9.3
Laminar Flat-Plate Boundary Layer: Exact Solution w-19
Laminar Flat-Plate 9.3
Boundary Layer: Exact Solution
The solution for the laminar boundary layer on a horizontal flat plate was obtained by
Prandtl’s student H. Blasius [2] in 1908. For two-dimensional, steady, incompressible flow
with zero pressure gradient, the governing equations of motion (Eqs. 5.27) reduce to [3]
@u @v
1
50
@x
@y
u
ð9:3Þ
@u
@u
@2u
1v
5ν 2
@x
@y
@y
ð9:4Þ
with boundary conditions
at y 5 0;
u 5 0;
at y 5 N;
u 5 U;
v50
@u
50
@y
ð9:5Þ
Equations 9.3 and 9.4, with boundary conditions Eq. 9.5 are a set of nonlinear, coupled, partial differential equations for the unknown velocity field u and v. To solve
them, Blasius reasoned that the velocity profile, u/U, should be similar for all values of
x when plotted versus a nondimensional distance from the wall; the boundary-layer
thickness, δ, was a natural choice for nondimensionalizing the distance from the wall.
Thus the solution is of the form
u
5 g ðηÞ
U
η~
where
y
δ
pffiffiffiffiffiffiffiffiffiffiffiffi
Based on the solution of Stokes [4], Blasius reasoned that δ~ νx=U and set
rffiffiffiffiffi
U
η5y
νx
ð9:6Þ
ð9:7Þ
We now introduce the stream function, ψ, where
u5
@ψ
@y
and
v 52
@ψ
@x
ð5:4Þ
satisfies the continuity equation (Eq. 9.3) identically. Substituting for u and v into
Eq. 9.4 reduces the equation to one in which ψ is the single dependent variable.
Defining a dimensionless stream function as
ψ
f ðηÞ 5 pffiffiffiffiffiffiffiffiffiffi
νxU
ð9:8Þ
makes f(η) the dependent variable and η the independent variable in Eq. 9.4. With ψ
defined by Eq. 9.8 and η defined by Eq. 9.7, we can evaluate each of the terms in Eq. 9.4.
The velocity components are given by
rffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi df U
@ψ
@ψ @η
df
5
5 νxU
u5
ð9:9Þ
5U
@y
@η @y
dη νx
dη
and
"
"
rffiffiffiffiffiffiffi #
rffiffiffiffiffiffiffi #
pffiffiffiffiffiffiffiffiffiffi @f
pffiffiffiffiffiffiffiffiffiffi df
@ψ
1 νU
1 1
1 νU
5 2 νxU
1
2 η
v 52
f 5 2 νxU
1
f
@x
@x 2
x
dη
2 x
2
x
w-20
Chapter 9 External Incompressible Viscous Flow
or
1
v5
2
rffiffiffiffiffiffiffi
νU df
2f
η
x
dη
ð9:10Þ
By differentiating the velocity components, it also can be shown that
@u
U d2f
52 η 2
@x
2x dη
pffiffiffiffiffiffiffiffiffiffiffiffi d2 f
@u
5 U U=νx 2
@y
dη
and
@2u
U 2 d3 f
5
@y2
νx dη3
Substituting these expressions into Eq. 9.4, we obtain
2
d3 f
d2 f
1
f
50
dη3
dη2
ð9:11Þ
with boundary conditions:
at η 5 0;
at η-N;
f 5
df
50
dη
ð9:12Þ
df
51
dη
The second-order partial differential equations governing the growth of the laminar
boundary layer on a flat plate (Eqs. 9.3 and 9.4) have been transformed to a nonlinear,
third-order ordinary differential equation (Eq. 9.11) with boundary conditions given
by Eq. 9.12. It is not possible to solve Eq. 9.11 in closed form; Blasius solved it using a
power series expansion about η 5 0 matched to an asymptotic expansion for η - N.
The same equation later was solved more precisely—again using numerical methods—
by Howarth [5], who reported results to 5 decimal places. The numerical values of f,
df/dη, and d2f/dη2 in Table 9.1 were calculated with a personal computer using fourthorder Runge-Kutta numerical integration.
The velocity profile is obtained in dimensionless form by plotting u/U versus η,
using values from Table 9.1. The resulting profile is sketched in Fig. 9.3b. Velocity
profiles measured experimentally are in excellent agreement with the analytical
solution. Profiles from all locations on a flat plate are similar; they collapse to a single
profile when plotted in nondimensional coordinates.
From Table 9.1, we see that at η 5 5.0, u/U 5 0.992. With the boundary-layer
thickness, δ, defined as the value of y for which u/U 5 0.99, Eq. 9.7 gives
5:0
5:0x
δ pffiffiffiffiffiffiffiffiffiffiffiffi 5 pffiffiffiffiffiffiffiffi
Rex
U=νx
The wall shear stress may be expressed as
#
#
pffiffiffiffiffiffiffiffiffiffiffiffi d2 f
@u
5 μU U=νx 2
τw 5 μ
@y
dη
y50
ð9:13Þ
η50
9.3
Laminar Flat-Plate Boundary Layer: Exact Solution w-21
Table 9.1
The Function f (η) for the Laminar Boundary Layer along a Flat Plate at Zero Incidence
rffiffiffiffiffi
U
η5y
f
f u 5 Uu
νx
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
0
0.0415
0.1656
0.3701
0.6500
0.9963
1.3968
1.8377
2.3057
2.7901
3.2833
3.7806
4.2796
4.7793
5.2792
5.7792
6.2792
0
0.1659
0.3298
0.4868
0.6298
0.7513
0.8460
0.9130
0.9555
0.9795
0.9915
0.9969
0.9990
0.9997
0.9999
1.0000
1.0000
fv
0.3321
0.3309
0.3230
0.3026
0.2668
0.2174
0.1614
0.1078
0.0642
0.0340
0.0159
0.0066
0.0024
0.0008
0.0002
0.0001
0.0000
Then
τw 5 0:332U
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0:332ρU 2
ρμU=x 5 pffiffiffiffiffiffiffiffi
Rex
ð9:14Þ
and the wall shear stress coefficient, Cf, is given by
Cf 5
τw
1
2
ρU
2
0:664
5 pffiffiffiffiffiffiffiffi
Rex
ð9:15Þ
Each of the results for boundary-layer thickness, δ, wall shear stress, τ w, and skin
friction coefficient, Cf, Eqs. 9.13 through 9.15, depends on the length Reynolds
number, Rex, to the one-half power. The boundary-layer thickness increases as x1/2,
and the wall shear stress and skin friction coefficient vary as 1/x1/2. These results
characterize the behavior of the laminar boundary layer on a flat plate.
E
xample
9.2
LAMINAR BOUNDARY LAYER ON A FLAT PLATE: EXACT SOLUTION
Use the numerical results presented in Table 9.1 to evaluate the following quantities for laminar boundary-layer flow
on a flat plate:
(a) δ*/δ (for η 5 5 and as η - N).
(b) v/U at the boundary-layer edge.
(c) Ratio of the slope of a streamline at the boundary-layer edge to the slope of δ versus x.
w-22
Given:
Find:
Chapter 9 External Incompressible Viscous Flow
Numerical solution for laminar flat-plate boundary layer, Table 9.1.
(a) δ*/δ (for η 5 5 and as η - N).
(b) v/U at boundary-layer edge.
(c) Ratio of the slope of a streamline at the boundary-layer edge to the slope of δ versus x.
Solution:
The displacement thickness is defined by Eq. 9.1 as
Z N
Z δ
u
u
δ* 5
12
12
dy dy
U
U
0
0
In order to use the Blasius exact solution to evaluate this integral,
we need
to convert
it ffifrom one involving
pffiffiffiffiffiffiffiffiffiffiffi
ffi
pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiuffi and y to
one involving fu (5 u/U) and η variables. From Eq. 9.7, η 5 y U=νx; so y 5 η νx=U and dy 5 dη νx=U
Thus,
rffiffiffiffiffi
rffiffiffiffiffi Z η
Z ηmax
max
νx
νx
δ* 5
ð1 2 f uÞ
ð1 2 f uÞdη
ð1Þ
dη 5
U
U 0
0
Note: Corresponding to the upper limit on y in Eq. 9.1, ηmax 5 N, or ηmax 5.
From Eq. 9.13,
5
δ pffiffiffiffiffiffiffiffiffiffiffiffi
U=νx
so if we divide each side of Eq. 1 by each side of Eq. 9.13, we obtain (with fu5df/dη)
Z
δ*
1 ηmax
df
5
12
dη
δ
5 0
dη
Integrating gives
δ*
1
η
5 ½η 2 f ðηÞ0max
δ
5
Evaluating at ηmax 5 5, we obtain
δ*
1
5 ð5:0 2 3:2833Þ 5 0:343 ß
δ
5
The quantity η 2 f(η) becomes constant for η . 7. Evaluating at ηmax 5 8 gives
δ*
ðη 5 5Þ
δ
δ*
ðη-NÞ
δ
*
δ
1
5 ð8:0 2 6:2792Þ 5 0:344 ß
5
δ
*
Thus, δη-N is 0.24 percent larger than δ*η-5 .
From Eq. 9.10,
rffiffiffiffiffiffiffi
rffiffiffiffiffiffiffi
1 νU
df
v
1
ν
df
1
df
2 f ; so
5
2 f 5 pffiffiffiffiffiffiffiffi η
2f
v5
η
η
2
x
dη
U
2 Ux dη
dη
2 Rex
Evaluating at the boundary-layer edge (η 5 5), we obtain
v
1
0:837
0:84
5 pffiffiffiffiffiffiffiffi ½5ð0:9915Þ 2 3:2833 5 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi
U
Rex
Rex
2 Rex
ß
v
ðη 5 5Þ
U
Thus v is only 0.84 percent of U at Rex 5 104, and only about 0.12 percent of U at Rex 5 5 3 105.
9.3
Laminar Flat-Plate Boundary Layer: Exact Solution w-23
The slope of a streamline at the boundary-layer edge is
dy
v
v
0:84
5
pffiffiffiffiffiffiffiffi
5
dx streamline
u
U
Rex
The slope of the boundary-layer edge may be obtained from Eq. 9.13,
rffiffiffiffiffi
5
νx
δ pffiffiffiffiffiffiffiffiffiffiffiffi 5 5
U
U=νx
so
dδ
55
dx
rffiffiffiffiffi
rffiffiffiffiffiffiffi
ν 1 21=2
ν
2:5
x
5 2:5
5 pffiffiffiffiffiffiffiffi
U2
Ux
Rex
Thus,
dy
dx
5
streamline
0:84 dδ
dδ
5 0:336
2:5 dx
dx
ß
dy
dx
streamline
This result indicates that the slope of the streamlines is about 13 of the slope of the boundary layer edge—the
streamlines penetrate the boundary layer, as sketched below:
This prob
lem illus
tra
ical data
from the tes use of numerBla
obtain oth
er inform sius solution to
U
U
ation on
laminar b
a fla
oundary
u
la
re
yer, inclu t plate
δ
sult that
ding the
the edge
layer is n
ot a stre of the boundary
amline.