2019-06-06

1

Hydraulics

Professor Sung-Uk Choi

Department of Civil & Environmental EngineeringYonsei University

4. Closed-Conduit Flows

Closed-Conduit Flow in Civil Engineering(1) 수도권 광역상수도

한국 상수도 보급률 98.8% (급수 인구 5,204만 명): 세계 최상위 수준

팔당댐 연간 광역상수도 공급량 13억 m3 (41 ton/sec)

한국 광역상수도 연간 공급량 (65억 m3)의 20%

수도권 물 공급량의 90% 담당

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Closed-Conduit Flow in Civil Engineering(2) 리비아 대수로공사

리비아의 대수로 공사는 사막에 물을 공급하는 인류 최대 役事 중의 하나이다. 1단

계 공사는 리비아 동부지역의 1,874 km를 송수하여 하루 2백만 톤의 물을 공급하

며, 2단계 공사는 서부지역으로 1,728 km를 연결해 하루 250만 톤을 공급한다. 중

력에 의한 자연 유하식과 펌프를 이용해 송수하며, 관은 프리스트레스 콘크리트 실

린더로 직경이 4 m에 달한다.

Storm Sewer Geyser

Geyser is the explosive release of water through vertical shafts connected to

a nearly horizontal pipeline.

Laboratory experiments revealed that the air can force water upward in the

shaft (Wright et al., 2011).

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Introduction

• Pipe flows vs. Open-channel flows

• How about sewer flows? (80%)

• Driving force: pressure difference (or pressure drop)

• Head loss (and minor loss)

22V

2g

Turbulent Flows in a Pipe: 1883 Reynolds’ Experiment

ReVL

Flow properties: V= characteristic velocity

L = characteristic length

Fluids property: ν = kinematic viscosity

• Show that Re is the ratio of inertia to viscous force.

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Repetition of Reynolds’ Dye Experiments 1

•A century later, the experiments were repeated using the same

apparatus at University of Manchester as Reynolds used.

transition

laminar

Repetition of Reynolds’ Dye Experiments 2fully-turbulent

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An Example (Re =4,160)

0.2

0.2

0.2

0.4

0.4

0.6

0.6

0.6

0.8

0.8

1.0

1.0

1.2

y / Hz

/H

0.00 0.25 0.50 0.75 1.000.00

0.25

0.50

0.75

1.000.02 UB

• It took about 60 days with the fastest PC 5 years ago. How long it would take if for Re = 41,600?

Basic Equations

• Continuity Equation:

• Momentum equation:

•Energy Equation:

2 21 1 2 2

1 22 2 l

V p V pz z h

g w g w

1 2 0sin 0F p A p A wAl Pl

0lh

lh

R w

• Head loss is proportional to wall shear stress and pipe length, but

inversely proportional to the diameter.

• We have the same equation for uniform open-channel flows.

1 1 2 2AV A V

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Darcy-Weisbach Formula

2

2l

l Vh f

D g

• Friction factor f = ( ε/D, Re): Moody diagram

• Effective roughness height ε

• Comparison with the foregoing relation

• Swamee and Jain’s (1976) relationship

• For laminar flows,

4 ff C

64 / Ref

Moody Diagram

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Shear Stress Distribution in a Pipe Flow

0( )r

rR

• This is derived using the force balance.

• The distribution is valid regardless of laminar and turbulent flows.

Velocity Distribution in a Pipe: Laminar Flows

0( )r

rR

• The velocity distribution is parabolic.

• Hagen-Poiseuille flow

• Newton’s law of viscosity, valid only for laminar flows, is used.

2 2

2( ) 1

4

dp R ru r

dx R

2

8

dp RV

dx

4

8

dp RQ

dx

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Velocity Profiles in a Pipe

• Velocity and shear stress profiles

• Unestablished flow and Established flow

• Boundary-layer

Velocity Distribution in a Pipe: Turbulent Flows 1

( ) t

duz

dz

• Fluid viscosity ignored.

• Valid in the vicinity of the wall.

• Mixing length theory used.

( ) 1b

zz

h

*

1ln constant

uz

u Log-Law

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Velocity Distribution in a Pipe: Turbulent Flows 2

• For hydraulically-smooth flows,

• For hydraulically rough flows,

*

*

1ln 5.5

u zu

u

*

1ln 8.5

s

u z

u k

• Viscous sublayer

• Wall units

• Note that the wall units are not

used for hydraulically-rough flows.

*

*

u zu

u

u z

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von Karman Constant

Power Law Velocity Distribution for Turbulent Flows

1/7

*

*

8.74u zu

u

1/7

0 *

*

8.74U u

u

1/7

0

u z

U

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Boundary Layer

Friction Loss and Minor Loss

• Friction loss:

• Minor loss: head loss other than the friction loss, i.e., inlet, outlet,

bend, contraction, and expansion

2

2l

l Vh f

D g

2

2l

Vh K

g

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A Table (1)

A Table (2)

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Mean Velocity Formulas

• Darcy-Weisbach formula

• Hazen-Williams formula

• Manning’s formula

0.5 0.5hV CR I

0.63 0.54H hV C R I

0.667 0.51hV R I

n

2

1/3

124.6nf

D

7. Single and Multiple Pipelines

7.1 A single pipeline from reservoir

7.2 A pipeline connecting two reservoirs

7.3 Series piping

7.4 Parallel piping

7.5 Branch piping

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7.1 Single Pipeline from Reservoir

2

2

1 e

gHV

lf f

D

7.2 A Pipeline connecting Two Reservoirs

2

2

e o

gHV

lf f f

D

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7.3 Pipelines in Series

2 21 1 2 2

1 21 22 2

l V l VH f f

D g D g

7.4 Parallel Piping

• Two types of problem

(1) Find Q with given H

(2) Find Qi with given Q

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Two Types of Problems in Parallel Piping

(1) Find Q with given H

- Calculate Qi and Sum over pipes

(2) Find Qi with given Q

- Assume Q1* for pipe 1

- Calculate hL1* with Q1*

- Estimate other Qi* with hL1*

- Redistribute the discharges

- Check hL1* with Q1

*

*i

ii

i

QQ Q

Q

7.5 Branch Piping

Ex.4.16

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Pipe Network

• The number of nodes n = 6 and the number of loops m = 2

• The Hardy-Cross method is to solve m equations (for head loss) to satisfy n

equations (for continuity).

• The solution procedure is based on that continuity must be maintained and the

head loss between any two nodes must be independent of the route take to get there.

1 2

1 4 5

5 6

2 3

3 4 7

6 7

0

0

0

0

0

0

A

B

C

D

E

F

Q Q Q Q

Q Q Q Q

Q Q Q

Q Q Q

Q Q Q Q

Q Q Q Q

2 2 2 21 1 4 4 3 3 2 2

1

2 2 2 25 5 6 6 7 7 4 4

2

0

0

l

l

h k Q k Q k Q k Q

h k Q k Q k Q k Q

Hardy-Cross Method

• This method is to solve m energy equations repeatedly with assumed

Qs in each pipe until n continuity equations at nodes are satisfied. Thus,

one has to solve m equations to obtain n Qs.

Ex.4.17

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Siphon

Ex.4.19

국내 사이펀 설치 현황

도순지

송석지

원형 철관 사이폰 사례

지평지

1.

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Hydropower

Ex.4.22

1,000e eP mgh QgH QH

e lH H h here

Drainage Time

2

1 e

ghV

lf f

D

dhQ aV A

dt

2 21 2

2

2

At H H

a g

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Bio-Fluid Mechanics

Annual Review of Fluid Mechanics, Vol.29 Blood Flow in Arteries (동맥)

Flow in Aorta and Arterial Branches

• 개의 동맥에서 혈의 유속과

혈압 동시 측정

• 혈류속이 감소하기 시작하

면서 혈압이 증가하는 양상

을 보임

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Flow through Carotid (경동맥)

• 경동맥 분기에서 hydrogen

bubble 기법을 이용한 혈류

가시화

• 혈류의 흐름상태는 층류이

며 분기에서 유선의 박리 발

생

• 유선 박리에 의한 이차흐름

관측

Velocity Profile at Flow Divider

• 심장 수축기 (a) 와 이완기

(b)에서 혈류 흐름 (CFD 계

산결과)

• 수축기에 혈류의 역류 현상

발생하지 않음

• 이완기 안쪽 벽을 따라 계

단형태의 유속분포 발생

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Stenosis

• 분기 이후 혈관의 협착증 (stenosis) 발생

• 이는 혈류량을 감소시켜 뇌의 압력을 증가시킴

Cincinnati Water Treatment Plant

최성욱 교수

토목환경공학과