viii. viscous flow and head loss. contents 1. introduction 2. laminar and turbulent flows 3....
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VIII. Viscous Flow and Head LossVIII. Viscous Flow and Head Loss
ContentsContents
1.1. IntroductionIntroduction
2.2. Laminar and Turbulent FlowsLaminar and Turbulent Flows
3.3. Friction and Head LossesFriction and Head Losses
4.4. Head Loss in Laminar FlowsHead Loss in Laminar Flows
5.5. Head Loss in Turbulent Flows Head Loss in Turbulent Flows
6.6. Head Loss of Steady Pipe FlowsHead Loss of Steady Pipe Flows
7.7. Minor LossesMinor Losses
8.8. ExamplesExamples
1. Introduction1. Introduction
Shear stress due to fluid viscosityShear stress due to fluid viscosity
uy
t m¶
=¶
0F =rD’Alembert D’Alembert
ParadoxParadox
2 2
1 22 2V p V p
z zg g
a ag g
é ù é ùê ú ê ú+ + = + +ê ú ê úë û ë û
2
2Vg
2
2Vg
pg
2
2Vg
p g
wh
whwh
2 2V g
pg
2
2Vg
pg
For real fluid flowsFor real fluid flows
2 2
upstream downstream
02 2 w
V p V pz z h
g g g ga a
r r
æ ö æ ö÷ ÷ç ç+ + ÷ - + + ÷ = >ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
Head Loss
Head Loss:
Losses due to friction
Minor Losses
entrance and exit
sudden change of cross sections
valves and gates
bends and elbows
……
2. Laminar and Turbulent Flows2. Laminar and Turbulent Flows
Reynolds’ ExperimentReynolds’ Experiment
Laminar Flows:
Movement of any fluid particle is regular
Path lines of fluid particles are smooth
Turbulent Flows:
Movement of any fluid particle is random
Path lines of fluid particles are affected by mixing
Transition from Laminar to Turbulent Flow:
for different fluid
for different diameter of pipe
Head Loss due to laminar and turbulent flowsHead Loss due to laminar and turbulent flows
fh
log V
log h f
Turbulent Flows:
Laminar Flows: fh Vµ
( )
( )
2
1.75
Rough wall
Smooth wall
fh V
V
µ
µ
Critical Condition
2300UdUd
Rr
n m= = =
Reynolds Number
3. Friction and Head Losses3. Friction and Head Losses
2
2Vg
1pg
2pg
1z 2z
wh
Momentum EquationMomentum Equation
1 2 sin 0pA pA AL PLg a t- - - =
A : area of the cross-A : area of the cross-
section section
P: wetted perimeterP: wetted perimeter
1 2 sin 0pA pA AL PLg a t- - - =
2 1sinz z
La
-=
( )1 22 1
p p PLz z
At
g g g- - - =
f
h
h PL A R
t tg g
= =
Hydraulic radius
2
2f
VC
rt =
f
h
h
L Rt
g=
2
2ffh
L Vh C
R g=
2
2f
L Vh f
D g= Darcy-Weisbach equation
4. Head Loss in Laminar Flows4. Head Loss in Laminar Flows
x
( )
0
0
u u r
v
w
=
=
=
cos
sin
x x
y r
z r
q
q
=
=
=
0u v wx y z
¶ ¶ ¶+ + =
¶ ¶ ¶
2 2 2
2 2 2
1u u u p u u uu v w
x y z x x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
%
2 2 2
2 2 2
1v v v p v v vu v w
x y z y x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
%
2 2 2
2 2 2
1w w w p w w wu v w
x y z z x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
%
0ux
¶=
¶
2 2
2 2
1 p u ux y z
nr
æ ö¶ ¶ ¶ ÷ç= + ÷ç ÷÷çç¶ ¶ ¶è ø
%
0py
¶=
¶%
0pz
¶=
¶%
2 2
2 2
sin sincos cos
cossin sin
r ry z y r y y r y
r rz r z z r z
r r r r
r r
q qq q
q qq q
q qq q
q q
q
æ öæ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶÷ ÷ç ç+ = + +÷ ÷ç ç÷ ÷÷ ÷ç çç ç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è øè ø
æ öæ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶÷ ÷ç ç+ + +÷ ÷ç ç÷ ÷ç çè øè ø¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶
æ öæ ö¶ ¶ ¶ ¶÷ ÷ç ç= + +÷ ÷ç ç÷ ÷ç çè øè ø¶ ¶ ¶ ¶
æ ö¶ ¶ ÷ç+ - ÷ç ÷çè ø¶ ¶
2
2 2
cos
1 1
r r
rr r r r
q
q
æ ö¶ ¶ ÷ç - ÷ç ÷çè ø¶ ¶
æ ö¶ ¶ ¶÷ç= +÷ç ÷çè ø¶ ¶ ¶
cos
sin
y r
z r
q
q
=
=
2 2
2 2
2
2 2
1 1
dp u udx y z
u ur
r r r r
ur
r r r
m
mq
m
æ ö¶ ¶ ÷ç= + ÷ç ÷÷çç¶ ¶è ø
é ùæ ö¶ ¶ ¶÷çê ú= +÷ç ÷çê úè ø¶ ¶ ¶ë ûæ ö¶ ¶ ÷ç= ÷ç ÷çè ø¶ ¶
%
21log
4dp
u r A r Bdxm
= + +%
( ) ( )2 2 2 214 4
dp Ju a r a r
dxg
m m= - - = -
%
( )
( )
0
finite 0
u r a
u r
= =
® ®
p p zg= +%
( )2 2 14 2r a r a
u Ja r J a
r rg
t m m gm= =
é ùé ù¶ ¶ ê ú= = - = -ê ú ê úê ú¶ ¶ë û ë û
f
h
h
L Rt
g= hJ Rt g=
4 2
128 32Q VJ
D Dm m
gp g= =
2
32fh VJ
L Dm
g= =
2642f
L Vh
VD D gm
r=
64f
R=
( )4
2 2
0 02 2
4 128
a a J J DQ ur dr a r r dr
g pgp p
m m
é ùê ú= = - =ê úë û
ò ò
2 4Q
VDp
=
gg r=
5. Head Loss in Turbulent Flows5. Head Loss in Turbulent Flows
Mean flow and fluctuationMean flow and fluctuation
t
B
B
1 t T
tB Bdt
T
+= ò
Mean flow and fluctuationMean flow and fluctuation B B B¢= +
B B=
0BB¢=
BB BB=
0B¢=
0B B¢ ¢¹
BB BB B B¢ ¢= +
B Bx x
æ ö¶ ¶÷ç =÷ç ÷çè ø¶ ¶
1 2 1 2B B B B+ = +
Basic Equations of Turbulent Flows: Basic Equations of Turbulent Flows:
2 2 2
2 2 2
1x
u u u u p u u uu v w f
t x y z x x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
2 2 2
2 2 2
1y
v v v v p v v vu v w f
t x y z y x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
2 2 2
2 2 2
1z
w w w w p w w wu v w f
t x y z z x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
0u v wx y z
¶ ¶ ¶+ + =
¶ ¶ ¶
2 2 2
2 2 2
1x
u u u u p u u uu v w f
t x y z x x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
0u v wx y z
¶ ¶ ¶+ + =
¶ ¶ ¶
2 2 2
2 2 2
1x
u u u u v u wu u v u w u
t x x y y z z
p u u uf
x x y zn
r
¶ ¶ ¶ ¶ ¶ ¶ ¶+ + + + + +
¶ ¶ ¶ ¶ ¶ ¶ ¶
æ ö¶ ¶ ¶ ¶ ÷ç= - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶è ø
2 2 2
2 2 2
1x
u uu uv uw p u u uf
t x y z x x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
Basic Equations of Turbulent Flows: Basic Equations of Turbulent Flows:
2 2 2
2 2 2
1x
u uu uv uw p u u uf
t x y z x x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
2 2 2
2 2 2
1y
v vu vv vw p v v vf
t x y z y x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
2 2 2
2 2 2
1z
w wu wv ww p w w wf
t x y z z x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
0u v wx y z
¶ ¶ ¶+ + =
¶ ¶ ¶
Reynolds’ Average Reynolds’ Average
0u v wx y z
¶ ¶ ¶+ + =
¶ ¶ ¶
0u v wx y z
¶ ¶ ¶+ + =
¶ ¶ ¶
Reynolds’ Average Reynolds’ Average
2 2 2
2 2 2
1x
u uu uv uw p u u uf
t x y z x x y zn
r
æ ö¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ÷ç+ + + = - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è ø
2 2 2
2 2 2
1x
u uu uu uv uv uw uwt x x y y z z
p u u uf
x x y zn
r
¢ ¢ ¢¢ ¢ ¢¶ ¶ ¶ ¶ ¶ ¶ ¶+ + + + + +
¶ ¶ ¶ ¶ ¶ ¶ ¶
æ ö¶ ¶ ¶ ¶ ÷ç= - + + + ÷ç ÷÷çç¶ ¶ ¶ ¶è ø
Reynolds Stresses Reynolds Stresses
xx
yx
zx
R uu
R uv
R uw
r
r
r
¢ ¢= -
¢¢= -
¢ ¢= -
Reynolds Stresses Reynolds Stresses
v¢( )u u y=
x
y
( )xF v u uvr r¢ ¢¢= =Mean flux of horizontal momentum:
Equivalent Shear Stress: yxR-
Reynolds Equations: Reynolds Equations:
2 2 2
2 2 2
1 1 yxxx zxx
u u u uu v w
t x y z
Rp u u u R Rf
x x y z x y zn
r r
¶ ¶ ¶ ¶+ + +
¶ ¶ ¶ ¶
æ ö æ ö¶¶ ¶ ¶ ¶ ¶ ¶÷ ÷ç ç= - + + + + + +÷ ÷ç ç÷ ÷÷ ÷ç çç ç¶ ¶ ¶ ¶ ¶ ¶ ¶è ø è ø
2 2 2
2 2 2
1 1 xy yy zyy
v v v vu v w
t x y z
R R Rp v v vf
y x y z x y zn
r r
¶ ¶ ¶ ¶+ + +
¶ ¶ ¶ ¶
æ ö æ ö¶ ¶ ¶¶ ¶ ¶ ¶ ÷ ÷ç ç= - + + + + + +÷ ÷ç ç÷ ÷÷ ÷ç çç ç¶ ¶ ¶ ¶ ¶ ¶ ¶è ø è ø
2 2 2
2 2 2
1 1 yzxz zzz
w w w wu v w
t x y z
Rp w w w R Rf
z x y z x y zn
r r
¶ ¶ ¶ ¶+ + +
¶ ¶ ¶ ¶
æ ö æ ö¶¶ ¶ ¶ ¶ ¶ ¶÷ ÷ç ç= - + + + + + +÷ ÷ç ç÷ ÷÷ ÷ç çç ç¶ ¶ ¶ ¶ ¶ ¶ ¶è ø è ø
2 2 2
2 2 2
1 1 yxxx zxx
u u u uu v w
t x y z
Rp u u u R Rf
x x y z x y zn
r r
¶ ¶ ¶ ¶+ + +
¶ ¶ ¶ ¶
æ ö æ ö¶¶ ¶ ¶ ¶ ¶ ¶÷ ÷ç ç= - + + + + + +÷ ÷ç ç÷ ÷÷ ÷ç çç ç¶ ¶ ¶ ¶ ¶ ¶ ¶è ø è ø
1 yxxx zx
x y z
ts tr
æ ö¶¶ ¶ ÷ç + + ÷ç ÷÷çç ¶ ¶ ¶è ø
12
12
xx
yx
zx
ux
v ux y
w ux z
s rn
t rn
t rn
¶=
¶
æ ö¶ ¶ ÷ç= + ÷ç ÷÷çç¶ ¶è ø
æ ö¶ ¶ ÷ç= + ÷ç ÷çè ø¶ ¶
12
12
xx e
yx e
zx e
uR
x
v uR
x y
w uR
x z
rn
rn
rn
¶=
¶
æ ö¶ ¶ ÷ç= + ÷ç ÷÷çç¶ ¶è ø
æ ö¶ ¶ ÷ç= + ÷ç ÷çè ø¶ ¶
Theory of Mixing Length Theory of Mixing Length
( )u u y=
x
y
l¢
duu l
dy¢ ¢= v u¢ ¢µ
2
2
uv
ducl
dy
t r
r
¢¢= -
æ ö÷ç¢= ÷ç ÷÷ççè ø
2yx
du dul
dy dyt r=
Logarithmic Velocity Distribution Logarithmic Velocity Distribution
( )u u y=
y
( )0.4l yk k= =
2
2 20
duy
dyt t rk
æ ö÷ç= = ÷ç ÷÷ççè ø
0t t=
0*
duy vdy
tk
r= º
* logv
u y Ck
= +
6. Head Loss of Steady Pipe Flows6. Head Loss of Steady Pipe Flows
Logarithmic Velocity Distribution Logarithmic Velocity Distribution
* logv
u y Ck
= +
*
1log
uy C
v k¢= +
*
*
1log
u vyC
v k n¢¢= +
*R
0
t
l
y
Logarithmic Overlap Layer
Logarithmic Velocity Distribution in a Pipe Logarithmic Velocity Distribution in a Pipe
y
2 0
3 0
1 0
01 01 01 0 1 0 0 0 01 0 0 01 0 01
*vy n
*
uv
*
*
2.5log 5.5u vyv n
= +
*
*
u vyv n
=
Viscous Turbulent
Viscous sublayer:Viscous sublayer: *0 5vyn
< £
Turbulent zone:Turbulent zone:* 70vyn
³
Transition Transition zone:zone:
*5 70vyn
< <
Velocity Distribution in Viscous SublayerVelocity Distribution in Viscous Sublayer
dudy
t m=
20 *vu y y
t rm m
= =
*
*
u vyv n
=
Velocity Distribution in a Pipe Velocity Distribution in a Pipe
y
Blasius’ 7th-root law Blasius’ 7th-root law
17
max 0
u yu r
æ ö÷ç ÷= ç ÷ç ÷çè ø
Valid for R = 3000 Valid for R = 3000 101055
Wall RoughnessWall Roughness
sk
* 5svkn
£
* 70svkn
³
Hydraulically smooth wall:Hydraulically smooth wall:
Roughness height is smaller than the Roughness height is smaller than the
thickness of the viscous sublayerthickness of the viscous sublayer
Hydraulically rough wall:Hydraulically rough wall:
Roughness height is larger than the lower Roughness height is larger than the lower
boundary of the turbulent zoneboundary of the turbulent zone
Hydraulically smooth pipe:Hydraulically smooth pipe:
Hydraulically rough pipe:Hydraulically rough pipe:
*
*
2.5log 5.5u vyv n
= +
*
2.5log 8.5s
u yv k
= +
Velocity Distribution in a PipeVelocity Distribution in a Pipe
Mean velocity in hydraulically smooth pipe:Mean velocity in hydraulically smooth pipe:
( )* *
*
2.5log 5.5 2.5log 5.5u vy v a rv n n
-= + = +
( )* * *2 2 0
22.5log 5.5
a
S
v v v a rV udS rdr
a ap
p p n-é ù
= = +ê úê úë ûò ò
*
*
2.5log 1.75V vav n
= +
Mean velocity in hydraulically rough pipe:Mean velocity in hydraulically rough pipe:
( )
*
2.5log 8.5 2.5log 8.5s s
u y a rv k k
-= + = +
( )* *2 2 0
22.5log 8.5
a
Ss
v v a rV udS rdr
a a kp
p p
é ù-ê ú= = +ê úë ûò ò
*
2.5log 4.75s
V av k
= +
Relation among mean velocity, friction velocity Relation among mean velocity, friction velocity
and friction factor:and friction factor:
2
2f
L Vh f
D g= 0 0
14
f
h
h
L R Dt tg g
= =
20 *
2 2
1 88
vf
V Vtr
= =
* 8V v f=
Friction factor in hydraulically smooth pipe:Friction factor in hydraulically smooth pipe:
( )10
10.884log 0.91
2.04log R 0.91
VDf
f
f
næ ö÷ç= -÷ç ÷çè ø
= -
*
*
2.5log 1.75V vav n
= +
Friction factor in hydraulically rough pipe:Friction factor in hydraulically rough pipe:
( )
( )
2
2
10
1
0.884log 1.68
1
2.04log 1.68
s
s
fa k
a k
=é ù+ê úë û
=é ù+ê úë û
*
2.5log 4.75s
V av k
= +
Experiment of NikuradseExperiment of Nikuradse
sk
Modified friction factor in hydraulically smooth pipe:Modified friction factor in hydraulically smooth pipe:
( )10 10
R12log R 0.8 2log
2.51f
ff
æ ö÷ç ÷= - = ç ÷ç ÷çè ø
10
1 R1.8log
6.9fæ ö÷ç= ÷ç ÷çè ø
0.25
0.316R
f =
( )84000 10R£ £
( )53000 10R£ £
( )Rff=
Modified friction factor in hydraulically rough pipe:Modified friction factor in hydraulically rough pipe:
( )
( )
2
10
2
10
1
2log 1.74
1
2log 3.7
s
s
fa k
D k
=é ù+ê úë û
=é ùê úë û
skffa
æ ö÷ç= ÷ç ÷çè ø
Colebrook Equation:Colebrook Equation:
( )
10
10
10
1 2.512log 0.27
R
2.512log 0, smooth
R
12log R , rough
3.7
s
s
s
kf D f
kf D
kD
æ ö÷ç ÷= - +ç ÷ç ÷çè ø
æ ö æ ö÷ç ÷ç÷® - ®ç ÷ç÷ ÷çç ÷ è øçè ø
æ ö÷ç® - ® ¥÷ç ÷çè ø
Head loss in hydraulically smooth pipe:Head loss in hydraulically smooth pipe:
( )2 2
1.750.25
0.3162 2f
L V L Vh f V
D g D gVD n= = µ
Head loss in hydraulically smooth pipe:Head loss in hydraulically smooth pipe:
2 22
2 2s
f
L V k L Vh ff V
D g D D gæ ö÷ç= = µ÷ç ÷çè ø
Practical pipe: equivalent roughnessPractical pipe: equivalent roughness
sk
7. Minor Losses7. Minor Losses
Head Loss due to Sudden ExpansionHead Loss due to Sudden Expansion
lxh
pz
g+
2
2Vg
Head Loss due to Sudden ExpansionHead Loss due to Sudden Expansion
1 2
1 1 2 2V A V A=( )
2 21 1 2 2
2 21 2 1 2
2
1 2
2 21 1
2
21
2 2
2
2
12
2
lx
x
p V p Vh
g g
p p V Vg g
V V
g
A VA g
Vg
g g
r
V
æ ö æ ö÷ ÷ç ç= + - +÷ ÷ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
- -= +
-=
æ ö÷ç ÷= -ç ÷ç ÷çè ø
=
( ) ( )2 22 2 1 1 1 1 1 2 1 2 2AV AV pA p A A pAr ¢- = + - -
( )2 22 2 1 1 21 2
2 1 22
AV AVp pV VV
Ar
--= = -
1p¢
Head Loss due to Sudden ContractionHead Loss due to Sudden Contraction
ch
2
2p V
zgg
+ +
( )
22
2 1
2lc c
c c
Vh
g
D D
V
V V
=
=
Head Loss at EntranceHead Loss at Entrance
eh
2
2p V
zgg
+ +
2
2le e
Vh
gV=
Head Loss at Bell-Mouthed EntranceHead Loss at Bell-Mouthed Entrance
2
2e e
Vh
gV=
Head Loss in BendHead Loss in Bend
( )
2
2b b
b b
Vh
g
r D
V
V V
=
=
8. An Example8. An Example
20 mH =
1 0.2 mD =2 0.4 mD =
1 80 mL = 2 50 mL =
?Q =
0.05 mmsk =0.05 mmsk =
2 2
2 2 w
A B
p V p Vz z H h
g gg g
æ ö æ ö÷ ÷ç ç+ + - + + = =÷ ÷ç ç÷ ÷ç ç÷ ÷ç çè ø è øå
2 2 2 2 21 1 1 1 2 2 2
1 21 22 2 2 2 2w e x o
V l V V l V Vh ff
g D g g D g gV V V= + + + +
0.5eV =
2
1
2
1 0.5625x
AA
Væ ö÷ç ÷= - =ç ÷ç ÷çè ø
1.0oV =
( )1 2
10 1
10.01438
2log 3.7 s
fD k
= =é ùê úë û
( )2 2
10 2
10.01250
2log 3.7 s
fD k
= =é ùê úë û
2 21 1 2 2
1 21 2
2 21 2
2 21 2
2 2
80 500.01438 0.5 0.5625 0.01250 1.0
0.2 2 0.4 2
6.8145 2.56252 2
e x o
l V l VH ff
D g D g
V Vg g
V Vg g
V V Væ ö æ ö÷ ÷ç ç÷ ÷= + + + +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
æ ö æ ö÷ ÷ç ç= ´ + + + ´ +÷ ÷ç ç ÷÷ çç è øè ø
= +
1 1 2 2 2 10.25V A V A V V= Þ =
1 2
27.50 m s
6.8145 0.25 2.5625gH
V = =+ ´
52 22 6
3 42
1.875 0.47.5 10
10
0.05 10 0.4 1.25 10s
V DR
k D
n -
- -
´= = = ´
= ´ = ´
1 7.50m sV =
2 10.25 1.875 m sV V= =
31 1 2 2 0.2356 m sQ V A V A= = =
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