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Chapter 10: Problem 79

A rectangular channel \(3.0 \mathrm{m}\) wide has a flow rate of 5.0 \(\mathrm{m}^{3} / \mathrm{s}\) with a normal depth of \(0.50 \mathrm{m} .\) The flow then encounters a dan that rises \(0.25 \mathrm{m}\) above the channel bottom. Will a hydraulic jump occur? Justify your answer.

Short Answer

Expert verified
The answer will depend on the calculated Froude Number and the comparison mentioned in step 5. If the Froude Number is more than 1 and the new flow depth is greater than the critical depth, then a hydraulic jump will occur.

Step by step solution

01

Calculate the Froude Number

The Froude number is a dimensionless number that determines if a flow is subcritical (\(Fr < 1\)), critical (\(Fr=1\)), or supercritical (\(Fr > 1\)). It's calculated as \(Fr = \sqrt{Q^2*g/ (b^2*y_n^3)}\), where \(g\) is the acceleration due to gravity which is approximately equal to 9.8 \(\mathrm{m}/\mathrm{s}^2\). When \(Q = 5.0 \mathrm{m}^3/\mathrm{s}\), \(b = 3.0 \mathrm{m}\), and \(y_n = 0.50 \mathrm{m}\), substitute these values into the formula to find the Froude Number.
02

Interpret the Froude Number

If the Froude Number is more than 1, the flow is supercritical, implying it can transition to subcritical, causing a hydraulic jump. If it's less than or equal to 1, then there is no hydraulic jump.
03

Consider the Dam Height

The dam will cause the flow depth (\(y_n\)) to increase by its height (\(h\)), hence the new flow depth is \(y_n + h = 0.50 m + 0.25 m = 0.75 m\). If this new flow depth is greater than the critical depth (\(y_c\)), the flow becomes subcritical and a hydraulic jump will occur.
04

Calculate the Critical Depth

The critical depth (\(y_c\)) is calculated using the formula \(y_c = (Q^2/g)^{1/3}\). Substitute the given values to find the critical depth.
05

Compare the Flow Depth and Critical Depth

With the new flow depth and the calculated critical depth, compare these two. If the new flow depth is greater than the critical depth, then a hydraulic jump will occur.

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Most popular questions from this chapter

Water flows in a 10 -m-wide open channel with a flowrate of \(5 \mathrm{m}^{3} / \mathrm{s}\). Determine the two possible depths if the specific energy of the flow is \(E=0.6 \mathrm{m}\)

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