Question

For a hydraulic jump formed in a rectangular horizontal channel, the sequent depth ratio is 2. The Froude number of supercritical stream is

• A. \(\sqrt{3}\)
• B. \(\sqrt{5}\)
• C. \(\sqrt{6}\)
• D. \(\sqrt{8}\)

Hint:

Use the Belanger sequent-depth equation \(y_2/y_1=\frac{1}{2}(\sqrt{1+8Fr_1^2}-1)\) and substitute the given ratio of 2 to solve for \(Fr_1\).

Answer 1

The Correct Option is A

Step 1: Recall the sequent depth relation for a hydraulic jump.
For a hydraulic jump in a rectangular horizontal channel, the ratio of the depth after the jump (\(y_2\)) to the depth before the jump (\(y_1\)) is given by the Belanger equation:
\[ \frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1+8Fr_1^2}-1\right) \]
where \(Fr_1\) is the Froude number of the supercritical (fast, shallow) approach flow before the jump.

Step 2: Substitute the given sequent depth ratio.
We are given \(\dfrac{y_2}{y_1}=2\). Substitute:
\[ 2=\frac{1}{2}\left(\sqrt{1+8Fr_1^2}-1\right) \]
Multiply both sides by 2:
\[ 4=\sqrt{1+8Fr_1^2}-1 \]

Step 3: Solve for \(Fr_1\).
Add 1 to both sides: \(\sqrt{1+8Fr_1^2}=5\).
Square both sides: \(1+8Fr_1^2=25\), so \(8Fr_1^2=24\), giving \(Fr_1^2=3\), hence \(Fr_1=\sqrt3\) (taking the positive root, since a Froude number is always positive).

Step 4: Check the result and rule out the other options.
Check: with \(Fr_1^2=3\), \(\sqrt{1+8(3)}=\sqrt{25}=5\), and \(\frac{1}{2}(5-1)=2\), which matches the given ratio.
If \(Fr_1=\sqrt5\) (option B), \(Fr_1^2=5\), giving a ratio of \(\frac{1}{2}(\sqrt{41}-1)\approx2.7\), not 2. If \(Fr_1=\sqrt6\) (option C), \(Fr_1^2=6\), giving a ratio of \(\frac{1}{2}(\sqrt{49}-1)=3\), not 2. If \(Fr_1=\sqrt8\) (option D), \(Fr_1^2=8\), giving a ratio of \(\frac{1}{2}(\sqrt{65}-1)\approx3.5\), not 2. So only \(Fr_1=\sqrt3\) fits.

Final Answer:
\[ \boxed{Fr_1=\sqrt3} \]