Introduction
This page concerns fluid flows down channels and pipes which are not full. The fluid has a free surface
which is subject to atmospheric pressure. This naturally occurs with rivers, and canals, and drainage ditches. The notes also include
fluid flowing over weirs and notches.
Symbols
A = Area (m^{2})
F1 = Force of fluid down channel (N)
F2 = Force up fluid down channel (N)
g = acceleration due to gravity (m/s^{2} )
h = fluid head (m)
i = incline
l = length down slope (m)
l_{h} = length  horizontal (m)
m = wetted mean length (m)
p = fluid pressure (N /m^{2} )
p _{s}= surface pressure (N /m^{2} )
P = perimeter (m)
ρ = fluid density (kg /m^{2} )

s wetted surface length (m)
u = velocity (m/s)
v = velocity (m/s)
x = depth of centroid (m)
θ =slope (radians)
ρ = density (kg/m^{3})
τ _{o} = shear stress (N /m^{2})

Channel Flow
In an open channel, the flowing water has a free surface and flows by the
action of gravity. See figure below. The water flows with a velocity
v down a channel with an incline θ. The water depth is uniform and therefore
the downward force F1 is balanced by the upward force F2. The only force causing
motion is the weight component in the direction of motion ρgAl sin θ.
The fluid is not accelerating so the downward gravity force is balanced only by the
friction force between the fluid and the wall.
If the length of wetted perimeter = s and the shear stress at the wall = τ _{o}sl
ρgAl sin θ = τ _{o}sl
Now let the incline i be x / l _{h}. For small angles i = sin θ .
ρgAli = τ _{o}sl ..
and therefore ..τ _{o} = ρgAi / s
Now let m be the mean wetted depth (m = A/s) the resulting equation is .
τ _{o} = ρgmi
Note: The relationship between τ _{o} and f is proved at the bottom of this page..
The quantites 2g/f are combined as a single constant ( C^{2} ) yielding the equation known as Chezy 's formula
The value of C can be obtained using the GanguilletKutter equation: with the relevant n values provided in the table below
Mannings formula C = m^{1/6} /n also applies...Using the same tabled values of n
Table showing n coefficients for using in Mannings equation and GanguilletKutter equation:
  
Description 
n Normal 
n Range 



Glass 
0,010 
0,0090,013 

Concrete 
Culvert straight and free of debris 
0,011 
0,0100,013 
Culvert with bends, connections and some debris 
0,013 
0,0110,014 
Sewer with manholes, inlet etc straight 
0,015 
0,0130,017 
Unfinished steel form 
0,013 
0,0120,014 
Unfinished smooth wood form 
0,014 
0,012  0,016 
Finished wood form 
0,012 
0,0110,014 

Clay 
Drainage tile 
0,013 
0,0110,017 
Vitrified clay sewer 
0,014 
0,0110,017 
Vitrified clay sewer with manholes inlet etc 
0,015 
0,0130,017 
Vitrified sub drain with open joint 
0,016 
0,0140,018 

Brickwork 
Glazed 
0,013 
0,0110,015 
Lined with cement mortar 
0,015 
0,0120,017 
Sewer coated with slimes , with bends 
0,013 
0,0120,016 
Rubble masonary 
0,025 
0,0180,030 

Cast Iron 
Coated 
0,013 
0,0100,014 
Uncoated 
0,014 
0,0110,016 

Excavated or Drained Channels 
Earth after weathering straight or uniform 
0,022 
0,0180,025 
Gravel straight uniform 
0,025 
0,0220,030 
Earth winding clean 
0,025 
0,0230,030 
Earth with some grass, weeds 
0,030 
0,0250,033 
Earthe bottom rubble sides 
0,030 
0,0280,035 
Dragline excavated, no vegetation 
0,028 
0,0250,033 
Rock cut smooth uniform 
0,035 
0,0250,040 
Rock cut smooth irregular 
0,040 
0,0350,050 
Unmaintained channels dense weeds 
0,080 
0,0500,120 

Natural streams 
Clear straight, fullstage no rifts or deep pools 
0,030 
0,0250,033 
As above but with more stones and weeds 
0,035 
0,0300,040 
Clean, winding some pools and shoals 
0,040 
0,0350,045 
As above but some weeds and stones 
0,045 
0,0350,050 

Flood Plains 
Pasture short grass 
0,030 
0,0250,035 
Pasture high grass 
0,035 
0,0300,050 

Cultivated Areas 
No crop 
0,030 
0,0200,040 
Mature row crops 
0,035 
0,0250,045 
Mature field crops 
0,040 
0,0300,050 

Major Streams Width > 30m 
Regular section with no boulders or bush 

0,0250,060 
Irregular and rough 

0,0350,10 
Thin Plate Weirs
1) Full Width Weir
Flow Q = 0,66√(2g). C_{d} b h_{e} ^{1,5}
C_{D} = 0,602 + 0,083 h/p
h_{e} = h + 0,0012m (h = measured head)
2) Supressed Weir
Flow Q = 0,66√(2g). C_{d} b. h_{e} ^{1,5}
C_{D} = 0,616(1  0,1h/b)
h_{e} = h + 0,001 (h = measured head m)
3) Vee Notch Weir
Flow Q = (8/15)√(2g).C_{d} tan (θ /2 ). h_{e} ^{2,5}
h_{e} = h + h_{k} (h = measured head m)
θ
Degrees 
C_{d} 
h_{k} 
20  0,592  0,0027 
40  0,581  0,0018 
60  0,576  0,0011 
80  0,578  0,0010 
90  0,579  0,0009 
Notes showing relationship between τ_{o} and f...Provided in support of proof of Chezy Formula above..
Showing relationship between τ_{o} and f
Darcy conducted experiments and proved that for pipes of uniform cross section and roughness and fully developed flow
the head loss due to friction (h_{f} ) along a pipe is in accordance with the following formula.
The fluid shear stress (τ _{o} ) at the boundary wall is related to the pressure differential along the pipe
by the expression.
P = perimeter length, A = Area of section
The differential head along the pipe is related to the differential pressure as follows/
The equation for shear stress is modified as ..
Now for fully developed flow with no axial sudden changes the flow pattern along the
pipe is constant and dh/dl is equal to h / l therefore ..

