Chapter 5
THE SHEAR LAYER
The shear layer or mixing layer is a more delicate analytical problem
than those treated so far. The main application of this flow is at
the edge of a jet or flow over a cavity. See high-bypass jet engines.
An important special case is that of different densities. This is a
prototype problem for coherent structures.
Roshko and some other investigators prefer the term organized
structure to the term coherent structure, because of the meaning of
the word coherent in optics and other wave phenomena. My own
position is that coherent has another meaning, as in coherent speech,
that is quite appropriate.
5.1 Plane laminar shear layer
The flow shown in FIGURE X is a plane shear layer between two
parallel streams having constant velocities
u
1
and
u
2
, where
u
1
in a
standard notation denotes the upper, higher-speed stream. The two
streams are separated for
x <
0 by a thin splitter plate or septum
whose boundary layers are neglected in the analysis, although they
can be a source of difficulty in practice. Especially when the velocity
u
2
in the lower stream is small, the plane mixing layer is boundary-
367
368
CHAPTER 5.
THE SHEAR LAYER
layer-like in its upper portion and jet-like in its lower portion.
5.1.1
Equations of motion
The laminar boundary-layer approximation in rectangular coordi-
nates is
∂u
∂x
+
∂v
∂y
= 0
(5.1)
ρ
(
∂uu
∂x
+
∂uv
∂y
)
=
μ
∂
2
u
∂y
2
.
(5.2)
The boundary conditions are that the pressure is constant every-
where and that the external streams are uniform;
u
(
x,
∞
) =
u
1
,
u
(
x,
−∞
) =
u
2
.
(5.3)
The global parameters for the shear layer are
u
1
,
u
2
,
ρ
,
μ
.
These parameters provide two characteristic velocities,
u
1
and
u
2
,
and corresponding lengths,
ν/u
1
and
ν/u
2
, but no dimensionless
combination except the velocity ratio
u
2
/u
1
itself, so that the mixing
layer forms a one-parameter family of flows.
Within the boundary-layer approximation, the flow in the shear
layer is not fully determined by the global parameters just listed.
The root of the problem is the lack of symmetry in the boundary
conditions, and the resolution of the problem is in some degree still
open.
The standard first step is to test for the existence of an integral
invariant. Let the momentum equation (5.2) be integrated formally
from
−∞
to
∞
in
y
(this means from
−
y
to
y
, with
y
→∞
as a final
step). The form obtained is
d
d
x
∞
∫
−∞
uu
d
y
=
−
u
1
v
1
+
u
2
v
2
(5.4)
where
v
1
=
v
(
x,
∞
)
,
v
2
=
v
(
x,
−∞
)
.
(5.5)
5.1. PLANE LAMINAR SHEAR LAYER
369
The same operation on the continuity equation (5.1) yields
d
d
x
∞
∫
−∞
u
d
y
=
−
v
1
+
v
2
.
(5.6)
To avoid the difficulty that the integrals diverge, consider the identity
d
d
x
∞
∫
−∞
(
u
−
u
1
)(
u
−
u
2
)d
y
=
d
d
x
∞
∫
−∞
uu
d
y
−
(
u
1
+
u
2
)
d
d
x
∞
∫
−∞
u
d
y .
(5.7)
Substitution of equations (5.4) and (5.6) in (5.7) yields
d
d
x
∞
∫
−∞
(
u
−
u
1
)(
u
−
u
2
) d
y
=
−
u
1
v
2
+
u
2
v
1
.
(5.8)
The dimensionless form of this integral will be considered in SEC-
TION X.
A different but equivalent approach is to treat equations (5.4)
and (5.6) as linear algebraic equations for
v
1
and
v
2
. Solution gives
v
1
=
1
(
u
1
−
u
2
)
d
d
x
∞
∫
−∞
u
(
u
2
−
u
) d
y
(5.9)
and
v
2
=
1
(
u
1
−
u
2
)
d
d
x
∞
∫
−∞
u
(
u
1
−
u
) d
y .
(5.10)
Substitution of these expressions in equation (5.4) leads back to equa-
tion (5.7).
370
CHAPTER 5.
THE SHEAR LAYER
5.1.2
Similarity
The appropriate affine transformation is
x
=
a
̂
x
y
=
b
̂
y
ψ
=
c
̂
ψ
ρ
=
d
̂
ρ
μ
=
e
̂
μ
(5.11)
u
1
=
p
̂
u
1
u
2
=
q
̂
u
2
=
p
̂
u
2
u
=
r
̂
u
=
c
b
̂
u
v
=
s
̂
v
=
c
a
̂
v .
Some of the scaling factors in equation (5.4) are redundant,
as indicated in the third column. First, introduction of a stream
function defined by
~u
= grad
ψ
×
grad
z
or equivalently by
u
=
∂ψ
∂y
,
v
=
−
∂ψ
∂x
(5.12)
leads to
r
̂
u
=
c
b
∂
̂
ψ
∂
y
,
s
̂
v
=
−
c
a
∂
̂
ψ
x
(5.13)
and thus to the equalities
r
=
c/b
,
s
=
c/a
. Second, the boundary
conditions (5.3) are transformed to
r
̂
u
(
a
̂
x,
∞
) =
p
̂
u
1
,
r
̂
u
(
a
̂
x,
−∞
) =
q
̂
u
2
.
(5.14)
5.1. PLANE LAMINAR SHEAR LAYER
371
Because
a
̂
x
is read “any value of
̂
x
,” the value of
a
is immaterial, so
that
r
=
p
=
q
. These results are incorporated in the group (5.11).
They yield one invariant of the mapping, representing the boundary
conditions. I take
u
1
as fundamental because it is by definition never
zero, and rewrite the equivalence
r
=
c/b
=
p
as
c
bp
= 1
.
(5.15)
Transformation of the momentum equation (5.2) yields a second in-
variant,
bdc
ae
= 1
.
(5.16)
After isolation of
c
and
b
, these invariants become
c
2
d
aep
= 1
,
b
2
dp
ae
= 1
(5.17)
and thus lead to the preliminary ansatz
A
ψ
(
Uνx
)
1
/
2
=
f
[
B
(
U
νx
)
1
/
2
y
]
=
f
(
ξ
)
(5.18)
where the quantity
U
is a generic global velocity whose essential
property is that it transforms like
u
1
or
u
2
; that is,
U
=
p
̂
U
. The
term generic is appropriate because any value of
U
on either side of
equation (5.18) can be replaced by another value by a suitable choice
of the normalizing constants
A
and
B
.
Substitution of the ansatz (5.18) in the momentum equation
(5.2) leads to the Blasius differential equation
2
ABf
′′′
+
ff
′′
= 0
(5.19)
with the boundary conditions
f
′
(
∞
) =
A
B
u
1
U
,
f
′
(
−∞
) =
A
B
u
2
U
.
(5.20)
Note that only two boundary conditions have been established
for a third-order ordinary differential equation. For all of the other
372
CHAPTER 5.
THE SHEAR LAYER
laminar shear flows considered in this monograph, a natural origin
for the
y
-coordinate and a natural third boundary condition, usually
in the form
ψ
= 0 on
y
= 0, or
f
(0) = 0, are provided either by an
explicit symmetry condition or by an implicit symmetry condition
associated, for example, with the presence of a plane wall bounding
the flow on the high-speed side. For the shear layer, this condition
is still appropriate in the upstream region where the two fluids are
physically separated by the septum. Downstream from the trailing
edge, the symmetry condition is replaced by the concept of the divid-
ing streamline, defined as the locus where
ψ
= 0 or
f
= 0 for
x >
0.
The two fluids, although they are assumed to have identical physical
properties, are still separated by a hypothetical surface that begins
at the trailing edge.
(The problem of the shear layer for two
incompressible and immiscible fluids having different den-
sities and/or viscosities was treated by Keuligan (ref ) and
Lock (ref ), both thinking of wind over water.)
The lack of
symmetry makes it unlikely that the dividing streamline
ψ
= 0 coin-
cides with
y
= 0 or
ξ
= 0, where
ξ
is the argument of
f
in equation
(5.23). At the same time, similarity requires
ξ
to be constant on the
dividing streamline.
(Why?)
Successive differentiation of equation (5.18) shows that the cor-
respondence between physical variables and dimensionless similarity
variables is
ψ
∼
f
,
u
∼
f
′
,
v
∼
ξf
′
−
f
,
τ
∼
f
′′
,
∂τ/∂y
∼
f
′′′
. The
Blasius equation (5.19) requires
f
′′′
= 0 when
f
= 0. Consequently,
the dividing streamline coincides with a maximum in the shearing
stress and with an inflection point in the velocity profile, as was
first pointed out by (
ref
). On practical grounds, it therefore seems
preferable to move the origin for the dimensionless
y
-coordinate to
the dividing streamline. That is, put
η
=
ξ
+
C .
(5.21)
The ansatz (5.18) should be revised to read
A
ψ
(
Uνx
)
1
/
2
=
f
{
B
(
U
νx
)
1
/
2
y
+
C
}
=
f
(
η
)
.
(5.22)
The differential equation is still (5.19), the first two boundary con-
5.1. PLANE LAMINAR SHEAR LAYER
373
ditions are still (5.20), and the third boundary condition is now
f
(0) = 0
(5.23)
on the dividing streamline, which for
x >
0 is the parabola
B
(
U/νx
)
1
/
2
y
=
−
C
, with
C >
0. (Typical dependent variables
for one value of
u
2
/u
1
are shown in figure x). The boundary con-
dition (5.23) does not necessarily fix the position of the dividing
streamline in physical space, because there may not be enough infor-
mation to determine the constant
C
.
Point out somewhere the role of
C
in making a connection
with Stewartson’s limiting separating boundary layer and with the
blow-off condition for the boundary layer with mass transfer
.)
5.1.3
Normalization
The normalization used throughout this monograph for laminar plane
flows requires putting
2
AB
= 1
(5.24)
in equation (5.19) in order to obtain the standard Blasius operator
f
′′′
+
ff
′′
. One further condition is needed to determine
A
and
B
. The velocity parameter
U
stands in the way. Recall that the
scaling parameter
p
in the affine group (5.11) can refer to
u
1
or
u
2
,
which transform in the same way, or to any suitable combination,
not necessarily linear, of
u
1
and
u
2
. Consider the case of a moving
observer who starts a clock as he passes the station
x
= 0. The
basic diffusion process provides an estimate
δ
2
∼
νt
for the layer
thickness seen by the observer. A plausible choice for the velocity of
the observer is the arithmetic mean of
u
1
and
u
2
, and his position is
then
x
∼
(
u
1
+
u
2
)
t
. He therefore sees a thickness
δ
∼
(
νx
u
1
+
u
2
)
1
/
2
.
(5.25)
If the constant of proportionality does not depend on
u
2
/u
1
, the
rate of growth is decreased by a factor of
√
2, other things being
374
CHAPTER 5.
THE SHEAR LAYER
equal, as
u
2
increases from zero to
u
1
.
(Check the literature for
calculations.)
An independent condition on
δ
is implicit in the form of the
dimensionless variable
η
in equation (5.22). The thickness
δ
repre-
sents an increment in
y
, and there is a corresponding increment in
η
which is not dependent on the value of
C
. It is enough to write the
proportionality
δ
∼
1
B
(
νx
U
)
1
/
2
.
(5.26)
The last two equations are consistent if the equality holds,
U
=
u
1
+
u
2
.
(5.27)
The boundary conditions (5.20) keep their parallel form. Since now
B
= 1
/
2
A
,
f
′
(
∞
) = 2
A
2
(
u
1
u
1
+
u
2
)
, f
′
(
−∞
) = 2
A
2
(
u
2
u
1
+
u
2
)
.
(5.28)
Finally, for no better reason than that the values
f
′
(
∞
) = 1,
f
′
(
−∞
) =
0 seem well suited to the special case
u
2
= 0, I take
2
A
2
= 1
(5.29)
so that
A
=
B
= 2
−
1
/
2
.
With this normalization, the ansatz (5.22) becomes
ψ
[2(
u
1
+
u
2
)
νx
]
1
/
2
=
f
[
(
u
1
+
u
2
2
νx
)
1
/
2
y
+
C
]
=
f
(
η
)
.
(5.30)
The function
f
satisfies the Blasius equation
f
′′′
+
f f
′′
= 0
(5.31)
with two boundary conditions
f
′
(
∞
) =
u
1
u
1
+
u
2
,
f
′
(
−∞
) =
u
2
u
1
+
u
2
.
(5.32)
5.1. PLANE LAMINAR SHEAR LAYER
375
The solutions of equation (5.31) form a single-parameter family, and
the parameter,
u
2
/u
1
, say, appears explicitly only in the boundary
conditions (5.32).
Integration of equation (5.31) can proceed formally without
regard to the constant
C
or the global velocity
U
. To fix the ideas,
and to avoid clutter, suppose that
A
=
B
= 1 in the ansatz (5.22).
Then the function
f
satisfies the ordinary differential equation
2
f
′′′
+
f f
′′
= 0
(5.33)
with the boundary conditions
f
(0) = 0
,
f
′
(
∞
) =
u
1
U
,
f
′
(
−∞
) =
u
2
U
.
(5.34)
The solutions will form a one-parameter family in
u
2
/u
1
.
Integration, say by a shooting method, is complicated by the
fact that both
f
′
(0) and
f
′′
(0) must be properly chosen before the
conditions at
±∞
can be satisfied (
what about
U
?
). An ingenious
procedure was proposed by T ̈opfer, who observed (in the context
of the Blasius boundary-layer problem) that if
f
(
η
) is a solution of
equation (5.33), so is
g
(
η
) =
af
(
aη
), where
a
is a constant. This
property can be proved directly, but it can also be connected with
the normalization procedure. Relabel the dependent variable
f
in
equation (5.22) as
g
. Then
g
satisfies
2
AB g
′′′
+
g g
′′
= 0
(5.35)
with the boundary conditions
g
(0) = 0
,
g
′
(
∞
) =
A
B
u
1
U
,
g
′
(
−∞
) =
A
B
u
2
U
.
(5.36)
Equation (5.35) is the same as equation (5.33) if
AB
= 1. The
boundary conditions become
g
(0) = 0
,
g
′
(
∞
) =
a
2
u
1
U
,
g
′
(
−∞
) =
a
2
u
2
U
(5.37)
and the problem takes T ̈opfer’s form. Thus choose a value of
u
2
/u
1
and a value of
U
, which depends somehow on
u
1
and
u
2
. Choose also
376
CHAPTER 5.
THE SHEAR LAYER
a value of
g
′
(0) and iterate
g
′′
(0) until the condition
g
′
(
−∞
)
/g
′
(
∞
) =
u
2
/u
1
is satisfied. The parameter
a
then follows from the relation
a
2
=
u
1
U
1
g
′
(
∞
)
.
(5.38)
Since
f
′
(0) =
a
2
g
′
(0) and
f
′′
(0) =
a
3
g
′′
(0), the function
f
(
η
) can be
evaluated immediately by a final integration.
It remains to consider entrainment. The easiest way to deter-
mine
v
1
and
v
2
is through the definition
v
=
−
∂ψ/∂x
applied directly
to the ansatz (5.30). The result is
v
1
=
−
[
(
u
1
+
u
2
)
ν
2
x
]
1
/
2
lim
η
→∞
[
f
−
(
u
1
u
1
+
u
2
)
η
]
(5.39)
v
2
=
−
[
(
u
1
+
u
2
)
ν
2
x
]
1
/
2
lim
η
→−∞
[
f
−
(
u
2
u
1
+
u
2
)
η
]
.
(5.40)
Under certain common experimental conditions, the problem of eval-
uating the constant
C
solves itself. Suppose that
v
1
= 0. This will
be the case if there is a parallel wall above the shear layer, as shown
in FIGURE X. A related configuration is the round jet, for which
the core flow downstream from the exit can be expected to be uni-
form and at constant pressure. If the Reynolds number is large, the
shear layer will be thin, and the effect of lateral curvature can be
neglected, at least close to the exit. For such cases, equation (5.39)
gives (
check again
)
C
= lim
η
→∞
(
η
−
f
)
.
(5.41)
This result is noted in FIGURE X. Integration of the Blasius equation
will lead to a value for
C
(check)
. The effect is very like the effect
of the displacement thickness for a boundary layer, except for the
direction of the deflection.
(Integrate in both directions from
dividing streamline?)
The conclusion (5.41) can also be argued from the continuity
equation in a form that applies for both laminar and turbulent flow.
5.1. PLANE LAMINAR SHEAR LAYER
377
In FIGURE X, continuity requires for the contour
ABCD
, which
is bounded in part by the wall
y
=
Y
and in part by the dividing
streamline
η
= 0 (
rethink this
),
Y
∫
0
u
1
d
y
=
Y
∫
(
η
=0)
u
d
y
(5.42)
or
Y
∫
0
(
u
1
−
u
)d
y
=
0
∫
(
η
=0)
u
d
y .
(5.43)
Thus the two shaded regions in the figure have equal areas. In simi-
larity form, equation (5.43) is
(
Y
)
∫
C
(1
−
f
′
)d
η
=
C
∫
0
f
′
d
η
=
f
(
C
)
−
f
(0)
(5.44)
from which, if
f
(0) = 0,
lim
η
→∞
(
η
−
f
) =
C .
(5.45)
This conclusion does not depend on the parameter
u
2
/u
1
(
look at
argument by Dimotakis; point out connection with displace-
ment concept; upper wall may also be a parabola; note lines
x
= constant
are characteristics
).
The most challenging element of the analytical problem is the
subtlety of the required third boundary condition (check refs to see
who was clear about this first). As a practical matter, the laminar
mixing layer is very unstable, and any experimental information is
likely to be incidental to work on the instability (Sato). However,
the problem needs to be considered here because it also comes up
for turbulent flow, and should not be blamed on the presence of
turbulence.
Read Lu Ting and other papers. The presence of a wall or
of axial symmetry removes the difficulty. Note singularity at
x
= 0
378
CHAPTER 5.
THE SHEAR LAYER
because equations are parabolic and characteristics are
x
= constant
.
Note use of rectangular coordinates; mention Kaplun on optimal
coordinates. The essence is the final additive constant in
f
. Note
also the need for an additive constant in
η
to get the shear layer from
Stewartson’s solution at the origin. Represent flow by distributed
sources and/or doublets? Must avoid pressure force on splitter plate.
Is streamline displacement at infinity symmetric or antisymmetric or
neither? Do control-surface argument
.
The nonlinear equation (5.24) with the boundary conditions
(5.16) and (5.17) has no known solution in closed form, and was first
solved numerically by (
refs.
). The solution for
f
is fixed only within
an additive constant in
η
, pending resolution of the third boundary
condition.
5.2 Plane turbulent mixing layer
This flow is second only to the turbulent boundary layer in the vol-
ume of literature it has generated (
pipe flow?
). Much of the more
recent work has aimed at the problem of chemistry, including the
dominant role of coherent structure in turbulent mixing. The latter
work also contributes substantially to the body of information on
mean properties.
The turbulent mixing layer grows rapidly in the downstream
direction. For a fixed ratio of the two constant external velocities,
the growth is known to be very nearly linear and nearly independent
of Reynolds number (see SECTION X). I will therefore not consider
the laminar stresses, particularly since the problem of mixed simi-
larity rules is beyond the state of my art. However, I can and will
attempt to avoid some limitations of the usual boundary-layer ap-
proximation. The conical property suggests that cylindrical polar
coordinates could be used, but experimenters move their probes in
rectangular coordinates, and so will I. The equations of motion are
ρ
(
∂uu
∂x
+
∂uv
∂y
)
=
−
∂p
∂x
+
∂
∂x
(
−
ρ
u
′
u
′
) +
∂
∂y
(
−
ρ
u
′
v
′
)
(5.46)
5.2. PLANE TURBULENT MIXING LAYER
379
ρ
(
∂uv
∂x
+
∂vv
∂y
)
=
−
∂p
∂y
+
∂
∂x
(
−
ρ
u
′
v
′
) +
∂
∂y
(
−
ρ
v
′
v
′
)
.
(5.47)
No information can be obtained about the remaining Reynolds stress
−
ρ
w
′
w
′
, which is involved only indirectly in the dynamics of the mean
flow. I am obliged to use boundary conditions of boundary-layer type
in what amounts to a hybrid formulation,
u
(
x,
∞
) =
u
1
, u
(
x,
−∞
) =
u
2
,
(5.48)
p
(
x,
∞
) =
p
(
x,
−∞
) =
p
∞
.
(5.49)
In addition, the Reynolds stresses are all assumed to vanish outside
the shear layer.
(Is the pressure condition correct?)
Let these equations and boundary conditions be subjected to
380
CHAPTER 5.
THE SHEAR LAYER
the affine transformation
x
=
a
̂
x
y
=
b
̂
y
ψ
=
c
̂
ψ
ρ
=
d
̂
ρ
u
′
u
′
=
f
̂
(
u
′
u
′
)
u
′
v
′
=
g
̂
(
u
′
v
′
)
v
′
v
′
=
h
̂
(
v
′
v
′
)
(5.50)
p
=
i
̂
p
p
∞
=
j
̂
p
∞
=
i
̂
p
∞
u
=
p
̂
u
=
c
b
̂
u
v
=
q
̂
v
=
c
a
̂
v
u
1
=
r
̂
u
1
u
2
=
s
̂
u
2
=
r
̂
u
2
.
The definitions
u
=
∂ψ/∂y
and
̂
u
=
∂
̂
ψ/∂
̂
y
require
p
=
c/b
.
(
Mention
v
.
) Transformation of the boundary conditions on
u
leads,
as in the laminar problem, to the relation
p
=
r
=
s
and to an
5.2. PLANE TURBULENT MIXING LAYER
381
invariant which I take as
c
br
= 1
.
(5.51)
By inspection, the boundary condition (5.49) on
p
leads to the rela-
tion
i
=
j
. Invariance of the equations of motion implies
c
2
abg
= 1
,
bi
adg
= 1
,
fb
ag
= 1
,
(5.52)
c
2
a
2
h
= 1
,
i
dh
= 1
,
bg
ah
= 1
.
(5.53)
Note from the second of equations (5.53) that
i/d
=
h
. From
the second of equations (5.52) and the third of equations (5.53), it
follows that
h
=
g
and thus that
b
a
= 1
.
(5.54)
Finally, from the third of equations (5.52),
f
=
g
. Thus equations
(5.52) and (5.53), which are a necessary condition for similarity, im-
ply that the layer grows linearly if the pressure perturbation and the
three surviving Reynolds stresses all transform in the same way, and
conversely.
Two invariants of the transformation are given by equations (5.51)
and (5.54). But these and the first of equations (5.52) or (5.53) imply
g
r
2
= 1
(5.55)
and three similar equations, given
f
=
g
=
h
=
i/d
. These combina-
tions require the pressure and the Reynolds stresses
−
ρ
u
i
u
j
to scale
like
ρu
2
1
. Since these stresses must vanish when
u
2
=
u
1
, it is rea-
sonable to adapt this conclusion to read that the Reynolds stresses
must scale like
ρ
(
u
1
−
u
2
)
2
.
(Why not
u
2
1
−
u
2
2
?)
In full, the ansatz
382
CHAPTER 5.
THE SHEAR LAYER
for the plane turbulent shear layer is
(introduce U)
A
ψ
Ux
=
f
(
B
y
x
+
C
)
=
f
(
η
)
(5.56)
u
′
u
′
= (
u
1
−
u
2
)
2
F
(
η
)
(5.57)
u
′
v
′
= (
u
1
−
u
2
)
2
G
(
η
)
(5.58)
v
′
v
′
= (
u
1
−
u
2
)
2
H
(
η
)
(5.59)
p
−
p
∞
=
ρ
(
u
1
−
u
2
)
2
P
(
η
)
(5.60)
where the constant
C
, as in the laminar case, supports the boundary
condition
f
(0) = 0 by locating the dividing streamline
ψ
= 0 in the
downstream flow as the straight line
y/x
=
−
C/B
. The functions
F
and
H
are necessarily positive, and
G
is expected to be negative.
Substitution of the appropriate derivatives of equations (5.56)-
(5.60) into the momentum equations (5.46) and (5.47) gives
−
U
2
B
2
A
2
f f
′′
(
u
1
−
u
2
)
2
= (
η
−
C
)
P
′
+ (
η
−
C
)
F
′
−
BG
′
(5.61)
−
U
2
B
A
2
(
η
−
C
)
f f
′′
(
u
1
−
u
2
)
2
=
−
BP
′
+ (
η
−
C
)
G
′
−
BH
′
.
(5.62)
5.2.1
The boundary-layer approximation
A short digression is needed here to put into evidence the result
that would be obtained if the boundary-layer approximation had
been made in the beginning. The single momentum equation to be
transformed is a truncated form of equation (5.46),
ρ
(
∂uu
∂x
+
∂uv
∂y
)
=
∂
∂y
(
−
ρ
u
′
v
′
)
(5.63)
with the boundary conditions (5.48). The invariants of the mapping
are
c
2
abg
= 1
,
c
ar
= 1
.
(5.64)
5.2. PLANE TURBULENT MIXING LAYER
383
If the Reynolds shearing stress transforms like (
u
1
−
u
2
)
2
, equa-
tion (5.55) again applies;
g
r
2
= 1
.
(5.65)
When the first two relationships are used to isolate
b
and
c
, the main
invariants are unchanged;
c
br
= 1
,
b
a
= 1
,
g
r
2
= 1
.
(5.66)
Thus the appropriate ansatz is the subset (5.56) and (5.58). The im-
plied similarity equation can be derived directly or by dropping terms
multiplied by (
η
−
C
) in equation (5.61). The same approximation
in equation (5.62) gives
P
′
+
H
′
= 0 or
p
+
ρ
v
′
v
′
=
p
∞
= constant
.
(5.67)
The full equations yield a more complicated equation for the
pressure. Multiply equation (5.61) by (
η
−
C
) and equation (5.62)
by
B
and subtract to obtain
[
B
2
+ (
η
−
C
)
2
]
P
′
=
−
(
η
−
C
)
2
F
′
+ 2
B
(
η
−
C
)
G
′
−
B
2
H
′
.
(5.68)
With (
η
−
C
) =
By/x
, this becomes in physical variables
(
1 +
y
2
x
2
)
∂
(
p
−
p
∞
)
/ρ
∂y
=
−
∂
v
′
v
′
∂y
+ 2
y
x
∂
u
′
v
′
∂y
−
y
2
x
2
∂
u
′
u
′
∂y
.
(5.69)
The three Reynolds stresses have similar shapes and comparable
magnitudes (see section x). At least the second term on the right
in equation (5.69) is not negligible since the coefficient 2
y/x
is typi-
cally about 0
.
2 in regions where the derivatives are appreciable. Note
that equation (5.69) does not involve the constants
A
,
B
, and espe-
cially
C
in equation (5.56).
(Integrate by parts.)
Equation (5.69)
will be tested experimentally and compared with the conventional
boundary-layer approximation (5.67) in SECTION X.
(Note that
f
= 0
when
G
′
= 0
. Does
p
(
∞
) =
p
(
−∞
)
?)
To eliminate the pressure from the problem, multiply equa-
tion (5.61) by
B
and equation (5.62) by (
η
−
C
) and add to obtain
384
CHAPTER 5.
THE SHEAR LAYER
(give boundary-layer approximation, comment on eddy vis-
cosity)
[
B
2
−
(
η
−
C
)
2
]
G
′
=
U
2
(
u
1
−
u
2
)
2
B
A
2
[
B
2
+ (
η
−
C
)
2
]
f f
′′
+
B
(
η
−
C
)(
F
′
−
H
′
) (5.70)
This expression is useful for comparing values of
G
measured di-
rectly with those inferred from measurements of the other quantities;
namely
f, F
, and
H
. The effect of making the boundary-layer ap-
proximation in equation (5.70) is less conspicuous than in the case
of equation (5.68), because the difference (
F
′
−
H
′
) is much smaller
than
F
′
or
H
′
alone. In physical variables, equation (5.70) becomes
(
1 +
y
2
x
2
)
ψ
∂u
∂y
=
x
[
y
x
∂
∂y
(
v
′
v
′
−
u
′
u
′
)
+
(
1
−
y
2
x
2
)
∂
∂y
(
u
′
v
′
)
]
.
(5.71)
Within the boundary-layer approximation, the dividing mean stream-
line
ψ
= 0 corresponds, as in the laminar case, to the condition
∂τ/∂y
= 0, although not necessarily to an inflection point in the
mean velocity profile. This condition should not be much in error
for the full equations.
It remains to assign values to the scaling parameters
A
and
B
in the defining similarity equation (5.56), or better in its derivative
(note that
η
= 0 is the dividing streamline).
A
B
u
U
=
f
′
(
B
y
x
+
C
)
=
f
′
(
η
)
.
(5.72)
As in the laminar problem, it is reasonable to take
(explain)
A
B
= 1
(5.73)
and thereby to fix two of the boundary conditions as
f
′
(
∞
) =
u
1
U
, f
′
(
−∞
) =
u
2
U
.
(5.74)
Again, the physical parameter
u
2
/u
1
of the problem appears only in
the lower boundary condition. The normal component of velocity
5.2. PLANE TURBULENT MIXING LAYER
385
outside the shear layer is the limit of
−
∂ψ/∂x
, with
ψ
given by
equation (5.56). The boundary conditions (5.9) and (5.10) lead to
v
1
=
−
u
1
A
lim
η
→∞
[
f
−
(
η
−
C
)]
,
(5.75)
v
2
=
−
u
1
A
lim
η
→−∞
[
f
−
u
2
u
1
(
η
−
C
)
]
.
(5.76)
If the condition
v
1
= 0 is enforced by an upper wall or by axial
symmetry with the faster stream near the axis, then again
C
= lim
η
→∞
(
η
−
f
)
.
(5.77)
The present state of the art of normalization is based on a more
empirical similarity approach suggested by FIGURE X. A thickness
δ
, commonly called the vorticity thickness, can be tentatively defined
in terms of the maximum slope
∂u/∂y
, which no longer necessarily
occurs on the dividing streamline. Within the boundary-layer ap-
proximation, the shearing stress in turbulent flow should have a maxi-
mum on the dividing streamline, because
Du/Dt
= 0
(explain)
. For
a particular value of the parameter
u
2
/u
1
, the profile in FIGURE X
can be represented in another similarity form, starting with
u
−
u
2
u
1
−
u
2
=
g
′
(
y
∗
δ
)
(5.78)
where
y
∗
is measured from the dividing streamline. The relationship
between
g
′
and
f
′
follows from equations (5.53), (5.54), and (5.58);
u
u
1
=
f
′
(
By
∗
x
)
=
u
2
u
1
+
(
1
−
u
2
u
1
)
g
′
(
y
δ
)
.
(5.79)
To preserve the benefits of the affine argument, it is necessary to
have
B
=
x
δ
.
(5.80)
The notation
σ
for
x/δ
was introduced by G ̈ortler (
ref
). This
parameter
σ
, which is the same as my
B
, can be expected to depend
on
u
2
/u
1
. For the most thoroughly studied case
u
2
/u
1
= 0,
σ
is
386
CHAPTER 5.
THE SHEAR LAYER
about 11, or
δ/x
is about 1/9. I prefer not to use G ̈ortler’s notation
on the hard ground that a different definition of
δ
might prove more
useful, and on the softer ground that the parameter
σ
seems to me
to be defined upside down; the ratio
δ/x
has a greater graphic and
mnemonic value.
(Is there an integral of
(
u
1
−
u
)(
u
−
u
2
)
?)
The boundary conditions for
g
′
from equation (5.101)
1
are
g
′
(
∞
) = 1
,
g
′
(
−∞
) = 0
.
(5.81)
Integration of (5.79) yields
f
(
y
δ
)
=
u
2
u
1
y
δ
+
(
1
−
u
2
u
1
)
g
(
y
δ
)
(5.82)
with
g
(0) = 0 if
f
(0) = 0 (
where was the latter done?
).
The parameter
u
2
/u
1
has thus disappeared from the boundary
conditions and appeared in the defining equation (5.82). An associ-
ated result, completely empirical at present, is based on the fact that
the profile
g
′
(
y/δ
) is a monotonic transition from one constant value
to another. It is only a small step to the proposition that this profile
is for practical purposes universal; i.e., it is the same function of
y/δ
for all values of
u
2
/u
1
. This proposition will be tested experimentally
in section x. (
Look also at laminar case. Do entrained flow,
both cases. Calculate
v
(
−∞
)
.
)
The dependence of
B
or
δ/x
or
σ
on
u
2
/u
1
was first studied
by SABIN (1965), who worked with plane shear layers at quite low
Reynolds numbers and who chose to plot
x/δ
against
u
2
/u
1
(
check
).
It is now more common to see
δ/x
plotted against (
u
1
−
u
2
)
/
(
u
1
+
u
2
).
The reason is that the latter dependence is found to be very nearly
linear;
δ
x
∼
(
u
1
−
u
2
u
1
+
u
2
)
.
(5.83)
I suspect that this linear dependence is somehow implicit in the
equations, especially equation (5.79), but I have not found a valid
argument
.
1
Possibly an incorrect reference.
5.2. PLANE TURBULENT MIXING LAYER
387
5.2.2
Structure of the shear layer
Some guidance on normalization is provided by the device of the
moving observer. For the case of a turbulent mixing layer, this de-
vice is both real and important, because it introduces the subject of
coherent structure, and thus requires another digression.
At the level of eddy viscosity or mixing length, the turbulent
mixing layer was thought to be a featureless wedge of turbulence,
perhaps with a trivially irregular boundary. This view changed dras-
tically with the work of BROWN and ROSHKO (1971, 1974) and
WINANT and BROWAND (1974). It is now recognized that the tur-
bulent mixing layer is inhabited by, or more properly is constructed
from, large spanwise vortex structures that grow both by entrainment
and by coalescence during the evolution of the flow. The structures
originate in an inviscid Kelvin-Helmholtz instability that operates in
both laminar and turbulent flow. In this coherent-structure model
of turbulence, each structure is assumed to move as a unit, pre-
serving its geometry and operational properties between coalescence
events, while the ambient flow accommodates itself to the kinematic
and dynamic needs of the structure. According to this model, the
translational velocity of the structures in laboratory coordinates is
well defined. It is is often called convection velocity and occasionally
phase velocity, although I will use the term celerity and the notation
c
.
When the averaging process introduced by REYNOLDS (1895)
is stopped at second order, which is to say at the first revelation of
the closure problem, all information about scale and phase of the
turbulent motions is lost. Various methods are available to recover
some of this information. If data are available at two points, for ex-
ample, some phase information can be rescued by the technique of
space-time correlation. The condition of optimum time delay then
provides an imperfect measure of phase velocity or celerity. Without
filtering, the results indicate that the celerity is not constant through
the thickness of a given plane flow, but is biased in the direction of the
mean-velocity profile. The bias can be reduced by retaining only the
low-frequency or large-scale content of the signals, as demonstrated
388
CHAPTER 5.
THE SHEAR LAYER
for the boundary layer by FAVRE, GAVIGLIO, and DUMAS (Phys.
Fluids
10
, Supplement, S138-S145, 1967). Early experimenters who
used this technique were not in any doubt about the meaning of their
work, as is evident from the fact that Favre and Kovasznay chose the
French word c ́el ́erit ́e to describe their findings quantitatively, rather
than the more conventional French word vitesse. (The technique of
time-space correlation could in principle be applied to existing nu-
merical solutions of the Navier-Stokes equations.) Much more accu-
rate measures have been obtained by flow visualization, particularly
for the mixing layer, where the large structures are two-dimensional
in the mean.
FIGURE X is a sequence of frames from a motion picture of a
mixing layer (Roshko, private communication;
what flow?
). FIG-
URE Y is a corresponding
x
-
t
diagram showing trajectories of the
recognizable features marked by + symbols in the first frame. Co-
alescence events occur quickly, like punctuation marks in the text
of turbulence. Similar figures have been published by DAMMS and
K
̈
UCHEMANN (RAE Tech Rep 72139, 1972), BROWN and ROSHKO
(JFM
64
, 775, 1974), and ACTON (JFM
98
, 1, 1980). Each of these
observers chose a particular local feature, not necessarily the same
feature, in order to assign values to the variable
x
(
t
) and hence to the
celerity d
x/
d
t
, and each was successful in exposing the phenomenon
of coalescence in the mixing layer.
These observations play a central role in normalization. FIG-
URE X (COLES 1981) is a cartoon of the instantaneous mean stream-
lines in the turbulent mixing layer, as seen by an observer moving
with the celerity
c
.
(Look up Brown, thesis, Univ. Missouri,
1978.)
In the model, the vortices are stationary, the flow is inviscid,
and the layer grows in time rather than in space. The essence of
figure x is the topology, which consists of saddle points (stagnation
points) alternating with stable foci (vortices). Fluid flowing toward
each saddle point along the converging separatrices (instantaneous
streamlines) must arrive at the saddle point with the same stagnation
pressure. Suppose for the moment that Mach numbers are small but
the streams have different densities. Far from the mixing layer, where
the two streams have the same static pressure, Bernoulli’s equation
5.2. PLANE TURBULENT MIXING LAYER
389
requires
ρ
1
2
(
u
1
−
c
)
2
=
ρ
2
2
(
u
2
−
c
)
2
.
(5.84)
When this equation is solved for
c
, the result is
c
=
(
ρ
1
)
1
/
2
u
1
+ (
ρ
2
)
1
/
2
u
2
(
ρ
1
)
1
/
2
+ (
ρ
2
)
1
/
2
.
(5.85)
Note that if the two densities are very different, the celerity ap-
proaches the velocity of the denser stream. Thus equation (5.85)
contains more information about the mixing process than might have
been anticipated. If the fluids are compressible, still with the same
p
outside the mixing layer and the same
p
o
at the saddle points, it fol-
lows from equation (1.82) of the introduction that equations (5.84)
and (5.85) are unchanged, provided only that the flow along the
separatrices is isentropic. If the velocity of one or both streams is
supersonic relative to the large structures, shock waves and expan-
sion waves may appear and may intersect the separatrices. The effect
on equation (5.85) is at present an open question.
I first used the relation (5.85) in a survey paper (COLES 1985)
that attempted to collect some important results that can be ob-
tained using the concept of coherent structure and cannot be ob-
tained without it. The same equation was derived independently by
DIMOTAKIS (AIAA Paper 84-0368) and perhaps by others. There
is persuasive evidence (WANG) that equation (5.85) predicts quite
accurately the effect of density ratio on celerity for low-speed flow.
The calculation just made depends on the topological simplic-
ity of the mixing layer. No comparable result has so far been ob-
tained for any other turbulent shear flow, presumably because the
large structures in other flows arise from more complex instabilities,
are less dominant in the mixing process, and are almost certainly
three-dimensional. Moreover, in any morphology of coherent struc-
ture, an important distinction arises between flows containing large-
scale mean vorticity of only one sense and flows containing large-scale
mean vorticity of both senses. The mixing layer is unique among the
classical plane flows in that it is the only flow that is driven naturally
toward a two-dimensional structure.
390
CHAPTER 5.
THE SHEAR LAYER
Now return to the problem of normalization. For the case of
equal densities, the constant celerity of the large structures, and thus
the proper velocity of the observer, is
U
=
c
=
u
1
+
u
2
2
.
(5.86)
It is a little ironic that this estimate is less of a guess than the same
estimate (5.27) for laminar flow. It defines the global velocity
U
in
the ansatz (5.56) and gives the position of an observer moving with
the structures as
x
∼
(
u
1
+
u
2
)
t .
(5.87)
For the laminar problem, another measure for the parameter
t
was
obtained from the diffusive model
δ
∼
(
νt
)
1
/
2
, which does not ap-
ply when the flow is turbulent. DIMOTAKIS (1991) has proposed
a different relation that involves
δ
and meets the need. The vortic-
ity thickness
δ
, or more accurately the maximum-slope thickness, is
defined by
δ
=
u
1
−
u
2
(
∂u/∂y
)
max
.
(5.88)
Since
δ
varies like
x
and like
t
, the reciprocal of the quantity (
∂u/∂y
)
max
is a plausible time scale;
δ
∼
(
u
1
−
u
2
)
t .
(5.89)
Immediately, therefore,
δ
x
∼
(
u
1
−
u
2
)
(
u
1
+
u
2
)
.
(5.90)
Equation (5.90) is a genuine scaling law only if the implied con-
stant of proportionality is independent of the velocity ratio
u
2
/u
1
.
The derivation here supposes and suggests that it is, although the ar-
gument, like many arguments in science, illustrates the principle that
it helps to know the answer. Note that negative values for
u
2
/u
1
,
which can occur for base or cavity flows, are permitted by the scaling
law, with
δ/x
varying from zero to infinity as
u
2
/u
1
varies from +1 to
−
1. This question will be taken up in SECTION X. ABRAMOVICH
(19xx) proposed equation (5.90) for the mixing layer, but stipulated
5.2. PLANE TURBULENT MIXING LAYER
391
some further dependence on
u
2
/u
1
. SABIN (19xx) proposed equa-
tion (5.90) in a different form as a scaling law based on his own mea-
surements at relatively low Reynolds numbers. ROSHKO in 19xx
collected the experimental data available at that time and demon-
strated an essentially linear relationship between the two sides of
equation (5.90), with a constant of proportionality on the right of
about xxx. There was considerable scatter in the data, especially for
the case
u
2
/u
1
= 0.
The road to normalization is now open, as much as any road
in turbulence is ever open. If the argument
B y/x
of the function
f
in equation (5.56) is to be equivalent to
y/δ
, then it is necessary to
have
B
=
x
δ
=
b
(
u
1
+
u
2
u
1
−
u
2
)
(5.91)
where
b
is independent of
u
2
/u
1
. Finally, as in the laminar case, I
take
A
=
B
(5.92)
on esthetic rather than logical grounds. This normalization leads to
the ansatz
2
bψ
(
u
1
−
u
2
)
x
=
f
[
b
(
u
1
+
u
2
u
1
−
u
2
)
y
x
+
C
]
(5.93)
with the boundary conditions
f
(0) = 0
, f
′
(
∞
) =
2
u
1
u
1
+
u
2
, f
′
(
−∞
) =
2
u
2
u
1
+
u
2
.
(5.94)
The constant
C
is still unspecified.
From equation (5.94), with the boundary-layer result that
f
′′
is a maximum on the dividing streamline
η
= 0
(check)
,
(
∂u
∂y
)
max
=
b
(
u
1
+
u
2
)
2
2
x
(
u
1
−
u
2
)
f
′′
(0)
(5.95)
and thus, with equations (5.88) and (5.91),
f
′′
(0) =
2
b
(
u
1
−
u
2
u
1
+
u
2
)
.
(5.96)
392
CHAPTER 5.
THE SHEAR LAYER
This result will be tested experimentally in SECTION X.
FIGURE X shows the experimental evidence for equation (5.83)
as presented by Roshko, with a number of later measurements. The
scatter, especially for the case
u
2
= 0, is unreasonable. Several rea-
sons have been proposed for the scatter. One, due to BATT (1975) is
that non-uniqueness is a relict of varying initial conditions, especially
the laminar or turbulent state of the boundary layer at the trailing
edge of the splitter plate or septum. This conjecture has inspired a
number of detailed and difficult studies (
refs
). Another, which I fa-
vor as an equally plausible source of scatter, is three-dimensionality.
An easy measure is the aspect ratio, or the ratio of shear-layer thick-
ness to the distance between the side plates usually provided to con-
trol the entrainment process. The effect should not be present for
the axisymmetric shear layer, but another more systematic effect of
lateral curvature is likely to be present instead, along with any effect
of initial conditions.
Collect profile data for plane flow,
u
2
= 0
. Plot
dδ/dx
(assign
this to the mean x) against
δ/w
, where
w
is the spanwise width. Use
tanh for a fit to the central part of the profile. Do not use
x/δ
,
because the origin of
x
depends on linear growth, which is not yet
proved (thus argument is circular). For round jet, use
δ/πD
, where
D
is orifice diameter (neglect effect of growing diameter of dividing
surface).
For two-stream flows, comment on use of porous obstacle and
two tailored nozzles, with a very delicate design condition. The di-
rection of the dividing streamline may be seriously affected.
Note that the third boundary condition is still unspecified, as
in the laminar problem.
Note also the singularity when
u
2
=
−
u
1
; the layer becomes
infinitely thick. The origin for
x
is not unique.
(Who did crossed
flows?)
It might be more useful to work in terms of the variable (should