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Open Channel Flow - General
Hydromechanics VVR090
Open Channel Flow
Open channel: a conduit for flow which has a free surface
Free surface: interface between two fluids of different density
Characteristics of open channel flow:
• pressure constant along water surface
• gravity drives the motion
• pressure is approximately hydrostatic
• flow is turbulent and unaffected by surface tension
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Water Supply
Aqueduct, Pont du Gard, France
Water Power
ITAIPU power plant
Cross-section of power plant
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Zola dam, Aix-en-Provence
Spillway, ITAIPU dam
Transportation
Panama Canal
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Flow Control and Measurement
Flow Phenomena
Tidal bore, Hangzou, China
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Flooding
Yellow River, China
The History of Open Channel Flow
Main periods of development:
• ancient times (river cultures)
• roman times (aqueducts)
• renaissance (first theory)
• 17th century (experimental techniques + theory)
• 18th century (rise of hydrodynamics)
• 19th century (split between hydraulics and hydrodynamics)
• 20th century (boundary layer theory)
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Centers of early civilization around the large rivers
Ancient Times
The nilometer on the Island of Rhoda
The Nile Delta
The Nile River
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Indus civilization
Public bath
Drainage pipe
Yellow River
Levee construction
Sediment-laden river water
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Roman aqueducts
Aqua Claudia
Construction of an aqueduct
Characteristics of Aqueducts
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Frontinus (40-103 A.D.)
Vitruvius (55 B.C. – 14 A.D.)
ReservoarTop of aqueduct
A Roman fountain
Leonardo da Vinci (1452-1519)
Water flow
Renaissance
”When you put together the science of the motion of water, remember to include in each proposition its application and use, in order that these sciences may not be useless.”
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Evangelista Torricelli (1608-1647)
Galileo Galilei (1564-1642)
barometer
Experimental Techniques (17th century)
Rapid developments in mathematics
Blaise Pascal (1623-1662)
Isaac Newton (1642-1727)
Gottfried Leibniz (1646-1716)
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The Rise of Hydrodynamics
Daniel Bernoulli (1700-1782)
Experimental Hydraulics (18th Century)
Italy: Poleni, Venturi
France: Pitot, Chezy, Borda
England: Smeaton
Pitot tube on an airplane wing
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19th Century Developments
Main efforts:
• collect experimental data
• formulate empirical relationships
• derive general physical principles
Split into hydraulics and hydrodynamics
Hydraulics:
Germany: Hagen, Weisbach
France: Poiseuille, Darcy
England: Manning, Froude
Hydrodynamics:
France: Navier, Cauchy, Poisson, Boussinesq
England: Stokes, Reynolds
Germany: Helmholtz, Kirchoff
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Navier-Stokes Equations
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
1
1
1
u u u u p u u uu v w Pt x y z x x y z
v v v v p v v vu v w Qt x y z y x y z
w w w w p w w wu v w Rt x y z z x y z
⎛ ⎞∂ ∂ ∂ ∂ ∂ μ ∂ ∂ ∂+ + + = − + + + +⎜ ⎟∂ ∂ ∂ ∂ ρ ∂ ρ ∂ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂ ∂ ∂ ∂ μ ∂ ∂ ∂+ + + = − + + + +⎜ ⎟∂ ∂ ∂ ∂ ρ ∂ ρ ∂ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂ ∂ ∂ ∂ μ ∂ ∂ ∂+ + + = − + + + +⎜ ⎟∂ ∂ ∂ ∂ ρ ∂ ρ ∂ ∂ ∂⎝ ⎠
Increased gap between hydraulics and hydrodynamics
Bridged by the introduction of boundary layer theory by Ludwig Prandtl, the father of modern fluid mechanics.
Ludwig Prandtl (1875-1953)
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Flow Classification I
• steady – unsteady
• uniform – non-uniform
• varied flow (= non-uniform):
gradually varied – rapidly varied
Flow Classification II
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Flow Classification III
Laminar, transitional, and turbulent flow
Characterized by Reynolds number:
Re UL=
ν
L taken to be the hydraulic radius R=A/P
Re < 500 laminar
500 < Re < 12,500 transitional
12,500 < Re turbulent
Flow Classification IV
• homogeneous – stratified flow
depends on the density variation
• subcritical – supercritical flow
characterized by the Froude number
UFrgL
=
L taken to be the hydraulic depth D=A/T
Fr < 1 subcritical flowFr = 1 critical flowFr > 1 supercritical flow
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Gravity Wave I
Celerity of gravity wave:
c gL=
(denominator in Froude number)
Movement of impermeable plate
Gravity Wave II
Continuity equation:
( )( )cy y y c u= + Δ −Δ
(coordinate system moving with velocity c)
Simplifying:
uc yy
Δ=
Δ
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Gravity Wave III
Momentum equation:
( ) ( )( )221 12 2
y y y cy c u cγ − γ + Δ = ρ −Δ −
Simplifying:
u gy c
Δ=
Δ
c gy=
Gravity Wave IV
Interpretation:
1. Subcritical flow (Fr < 1): Velocity of flow is less than the celerity of a gravity wave. Gravity wave can propagate upstream. Upstream areas in hydraulic communication with downstream areas.
2. Supercritical flow (Fr > 1): Velocity of flow is greater than the celerity of a gravity wave. Gravity wave cannot propagate upstream. Upstream areas not in hydraulic communication with downstream areas.
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Channel Types
Natural channels: developed by natural processes (e.g., creeks, small and large rivers, estuaries)
Artificial channels: channels developed by human efforts (e.g., navigation channels, power and irrigation channels, drainage ditches)
Easier to treat artificial channels.
Artificial Channels
1. Prismatic (constant shape and bottom slope)
2. Canal (long channel of mild slope)
3. Flume (channel built above ground)
4. Chute and drop (channel with a steep slope)
5. Culvert (pipe flowing only partially full)
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Definition of Channel and Flow Properties I
Depth of flow (y): vertical distance from channel section to water surface
cosdy =θ
(d = depth of flow measured perpendicular to the channel bottom; q = slope angle of channel bottom)
Small slopes:
y d≈
Definition of Channel and Flow Properties II
Stage: elevation of the water surface relative to a datum
Top width (T): width of channel section at water surface
Flow area (A): cross-sectional area of the flow taken perpendicular to the flow direction
Wetted perimeter (P): length of the line which is the interface between the fluid and the channel boundary
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Definition of Channel and Flow Properties III
Hydraulic radius (R): ratio of flow area to wetted perimeter
ARP
=
ADT
=
Hydraulic depth (D): ratio of flow area to top width
For irregular channels: integrate and use representative values for above-discussed quantities
Definition of Channel and Flow Properties IV
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Governing Equations
Flow is turbulent in situation of practical importance (Re > 12,500) => Laminar flow is not discussed.
Description of turbulent flow:
´
´
´
u u u
v v v
w w w
= += += +
Average in time: Average in space:
0
1T
Tu udtT= ∫
1
A
u udAA
= ∫
Statistical Quantities
( )1/2
2
0
1( ') '
T
rms u u dtT
⎛ ⎞⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠∫
Root-mean-square (rms) value of velocity fluctuation:
Average kinetic energy (KE) of the turbulence per unit mass:
( ) ( ) ( )( )2 2 21' ' '
2KE
u v wmass
= + +
Reynolds stresses:
0
1' ' ' '
T
u v u v dtT= ∫
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Energy Equation
Bernoulli equation (along a streamline):
2
cos2
AA A
uH z dg
= + θ+
Small values of q:
2
2uH z yg
= + +
Fundamental Equations
Conservation of mass:
Q uA=
Conservation of momentum:
2 1( )F Q u u= ρ −∑
Conservation of energy:
2
2uH z yg
= + +
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Correction of Momentum Flux
True transfer of momentum:
2
A
u dAρ∫
Average transfer:
Quρ
Momentum correction coefficient:
2 2
2A A
u dA u dA
Qu u A
ρ ρβ = =
ρ ρ
∫ ∫
Correction of Energy Flux
True transfer of energy:
312A
u dAρ∫
Average transfer:
212
Quρ
Energy correction coefficient:
3 2
2 3A A
u dA u dA
Qu u A
ρ ρα = =
ρ ρ
∫ ∫
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Properties of a and b
• equal to unity for uniform flow (otherwise greater than 1)
• a is more sensititve to velocity variations than b
• a and b used only for complex cross-sectional shapes (e.g., compound sections)
Boundary Layers
Consider flat surface: boundary layer depends on U, r, m, and x. Laminar boundary layer thickness (Blasius):
5 at 0.99Rex
x uU
δ = =
Transition to turbulent boundary layer:
500,000 Re 1,000,000x< <
Turbulent boundary layer thickness:
0.2
0.37 at 0.99Rex
x uU
δ = =
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Observations Regarding Boundary Layers I
The following relationships exist:
, , ,, , ,
x Ux U↑ μ ↑ ρ ↓ ↓ ⇒ δ ↑
↓ μ ↓ ρ ↑ ↑ ⇒ δ ↓
Boundary layers can grow within boundary layers (e.g., change in channel shape or roughness)
Observations Regarding Boundary Layers II
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Observations Regarding Boundary Layers III
Boundary layers classified as hydraulically smooth or rough.
Hydraulically smooth: laminar sublayer cover the roughness elements
Hydraulically rough: roughness elements project through the laminar sublayer
*
*
*
0 5 smooth
5 70 transition
70 rough
s
s
s
k u
k u
k u
≤ ≤ν
≤ ≤ν
≤ν
Resistance Estimate
Chezy equation:
*
u C RS
u gRS
=
=
0 5 smooth
5 70 transition
70 rough
s
s
s
k u gC
k u gCk u g
C
≤ ≤ν
≤ ≤ν
≤ν