Gradually Varied Flow

1. Gradually Varied Flow

2. Gradually Varied Flow • Uniform flow requires a channel of constant cross-section and sufficient length for the gravitational forces to balance the frictional resistance.

3. L y1 WsinQ v1 y2 Q v2 Rf W For uniform flow SFs = 0 y1 = y2 v1 = v2 so

4. yn y yc Gradually Varied Flow • At changes in cross-section, slope or roughness, the two forces will not be balanced and the flow conditions will adjust toward equilibrium.

5. Gradually Varied Flow • Within the channel reach where this adjustment occurs, the flow is said to be varied or non-uniform. • If the change in flow conditions occurs gradually over relatively long channel reaches, the flow is said to be gradually varied flow.

6. hL1-2 EGL HGL H y1 y2 Channel Bottom z1 z2 Datum Energy Equation

7. Gradually Varied Flow Differentiating with respect to x, the distance along the channel

8. dH EGL HGL H y1 y2 Channel Bottom z1 z2 Datum dx

9. If we assume a wide channel and a per unit width consideration dH/dx represents the slope of the energy gradeline, S, which is positive downward. dz/dx is the channel slope, S0, also positive downward.

10. or so

11. so remember dy/dx is the slope of the water surface with respect to the channel bottom.

12. Gradually Varied Flow • If dy/dx is positive the water is getting deeper in the downstream direction. • If dy/dx is negative the water is getting shallower in the downstream direction. • If dy/dx is zero then the flow is uniform.

13. Normal depth, yn, is the depth of uniform flow for a given channel slope and roughness. • Critical depth, yc, is the depth of flow where the Froude number is equal to 1 and is independent of slope and channel roughness.

14. If • yn > yc the slope is termed mild and denoted with a subscript M. • yn < yc the slope is termed steep (S). • yn = yc the slope is termed critical slope, (C). • A slope that is negative or runs uphill in the downstream direction is termed adverse, (A). • A channel with no slope is said to be horizontal (H).

15. If • y > yn and yc flow is said to be in zone 1 and denoted with a subscript 1. • If y falls between yn and yc then flow is in zone 2. • If y is < yn and yc flow is said to be in zone 3. • Also • If yn > y, S0 < S • If yn < y, S0 > S • If S0 is less than or equal to 0, yn is not defined.

16. Figure 4.23 (Haan et al., 1994) and Figure 9-2 (Chow, 1959) depict the flow profiles or backwater curves for each type of slope and zone. • The slope of the water surface for the various situations can be deduced from : S can be approximated as the slope calculated from Manning’s equation using the actual depth of flow. S0 is the slope in Manning’s equation corresponding to yn.

17. } Zone 2 by definition y < yn so S0 < S y > yc so F < 1 Example 1 Flow Profiles : M-2 Profile yn > yc (Mild slope by definition) From Equation 4.53, the numerator will be negative, the denominator will be positive, so dy/dx will be negative and flow will be shallower in the downstream direction.

18. Example 2 Flow Profiles: • S-1 Profile • yc > yn (Steep slope by definition) • y > yn , y > yc (Zone 1 by definition) • S0 > S, F < 1 • From Equation 4.53, the numerator will be positive, the denominator will be positive, so dy/dx will be positive and flow will be getting deeper in the downstream direction.

19. hL EGL HGL E2 E1 H Channel Bottom z1 z2 Datum dx Calculating Flow Profiles • We can approximate the flow profiles by considering E1 + z1 = E2 + z2 + hL:

20. By definition: Where Dx is the length of the channel reach. Also: Where Sf is the friction slope or the slope of the energy grade line.

21. Combining these two equations we get: • Sf can be approximated from Manning’s equation using an average flow depth for the reach. • For subcritical flow, backwater curves should be determined in the upstream direction. • For supercritical flow curves should be determined in the downstream direction.

22. Start profile calculations at points of known water surface elevations, for example at overfalls from mild channels, the depth y is equal to the critical depth yc. • Application of this method is called the direct step method.