Uniform flows and gradually varied flows
5.1 Uniform flows 95
Notes
1. For flat channel the bed slope is:
S^ = sin0 = — ^ - tmO ds
2. For uniform or non-uniform flows thQ friction slope Sf is defined as:
dH 4 T .
^ f = — — o
ds pgD^
3. At any cross-section the relationship between the Darcy coefficient and the fiiction slope leads to:
s =f±-YL = ^M.
' ^ Dn2g SgA' 4. The uniform flow depth or normal depth is denoted CIQ.
5.1.2 Discussion Mild and steep slopes
A chamiel slope is usually 'classified' by comparing the uniform flow depth d^ with the critical flow depth dc. When the uniform flow depth is larger than the critical flow depth, the uniform equilibrium flow is tranquil and subcritical. The slope is called a mild slope. For do < d^, the uniform flow is supercritical and the slope is steep.
do > dc Mild slope Uniform flow: Fr^, < 1 (subcritical flow) do = dc Critical slope Uniform flow: Fr^, = 1 (critical flow) do < dc Steep slope Uniform flow: Fr^ > 1 (supercritical flow) Note
For a wide rectangular channel, the ratio djdc can be rewritten as:
^ = 3 / -
d^ p s i n ^
w h e r e / i s the fi*iction factor and 6 is the channel slope. The above result shows that the notion of steep and mild slope is not only a function of the bed slope but is also a function of the flow resist- ance: i.e. of the flow rate and roughness height.
Critical slope
A particular case is the situation where dQ = d^: i.e. the uniform equilibrium flow is critical. The channel slope, for which the uniform flow is critical, is called the critical slope and is denoted S^.
Critical slopes are seldom found in nature because critical flows and near-critical flows ^ are unstable. They are characterized by flow instabilities (e.g. free-surface undulations and waves) and the flow becomes rapidly unsteady and non-uniform.
^Near-critical flows are characterized by a specific energy only slightly greater than the minimum specific energy and by a Froude number close to unity (i.e. 0.7 <Fr< 1.5 typically). Such flows are unstable, as any small change in specific energy (e.g. bed elevation and roughness) induces a large variation of flow depth.
Note
In the general case, the critical slope satisfies:
S,= sme,= Sf=f- V^
2g(Ai)o
where VQ and (D^^ are the uniform flow velocity and hydrauHc diameter respectively which must sat- isfy also: V, = V, and (D^), = (Du)^.
For a wide rectangular channel, the critical slope satisfies:
S^ = sin e^= ^ f
Application: most efficient cross-sectional shape
Uniform flows seldom occur in Nature. Even in artificial channels of uniform section (e.g. Figs 5.1 and 5.2), the occurrence of uniform flows is not fi*equent because of the existence of con- trols (e.g. weirs and sluice gates) which rule the relationship between depth and discharge.
However, uniform equilibrium flow is of importance as a reference. Most channels are analysed and designed for uniform equilibrium flow conditions.
During the design stages of an open channel, the channel cross-section, roughness and bot- tom slope are given. The objective is to determine the flow velocity, depth and flow rate, given any one of them. The design of channels involves selecting the channel shape and bed slope to convey a given flow rate with a given flow depth. For a given discharge, slope and roughness, the designer aims to minimize the cross-sectional area A in order to reduce construction costs (e.g. Henderson, 1966: p. 101).
The most 'efficient' cross-sectional shape is determined for uniform flow conditions. Consider- ing a given discharge Q, the velocity Fis maximum for the minimum cross-section ^4. According to the Chezy equation (4.23) the hydraulic diameter is then maximum. It can be shown that:
1. the wetted perimeter is also minimum (e.g. Streeter and Wylie, 1981: pp. 450-452),
2. the semi-circle section (semi-circle having its centre in the surface) is the best hydraulic section (e.g. Henderson, 1966: pp. 101-102).
For several types of channel cross-sections the 'best' design is shown in Fig. 5.1 and summar- ized in the following table:
Cross-section Rectangular Trapezoidal Semi-circle
Optimum widths 2d
V3
2d
Optimum cross area^
2d'
f3cf
^d' 2
-sectional Optimum perimeter 4d 2j3d
ird
wetted Optimum hydraulic diameter DH 2d 2d
2d
5.1.3 Uniform flow depth in non-rectangular channels
For any charmel shape, the uniform flow conditions are deduced from the momentum equation (4.19a). The uniform flow depth must satisfy:
^o = -JQ— (5.2a) {PJ, 8gsin0
V : V
I of •••••.. / \ I c y •••••..
- - • • I . , - - - " / I " - - - • • i ,,-•-•""
Circular channel Trapezoidal channel Rectangular channel mum design'
Fig. 5.1 Cross-sections of maximum flow rate: i.e. 'opti-
Fig. 5.2 Examples of waterways, (a) Storm waterway with small baffle chute in background, Oxenford QLD, Gold Coast (April 2003) - looking upstream - concrete-lined low flow channel and grass-lined flood plains, (b) The McPartlan canal connecting Lake Pedler to Lake Gordon, Tasmania (20 December 1992). Trapezoidal concrete-lined channel - flow from the left to the top background.
Fig. 5.2 (c) Norman Creek in Brisbane on 13 IVlay 2002 - looking downstream at the trapezoidal low flow channel and grass-lined flood plains on each side - note the bridge in background (Fig. 5.2d), (d) Norman Creek in Brisbane on 31 December 2001 during a small flood (Q ~ 60m^/s) - looking upstream from the bridge seen in Fig. 5.2c - note the hydraulic jump, in the foreground, induced by the bridge constriction.
where the cross-sectional area AQ and wetted perimeter (P^)o for uniform flow conditions are func- tions of the normal flow depth CIQ. Typical examples of relationships between yl, P^ and J have been detailed in Section 3.4.
For a wide rectangular channel equation (5.2a) yields:
I Sgsin d Wide rectangular channel (5.2b) where q is the discharge per unit width. Equation (5.2b) applies only to rectangular channels such 2isB> lOd^.
5.1 Uniform flows 99
Notes
1. The calculation of the normal depth (equation (5.2)) is an iterative process (see next application).
In practice, the iterative method converges rapidly. The Chezy equation (4.23) may also be used (see Sections 4.4.2 and 4.5.2).
2. Calculations of the normal flow depth always converge to an unique solution for either laminar or turbulent flows. But near the transition between laminar and turbulent flow regimes, the calcu- lations might not converge. Indeed flow conditions near the transition laminar-turbulent flow are naturally unstable.
Applications
1. Considering a rectangular channel, the channel slope is 0.05°. The bed is made of bricks (ks = 10 mm) and the width of the channel is 3 m. The channel carries water (at 20°C). For a 2 w?/s flow rate at uniform equilibrium flow conditions, is the flow super- or subcritical?
Solution
We must compute both the normal and critical depths to answer the question.
For a rectangular channel, the critical depth equals (Chapter 3):
d = 3
It yields Jc = 0.357 m.
The uniform equilibrium flow depth (i.e. normal depth) is calculated by an iterative method. At the first iteration, we assume a flow depth. For that flow depth, we deduce the velocity (by continuity), the Reynolds number, relative roughness and Darcy friction factor. The flow velocity is computed from the momentum equation:
f; = j ^ ^ ^ s i n e (4.19b) Note that the Chezy equation (4.23) may also be used. The flow depth do is deduced by continuity:
where AQ is the cross-sectional area for normal flow conditions (i.e. AQ = Bd^ for a rectangular cross- sectional channel).
The process is repeated until convergence:
Iteration 1 2 Solution
d initialization (m) 0.357
0.903
Re
2.14 X 10^
1.65 X 10^
1.84 X 10^
kJDy, 0.0087 0.0044 0.0054
/ 0.0336 0.0284 0.0299
VoirnJs) 0.738 1.146 1.005
(io(m) 0.903 0.581 0.663
The normal flow is a fully rough turbulent flow {Re = 1.8 X 10^, Re* = 622). This observation justifies the calculation of the Darcy friction factor using the Colebrook-White correlation. The normal depth is 0.663 m.
As the normal depth is greater than the critical depth, the uniform equilibrium flow is subcritical.
2. For the same channel and the same flow rate as above, compute the critical slope Sc-
Solution
The question can be re-worded as: for g = 2 m^/s in a rectangular (B = 3 m) channel made of bricks (ks = 10 mm), what is the channel slope 6 for which the uniform equilibrium flow is critical?
The critical depth equals 0.357 m (see previous example). For that particular depth, we want to determine the channel slope such as do = dc = 0.357 m. Equation (4.19) can be transformed as:
v'
sin 6^ = (8g//)((Ai)o/4) in which dQ = dc = 0.357 m and Fo = Kc = 1.87m/s.
It yields S^ = sin 0^ = 0.00559. The critical slope is 0.32°.
Comments
For wide rectangular channels, the critical slope can be approximated as: ^Sc =^78. In our applica- tion, calculations indicate that, for the critical slope,/— 0.036 leading to: S^ ~ 0.0045 and d^ ~ 0.259°. That is, the approximation of a wide channel induces an error of about 25% on the critical slope Sc.
3. For a triangular channel (26.6° wall slope, i.e. 1 V:2H) made of concrete, compute the (centreline) flow depth for uniform flow conditions. The flow rate is 2m^/s and the channel slope is 0.05°. Is channel slope mild or steep?
Solution
The uniform flow depth is calculated by successive iterations. We assume ^s ~ 1 ^^^^^^ ^^ absence of further information on the quality of the concrete:
Iteration 1 2 Solution
d initialization (m) 0.7
0.97
Re 2.5 X 10^
9.2 X 10^
9.9 X 10^
kJDn 0.0008 0.0006 0.00062
/ 0.0187 0.0177 0.0180
^o(m/s) 1.07 1.29 1.62
do(m) 0.97 0.88 0.899 The normal (centreline) depth equals: do = 0.899 m. The uniform flow is turbulent (Re = 9.9 X 10^, Re* = 58.2). To characterize the channel slope as mild or steep, we must first calculate the critical depth. For a non-rectangular channel, the critical depth must satisfy a minimum specific energy. It yields (Chapter 3):
— - ^ (3.34a) B g
where A is the cross-sectional area and B is the fi*ee-surface width. For a triangular channel, A = O.SBd and A = 2d cot 8 where 8 is the wall slope (8 = 26.6°). Calculations indicate that the crit- ical depth equals: d^ = 0.728 m. As the normal depth is larger than the critical depth, the channel slope is mild.
5.2 NON-UNIFORM FLOWS