Chapter 8
THE ROUND JET
The classical laminar or turbulent round jet issues from a point source
into a stagnant fluid. An extreme case is shown in the photograph
(Figure 8.1). An important reason for the popularity of the turbu-
lent round jet as a subject for fundamental experimental study is
economy. The geometry is simple, and a large part of the energy
supplied to the fluid appears as turbulent motion before being dissi-
pated in heat. An easily accessible and important variation on the
basic problem of momentum transport is simultaneous transport of
heat or mass.
A jet may be surrounded by a moving fluid, as in a rocket or
jet engine. In such cases the problem can sometimes be linearized,
giving a point of contact with the round wake. Jets are components
of many practical devices such as torches and sprays. A jet may
also be enclosed in a shroud or housing to produce a jet pump or
ejector. Several jets may interact, or a non-circular jet may relax
toward axial symmetry, or a jet may have a component normal to an
ambient flow, as in thrust-control devices. Swirl is sometimes used
to enhance mixing. Finally, a jet of one fluid into another is a nice
problem in entrainment, mixing, and relaxation.
399
400
CHAPTER 8.
THE ROUND JET
Figure 8.1: Photo from JOHNSON and HOUK (1968, Los
Angeles, Portrait of an extraordinary city: Menlo
Park, Calif., Lane Magazine and Book Co., caption by
K. Coles).
8.1. LAMINAR ROUND JET INTO FLUID AT REST
401
8.1 Laminar round jet into fluid at rest
8.1.1
Preview
The steady laminar flow associated with a point momentum source in
a viscous incompressible fluid is one of the few known exact solutions
of the Navier-Stokes equations. The reason that this solution came
long ago to be known is not the usual one, which is that a particularly
simple geometry has reduced the number of independent variables,
as in pipe flow. The reason is dimensional in a different way, having
to do with the number and nature of global parameters. A kindred
case is the laminar sink flow in a wedge-shaped channel, treated ear-
lier in SECTION X. These two flows have in common that the exact
solution is known in closed form for all Reynolds numbers (if elliptic
functions qualify as closed form). An important difference is that
entrainment is an essential feature of the boundary-layer approxima-
tion for the round jet, but not for the channel flow. Hence the round
jet is unique in providing an opportunity to practice the technique of
matched asymptotic expansions to arbitrarily high order. As far as
I know, this opportunity has never been exploited, and I will not at-
tempt here to rise above the level of first-order boundary-layer theory
in SECTIONS X, Y, and Z.
The organization of the next (
four
) sections is shown schemat-
ically in Figure 8.2. The box at the left, called “NS (Navier-Stokes)
equations,” is the core element for the diagram. My first objective is
to show that the two paths from “NS equations” to “BL (boundary-
layer) solution” are precisely equivalent, as are the two paths from
“NS equations” to “potential flow.” My second objective is to il-
lustrate by example the use of the powerful technique of matched
asymptotic expansions to construct a “composite expansion” that is
not exact, but cannot for practical purposes be distinguished from the
exact solution at the Reynolds number of the present exercise. These
derivations lay a foundation for operations that build on boundary-
layer theory in cases where no exact solution is known; e.g., the plane
laminar jet, or any turbulent flow.
After some dimensional preliminaries, the exact solution is de-
402
CHAPTER 8.
THE ROUND JET
Figure 8.2: Caption for Figure with label Fig15-7
(missing. Evidently this was concept map or diagram
connecting ideas in this chapter. - K. Coles)
rived in SECTION 8.1.3. Most of the material there is not original,
and can be found in ROSENHEAD, “Laminar Boundary Layers”
(1963) or in BATCHELOR, “An Introduction to Fluid Dynamics”
(1967). The original sources are papers by SLEZKIN (1934), LAN-
DAU (1944), and SQUIRE (1951).
8.1.2
Dimensional argument
An appropriate coordinate system for the problem of the laminar
round jet is the spherical polar system (
r
,
θ
,
φ
) shown at the left in
FIGURE 8.3. For comparison and contrast, the corresponding plane
jet from a line momentum source is also sketched in a cylindrical po-
lar system, (
R
,
θ
,
z
) at the right. These coordinates are deliberately
chosen so that one coordinate in the two-dimensional reduced system
is dimensionless.
The global parameters for the point momentum source (round
jet) or the line momentum source (plane jet) are the fluid properties
ρ
and
μ
(or
ρ
and
ν
=
μ/ρ
) and the specified momentum flux
J
,
whose dimensions for the round jet are momentum per unit time
and for the plane jet are momentum per unit time per unit length.
8.1. LAMINAR ROUND JET INTO FLUID AT REST
403
Figure 8.3: Caption for Figure with label Fig15-6
(missing)
This small descriptive difference has large consequences. In terms
of mass, length, and time as fundamental units, the three global
parameters have dimensions
1
Point source
Line source
[
J
] =
ML
T
2
,
[
J
] =
M
T
2
,
[
ρ
] =
M
L
3
,
[
ρ
] =
M
L
3
,
(8.1)
[
ν
] =
L
2
T
,
[
ν
] =
L
2
T
.
Rearrangement to isolate the characteristic scales
M
,
L
, and
T
in
each column yields quite different results for the two problems;
[
J
ρν
2
]
= 0
.
[
ρ
4
ν
6
J
3
]
=
M
,
[
ρν
2
J
]
=
L
,
(8.2)
[
ρ
2
ν
3
J
2
]
=
T
.
1
In this section equations on the left refer to the point source (round jet) and
on the right the line source (plane jet).
404
CHAPTER 8.
THE ROUND JET
For the point source on the left, no characteristic scales can be de-
fined. It is this fact that makes an exact solution possible. Such a
solution can be expected to depend on one dimensionless parameter,
(
J/ρν
2
)
1
/
2
, having the nature of a Reynolds number. For the line
source on the right, the situation is quite otherwise. The character-
istic scales are well defined, but there is no dimensionless parameter.
Hence there can be only one solution. Note that this argument is
not based on the equations of motion, but only on the physical pa-
rameters for each flow.
Let a suitable solution be anticipated in terms of a single
stream function
ψ
, where
[
ψ
] =
L
3
T
.
[
ψ
] =
L
2
T
.
(8.3)
If
L
and
T
can be defined, these relations can be written in dimen-
sionless form as equalities. To mark the profound change in content
for the symbols, a different font is used;
ψT
L
3
= fn
(
r
L
, θ
)
.
ψT
L
2
= fn
(
R
L
, θ
)
.
(8.4)
When
T
is eliminated using
T
=
L
2
/ν
, these become
ψ
νL
= fn
(
r
L
. θ
)
.
ψ
ν
= fn
(
R
L
. θ
)
.
(8.5)
In the left column,
L
can not be defined and therefore cannot appear.
The only rational action is to replace
L
by
r
. In the right column,
L
can be defined as
ρν
2
/J
. Consequently, from one of equations (8.2).
ψ
νr
= fn (1
, θ
)
.
ψ
ν
= fn
(
RJ
ρν
2
, θ
)
.
(8.6)
The result for the round jet on the left is much more than an ac-
cidental separation of variables. The result states that
ψ/νr
depends
only on
θ
, and must therefore be obtainable by solving an ordinary
differential equation (with
J/ρν
2
as parameter). Such a conclusion
normally requires a much more elaborate similarity argument based
8.1. LAMINAR ROUND JET INTO FLUID AT REST
405
on the transformation properties of the equations of motion. It may
also require good judgment in choosing an appropriate system of
coordinates.
No corresponding reduction appears for the line source on the
right. This flow will be discussed in SECTION 9.1.2.
8.1.3
The exact solution
Take the velocity for the point momentum source to be
~u
= (
u, v, w
)
in spherical polar coordinates (
r
,
θ
,
φ
). Consider steady axisymmet-
ric flow without swirl; i.e.,
∂/∂t
=
∂/∂φ
=
w
= 0. The equations of
motion (
reference
) are then
1
r
2
∂ur
2
∂r
+
1
r
sin
θ
∂v
sin
θ
∂θ
= 0
,
(8.7)
u
∂u
∂r
+
v
r
∂u
∂θ
−
v
2
r
=
−
1
ρ
∂p
∂r
+
ν
(
∇
2
u
−
2
u
r
2
−
2
r
2
sin
θ
∂v
sin
θ
∂θ
)
,
(8.8)
u
∂v
∂r
+
v
r
∂v
∂θ
+
uv
r
=
−
1
ρr
∂p
∂θ
+
ν
(
∇
2
v
+
2
r
2
∂u
∂θ
−
v
r
2
sin
2
θ
)
,
(8.9)
where the Laplace operator is
∇
2
α
=
1
r
2
∂
∂r
r
2
∂α
∂r
+
1
r
2
sin
θ
∂
∂θ
sin
θ
∂α
∂θ
.
(8.10)
It is convenient first to eliminate the pressure by working with
the vorticity,
~
Ω = curl
~u
= (
ξ, η, ζ
), which has only a
φ
-component;
ζ
=
1
r
∂rv
∂r
−
1
r
∂u
∂θ
.
(8.11)
The velocity components are derivable from a stream function
ψ
using the definition
~u
= grad
ψ
×
grad
φ
;
u
=
1
r
2
sin
θ
∂ψ
∂θ
,
v
=
−
1
r
sin
θ
∂ψ
∂r
,
(8.12)
and consequently, from equation (8.11),
ζ
=
−
1
r
sin
θ
∇
2
ψ
+
2
r
sin
θ
(
u
cos
θ
−
v
sin
θ
)
.
(8.13)
406
CHAPTER 8.
THE ROUND JET
The equation satisfied by
ζ
, from equations (8.8) and (8.9), is
u
∂ζ
∂r
+
v
r
∂ζ
∂θ
=
ζ
r
sin
θ
(
u
sin
θ
+
v
cos
θ
) +
ν
(
∇
2
ζ
−
ζ
r
2
sin
2
θ
)
.
(8.14)
The first term on the right-hand side represents vortex stretching of
a trivial kind. The meaning of this term emerges if it is noted that
r
sin
θ
=
R
, say, is the perpendicular distance from any point in the
flow to the polar axis of symmetry. Moreover,
(
u
sin
θ
+
v
cos
θ
) =
~u
·∇
R
=
DR/D .
(8.15)
In the absence of diffusion due to viscosity, therefore, the vorticity
obeys the equation (
need to display formula for grad?
)
1
ζ
Dζ
Dt
=
1
R
DR
Dt
.
(8.16)
Hence
ζ/R
is constant following an element of the fluid; the strength
of any (circular) vortex filament varies directly with the filament
diameter. This observation is obviously not limited to the problem
of the point momentum source, but is valid for any axially symmetric
motion.
It has already been argued in equation (8.6) that the stream
function must have the form
ψ
=
ν rf
(
θ
). A convenient variant is
ψ
=
ν rf
(cos
θ
) =
ν rf
(
ξ
)
,
(8.17)
where
ξ
= cos
θ
. It follows from equations (8.12) and (8.11) that
u
=
−
ν
r
f
′
,
(8.18)
v
=
−
ν
r
f
sin
θ
,
(8.19)
ζ
=
−
ν
r
2
f
′′
sin
θ ,
(8.20)
where
f
′
means d
f/
d cos
θ
= d
f/
d
ξ
. When these are substituted in
equation (8.14),
f
satisfies
3
f
′
f
′′
+
ff
′′′
+ 4
ξ f
′′′
−
(1
−
ξ
2
)
f
′′′′
= 0
.
(8.21)
8.1. LAMINAR ROUND JET INTO FLUID AT REST
407
This equation is fourth-order because the pressure is still a
variable, although it has been formally suppressed by using the curl
operator. Two successive integrals of equation (8.21) are
f
′
f
′
+
f f
′′
−
2
f
′
+ 2
ξ f
′′
−
(1
−
ξ
2
)
f
′′′
=
C
1
,
(8.22)
f f
′
−
2
f
−
(1
−
ξ
2
)
f
′′
=
C
1
ξ
+
C
2
.
(8.23)
The boundary condition of axial symmetry is expressed by taking
f
= 0 on
θ
= 0 and
θ
=
π
, or
f
(1) =
f
(
−
1) = 0. At
ξ
= 1, equation
(8.23) requires
C
1
+
C
2
= 0, and at
ξ
=
−
1 it requires
−
C
1
+
C
2
= 0.
Hence
C
1
=
C
2
= 0. A further integration gives
f
2
−
4
ξ f
−
2(1
−
ξ
2
)
f
′
=
C
3
= 0
,
(8.24)
where the same symmetry condition requires
C
3
= 0. Finally, the
substitution
h
(
ξ
) =
f
(
ξ
)
(1
−
ξ
2
)
(8.25)
transforms equation (8.24) into
h
′
=
h
2
2
.
(8.26)
The final exact solution (retransformed) is therefore
f
(
ξ
) =
2(1
−
ξ
2
)
1
−
ξ
+
c
(8.27)
or
f
(cos
θ
) =
2 sin
2
θ
1
−
cos
θ
+
c
,
(8.28)
where (1 +
c
)
/
2 is a constant of integration. The stream function, ve-
locity components, and azimuthal vorticity are obtained from equa-
408
CHAPTER 8.
THE ROUND JET
Figure 8.4: Caption for Figure with label Fig15-8
(missing)
tions (8.17)–(8.20) as
ψ
= 2
νr
sin
2
θ
1
−
cos
θ
+
c
,
(8.29)
u
=
2
ν
r
[
2
c
cos
θ
−
(1
−
cos
θ
)
2
]
(1
−
cos
θ
+
c
)
2
,
(8.30)
v
=
−
2
ν
r
sin
θ
1
−
cos
θ
+
c
,
(8.31)
ζ
=
4
ν
r
2
c
(
c
+ 2)
sin
θ
(1
−
cos
θ
+
c
)
3
.
(8.32)
Some typical streamline patterns are shown in FIGURE 8.4
[for?] values of the parameter
c
. The sense of the figure is borrowed
from Batchelor, p. 208. The chief difference is that the stream
function and other variables are here put in dimensionless form with
the aid of a trick, which is the introduction of a length
L
that is never
defined. Lagerstrom used to call
L
the length of the blackboard.
8.1. LAMINAR ROUND JET INTO FLUID AT REST
409
Thus write, in what might be called virtual variables,
Ψ =
ψ
4
ν
L
,
(8.33)
R
=
r
L
,
(8.34)
Θ =
θ .
(8.35)
so that equation (8.29) becomes
Ψ =
R
sin
2
Θ
2(1
−
cos Θ +
c
)
.
(8.36)
(
Define coordinates used in figure
). Since
ψ
varies linearly with
r
at constant
θ
, one streamline suffices to define each flow, with other
streamlines obtained by a zoom transformation. Also at constant
θ
,
the velocities
u
and
v
vary inversely with
r
, according to equations
(8.30) and (8.31). As the Reynolds number increases (the parame-
ter
c
decreases toward zero) on the one hand, a strong narrow jet
emerges along the polar axis. The remainder of the flow represents
fluid motion induced by this jet. At values of
c
that are large com-
pared with unity, on the other hand,
c
dominates the denominator
of equation (8.27), giving a Stokes flow with streamlines that are
symmetrical upstream and downstream.
There is no mass flux from the singularity at the origin. Con-
sider an integral over a sphere of fixed radius
r
about the origin, with
d
S
= 2
π r
sin
θr
d
θ
. The net flux for the sphere is
∫∫
ρ~u
·
~n
d
S
= 2
πρr
2
π
∫
0
u
sin
θ
d
θ
= 2
πρνr
1
∫
−
1
f
′
(
ξ
) d
ξ
= 0
.
(8.37)
To compute the corresponding momentum flux, observe that the
component of any vector
~a
= (
a
1
, a
2
,
0) along the polar axis (the
410
CHAPTER 8.
THE ROUND JET
x
-axis in FIGURE 8.3) is (
a
1
cos
θ
−
a
2
sin
θ
) =
a
x
(say). Then
J
=
[
∫∫∫
ρ
~
F
d
V
]
x
=
[
∫∫
ρ~u
(
~u
·
~n
) d
S
]
x
−
[
∫∫
(
−
pI
+
μ
def
~u
~n
d
S
]
x
=
∫∫
ρ
(
u
cos
θ
−
v
sin
θ
)
u
d
S
+
∫∫
p
cos
θ
d
S
(8.38)
−
μ
∫∫
[
2
∂u
∂r
cos
θ
−
(
1
r
∂u
∂θ
+
∂v
∂r
−
v
r
)
sin
θ
]
d
S .
A brief digression is necessary to calculate the pressure
p
from the
two non-zero components of the momentum equations (8.8) and (8.9)
above. For the exact solution
ψ
=
νrf
(
ξ
), these become
1
ρ
∂p
∂r
=
ν
2
r
3
[
2
ξf
′′
−
(1
−
ξ
2
)
f
′′′
+
f
′
f
′
+
f f
′′
+
f
2
(1
−
ξ
2
)
]
=
ν
2
r
3
[
2
f
′
+
f
2
(1
−
ξ
2
)
]
(8.39)
and
1
ρr
∂p
∂θ
=
−
ν
2
r
3
[
f
′′
+
f f
′
(1
−
ξ
2
)
+
ξf
2
(1
−
ξ
2
)
2
]
,
(8.40)
after use of equation (8.22) with
C
1
= 0 in equation (8.39). The first
equation can be integrated to obtain
p
ρ
=
−
ν
2
2
r
2
[
2
f
′
+
f
2
(1
−
ξ
2
)
]
+
g
(
ξ
)
.
(8.41)
Differentiation then shows that the equation (8.40) is satisfied if
g
(
ξ
) = constant =
p
0
/ρ
(say). Given that
ξ
,
f
(
ξ
), etc. are of or-
der unity, the difference between
p/ρ
and
p
0
/ρ
for large
r
is of order
(
ν/r
)
2
, as are the squared velocities from equations (8.30) and (8.31).
Another form of equation (8.41) that may be useful is
p
ρ
+
v
2
2
−
νu
r
=
p
0
ρ
.
(8.42)
On the jet axis, where
f
(1) = 0 and
f
′
(1) =
−
4
/c
from equa-
tion (8.27), it follows from equation (8.41) that the static pressure
8.1. LAMINAR ROUND JET INTO FLUID AT REST
411
slightly exceeds the stagnation pressure
p
0
/ρ
(why not Bernoulli
equation far from jet?); (
is this so?
)
p
ρ
=
p
0
ρ
+
4
ν
2
r
2
c
.
(8.43)
This expression already suggests that
c
should approach zero like
ν
2
as the Reynolds number increases.
Use of equation (8.41) for
p
and equations (8.30) and (8.31) for
u
and
v
in equation (8.38) leads eventually to
J
= 2
πρν
2
∫
1
−
1
[
f
′
f
′
−
f
2
2(1
−
ξ
2
)
−
3
f
′
]
ξ
d
ξ .
(8.44)
After substitution for
f
(
ξ
) from equation (8.27) and evaluation
of the integrals, there is obtained
J
8
πρν
2
=
8
3
(
c
+ 1)
c
(
c
+ 2)
+ 2(
c
+ 1)
−
(
c
+ 1)
2
ln
(
c
+ 2
c
)
,
(8.45)
which shows precisely how the exact solution depends on the single
parameter
J/ρν
2
. Particularly useful for what follows is an expansion
for small
c
,
J
8
πρν
2
=
4
3
c
+ ln
c
+
(
8
3
−
ln 2
)
+ 2
c
ln
c
+ 2
c
(
7
12
−
ln 2
)
+
+
c
2
ln
c
−
c
2
(
17
24
+ ln 2
)
+
... ,
(8.46)
where
...
stands for the third and higher powers of
c
. There are
no more logarithms. The series evidently converges for 0
< c <
2.
(
check
) The leading term represents the boundary-layer approxima-
tion (see below);
c
=
32
3
π
ρν
2
J
.
(8.47)
(
Jet out of wall? Do
θ
0
here?
)
412
CHAPTER 8.
THE ROUND JET
8.1.4
The boundary-layer approximation
Suppose now that the exact solution of the Navier-Stokes equations
is not known. The round jet into fluid at rest can also be approached
from the outset as a boundary-layer problem of classical type, and
was approached in this way by SCHLICHTING (1933,
check
) be-
fore the exact solution was discovered. The essential assumption is
that the jet is concentrated near the polar axis, as indicated in FIG-
URE 8.3. A suitable magnified boundary-layer variable in spherical
polar coordinates is evidently
θ
=
θ
,
(8.48)
where the small quantity
is specified to be dimensionless and inde-
pendent of
r
and
θ
, with a magnitude chosen to make
θ
=
O
(1) in
the body of the jet. By assumption,
→
0 as
ν
→
0 or Re
→ ∞
.
The continuity equation (8.7) with sin
θ
∼
θ
becomes
1
r
2
∂
∂r
ur
2
+
1
r
θ
∂
∂
θ
v
θ
= 0
.
(8.49)
The two terms must be of the same order, and the
v
-velocity must
also be magnified by a factor 1
/
,
v
=
v
,
(8.50)
to give
1
r
2
∂
∂r
ur
2
+
1
r
θ
∂
∂
θ
v
θ
= 0
.
(8.51)
Now introduce a stream function in the usual way, putting
u
=
1
r
2
θ
∂
ψ
∂
θ
,
v
=
−
1
r
θ
∂
ψ
∂r
,
(8.52)
where the stream function
ψ
is magnified according to its own rule;
ψ
=
ψ
2
.
(8.53)
8.1. LAMINAR ROUND JET INTO FLUID AT REST
413
The azimuthal vorticity, defined by equation (8.11) becomes
ζ
=
r
∂r
v
∂r
−
1
r
∂u
∂
θ
(8.54)
and leads to
ζ
=
ζ
.
(8.55)
After these preliminaries, the radial and azimuthal momentum
equations (8.8) and (8.9) become, respectively (
check these care-
fully
),
u
∂u
∂r
+
v
r
∂u
∂
θ
−
2
v
2
r
=
−
1
ρ
∂p
∂r
+
ν
2
(
2
r
2
∂
∂r
r
2
∂u
∂r
+
1
r
2
θ
∂
∂
θ
θ
∂u
∂
θ
−
2
2
u
r
2
−
2
2
r
2
θ
∂
v
θ
∂θ
)
,
(8.56)
2
(
u
∂
v
∂r
+
v
r
∂
v
∂
θ
+
u
v
r
)
=
−
1
ρr
∂p
∂
θ
+
ν
2
(
4
r
2
∂
∂r
r
2
∂
v
∂r
+
2
r
2
θ
∂
∂
θ
θ
∂
v
∂
θ
+
2
2
r
2
∂u
∂
θ
−
2
v
r
2
θ
2
)
.
(8.57)
The equations of motion are now ready for the boundary-layer
approximation or inner limit. This is the limit
→
0 with
r
and
θ
=
θ/
fixed and
O
(1), so that points in the body of the jet remain
in the jet, even in the limit as the body of the jet becomes the polar
axis.
Because at least one of the viscous terms in the first equation
(8.56) must survive in the limit
→
0, it is necessary that
∼
ν
1
/
2
.
(8.58)
Each of the terms in the second equation (8.57), except possibly the
pressure term, is at most
O
(
2
), and this must therefore also be true
of the pressure term. Hence
∂p/∂
θ
=
O
(
2
), or
∂p
∂θ
=
O
(
) =
O
(
ν
1
/
2
)
.
(8.59)
414
CHAPTER 8.
THE ROUND JET
This estimate can be confirmed from equation (8.41) (
check
). To
this order, the pressure is constant across the body of the jet, al-
though the constant may depend on
r
. However, the ambient fluid
has been stipulated to be at rest. Hence the centripetal-acceleration
term and the pressure-gradient term can be dropped entirely. When
physical variables are restored, the boundary-layer problem is defined
by
u
∂u
∂r
+
v
r
∂u
∂θ
=
ν
(
1
r
2
θ
∂
∂θ
θ
∂u
∂θ
)
(8.60)
with
u
=
1
r
2
θ
∂ψ
∂θ
, v
=
−
1
rθ
∂ψ
∂r
.
(8.61)
The pressure is no longer a dependent variable, but is determined as
part of the boundary conditions. The order of the governing equa-
tions is reduced by one.
As is usual with boundary-layer problems, the validity of the
argument just given can be tested
a posteriori
by using the boundary-
layer solution in the full equations.
(Expand on this.)
8.1.5
The boundary-layer Solution
The original dimensional argument for the form of the solution made
no use of equations and is unchanged by the boundary-layer approx-
imation leading to equation (8.60). It should be possible to argue
this form using an affine transformation together with the associated
invariants (a scheme which is equivalent to a dimensional argument),
but I am not satisfied with the analysis at present.
(Do this)
. The
absence of characteristic scales again requires the ansatz
ψ
=
νrg
(
θ
)
(8.62)
where
θ
now means
θ
and not cos
θ
as in the exact solution. The
velocity components from equation (8.12) become
u
=
ν
rθ
g
′
,
v
=
−
ν
rθ
g ,
(8.63)
8.1. LAMINAR ROUND JET INTO FLUID AT REST
415
with
ζ
=
−
1
r
∂u
∂θ
=
−
ν
r
(
g
′′
θ
−
g
′
θ
2
)
.
(8.64)
Substitution for
u
and
v
in equation (8.60) yields
−
g
′
g
′
θ
+
gg
′
θ
2
−
gg
′′
θ
−
g
′
θ
2
+
g
′′
θ
−
g
′′′
= 0
.
(8.65)
This equation is third order (where equation (8.21) was fourth order)
because the absence of the pressure in the boundary-layer approx-
imation makes it unnecessary to take the curl. A first integration
gives
−
gg
′
θ
+
g
′
θ
−
g
′′
=
C
1
= 0
.
(8.66)
To show that the constant
C
1
is zero, note from the first of equa-
tions (8.63) that a power-series expansion for
g
(
θ
) must begin with
a term in
θ
2
. A second and non-trivial integration, with the same
condition at
θ
= 0, gives
−
g
2
2
θ
+
2
g
θ
−
g
′
=
C
2
= 0
(8.67)
and finally
g
=
4
θ
2
θ
2
+ 2
c
,
(8.68)
where 2
c
is an undetermined constant of integration.
The boundary-layer approximation to the momentum integral
(8.38) is
(connect this with momentum integral for plane jet)
J
= 2
πρ
2
π
∫
0
u
2
r
2
θ
d
θ .
(8.69)
Note that the axial or
x
-component of velocity,
u
cos
θ
−
v
sin
θ
=
u
−
2
v
θ
, is indistinguishable from the radial component for
→
0.
A similar statement holds for the radial and axial coordinates
r
and
z
. Consequently,
J
=
2
2
πρ
ν
2
∫
∞
0
g
′
g
′
θ
d
θ .
(8.70)
416
CHAPTER 8.
THE ROUND JET
Substitution from equation (8.68) for
g
and integration give a relation
between
J
and
c
,
J
2
πρν
2
=
16
3
c
.
(8.71)
8.1.6
The inner limit
The operational diagram in FIGURE X
2
identifies the boundary-
layer solution with the inner limit of the exact solution. With the
preliminary approximation sin
θ
∼
θ
, cos
θ
∼
1
−
θ
2
/
2 for small
θ
,
the exact solution from equations (8.29)–(8.31) is reduced to
ψ
= 4
νr
θ
2
(
θ
2
+ 2
c
)
,
(8.72)
u
=
16
νc
r
1
(
θ
2
+ 2
c
)
2
,
(8.73)
v
=
−
4
ν
r
θ
(
θ
2
+ 2
c
)
,
(8.74)
ζ
=
4
ν
r
2
c
(
c
+ 2)
θ
(
θ
2
/
2 +
c
)
3
.
(8.75)
In the virtual variables defined by equations (8.33)–(8.35), the
boundary-layer approximation for the stream function is
Ψ =
R
Θ
2
Θ
2
+ 2
c
.
(8.76)
These streamlines are plotted in FIGURE 15.x for a value
c
= 0
.
005
(
check
).
3
Again there is only one streamline, with others derived
from this by a zoom transformation. The fictitious spherical stream
surfaces at the left are generated by equation (8.76) when Θ
c
, so
that Ψ =
R
, approximately. This behavior is an artifact of the spher-
ical polar coordinate system and the fact that the boundary-layer
2
It is not known what figure this is meant to refer to.
3
Possibly refers to a missing Figure 8.4
8.1. LAMINAR ROUND JET INTO FLUID AT REST
417
solution has no meaning outside the boundary layer. The charac-
teristics of the boundary-layer equations are the lines
r
= constant.
There is no upstream diffusion of vorticity, and upstream here means
directed inward toward the origin along a radius in FIGURE 15.x.
SQUIRE (1955) encountered this behavior in his third paper, whose
subject was conical laminar jets in spherical polar coordinates. He
questioned the behavior, but did not resolve it.
Equations (8.xx) are identical with the boundary-layer solution
given by equations (8.62), (8.63), and (8.68). Finally, equation (8.71)
in the form
J
2
πρν
2
=
16
3
c
(8.77)
is seen to be the leading term in the expansion (8.46). In short,
the inner limit of the exact solution coincides in all respects with
the solution of the inner limit of the exact equations, as originally
claimed for the operational diagram, FIGURE 8.2.
8.1.7
The outer limit
As a fluid element is entrained in the jet, it first undergoes a rapid
acceleration. This is followed by a slow deceleration, as the element
finds itself close to the jet axis, where
u
∼
1
/r
. Each stream tube
therefore first converges and then diverges. A convenient measure for
the angle
θ
, say
θ
=
θ
0
, is provided by the point of closest approach
to the axis (other nearly equivalent measures can be defined). In
boundary-layer variables, as indicated in the sketch (
comment on
outer flow
),
ψ
=
4
νr
θ
2
θ
2
+ 2
c
=
4
νz
R
2
R
2
+ 2
cz
2
.
(8.78)
Along a streamline, therefore,
d
R
d
z
=
R
4
cz
3
(
2
cz
2
−
R
2
)
.
(8.79)
The derivative vanishes when
R
2
0
z
2
0
=
θ
2
0
= 2
c .
(8.80)
418
CHAPTER 8.
THE ROUND JET
The parameter
θ
0
= (2
c
)
1
/
2
=
(
32
3
2
πρν
2
J
)
1
/
2
(8.81)
suggests itself
a posteriori
as a suitable quantitative choice for the di-
mensionless parameter
, with all of the correct properties, beginning
with the property
̄
θ
= O(1). In fact, according to equation (8.48),
where
was first introduced, this choice amounts to putting
θ
0
= 1.
(
Is there an equivalent for other flows? Why stick to spherical polar
coordinates? Note that the thickness of the laminar round jet varies
like
ν
, not
ν
1
/
2
: mention subcharacteristics. Figure out what this
means for
J
and
ν
.)
(
Need to consider laminar round jet out of wall; see p. 19 of
1981 notes and paper by Squire.
)
Outside the jet; i.e., for
θ
θ
0
, the stream function and ve-
locity components from equations (8.72)–(8.74) approach (
mention
circle, Squire
)
ψ
= 4
νr
or
ψ
= 4
νr ,
(8.82)
u
=
16
ν
c
θ
4
or
u
=
16
νc
θ
4
,
(8.83)
v
=
−
4
ν
r
θ
or
v
=
−
4
ν
rθ
=
−
4
ν
R
.
(8.84)
It is worth noting that the streamwise velocity
u
approaches zero for
large
θ
algebraically, like
θ
−
4
, rather than exponentially; and also
that a power series expansion for
u
(
θ
) converges only for
θ < θ
0
,
and thus for
u
(
θ
)
/u
(0)
>
1
/
4 from equation (8.73). The reason is
the existence of simple poles at
θ
=
±
iθ
0
in the complex
θ
-plane
(
mention Weyl?
). (
Exact solution has same property?
)
The message of equation (8.74) is that
rv
outside the boundary
layer is independent of
r
, and therefore that the action of the jet
on the ambient fluid is essentially like that of a uniform line sink
8.1. LAMINAR ROUND JET INTO FLUID AT REST
419
along the positive
z
-axis. To calculate the associated potential flow,
first use the representation for a sink of strength
Q
at the origin in
spherical polar coordinates;
u
=
−
Q
4
πρr
2
.
(8.85)
For a distribution of such sinks along the positive
z
-axis the velocity
becomes
u
=
∫
d
u
=
∫
cos (
θ
′
−
θ
) d
u
′
=
−
∫
cos (
θ
′
−
θ
)d
Q
4
πρr
′
2
(8.86)
in the notation shown in the sketch. For a uniform sink, put
(explain
why there should be a constant
A
)
d
Q
= 4
πρA
d
z
′
(8.87)
where
A
is a constant to be determined from the boundary condi-
tions. Note from the sketch that
z
−
z
′
r
sin
θ
= cot
θ
′
,
d
z
′
=
r
sin
θ
d
θ
′
sin
2
θ
′
.
(8.88)
Finally, therefore,
u
=
−
A
r
sin
θ
∫
π
θ
cos(
θ
′
−
θ
) d
θ
′
=
−
A
r
.
(8.89)
The outer stream function in spherical polar coordinates follows from
the first of equations (8.12),
ψ
=
Ar
cos
θ
+
B
(
r
)
.
(8.90)
The boundary conditions are
ψ
= 4
νr
at
θ
= 0 and
ψ
= 0 at
θ
=
π
.
Consequently
A
= 2
ν
and
B
(
r
) = 2
νr
, and
ψ
= 2
νr
(1 + cos
θ
)
.
(8.91)
This expression is evidently the outer limit (
→
0 or
c
→
0
with
r
and
θ
fixed) of the exact solution (8.29). The streamlines
described by equation (8.91) are confocal paraboloids of revolution,
420
CHAPTER 8.
THE ROUND JET
as shown in the sketch.
4
Write
ψ/
2
ν
=
r
+
z
, take the square, and
use
r
2
=
R
2
+
z
2
to obtain
R
2
=
ψ
ν
(
ψ
4
ν
−
z
)
.
(8.92)
It is worth noting that the streamlines from the boundary-layer so-
lution outside the boundary layer,
ψ
= 4
νr
, are concentric circles
(see Squire). If the boundary-layer approximation had been made
in cylindrical polar coordinates, there would have been obtained
ψ
= 4
νz
= 4
νr
cos
θ
. These streamlines are straight lines normal to
the
z
-axis. Neither result is useful, because the boundary-layer ap-
proximation should not be relied on outside the boundary layer. The
correct streamlines are the paraboloids (?) given by equation (8.91),
ψ
= 2
νr
(1 + cos
θ
).
(Comment on the outer limit of the exact
solution.)
The rule for constructing the composite expansion (see Van
Dyke 1975 and the sketch) is to add the inner and outer approxima-
tions and subtract the common part. The procedure is illustrated in
the sketch. From equations (8.72) and (8.91),
ψ
c
= 4
νr
θ
2
θ
2
+
θ
2
0
+ 2
νr
(1 + cos
θ
)
−
4
νr .
(8.93)
For
θ
∼
θ
0
, the third term essentially cancels the second, and the
first term dominates. For
θ
θ
0
, the third term essentially cancels
the first, and the second term dominates. It is plausible that the
expression (8.93) is a uniformly valid approximation to the exact
solution of equation (8.29),
ψ
e
= 2
νr
sin
2
θ
1
−
cos
θ
+
c
.
(8.94)
That is, the quantity
∣
∣
∣
∣
ψ
e
−
ψ
c
2
νr
∣
∣
∣
∣
(8.95)
should be a bounded function of
θ
for sufficiently small
c
(actually
for all
c
).
4
It is not known what sketch this refers to.
8.1. LAMINAR ROUND JET INTO FLUID AT REST
421
8.1.8
Miscellaneous remarks
CHIN (1981) recently showed that confocal paraboloidal coordinates
are optimal for the Squire-Landau problem in the sense defined by
KAPLUN (1954); the boundary-layer solution includes the outer so-
lution, although the boundary-layer solution is not exact. In my
review of Chin’s paper for another journal, I objected (unsuccess-
fully) that it is not necessary to derive and solve the boundary-layer
equations in the paraboloidal system, since Kaplun’s substitution
theorem is more efficient.
WYGNANSKI (1970) has extended the original exact Squire-
Landau solution in equations (8.29)–(8.32) to the case of flow with
swirl by resorting to numerical methods. The minimum in the axial
velocity on the axis for large swirl, the increased entrainment, and
the approach of the outer flow to a viscous core/potential vortex
motion are clearly brought out.
SQUIRE in a second paper (1952) considered the exact prob-
lem when there is a conical wall at
θ
= Θ, particularly Θ =
π/
2,
with a slip boundary condition at the wall. The issue is mainly the
evaluation of the constants
c
1
,
c
2
,
c
3
in equations (8.22)–(8.24) when
these are not all zero, as well as the complications that set in dur-
ing the final integration step. SCHNEIDER (1981) claims to find an
exact solution for this case with a no-slip condition, thus taking into
account the displacement effect on the outer flow of the boundary
layer on the wall. I have not studied this paper closely enough to
understand in what sense the solution is exact. GINEVSKII (1966)
carried out the outer-flow approximation for a turbulent jet with a
wall at
θ
= Θ, in the manner used to obtain equation (8.91). The
entrainment velocity was estimated by using the polynomial mean-
velocity profile. The two-dimensional case is treated similarly. In no
case was a no-slip condition applied at the wall.
A third paper by SQUIRE (1955) attempted to treat a conical
jet lying along a surface
θ
=
θ
∗
, with the radial jet (
θ
∗
=
π/
2)
as an important special case. The analysis uses a boundary-layer
approximation, and the streamlines outside the jet show the same
pathological behavior shown by equations (8.72) for
θ
2
2
c
. The
422
CHAPTER 8.
THE ROUND JET
outer flow and the composite expansion are not considered.
A few other references are cited in ROSENHEAD (1963), WYG-
NANSKI (1970), and SCHNEIDER (1981).
8.2 Laminar round jet into moving fluid
8.3 Transition
ANDRADE (1937)
DOMM et al (1955)
BECKER and MASSARO (1968)
SYMONS and LABUS (1971)
ZAUNER (1985)
TUCKER and ISLAM (1986)
PETERSEN et al (1988)
MEIBURG et al (1989)
LIEPMANN (1991)
BROZE and HUSSAIN (1994)
TONG and WARHAFT (1994)
8.3. TRANSITION
423
References
Batchelor, G. K. 1967 “An Introduction to Fluid Dynamics,” Cam-
bridge, 205–211.
Chin, W. C. 1981
Optimal Coordinates for Squire’s Jet
. AIAA
J.
19
, 123-124.
Kaplun, S. 1954
The Role of Coordinate Systems in Boundary-
Layer Theory
. ZAMP
5
, 111–135.
Landau, L. F 1944
A New Exact Solution of the Navier Stokes
Equations
. Doklady Acad. Sci.
43
, 286–288.
Rosenhead, L. (ed.) 1963
Laminar Boundary Layers,
Oxford,
150–155.
Schneider, W. 1981
Flow Induced by Jets and Plumes
. J. Fluid
Mech.
108
, 55–65.
Slezkin, N. A. 1934
On an Exact Solution of the Equations of
Viscous Flow
. Scientific Papers, Moscow State University, No. 2 (I
have not seen this paper).
Squire, H. B. 1951
The Round Laminar Jet
. Quart. J. Mech.
Appl. Math.
4
, 321–329.
Squire, H. B. 1952
Some Viscous Fluid Flow Problems. I: Jet
Emerging From a Hole in a Plane Wall
. Phil. Mag. (7)
43
, 942–945.
Squire, H. B. 1955
Radial Jets
. In
50 Jahre Grenzschichtforschung,
Vieweg, Braunschweig, 47–54.
Van Dyke, M. 1975 “Perturbation Methods in Fluid Mechan-
ics,” Parabolic Press.
Wygnanski, I. 1970
Swirling Axisymmetrical Laminar Jet
. Phys.
Fluids
13
, 2455–2460.
424
CHAPTER 8.
THE ROUND JET
8.4 Turbulent round jet into fluid at rest
[This section was found in a separate file and appears to belong here,
though in its found form it repeated the first two paragraphs of the
start of this chapter]
For the round jet, cylindrical polar coordinates (
r,θ,z
) are
appropriate because experimenters move their probes in a plane
z
= constant. The velocity components are (
u,v,w
), and the jet
motion is along the positive
z
-axis. The ambient fluid, which is here
taken to be the same as the fluid in the jet, is nominally at rest, so
that the pressure is nominally constant. If the mean motion is steady,
axially symmetric, and free of swirl, the boundary-layer equations of
motion are
1
r
∂ru
∂r
+
∂w
∂z
= 0
,
(8.96)
u
∂w
∂r
+
w
∂w
∂z
=
1
ρr
∂rτ
∂r
,
(8.97)
τ
=
μ
∂w
∂r
−
ρ
u
′
w
′
.
(8.98)
The first and second equations can be combined in the form
∂ruw
∂r
+
∂rww
∂z
=
1
ρ
∂rτ
∂r
(8.99)
and this expression can be integrated over a plane
z
= constant, with
boundary conditions
u
= 0 at
r
= 0 and
w
=
τ
= 0 at
r
=
∞
, to
obtain the momentum integral
J
= 2
πρ
∫
∞
0
rwwdr
= constant
.
(8.100)
The parameters of the problem are
J
and
ρ
, with, from equa-
tion (8.100),
[
J
ρ
]
=
L
4
T
2
=
L
2
U
2
.
(8.101)
8.4. TURBULENT ROUND JET INTO FLUID AT REST
425
There is nothing else to work with. The absence of a characteris-
tic length means that the jet must grow conically. When a stream
function is introduced such that
u
=
−
1
r
∂ψ
∂z
,
w
=
1
r
∂ψ
∂r
(8.102)
it is seen that
[
ψ
] =
L
2
U
=
L
[
J
ρ
]
1
/
2
.
(8.103)
(Profile is
f
′
/η
, not
f
′
. Can this be fixed?)
The profile sim-
ilarity assumption is therefore to use
z
and
J/ρ
to form the non-
dimensionalizing combination; thus
Aψ
=
z
(
J
ρ
)
1
/
2
f
(
B
r
z
)
(8.104)
where
A
and
B
are disposable dimensionless constants.
(Why not
use spherical polar coordinates?)
The mean velocity components are
u
=
−
1
Ar
(
J
ρ
)
1
/
2
(
f
−
ηf
′
)
,
(8.105)
w
=
B
Ar
(
J
ρ
)
1
/
2
f
′
.
(8.106)
Papers with profiles can give both the growth rate and the cen-
terline velocity decay. To begin with, use only data that include both
sides of the profile (denoted by sym). Try 3 functions for fit near
plane of symmetry. Look for constant in growth rate and distance re-
quired to achieve it after arbitrary initial condition. Apparent origins
from
δ
(
x
)
and
u
o
/u
c
(
x
)
should be the same.
Papers with profiles:
VOORHEIS (1940)
REICHARDT (1942) 9A (sym)
ALBERTSON et al. (1948)