# Open Channel Hydraulics and Uniform Flow

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```					A review

OPEN CHANNEL HYDRAULICS AND
UNIFORM FLOW
OPEN CHANNEL HYDRAULICS

   Three basic relationships
 Continuityequation
 Energy equation

 Momentum equation
CONTINUITY EQUATION

3a
Inflow                3        A

Change in Storage

3b

Outflow
1                      A      2
Section AA
Inflow – Outflow = Change in Storage

Q1 – Q2 = Change in storage rate
GENERAL FLOW EQUATION

Q = vA
Area of the cross-
section
Avg. velocity of flow   (ft2) or (m2)
Flow rate (cfs) or   at a cross-section
(m3/s)               (ft/s) or (m/s)
ENERGY EQUATION (BERNOULLI’S)

2                          2
v             p1 v              p2
1
 y1  z1      y 2  z 2   h L,1 2
2
2g            γ 2g              γ
Energy loss between
sections 1 and 2
OPEN CHANNEL ENERGY EQUATION
   In open channel flow (as opposed to pipe flow)
the free water surface is exposed to the
atmosphere so that p/g is 0, leaving:

2                2
v              v
1
 y1  z1      y 2  z 2  h L ,1 2
2
2g             2g
2
v1
2g                         hL1-2

v2
2
y1                         2g

y2

z1
z2
Datum

1                2
SPECIFIC ENERGY DIAGRAMS
y

y2

yc
y1

Ec   Eo   E
q2
2
yE
2gy

*Note q is constant.
CRITICAL DEPTH, YC

   The depth of flow corresponding to the
minimum E is the critical depth, yc

yc  q g
3   2

or
v
1
gyc
FROUDE NUMBER, F

F  v gyc           is known as the Froude Number, F

   If F = 1, y = yc and flow is critical.
   If F < 1, y > yc and flow is subcritical.
   If F > 1, y < yc and flow is supercritical.
   F is independent of the slope of the channel, yc
dependent only on Q.
v
F              (for non-rectangular channels)
gd h
WHAT IS UNIFORM FLOW?
 If flow characteristics at a point are unchanging
with time the flow is said to be steady.
 If flow properties are the same at every location
along the channel, the flow is uniform.
 The energy line, water surface and channel
bottom are all parallel in uniform flow.
NATURAL CHANNELS

 In natural flow situations flow is generally non-
 In designing most channels steady, uniform
flow is assumed with the channel design being
based on some peak or maximum discharge.
MOMENTUM EQUATION

L

WsinQ
P1

v1
Q
W             P2
v2

Rf
1
2

 Fs  P1  W sin   P2  R f
MANNING’S EQUATION
   In 1889 Irish Engineer, Robert Manning
presented the formula:


v         23 12
R S
n
 is a constant, 1.0 for SI units and 1.49 for B.G.
units.

n is known as Manning’s n and is a coefficient of
roughness.
DETERMINING MANNING’S N

   4 approaches to estimating:
1)   Understand the factors that affect the value of n
and acquire a basic knowledge of the problem,
narrowing the wide range of guesswork.
2)   Consult a table of typical n values for channels of
various types. (Table 4.1 (Haan et al., 1994);
Table 5-6 (Chow, 1959)).
DETERMINING MANNING’S N CONT.
3)   Examine and become familiar with the
appearance of some typical channels whose n
coefficients are known (Fig. 5-5, Chow(1959)).
4)   Determine n by an analytical procedure based on
the theoretical velocity distribution in the channel
cross-section and on the data of either velocity or
roughness measurements.
FACTORS AFFECTING MANNING’S N
A.   Surface Roughness
B.   Vegetation
C.   Channel Irregularity
D.   Channel Alignment
E.   Silting and Scouring
F.   Obstruction
G.   Size and Shape of the Channel
H.   Stage and Discharge
I.   Seasonal Change
PROPERTIES OF TYPICAL CHANNELS

 Figure 4.9, Haan et al. (1994) shows how to
compute properties for typical channels.
 For very wide trapezoidal channels the
trapezoid can be approximated by a
rectangular channel where        bd
R
b  2d
   If b>>d then 2d can be ignored and:

R  bd / b  d
PROPERTIES OF TYPICAL CHANNELS

   For a parabolic channel, if t>>d, 4d2 can
be ignored leaving:
2
t d    2
R      2
 d
1.5t    3
These approximations can serve as estimates for d
in trial and error solutions that often occur in open
channel hydraulics.
FLOW IN CIRCULAR CONDUITS

   For a circular conduit of diameter, D.

2
A
D
  sin  
8
D  sin  
R  1      
4     
PIPE LESS THAN HALF FULL (0<Y<D/2)

 D 2          2 
  D      
     Y 
1   2    2  
  2 tan                   
D
         Y     
       2        
                
D is diameter of pipe
Y is depth of flow
PIPE FLOWING GREATER THAN HALF FULL
(D/2<Y<D)

 D 2          2 
        D 
    Y  
1   2      2 
  2  2 tan                   
D
         Y     
       2        
                
Pipe is flowing greater than ½ full but less than full.
PIPE FLOWING HALF FULL (Y=D/2)

 
FLOW IN COMPOUND CHANNELS

   Natural channels often have a main channel
and an overbank section.

Overbank Section

Main Channel
FLOW IN COMPOUND CHANNELS
 Most flow occurs in main channel; however
during flood events overbank flows may occur.
 In this case the channel is broken into cross-
sectional parts and the sum of the flow is
calculated for the various parts.
FLOW IN COMPOUND CHANNELS

23
1.49 1 / 2  Ai 
Vi      S    P
ni         i
n
Q  Vi Ai
i 1
In determining R only that part of the wetted
perimeter in contact with an actual channel
boundary is used.

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