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## Gradually Varied Flow

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**1. **Gradually Varied Flow Steady flow – no change with time
Flow varies slowly along a channel
Used to describe flow profiles upstream or downstream of structures, obstructions etc.
Bed slope and friction are important

**2. **Derivation of Equations

**3. **Assumptions Streamlines have negligible curvature, i.e. no noticeable convergence or divergence, and so the pressure distribution remains hydrostatic;
Velocity is uniformly distributed over the channel cross section;
The slope of the energy line may be found from uniform flow formulae such as Manning, using the velocity and the hydraulic mean depth which exist at the section under consideration ( or the slope of the energy line at a particular depth at a particular section, is the same as the slope of the bed which would sustain uniform flow at that depth);
The slope of the channel is so small that any difference between depths measured vertically and perpendicular to the channel bed may be neglected.

**4. **GVF is still steady flow so
Total head at section 1:
(no cos ? and no a – assumptions 4 and 2 above)
Differentiating along channel:

**5. **Now and so
Also so
and
Hence

**6. **Back substituting:
and and
So:

**7. **GVF Equation
Equation is for change in surface level along channel. S0 = bed slope, Sf = friction slope.
K0 is the conveyance you would have at normal depth in the channel, determined from S0; K is the conveyance you have at the actual observed depth at any point determined from Manning knowing Q, y and B.

**8. **Notes There are many solutions, some analytical (for prismatic channels)
Usually solve numerically – Direct Step or Standard Step methods
Application software such as HEC-RAS
You need a known depth to start from. This is typically a structure such as a weir

**9. **Direct Step Method
Start at known level
Accuracy depends on size of step ?y

**10. **Direct Step Example A trapezoidal channel has a bed width of 7m, bed slope = 0.0247, side slopes 1:2 (vert.:horiz.), Manning's n=0.03 and carries a flow of 12 cumecs. The water depth behind an underflow sluice placed across the channel is 1.2m. Find the maximum distance upstream from the sluice where irrigation water can be drawn off if this requires at least 0.8m water depth in the channel.

**11. **Standard Step Method
Again, start from known depth
Compute ?y at defined intervals ?x
Iterative solution, as both E and Sf are unknown at x + ?x

**12. **Standard Step Example Repeat previous example
Use a spreadsheet solution

**13. **Standard Step Complexities Can deal with natural channels – A, P etc can vary between each station
Can deal with flood plains – though have to check with each iteration if you are in the flood plain or the main channel only
Often implemented in commercial software

**14. **Reference Chadwick and Morfett
Derivation pages 154-155
Direct Step Method 160-166
Standard Step Method 160-169

**15. **HEC-RAS River Analysis System

**16. **Web Link http://www.hec.usace.army.mil/
2 Useful programmes
HEC-RAS River Analysis System
Also HEC-HMS Hydrologic Modelling System

**17. **Brief Demonstration of HEC-RAS Tweed Study
Meaning of cross-section data

**18. **Getting Started Start software from School of Engineering Applications in computer labs
… or download to your own PC
Download and follow the tutorials from the HEC site

**19. **More on Solving the GVF equation Sometimes work downstream and sometimes up depending on known boundary condition.
In each case the other direction won’t work
Need to classify flow to start solving the problem

**21. **Types of Profile in Prismatic Channels
Direction and relative magnitude of bed slope
Actual depth relative to normal depth and critical depth

**22. **Sustaining or positive slope (S)
Mild – subcritical flow (at given Q)
Critical – critical flow
Steep – supercritical flow
Horizontal slope (H)
Adverse slope (A)

**23. **Three depth classifications:
Region 1 y > (yo & ycr)
Region 2 y lies between yo & ycr
Region 3 y < (yo & ycr)

**25. **Key Points on Flow Classification and GVF Flow will always try to revert to Normal Depth y0
If in moving towards normal depth, the flow passes through critical depth, a hydraulic jump or hydraulic drop will occur
Essential first steps are to calculate the critical depth and normal depth for each reach of channel

**26. **Summary You need to know what to expect before applying equations, or you might get it wrong!

**27. **Reference Chadwick and Morfett pages 149-159
Software – HEC-RAS
US Army Corps of Engineers
http://www.hec.usace.army.mil/

**28. **Channel with Free Inflow
Inflow dependent on head differential
Head differential dependent on flow in channel

**29. **Neglecting friction and change in bed level from 1 to 2:
Hence:
Also for a rectangular channel:
So:
But if it is a free outfall and there is normal depth in the channel:
Thus we have:
Which can be solved for y2

**30. **Inflow Plot of y2 against q for fixed y1

**31. **Flow in Channel Plot of y2 against q

**32. **Combination Superimpose the two plots for fixed y1.
Max q at critical depth in the channel.

**33. **Example A large reservoir discharges into a wide rectangular channel that has a bed slope of 0.0005, and a Manning’s ‘n’ of 0.025. The reservoir level is 100 maod, and the elevation of the bed of the channel at the reservoir is 98.0 maod. Determine the free discharge per unit width from the reservoir.

**34. **Reference None for this section.