1. Fluvial Hydraulics CH-3
Uniform Flow –
Flow in Curves/Bendways

2. Fundamentals
With a curve or bend, a constant Q corresponds to a constant U and wetted area, A
Distribution of flow depth, h(y), at a cross-section
Transversal water slope and super-elevation, Dz
ro = radius of curvature
Where is the velocity a maximum?
Explain the flow distribution across the cross-section.
Why do they show erosion at the outer wall?

3. Velocity Distribution
Kozeny (1953) derived a simplified expression for velocity distribution due to turbulent flow:
Assumes that all streamlines have a radius of curvature rc:

4. Super-Elevation
Super-Elevation estimated as difference in two velocity heads (outside vs. inside):

5. Super-Elevation
Other researchers have proposed use of a coefficient of super-elevation:
Super-elevation can be used to determine the discharge:

6. Cross-Waves Usually found in supercritical flows
Caused by turning effect of the curved walls, which does not act equally on all streamlines
Outer wall turns inward to the flow
Inner wall turns away from the flow

7. Cross-Waves
Small disturbance caused by curvature of the outer wall at A and is propagated along line AB
Angle b with tangent extended beyond point A
Initial disturbance by inner wall propagated along the line A’B
Two propagation fronts meet at B
Upstream from ABA’ the flow is unaffected by the curve
Beyond point B, two wavefronts AB and A’B affect each other and are no longer propagated in straight lines but in curved paths BD and BC
Outer concave wall tends to deflect the flow – water surface is raised to a maximum at C
After C, the effect of the inner wall, which is to lower the the water surface, begins to operate – water surface on outer wall starts to drop

8. Cross-Waves
Along inner wall, water surface is depressed further along A’D until point D
After D the effect of the outer wall starts and water surface begins to rise again
Reflection of disturbance waves will not stop when they meet near the center
Waves continue to be reflected back and forth across the channel
Leads to a series of minimum and maximum water surface elevations at q, 3q, 5q, etc.
Can continue beyond bend way

9. Cross-Waves
Ippen (1950) proposed a method for calculating maximum and minimum flow depth:
Graf gives an estimate for the maximum super-elevation (Dz’ + Dz):

10. Example 3.E - Graf
Assume a channel with uniform flow at a depth of 5.03 m. Channel is rectangular with a width of 9 m and average velocity of 12 m/s. After a straight reach, the channel makes an a = 60o curve with ro = 100 m. How much super-elevation is expected?

11. Solution Methodology Note that for a rectangular channel:
Dh = A/B = (Bh)/B = h
Need the determine if flow is supercritical:
Determine b:

12. Solution Methodology Determine q:
Calculate maximum and minimum water depths:

13. Solution Methodology
Another solution for super-elevation:

14. Solution Methodology