Chapter 9
THE PLANE JET
Consider a plane jet issuing into a stagnant fluid from a slit in a plane
wall, as shown schematically in FIGURE 9.1. This configuration is
often used in experimental work for the sake of its standard geometry.
The most important quantity in any description of such a jet flow,
whether laminar or turbulent, is
J
, the initial flux of momentum per
unit span. As the flow develops in the downstream direction, this ini-
tial momentum is conserved as it is gradually transferred from the jet
fluid to the ambient fluid by shearing stresses. The rate of momentum
transfer will depend not only on the nature of these stresses, but also
on the relative densities of the two fluids, if these are different, and
on the effect of real rather than ideal initial and boundary conditions.
In any case, the total rate of fluid flow in the jet will increase contin-
uously in the downstream direction as external fluid is entrained. It
is this entrainment process that dominates most practical problems.
For laminar flow, the solution of the boundary-layer problem is
known in closed form, and the absence of a dimensionless parameter
implies that all laminar plane jets are equivalent. So are all turbulent
jets, for the same reason. However, for turbulent flow the growth rate
is relatively rapid, and a boundary-layer approximation may not be
appropriate.
A simple generalization of the classical flow is obtained if the
wall is absent and the jet issues from a line momentum source into
433
434
CHAPTER 9.
THE PLANE JET
Figure 9.1: A schematic representation of the laminar
plane jet out of a vertical wall.
J
is the rate of
momentum flux per unit span.
a parallel moving stream. A small enough velocity difference, with
or without a pressure gradient, may allow the jet problem to be
linearized, thus providing a link with the problem of the plane wake.
Even if the velocity difference is not small near the origin, it is likely
to become small farther downstream, unless the pressure gradient is
specially tailored to maintain a state of overall similarity.
In some practical applications, the jet may issue into a confined
channel and act primarily as a jet pump or as a device for thrust
augmentation. In other geometries, the objective may be to shield
a region by means of a jet curtain, or to modify the flow around a
lifting surface by means of a jet flap, or to exploit the Coanda effect,
which is the tendency for jet entrainment to evacuate an unvented
region on one side or the other to a point where the jet sheet must
curve toward the unvented region in order to realize the required
pressure gradient normal to the local flow direction.
Finally, multiple jets are known to interact strongly with each
other under certain conditions. The flow immediately downstream of
9.1. LAMINAR PLANE JET INTO FLUID AT REST
435
a monoplane grid or cascade, for example, might be considered as an
array of plane wakes or as an array of plane jets, depending on the
solidity of the grid. In the latter case, large-scale instabilities may
occur if entrainment by a jet tends to favor one unvented region over
another, or if jets compete with each other in entraining fluid from
the same unvented or partially vented region. FIGURE 9.2 shows a
striking example of the second circumstance.
9.1 Laminar plane jet into fluid at rest
9.1.1
The equations of motion
The classical laminar plane jet in an incompressible fluid is an unex-
ceptional flow in terms of similarity arguments. Appropriate velocity
components are (
u, v
) in rectangular coordinates (
x, y
). The mo-
tion is two-dimensional in the
xy
-plane and is symmetrical about
the plane
y
= 0. The pressure and density are constant everywhere.
The laminar boundary-layer equations are
∂u
∂x
+
∂v
∂y
= 0
,
(9.1)
ρ
(
u
∂u
∂x
+
v
∂u
∂y
)
=
μ
∂
2
u
∂y
2
.
(9.2)
Boundary conditions that require the jet flow to be symmetric about
the plane
y
= 0 and to vanish for large
y
are
ψ
(
x,
0) = 0
or
v
(
x,
0) = 0
,
(9.3)
u
(
x,
±∞
) = 0
,
(9.4)
where
ψ
is a stream function that links the velocities
u
and
v
through
the relations
u
=
∂ψ
∂y
, v
=
−
∂ψ
∂x
,
(9.5)
and thus satisfies the continuity equation (9.1) identically.
Because pressure forces are neglected in the boundary-layer
approximation, the momentum flux in the body of the jet must be
436
CHAPTER 9.
THE PLANE JET
Figure 9.2: Visualization, using smoke filaments, of
flow through a monoplane grid of cylindrical rods.
The blockage is large enough to excite an entrainment
instability (figure 3, plate 1, of Bradshaw 1965).
Photograph courtesy of Cambridge University Press.
9.1. LAMINAR PLANE JET INTO FLUID AT REST
437
conserved during the mixing. If the continuity equation (9.1) is mul-
tiplied by
ρu
and added to the momentum equation (9.2), the result
is
ρ
(
∂uu
∂x
+
∂uv
∂y
)
=
μ
∂
2
u
∂y
2
.
(9.6)
Formal integration with respect to
y
and use of the boundary condi-
tions yields
ρ
d
d
x
∞
∫
−∞
uu
d
y
= 0
(9.7)
or
ρ
∞
∫
−∞
uu
d
y
= constant =
J .
(9.8)
The last expression is one form of the momentum-integral equa-
tion, which plays a role in every kind of boundary-layer problem.
Equation (9.8) is derived from equation (9.2), but contains new in-
formation because it generates the conserved quantity
J
as an im-
portant dimensional parameter. Equation (9.8) also incorporates the
boundary conditions at infinity, a point that will be developed later.
Physically, the integral
J
represents the flux of momentum per unit
time per unit span, or equally the reaction force of the jet at the
origin per unit span. I have deliberately written the integrand as
uu
rather than
u
2
because the two velocities have different physical
meanings. One is momentum per unit mass, and the other is volume
flux per unit area per unit time.
9.1.2
Dimensional properties
Intrinsic scales.
If geometric details near the origin are ignored,
the important physical parameters for the laminar plane jet are
J
,
ρ
, and
μ
. In terms of fundamental units
M
(mass),
L
(length), and
T
(time), denoted here and elsewhere by boldface symbols, these
parameters have the dimensions
[
J
] =
M
T
2
,
[
ρ
] =
M
L
3
,
[
μ
] =
M
LT
,
(9.9)
438
CHAPTER 9.
THE PLANE JET
where “ [
...
] = ” means “the dimensions of
...
are,” and where force
is replaced for dimensional purposes by mass times acceleration. Let
these statements now be interpreted as defining equations for intrin-
sic scales
M
,
L
, and
T
. That is, write
M
T
2
≡
J ,
M
L
3
≡
ρ ,
M
LT
≡
μ .
(9.10)
These three definitions form an algebraic system that can be solved
uniquely (in this particular instance) for the three quantities
M
,
L
,
and
T
;
M
=
ρ
4
ν
6
J
3
,
L
=
ρν
2
J
,
T
=
ρ
2
ν
3
J
2
,
(9.11)
where
ν
=
μ/ρ
, and for the derived quantity
U
;
U
=
L
T
=
J
ρν
.
(9.12)
That the parameters
μ
and
J
should appear only in the kinematic
combinations
μ/ρ
=
ν
and
J/ρ
is implied by the form of equations
(9.2) and (9.8), respectively. Note from equations (9.11) and (9.12)
that the intrinsic length and velocity scales
L
and
U
have small mag-
nitudes, in the sense that their product corresponds to unit Reynolds
number;
UL
ν
= 1
.
(9.13)
Equations (9.11) define dimensional scales that can be used to
make the problem dimensionless at the outset, without regard to the
question of similarity. Such an exercise serves little purpose except to
confirm, as one consequence of a dimensional inspection at the lowest
possible level, that there is no dimensionless combination of
J
,
ρ
, and
ν
that can differ from one experiment to another.
(cf. an appeal to
the Buckingham
Π
theorem. How is round jet different?)
. In
other words, there is only one laminar plane jet.
Local scales.
Some standard variations on the theme of di-
mensions now follow. Ignore temporarily the intrinsic scales
L
and
U
just defined, and suppose instead that the flow has local length and
9.1. LAMINAR PLANE JET INTO FLUID AT REST
439
velocity scales
L
(
x
) and
U
(
x
), whose nature has to be determined.
Similarity implies that the streamwise velocity should have the form
u
U
=
g
(
y
L
)
.
(9.14)
The form of the stream function follows on integration of the first of
equations (9.5);
ψ
UL
=
1
L
y
∫
0
g
d
y
=
f
(
y
L
)
=
f
(
η
)
.
(9.15)
Because
L
(
x
) and
U
(
x
) are designed to represent the local physical
extent and the local velocity magnitude in the jet flow, the func-
tion
f
and its derivatives will be of order unity. Substitution in the
momentum equation (9.2) gives eventually
f
′′′
+
L
ν
d
UL
d
x
ff
′′
−
L
2
ν
d
U
d
x
f
′
f
′
= 0
,
(9.16)
where primes indicate differentiation with respect to
y/L
=
η
. If
f
is
to depend only on
η
, the coefficients must be independent of
x
;
L
ν
d
UL
d
x
= constant
,
L
2
ν
d
U
d
x
= constant
.
(9.17)
A brief trial shows that these equations are not sufficient to determine
U
(
x
) and
L
(
x
). The reason is that the boundary conditions have so
far not been taken into account. The further relationship that is
needed is contained in the momentum-integral equation (9.8), which
incorporates the boundary conditions at infinity;
J
=
ρU
2
L
∞
∫
−∞
f
′
f
′
d
η .
(9.18)
Hence
U
2
L
= constant
.
(9.19)
It now follows easily, on integration of either of the two equations
(9.17) with (9.19), that
U
(
x
)
∼
x
−
1
/
3
, L
(
x
)
∼
x
2
/
3
, UL
∼
x
1
/
3
.
(9.20)
440
CHAPTER 9.
THE PLANE JET
Note that the power-law dependence of
U
and
L
on
x
is derived
rather than assumed.
A different scheme allows the exponents to be established cor-
rectly, without recourse to the governing momentum equation (9.2)
except through its integral (9.8). The scheme uses the device of a
moving observer in the style of H. W. Liepmann. An observer trav-
eling with the fluid on the jet centerline moves at a variable speed
d
x/
d
t
=
U
(
x
). If a power law is appropriate,
U
∼
x
n
. Integration
gives
x
∼
t
1
/
(1
−
n
)
.
(9.21)
A local diffusion approximation yields an estimate of the local jet
thickness;
L
(
x
)
∼
(
νt
)
1
/
2
∼
x
(1
−
n
)
/
2
.
(9.22)
The product
U
2
L
being constant, according to equation (9.19), it
follows that
U
(
x
)
∼
x
(
n
−
1)
/
4
∼
x
n
.
(9.23)
Hence
n
=
−
1
/
3, and the local scales follow as before.
The three proportionalities (9.20) can be converted into equal-
ities by incorporating the intrinsic scales of the problem. Thus put
U
=
UL
1
/
3
x
−
1
/
3
=
(
J
2
ρ
2
νx
)
1
/
3
,
(9.24)
L
=
L
1
/
3
x
2
/
3
=
(
ρν
2
x
2
J
)
1
/
3
,
(9.25)
UL
=
UL
2
/
3
x
1
/
3
=
(
Jνx
ρ
)
1
/
3
.
(9.26)
Equation (9.15) now becomes a complete dimensionless ansatz,
(
ρ
Jνx
)
1
/
3
ψ
=
f
[
(
J
ρν
2
x
2
)
1
/
3
y
]
.
(9.27)
Affine transformation.
The ansatz (9.27) is unique, al-
though it can be derived by more than one method. I usually prefer
9.1. LAMINAR PLANE JET INTO FLUID AT REST
441
a different approach to similarity that has a stronger mathematical
flavor, although it also depends implicitly on the physical premise
that all of the terms in the momentum equation (9.2) must have
the same dimension. Require this equation, together with its bound-
ary conditions, to be invariant under an affine transformation
(see
Bluman and Cole)
. Let all of the variables and parameters of the
problem, taken in a standard order
x
,
y
,
ψ
,
ρ
,
μ
followed by any spe-
cial parameters—everything in sight—be rescaled according to the
rules
x
=
a
̂
x ,
y
=
b
̂
y ,
ψ
=
c
̂
ψ ,
ρ
=
d
̂
ρ ,
μ
=
e
̂
μ ,
(9.28)
J
=
j
̂
J ,
u
=
r
̂
u
=
c
b
̂
u ,
v
=
s
̂
v
=
c
a
̂
v ,
where
a, b, c
,
...
are positive constant numbers that stretch or shrink
the scales used to measure the various quantities. The last two lines
of this table are required if the equations to be transformed are taken
initially in the form (9.1)–(9.2), in which
u
and
v
appear rather than
ψ
. Transformation of equations (9.5) then gives
r
̂
u
=
c
b
∂
̂
ψ
∂
̂
y
, s
̂
v
=
−
c
a
∂
̂
ψ
∂
̂
x
.
(9.29)
Invariance of the continuity equation thus requires
r
=
c/b
and
s
=
442
CHAPTER 9.
THE PLANE JET
c/a
, so that
u
transforms like
ψ/y
and
v
like
ψ/x
.
The result of the transformation when applied to the momen-
tum equation (9.2) is
c
2
d
ab
2
̂
ρ
(
̂
u
∂
̂
u
∂
̂
x
+
̂
v
∂
̂
u
∂
̂
y
)
=
ce
b
3
̂
μ
∂
2
̂
u
∂
̂
y
2
.
(9.30)
The same procedure transforms the momentum integral (9.8) into
c
2
d
b
̂
ρ
∞
∫
−∞
̂
u
̂
u
d
̂
y
=
j
̂
J .
(9.31)
No useful information is obtained by transforming the null bound-
ary conditions (9.3)–(9.4). For example, the first of equations (9.3)
becomes
c
̂
ψ
(
a
̂
x,
0) = 0
.
(9.32)
If the original boundary condition is read as “
ψ
is zero for
y
= 0
and all
x
,” then the transformed boundary condition is read as “
̂
ψ
is zero for
̂
y
= 0 and all
̂
x
,” whatever the values of
a
and
c
might be.
If the problem defined by equations (9.30) and (9.31) is to be
invariant under the affine transformation, it is necessary to put
bcd
ae
= 1
,
c
2
d
bj
= 1
.
(9.33)
Because the jet has a strongly preferred direction, it is most pro-
ductive to work with
ψ
and
y
as dependent variable and primary
independent variable, respectively (other strategies will work, but
not as directly and efficiently). When equations (9.33) are solved to
separate the corresponding scaling factors
c
and
b
, they become
c
3
d
2
aej
= 1
,
b
3
dj
a
2
e
2
= 1
.
(9.34)
These two conditions have magical properties. When
a
,
b
,
c
,
...
are
replaced by the corresponding ratios from the table (9.28), the in-
variant combinations of the transformation appear in physical dress.
9.1. LAMINAR PLANE JET INTO FLUID AT REST
443
Figure 9.3: The final step of a similarity argument
based on an affine transformation. For the example in
the text, the curve on the right is
̂
y
∼
̂
x
2
/
3
.
At corresponding points of the affine transformation,
f
=
(
ρ
Jνx
)
1
/
3
ψ
=
(
̂
ρ
̂
J
̂
ν
̂
x
)
1
/
3
̂
ψ
=
̂
f ,
(9.35)
η
=
(
J
ρν
2
x
2
)
1
/
3
y
=
(
̂
J
̂
ρ
̂
ν
2
̂
x
2
)
1
/
3
̂
y
=
̂
η .
(9.36)
This equivalence does not quite establish that the combination
(9.35) must be a function only of the combination (9.36), although
this conclusion is in fact correct, according to equation (9.27). This
question was raised but not resolved in SECTION 1.3.5 of the intro-
duction. The essence of the required argument is shown geometrically
in FIGURE 9.3. The reasoning is symmetric, and could equally well
proceed by interchanging the roles of the two flows (hence the term
“symmetry analysis” sometimes used for the procedure). Suppose
that the physical parameters
J
,
ρ
, and
ν
are known for the given
flow at the left in FIGURE 9.3, and similarly that
̂
J
,
̂
ρ
, and
̂
ν
are
known for the transformed flow at the right. Suppose also that
ψ
is
known, analytically or experimentally, for one fixed point (
x, y
) in
444
CHAPTER 9.
THE PLANE JET
the given flow. Then
η
and
f
are known at this point from equations
(9.35) and (9.36). So also are
̂
η
=
η
and
̂
f
=
f
. But the transformed
point (
̂
x,
̂
y
) is not a point but a curve,
̂
y/
̂
x
2
/
3
= constant, which is
defined through equation (9.36). Along this curve both
̂
η
and
̂
f
are
constant, and it follows that
̂
f
depends only on
̂
η
.
A final point is that the combinations (9.35) and (9.36) are
guaranteed to be dimensionless by the terms of their construction.
They are also guaranteed to be complete, since they incorporate all
of the independent and dependent variables and all of the parame-
ters that appear in the equations and the boundary conditions. If the
frequent application in this monograph of the method of the affine
transformation is viewed as a process of turning a crank, the crank
is a handsome one. I should say that I developed the procedure just
described about 1975, without realizing until later that I had repro-
duced a small portion of group theory (which was not a part of my
early mathematical education). The system (9.28) evidently defines a
group, since it contains the identity transformation (
a
=
b
=
...
= 1),
the inverse transformation (
̂
x
=
x/a
=
αx,...
), and the product
transformation (
x
=
a
̂
x
,
̂
x
=
A
̂
̂
x
,
x
=
aA
̂
̂
x
=
α
̂
̂
x,...
). The two re-
lations (9.34) are invariants of the group for the particular problem
being considered here. A closely related system, called by its author
a dilation group, is introduced early in the book on symmetry by
CANTWELL (2002).
Similarity.
Both of the similarity arguments just given have
succeeded in reducing the number of independent variables from two
to one. The ansatz for the stream function must have the form first
discovered in equation (9.27),
A
(
ρ
Jνx
)
1
/
3
ψ
=
f
[
B
(
J
ρν
2
x
2
)
1
/
3
y
]
=
f
(
η
)
,
(9.37)
where the new quantities
A
and
B
are positive numerical constants
of order unity whose function is to support a final normalization
of the variables
f
and
η
. The constant
A
adjusts the dimensionless
volume flux (i.e., the stream function), and the constant
B
adjusts
9.1. LAMINAR PLANE JET INTO FLUID AT REST
445
the dimensionless thickness of the layer. Differentiation yields
u
=
∂ψ
∂y
=
B
A
(
J
2
ρ
2
νx
)
1
/
3
f
′
,
(9.38)
v
=
−
∂ψ
∂x
=
1
3
A
(
Jν
ρx
2
)
1
/
3
(
2
ηf
′
−
f
)
,
(9.39)
τ
=
μ
∂u
∂y
=
B
2
A
J
x
f
′′
,
(9.40)
∂τ
∂y
=
B
3
Ax
(
J
4
ρν
2
x
2
)
1
/
3
f
′′′
,
(9.41)
and so on, where the primes indicate differentiation with respect to
η
. Substitution of these expressions into equation (9.2) leads to the
ordinary differential equation
3
ABf
′′′
+
ff
′′
+
f
′
f
′
= 0
.
(9.42)
The boundary conditions, from equations (9.3)–(9.4), are
f
(0) =
f
′
(
±∞
) =
f
′′
(
±∞
) = 0
.
(9.43)
These homogeneous relationships might seem to define an eigen-
value problem, but do not, because the constant-momentum con-
dition (9.8) also has to be satisfied. Insertion of equation (9.38) for
u
in equation (9.8) yields the constraint
∞
∫
−∞
f
′
f
′
d
η
=
A
2
B
.
(9.44)
9.1.3
The boundary-layer solution
Equation (9.42) can be solved in closed form. Three successive inte-
grations, with the appropriate symmetry conditions, give
3
ABf
′′
+
ff
′
= constant = 0 ;
(9.45)
446
CHAPTER 9.
THE PLANE JET
6
ABf
′
+
f
2
= constant =
C
2
;
(9.46)
f
=
C
tanh
(
C
6
AB
η
)
;
(9.47)
where
C >
0 is a constant of integration. The velocity profile is
f
′
=
C
2
6
AB
sech
2
(
C
6
AB
η
)
.
(9.48)
Equation (9.42) was first derived by SCHLICHTING (1933), who did
not notice that it could be integrated in closed form, and therefore
resorted to numerical integration. The analytical integral (9.47) was
supplied by BICKLEY (1937).
Compact variables.
The boundary-layer solution emerges
from equations (9.37) and (9.47) as
A
C
(
ρ
Jνx
)
1
/
3
ψ
= tanh
[
C
6
A
(
J
ρν
2
x
2
)
1
/
3
y
]
= tanh
(
C
6
AB
η
)
.
(9.49)
The streamline pattern contained in this equation is most instructive
when it is expressed in what will shortly be defined as compact outer
variables, with
x
and
y
made dimensionless in the same way. An
intrinsic length scale is available for this purpose in the second of
equations (9.11); namely,
L
=
ρν
2
/J
. When
J
is eliminated in favor
of
L
in equation (9.49), this becomes
A
C
(
L
x
)
1
/
3
ψ
ν
= tanh
[
C
6
A
(
y
3
L
x
2
)
1
/
3
]
.
(9.50)
Compact dimensionless outer variables thus reveal themselves as
Ψ =
ψ
6
ν
, X
=
(
C
6
A
)
3
x
L
, Y
=
(
C
6
A
)
3
y
L
.
(9.51)
In these variables, equation (9.49) takes the simple form
Ψ =
X
1
/
3
tanh
(
Y
X
2
/
3
)
,
(9.52)
9.1. LAMINAR PLANE JET INTO FLUID AT REST
447
which does not depend formally on the values chosen for the nor-
malizing constants
A
,
B
, and
C
. However, these constants are not
themselves independent. The integral in equation (9.44) applies for a
general function
f
(
η
). When the specific function of equation (9.47)
is inserted, a specific value involving
C
is obtained for the integral;
∞
∫
−∞
f
′
f
′
d
η
=
2
C
3
9
AB
=
A
2
B
,
(9.53)
from which
(
C
6
A
)
3
=
1
48
.
(9.54)
Hence the proper dimensionless outer variables in equation (9.52) are
Ψ =
ψ
6
ν
, X
=
x
48
L
, Y
=
y
48
L
,
(9.55)
which are clearly independent of
A
,
B
, and
C
because
L
depends
only on
ρ
,
ν
, and
J
. The first of these equations could equally well
be written as Ψ =
ψ/
6
UL
.
Streamlines Ψ (
X, Y
) = constant for the boundary-layer so-
lution in compact outer variables are shown in FIGURE 9.4. Such
a figure was first constructed, absent scales on the axes, by Bickley.
Note that the ranges for
X
and
Y
in the figure extend to large nu-
merical values, because the quantities
U
and
L
refer to a Reynolds
number of unity. The local scales
U
(
x
) and
L
(
x
) are a different mat-
ter. These two scales are defined by equations (9.24) and (9.25) as
U
=
(
J
2
ρ
2
νx
)
1
/
3
, L
=
(
ρν
2
x
2
J
)
1
/
3
.
(9.56)
Their product, which is one form of local Reynolds number, can be
expressed in terms of
X
alone;
Re
(
x
) =
UL
ν
=
(
Jx
ρν
2
)
1
/
3
=
(
x
L
)
1
/
3
= (48
X
)
1
/
3
.
(9.57)
FIGURE 9.4 therefore displays the flow in a laminar plane jet up to
a local Reynolds number of about 36. There is only one jet, and only
448
CHAPTER 9.
THE PLANE JET
500
-500
0
Y
500
1000
0
X
Figure 9.4: Streamlines
Ψ
= constant of the
boundary-layer approximation for the laminar plane jet
according to equation (9.52). The range of
Ψ
is
−
9 (1)
9
.
9.1. LAMINAR PLANE JET INTO FLUID AT REST
449
one figure. A change in displayed Reynolds number amounts to use of
a larger or smaller mask for this figure. The local Reynolds number
increases in the downstream direction like
x
1
/
3
, and the laminar plane
jet can therefore be expected to become unstable at some point in
its evolution.
The boundary-layer solution (9.52) is silent about the flow for
x <
0 and is indifferent to the presence or absence of walls, whether
along the
y
-axis or elsewhere. The reason is that the line
x
= 0 is a
characteristic of the parabolic boundary-layer equations (9.1)–(9.2)
and is the locus of a discontinuity in
v
. This property is entirely an
artifact of the choice of coordinates, as explained by KAPLUN (
)
in his beautiful paper on the role of coordinate systems in boundary-
layer theory. It is possible that some early experimenters, beginning
with ANDRADE (1939), may have been inspired to place a wall at
x
= 0 in order to guide the relatively slow induced flow in the proper
direction in a drafty facility. A more cogent reason might be that the
geometry is simple and easily reproduced. In any event, the stream-
line pattern in FIGURE 9.4 cannot be correct outside the body of
the jet, because it represents a rotational flow. The derivative
∂u/∂y
is zero, but the derivative
∂v/∂x
is not. However, knowledge of the
streamline pattern does allow a useful conclusion about the veloc-
ity history of fluid moving along a given streamline. It is clear from
the streamline spacing in FIGURE 9.4 that this velocity starts at a
low value, increases rapidly near the axis of symmetry, and then de-
creases slowly toward the right edge of the figure. The
x
-component
of velocity has the same property and is easier to describe. The pres-
sure being constant, the acceleration
Du/Dt
along a streamline from
equation (9.2) is proportional to
∂τ/∂y
, which in turn is proportional
to
f
′′′
from equation (9.41). The maximum
x
-velocity therefore oc-
curs when
f
′′′
(
η
) = 0, and the shearing stress
τ
is a maximum there.
Differentiation of equation (9.48) shows that
f
′′′
(
η
) vanishes when
η
=
η
m
, with
cosh
2
(
C
6
AB
η
m
)
=
3
2
.
(9.58)
450
CHAPTER 9.
THE PLANE JET
Substitution in equation (9.48) yields
f
′
(
η
m
) =
2
3
C
2
6
AB
=
2
3
f
′
(0)
.
(9.59)
Within the boundary-layer approximation, therefore, the maximum
streamwise velocity on a streamline occurs at the inflection point of
the profile and is two thirds of the velocity in the plane of symmetry
at the same value of
x
.
Mathematically speaking, the jet described here entrains the
whole universe (more accurately, half of it). It may be useful to think
of a fluid element upstream from its velocity maximum as not yet
entrained, because it is still acquiring momentum from fluid closer
to the axis of the jet, whereas a fluid element downstream from its
maximum is losing momentum to fresh fluid being entrained in its
vicinity and is effectively part of the body of the jet. The sharp
change in direction of the streamlines in FIGURE 9.4, incidentally,
serves notice that static pressure variations, although small at this
order, may be relatively greater in a jet than in a more unidirectional
flow such as a boundary layer.
9.1.4
Normalization
So far, I have deliberately left the normalizing constants
A
and
B
and the constant of integration
C
in the boundary-layer solu-
tion (9.46) unspecified, for the sake of generality. Most but not all
authors have followed Bickley in putting
f
′
(0) = 1,
f
(
∞
) = 1, and
∫
∞
−∞
f
′
f
′
d
η
= 4
/
3. According to equations (9.48), (9.47), and (9.44),
this amounts to putting
A
= (2
/
9)
1
/
3
,
B
= (1
/
48)
1
/
3
, and
C
= 1.
However, I prefer a slightly different normalization that aims primar-
ily at consistency among the various flows treated in this monograph.
The present analysis has provided one relation between
A
and
C
in
equation (9.54);
C
A
=
(
9
2
)
1
/
3
,
(9.60)
which is consistent with Bickley’s selection. Another condition is sug-
gested by the form of the differential equation (9.42). In favor of the
9.1. LAMINAR PLANE JET INTO FLUID AT REST
451
condition 3
AB
= 1 is the fact that the reduced operator
f
′′′
+
ff
′′
is common to all of the plane laminar boundary-layer problems con-
sidered here, with a numerical coefficient for the term
f
′′′
that varies
from one flow to another. The quantities
f
′′′
and
ff
′′
represent, re-
spectively, viscous diffusion and transport (of vorticity). They are
commensurate in the physics, and I propose that they should also be
commensurate in the mathematics; i.e., not weighted differently for
different flows. Consequently, I take as a second condition
3
AB
= 1
.
(9.61)
The constant
C
.
Equation (9.61) allows several properties
derived above for the boundary-layer solution to be expressed in
terms of the undefined constant of integration
C
alone. In particular,
f
′
(0) =
C
2
2
;
(9.62)
∫
∞
0
f
′
(
η
)d
η
=
f
(
∞
) =
C .
(9.63)
Now let the functions
f
(
η
) and
f
′
(
η
) be plotted against
η
, as
in FIGURES 9.5 and 9.6.
The numerical magnitudes needed to plot these curves have
been suppressed in each case. As one possible scale, consider the
maximum-slope thickness indicated in FIGURE 9.5. The tangent
line is defined by
f
=
f
′
(0)
η
. This line can be terminated when
f
=
f
(
∞
) =
C
and
η
=
η
s
, say. Then
f
(
η
s
) =
C
=
f
′
(0)
η
s
=
C
2
2
η
s
.
(9.64)
If the characteristic scale
η
s
is arbitrarily put equal to unity, then
C
= 2
.
(9.65)
Next, define an integral scale
η
i
(reminiscent of the displace-
ment thickness for a boundary layer) as shown in the right half of
FIGURE 9.6, with
∫
∞
0
f
′
d
η
=
C
=
f
′
(0)
η
i
=
C
2
2
η
i
.
(9.66)
452
CHAPTER 9.
THE PLANE JET
s
Figure 9.5: Graphical definition of the maximum-slope
thickness
η
s
for the laminar plane jet.
Figure 9.6: Graphical definition of the half-integral
scale
η
i
for the laminar plane jet, on the right, and the
half-width
η
c
for the osculating parabola, on the left.
9.1. LAMINAR PLANE JET INTO FLUID AT REST
453
If
η
i
is put equal to unity, then again
C
= 2. The areas of the two
cross-hatched regions are equal.
Finally, define a curvature scale
η
C
(reminiscent of the Taylor
microscale in turbulence) as the intersection at
f
′
= 0,
η
=
η
C
of the
osculating parabola in the left half of FIGURE 9.6. The parabola is
defined by the first two terms of a power series,
f
′
(
η
) =
f
′
(0) +
f
′′′
(0)
η
2
2
.
(9.67)
Put
f
′
(0) =
C
2
/
2 and
f
′′′
(0) =
−
[
f
′
(0)]
2
, the latter from equation
(9.42). Thus
f
′
(0) =
C
2
2
=
[
f
′
(0)
]
2
η
2
C
2
=
C
4
8
η
2
C
.
(9.68)
If
η
C
is put equal to unity, then once more
C
= 2. Note that these
three arguments are related, in the sense that they all involve
f
′
(0).
These results are persuasive, even without the observation that
there is no obvious reason why the argument of
f
should not be taken,
strictly for simplicity, as
η
rather than
Cη/
2 with
C
6
= 2. In this con-
nection, I should mention a condition sometimes used to define a
scale for turbulent shear flows. This is to use the point where the ve-
locity is half of some suitable characteristic velocity, say the velocity
on the centerline of the present jet (
cf
. the treatment of the wake
function in the turbulent boundary layer). If this point has the value
η
= 1, then the constant
C
is the root of the transcendental equation
cosh(
C/
2) =
√
2. Other authors may prefer a different normalization
from the ones proposed here.
After some reflection, I believe that the weight of the evidence
just described is on the side of taking the integral scale as fundamen-
tal. The three equations (9.60), (9.61), and (9.65) then imply
A
=
(
16
9
)
1
/
3
, B
=
(
1
48
)
1
/
3
, C
= 2
.
(9.69)
The effect of this normalization is to set the boundary-layer problem
454
CHAPTER 9.
THE PLANE JET
and its solution for the laminar plane jet in the form
(
16
ρ
9
Jνx
)
1
/
3
ψ
=
f
[
(
J
48
ρν
2
x
2
)
1
/
3
y
]
=
f
(
η
)
,
(9.70)
f
′′′
+
ff
′′
+
f
′
f
′
= 0
,
(9.71)
f
(
η
) = 2 tanh
η ,
(9.72)
f
′
(
η
) = 2 sech
2
η ,
(9.73)
f
′
(0) = 2
,
(9.74)
f
′′
(
η
) = 4
sinh
η
cosh
3
η
=
f
(
η
)
f
′
(
η
)
,
(9.75)
1
2
∞
∫
−∞
f
′
d
η
=
f
(
∞
) = 2
,
(9.76)
∞
∫
−∞
f
′
f
′
d
η
=
16
3
,
(9.77)
where the equation for
f
′′
takes account of equation (9.45). The nor-
malized functions
f
(
η
) and
f
′
(
η
) for the boundary-layer solution are
displayed as solid curves in FIGURE 9.7. The connection with the
physical variables
ψ
,
u
,
v
,
τ
is given by properly normalized versions
of equations (9.37)–(9.40);
f
(
η
) =
(
16
ρ
9
Jνx
)
1
/
3
ψ
= 2 tanh
η ,
(9.78)
f
′
(
η
) =
(
256
ρ
2
νx
3
J
2
)
1
/
3
u
=
2
cosh
2
η
,
(9.79)