In open-channel flow theory, the specific energy at a section (per unit weight, measured relative to the channel bed) is equal to which expression?

Introduction / Context:
Specific energy is central to analyzing critical flow, alternate depths, and transitions in open channels. It differs from total head by using the channel bed as the datum and excluding pressure head (since pressure distribution is approximately hydrostatic across the depth for subcritical flows).

Given Data / Assumptions:

• Prismatic open channel with a free surface.
• Hydrostatic pressure distribution across depth.
• Depth of flow y and mean velocity V at the section are known.

Concept / Approach:

Specific energy E is defined per unit weight, measured above the channel bed: E = y + V^2/(2g). Here y is potential (elevation) above bed and V^2/(2g) is kinetic energy head. It is not the same as the total head H = z + p/γ + V^2/(2g) used in pipe flow (with z any external datum).

Step-by-Step Solution:

Verification / Alternative check:

At critical flow, dE/dy = 0 yields the classic relation Fr = 1 and minimum E for a given discharge, confirming consistency with open-channel critical-depth theory.

Why Other Options Are Wrong:

(b) is the total head, not specific energy relative to bed. (c) and (d) omit necessary terms. (e) incorrectly mixes hydrostatic head with y; p/γ inside a free-surface stream at the surface is atmospheric and not included in E.

Common Pitfalls:

Confusing “specific energy” with “specific head” in pressurized flow; ignoring that the datum for E is the channel bed.

Final Answer:

E = y + V^2/(2g)