Upstream depth is to stay unchanged while the contraction just produces critical flow. What condition is used?

The minimum specific energy at the contraction equals the upstream specific energy E1

The upstream Froude number equals 1

The critical depth equals the upstream depth 1.8 m

The discharge per unit width stays at 2.7 m^2/s

With E1 = 1.9147 m, what is the critical depth in the contraction and why?

1.2765 m, since yc = (2/3)E1 for a rectangular section

1.8 m, since the depth is unchanged

After finding yc, how is the contracted width b2 obtained?

b2 = Q / q2, where q2 = sqrt(g yc^3)

What happens if the contraction is made narrower than the computed 1.79 m?

The flow chokes and the upstream depth rises

Flow becomes supercritical upstream

Nothing changes since depth is fixed

2072 Magh · 3 marks

Contraction width that just gives critical flow

Short answer

Set E1 = 1.915 m as the critical energy in the contraction. Then yc = 1.277 m, q2 = √(g yc³) = 4.517 m²/s, and the required width b2 = Q/q2 = 1.79 m. A narrower width would choke the flow.

Water flows at a depth of 1.8 m and velocity of 1.5 m/s in a 3 m wide rectangular channel. Find the width at contraction which just causes critical flow without a change in the upstream depth.

Given data

$b_1=3$ m, $y_1=1.8$ m, $V_1=1.5$ m/s, no change in bed level.

Q=3\times1.8\times1.5=8.10\ \text{m}^3/\text{s},\qquad q_1=2.7\ \text{m}^2/\text{s}

E_1=1.8+\frac{1.5^2}{19.62}=1.9147\ \text{m}

Condition

The upstream depth does not change, so $E_1$ is just enough for critical flow in the contraction: $E_{min2}=E_1$ .

y_c=\frac23E_1=1.2765\ \text{m}

q_2=\sqrt{gy_c^3}=\sqrt{9.81\times1.2765^3}=4.5169\ \text{m}^2/\text{s}

b_2=\frac{Q}{q_2}=\frac{8.10}{4.5169}=1.793\ \text{m}

Answer: contracted width $=1.79$ m (critical depth $1.28$ m). A narrower width would choke the flow and raise the upstream depth.

✓ Answer checked by Claude Opus 5.5 (high effort)

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Contraction width that just gives critical flow

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