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Fluvial Hydraulics CH-3

Fluvial Hydraulics CH-3. Uniform Flow – Flow in Curves/Bendways. 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, D z r o = radius of curvature

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Fluvial Hydraulics CH-3

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  1. Fluvial HydraulicsCH-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

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