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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