Representative velocity to use in hydraulic engineering calculations When solving flow problems in pipes and open channels, which velocity is typically used for continuity, energy, and momentum equations?

Introduction / Context:
Hydraulic equations (continuity A V, energy head V^2/2g, momentum ρ Q V) rely on a representative velocity. Because real velocity profiles vary across a section, a defined average is used to ensure mass and momentum balances are satisfied.

Given Data / Assumptions:

• Fully developed or developing internal/open-channel flows.
• Use of control-volume formulations in design practice.

Concept / Approach:
The correct representative velocity is the cross-sectional average, defined as V_avg = Q / A = (1/A) ∫ V_local dA. This ensures that continuity, energy, and momentum equations remain consistent. Centerline or wall velocities do not conserve mass or momentum when substituted directly without profile factors.

Step-by-Step Solution:

Verification / Alternative check:
For laminar pipe flow, centerline velocity = 2 V_avg; using centerline value would double the implied discharge, violating continuity.

Why Other Options Are Wrong:
Centerline/wall values are point velocities; their mean is arbitrary and not physically founded for balances.

Common Pitfalls:
Ignoring kinetic energy/momentum correction factors in strongly non-uniform profiles.

Final Answer:
average velocity of flow over the cross-section