Gradually Varied Flow • Uniform flow requires a channel of constant cross-section and sufficient length for the gravitational forces to balance the frictional resistance.
L y1 WsinQ v1 y2 Q v2 Rf W For uniform flow SFs = 0 y1 = y2 v1 = v2 so
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.
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.
hL1-2 EGL HGL H y1 y2 Channel Bottom z1 z2 Datum Energy Equation
Gradually Varied Flow Differentiating with respect to x, the distance along the channel
dH EGL HGL H y1 y2 Channel Bottom z1 z2 Datum dx
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.
so remember dy/dx is the slope of the water surface with respect to the channel bottom.
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.
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.
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).
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.
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.
} 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.
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.
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:
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.
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.
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.