Understanding Shear Stress in Fluvial Hydraulics: Stable Channel Design

1. Fluvial HydraulicsCH-3 Uniform Flow – Stable Channels

2. Shear Stress on Bed • Shear stress defined earlier as… • How does this equation simplify for channels with a large width? • Assume a trapezoidal channel… • Will shear stress on bed be larger or smaller than the shear stress on the sides of the channel?

3. Distribution of Shear Stress • Shear stress is distributed over the wetted perimeter, P • Graf gives the typical distribution for a trapezoidal channel (Chow, 1959) – derived from analytical and finite-difference methods • Pattern of distribution varies with shape of the section but unaffected by size of the section

4. Forces - Bottom of the Channel • Particles at bottom of channel resist shear stress of moving fluid… • Write a force balance equation at the moment of motion (incipient motion): David Chin, Water Resources Engineering

6. Forces on the Side of the Channel • Particles at side of channel resist shear stress of moving fluid and particle weight that acts down the side of the channel… • Total force tending to move particle: • Total force resisting motion:

7. Forces on the Side of the Channel • When motion is incipient: • Based on this equation, what is the requirement for stable side walls?

8. Example 3.C A channel excavated in earth should convey a water discharge of Q = 57 m3/s at an average temperature of 14oC. The bed slope is 0.001; the banks have side slopes of 1.5:1 (H:V). A grain size analysis yielded d50 = 37 mm, the angle of repose is 37o, ss = 2.65, and n = 0.02. What should be the dimensions of this channel, if no erosion is allowed either at the bottom or on the banks?

9. Solution Methodology • Stability of banks requires q < j • Check using m… q = 33.7o • Calculate critical bed-shear stress on walls: • Need the critical shear stress – How?

10. Solution Methodology • Calculate a flow depth not to exceed these critical values: • Use minimum value of h based on shear stress on side walls • Actually should use an h smaller than this critical value

11. Solution Methodology • Last step is to solve for b: • Use Manning’s Equation with given Q • Use Table 1.1 for A, Rh

12. Stable Section • Stable cross section – section in a channel with a mobile bed where there is no erosion over the entire wetted perimeter • Ideal stable cross section – stable cross section with a maximum discharge and a minimal wetted perimeter • Minimum water area, minimum top width, and maximum mean velocity  minimum excavation

13. Stable Section • Assume a channel with a side wall angle at the water surface equal to the angle of repose: • To design a stable hydraulic section for maximum efficiency, it is necessary to create a condition of impending motion everywhere on channel bed

14. Stable Section(US Bureau of Reclamation) • Tractive force acting on a particle on the sloping wall:

15. Stable Section(US Bureau of Reclamation) • Condition of impending motion everywhere on bed: • Note the difference between hand h’

16. Stable Section(US Bureau of Reclamation) • The following differential equation is derived:

17. Stable Section(US Bureau of Reclamation)

18. Stable Section(US Bureau of Reclamation) • Other hydraulic parameters of the ideal cross section:

19. Stable Section(US Bureau of Reclamation) • If Q which must be conveyed through the channel is different than Qi=UA: • Q < Qi : Replace B with reduced B’ • Q > Qi : Replace B with increased B”

20. Example 3.C A channel excavated in earth should convey a water discharge of Q = 57 m3/s at an average temperature of 14oC. The bed slope is 0.001; the banks have side slopes of 1.5:1 (H:V). A grain size analysis yielded d50 = 37 mm, the angle of repose is 37o, ss = 2.65, and n = 0.02. What should be the dimensions of this channel, if no erosion is allowed either at the bottom or on the banks?

25. Example 3.D An artificial channel is constructed in a mountainous region and should convey 30 m3/s at T=14oC without causing erosion. The slope of the channel will be 0.01 and n = 0.025. The grain size analysis has shown that the granular material is non-cohesive with d50=50 mm, j = 37o, and ss = 2.65. (a) Determine dimensions of a rectangular channel with sides of wooden boards. (b) Determine the dimensions of an ideal stable cross section with the channel constructed of its bed material.

26. Solution Methodology • Use critical shear stress criteria: • Use Fig. 3.13 (Shields diagram) to determine t*cr • Use t*cr to solve for tocr • Use the bed shear stress equation to solve for the hydraulic radius: • Use Manning’s equation to calculate U • Solve for A, P, b and h

27. Solution Methodology • We need to first solve for h (maximum depth in the middle of the ideal cross-section): • Use critical shear stress from before to solve for tocr • Solve for h using:

28. Solution Methodology • Use equations for ideal cross-section to solve for h’, A, B, and U • Check the ideal discharge versus the actual discharge and adjust B if necessary

29. Example 3.D An artificial channel is constructed in a mountainous region and should convey 30 m3/s at T=14oC without causing erosion. The slope of the channel will be 0.01 and n = 0.025. The grain size analysis has shown that the granular material is non-cohesive with d50=50 mm, j = 37o, and ss = 2.65. (a) Determine dimensions of a rectangular channel with sides of wooden boards. (b) Determine the dimensions of an ideal stable cross section with the channel constructed of its bed material.