Encyclopedia of Environmental Science and Engineering, Volume I and II

1284 WATER FLOW

Equation (6) can be written as:

D

D

(^2) F
1
1
(^12)
2
(), 18 1 (63)
where
FV D 11  /( )g 1
is the upstream Froude number.
The energy loss E j across the hydraulic jump on a
horizontal floor can be obtained by coming Eqs. (61), (62)
and the energy equation:
D
V
D
V
1 1 Ej
2
2
2
2
22

gg
(64)
to give:
E
DD
j DD

()
(^21).
3
(^412)
(65)
The head loss E j is graphically shown in the specific energy
and flow diagrams (Figure 8).
Surface Water Profiles
Non-uniform Differential Equation Using the notation given
in Figure 9, the energy relations can be expressed as:
iL D
V
SL D D
VV
     
222
222 ggg
dd() d
⎛
⎝⎜
⎞
⎠⎟
(66)
d
d/
L
DV
iS



(
()
(^22) g)
(67)
d
d
E
L
().iS (68)
For a finite length, L, Eq. (67) becomes:
LLL
DV D V
iQNAR




()
()( )
,
21
11
2
22
2
22 243
// 22
/ /
⎛ gg
⎝⎜
⎞
⎠⎟
(69)
where the Manning equation is used to calculate the energy
slope. Flow computation must start at a control section
where all the flow parameters are known. The calculation
proceeds upstream for subcritical and downstream for super-
critical flow. In Eq. (69), the solution of the reach length,
L, is direct if the immediately upstream depth, D 2 , is given
a value. If L is given a value, D 2 has to be solved by trial.
This method of computing surface water profiles is suitable
for regular channels.
Classification of Flow Profiles Twelve distinct types of
non- uniform profiles have been systematically classified
(Figure 10).

1. Firstly, the curves are identified according to bed
   slopes as mild (M), steep (S), horizontal (H), criti-
   cal (C) and adverse (A).


2. Secondly, numbers are assigned to flow regions.
   The numerical 1 refers to actual flow depths
   exceeding both critical ( D c ) and normal ( D ) depths.
   For flow depths less than both critical and normal,
   the number 3 is affixed to it. The numeral 2 is for
   depths intermediate between critical and normal.
   Water Profiles in Irregular Channels The river channel has
   to be divided into panels (Figure 11) with the side panels
   conveying overbank flow. Let Q  total flow, Q c  central
   channel discharge, Q l  left overbank flow,
   Q r  right overbank flow. The continuity condition
   requires that:
   QQ Q Qclr, (70)
   By Manning’s Equation:
   Q
   N
   AR S
   N
   AR S
   N
   AR S
   c
   cc
   l
   l
   r
   rr
   (^111) 23 12 23 12 23 12// // //

. (71)

The energy slope, S, has been taken as the same for Q c ,

Q l , Q r ; this assumption seemed to be justified in practice.

Due to different channel roughness, vegetative and other

obstructions, Manning’s N for the three flow panels would

(1) (2)

TEL

S

i

dL

idL

D

(D + dD)

V^2

2g

+ d (V

2

2g

)

V S dL

2

2g

FIGURE 9 Non-uniform flow derivation.

C023_003_r03.indd 1284C023_003_r03.indd 1284 11/18/2005 11:12:14 AM11/18/2005 11:12:14 AM