Chapter 8: Flow in Pipes Ship Hydrodynamics 2 - CNU FINCLfincl.cnu.ac.kr/lecture/shiphydrodynamics/Chapter_08B… · · 2012-02-12Friction force of wall on fluid Chapter 8:

Chapter 8. Flow in PipesChapter 8. Flow in Pipes

Prof. Byoung-Kwon Ahn

bkahn@cnu ac kr http//fincl cnu ac [email protected] http//fincl.cnu.ac.kr

Dept. of Naval Architecture & Ocean EngineeringCollege of Engineering, Chungnam National University

Objectives

1 Have a deeper understanding of laminar and turbulent1. Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow.

2. Calculate the major and minor losses associated with pipe flow in piping networks and determine the p p p p gpumping power requirements.

3. Understand the different velocity and flow rate measurement techniques and learn their advantages and disadvantages.

Chapter 8: Flow in PipesShip Hydrodynamics 2

8.1 Introduction

1) Average velocity in a pipe① Recall - because of the no-slip

condition, the velocity at the walls of a pipe or duct flow is zeroof a pipe or duct flow is zero

② We are often interested only in Vavg, which we usually call just V (drop th b i t f i )the subscript for convenience)

③ Keep in mind that the no-slip condition causes shear stress and friction along the pipe wallsFriction force of wall on fluid


8.1 Introduction

1) For pipes of constant di t ddiameter and incompressible flow

2) V sta s the same2) Vavg stays the same down the pipe, even if the velocity profile

Vavg Vavg

the velocity profile changes

3) Why? Conservation of ) yMass

samesame

same


8.1 Introduction

For pipes with variable diameter, m is still the same due to conservation of mass, but V1 ≠ V2

D1

D2

1

V2V1 m m

2

11


8.1 Introduction

• Osborne Reynolds (1842 ~ 1912)1842 b B lf I l d• 1842: born at Belfast, Ireland

• 1867: graduate at Queens’ College, Univ. of Cambridge, in mathematics• 1868: 1st professor of engineering at Owens College, Manchester• 1878: `Improvements in Machinery for Propelling Ships or Vessels’1878: Improvements in Machinery for Propelling Ships or Vessels

Intertial forceViscous forceReynolds number ⎡ ⎤⎣ ⎦

2 2⎡ ⎤⎡ ⎤ ⎡ ⎤2 2

I

V

F V L VLe V

VL

LFR υ

ρ ρμ μ

⎡ ⎤⎡ ⎤ ⎡ ⎤= = = =⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎡ ⎤⎣ ⎦


8.1 Introduction


8.2 Laminar and Turbulent Flows

1) Critical Reynolds number (Recr) for flow in a round pipeflow in a round pipe

Re < 2300 ⇒ laminar2300 ≤ Re ≤ 4000 ⇒ transitional

Definition of Reynolds number

Re > 4000 ⇒ turbulent

2) Note that these values are2) Note that these values are approximate.

3) For a given application, Re depends uponupon①Pipe roughness②Vibrations③Upstream fluctuations,

disturbances (valves, elbows, etc. that may disturb the flow)


8.2 Laminar and Turbulent Flows

1) For non-round pipes, define the h dra lic diameterhydraulic diameter Dh = 4Ac/P

Ac = cross-section areaP tt d i tP = wetted perimeter

2) Example: open channelAc = 0.15 * 0.4 = 0.06m2

P = 0.15 + 0.15 + 0.4 = 0.7mP 0.15 0.15 0.4 0.7mDon’t count free surface, since it does not

contribute to friction along pipe walls!Dh = 4Ac/P = 4*0.06/0.7 = 0.34mDh 4Ac/P 4 0.06/0.7 0.34mWhat does it mean? This channel flow is

equivalent to a round pipe of diameter 0.3m (approximately).


8.3 The Entrance Region

Consider a round pipe of diameter D. The flow can be laminar or turbulent. In either case, the profile develops downstream over several diameters called th t l th L L /D i f ti f Rthe entry length Lh. Lh/D is a function of Re.

Lh


8.3 The Entrance Region

,laminar

1/4

0.05 RehL D≅1/4

,turbuent

,turbuent

1.359 Re

10h

h

L D

L D

≅

≈


,

8.4 Laminar Flow in Pipes

Average velocity:

( ) ( )(2 ) (2 ) (2 ) (2 ) 0x x dx r r dr

d d

rdrP rdrP rdx rdx

r rP P

π π π τ π τ

τ τ+ +− + − =

−− ( ) ( )0

, 0

x dx x r dr rP Pr

dx drdx dr

+ ++ =

→

( ),

0d rdP

rdx dr

τ+ =

dx drdu

drτ μ≡ −

drd du dP

rr dr dr dx

μ ⎛ ⎞ =⎜ ⎟⎝ ⎠


⎝ ⎠


2

1 2( ) ln 4

r dPu r C r C

dxμ⎛ ⎞= + +⎜ ⎟⎝ ⎠

2 wdP

dx R

τ= −

2 2

2( ) 1

4

R dP ru r

dx Rμ⎡ ⎤⎛ ⎞= − −⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

⎡ ⎤2 2 2

2 2 20 0

2

2 2( ) 1

4 8

R R

avg

R dP r R dPV u r rdr rdr

R R dx R dxμ μ⎡ ⎤⎛ ⎞ ⎛ ⎞= = − − − = −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤

∫ ∫2

2( ) 2 1

at 0

avg

ru r V

R

r

⎡ ⎤= −⎢ ⎥

⎣ ⎦=

02 2

( ) ( )22

( )c

R

c RA

avg

u r dA u r rdr

V u r rdrA R R

ρ ρ π= = =∫ ∫

∫max

at 0

2 avg

r

u V

==

The average velocity in fully developed laminar pipe flow is one half of the maximum velocity

2 20

( )avgcA R Rρ ρπ ∫


y


1) There is a direct connection between the pressure drop in a pipe1) There is a direct connection between the pressure drop in a pipe and the shear stress at the wall

2) Consider a horizontal pipe, fully developed, and incompressible floflow

τw

P1 P2VTake CV inside the pipe wall

1 2L

2

1



Pressure Drop: 2

avg

R dPV ⎛ ⎞= − ⎜ ⎟

⎝ ⎠2 1

8 32

dP P P

dx LLV LVμ μ

−=

8avgVdxμ ⎜ ⎟

⎝ ⎠

1 2 2 2

8 32avg avgLV LVP P P

R D

μ μΔ = − = =

2avgVL

P fρ

Δ

2

28

: Darcy friction factor

avgL

w

P fD

fV

ρ

τ

Δ =

≡2

y

64 64

Re

avg

fV

fDV

ρ

μρ

≡ =ReavgDVρ

In laminar flow, the friction factor is a function of the Reynolds number only and is independent of the roughness of the pipe surface



Head Loss: equivalent fluid column height

2

P gh

VP L

ρΔ =

ΔPoiseuille’s law: pressure drop and requiring power is proportional to the length of the pipe and the

2avg

L

VP Lh f

g D gρΔ

= =is proportional to the length of the pipe and the viscosity of the fluid, inversely proportional to the fourth power of the radius of the pipe

2 2 21 2 1 2( ) ( )

pump L LW V P V gh mgh

P P R P P D PDV

ρ= Δ = =

− − Δ= = =

2 4 421 2 1 2

8 32 32

( ) ( )

avg

avg c

VL L L

P P R P P D P DV V A R

μ μ μπ ππ− − Δ

= = = =

4

8 128 128

128

avg c

avg c

L L L

LP V A

D

μ μ μμ

Δ =


4g Dπ

8.5 Turbulent Flow in Pipes

u u u′= +Eddy Fluctuation

InstantaneousInstantaneousprofiles



τw = shear stress at the wall, acting on the fluid

Laminar Turbulent

w g

τw τw

du

dτ μ=

lam tur

drμ

τ τ<



Turbulent shear stress:Tangential(Shear) Force:

( )( )F v dA u u v dA

F

δ ρ ρδ

′ ′ ′ ′= − = −

Fu v

dA

δ ρ ′ ′= −

′ ′turb u vτ ρ ′ ′≡ −Reynolds stress or Turbulent stress

0, 0, 0, 0u v u v u v′ ′ ′ ′ ′ ′= = = ≠

total lam tur

duu v

dyτ τ τ μ ρ

⎛ ⎞′ ′= + = − +⎜ ⎟⎝ ⎠


y⎝ ⎠


u∂Joseph Boussinesq:

: eddy viscosity or turbulent viscosity

turb t

t

uu v

yτ ρ μ

μ

∂′ ′= − =∂

2

2u ulτ μ ρ⎛ ⎞∂ ∂

= = ⎜ ⎟

L. Prandtl:

∂ ∂: mixing length

turb t mly y

l

τ μ ρ= = ⎜ ⎟∂ ∂⎝ ⎠

( ) ( )

/ : kinematic eddy viscosity

toal t t

u u

y yτ μ μ ρ ν ν

ν μ ρ

∂ ∂= + = +

∂ ∂≡ / : kinematic eddy viscosity

kinematic turbulent viscosity

eddy diffusivity of momentum

t tν μ ρ≡


eddy diffusivity of momentum




Viscous sub-layer:

ww

du u u

dy y y

τ ντ μ ρνρ

= = → =

* /

: friction velocitywu

u

τ ρ=

*

*

: friction velocity

law of the wall

u

u yu

u ν=

*

*5 25y= 0 5sublayer

u

yu

νν νδ = = ≤ ≤*

y sublayer u uδ νThe thickness of the viscous sub-layer is proportional to the kinematic viscosity and inversely proportional to the friction velocity


y p p y


N di i l i bl

i l hν

Nondimensional variables:

*

: viscous lenghtu

ν

*y =yu

ν+ ⎫

⎪⎪

=u y

uu

ν + +

+

⎪ =⎬⎪⎪

Nomalized law of the wall:

*

uu ⎪⎭



Overlap layer:

*

the logarithmic law

1l

u yuB+

*

ln y

Bu κ ν

= +

( 0.4, 5.0)Bκ = =

*

*

2.5ln 5.0 u yu

u ν= +

u 2.5ln 5.0y+ += +



Outer turbulent layer:

max

velocity defect law:

2 5lnu u R−

=*

2.5ln

power-law velocity profile:

u R r=

−

1/

max

nu y

u R⎛ ⎞= ⎜ ⎟⎝ ⎠

1/

max

1n

u r

u R⎛ ⎞= −⎜ ⎟⎝ ⎠max

one-seventh power-law


8.5 Moody chart

Colebrook:Colebrook:

1 / 2.512.0log

3 7 Re

D

f f

ε⎛ ⎞= − +⎜ ⎟⎜ ⎟

⎝ ⎠3.7 Ref f⎜ ⎟⎝ ⎠

1 11

Haaland:

⎛ ⎞⎛ ⎞1.11

1 6.9 /1.8log

Re 3.7

D

f

ε⎛ ⎞⎛ ⎞≅ − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠


8.5 Moody chart

Types of Pipe Flow Problems

2VL2

4

2avgVL

P fD

P D

ρ

π

Δ =

Δ128

P DV

L

πμ

Δ=


8.5 Moody chart

1 / 2.51 1 / 2.512 0l ( ) 2 0l 0

D Df

ε ε⎛ ⎞ ⎛ ⎞+ → + +⎜ ⎟ ⎜ ⎟2.0log ( ) 2.0log 0

3.7 3.7Re Reg f

f f f f= − + → = + + =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

6 3 510 , 10 , 10eD Rε − −= = =


8.8 Flow Rate and Velocity Measurement

1) Pitot and Pitot-Static Probes2) Obstruction Flow meters:2) Obstruction Flow-meters: ① Orifice② Venturi Meter③ N l M t③ Nozzle Meter

3) Positive Displacement Flow-meters4) Turbine Flow-meters5) Paddlewheel Flow-meters6) Variable Area Flow-meters7) Ultrasonic Flow-meters)8) Doppler-Effect Ultrasonic Flow-meters9) Electromagnetic Flow-meters10) Vortex Flow-meters10) Vortex Flow meters11) Thermal (Hot-wire) Anemometers12) Laser Doppler Velocimetry (LDV)13) Particle Image Velocimetry (PIV)


13) Particle Image Velocimetry (PIV)

In Chap.5: Bernoulli Equation

s sF ma=∑( ) sinsF PdA P dP dA W θ= − + −∑

s s∑

2 / 0 ( )n na V R a if R

W gdAdsρ= → = →∞

∑

dz dVdPdA dAd dAd V⎛ ⎞

⎜ ⎟ W gdAdsρ=

1 1 1

dPdA gdAds dAds Vds ds

dP gdz VdV

d

ρ ρ

ρ ρ

⎛ ⎞∴− − = ⎜ ⎟⎝ ⎠

− − = 2P Vgz C+ + =

2 21 1 1( ) 0

2 2

dPd V gdz dP dV g dz C

ρ ρ+ + = → + + =∫ ∫ ∫ 2

gρ


In Chap. 5: Euler & Bernoulli Equation

1) The Bernoulli equation was first stated in words by the Swiss mathematician Daniel Bernoulli (1700~1782) in a text written in 1738. It was later derived in general equation from by Leonhard Euler (1707 1783) in 1755from by Leonhard Euler (1707~1783) in 1755.

2) The Bernoulli equation is derived assuming incompressible & inviscid flow, and thus is should not be used fro flows with significant compressibility effects.

3) Unsteady, Compressible Flow:

4) St d C ibl Fl

2

2

dP V Vds gz C

tρ∂

+ + + =∂∫ ∫

4) Steady, Compressible Flow:2

2

dP Vgz C

ρ+ + =∫

5) Steady, Incompressible Flow:

2P V 2

[J]2

P Vgz C

ρ+ + =

The sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during


constant along a streamline during steady flow

In Chap. 5: Static, Dynamic & Stagnation Pressures

1) Ke and Pe can be converted to flow energy (vice versa) during flow:versa) during flow:

① Static pressure (정압)

2

[Pa]2

VP gz Cρ ρ+ + =

② Dynamic pressure (동압)③ Hydrostatic pressure (정수압)

2) Stagnation press re (정체압)2) Stagnation pressure (정체압)= 정압(static pressure)+ 동압(dynamic pressures)

1) 유동압력 측정기구: Pitot tube, Piezometer, U type manometer, Pitot-static probe etc.

2V 2

2stag

VP P ρ= +


Documents

Chapter 8: Flow in Pipes Ship Hydrodynamics 2 - CNU FINCLfincl.cnu.ac.kr/lecture/shiphydrodynamics/Chapter_08B… · · 2012-02-12Friction force of wall on fluid Chapter 8: