Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 13, Problem 85P

During a hydraulic jump in a wide channel, the flow depth increases from 1.1 to 3.3 in. Determine the velocities and Froude numbers before and after the jump, and the energy dissipation ratio.

Expert Solution & Answer
Check Mark
To determine

The velocity before the jump.

The velocity after the jump.

The Froude number before the jump.

The Froude number after the jump.

The energy dissipation ratio.

Answer to Problem 85P

The velocity before the jump is 8.402m/s.

The velocity after the jump is 2.80m/s.

The Froude number before the jump is 2.449.

The Froude number after the jump is 0.492.

The energy dissipation ratio is 0.212.

Explanation of Solution

Given Information:

The flow depth after the jump is 3.3m and the flow depth before the jump is 1.1m.

Write the expression for the flow depth after the hydraulic jump.

  y2=0.5y1(1+1+8Fr12)...... (I)

Here, the Froude number before jump is Fr1, the flow depth before the jump is y1.

Write the expression for the Froude number before jump.

  Fr1=V1gy1...... (II)

Here, the velocity before the jump is V1 and the acceleration due to gravity is g.

Write the expression for the average velocity after the hydraulic jump.

  y1V1=y2V2V2=y1V1y2...... (III)

Write the expression for the Froude number after jump.

  Fr2=V2gy2...... (IV)

Here, the velocity after the jump is V2.

Write the expression for the head loss associated with the hydraulic jump.

  hL=y1y2+V12V222g...... (V)

Here, the average velocity before hydraulic jump is V1, the velocity after hydraulic jump is V2, the flow depth before the jump is y1, the flow depth after the hydraulic jump is y2 and the acceleration due to gravity is g.

Write the expression for the specific energy of water before the jump.

  Ec1=y1+V122g....... (VI)

Write the expression for the dissipation ratio.

  Dr=hLEc1...... (VII)

Calculation:

Substitute 3.3m for y2 and 1.1m for y1 in Equation (I).

  3.3m=0.5(1.1m)(1+ 1+8 ( F r 1 ) 2 )1+8 ( F r 1 )2=3.3m( 5.5m)+1(F r 1)2= ( 1.6 )212Fr1=2.449

Substitute 9.81m/s2 for g, 2.449 for Fr1 and 1.2m for y1 in Equation (II).

  2.449=V1 ( 9.81m/ s 2 )( 1.2m )V1=(2.449)( 11.772 m 2 / s 2 )V1=(2.449)(3.43m/s)V1=8.402m/s

Substitute 8.402m/s for V1, 3.3m for y2 and 1.1m for y1 in Equation (III).

  V2=( 1.1m)( 8.402m/s )( 3.3m)=( 9.24m/s )( 3.3)=2.80m/s

Substitute 9.81m/s2 for g, 2.80m/s for V2 and 3.3m for y2 in Equation (IV).

  Fr2=2.80m/s ( 9.81m/ s 2 )( 3.3m )=2.80m/s ( 32.37 m 2 / s 2 )=2.805.68=0.492

Substitute 9.81m/s2 for g, 2.80m/s for V2, 8.402m/s for V1, 3.3m for y2 and 1.1m for y1 in Equation (V).

  hL=(1.1m)(3.3m)+ ( 8.402m/s )2 ( 2.80m/s )22( 9.81m/ s 2 )=(2.2m)+( 62.75 m 2 / s 2 )( 19.62m/ s 2 )=(2.2m)+3.198m=0.998m

Substitute 9.81m/s2 for g, 8.402m/s for V1 and 1.1m for y1 in Equation (VI).

  Ec1=1.1m+ ( 8.402m/s )22( 9.81m/ s 2 )=1.1m+( 70.5936 m 2 / s 2 )( 19.62m/ s 2 )=1.1m+3.59m=4.69m

Substitute 4.69m for Ec1 and 0.998m for hL in Equation (VII).

  Dr=0.998m4.69m=0.212

Conclusion:

The velocity before the jump is 8.402m/s.

The velocity after the jump is 2.80m/s.

The Froude number before the jump is 2.449.

The Froude number after the jump is 0.492.

The energy dissipation ratio is 0.212.

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Chapter 13 Solutions

Fluid Mechanics: Fundamentals and Applications

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