23/09/2014
Hydraulics 3: Topics
David Apsley (weeks 1-4 and 7-8)
• Open-channel flow
• Sediment transport
Tim Stallard (weeks 5-6 and 8-12)
• Waves
Hydraulics 3: Assessment
• Exam (80%)
– same format as previous years
– 4 questions from 6; made up of ...
– 2 questions from 3 in each of Sections A, B
• Coursework / labs (20%)
– my part: 5% written coursework; 5% lab
Content (DDA)
Open-Channel Flow
•
•
•
•
Introduction
Rapidly-varied flow
Gradually-varied flow
Wave speed and analogy with compressible flow
Sediment Transport
•
•
•
•
Overview
Threshold of motion
Bed load
Suspended load
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23/09/2014
Recommended Books (DDA)
• Chanson, Hydraulics of Open Channel Flow:
An Introduction
• Massey, Mechanics of Fluids
• White, Fluid Mechanics
• Chadwick, Morfett and Borthwick, Hydraulics in
Civil and Environmental Engineering
• Hamill, Understanding Hydraulics
1. Introduction
Characteristics of Open-Channel Flow
• Free surface (p = 0)
• Balance between gravity and friction
• Variable depth
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Classification of Open-Channel Flow
•
Steady vs unsteady
GVF
RVF
GVF
sluice
gate
•
hydraulic
jump
RVF
weir
GVF
RVF
GVF
UF
change
of slope
requires a uniform channel!
limiting behaviour given sufficient fetch
Rapidly-varied flow
–
–
•
GVF
Uniform flow (steady uniform flow = normal flow)
–
–
•
RVF
short fetch; bed friction unimportant;
examples: hydraulic jump, weir, venturi, sluice, …
Gradually-varied flow
–
–
long fetch; depth adjustment following disturbance
result of imbalance between bed friction and component of weight
Normal Flow: Balance of Forces
L
A
P
τb  ( PL)  (ρAL)  g sin θ
A
τ b  ρ g sin θ
P
A
Rh 
τ b  ρgRh S
P
τ b  c f ( 12 ρV 2 )
mg
b

hydraulic radius
definition of skin-friction coefficient
V
c f ( 12 ρV 2 )  ρgRh S
2g
Rh S
cf
Normal Flow: Friction Laws
V
2g
Rh S
cf
8g
Rh S
λ
Darcy
λ  4c f
V
Chézy
C  2g / c f
V  C Rh S
Manning
2g / c f 
1 1/ 6
Rh
n
V
1 2 / 3 1/ 2
Rh S
n
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23/09/2014
Typical Values of Manning’s n
Artificial lined channels
Excavated earth channels
Natural channels
Floodplains
Glass
Brass
Steel, smooth
painted
riveted
Cast iron
Concrete, finished
unfinished
Planed wood
Clay tile
Brickwork
Asphalt
Corrugated metal
Rubble masonry
Clean
Gravelly
Weedy
Stony, cobbles
Clean and straight
Sluggish, deep pools
Major rivers
Pasture, farmland
Light brush
Heavy brush
Trees
n (m–1/3 s)
0.01
0.011
0.012
0.014
0.015
0.013
0.012
0.014
0.012
0.014
0.015
0.016
0.022
0.025
0.022
0.025
0.03
0.035
0.03
0.04
0.035
0.035
0.05
0.075
0.15
Normal Flow: Calculation Formulae
Rh 
Hydraulic radius:
A
P
A
P
1 2 / 3 1/ 2
Rh S
n
Manning’s equation:
V
Discharge:
Q  VA
Particular Channel Shapes
Rectangular: Rh 
bh
b  2h
Wide:
Rh  h
Trapezoidal:
Rh 
Circular:
Rh 

h
1  2h / b
h
b
h(b  mh)
h
2( 12 R 2 θ  12 R cos θ R sin θ)
2 Rθ
1
m
b  2h 1  m 2
b

R  sin 2θ 
1 

2
2θ 
 R
h
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23/09/2014
Example
The discharge in a channel with bottom width 3 m is
12 m3 s–1. If Manning’s n is 0.013 m–1/3 s and the streamwise
slope is 1 in 200, find the normal depth if:
(a) the channel has vertical sides (i.e. rectangular channel);
(b) the channel is trapezoidal with side slopes 2H:1V.
Fluid Head
2
Total pressure: p  ρgz  12 ρV
p
V2
z
ρg
2g
Total head (H):
zs(x)
h
zb(x)
If hydrostatic:
p  ρgz is constant along a vertical line
 p

p
 z    z 
 zs
ρg
 ρg
 surface
Total head in (gradually-varied) open-channel flow: H  z s 
V2
2g
Froude Number
Fr 
V
gh
Wide or rectangular channel: h  depth
Non-rectangular channel:
h  mean depth 
A
bs
bs
A
h
Fr < 1: subcritical (tranquil)
Fr > 1: supercritical (rapid)
Fr = 1: critical
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23/09/2014
Interpretations of Froude Number
Fr 
V
gh
1. (Square root of) ratio of inertial to gravitational forces
2. Ratio of water velocity V to wave speed gh
3. Critical depth (Fr = 1)  minimum specific energy
Separates:


deep, slow, subcritical flow (Fr < 1)
shallow, fast, supercritical flow (Fr > 1)
Occurs at a control point in critical-flow devices such as
broad-crested weirs and venturi flumes.
Example
The discharge in a rectangular channel of width 6 m with
Manning’s n = 0.012 m–1/3 s is 24 m3 s–1. If the streamwise
slope is 1 in 200 find:
(a) the normal depth;
(b) the Froude number at the normal depth;
(c) the critical depth.
State whether the normal flow is subcritical or supercritical.
6

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