2019-06-06

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Hydraulics

Professor Sung-Uk Choi

Department of Civil & Environmental EngineeringYonsei University

5. Open-Channel Hydraulics

Geometric Elements of Channel Section

• water depth (水深): h

• stage (水位)

• top width (水面幅): T

• water area (流水斷面積): A

• wetted perimeter (潤邊): P

• hydraulic radius (動水半徑): R

• hydraulic depth (水理水深): D

• prismatic channel (對象斷面)

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Stage: 수위

Laminar and Turbulent Flows in Open-Channel

• Understand 1883 Reynolds Experiment

• What are characteristic length and

velocity in Reynolds experiments?

• What do you mean by “characteristic”?

• How do you obtain Rec = 500 for open-

channel flows?

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1883 Reynolds’ Experiment

ReVL

Flow properties: V= characteristic velocity

L = characteristic length

Fluids property: ν = kinematic viscosity

Velocity Distribution of Open-Channel Flow

• velocity dip

• secondary currents

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Classifications of Open-Channel Flows

• 등류 (uniform flow)

• 부등류 (non-uniform flow)

점변류 (gradually varied flow)

급변류 (rapidly varied flow)

• 부정류 (unsteady flow)

Characteristics of Uniform Flows

• The discharge, water area, velocity, and depth are constant with distance.

velocity: velocity averaged over cross section and time

• The bottom slope (S0), the slope of the water surface (Sw), and the slope of the energy line (Sf) are the same.

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Uniform Flow: Continuity Equation

0A Q

t x

• Q = Q1 = Q2 = constant

• A1 = A2 = constant

• H1 = H2 = constant

Uniform Flow: Momentum Equation

• A balance between the gravity & the bottom shear (friction)

Fg = w A dx sinθFf = τ0 P dx

τ0 = w Rh S

This is valid regardless of the shape of the cross section.

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Uniform Flow Formulas

• Chezy formula

• Manning Formula

• Darcy-Weisbach formula

hV C R S

2/3 1/21hV R S

n

8h

gV R S

f

Dimensional Non-homogeneity of Manning’s Formula

• Dimensional non-homogeneous

• Why is it?

• Then, what is the dimension of roughness

coefficient?

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Comparisons of Uniform Flow Formulas

• Chezy C has a dimension of

• It is reasonable to think that and

• However, in practice, we take n dimensionless

and

1/68 m hC RC

f ng g

g

mC g 1/6n L

1/6m

gC

L

What happens if n is dimensional (L1/6) ?

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• From Chezy’s formula,

Conveyance: 通水能

Q K S

/K Q S

/ hK CA R

• From Manning’s formula,2/3mh

CK AR

n

• Definition: The cross section with the minimum wetted

perimeter (or maximum hydraulic radius) for a given area

• Rectangular section: Half of a square

• Trapezoidal section: Half hexagon

• Triangular section: Half square on its apex

Best Hydraulic Section

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Composite Roughness (複合粗度)(1) Channel with Different Roughnesses

2/3

3/2

2/3

i ii

Pn

nP

•Yen (2002), JHE 128(1). Open-channel flow resistance.

Composite Roughness(2) Compound Channel Section (複斷面)

2/3

1/21 ii

i i

AV S

n P

i ii

Q AV

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Specific Energy: 比에너지

• Since pressure is not constant across the section, use the

piezometer head or flow depth by assuming that S = 0. If one

further assumes α=1 and hL =0,

• Thus, we have

which is the total energy per unit weight.

2 21 2

1 22 2

V Vh h

g g

2 2

22 2s

V QE h h

g gA

2 21 1 2 2

1 22 2 L

p V p Vz z h

w g w g

• Energy equation at two sections

Specific Energy vs. Flow Depth

• Critical depth is the flow depth when the specific energy is minimum (for a fixed Q).

• Two depths h1 and h2 for the same Es are alternate depths

(대응수심).

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Critical Depth

• The specific energy has a minimum value under which

the given Q cannot occur.

Note that dA/dh ≈ T. Then, we have

where D = hydraulic depth.

• Critical flow occurs when dEs/dh = 0. That is,

•

2 2

31 1sdE Q dA V dA

dh gA dh gA dh

2 2

1 1sdE V T V

dh gA gD

V gD

Froude Number

• The Froude number is defined by

• Square root of the ratio of the inertia force to the

gravity force (Can you show this?)

• The most important dimensionless number in the open-

channel flow.

• Note that the characteristic length is the hydraulic

depth. (How about for the Reynolds number?)

VFr

gD

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Froude Number

• Classification of open-channel flows

- subcritical flow: Fr < 1

- critical flow: Fr =1

- supercritical flow: Fr > 1

Discharge vs. Flow Depth(for a rectangular channel)

2 22s

gQ E h B h

• Critical flow is made when the discharge is maximum (for a fixed Es).

• For the critical flow, Es = 3/2hc

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Critical Slope

• Critical Slope: The slope of the channel that

sustains a given discharge at a uniform and critical

depth.

• Mild and Steep Slope Conditions

Mild slope condition: S < Sc; hn > hc

Steep slope condition: S > Sc; hn < hc

• Note that the uniform flow on a mild slope is

subcritical, but the uniform flow on a steep slope is

supercritical.

2

3/4ch

n gDS

R

Specific Force: 比力

• The force per unit weight of water is conserved.

2 1F Q V V

1 2 sin fF P P W F 2 2

1 1 2 21 2

G G

Q Qh A h A

gA gA

2

G

QM h A

gA

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Conjugate Depths

• Two depths are present for one specific force. They are conjugate

depths.

• Critical flow occurs for minimum M.

Hydraulic Jump 0

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Hydraulic Jump 1

• Continuity:

• Momentum:

• Energy:

Hydraulic Jump 2

21 12 11 8

2 2

h hh Fr 32 1

1 24l

h hh

h h

• If the hydraulic jump takes place after the apron, the supercritical flow

erode the channel bottom, which may result in destroying stream stability.

• Two depths before and after the jump are conjugate depths (공액수심).

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Hydraulic Jump 3

• The use of the inclined long stilling basin apron will induce the hydraulic

jump to occur within the reach of the apron.

Hydraulic Jump 4

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Gradually Varied Flows 1

• Features

- steady flow

- streamlines are parallel: hydrostatic pressure dist.

Gradually Varied Flows 2

• Features

- steady flow

- streamlines are parallel: hydrostatic pressure dist.

• Governing equation:

here S0 = bed slope, Sf = slope of energy line, and

0

21fS Sdh

dx Fr

2 22

3( )

V Q TFr fn h

gD gA

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Backwater Equation 1

• Non-linear ODE

• dh/dx = 0 when S0 = Sf, indicating the uniform flow.

• Valid for arbitrarily-shaped channel section

• 시간에 따라 급격히 유량 혹은 수위가 변하는 경우를 제외하고는 (실

시간 홍수예경보), 대부분의 홍수 문제는 배수곡선식으로 해결된다.

0

21fS Sdh

dx Fr

Backwater Equation 2• For a wide rectangular channel,

• Note that Manning equation (uniform flow formula) is

used to estimate the slope of the energy line. How?

10/3

0 3

1 /

1 /

n

c

h hdhS

dx h h

2 2

0 2 10/3n

n QS

b h

2 2

2 10/3f

n QS

b h

32 ch

Frh

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Gradually Varied Flows: M & S Curves

Gradually Varied Flows: C, H, & A Curves

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Examples of Water Surface Profiles