Question

In a rectangular open channel, a hydraulic jump occurs. The ratio of the post-jump depth (\(y_2\)) to the pre-jump depth (\(y_1\)) is measured to be 2. What is the Froude number (\(Fr_1\)) of the flow immediately before the jump?

• A. \(\sqrt{2}\)
• B. 1.5
• C. \(\sqrt{3}\)
• D. 2.5

Hint:

Memorize the Bélanger equation for hydraulic jumps in rectangular channels, as it's frequently tested. It directly links the upstream Froude number (\(Fr_1\)) with the ratio of depths (\(y_2/y_1\)). Knowing this formula allows for a direct and quick calculation.

Answer 1

The Correct Option is C

Step 1: Identifying the Governing Equation:
In a rectangular channel, the relationship between the sequent depths (\(y_1, y_2\)) and the upstream Froude number (\(Fr_1\)) for a hydraulic jump is defined by the Bélanger equation: \[ \frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 Fr_1^2} - 1 \right) \]
Step 2: Substituting Given Values:
We are given the depth ratio \(\frac{y_2}{y_1} = 2\). Substituting this into the formula: \[ 2 = \frac{1}{2} \left( \sqrt{1 + 8 Fr_1^2} - 1 \right) \]
Step 3: Solving for \(Fr_1\):
1. Multiply by 2: \[ 4 = \sqrt{1 + 8 Fr_1^2} - 1 \] 2. Add 1 to both sides: \[ 5 = \sqrt{1 + 8 Fr_1^2} \] 3. Square both sides: \[ 25 = 1 + 8 Fr_1^2 \] 4. Isolate the Froude number term: \[ 24 = 8 Fr_1^2 \] \[ Fr_1^2 = \frac{24}{8} = 3 \] 5. Take the square root: \[ Fr_1 = \sqrt{3} \] The upstream Froude number is \(\sqrt{3}\), which corresponds to option (C).