Critical flow in open channels: At the critical depth (y_c), what happens to the discharge for a given specific energy or channel control condition?

Introduction / Context:
Critical flow is a cornerstone idea in open-channel hydraulics. It occurs at the critical depth y_c where the Froude number equals 1. For a given specific energy or at a control (such as a sluice or section where energy is fixed), the discharge reaches an extremum at y_c. Understanding this relationship helps in designing flumes, measuring structures, and recognizing flow regime transitions.

Given Data / Assumptions:

• Prismatic channel, typically analyzed as rectangular for unit-width reasoning.
• Steady, incompressible flow with negligible losses at the control section.
• Specific energy E is considered given (e.g., by an upstream control).

Concept / Approach:
For a rectangular channel of unit width, discharge per unit width is q = v * y. Specific energy is E = y + v^2/(2g) = y + (q^2)/(2g y^2). For fixed E, differentiate q with respect to y and set derivative to zero to locate the extremum. The condition for extremum yields v^2 = g y, i.e., critical state (Fr = 1). Evaluating the second-derivative or using standard results shows that the extremum is a maximum, not a minimum.

Step-by-Step Solution:

Verification / Alternative check:
Dimensionally, the well-known formula for a rectangular channel is y_c = (q^2/g)^{1/3}. Rearranged, q = (g y_c^3)^{1/2}. For any other y at the same E, q is smaller, confirming maximality at y_c.

Why Other Options Are Wrong:

• Zero: Critical flow is a finite, nonzero discharge state.
• Minimum: Minimum specific energy occurs at y_c, not minimum discharge for given E.
• Indeterminate: It is determinate and known to be a maximum.

Common Pitfalls:
Confusing “minimum specific energy” at y_c with “minimum discharge”; mixing up conditions “for given E” versus “for given q”.

Final Answer:
maximum