1. Glenn E. Moglen Department of Civil & Environmental Engineering Virginia Tech Review/Prepare for Exam 1 & Start Momentum CEE 4324/5984 –Open Channel Flow – Lecture 9

2. Questions from Lecture 8? Review/Prepare for 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. Example 14: Using Figures 2-15 and 2-16 Setting: A lake discharges at a trapezoidal channel outfall, b = 5 feet, m = 2.0. Outfall is at head of a steep slope. Surface elevation of the lake relative to the outfall is 2.0 feet. Find the discharge, Q. >

4. Which Figure? Figure 2-15 or 2-16? Step 1 is ALWAYS to first determine if the given information points to known critical depth (Figure 2-15) or critical energy (Figure 2-16). For this problem, which is it?

6. Solving Example 14 w/o Figure 2-16 Re-arrange Specific Energy equation, solving for Q : Note that A is f(y) Build spreadsheet with trial y values, A(y), and resultant Q (from above eq’n) Choose maximum resultant Q

7. Solving Example 14 w/o Figure 2-16 Maximum Q value Note that solution points to 68 ft 3 /s (Figure 2-16 gave us 67 ft 3 /s) Looks like we can really only read Figure 2-16 to one significant figure So Figure 2-16 answer is more accurately, 70 ft 3 /s

8. Example 15: Using Figures 2-15 and 2-16 Settings: Trapezoidal Channel, b = ? feet, m = 2.0 Outfall to steep slope Depth of flow at outfall is 3.0 feet at outfall invert Discharge is 260 ft 3 /s Find b Which Figure? >

10. Review of Approach to Solving Critical Flow in Circular or Trapezoidal Channel Identify Figure. Does the problem pertain to critical depth or energy? Identify what problem pertains to: 1. Discharge? 2. Depth or Energy? 3. Geometry ( D or b )? If #1 or #2 above, if high precision is needed, spreadsheet approach may be indicated.

11. Non-Rectangular Channel Relationships – Discharge Previously (unit discharge [=] ft 2 /s): Now (discharge [=] ft 3 /s):

12. Non-Rectangular Channel Relationships – Specific Energy Previously: Now: Always good:

13. Non-Rectangular Channel Relationships – Critical Depth Previously: Now (use Figure 2-15):

14. Non-Rectangular Channel Relationships – Critical Energy Previously: Now (use Figure 2-16):

15. Non-Rectangular Channel Relationships – Froude Number Previously: Now:

16. Non-Rectangular Channel Relationships – Alternate Depth Previously: Now:

17. Reminder: “Summary…” under “Help Files and Useful Links”

18. Preparation for Exam 1 Mechanics: Probably 3-4 multi-part problems You will be provided with any figures (e.g. Fig. 2-15) or tables (e.g. circular properties) you need You are allowed 1 sheet 8.5”x11” (both sides) on which to write equations – but NO worked problems. Sheet is turned in with exam. Content: All energy material up to and including Homework #2

19. Preparation for Exam 1 – General Advice Be prepared for any double combination of gate/step/constriction for both rectangular and non-rectangular cross- sections. Understand the underlying principles so you can apply those principles in a situation you’ve never seen. Be comfortable with E-y diagram so you can solve problems graphically.

20. Preparation for Exam 1 – General Advice Be able to identify choke conditions and solve for initial quantities during transient conditions. In choke problems, how does the flow overcome the choke? (i.e. how does the flow gain energy?) Does a sub-critical flow have a pathway to gain energy – how? Does a super-critical flow have a pathway to gain energy – how?

21. Sample Energy Problems to Consider – Build your own Rectangular cross-section with width, w 1 Q = xx ft 3 /s Initial depth is y 1 Flow encounters an upward step of height, z Find: Is it a choke? If yes, Find the transient initial discharge. Find the final upstream depth. Find a second width, w 2 so it’s not a choke If no, find Fr 1, Fr 2, E 1, E 2, y 2

22. Sample Energy Problems to Consider – Build your own (cont.) Take problem from previous slide and change: z (make it bigger of smaller to force or remove choke) y 1 (make it bigger or smaller to change energy levels and also change if initial depth is sub- or super-critical) w 1 (make it bigger of smaller to change q and influence specific energy)

23. Static and Dynamic Forces Static Force: Dynamic Force:

24. Derive Rectangular Momentum Function >

25. Contrast Energy vs. Momentum Energy Momentum

26. Contrast Energy vs. Momentum

27. Hydraulic Jump Derivation >

28. Conjugate Depths in a Rectangular Channel >

29. Return to Sluice Gate/Hydraulic Jump

30. Return to Sluice Gate/Hydraulic Jump (cont.) >

31. Return to Sluice Gate/Hydraulic Jump (cont.) 1 2e2m 3m 3e EE