Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
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6 Newtonian Flow
Fig. 6.28 Boundary values in turbulent boundary layer
Equations (6.6.85) and (6.6.86) now can be integrated over each layer
separately, referring each boundary value as indicated in Fig. 6.28, to give
Uu
TT
U
k
u
TTkq
l
wlc
l
wlc
w
w
/
0
0
KKW
(6.6.87)
and
Uu
TT
c
q
l
l
p
f
W
(6.6.88)
It is reasonable, therefore, to assume that the ratio
ww
q
W
/
and
W
/q
(similarly
ww
q
W
/ and
ty
q
W
/ for each layer in the integration) remains
constant across the whole width
G
of the boundary layer. Furthermore,
since 1
|P
r
, we can set the similarity of the velocity and temperature pro-
file as
w
TT
TT
U
u
f
f
(6.6.89)
Equating Eqs. (6.6.87) and (6.6.89), we can now obtain a relationship be-
tween the velocity and temperature where
»
¼
º
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
f
)( 11 Pr
U
u
u
U
Pr
TT
TT
l
l
wl
w
(6.6.90)
In many situations in engineering, it proves useful to consider the local
heat transfer coefficient
x
h that can be defined as
384
Exercise
f
TT
q
h
w
w
x
(6.6.91)
This can be further reduced to a more convenient form with Eqs. (6.6.91)
and (6.6.87) where
w
wl
l
l
x
q
TT
Uu
PrUu
Pr
h
»
¼
º
«
¬
ª
»
¼
º
«
¬
ª
/
11
w
p
l
U
c
PrUu
W
11
1
(6.6.92)
As an alternative expression to the local heat coefficient
x
h , the non-
dimensional parameter, the local Nusselt number
x
Nu (as defined in Ex-
ercise 6.5.2) can be written by other flow field parameters
11
2
1
PrUu
PrRec
Nu
l
xf
x
(6.6.93)
Further, the Stanton number is similarly given where
11
2
1
PrUu
c
Sn
l
f
x
(6.6.94)
The expression in Eq. (6.6.93) is derived from Prandtl and G.I. Taylor
independently as the extension of the Reynolds’ analogy to the turbulent
heat transfer. The case, when 1z
t
Pr , was studied more by Ambrok (1957).
Numerical treatment of the turbulent flow problem in the k -
H
equation is
found in Jones and Lunder (1972).
Exercise
Exercise 6.6.1 Estimation of Drag Coefficient on Flat Plate for
Turbulent Flow
Based on the momentum integral analysis, estimate the drag coefficient
for turbulent flow over a flat plate with a zero attack angle. Let us assume
here, for simplicity, that the turbulent boundary layer grows from the lead-
ing edge of the flat plate. The flow is a steady flow over a flat plate with a
385
6 Newtonian Flow
zero pressure gradient so that the stream velocity U is kept constant. Con-
sider a case of the mean velocity distribution for the turbulent boundary
layer to be the power law
7
11
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
GG
yy
U
u
n
(1)
Ans.
By the Kàrmàn integral equation, the local friction coefficient
x
f
c can
be written by
dx
d
U
c
w
f
x
T
U
W
2
2
1
2
(2)
T
is the momentum thickness, which is given where
³
f
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
0
1dy
U
u
U
u
T
(3)
Setting
G
K
y , the velocity distribution is expressed by the power law,
as assumed where
7
1
K
U
u
(4)
The wall of shear stress
w
W
of pipe given by Blasius’ formula in Eq.
(6.4.40) is transformed to the flat plate coordinates by setting
2
0
/dr
G
, and
4960
max
UU
with reference to Eq. (6.4.54), so that
for the boundary layer over a flat plate, we have
4
1
2
02330
XGUW
UU
w
.
(5)
The integral equation for Eq. (2) can be rewritten by using Eqs. (3) and (4),
which is further equated with Eq. (5) as follows
4
1
2
1
0
7
1
7
1
2
022301
°
¿
°
¾
½
°
¯
°
®
¸
¸
¹
·
¨
¨
©
§
³
XGKKK
G
UUd
dx
d
dx
d
U .
(6)
Equation (6) is an ordinary differential equation with respect to
x
G
, and
is solved by assuming 0
G
for 0 x to give
386
Exercise
dxUd
4
1
4
1
240 /.
XGG
(7)
Resultantly, it follows
5
1
5
1
3820
3820
x
xRe
xU
x
.
/
.
X
G
(8)
It is noted here that the boundary layer thickness
x
G
of the turbulent
flow
54
xx |
G
is far thicker than that of a laminar boundary layer
21
xx |
G
, as is evident by comparison between Eq. (8) and Eq. (6.5.25).
Now the local value of the shear stress
w
W
of Eq. (5) can be evaluated by
using
x
G
in Eq. (8), so that
5
1
2
02960
XUW
xUU
w
.
5
1
2
02960
x
ReU
U
.
(9)
which gives
5
1
05920
xf
Rec
x
. for
7
10
x
Re
(10)
Thus, the required drag force
D
F on one side of the flat plate is calculated,
according to Eq. (10) in Exercise 6.5.1, as
blU
dxbF
l
wD
5
4
5
1
5
9
0
0370
XU
W
.
³
(11)
The drag coefficient is
¸
¹
·
¨
©
§
2
2
1
ubl
F
c
D
f
U
5
1
0740
l
Re. for
75
10105 ddu
l
Re
(12)
x
f
c
and
f
c
given in Eqs. (10) and (12) respectively give good estimate
values for the Reynolds number range indicated in the equations.
For reference, Schlichting (1955) proposed
x
f
c and
f
c , using the log-
arithmic law of the wall, and those are in good agreement with experi-
mental data in the Reynolds number range,
97
10Re10 dd
l
, which is
387
6 Newtonian Flow
32
650log2
.
.
x
f
Rec
x
(13)
and
582
log
4550
.
.
l
f
Re
c
(14)
Note that
f
c shown in Eqs. (12) and (14) is subjected to the turbulent
boundary layer developed from the leading edge. However, in practice, the
laminar boundary is usually developed immediately after the leading edge
toward the position of the transition region that usually lies in the range,
65
103103 uddu
x
Re , so that, when the transition is assumed to occur at
5
105u|
x
Re , it is convenient to write
f
c
for a fully developed turbulent
boundary flow as
xffp
Recc 1700 .
975
1010105 orRe
x
ddu
(15)
where
f
c is given either by Eq. (12) or (14), depending upon the upper
range of
x
Re
.
Exercise 6.6.2 The Temperature Law of the Wall
The total rate of heat transfer
w
q
from a flat plate is written by using the
turbulent Prandtl number with reference to Eq. (6.6.84) where
dy
Td
PrPr
cq
t
t
pw
¸
¸
¹
·
¨
¨
©
§
KK
0
(1)
dyTd is taken for the negative gradient from the wall. If
0
K
K
t
is given
via a linear function of
y where
ky
t
0
K
K
(2)
Show that
T
is a logarithmic function of
y . Note that
y is defined by
the friction velocity
W
u , such that
0
K
U
U
W
X
W
w
y
yu
y
(3)
388
Exercise
Ans.
Equation (1) can be integrated to give
³
¿
¾
½
¯
®
¸
¹
·
¨
©
§
y
t
t
t
w
PrPr
dy
T
TT
T
0
0
11
K
K
(4)
where
t
T is defined analogous to the friction velocity, such that
W
U
uc
q
T
p
w
t
(5)
T
given in Eq. (4) is the so-called temperature law of the wall, analogous
ing expression:
PrBy
a
Pr
T
t
ln
(6)
where
a is an experimental constant (such as in Eq. (6.6.67)) and
B
is an
integration constant given at a boundary of
y . The expression is the loga-
rithmic expression for the distribution of temperature in the turbulent
boundary layer. The similarity of velocity distribution in the temperature is
evident. Note that very near to the wall, where
0
K
K
t
, the thermal
sublayer has the following linear temperature distribution, that is repre-
sented by
yPrT
(7)
There are some expressions for
PrB
that are obtained from experi-
mental verification. For example, by referring Kader (1981) where
PrPrPrB ln12.23185.3
2
3
1
¸
¸
¹
·
¨
¨
©
§
| .
(8)
Equation (8) is valid for air, water and etc. for 1707.0 dd Pr .
to the law of the wall variable
y as defined in Equation (3). Equation (4)
together with the relation of (
0
K
K
t
), as given in Eq. (2), yields the follow-
389
6 Newtonian Flow
Problems
6.6-1. Derive the relationship to describe a turbulent velocity profile in a
pipe with radius
0
r from the definition of the eddy viscosity
t
K
and
of the eddy diffusivity
U
K
X
/
tt
in the turbulent boundary layer of
a flat plate.
Ans.
»
»
¼
º
«
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
dy
du
r
y
t
X
X
11
0
6.6-2. The velocity profile in the turbulence core for a smooth pipe can be
expressed in the logarithmic form
5.5ln5.2
yu
Find
X
X
/
t
with the functional form, using the result from previous
question.
Ans.
»
»
¼
º
«
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
11
52
0
r
yy
.
6.6-3. Air is flowing on a flat plate with a zero attack angle. Sketch how
the boundary layer is developed on the plate, assuming that, at the
same distance,
x
from the leading edge, where the flow is reached
to be a turbulent flow from a laminar flow at
X
xURe
critical
, the
turbulent boundary layer has started to be developed onward.
Ans.
>@
example for 18Ref. Fig. 6.
6.6-4. In problem 6.5-4, denote that the free stream velocity is sm5 and
the kinematic viscosity of air is
sm1061
25
u. . The length and
width of the plate is m4 and m2 respectively. Estimate the bound-
ary layer thickness at the end of the plate surface, and calculate the
total drag force acting on one side of the plate. Assume that the criti-
cal Reynolds number from laminar to turbulent is approximately
5
105 u
critical
Re .
Ans.
>@
0.098 Nm110
D
end
F,.
G
6.6-5 Compare the velocity profile and temperature profile over the turbu-
lent boundary layer with a heat transfer from a flat plate with a zero
attack angle. Assume 01.|Pr and
01.
t
Pr
, and use Kader’s ex-
pression with reference to Exercise 6.6.2, Eq. (8).
390
Nomenclature
Nomenclature
A
configuration parameter
cba ,,
constant in a turbulent flow
0
a
speed of sound
b
internal heat generation
Br
Brinkman number
fD
cc ,
drag coefficients and friction coefficient
vp
cc ,
specific heat
n
cccc ...,,
210
constants
d
diameter
0
2rd
h
d
hydraulic diameter
e
rate of strain tensor
i
e
ˆ
unit vectors
Ec
Eckert number
Eu
Euler number
F
,
F
force, vector
D
F
drag force
Fr
Froude number
f
frequency
g
body force
g
gravitational acceleration
h
height or thickness or enthalpy per unit mass or heat trans-
fer coefficient
w
h
wall heat transfer coefficient
kji, ,
unit vectors in Cartesian coordinates system
I
unit tensor
K
constant in a turbulent
0
K
bulk modulus
k
average kinetic energy of turbulence or specific heat ratio
c
k
thermal conductivity
B
k
Boltzmann constant
r
k
radius ratio
D
k
thermal diffusivity
lL,
length or l ;length scale of an eddy or ;mixing length
c
L
characteristic length
391
6 Newtonian Flow
p
l
wetted perimeter
m
l
mean free path
m
mass
n
ˆ
unit normal
P
power
p
pressure
Pe
Peclet number
P
r
Prandtl number
t
Pr
turbulent Prandtl number
q
, q
heat flux, vector
w
q
wall heat flux, wall heat transfer rate
Q
volumetric flow rate
R
(specific) gas constant
Ra
Rayleigh number
Re
Reynolds number
0
r
radius
h
r
hydraulic radius
S
area
Sn
Stanton number
St
Strouhal number
s
specific entropy
t
,
0
t
time, reference time or starting time
t
stress vector
T
total stress tensor
T
temperature or averaging time
r
T
torque
u
velocity vector
W
u
friction velocity
U
speed or potential core speed
U
average velocity
w
v
u ,,
velocity components in Cartesian coordinates system
V
volume
We
Weber number
zy
x
,,
Cartesian coordinates system
zr ,,
T
cylindrical coordinates system
I
T
,,r
spherical coordinates system
J
E
D
,,
matching constants
D
D
damping factor
392
Nomenclature
41
0
D
D
D
,...,,
constants
T
E
coefficient of thermal expansion
J
share rate
G
boundary layer thickness or shear layer thickness
G
c
displacement thickness
h
G
enthalpy thickness
T
G
thermal boundary layer thickness
H
roughness or dissipation energy of turbulence
]
pressure loss coefficient
9
boundary thickness ratio
K
similarity variable
0
K
Newtonian viscosity
t
K
eddy viscosity
T
momentum thickness
d
N
dilatational viscosity
O
friction factor
k
O
Kolmogorov scale of length
T
O
Taylor microscale
0
O
second viscosity coefficient
U
density
V
surface tension
IJ
stress tensor
w
W
wall stress
t
W
turbulent (shear) stress
k
W
Kolmogorov scale of time
R
IJ
Reynolds stress tensor
X
kinematic viscosity
1
X
kinematic eddy viscosity
0
I
gravitational potential
)
dissipation function
\
stream function
Z
angular frequency
Ȧ
spin tensor
Ȧ
vorticity (vector)
Superscripts
dimensionless variable
values based on law of wall
393