Open-channel flow — alternate (conjugate) depth in a rectangular channel In a rectangular channel, water flows with depth y1 = 0.10 m and velocity v1 = 6 m/s. Compute the alternate (conjugate) depth y2 corresponding to a hydraulic jump for the same discharge per unit width. Provide the closest value.

Introduction / Context:
When supercritical flow transitions to subcritical flow in a rectangular channel, a hydraulic jump forms. The upstream and downstream depths satisfying momentum conservation are called conjugate (or alternate) depths. Determining the downstream conjugate depth y2 from the upstream conditions is essential for designing stilling basins and estimating energy dissipation.

Given Data / Assumptions:

• Rectangular, horizontal channel; width b cancels in unit-width formulation.
• Upstream depth y1 = 0.10 m, velocity v1 = 6 m/s.
• Assume hydrostatic pressure at sections outside the roller and negligible friction over the jump length.

Concept / Approach:
Use continuity and momentum (not energy, as the jump dissipates energy). First compute the upstream Froude number: Fr1 = v1 / sqrt(gy1). For a rectangular channel, the conjugate depth ratio is determined by:
y2 / y1 = (1/2) * (sqrt(1 + 8Fr1^2) - 1). This follows from the specific force balance for a control volume spanning the jump.

Step-by-Step Solution:
Compute Fr1 = 6 / sqrt(9.81 * 0.10) ≈ 6 / 0.990 ≈ 6.06.Evaluate y2/y1 = 0.5 * (sqrt(1 + 8 * 6.06^2) - 1) ≈ 0.5 * (sqrt(1 + 8 * 36.7) - 1) ≈ 0.5 * (sqrt(294.6) - 1).sqrt(294.6) ≈ 17.16 → y2/y1 ≈ 0.5 * (17.16 - 1) = 8.08.Thus y2 ≈ 8.08 * 0.10 m ≈ 0.81 m (rounded).

Verification / Alternative check:
The computed y2 is much larger than y1 (as expected for a strong jump with Fr1 » 1). Using the same discharge per unit width, the downstream velocity v2 = q / y2 will be small and subcritical, consistent with momentum conservation.

Why Other Options Are Wrong:
0.30 m, 0.40 m, 0.60 m: Too small for such a high Froude number; do not satisfy the momentum relation.
0.86 m: Close, but 0.81 m is the more accurate rounded value from the formula.

Common Pitfalls:

• Using specific energy equality (specific-energy alternate depth) instead of momentum conjugate depth; the hydraulic jump is governed by momentum, not energy conservation.
• Forgetting to compute Fr1 with depth y1 rather than hydraulic radius.

Final Answer:
0.81 m