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Channel Flow

Introduction..... Symbols..... Channel flow..... Table of Manning n Coefficients..... Thin Plate weirs.....


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 (m2)
F1 = Force of fluid down channel (N)
F2 = Force up fluid down channel (N)
g = acceleration due to gravity (m/s2 )
h = fluid head (m)
i = incline
l = length down slope (m)
lh = length --- horizontal (m)
m = wetted mean length (m)
p = fluid pressure (N /m2 )
p s= surface pressure (N /m2 )
P = perimeter (m)
ρ = fluid density (kg /m2 )
s wetted surface length (m)
u = velocity (m/s)
v = velocity (m/s)
x = depth of centroid (m)
θ =slope (radians)
ρ = density (kg/m3)
τ o = shear stress (N /m2)




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 osl

ρgAl sin θ = τ osl

Now let the incline i be x / l h.  For small angles i = sin θ

.

ρgAli = τ osl .. 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 ( C2 ) yielding the equation known as Chezy 's formula

The value of C can be obtained using the Ganguillet---Kutter equation: with the relevant n values provided in the table below

Mannings formula C = m1/6 /n also applies...Using the same tabled values of n


Table showing n coefficients for using in Mannings equation and Ganguillet---Kutter equation:

Description n
Normal
n
       Range         
     
Glass 0,010 0,009---0,013
 
Concrete
Culvert straight and free of debris 0,011 0,010---0,013
Culvert with bends, connections and some debris 0,013 0,011---0,014
Sewer with manholes, inlet etc straight 0,015 0,013---0,017
Unfinished steel form 0,013 0,012---0,014
Unfinished smooth wood form 0,014 0,012 --- 0,016
Finished wood form 0,012 0,011---0,014
 
Clay
Drainage tile 0,013 0,011---0,017
Vitrified clay sewer 0,014 0,011---0,017
Vitrified clay sewer with manholes inlet etc 0,015 0,013---0,017
Vitrified sub drain with open joint 0,016 0,014---0,018
 
Brickwork
Glazed 0,013 0,011---0,015
Lined with cement mortar 0,015 0,012---0,017
Sewer coated with slimes , with bends 0,013 0,012---0,016
Rubble masonary 0,025 0,018---0,030
 
Cast Iron
Coated 0,013 0,010---0,014
Uncoated 0,014 0,011---0,016
 
Excavated or Drained Channels
Earth after weathering ---straight or uniform 0,022 0,018---0,025
Gravel straight uniform 0,025 0,022---0,030
Earth winding clean 0,025 0,023---0,030
Earth with some grass, weeds 0,030 0,025---0,033
Earthe bottom rubble sides 0,030 0,028---0,035
Dragline excavated, no vegetation 0,028 0,025---0,033
Rock cut smooth uniform 0,035 0,025---0,040
Rock cut smooth irregular 0,040 0,035---0,050
Unmaintained channels dense weeds 0,080 0,050---0,120
 
Natural streams
Clear straight, fullstage no rifts or deep pools 0,030 0,025---0,033
As above but with more stones and weeds 0,035 0,030---0,040
Clean, winding some pools and shoals 0,040 0,035---0,045
As above but some weeds and stones 0,045 0,035---0,050
 
Flood Plains
Pasture short grass 0,030 0,025---0,035
Pasture high grass 0,035 0,030---0,050
 
Cultivated Areas
No crop 0,030 0,020---0,040
Mature row crops 0,035 0,025---0,045
Mature field crops 0,040 0,030---0,050
 
Major Streams Width > 30m
Regular section with no boulders or bush   0,025---0,060
Irregular and rough   0,035---0,10



Thin Plate Weirs
1) Full Width Weir

Flow Q = 0,66(2g). Cd b he 1,5
CD = 0,602 + 0,083 h/p
he = h + 0,0012m (h = measured head)

2) Supressed Weir

Flow Q = 0,66(2g). Cd b. he 1,5
CD = 0,616(1 --- 0,1h/b)
he = h + 0,001 (h = measured head ---m)

3) Vee Notch Weir

Flow Q = (8/15)(2g).Cd tan (θ /2 ). he 2,5

he = h + hk (h = measured head ---m)

θ
Degrees
Cd hk
200,5920,0027
400,5810,0018
600,5760,0011
800,5780,0010
900,5790,0009




Notes showing relationship between τo and f...Provided in support of proof of Chezy Formula above..

Showing relationship between τo 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 (hf ) 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 ..





Useful Links
  1. Estimating Flow in Streams .. Page of useful Notes
  2. Mannings Formula Calculator.. Useful calculatore
  3. Section 2 Open Channel Hydraulics.. pdf download from University of Leeds... Very useful and detailed notes
  4. Open channel flow resistance.. pdf download : A very detailed and informative paper


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Last Updated 28/01/2013