# Channel Flow

### Channel Flow

#### Table of Contents

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 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 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