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: Understanding the Question:
The question describes a hydraulic jump in a rectangular channel. We are given the ratio of the sequent depths (downstream depth \(y_2\) to upstream depth \(y_1\)) and asked to find the Froude number of the flow before the jump (\(Fr_1\)). A hydraulic jump can only occur when the upstream flow is supercritical, i.e., \(Fr_1>1\).
Step 2: Key Formula or Approach:
The relationship between the sequent depth ratio and the upstream Froude number for a hydraulic jump in a rectangular channel is given by the Bélanger equation: \[ \frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 Fr_1^2} - 1 \right) \] Step 3: Detailed Explanation:
We are given that the ratio of post-jump depth to pre-jump depth is 2. \[ \frac{y_2}{y_1} = 2 \] Now, we substitute this value into the Bélanger equation and solve for \(Fr_1\). \[ 2 = \frac{1}{2} \left( \sqrt{1 + 8 Fr_1^2} - 1 \right) \] Multiply both sides by 2: \[ 4 = \sqrt{1 + 8 Fr_1^2} - 1 \] Add 1 to both sides: \[ 5 = \sqrt{1 + 8 Fr_1^2} \] Square both sides to remove the square root: \[ 5^2 = 1 + 8 Fr_1^2 \] \[ 25 = 1 + 8 Fr_1^2 \] Subtract 1 from both sides: \[ 24 = 8 Fr_1^2 \] Divide by 8: \[ Fr_1^2 = \frac{24}{8} = 3 \] Take the square root of both sides: \[ Fr_1 = \sqrt{3} \] Step 4: Final Answer:
The Froude number of the flow immediately before the jump (\(Fr_1\)) is \(\sqrt{3}\). This corresponds to option (C).