Fluvial Hydraulics CH-3 Uniform Flow – Stable Channels

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?

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

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

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:

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

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?

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?

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

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

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

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

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

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

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

Stable Section (US Bureau of Reclamation)

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

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”

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?

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.

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

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:

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

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.