Chapter 4
THE BOUNDARY
LAYER
4.1 Generalities
Since its beginning in an inspired paper by PRANDTL (1905) the lit-
erature of boundary-layer theory and practice has become probably
the largest single component of the literature of fluid mechanics. The
most important and practical cases are often the least understood,
because there are few organizing principles for turbulent flow. In fact,
a list of these principles almost constitutes a history of the subject.
They all are based on experimental results or on particular insight
into the meaning of experimental results, sometimes with a strong
element of serendipity. They have often also required development
of new instrumentation. For example, the first competent measure-
ments in turbulent boundary layers at constant pressure were made
in Prandtl’s institute at G ̈ottingen by SCHULTZ-GRUNOW (1940).
These data extended the validity of the wall law and defect law from
pipe flow to boundary-layer flow. The means to this end was the first
use of a floating element on a laboratory scale to measure directly
the local wall friction. An associated major advance at G ̈ottingen
was the development and use of a wind tunnel designed with sound
understanding of the relevant fluid mechanics.
207
208
CHAPTER 4.
THE BOUNDARY LAYER
Perhaps the most important early use of the boundary-layer
concept was the application of KARMAN’s momentum-integral equa-
tion (1921) to turbulent flows approaching separation, say on airfoils
or in diffusers. However, the available data seemed to show the
wall friction in a positive pressure gradient increasing rapidly in the
downstream direction as separation was approached, rather than de-
creasing toward zero. This result was not credible, but it could only
be displaced by a direct attack based on some reliable method for
measuring the wall friction in a pressure gradient. The instrument
devised for this purpose at G ̈ottingen by LUDWIEG (1949) was the
heated element, which depends for its operation on a link between
the equations describing transfer of heat and momentum in low-speed
flow near a wall, as described in SECTION 1.2.6 of the introduction.
The measured results (LUDWIEG and TILLMANN 1949) showed
the friction decreasing in a positive pressure gradient as expected,
and the anomalous increase was traced in the thesis by TILLMANN
(1947) to three-dimensional mean flow. What was not expected from
this research was the observation that the similarity law called the
law of the wall proved to be insensitive to pressure gradient, either
positive or negative, and also to free-stream turbulence level. This
observation made it possible to infer the wall friction very cheaply, by
fitting part of the mean-velocity profile to the standard logarithmic
formula near the wall.
A few years later, CLAUSER (1954) used this result to sup-
port a corresponding generalization of the defect law to certain spe-
cial boundary-layer flows with pressure gradient. Since the friction
velocity is a required element of the defect law, say through demon-
stration of a constant value for the combination
δ
∗
u
∞
/δu
τ
, Clauser’s
method depended on a cut and try adjustment of the pressure varia-
tion. Clauser chose the term “equilibrium” to define the class of flows
involved. It is important to appreciate that Clauser’s contribution
was not a discovery, but an invention.
The next step fell to me when I used Clauser’s work to gener-
alize the profile formula to flows not in equilibrium;
i.e.,
not having
a defect law, by introducing a second universal component that I
called the law of the wake (COLES 1956). I provided a rough phys-
4.1. GENERALITIES
209
ical interpretation for the formula as a description of a wake-like
flow modified by the no-slip condition at the wall. A new profile
parameter, which I called Π, measures the relative magnitude of the
wake and wall components and is constant for any one of Clauser’s
equilibrium flows, including the flow at constant pressure.
These concepts involving similarity for turbulent flow will be
developed in more detail in this chapter. They are in place and are ac-
cepted by most but not all of the fluid-mechanics community. It may
or may not be significant that they were almost all contributed by
relatively young professionals working at academic or near-academic
institutions. Except for the concept of coherent structure, little has
been added in the last fifty years, which have seen a steady accu-
mulation of experiments and a gradual and valuable proliferation of
turbulence models and direct solutions based on the increasing power
of digital computers. In particular, plausible similarity laws for flows
with mass transfer, compressibility, and other features are still not
available.
In 1957 I proposed a model for equilibrium turbulent boundary-
layer flow that was based on an analogy between the laminar equilib-
rium flows of Falkner and Skan and the turbulent equilibrium flows
of Clauser. This model has not been accepted. However, I still see
value in it, and I therefore have described it in SECTION 4.10.1 of
this chapter. If it is ever to be useful, the model will eventually have
to be developed like the Thwaites method for laminar flow, using
similarity laws that are compound rather than simple.
My strategy here is first to discuss thoroughly the laminar
problem, in the hope that some qualitative analogy with turbulent
flow will emerge and will suggest questions whose answers, if found,
may lead to better understanding. The first part of this chapter will
therefore be concerned with two-dimensional laminar boundary lay-
ers. They will be treated within the boundary-layer approximation
of PRANDTL (1905), which was derived in CHAPTER 1.
210
CHAPTER 4.
THE BOUNDARY LAYER
4.1.1
The momentum-integral equation
Karman.
An important tool in the boundary-layer trade is an in-
tegral relationship first derived by KARMAN (1921) and applied to
laminar flow in a companion paper by K. POHLHAUSEN (1921).
It may have been this example of practical application that finally
brought about the acceptance of the boundary-layer concept by the
fluid-mechanics community outside Germany. Incidentally, Karman’s
powerful paper deserves a high place in the literature of the subject
for another and quite different reason, which is that it derived the
boundary-layer approximation as a formal limit of the Navier-Stokes
equations for small viscosity and it anticipated quite accurately the
essence of the method of matched asymptotic expansions, including
inner and outer expansions, the matching condition, and the com-
posite expansion.
The raw material needed here consists of the continuity and
momentum equations in boundary-layer form, as set out in SEC-
TION 1.3.2. These are written for steady two-dimensional flow of
a compressible fluid, since there is no extra cost at this stage for
including compressibility and mass transfer. For the present there is
also no need to distinguish between laminar and turbulent flow, and
both are included in the equations
∂ρu
∂x
+
∂ρv
∂y
= 0
,
(4.1)
∂ρuu
∂x
+
∂ρuv
∂y
=
−
d
p
d
x
+
∂τ
∂y
,
(4.2)
where
τ
is the shearing stress, laminar or turbulent. The second of
these equations can also be written, with the aid of the first, as
ρu
∂u
∂x
+
ρv
∂u
∂y
=
−
d
p
d
x
+
∂τ
∂y
.
(4.3)
It is part of the boundary-layer approximation that the pres-
sure
p
is no longer treated as a dependent variable, but enters through
the boundary conditions imposed by the external flow. A relation-
ship between
p
and
u
∞
follows on evaluating equation (4.3) outside
4.1. GENERALITIES
211
the boundary layer, where
∂u/∂y
=
τ
= 0,
u
=
u
∞
, and
ρ
=
ρ
∞
.
Thus
ρ
∞
u
∞
d
u
∞
d
x
=
−
d
p
d
x
.
(4.4)
There remain two equations for two dependent variables,
u
and
v
(and
τ
, if the flow is turbulent). Begin by integrating both
equations from the wall to some value of
y
within the local boundary
layer. Use of the boundary conditions
u
= 0,
v
=
v
w
,
τ
=
τ
w
at
y
= 0 yields
ρv
−
ρ
w
v
w
=
−
y
∫
0
∂ρu
∂x
d
y
;
(4.5)
y
∫
0
∂ρuu
∂x
d
y
+
ρuv
=
y
∫
0
ρ
∞
u
∞
d
u
∞
d
x
d
y
+
τ
−
τ
w
.
(4.6)
These can be combined in the single equation
y
∫
0
∂ρuu
∂x
d
y
+
u
ρ
w
v
w
−
y
∫
0
∂ρu
∂x
d
y
=
y
∫
0
ρ
∞
u
∞
d
u
∞
d
x
d
y
+
τ
−
τ
w
.
(4.7)
The integrals in this expression diverge as
y
→ ∞
, unless the range
of
y
is artificially limited by introducing a boundary-layer thickness.
The integrals were left in this condition by Karman, who was con-
cerned with larger issues. Pohlhausen disposed of the divergence by
adding to equation (4.7) the identity
−
y
∫
0
∂ρuu
∞
∂x
d
y
+
y
∫
0
u
∞
∂ρu
∂x
d
y
=
−
y
∫
0
ρu
d
u
∞
d
x
d
y
(4.8)
to obtain
∂
∂x
y
∫
0
ρu
(
u
−
u
∞
) d
y
+
ρ
w
v
w
u
+ (
u
∞
−
u
)
∂
∂x
y
∫
0
ρu
d
y
+
+
d
u
∞
d
x
y
∫
0
(
ρu
−
ρ
∞
u
∞
) d
y
=
τ
−
τ
w
.
(4.9)
212
CHAPTER 4.
THE BOUNDARY LAYER
Now let
y
→∞
. Provided that (
u
∞
−
u
) and (
ρ
∞
u
∞
−
ρu
) approach
zero sufficiently rapidly, there is now convergence;
τ
w
=
d
d
x
∞
∫
0
ρu
(
u
∞
−
u
) d
y
−
ρ
w
v
w
u
∞
+
d
u
∞
d
x
∞
∫
0
(
ρ
∞
u
∞
−
ρu
) d
y .
(4.10)
Equation (4.10) is the momentum-integral equation in rectangular
coordinates for a compressible fluid for laminar or turbulent flow.
Other coordinate systems can be treated in the same way.
Gruschwitz.
The notation in equation (4.10) can be simpli-
fied by defining an integral displacement thickness
δ
∗
as
ρ
∞
u
∞
δ
∗
=
∞
∫
0
(
ρ
∞
u
∞
−
ρu
) d
y
(4.11)
and an integral momentum thickness
θ
as
ρ
∞
u
2
∞
θ
=
∞
∫
0
ρu
(
u
∞
−
u
) d
y .
(4.12)
These definitions, together with the notation
δ
∗
,
θ
, were introduced
by GRUSCHWITZ (1931). Equation (4.10) becomes
τ
w
=
d
d
x
ρ
∞
u
2
∞
θ
+
ρ
∞
u
∞
δ
∗
d
u
∞
d
x
−
ρ
w
v
w
u
∞
.
(4.13)
A different form of this equation is sometimes preferable. Expand
the first term on the right into three derivatives, and replace d
ρ
∞
by
the isentropic equivalent d
p/a
2
∞
,
a
being the speed of sound. When
terms are collected, equation (4.13) is replaced by
τ
w
=
ρ
∞
u
2
∞
d
θ
d
x
−
θ
(
2 +
δ
∗
θ
−
M
2
∞
)
d
p
d
x
−
ρ
w
v
w
u
∞
,
(4.14)
4.1. GENERALITIES
213
where
M
∞
=
u
∞
/a
∞
. Experiments in adiabatic turbulent flow at
nominally constant pressure have been collected by FERNHOLZ and
FINLEY (1977). They show that the ratio
δ
∗
/θ
increases with in-
creasing Mach number, but less rapidly than
M
2
∞
, and that the quan-
tity in parentheses in equation (4.14) changes sign at a Mach num-
ber near 2.5 at laboratory Reynolds numbers. Consequently, flow
irregularities might be expected to have the least influence on flow
description at this Mach number.
The thicknesses
δ
∗
and
θ
are sometimes denoted by
δ
∗
and
δ
∗∗
or by
δ
1
and
δ
2
in the European literature. The latter forms are
most useful if higher moments are needed. They have simple physical
meanings. In the sketch
1
, let
y
=
Y
be a value of
Y
well outside the
boundary layer. Then the mass flux past the station in the sketch is
defined by the stream function
ψ
(
Y
) =
Y
∫
0
ρu
d
y
=
ρ
∞
u
∞
Y
−
Y
∫
0
(
ρ
∞
u
∞
−
ρu
) d
y
=
ρ
∞
u
∞
(
Y
−
δ
∗
)
.
(4.15)
Thus
δ
∗
is the position of a wall bounding a hypothetical flow that
coincides with the given flow at infinity but has the density
ρ
∞
and
velocity
u
∞
throughout. Appropriately,
δ
∗
is called the displacement
thickness. It can also be viewed as an integral scale for the profile
(need equation). (
Need corresponding argument for
θ
. See
Bradshaw.
)
One potentially valuable property of the momentum-integral
equation (4.13) is that it provides a means for determining the wall
stress
τ
w
experimentally, provided that all other factors that control
the momentum flux near a wall can be measured. This property
is less valuable if d
u
∞
/
d
x
is large and negative, because
τ
w
is then
obtained as a small difference between two large terms, and the cal-
culation is degraded by even slight three-dimensionality.
The momentum-integral equation is at the center of several
calculation schemes that represent a boundary-layer flow in terms of
a specified family of mean-velocity profiles. The first such scheme
1
This sketch has not been found.
214
CHAPTER 4.
THE BOUNDARY LAYER
was proposed in K. Pohlhausen’s original paper on laminar flow.
A variation is discussed in SECTION X. Possible applications to
turbulent flow are discussed in SECTION X.
2
4.2 Laminar equilibrium flow
4.2.1
The affine transformation
For the rest of this chapter, the fluid will be assumed to be incom-
pressible. The two-dimensional laminar boundary-layer equations for
steady flow are
∂u
∂x
+
∂v
∂y
= 0 ;
(4.16)
ρ
(
u
∂u
∂x
+
v
∂u
∂y
)
=
−
d
p
d
x
+
μ
∂
2
u
∂y
2
.
(4.17)
If there is no mass transfer, the boundary conditions are the no-slip
and streamline conditions at the wall,
u
(
x,
0) = 0
, v
(
x,
0) = 0 or
ψ
(
x,
0) = 0
,
(4.18)
together with the specification of a free-stream pressure or velocity,
p
=
p
(
x
)
or
u
(
x,
∞
) =
u
∞
(
x
)
.
(4.19)
Similarity.
The topic to be developed in the next few sections
is the topic of laminar similarity. When the equations and boundary
conditions that govern a problem are known, as they are here, the
purpose of a similarity argument is to decrease the number of inde-
pendent variables through what usually amounts to an application
of group theory. A more physical statement in the present context
is that flows with similarity are understood to be in equilibrium, in
the sense that all of the processes represented by the terms in the
momentum equation (4.17) have the same relative importance ev-
erywhere and always. Acceleration, pressure force, and viscous force
2
It is not clear what section(s), if any, cover these topics.
4.2. LAMINAR EQUILIBRIUM FLOW
215
are all acting in concert and in continuous equilibrium. In laminar
flows with pressure gradient, in particular, vorticity is continuously
being generated at the wall and diffusing outward as it is transported
downstream. In an equilibrium flow, diffusion cooperates with pro-
duction and transport to keep the vorticity and velocity profiles in
a state of similarity. Finally, there is persuasive evidence that equi-
librium flows represent limits to which flows tend provided that the
boundary conditions are propitious.
(See “Remarks” paper; cite
Libby and Narasimha; mention eigenvalues)
.
Another important property of similarity arguments is that
they provide a unifying principle for associating different solutions
of the two-dimensional Navier-Stokes equations or their boundary-
layer approximation. In this chapter, these solutions include the
boundary layer at constant pressure (SECTION 4.4.1), which first
demonstrated the power of Prandtl’s boundary-layer approximation;
the stagnation-point flow (SECTION 4.4.3), which showed that the
boundary-layer approximation can sometimes be exact; the sink flow
(SECTION 4.4.4), for which the Navier-Stokes equations can be
solved exactly for any Reynolds number, allowing the nature of the
boundary-layer approximation to be analyzed in arbitrary detail; and
the continuously separating boundary layer (SECTION 4.4.5), which
raises new and complex issues in boundary-layer theory. These flows
are all special solutions of an equation called the Falkner-Skan equa-
tion, and they will be discussed separately after this equation has
been derived.
The immediate objective is to seek solutions of equations (4.16)
and (4.17) in similarity form. To simplify the boundary conditions
and avoid some awkward algebra, the dependent variables
u
and
v
are
first fused in a stream function,
ψ
, defined in rectangular coordinates
by the equation
~u
(
x,y
) = grad
ψ
×
grad
z .
(4.20)
Two vector identities from the introduction,
div (
~a
×
~
b
)
≡
~
b
·
curl
~a
−
~a
·
curl
~
b
(4.21)
and
curl grad
c
≡
0
,
(4.22)
216
CHAPTER 4.
THE BOUNDARY LAYER
then imply
div
~u
= 0
,
(4.23)
so that the continuity equation (4.16) is automatically satisfied. The
velocity vector lies in the intersection of the surfaces
ψ
= constant
and
z
= constant, and the velocity components calculated from equa-
tion (4.20) are
(
u, v, w
) =
(
∂ψ
∂y
,
−
∂ψ
∂x
,
0
)
.
(4.24)
The use of a stream function reduces equation (4.17) to a third-order
equation for a single dependent variable
ψ
;
ρ
(
∂ψ
∂y
∂
2
ψ
∂x∂y
−
∂ψ
∂x
∂
2
ψ
∂y
2
)
=
ρu
∞
d
u
∞
d
x
+
μ
∂
3
ψ
∂y
3
,
(4.25)
where
u
∞
(
x
) is a given function that has to be specified in each case.
Conventional boundary conditions at a solid wall are
ψ
= 0
,
∂ψ
∂x
= 0
at
y
= 0
.
(4.26)
Equation (4.25) is of parabolic type, with real characteristics
x
=
constant. Integration proceeds in the direction of increasing
x
, and
there is no upstream effect of downstream boundary conditions (see
SECTION 4.4.1 and FIGURE 4.1).
Throughout fluid mechanics, sufficient conditions for the prop-
erty of similarity can often be established by dimensional arguments,
provided that the number of global parameters in the problem is not
too large or too small. For example, assume the existence of local
scales
U
(
x
) and
L
(
x
), usually representing local free-stream veloc-
ity and local layer thickness, respectively, for equation (4.25). When
solutions are sought in the form
ψ/UL
=
f
(
y/L
), and no separate de-
pendence on
x
is allowed, sufficient conditions usually emerge to de-
termine the functions
U
(
x
) and
L
(
x
) in adequate detail. The method
is powerful, and its results are not limited to power laws. Examples
can be found in SECTIONS 4.4.6 and 10.1.1.
4.2. LAMINAR EQUILIBRIUM FLOW
217
In this monograph I will usually prefer another more formal
and more efficient means to the same end. This method is to deter-
mine the affine transformation group for the equations and boundary
conditions of a problem. In fact, the theory of Lie groups was orig-
inally developed and applied as a method for the solution of partial
differential equations, (
Lie, Ovsiannikov, Barenblatt, Bluman
and Cole, others
).
4.2.2
The Falkner-Skan equation
What is wanted is an affine transformation that transforms equation
(4.25) and its boundary conditions (4.19) and (4.26) into themselves.
To discover this transformation, every variable or parameter is mind-
lessly scaled, or transformed, or mapped, according to a standard
program;
x
=
a
̂
x
;
y
=
b
̂
y
;
ψ
=
c
̂
ψ
;
(4.27)
ρ
=
d
̂
ρ
;
μ
=
e
̂
μ
;
u
∞
=
f
̂
u
∞
;
where
a
,
b
,
c
,
...
are dimensionless scaling constants that are finite
and positive. Some authors
(cite Birkhoof, Sedov)
seek to ex-
press all of these constants in terms of one of them through some
tedious algebra, but I find this variation to be neither as simple nor
as instructive as the one used here.
The result of the mapping (4.27) in the case of equation (4.25)
is
dc
2
ab
2
̂
ρ
(
∂
̂
ψ
∂
̂
y
∂
2
̂
ψ
∂
̂
x∂
̂
y
−
∂
̂
ψ
∂
̂
x
∂
2
̂
ψ
∂
̂
y
2
)
=
df
2
a
̂
ρ
̂
u
∞
d
̂
u
∞
d
̂
x
+
ec
b
3
̂
μ
∂
3
̂
ψ
∂
̂
y
3
.
(4.28)
Given that
u
transforms like
∂ψ/∂y
, the corresponding statement for
the boundary condition (4.19) is
c
b
̂
u
(
a
̂
x,
∞
) =
f
̂
u
∞
(
a
̂
x,
∞
)
.
(4.29)
218
CHAPTER 4.
THE BOUNDARY LAYER
The other two boundary conditions (4.18) yield no information. In-
variance of the equation and boundary conditions evidently requires
bcd
ae
= 1
,
b
3
df
2
ace
= 1
,
c
bf
= 1
.
(4.30)
Division of the first of these equations by the second yields the third,
which is therefore redundant;
u
∞
and
u
transform according to the
same rule.
Although the dependent variable is
ψ
(
x, y
), the essence of
most descriptive processes is the functional relationship between
u
=
∂ψ/∂y
and
y
at constant
x
. This observation suggests that the first
two of equations (4.30) should be revised to isolate
c
and
b
, which
are the scaling factors for
ψ
and
y
, respectively. Other strategies will
work, but may be awkward. The result can be stated as
c
2
d
aef
= 1
,
b
2
df
ae
= 1
.
(4.31)
Now let the affine-transformation table (4.27) be interpreted as a
group of definitions for
a
,
b
,
c
,
...
, and let these definitions be sub-
stituted into equations (4.31). Two invariants of the transformation
appear;
ψ
(
νu
∞
x
)
1
/
2
=
̂
ψ
(
̂
ν
̂
u
∞
̂
x
)
1
/
2
,
(
u
∞
νx
)
1
/
2
y
=
(
̂
u
∞
̂
ν
̂
x
)
1
/
2
̂
y .
(4.32)
These combinations are dimensionless by construction, since each
term in equation (4.25) must have the same physical dimensions.
A non-trivial reasoning process described in SECTION 1.3.2
of the introduction implies finally that similarity solutions are to be
found by adopting the ansatz
A
ψ
(
νu
∞
x
)
1
/
2
=
f
[
B
(
u
∞
νx
)
1
/
2
y
]
=
f
(
η
)
,
(4.33)
in which two dimensionless scaling constants
A
and
B
are included
for later use in normalizing the final differential equation and its
boundary conditions. No harm is done if the step from equations
4.2. LAMINAR EQUILIBRIUM FLOW
219
(4.32) to equation (4.33) is treated as intuitive rather than exact.
The ansatz (4.33) shows explicitly that the variables
ψ
,
x
, and
y
should be combined as
ψ/x
1
/
2
and
y/x
1
/
2
. The appearance of
ν
1
/
2
is a more subtle question already discussed in SECTION 1.3.2.
When the stream function from equation (4.33) is inserted in
the momentum equation (4.25), four transport terms are generated,
but two of these cancel. This property is typical of similarity analyses
in the boundary-layer theory of viscous shear flows. The result can
be written
2
AB f
′′′
+
(
1 +
x
u
∞
d
u
∞
d
x
)
ff
′′
+ 2
x
u
∞
d
u
∞
d
x
(
A
2
B
2
−
f
′
f
′
)
= 0
,
(4.34)
where primes indicate differentiation with respect to
η
, the argument
of
f
in equation (4.33). From left to right, the original source of the
four terms in
f
is viscous (highest derivative), transport (nonlinear),
pressure force (lacking
f
), and again transport. If
η
is to be the only
independent variable, the coefficients cannot depend on
x
, and it is
necessary to require
x
u
∞
d
u
∞
d
x
= constant =
m
(say)
.
(4.35)
The external flow condition is therefore specified by the power law
u
∞
∼
x
m
.
(4.36)
This conclusion ignores the question of algebraic signs for quantities
inside the square roots in equation (4.33), a question that will be
taken up in SECTION 4.3.1. With this reservation, equation (4.34)
becomes
2
AB f
′′′
+ (1 +
m
)
ff
′′
+ 2
m
(
A
2
B
2
−
f
′
f
′
)
= 0
.
(4.37)
4.2.3
Normalization
It remains to choose the constants
A
and
B
, mainly on the basis of
esthetic considerations. On the way from equation (4.25) to equation
220
CHAPTER 4.
THE BOUNDARY LAYER
(4.37), use was made of the derivative relationship
u
=
∂ψ
∂y
=
B
A
u
∞
f
′
.
(4.38)
A convenient condition on
A
and
B
is therefore
7
A
=
B ,
(4.39)
because it yields the simple result
u
u
∞
=
f
′
(
η
)
.
(4.40)
This same condition (4.39) is also suggested by the form of the third
term in the differential equation (4.37). The boundary conditions
(4.18) and (4.19) become
f
(0) =
f
′
(0) = 0
,
f
′
(
∞
) = 1
.
(4.41)
An operator.
It is typical of similarity arguments for lam-
inar flow in rectangular coordinates that they lead to an operator
f
′′′
+
ff
′′
, sometimes with numerical coefficients that differ from one
problem or one author to another. I propose to give this opera-
tor
f
′′′
+
ff
′′
a uniform aspect, free of coefficients, throughout this
monograph. I therefore take in equation (4.37)
1 +
m
2
AB
=
1 +
m
2
A
2
= 1
,
(4.42)
so that, still without regard to signs,
A
=
B
=
(
1 +
m
2
)
1
/
2
.
(4.43)
Equation (4.37) is reduced to a one-parameter equation,
f
′′′
+
ff
′′
+
2
m
1 +
m
(
1
−
f
′
f
′
)
= 0
,
(4.44)
and the perfected form of the ansatz (4.33) is
(
1 +
m
2
νu
∞
x
)
1
/
2
ψ
=
f
[
(
(1 +
m
)
u
∞
2
νx
)
1
/
2
y
]
=
f
(
η
)
.
(4.45)
4.2. LAMINAR EQUILIBRIUM FLOW
221
The equation (4.44) with the boundary conditions (4.41) is
called the Falkner-Skan equation, after the authors of the fundamen-
tal paper on the subject (FALKNER and SKAN 1931). When related
equations appear in other parts of this book, the term Falkner-Skan
will sometimes be used to emphasize the community of similarity
formulations for various laminar shear flows.
Hartree.
HARTREE (1937) introduced another notation
β
for the single parameter of the problem,
β
=
2
m
1 +
m
or
m
=
β
2
−
β
,
(4.46)
so that equation (4.45) can also be written
(
1
(2
−
β
)
νu
∞
x
)
1
/
2
ψ
=
f
[
(
u
∞
(2
−
β
)
νx
)
1
/
2
y
]
=
f
(
η
)
.
(4.47)
Neither notation,
β
or
m
, has any particular mnemonic value, but
both are established by long usage. The change from
m
to
β
in
equation (4.44) yields the commonly accepted form of the Falkner-
Skan equation,
f
′′′
+
ff
′′
+
β
(1
−
f
′
f
′
) = 0
,
(4.48)
with a single parameter
β
(or
m
). Both notations can be useful
for identifying a particular member of the family of Falkner-Skan
flows, depending on the circumstances. These circumstances will
be developed in SECTION 4.4.6 below. For the present, note from
equations (4.46) that there are two exceptional cases. The first is
the case
m
=
−
1,
β
=
±∞
, and the second is the case
β
= 2,
m
=
±∞
. These cases are discussed in detail in SECTION 4.4.4 and
SECTION 4.4.6, respectively.
Displacement.
The notation and normalization adopted here
are the ones often found in the literature. For later use (see SEC-
TION 4.6.1), expressions for the surface friction and the two impor-
tant thickness scales of the boundary layer are
δ
∗
=
∞
∫
0
(
1
−
u
u
∞
)
d
y
=
(
2
νx
(1 +
m
)
u
∞
)
1
/
2
∞
∫
0
(1
−
f
′
)d
η
;
(4.49)
222
CHAPTER 4.
THE BOUNDARY LAYER
---:
----'::===:
=
: :==--:
=
:::=
::
=
y
:
:
:==--
.
:
::;£7~
x
Figure 4.1: The displacement effect in the free
stream for the Blasius boundary layer in rectangular
coordinates. The hatched area extends to
y
=
δ
∗
θ
=
∞
∫
0
u
u
∞
(
1
−
u
u
∞
)
d
y
=
(
2
νx
(1 +
m
)
u
∞
)
1
/
2
∞
∫
0
f
′
(1
−
f
′
)d
η
;
(4.50)
τ
w
=
u
(
∂u
∂y
)
w
=
(
(1 +
m
)
ρ
2
νu
3
∞
2
x
)
1
/
2
f
′′
(0)
.
(4.51)
With
u
∞
∼
x
m
, these imply
δ
∗
∼
x
(1
−
m
)
/
2
;
(4.52)
θ
∼
x
(1
−
m
)
/
2
;
(4.53)
τ
w
∼
x
(3
m
−
1)
/
2
.
(4.54)
The displacement thickness
δ
∗
is an effective thickness for the bound-
ary layer profile, as shown earlier in FIGURE 4.1. This thickness
defines the apparent shape of a body plus boundary layer. Rewrite
equation (4.49) as
(see SECTION 4.4.1)
C
=
∞
∫
0
(1
−
f
′
)d
η
=
δ
∗
(
(1 +
m
)
u
∞
2
νx
)
1
/
2
,
(4.55)
4.2. LAMINAR EQUILIBRIUM FLOW
223
and note also that
lim
η
→∞
η
∫
0
(1
−
f
′
)d
η
= lim
η
→∞
(
η
−
f
) = constant =
C
(4.56)
outside the boundary layer, where
η
is large enough so that the in-
tegrand is effectively zero. The constant
C
is defined by equation
(4.55), and the dimensionless relation
f
=
η
−
C
becomes in physical
coordinates
ψ
=
u
∞
(
y
−
δ
∗
)
.
(4.57)
This expression does not describe an irrotational flow, except in one
special case
m
= 1,
β
= 1, for which
δ
∗
does not depend on
x
. This
special case is taken up in SECTION 4.4.3.
4.2.4
Potential flow past a wedge
The physical significance of
m
is clear enough in equation (4.36). The
physical significance of
β
was pointed out by Falkner and Skan, who
recognized that their flows corresponded to plane potential flow past
a wedge of included angle
πβ
, as shown at the right in FIGURE 4.2.
The flow in the upper half plane is most easily found by a conformal
mapping, with the flow in the lower half plane obtained by reflection
in the
x
-axis. For uniform flow in the
ζ
-plane at the left, the complex
potential is
F(
ζ
) =
φ
+
iψ
=
U
0
ζ
(say)
.
(4.58)
Assume a mapping of the form
ζ
=
L
1
−
p
0
z
p
e
iσ
,
(4.59)
where
L
0
has the dimension of length, and the parameter
σ
allows a
possible rotation. Put
ζ
=
Re
i
Θ
and
z
=
re
iθ
to obtain
Θ =
pθ
+
σ .
(4.60)
The mapping in the upper half plane in the sketch requires
θ
=
πβ/
2
when Θ = 0, and
θ
=
π
when Θ =
π
. These two conditions imply
for
p
and
σ
p
=
2
2
−
β
, σ
=
−
πβ
2
−
β
.
(4.61)
224
CHAPTER 4.
THE BOUNDARY LAYER
y
x
Figure 4.2: Potential flow past a wedge of total
angle
πβ
.
The complex potential in the
z
-plane is
F(
z
) =
U
0
ζ
(
z
) =
U
0
L
−
β
(2
−
β
)
0
r
2
2
−
β
e
i
(
2
θ
−
πβ
2
−
β
)
=
φ
+
iψ .
(4.62)
The stream function,
ψ
=
U
0
L
−
β
(2
−
β
)
0
r
2
2
−
β
sin
(
2
θ
−
πβ
2
−
β
)
,
(4.63)
vanishes when
θ
=
π
and when
θ
=
πβ/
2, as desired. The radial
velocity is
u
r
=
1
r
∂ψ
∂θ
=
2
U
0
L
−
β
(2
−
β
)
0
2
−
β
r
β
2
−
β
cos
(
2
θ
−
πβ
2
−
β
)
.
(4.64)
On the wall, where
θ
=
πβ/
2, this is
u
r
=
2
U
0
L
−
β
2
−
β
0
2
−
β
r
β
2
−
β
.
(4.65)