In open-channel hydraulics, the hydraulic radius R is defined as which ratio of geometric properties of the flowing section?

Introduction / Context:
Hydraulic radius (R) is a key geometric parameter in open-channel and pipe flow used in Manning’s and Chezy’s equations. It condenses the cross-sectional shape into a single length scale that reflects the balance of area conveying flow and perimeter creating frictional resistance.

Given Data / Assumptions:

• Steady internal flow fully wetting the considered perimeter.
• Prismatic channel or uniform pipe cross-section.
• Definitions: flow area A, wetted perimeter P (only the boundary in contact with water).

Concept / Approach:

By definition, hydraulic radius is R = A / P. The area A promotes discharge capacity, while the wetted perimeter P is where shear develops. A larger R generally implies less relative boundary friction for a given area, improving conveyance efficiency.

Step-by-Step Solution:

Verification / Alternative check:

For a wide channel, P ≈ b (width) and A ≈ b * y (depth), so R ≈ (b*y)/b = y, matching the rule-of-thumb that R ~ depth for very wide channels.

Why Other Options Are Wrong:

(a) R is not A/P^2; (c) inverts the correct ratio; (d) sqrt(A) lacks perimeter information and is dimensionally inconsistent for R’s physical meaning.

Common Pitfalls:

Including dry boundaries in P; confusing hydraulic radius (A/P) with hydraulic diameter (4A/P) used in closed conduits.

Final Answer:

Area divided by wetted perimeter