard descriptor of turbulent pipe profiles.

Common Pitfalls:

• Overgeneralizing parabolic profiles to all internal flows.

Final Answer:
logarithmic