Critical velocity in terms of critical depth in open-channel flow If h_c is the critical depth, what is the expression for the critical velocity V_c in a wide channel (hydraulic depth ≈ depth)?

Introduction / Context:Critical flow concepts are used to size control sections and predict transitions such as hydraulic jumps. The relation between critical depth and critical velocity provides a quick check in design and analysis for wide, prismatic channels.

Given Data / Assumptions:

• Wide channel so hydraulic depth D ≈ flow depth y.
• Critical state satisfies Fr = 1.
• Acceleration due to gravity = g.

Concept / Approach:

Froude number Fr = V / sqrt(g * D). At critical flow, Fr = 1, hence V_c = sqrt(g * D_c). For wide channels, D_c ≈ h_c, yielding the familiar square-root relation.

Step-by-Step Solution:

Verification / Alternative check:

The result is consistent with specific energy theory where minimum specific energy occurs at Fr = 1 and V_c relates to depth through the same relation.

Why Other Options Are Wrong:

g * h_c or h_c / sqrt(g) have incorrect dimensions; sqrt(g / h_c) is dimensionally inconsistent for velocity.

Common Pitfalls:

Using hydraulic radius or area/width inconsistently; the wide-channel simplification avoids those complications.

Final Answer:

V_c = sqrt(g * h_c)