34;"K J TfLI
4.4.2 Flow resistance of open channel flows
Momentum equation in steady uniform equilibrium open channel flow
The fiindamental problem of steady uniform equilibrium flow is determining the relation between the flow velocity, the uniform flow depth, the channel slope and the channel geometry.
For a steady and uniform equilibrium flow the flow properties are independent of time and of position along the flow direction. That is:
3^
and
dV_
ds 0
where t is the time and s is the co-ordinate in the flow direction.
The momentum equation along a streamline states the exact balance between the shear forces and the gravity component. Considering a control volume as shown in Fig. 4.11, the momentum equation yields:
^ o ^ w ^ ^ pgA/iis sind (4.19a)
where TQ is the bottom shear stress, P^ is the wetted perimeter and A^ is the length of the con- trol volume, A is the cross-sectional area and 6 is the channel slope. Replacing the bottom shear stress by its expression (equation (4.14)), the momentum equation for uniform equilibrium flows becomes:
K^JMjCAiksin.
(4.19b)where VQ is the uniform (equilibrium) flow velocity and (Z)H)O is the hydraulic diameter of uniform equilibrium flows.
The momentum equation for steady uniform open channel flow (equation (4.19a)) is rewrit- ten usually as:
Sf=S, (4.19c) where Sf is called the fiiction slope and 5*0 is the channel slope defined as:
Sr = -dH 4 T .
3^ pgD^ (4.20)
Bottom shear stress
Fig. 4.11 Application of the momentum equation to uni- form equilibrium open channel flow.
4.4 Flow resistance in open channels 77
So = - 3z ^ =sme (4.21)
ds
where H is the mean total head and ZQ is the bed elevation. Note that the definitions of the fiiction and bottom slope are general and applied to both uniform equilibrium and gradually varied flows.
Notes
1. In sediment transport calculations, the momentum equation is usually expressed as equation (4.19a). Equation (4.19b) is more often used in clear-water flows.
2. For flat channel, the channel slope SQ (or bed slope) is almost equal to the slope tangent:
S^ = —^ = sin ^ - tan 6 as
SQ might be denoted '/' (Comolet, 1976).
3. For uniform equilibrium and gradually varied flows the shear stress TQ can be expressed as:
4. Equations (4.19)-(4.21) are valid for any shape of channel cross-section.
Combining the definitions of bottom shear stress (equation (4.14)) and of fi*iction slope, the momentum equation can be rewritten as:
(4.22) This relationship (i.e. equation (4.22)) is valid for both uniform equilibrium and gradually varied flows.
Chezy coefficient
The dependence of the flow velocity on channel slope and hydraulic diameter can be deduced fi-om equations (4.14) and (4.19). Replacing TQ by its expression (equation (4.14)), the momentum equation (equation (4.19b)) can be transformed to give the Chezy equation:
V=Ccu,^^^sme (4.23) where Qhezy is the Chezy coefficient (units m^^^/s), D^ is the hydraulic diameter and 6 is the
channel slope (e.g. Henderson, 1966: pp. 91-96; Streeter and Wylie, 1981: p. 229).
The Chezy equation (4.23) was first introduced in A.D. 1768 as an empirical correlation.
Equation (4.23) is defined for uniform equilibrium and non-uniform gradually varied flows. The Chezy coefficient ranges typically from 30m^^^/s (small rough channel) up to 90m^^^/s (large smooth channel). Equations (4.19b) and (4.23) look similar. But it must be emphasized that equa- tion (4.19b) was deduced from the momentum equation for uniform equilibrium flows. Equation (4.19b) is not valid for non-uniform equilibrium flows for which equation (4.22) should be used.
Notes
1. The Chezy equation was introduced in 1768 by the French engineer A. Chezy when designing canals for the water supply of the city of Paris.
2. The Chezy equation applies for turbulent flows. Although A. Chezy gave several values for Qh^zyj several researchers assumed later that Cchezy was independent of the flow conditions. Further stud- ies showed clearly its dependence upon the Reynolds number and channel roughness.
3. The Chezy equation is valid for uniform equilibrium and non-uniform (gradually varied) turbu- lent flows.
4. For uniform equilibrium flows the boundary shear stress To can be rewritten in term of the Chezy coeflficient as:
Pg V'
(^Chezy) hezy/
The Chezy equation in uniform equilibrium flows can be rewritten in term of the shear velocity as:
V _ ^Chezy
and the Chezy coefficient and the Darcy friction factor are related by:
c = &
^ Chezy -» ^
5. At uniform equilibrium and in fully rough turbulent flows (e.g. natural streams), the Chezy coefficient becomes:
^chezy ~ ^^-^ ^^Sio " ^ +10.1 Uniform equilibrium fully rough turbulent flow This expression derives firom Colebrook-White formula and it is very close to Keulegan formula (see below).
6. In non-uniform gradually varied flows the combination of equations (4.22) and (4.23) indicates that the Chezy coefficient and the Darcy friction factor are related by:
r = M Ui-
7. Empirical estimates of the Chezy coefficient include the Bazin and Keulegan formulae:
Qhezy ~ , Bazin formula 87
l+^Bazin/VAl/4
where ^Bazin — 0.06 (very smooth boundary: cement and smooth wood), 0.16 (smooth boundary:
wood, brick, freestone), 0.46 (stonework), 0.85 (fine earth), 1.30 (raw earth), 1.75 (grassy bound- ary, pebble bottom, and grassed channel) (e.g. Comolet, 1976); and
Qhezy = 18.0 logio — + 8.73 Keulegan formula
which is valid for fully rough turbulent flows and where k^ (in mm) = 0.14 (cement), 0.5 (planed wood), 1.2 (brick), 10-30 (gravel) (Keulegan, 1938).
Interestingly Keulegan formula was validated with Bazin's data (Bazin, 1865a).
4.4 Flow resistance in open channels 79
The Gauckler-Manning coefficient
Natural channels have irregular channel bottom, and information on the channel roughness is not easy to obtain. An empirical formulation, called the Gauckler-Manning formula, was developed for turbulent flows in rough channels.
The Gauckler-Manning formula is deduced from the Chezy equation by setting:
^Manning V
and equation (4.23) becomes:
^Manning
sin e (4.25) where ^Manning is the Gauckler-Manning coefficient (units slvo}'^), Dn is the hydraulic diameter
and 6 is the channel slope.
The Gauckler-Manning coefficient is an empirical coefficient, found to be a characteristic of the surface roughness alone. Such an approximation might be reasonable as long as the water is not too shallow nor the channel too narrow.
Notes
1. Equation (4.25) is improperly called the 'Manning formula'. In fact it was first proposed by Gauckler (1867) based upon the re-analysis of experimental data obtained by Darcy and Bazin (1865).
2. Philippe Gaspard Gauckler (1826-1905) was a French engineer and member of the French 'Corps des Ponts-et-Chaussees'.
3. Robert Manning (1816-1897) was chief engineer at the Office of Public Works, Ireland. He presented two formulas in 1890 in his paper 'On the flow of water in open channels and pipes' (Manning, 1890). One of the formula was the 'Gauckler-Manning' formula (equation 4.25) but Robert Manning did prefer to use the second formula that he gave in the paper. Further informa- tion on the 'history' of the Gauckler-Manning formula was reported by Dooge (1991).
4. The Gauckler-Manning equation is valid for uniform equilibrium and non-uniform (gradually varied) flows.
5. Equation (4.25) is written in SI units. The units of the Gauckler-Manning coefficient ^Manning is s/m^^^. A main critic of the (first) Manning formula (equation (4.25)) is its dimensional aspect:
i.e. ^Manning is not dimcnsionlcss (Dooge, 1991).
6. The Gauckler-Manning equation applies for fully rough turbulent flows and water flows. It is an empirical relationship but has been found reasonably reliable.
7. Typical values of ^Manning (ill SI Units) are:
'^Manning
0.010 0.012 0.013 0.012 0.014 0.025 0.029 0.05 0.15
Glass and plastic Planed wood Unplanned wood Finished concrete Unfinished concrete Earth
Gravel
Flood plain (light brush) Flood plain (trees)
Yen (1991b) proposed an extensive list of values for a wide range of open channels.
The Strickler's coefficient
In Europe, the Strickler equation is used by defining:
vl/6
Q h e z y ^ ^Strickler! ~7~ I ( 4 . 2 6 )
and equation (4.23) becomes the Strickler's equation:
^ ~ ^strickler
^D V ' /
V 4 y (4.27)
where A:strickier is only a function of the surfaces.
Notes
1. The Gauckler-Manning and Strickler coefficients are related as:
^strickler ~ ^'^Manning
2. The Strickler equation is used for pipes, gallery and channel carrying water This equation is preferred to the Gauckler-Manning equation in Europe.
3. The coefficient ^strickler varies from 20 (rough stone and rough surface) to 80m^^^/s (smooth concrete and cast iron).
4. The Strickler equation is valid for uniform and non-uniform (gradually varied) flows.
Particular flow resistance approximations
In Nature, rivers and streams do not exhibit regular uniform bottom roughness. The channel bed consists often of unsorted sand, gravels and rocks. Numerous researchers attempted to relate the equivalent roughness height k^ to a characteristic grain size (e.g. median grain size d^o). The analysis of numerous experimental data suggested that:
K ^ dso (4.28) where the constant of proportionality kjd^o ranges from 1 to well over 6 (see Table 12.2)!
Obviously it is extremely difficult to relate grain size distributions and bed forms to a single parameter (i.e. k^.
For gravel-bed streams Henderson (1966) produced a relationship between the Gauckler- Manning coefficient and the gravel size:
"Manning = 0 . 0 3 8 < ( 4 . 2 9 )
where the characteristic grain size d-j^ is in metres and ^Manning is in s/m^^^. Equation (4.29) was developed for kJD^ < 0.05.
Vox flood plains the vegetation may be regarded as a kind of roughness. Chow (1973) presented several empirical formulations for grassed channels as well as numerous photographs to assist in the choice of a Gauckler-Manning coefficient.
4.5 Flow resistance calculations in engineering practice 81
Notes
1. Strickler (1923) proposed the following empirical correlation for the Gauckler-Manning coefficient of rivers:
^Manning = 0 . 0 4 1 4 ^
where ^50 is the median grain size (in m).
2. In torrents and mountain streams the channel bed might consists of gravels, stones and boulders with size of the same order of magnitude as the flow depth. In such cases, the overall flow resistance results fi-om a combination of skin friction drag, form drag and energy dissipation in hydraulic jumps behind large boulders. Neither the Darcy fiiction factor nor the Gauckler-Manning coefficient
should be used to estimate the friction losses. Experimental investigations should be performed to estimate an overall Chezy coefficient (for each discharge).
4.5 FLOW RESISTANCE CALCULATIONS IN ENGINEERING PRACTICE