Encyclopedia of Environmental Science and Engineering, Volume I and II

WATER FLOW 1281

principally by channel friction, this phenomenon is known as

gradually varied flow. For the type of flow in which the water

surface changes substantially within a very short channel

length due to a sudden variation in bed slope or cross-section,

this category is referred to as rapidly varied flow.

Open channels fabricated from concrete are often rect-

angular or trapezoidal in shape. Canals excavated in erodible

material have trapezoidal cross-sections. Although sewer

pipes are closed sections, they are still considered as open

channels so long as they are not flowing full; these cross-

sections are usually circular.

Channel Friction Equation

The most widely used open channel friction formula is the

Manning equation as mentioned earlier in pressure flow:

Q

N

 AR S

(^1) 23 12// (42)
S
N
AR

Q^22
243 /. (17)
Manning’s equation in hydraulic engineering is used for fully
turbulent flow and, as such, the values of Manning’s N apply
to this flow regime.
In a natural tortuous stream channel, the mean value
of Manning’s N can be obtained from the following
considerations:

1. estimate an equivalent basic N s , for a straight chan-
   nel of that material,


2. select modifying values of N m for non-uniform
   roughness, irregularity, variation in shape of cross-
   section, vegetation, and meandering,


3. sum the basic, N s together with the modifying
   values to obtain the total mean N.
   Normal values of Manning’s N for straight channels
   and various modifying values are given in Table 8. The total
   mean N value for the channel is obtained from the relation:
   NNsm∑N. (43)
   Energy Principles
   In deriving the energy relationships for open channel flow,
   the following assumptions are normally used:


4. a uniform velocity distribution over the cross-
   section is assumed, that is, the velocity coefficient,
   a, in the velocity head term, aV^2 /2 g, is taken as
   unity. In practice, the value of depends on the
   shape of the stream channel and has an average
   value of about 1.02 which makes this assumption
   sufficiently valid.


5. streamlines are essentially parallel,


6. channel slopes are small.
   Consider the water particle of mass, m, and of weight, W
   (Figure 5). The elevation and pressure energies of the parti-
   cle are Wh 1 ; and Wh 2 respectively. Thus, the potential energy
   of the water particles is, W ( h 1  h 2 ) and is independent of its
   elevation over the flow cross-section. As the kinetic energy
   is WV^2 /2 g the total energy of the water particles, e is:
   eWh h
   V
    12
   2
   2 g
   ⎛
   ⎝⎜
   ⎞
   ⎠⎟

. (44)

ZDh h^12 (45)

and noting that the total flow passing the cross-section is gQ

the total energy of the water passing the cross-section per

second, E t is given by:

EQZD

V

tg

2

2 g

⎛

⎝⎜

⎞

⎠⎟

. (46)

TABLE 8

Values of Manning’s N

Basic NS for straight channels

Type of channel Ns

Earth 0.010

Sand 0.012

Fine gravel 0.014

Rock 0.015

Coarse gravel 0.028

Cobbles and boulders 0.040

Modifying values of Nm Ns

Irregularity 0.005 to 0.020

Changes in shape 0.005 to 0.020

Vegetation 0.005 to 0.100

Meander 0.10 Ns to 0.40 Ns

V1/2g^2

(1) (2)

V 1

D 1 Dh^2

Z 1 Z

L

TEL

hf

V 22 /2g

h 1

i V 2

D 2

Z 2

FIGURE 5 Energy principles.

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