1. Glenn E. Moglen Department of Civil & Environmental Engineering Virginia Tech Momentum (continued) CEE 4324/5984 –Open Channel Flow – Lecture 11

2. Questions from Lecture 10? Brief Comments on Exam 1 Begin Momentum Static and Dynamic Forces Derive Rectangular Momentum Function Contrast Momentum vs. Energy Hydraulic Jump Derivation Conjugate Depths in a Rectangular Channel Return to Sluice Gate/Hydraulic Jump Today’s Agenda:

3. Contrast Energy vs. Momentum Energy Momentum

4. Contrast Energy vs. Momentum

5. Momentum/Hydraulic Jumps in a Rectangular Channel Momentum From which we derived (for a rectangular channel):

6. Let’s look at a few different hydraulic jumps…

7. Example 16: Conjugate Depths in a Rectangular Channel q = 10 ft 2 /s y 1 = 0.24 feet Find: y 2 ? Show that M 1 and M 2 are the same value for y 1 and y 2. >

8. Example 16: Conjugate Depths in a Rectangular Channel y1y1 y2y2

9. Return to Sluice Gate/Hydraulic Jump

10. Example 17: Return to Sluice Gate/Hydraulic Jump (cont.) > q = 10 ft 2 /s y 1 = 8.0 feet Find: y 2 ? (alternate depth to y 1 ) y 3 ? (conjugate depth to y 2 ) Energy lost in jump? Momentum (thrust) on gate? Show results graphically.

11. Return to Sluice Gate/Hydraulic Jump (cont.) 1 2e 2m 3m 3e EE y 1 =8.0ft MM

12. Example 17: Take-away facts This example shows the graphical and numerical solution of a sluice- gate/hydraulic jump system. Energy is conserved at a sluice gate, but there is a loss of momentum (net thrust) on the gate by the flow. Momentum is conserved in a hydraulic jump, but there is a net energy loss in the jump.

13. Momentum in Non-Rectangular Channels Momentum in Rectangular Channel Momentum in Non-Rectangular Channel

14. y Centroid

15. Hydraulic Jumps in Non-Rectangular Channels There is no equivalent analytical relationship between conjugate depth pairs in non- rectangular channels. There are graphical solutions to conjugate depth pairs for trapezoidal and circular cross-sections. Such graphical solutions do NOT seem to be presented in our class text. PDF files for these graphical solutions are presented at the class web page and are copied on the following two slides. Examples based on these figures will follow…

16. Hydraulic Jumps in Non-Rectangular Channels: Trapezoidal Cross-Section

17. Hydraulic Jumps in Non-Rectangular Channels: Circular Cross-Section

18. Example 18: Hydraulic Jump in a Trapezoidal Channel Q = 640 ft 3 /s b = 20 feet, m = 2 y 1 = 2.0 feet Find: Show that y 1 corresponds to upstream, supercritical conditions Find y 2 ? (downstream conjugate depth to y 1 ) Show that momentum is conserved in jump Energy lost in jump? >

19. Example 18: Hydraulic Jump in a Trapezoidal Channel Z=10 y 2 /y 1 =2

20. Example 18: Take-away facts A hydraulic jump in a trapezoidal channel conserves momentum A hydraulic jump in a trapezoidal channel loses energy Figure is helpful in solving problem It would be difficult to read the figure with high precision if not exactly on curve or on one of the provided grid lines.