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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:

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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