Acta Polytechnica Hungarica
Vol. 1, No. 2, 2004
Computation of Boundary Layers
József Dénes, István Patkó
Budapest Tech
Doberdó út 6, H-1034 Budapest, Hungary, E-mail:
denes.jozsef@nik.bmf.hu
patko@bmf.hu
Abstract:This paper is the first part of a series of studies where we examine several
methods for the solution of the boundary layer equation of the fluid mechanics. The first of
these is the analytical or rather quasi analytical method due to Blasius. This method
reduces a system of partial differential equations to a system of ordinary differential
equations and these in turn are solved by numerical methods since no exact solution of the
Blasius type equations is known. We determind all the Blasius equation neccessary for up
to 11-th order approximation. Our further aim to study the finite difference numerical
solutions of the boundary layer equation and some of the methods applying weighted
residual principles and by comparing these with the ”exact” solutions arrived at by Blasius
method develop a quick reliable method for solving the boundary layer equation.
Keywords: Boundary Layer, Blasius Method, Boundary Layer equation
1
Boundary Layer
The motion of a fluid around a solid body according to Prandtl (1904) can be
described by the Euler equation of the perfect (that is nonviscous) fluid motion
except in a thin layer near the surface of the solid body where the speed of the
motion increases from zero to the speed that would be in case if the fluid had no
viscosity at all. Outside the boundary layer the fluid may be considered as
nonviscous. This is the case when the velocity of the fluid in the direction of the
flow around the body increases. When it decreases that is the pressure increases,
often the fluid motion unable to follow the bodies’ surface and it gets detached
and the space between the surface of the solid and the detached fluid is filled with
irregularly moving fluid. Prandt’s theory of boundary layer, more precisly his
equations describing the motion within the boundary layer can predict the point(s)
of detachment accurately. The detachment begins where the curve of the velocity
profile starts out perpendiclar to the surface of the solid. After this point a
backward flow develops. The typical values used for describing the boundary
layer are:
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J. Dénes et al.
Computation of Boundary Layers
δ : Boundary layer thickness is the distance measured from the surface of the
solid where the speed of the fluid is within 1% of the speed outside of the
boundary layer.
∞
1
δ 1 : Displacement thickness δ 1 = ∫ (U − u )dy
U 0
∞
1
δ 2 : Impulseloss thickness δ 2 = 2 ∫ (U − u)udy
U 0
δ
**
: Energyloss thickness
Profile parameter H 1, 2 =
2
δ
∞
**
1
= 3 ∫ (U − u) 2 udy
U 0
δ **
δ2
The Eqations of the Boundary Layer Flow
Inside the boundary layer that is in the vicinity of the body the forces due to
viscosity are comparable in magnitude with the forces of inertia they can however
be neglected outside of it. The pressure in the boundary layer could be taken as
constant and its value equal to the pressure belonging to the corresponding perfect
fluid flow, that is the pressure outside of the boundary layer. Without going into
more details we give the equations of the boundary layer motion in case of two
dimensional staionary incompressible fluid flow:
∂ 2u
∂u
∂u
dU
=U
+ν
+v
∂y
∂y
∂x
dx
∂u ∂v
+
=0
∂x ∂y
u
where u is the velocity of the fluid in the boundary layer parallel to the tangent of
the surface of the solid v is perpendicular to it and U is the velocity outside of the
boundary layer. u and v must also satisfy the boundary conditions: y = 0 :
u = 0 v = 0 and at y = ∞ u = U (x) .
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Acta Polytechnica Hungarica
3
Vol. 1, No. 2, 2004
Blasius’ Method for Solving the Boundary Layer
Equation
The boundary layer equations can be reduced to an infinite system of ordinary
differential equations with the following method due to Blasius. By substituting
for the velocity components
u=
∂ψ
∂ψ
and v = −
, where ψ is the stream
∂y
∂x
function of Lagrange the second of the two boundary layer equations is
automaticly satisfied and the first one becomes:
ψ yψ xy − ψ xψ yy = U
dU
+ νψ yyy .
dx
This equation then can be reduced to a set of ordinary differential equation if for
Ψ in case of symmetric bodies the following power series expansion is
substituted:
ψ=
ν
u1
{u1 xf 1 (η ) + 4u 3 x 3 f 3 (η ) + 6u 5 x 5 f 5 (η ) + 8u 7 x 7 f 7 (η ) +
+ 10u 9 x 9 f 9 (η ) + 12u11 x 11 f11 (η ) + ....
where
η=y
u1
ν
.and u1 , u 3 ,⋅ ⋅ ⋅ ⋅ ⋅ are the coefficients in the power series
expansion of U ( x ) , that is:
U = u1 x + u 3 x 3 + u 5 x 5 + u 7 x 7 + u 9 x 9 + u11 x11 + u13 x13 + u15 x15 +
+ u17 x17 + u19 x19 + ........
From the last two equations it follows that:
(
)
dU
= u12 x + 4u1u 3 x 3 + 6u1u 5 + 5u 32 x 5 + (8u1u 7 + 8u 3u 5 )x 7 +
dx
10u1u 9 + 10u 3u 7 + 5u 52 x 9 + (12u1u11 + 12u 3 u 9 + 12u 5 u 7 )x 11 + ........
U
(
)
– 81 –
J. Dénes et al.
ψx =
ν
u1
Computation of Boundary Layers
{u1 f 1 (η ) + 12u 3 x 2 f 3 (η ) + 30u 5 x 4 f 5 (η ) + 56u 7 x 6 f 7 (η ) + 90u 9 x 8 f 9 (η ) +
+ 132u11 x10 f11 (η ) + .....}.
ψ y = u1 xf1' (η ) + 4u 3 x 3 f 3', (η ) + 6u 5 x 5 f 5' (η ) + 8u 7 x 7 f 7' (η ) + 10u 9 x 9 f 9' (η ) +
+ 12u11 x 11 f 11' (η ) + .......
ψ xy = u1 f 1' (η ) + 12u 3 x 2 f 3' (η ) + 30u 5 x 4 f 5' (η ) + 56u 7 x 6 f 7' (η ) + 90u 9 x 8 f 9' (η ) +
+ 132u11 x 10 f 11' (η ) + .....
ψ yy =
u1
ν
{u1 xf 1'' (η ) + 4u 3 x 3 f 3'' (η ) + 6u 5 x 5 f 5'' (η ) + 8u 7 x 7 f 7'' (η ) +
+ 10u 9 x 9 f 9'' (η ) + 12u11 x 11 f 11'' (η ) + .......
ψ yyy =
u1
(u1 xf 1''' (η ) + 4u 3 x 3 f 3''' (η ) + 6u 5 x 5 f 5''' (η ) + 8u 7 x 7 f 7''' (η ) +
ν
+ 10u 9 x 9 f 9''' (η ) + 12u11 x 11 f11''' (η ) + .......)
and
ψ yψ xy = u12 xf 1'2 + 12u1u 3 f 1' f 3' x 3 + (36u1u 5 f 1' f 5' + 48u 32 f 3'2 ) x 5 +
(64u1u 7 f 1' f 7' + 192u 3 u 5 f 3' f 5' ) x 7 +
(100u1u 9 f 1' f 9' + 320u 3 u 7 f 3' f 7' + 180u 52 f 5' 2 ) x 9 +
(144u1u11 f 1' f 11' + 480u 3 u 9 f 3' f 9' + 576u 5 u 7 f 5' f 7' ) x 11 + ........
ψ xψ yy = u12 f 1 f 1'' x + (4u1u 3 f 1 f 3'' + 12u1u 3 f 3 f 3'' ) x 3 +
(6u1u 5 f 1 f 5'' + 48u 32 f 3 f 3'' + 30u1u 5 f 5 f 1'' ) x 5 +
(8u1u 7 f 1 f 7'' + 72u 3 u 5 f 3 f 5'' + 120u 3 u 5 f 5 f 3'' + 56u1u 7 f 7 f 1'' ) x 7 +
(10u1u 9 f 1 f 9'' + 96u 3 u 7 f 3 f 7'' + 180u 52 f 5 f 5'' + 224u 3 u 7 f 7 f 3'' + 90u1u 9 f 9 f 1'' ) x 9 +
(12u1u11 f 1 f 11'' + 120u 3 u 9 f 3 f 9'' + 240u 5 u 7 f 5 f 7'' + 336u 5 u 7 f 7 f 5'' +
360u 9 u 3 f 9 f 3'' + 132u11u1 f 9 f 1'' ) x 11 + ......
finally substituting these into the bondary layer equation:
ψ yψ xy − ψ xψ yy = UU x + ψ yyy
and comparing the coefficients of the powers of x we get the differential equations
of Blasius:
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Acta Polytechnica Hungarica
Vol. 1, No. 2, 2004
Comparing the coefficients of x yields:
u12 f 1'2 − u12 f 1 f 1'' = u12 + u12 f 11'''
and from here we get:
f1'2 − f 1 f1'' = 1 + f 1'''
3
From the coefficients of x we get:
12u1u 3 f 1' f 3' − (4u1u 3 f 1 f 3'' + 12u1u 3 f 1'' f 3 ) = 4u1u 3 + 4u1u 3 f 3'''
that is:
3 f 1' f 3' − f 1 f 3'' − 3 f 1'' f 3 = 1 + f 3'''
5
As for x :
36u1u 5 f1' f 5' + 48u 32 f 3'2 − (6u1u 5 f 1 f 5'' + 48u 32 f 3 f 3'' + 30u1u 5 f1'' f 5 ) =
6u1u 5 + 3u 32 + 6u1u 5 f 5'''
Dividing by 6u1u 5 :
6 f 1' f 5' + 8
u 32
u2
1 u 32
+ f 5'''
f 3'2 − ( f 1 f 5'' + 8 3 f 3 f 3'' + 5 f 1'' f 5 ) = 1 +
u1 u 5
u1 u 5
2 u1 u 5
If we seek f 5 in the form
f5 = g5 +
u 32
h5 we get for g 5 and h5 the following differentialequations:
u1 u 5
6 f 1' g 5' − f 1 g 5'' − 5 f 1'' g 5 = 1 + g 5'''
6 f1' h5' + 8 f 3'2 − f1 h5'' − 5 f1'' h5 − 8 f 3 f 3'' =
1
+ h5'''
2
7
Case x . This case is sufficiently complex to demonstrate the method of finding
the Blasius type diffrential equations for the general case that is for x
is an arbitrary odd integer. By comparing the coefficients of
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7
n
x we get:
where n
J. Dénes et al.
Computation of Boundary Layers
(64u1u 7 f 1' f 7' + 192u 3 u 5 f 3' f 5' ) −
(8u1u 7 f1 f 7'' + 72u 3u 5 f 3 f 5'' + 120u 5 u 3 f 5 f 3'' + 56u 7 u1 f 7 f 1'' ) =
(8u1u 7 + 8u 3 u 5 ) + 8u1u 7 f 7'''
Dividing by 8u1u 7 gives:
(8 f 1' f 7' + 24
(1 +
u 3u 5 ' '
uu
uu
f 3 f 5 ) − ( f1 f 7'' + 9 3 5 f 3 f 5'' + 15 3 5 f 5 f 3'' + 7 f 7 f1'' ) =
u1u 7
u1u 7
u1u 7
u 3u 5
) + f 7'''
u1u 7
Let us seek f 7 in the form f 7 = g 7 +
u3u5 ~
h 7 . Substituting this into the last
u1u 7
equation yields for g 7 the following ordinary differential equation:
8 f 1' g 7' − f 1 g 7'' − 7 f 7 f 1'' = 1 + g 7'''
~
and for
h7
u3u5 ' ~ '
u3u5 ' '
u 32 '
u 3u 5 ~ ''
8
f1 h 7 + 24
f3 (g5 +
h5 ) −
f1 h 7 −
u1u 7
u1u 7
u1u 5
u1u 7
9
~
u 3u 5
u2
uu
u2
f 3 ( g 5'' + 3 h5'' ) − 15 3 5 f 3'' ( g 5 + 3 h5 ) − 7 h 7 f1'' =
u1u 7
u1u 5
u1u 7
u1u 5
u 3 u 5 u 3 u 5 ~ '''
+
h7
u1u 7 u1u 7
~
Substituting for
~
h 7 h 7 = h7 +
u 32
k 7 yields for h7
u1u 7
8 f 1' h7' + 24 f 3' g 5' − f 1 h7'' − 9 f 3 g 5'' − 15 f 3'' g 5 − 7 f 1'' h7 = 1 + h7'''
and for
k7
8 f 1' k 7' + 24 f 3' h5' − f1 k 7'' − 9 f 3 h5'' − 15 f 3'' h5 − 7 f1'' k 7 = k 7'''
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Acta Polytechnica Hungarica
Vol. 1, No. 2, 2004
With the same method we can arrive at the equations for all the
functions. With n increasing
terms. Here the forms of
f n Blasius
f n has to be broken down into more and more
f n and the corresponding differential equations are
given up to the order of 11.
f5 = g5 +
u 32
h5
u1 u 5
f7 = g7 +
u 3u 5
u2
h7 + 2 3 k 7
u1u 7
u1 u 7
f9 = g9 +
u 3u 7
u2
u 2u
u4
h9 + 5 k 9 + 32 5 j9 + 3 3 q9
u1u 9
u1u 9
u1 u 9
u1 u 9
f11 = g11 +
u 3u
u u2
u 3u 9
uu
u 2u
h11 + 5 7 k11 + 23 7 j11 + 23 5 q11 + 33 5 m11 +
u1u11
u1u11
u1 u11
u1 u11
u1 u11
u 35
n11
u14 u11
f 3 , g 5 , h 5 , g 7 , h7 , k 7 , g 9 ,
h9 , k 9 , j9 , q9 , g11 , h11 , k11 , j11 , q11 , m11 , n11 have to satisfy are:
The differential equatios that the functions f 1 ,
f1'2 − f 1 f1'' = 1 + f 1'''
3 f 1' f 3' − f 1 f 3'' − 3 f 1'' f 3 = 1 + f 3'''
6 f 1' g 5' − f 1 g 5'' − 5 f 1'' g 5 = 1 + g 5'''
6 f1' h5' + 8 f 3' 2 − f1h5'' − 5 f1'' h5 − 8 f 3 f 3'' =
1
+ h5'''
2
8 f 1' g 7' − f 1 g 7'' − 7 f 7 f 1'' = 1 + g 7'''
8 f 1' h7' + 24 f 3' g 5' − f 1 h7'' − 9 f 3 g 5'' − 15 f 3'' g 5 − 7 f 1'' h7 = 1 + h7'''
8 f 1' k 7' + 24 f 3' h5' − f1 k 7'' − 9 f 3 h5'' − 15 f 3'' h5 − 7 f1'' k 7 = k 7'''
10 f1' g 9' − f 1 g 9'' − 9 f 1'' g 9 = 1 + g 9'''
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J. Dénes et al.
Computation of Boundary Layers
10 f1' h9' + 32 f 3' g 7' − f 1 h9'' − 9,6 f 3 g 7'' − 22,4 f 3'' g 7 − 9 f 1'' h9 = 1 + h9'''
10 f1' k 9' + 18 g 5' 2 − f1k 9'' − 18 g 5 g 5'' − 9 f1'' k 9 =
1
+ k 9'''
2
10 f 1' j9' + 32 f 3' h7' + 36 g 5' h5' − f1 j9'' − 9,6 f 3 h7'' − 18 g 5 h5'' − 18 g 5'' h5 −
22,4 f 3'' h7 − 9 f 1'' j9 = j9'''
10 f1' q 9' + 32 f 3' k 7' + 18h5' 2 − f1 q9'' − 9,6 f 3 k 7'' − 18h5 h5'' − 22,4 f 3'' k 7 −
9 f1'' q 9 = q9'''
12 f1' g11' − f 1 g11'' − 11 f 1'' g11 = 1 + g11'''
12 f1' h11' + 40 f 3' g 9' − f1 h11'' − 10 f 3 g 9'' − 30 f 3'' g 9 − 11 f1'' h11 = 1 + h11'''
12 f1' k11' + 48 g 5 g 7' − f1 k11'' − 20 g 5 g 7'' − 28 g 5'' g 7 − 11 f 1'' k11 = 1 + k11'''
12 f1' j11' + 40 f 3' h9' + 48h5' g 7' − f1 j11'' − 10 f 3 h9'' − 20h5 g 7'' − 28h5'' g 7 −
30 f 3'' h9 − 11 f1'' j11 = j11'''
12 f1' q11' + 40 f 3' k 9' + 48 g 5' h7' − f1 q11'' − 10 f 3 k 9'' − 20 g 5 h7'' − 28 g 5'' h7 −
30 f 3'' k 9 − 11 f 1'' q11 = q11'''
12 f1' m11' + 40 f 3' j9' + 48h5' k 7' + 48 g 5' h7' − f1 m11'' − 10 f 3 j9'' − 20h5 h7'' −
20 g 5 k 7'' − 28 g 5'' k 7 − 28h5'' h7 − 30 f 3'' j9 − 11 f1'' m11 = m11'''
12 f1' n11' + 40 f 3' q9' − f 1 n11'' − 10 f 3 q9'' − 20h5 k 7'' − 28h5'' k 7 − 30 f 3'' q9 −
11 f 1'' n11 = n11'''
The
u ( x, y ) = 0 at y = 0 and lim u ( x, y ) = U ( x) boundary conditios in
y →∞
f 3 , g 5 , h 5 , g 7 , h7 , k 7 , g 9 , h9 , k 9 , j9 , q9 , g11 , h11 , k11 ,
j11 , q11 , m11 , n11 become
terms of f 1 ,
at η
=0
f1 = f 1' = 0; f 3 = f 3' = 0; g 5 = g 5' = 0; h5 = h5' = 0; g 7 = g 7' = 0;
h7 = h7' = 0; k 7 = k 7' = 0; g 9 = g 9' = 0; h9 = h9' = 0; k 9 = k 9' = 0;
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Acta Polytechnica Hungarica
Vol. 1, No. 2, 2004
j9 = j9' = 0; q9 = q9' = 0; g11 = g11' = 0; h11 = h11' = 0; k11 = k11' = 0;
j11 = j11' = 0; q11 = q11' = 0; m11 = m11' = 0; n11 = n11' = 0;
and at η = ∞
1
1
1
1
; g 5' = ; h5' = 0; g 7' = ; h7' = 0; k 7' = 0; g 9' = ;
4
6
8
10
1
h9' = 0; k 9' = 0; j9' = 0; q9' = 0; g11' = ; h11' = 0; k11' = 0; j11' = 0;
12
'
'
'
q11 = 0; m11 = 0; n11 = 0;
f1' = 1; f 3' =
Conclusion
We have derived the ordinary differential equations necessary to carry out
numerical approximation to the solution of the boundary layer equation by 11-th
order Blasius method.
References
[1]
L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung.
Proceedings 3rd Intern. Math. Congr. Heidelberg 1904, 484-491. Reprinted
in Coll. Works II 575-584.
[2]
H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math.
u. Phys. 56, 1-37 (1908).
[3]
L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung.
Proceedings 3rd Intern. Math. Congr. Heidelberg 1904, 484-491. Reprinted
in Coll. Works II 575-584.
[4]
H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math.
u. Phys. 56, 1-37 (1908).
– 87 –