of 32
Chapter 10
THE WALL JET
A plane jet flowing parallel to an adjacent wall—a wall jet—is a
configuration often encountered in ejector design, in film-cooling ap-
plications, and in boundary-layer control. The radial wall jet is a
variation that is important in problems of heat and mass transfer, as
in heating by a torch or drying by an impinging jet. The situations
of interest are almost always turbulent. The latter flows are sensitive
to residual effects of transition, and the approach to experimental
similarity is awkward because a simple displacement of the origin is
not compatible with the radial geometry.
In CHAPTER 4, the model for the turbulent boundary layer
is a continuously evolving turbulent wake, modified in a definite way
by the insertion of a wall along the plane of symmetry. The no-
slip condition reduces the velocity to zero at the wall and strongly
affects the flow near the original plane of symmetry. In particular, the
presence of the wall radically changes the normal or
v
′
fluctuations,
which are now reduced to zero at
y
= 0. The no-slip condition also
changes the other two fluctuations
u
′
and
w
′
in a more complicated
way, as discussed in various places in this monograph. As far as I am
aware, the corresponding model has never been considered seriously
for the wall jet. This model might be expected to lead to something
called the law of the jet and to further development of the concept
of equilibrium flow, and it will be addressed in SECTION 10.3.2.
527
528
CHAPTER 10.
THE WALL JET
Figure 10.1: Schematic connection between the laminar
plane free jet and wall jet.
The laminar wall jet can be visualized as a laminar plane jet
with a thin plate inserted on the plane of symmetry, as shown in
FIGURE 10.1. The main mathematical consequence of the loss of
symmetry for the wall-jet flow is a qualitative change in the sim-
ilarity argument, which now leads to an eigenvalue problem. It is
therefore more important than usual to practice technique with the
problem of laminar flow. Part of this technique is an application
in SECTION 10.1.6 of the Mangler transformation, which relates a
plane flow and a radial flow in the manner shown earlier for the free
jet. Many of the other operations carried out in this chapter have al-
ready been encountered in CHAPTER 9 on the free jet, where they
are described in somewhat greater detail and supported by more ex-
tensive arguments.
10.1. LAMINAR PLANE WALL JET
529
10.1 Laminar plane wall jet into fluid at rest
10.1.1
The eigenvalue problem
Priority in solving the problem of the laminar plane wall jet with
similarity is generally assigned to GLAUERT (1956), although an
essentially complete account was published earlier by TETERVIN
(1948). The problem is more subtle than the problem of the plane
free jet, and the subtleties were fully appreciated by Glauert. The mo-
mentum equation in the boundary-layer approximation is the same
as for the free jet;
ρ
(
∂uu
∂x
+
∂uv
∂y
)
=
μ
∂
2
u
∂y
2
=
∂τ
∂y
.
(10.1)
The boundary conditions are suitably chosen from
ψ
=
u
=
v
= 0 at
y
= 0
, u
=
τ
= 0 at
y
=
∞
.
(10.2)
The momentum-integral equation is easily written down by
inspection of equation (10.1);
ρ
d
d
x
∞
∫
0
u u
d
y
=
d
J
d
x
=
−
τ
w
,
(10.3)
where
τ
w
=
μ
(
∂u/∂y
)
w
and
J
is the momentum integral previously
defined for the plane free jet by equation (9.8). The fact that
J
is
no longer a constant, as it was for the free jet, prevents the intro-
duction at the outset of intrinsic scales for mass, length, and time.
The friction at the wall continuously removes momentum from the
wall jet, beginning at the origin of the flow at
x
= 0, at a rate that
is slow but significant. It will be found for the case of laminar flow
that similarity requires the terms in equation (10.3) to behave like
x
−
5
/
4
near the origin (see equations (10.60) and (10.61) below). Thus
the singularity in
τ
w
at
x
= 0 is not integrable. Moreover, since the
integral in equation (10.3) behaves like
x
−
1
/
4
near
x
= 0, the initial
momentum flux
J
in the similarity formulation is infinite. I empha-
size these points because some experimenters have assumed that the
530
CHAPTER 10.
THE WALL JET
momentum flux
J
measured at the jet exit or elsewhere in a laminar
laboratory flow has some important role to play in similarity formu-
lations of their data. Similar problems with turbulent flow are taken
up in SECTION 10.3.1.
One primitive but popular version of dimensional analysis is to
assume a power-law behavior and to determine the exponents for two
local scales
U
(
x
)
∼
x
p
and
L
(
x
)
∼
x
q
by substitution of a suitable
ansatz in the momentum equation (10.1) and its integral (10.3). This
approach is demonstrated for several different flows by BIRKHOFF
and ZARANTONELLO (1957), for example. These authors did not
anticipate the problem of the wall jet, but did comment on an eigen-
value problem for a different flow, the momentumless wake. The new
feature in the case of the momentumless wake is that the global
constant defined by the momentum integral, the drag, vanishes iden-
tically. The new feature in the case of the wall jet is that the two
equations (10.1) and (10.3) have essentially the same dimensional
structure. In either case, only one condition can be found for the two
exponents
p
and
q
unless the problem is attacked at a deeper level.
Glauert.
To arrive at a dimensionless ansatz, Glauert assumed
power-law behavior. He took the free jet as a model and a point of
departure. The analysis that follows is faithful in spirit to Glauert’s
presentation, but the notation and certain details have been changed
to suit the style of this monograph. I have also chosen to begin with
plane flow rather than radial flow. Glauert postulated the existence
of a local velocity scale
U
and a corresponding local length scale
ν/U
,
and assumed a solution of the form
ψ/ν
(
Ux/ν
)
a
=
f
[
(
Uy/ν
)
(
Ux/ν
)
b
]
=
f
(
η
)
,
(10.4)
where
u
=
∂ψ
∂y
, v
=
−
∂ψ
∂x
,
(10.5)
as usual. Substitution in equation (10.1) yields
(
a
−
b
)
f
′
f
′
−
aff
′′
= (
Ux/ν
)
1
−
a
−
b
f
′′′
,
(10.6)
where primes indicate differentiation with respect to
η
. If
f
is re-
quired to depend only on
η
and not separately on
x
, this equation
10.1. LAMINAR PLANE WALL JET
531
supplies one relation between the exponents
a
and
b
;
a
+
b
= 1
,
(10.7)
together with an ordinary differential equation for
f
,
f
′′′
+ (1
−
b
)
ff
′′
−
(1
−
2
b
)
f
′
f
′
= 0
,
(10.8)
whose boundary conditions, from equations (10.2), are
f
(0) =
f
′
(0) =
f
′
(
∞
) = 0
.
(10.9)
Glauert’s first major contribution was to establish that there
exists at least one non-trivial similarity solution of equation (10.8),
satisfying the null boundary conditions (10.9), provided that the ex-
ponent
b
has the eigenvalue 3/4. The analysis begins with an inte-
gration whose purpose is to examine the shearing stress
f
′′
and to
deal with the absence of symmetry. Replace
ff
′′
by (
ff
′
)
′
−
f
′
f
′
and
integrate equation (10.8) formally from some arbitrary positive value
of
η
to
η
=
∞
to obtain
f
′′
+ (1
−
b
)
ff
′
+ (2
−
3
b
)
g
= 0
,
(10.10)
where
g
(
η
) =
∞
∫
η
f
′
f
′
d
η .
(10.11)
The range of integration is evidently chosen to exploit the fact that
f
′
and
f
′′
vanish at infinity for both the free jet and the wall jet. In
particular,
f
′′
(0) = (3
b
−
2)
g
(0)
,
(10.12)
where
g
(0) =
∞
∫
0
f
′
f
′
d
η .
(10.13)
A brief digression disposes of the symmetric problem (the free
jet). The boundary conditions in the plane of symmetry are then
f
(0) = 0 and
f
′′
(0) = 0, corresponding to
ψ
(
x,
0) = 0 and
τ
(
x,
0) =
532
CHAPTER 10.
THE WALL JET
0, with
f
′
(0)
∼
u
c
(
x
) left unspecified. Since
g
(0) is a positive con-
stant, it follows from equation (10.12) that the boundary condition
f
′′
(0) = 0 can be satisfied only if
b
= 2
/
3,
a
= 1
/
3, in agreement
with the result obtained more directly in SECTION 9.1.2 above.
Now return to the unsymmetric problem, the wall jet. The
boundary conditions at the wall are
f
(0) = 0 and
f
′
(0) = 0, cor-
responding to
ψ
(
x,
0) = 0 and
u
(
x,
0) = 0, with
f
′′
(0)
∼
τ
w
(
x
)
left unspecified. Nothing can be learned from equation (10.12), and
something more is required. Glauert eliminated
f
′′
by multiplying
equation (10.10) by
f
′
and integrating through the thickness of the
wall jet. After some integration by parts and use of the identity
g
′
=
−
f
′
f
′
and the boundary condition
g
(
∞
) = 0, the result is
(3
−
4
b
)
∞
∫
0
f
′
g
d
η
= 0
.
(10.14)
The integral in equation (10.14) is a positive constant, provided that
the velocity
f
′
is non-negative everywhere, and the equation can
therefore be satisfied only for the exponents
b
=
3
4
, a
=
1
4
.
(10.15)
This value for
b
requires, from equation (10.12),
f
′′
(0) =
1
4
g
(0)
.
(10.16)
It reduces the differential equation (10.8) to
4
f
′′′
+
ff
′′
+ 2
f
′
f
′
= 0
(10.17)
and also provides the necessary invariant, which can have different
forms;
∞
∫
0
f
′
g
d
η
=
∞
∫
0
f
′
∞
∫
η
f
′
f
′
d
η
d
η
=
−
∞
∫
0
fg
′
d
η
=
=
∞
∫
0
ff
′
f
′
d
η
= constant
.
(10.18)
10.1. LAMINAR PLANE WALL JET
533
Of these, the two dominant forms in physical variables are the second
and the fourth;
ρ
∞
∫
0
u
∞
∫
y
uu
d
y
d
y
=
ρ
∞
∫
0
ψuu
d
y
=
F
= constant
.
(10.19)
Like Glauert, I have some difficulty in assigning a physical meaning to
the quantity
F
. His best effort produced the phrase “flux of exterior
momentum flux.”
Having established the structure of his problem, Glauert re-
peated his derivation from the beginning in physical variables for
readers who do not object to a strong element of
deus ex machina
.
Note, as did Glauert, that this second derivation does not require
the assumption of similarity or of power-law behavior. First, write
an incomplete integral corresponding to equation (10.3) in the form
∂
∂x
ρ
∞
∫
y
uu
d
y
−
ρuv
+
τ
= 0
.
(10.20)
Denote the integral by
W
, say;
W
=
ρ
∞
∫
y
uu
d
y ,
(10.21)
and observe that good things happen if the equation
∂W
∂x
−
ρuv
+
τ
= 0
(10.22)
is multiplied by the streamwise velocity
u
and if it is noticed that
−
ρuu
=
∂W/∂y
from equation (10.21). Thus
u
∂W
∂x
+
v
∂W
∂y
+
τu
= 0
.
(10.23)
Add to this the continuity equation multiplied by
W
to obtain
∂uW
∂x
+
∂vW
∂y
+
τu
= 0
.
(10.24)
534
CHAPTER 10.
THE WALL JET
Finally, integrate over the thickness of the layer and use the boundary
conditions
v
(0) = 0,
W
(
∞
) = 0. The result is
d
d
x
∞
∫
0
uW
d
y
+
∞
∫
0
τu
d
y
= 0
.
(10.25)
A last crucial step can be carried out provided that the flow is lam-
inar, with
τ
=
μ∂u/∂y
. Then the second term in equation (10.25)
drops out;
∞
∫
0
τu
d
y
=
μ
∞
∫
0
∂u
2
/
2
∂y
d
y
= 0
,
(10.26)
since
u
is zero at both limits. For laminar flow, this procedure has
reproduced the conserved quantity (10.19);
∞
∫
0
uW
d
y
=
ρ
∞
∫
0
u
∞
∫
y
uu
d
y
d
y
=
F
= constant
.
(10.27)
For turbulent flow, neither equation (10.26) nor equation (10.27) is
valid.
(Interpret this process in terms of work done on fluid? Minimize
the integral of
τu
?
W
is the momentum flux outboard of a particular
point in the flow. Equation (10.23), written as
DW
Dt
+
τu
= 0
,
(10.28)
suggests that the rate of change of this quantity following a stream-
line is given by the rate that work is done by the shearing stress
(this needs work). Look at the difference between
F
and the con-
served quantity
J
for the free jet. Interpret as divergence. Look at
energy. Comment on vorticity as variable, with no symmetry and
zero integral. See the Rayleigh problem in the introduction.)
Intrinsic scales.
Given the existence of the integral invariant
F
, it is now a simple matter to work out intrinsic scales for the
laminar wall jet. The dimensional statements
[
F
] =
ML
2
T
3
,
[
ρ
] =
M
L
3
,
[
μ
] =
M
LT
(10.29)
10.1. LAMINAR PLANE WALL JET
535
imply, in their alternative role as definitions,
M
=
ρ
4
ν
9
F
3
,
L
=
ρν
3
F
,
T
=
ρ
2
ν
5
F
2
,
(10.30)
with
U
=
L
T
=
F
ρν
2
(10.31)
and, as for the free jet,
UL
ν
= 1
.
(10.32)
The relation (10.31) provides
a posteriori
justification for Glauert’s
original ansatz (10.4), because
U
is now precisely defined. In fact,
substitution for
U
yields immediately
(
ρ
Fνx
)
1
/
4
ψ
=
f
[
(
F
ρν
3
x
3
)
1
/
4
y
]
.
(10.33)
Another brief calculation shows that this expression is equivalent to
ψ
UL
3
/
4
x
1
/
4
=
f
(
y
L
1
/
4
x
3
/
4
)
.
(10.34)
Tetervin.
Tetervin’s earlier approach to the same problem
was handicapped by a dreadful notation and by failure to introduce
a stream function until the last possible moment. What follows is a
radical paraphrase of his argument. In effect, he assumed similarity
in terms of local scales for velocity
U
and layer thickness
L
;
ψ
UL
=
f
(
y
L
)
=
f
(
η
)
,
(10.35)
where
U
(
x
) and
L
(
x
) have to be determined. Substitution in the
momentum equation (10.1) gives, just as in the case of the laminar
free jet (see SECTION 9.1.2),
f
′′′
+
L
ν
d
UL
d
x
ff
′′
−
L
2
ν
d
U
d
x
f
′
f
′
= 0
.
(10.36)
536
CHAPTER 10.
THE WALL JET
Substitution in the momentum-integral equation (10.3) gives
L
νU
d
U
2
L
d
x
∞
∫
0
f
′
f
′
d
η
=
−
f
′′
(0)
.
(10.37)
Tetervin eventually normalized the integral to unity;
∞
∫
0
f
′
f
′
d
η
=
g
(0) = 1
,
(10.38)
so that
L
νU
d
U
2
L
d
x
=
−
f
′′
(0)
.
(10.39)
Only two of the three constant coefficients involving
U
and
L
in
equations (10.36) and (10.39) are independent, and these two are
not sufficient to determine
U
(
x
) and
L
(
x
) explicitly. Neither is the
device of the moving observer useful for resolving the question of
exponents. Tetervin, like Glauert, found another way.
When
ν
is eliminated between equations (10.36) and (10.39),
and the variables depending on
x
and on
η
are separated, the result
is
L
d
U/
d
x
U
d
L/
d
x
=
−
f
′′′
+
f
′′
(0)
ff
′′
2
f
′′′
−
f
′′
(0)(
ff
′′
−
f
′
f
′
)
=
−
k ,
(10.40)
where
k
must be a positive constant because
x
and
η
are arbitrary
and d
U/
d
x <
0, d
L/
d
x >
0. This expression strongly suggests that
power laws are appropriate for
U
(
x
) and
L
(
x
), and guarantees in any
case that
UL
k
= constant
.
(10.41)
Tetervin noted in passing that the boundary condition
f
′′
(0) = 0 in
equation (10.40) implies
k
= 1
/
2 and thus
U
2
L
= constant, so that
the case of the plane free jet is accounted for. The present interest
is in the case of lost symmetry with its eigenvalue
k
. This eigenvalue
appears along with
f
′′
(0) in the differential equation obtained from
the second part of equation (10.40);
f
′′′
+
(
1
−
k
2
k
−
1
)
f
′′
(0)
ff
′′
+
(
k
2
k
−
1
)
f
′′
(0)
f
′
f
′
= 0
.
(10.42)
10.1. LAMINAR PLANE WALL JET
537
At this point, Tetervin’s argument becomes opaque. The essence
of his procedure, suitably revised to leave open the question of nor-
malization, is to multiply equation (10.42) by
f
and integrate over
the thickness of the wall jet. After the usual integration by parts and
use of the boundary conditions, the result is
(
3
k
−
2
2
k
−
1
)
f
′′
(0)
∞
∫
0
f f
′
f
′
d
η
= 0
.
(10.43)
Both the integral and the factor
f
′′
(0) are necessarily positive, so
that the desired invariant emerges from this equation together with
the eigenvalue
k
=
2
3
.
(10.44)
Equation (10.42) becomes
f
′′′
f
′′
(0)
+
ff
′′
+ 2
f
′
f
′
= 0
.
(10.45)
Equation (10.41) becomes
U
3
L
2
= constant
,
(10.46)
and it follows from this result and equation (10.39) that
U
∼
x
−
1
/
2
,
L
∼
x
3
/
4
.
(10.47)
Tetervin integrated equation (10.45) numerically for the par-
ticular initial conditions
f
(0) =
f
′
(0) = 0 and
f
′′
(0) = 1
/
4. His
conversion of a two-point boundary-value problem to an initial-value
problem was successful, although he may not have been aware of the
reason, which involves a property first pointed out for the Blasius
equation by T
̈
OPFER (1912). The argument is easily extended by
inspection to equation (10.45), which also has no pressure-gradient
term. If
f
(
η
) is a solution, so is
φ
(
η
) =
αf
(
αη
), where
α
is any
constant. It follows that
f
′′
(0) can be chosen arbitrarily, with
φ
(
∞
)
adjusted later to any desired value by a proper choice of
α
(see SEC-
TION X).
538
CHAPTER 10.
THE WALL JET
10.1.2
Similarity
The affine transformation.
Discovery of the integral invariant
F
allows the problem of the laminar plane wall jet to be treated by
the method of the affine transformation. Let a stream function
ψ
be introduced in the usual way to satisfy the continuity equation.
Rewrite equation (10.1) as
ρ
(
∂ψ
∂y
∂
2
ψ
∂x∂y
−
∂ψ
∂x
∂
2
ψ
∂y
2
)
=
μ
∂
3
ψ
∂y
3
(10.48)
and apply the affine transformation
x
=
a
̂
x ,
y
=
b
̂
y ,
ψ
=
c
̂
ψ ,
(10.49)
ρ
=
d
̂
ρ ,
μ
=
e
̂
μ ,
F
=
f
̂
F .
This is the same group as equations (9.28) for the plane free jet,
except that
F
replaces
J
. The result is
c
2
d
ab
2
̂
ρ
(
∂
̂
ψ
∂
̂
y
∂
2
̂
ψ
∂
̂
x∂
̂
y
−
∂
̂
ψ
∂
̂
x
∂
2
̂
ψ
∂
̂
y
2
)
=
ce
b
3
̂
μ
∂
3
̂
ψ
∂
̂
y
3
.
(10.50)
Invariance of equation (10.48) thus requires
bcd
ae
= 1
,
(10.51)
just as in the case of the plane free jet. Transformation of equation
(10.19),
ρ
∞
∫
0
ψ
∂ψ
∂y
∂ψ
∂y
d
y
=
F ,
(10.52)
10.1. LAMINAR PLANE WALL JET
539
yields
c
3
d
b
̂
ρ
∞
∫
0
̂
ψ
∂
̂
ψ
∂
̂
y
∂
̂
ψ
∂
̂
y
d
̂
y
=
f
̂
F ,
(10.53)
and requires for invariance
c
3
d
bf
= 1
.
(10.54)
As usual, I take the primary variables to be
ψ
and
y
. When equations
(10.51) and (10.54) are revised to isolate for
c
and
b
, the result is
c
4
d
2
aef
= 1
,
b
4
d
2
f
a
3
e
3
= 1
.
(10.55)
Hence the proper ansatz, including constants
A
and
B
for later nor-
malization, is again equation (10.33),
A
(
ρ
Fνx
)
1
/
4
ψ
=
f
[
B
(
F
ρν
3
x
3
)
1
/
4
y
]
=
f
(
η
)
.
(10.56)
Substitution of this ansatz in the momentum equation (10.48) yields
4
ABf
′′′
+
ff
′′
+ 2
f
′
f
′
= 0
,
(10.57)
with boundary conditions
f
(0) =
f
′
(0) = 0
, f
′
(
∞
) = 0
,
(10.58)
corresponding to
ψ
=
u
= 0 at
y
= 0 and
u
= 0 at
y
=
∞
. If
AB
= 1, equation (10.57) is identical with my version of Glauert’s
result, equation (10.17). Substitution of equation (10.56) in equation
(10.52) for
F
gives
∞
∫
0
ff
′
f
′
d
η
=
A
3
B
.
(10.59)
The singular behavior of the flow at the origin, mentioned earlier, is
demonstrated by the relations
J
=
ρ
∞
∫
0
uu
d
y
=
B
A
2
(
F
3
ρ
νx
)
1
/
4
∞
∫
0
f
′
f
′
d
η
(10.60)
540
CHAPTER 10.
THE WALL JET
and
τ
w
ρ
=
B
2
A
(
F
3
ρ
3
νx
5
)
1
/
4
f
′′
(0)
.
(10.61)
10.1.3
The boundary-layer solution
Glauert’s second major contribution was to obtain the eigenfunc-
tion
f
(
η
) in closed form. First, multiply equation (10.57) by
f
and
integrate to obtain
4
AB
(
ff
′′
−
f
′
f
′
2
)
+
fff
′
= 0
,
(10.62)
where the constant of integration vanishes by virtue of the first two
boundary conditions (10.58). Multiply this result by
f
−
3
/
2
and inte-
grate again, to obtain
4
AB
f
′
f
1
/
2
+
2
3
(
f
3
/
2
−
C
3
/
2
)
= 0
,
(10.63)
where
C >
0 is a constant of integration. The boundary condition
(10.58) at infinity requires
C
=
f
(
∞
)
.
(10.64)
Finally, integrate equation (10.63) with the aid of the change of vari-
able
f
=
Ch
2
=
CH
(10.65)
and the method of partial fractions. An intermediate result is
C
4
AB
d
η
=
d
h
(1
−
h
)
+
2d
h
(1 +
h
+
h
2
)
+
h
d
h
(1 +
h
+
h
2
)
.
(10.66)
The final result in terms of
h
, after use of the boundary condition
f
(0) =
h
(0) = 0 to evaluate the constant of integration, can be
written
C
2
AB
η
= ln
(1
−
h
3
)
(1
−
h
)
3
+ 2
√
3 tan
−
1
(
√
3
h
2 +
h
)
.
(10.67)
10.1. LAMINAR PLANE WALL JET
541
Equations (10.65) and (10.67) are a parametric system for
f
(
η
), with
h
as parametric variable. Note that
h
depends not directly on
η
but
on
Cη/
2
AB
.
Pause here to look at experimental data for the laminar
profile; see
BAJURA and SZEWCZYK (1970)
BAJURA and CATALANO (1975)
TSUJI et al. (1977)
TSUJI and MORIKAWA (1977)
HORNE and KARAMCHETI (1979)
SCIBILIA and DUROX (1980)
PAIGE (1988)
ZHOU et al. (1992)
It remains to consider the streamlines of the boundary-layer
flow in compact outer variables (
x, y
) having equal scales. A unique
representation of the flow can again be found, without regard for the
values of the three constants
A
,
B
, and
C
. Rewrite equation (10.56)
in terms of
H
as
A
C
(
ρ
Fνx
)
1
/
4
ψ
=
f
(
η
)
C
=
H
(
C
2
AB
η
)
=
H
[
C
2
A
(
F
ρν
3
x
3
)
1
/
4
y
]
.
(10.68)
In the combinations containing
ψ
and
y
, use the second of equations
(10.30) to eliminate the quantity
F
in favor of
L
=
ρν
3
/F
. Thus
write
A
C
(
L
x
)
1
/
4
ψ
ν
=
H
[
C
2
A
(
y
4
L
x
3
)
1
/
4
]
.
(10.69)
Compact outer variables define themselves immediately as
Ψ =
ψ
2
ν
, X
=
(
C
2
A
)
4
x
L
, Y
=
(
C
2
A
)
4
y
L
,
(10.70)
and equation (10.68) takes the form
Ψ =
X
1
/
4
H
(
Y
X
3
/
4
)
.
(10.71)
542
CHAPTER 10.
THE WALL JET
Note that
H
=
h
2
=
f
(
η
)
/C
, but that the argument of
H
is the
quantity
Cη/
2
AB
on the left in equation (10.67). The example of
the free jet suggests that a useful relation involving the constants
A
and
C
should emerge when the integral invariant (10.59) is evalu-
ated for Glauert’s closed-form solution. Use equation (10.65) and its
derivative, together with equation (10.63), to replace the variable
f
by
h
. The result is
∞
∫
0
ff
′
f
′
d
η
=
C
4
3
AB
1
∫
0
h
4
(1
−
h
3
)d
h
=
C
4
40
AB
=
A
3
B
,
(10.72)
from which
(
C
2
A
)
4
=
5
2
.
(10.73)
The variables in equation (10.71) can therefore be written
Ψ =
ψ
2
ν
, X
=
5
x
2
L
, Y
=
5
y
2
L
.
(10.74)
Streamlines for the boundary-layer approximation (10.71) are
shown in FIGURE 10.2
1
for the case of a laminar wall jet flowing
from the origin along a plane wall that extends to infinity in the
positive x-direction. Rather than calculate Ψ on a large rectangular
array (
X, Y
) and find level curves on which Ψ is constant, it is
simpler here to define each streamline separately. The algorithm is:
fix Ψ, vary
X
. Calculate
H
= Ψ
/X
1
/
4
=
h
2
. Calculate
h
. Calculate
Cη/
2
AB
=
Y/X
3
/
4
from equation (10.67). Calculate
Y
.
A local Reynolds number can be expressed in compact outer
variables by beginning with dimensionless versions of equations (10.47);
U
(
x
) =
UL
1
/
2
x
−
1
/
2
, L
(
x
) =
L
1
/
4
x
3
/
4
.
(10.75)
Use of equations (10.30) and (10.31) leads to
U
=
(
F
2
ρ
2
ν
2
x
2
)
1
/
4
, L
=
(
ρν
3
x
3
F
)
1
/
4
,
(10.76)
1
A longer handwritten version of the caption for this figure in the 1996 ms.
reads ”Streamlines Ψ =
ψ/
2
ν
=
constant
of the boundary-layer approximation
for the laminar plane wall jet according to equation (10.71). The range of Ψ is
0(1)10 (
check
).
10.1. LAMINAR PLANE WALL JET
543
Figure 10.2: Streamlines of the boundary-layer model
for the laminar plane wall jet according to equation
(10.71). The range of........
and thus to
Re
(
x
) =
UL
ν
=
(
Fx
ρν
3
)
1
/
4
=
(
x
L
)
1
/
4
=
(
2
5
X
)
1
/
4
.
(10.77)
10.1.4
Normalization
The three constants
A
,
B
, and
C
for the plane wall jet can be as-
signed sensible values by operations that run in parallel with similar
operations for the plane free jet in SECTION 9.1.4. The condition
4
AB
= 1
(10.78)
establishes the standard operator
f
′′′
+
ff
′′
in equation (10.57). A
second and mandatory condition, just derived, is
C
A
= (40)
1
/
4
.
(10.79)
(See end of Part A of this chapter. This is too messy. No tidy
normalization seems to be in view. Sort through this material to find
544
CHAPTER 10.
THE WALL JET
something simple, elegant, and redundant. Consider equation (10.67)
for
h
near 1 and for
h
large and negative. Note that for
h
= 1
the
angle is 30 degrees. Should square root be
±
? Need to match to plane
jet at infinity. Try osculating parabola.)
The third condition determining the constants
A
,
B
, and
C
requires definition of a length or velocity scale. The simplest choice,
suggested by the example of the free jet (see SECTION 9.1.4), is to
set the argument of
h
; i.e., the left-hand side of equation (10.67),
equal to
η
itself, so that
C/
2
AB
= 1 or
C
= 1
/
2.
The maximum velocity
U
=
u
m
(
m
for maximum) is easily
worked out;
Re
=
UL
ν
=
1
ν
∞
∫
0
u
d
y
=
ψ
(
x,
∞
)
ν
=
1
A
(
Fx
ρν
3
)
1
/
4
f
(
∞
) = 2
X
1
/
4
.
(10.80)
This Reynolds number is small compared with the corresponding
value
Re
= 12
X
1
/
3
for the plane free jet. In FIGURE 10.2, which
extends
(ten)
times farther than FIGURE 9.4, the Reynolds number
at the right boundary is
(six)
times smaller. Comparison of equa-
tion (10.80) with equation (9.57) for the free jet suggests that the
two measures just cited are associated with the exponent and the
coefficient, respectively.
The maximum streamwise velocity
η
m
occurs when
∂u/∂y
∼
f
′′
= 0. With this condition, equations (10.62) and (10.63) can be
restated in terms of
h
and
h
′
and solved algebraically to produce
(see Tetervin)
f
(
η
m
) =
(
1
4
)
2
/
3
C
;
(10.81)
f
′
(
η
m
) =
(
1
4
)
1
/
3
C
2
8
AB
.
(10.82)
Thus if
f
′
(
η
m
) = 1, then
C
2
/
4
AB
or
C
= (32)
1
/
6
. Conversely, if
C
= 1
/
2, then
f
′
(
η
m
) = (2)
1
/
3
/
16.
Several other choices suggest themselves, chief among them the
integral scale
L
for the profile. Define this integral scale in terms of
10.1. LAMINAR PLANE WALL JET
545
the maximum velocity
u
m
(
m
for maximum) by
u
m
L
=
∞
∫
0
u
d
y
=
ψ
(
x,
∞
)
.
(10.83)
After use of the ansatz (10.56) and the second of conditions (10.82),
this turns into
̃
η
=
B
(
F
ρν
3
x
3
)
1
/
4
L
=
∞
∫
0
f
′
(
η
)
f
′
(
η
m
)
d
η
= (4)
1
/
3
8
AB
C
.
(10.84)
Hence if
̃
η
= 1,
C/
4
AB
= 2(4)
1
/
3
.
A similar calculation, with
h
replacing
f
, leads from the defi-
nition (10.13) to
g
(0) =
∞
∫
0
f
′
f
′
d
η
=
1
18
C
3
AB
.
(10.85)
Finally, integration of the primary differential equation (10.57) be-
tween the limits zero and infinity leads to
f
′′
(0) =
g
(0)
4
AB
=
1
72
C
3
A
2
B
2
.
(10.86)
Either of these relationships, as well as
f
(
∞
) =
∞
∫
0
f
′
d
η
=
C ,
(10.87)
could provide a third condition if its right-hand side is arbitrarily
set equal to unity, say. The results, respectively, are
C
3
= 9
/
2 if
g
(0) = 1,
C
3
= 9
/
2 if
f
′′
(0) = 1, and
C
= 1 if
f
(
∞
) = 1, where 4
AB
is read as unity.
The inflection point in the profile at
η
=
η
i
, say, is found by
putting
f
′′′
= 0. This point also marks the maximum velocity along
a streamline, since
Du/Dt
=
ν∂
2
u/∂y
2
= 0 for laminar flow. Then
546
CHAPTER 10.
THE WALL JET
equation (10.57) becomes
ff
′′
+ 2
f
′
f
′
= 0, and other derivatives
at the inflection point can be calculated from this truncated form
together with equations (10.62) and (10.63). The results are
f
(
η
i
) =
(
5
8
)
2
/
3
C
;
(10.88)
f
′
(
η
i
) =
1
16
(
5
8
)
1
/
3
C
2
AB
;
(10.89)
f
′′
(
η
i
) =
−
1
128
C
3
A
2
B
2
.
(10.90)
Hence if
f
′
(
η
i
) = 1, then
C
2
/
4
AB
= 8
/
5
1
/
3
.
The vorticity thickness
η
ζ
is defined graphically in FIGURE X
and is defined algebraically by
η
ζ
=
−
f
′
(
η
m
)
f
′′
(
η
i
)
= 16
(
1
4
)
1
/
3
AB
C
,
(10.91)
where
f
′
(
η
m
) is given by equation (10.82). Hence if
η
ζ
= 1,
C/
4
AB
=
(4)
2
/
3
.
The normalizations used by Glauert and Tetervin can be in-
ferred by using my notation in the ansatz (10.56) and the relations
that follow. Glauert put
C
= 1 and also 4
AB
= 1, according to his
equation (4.1). It follows that
A
= (1
/
40)
1
/
4
,
B
= (5
/
32)
1
/
4
; and
these are the numbers that appear in Glauert’s equations (4.9) for
the plane case. Tetervin’s final normalization can be deduced from
his equation (20), in which his
G
(
ξ
) is the same as my
f
(
η
). He
put 4
AB
= 4
/
3 and
f
′′
(0) = 1
/
4, and these together with equation
(10.86) above imply
C
=
f
(
∞
) = (2)
1
/
3
= 1
.
259921. His numerical
result for large
η
, namely
f
(14
.
95) = 1
.
259916, is evidence that his
integration was carried out with remarkable accuracy.
(Give normalized
f
,
f
′
, etc.)
10.1. LAMINAR PLANE WALL JET
547
y
77T777777'77T,'777?'7777777T,TT
~
~~~~ge~~e
,
//!
(
.
··-'··
--~--
.-._Physic~ne
FIGURE
19.7.
Mapping
of
the
outer
entrained
flow
for
the
laminar
plane
wall
jet.
..
Figure 10.3: Mapping of the outer entrained flow for
the laminar plane wall jet.
10.1.5
Entrainment and composite flow
The outer or entrained flow associated with the boundary-layer solu-
tion in FIGURE 10.2 can again be obtained by the method of confor-
mal mapping, as indicated in FIGURE 10.3. The complex potential
for uniform flow in the
ζ
-plane is
F
(
ζ
) =
φ
+
iψ
=
U
0
ζ .
(10.92)
The mapping can be assumed to be of the form
ζ
=
L
3
/
4
0
z
1
/
4
e
iα
,
(10.93)
where
ζ
=
Re
iω
and
z
=
re
iθ
, with
U
0
,
L
0
, and
α
to be determined.
Angles are related by
ω
=
θ
4
+
α .
(10.94)
Consequently, if
θ
=
π
when
ω
=
π
on
OA
,
α
=
3
π
4
.
(10.95)
Since now
θ
= 0 when
ω
= 3
π/
4 on
OB
, streamlines of the outer
flow will intersect the
x
-axis at an angle of 45 degrees. The complex
548
CHAPTER 10.
THE WALL JET
potential in the
z
-plane becomes
F
(
z
) =
φ
o
+
iψ
o
=
U
0
ζ
(
z
) =
U
0
L
3
/
4
0
r
1
/
4
e
i
(
θ
+3
π
4
)
,
(10.96)
and the outer stream function is
ψ
o
(
r, θ
) =
U
0
L
3
/
4
0
r
1
/
4
sin
(
θ
+ 3
π
4
)
.
(10.97)
At the wall, where
θ
= 0 and
r
=
x
,
ψ
o
(
x,
0) =
1
√
2
U
0
L
3
/
4
0
x
1
/
4
.
(10.98)
This outer flow on the positive
x
-axis in the physical plane is to
be matched to the inner stream function at infinity, from equation
(10.56) with
f
(
∞
) =
C
;
ψ
i
(
x,
∞
) =
C
A
(
Fνx
ρ
)
1
/
4
.
(10.99)
Matching therefore requires
1
√
2
U
0
L
3
/
4
0
=
C
A
(
Fν
ρ
)
1
/
4
=
C
A
UL
3
/
4
,
(10.100)
where the last equality makes use of equations (10.30). Finally, there-
fore,
ψ
o
(
r, θ
) =
√
2
C
A
(
Fνr
ρ
)
1
/
4
sin
(
θ
+ 3
π
4
)
.
(10.101)
(Work out pressure here.)
The composite stream function
ψ
c
is the sum of the inner com-
ponent (10.56) and the outer component (10.101) with the common
part (10.99) subtracted,
ψ
c
=
C
A
(
Fνr
ρ
)
1
/
4
{
(
x
r
)
1
/
4
[
f
(
η
)
C
−
1
]
+
√
2 sin
(
θ
+ 3
π
4
)}
.
(10.102)
10.1. LAMINAR PLANE WALL JET
549
Figure 10.4: Streamlines
Ψ
= constant of the composite
model for the laminar plane wall jet into a stagnant
fluid according to equation (10.103).
In terms of the reduced similarity variables
X
and
Y
defined by
equations (10.70), with
R
= (
X
2
+
Y
2
)
1
/
2
and tan Θ =
Y/X
, this is
Ψ
c
=
R
1
/
4
{
I
(Θ)
[
H
(
Y
X
3
/
4
)
−
1
]
+
√
2 sin
(
Θ + 3
π
4
)}
,
(10.103)
where
I
(Θ) = (cos Θ)
1
/
4
for
x >
0
,
= 0
for
x <
0
.
(10.104)
Streamlines for the composite flow are shown in FIGURE 10.4. The
calculation here requires an iteration for
h
(
η
) and a contour sub-
routine. The figure can be viewed as a conceptual model for flow
near the nozzle of a plane wall ejector with small induced flow (see
SECTION X).
(Want
S
,
U
,
V
,
T
; see free jet. Plot corrected profile, etc.)