J.
Fluid
Meeh.
(1974).
vol.
64,
part 4,
pp.
775-816
Printed
in
Great
Britain
775
On density effects and large structure in
turbulent mixing layers
By GARRY L.BROWN
AND
ANATOL ROSHKO
University of Adelaide
California Institute of Technology
(Received
15
January
1974)
Plane turbulent mixing between two streams of different gases (especially
nitrogen and helium) was studied in
a
novel apparatus. Spark shadow pictures
showed that, for all ratios of densities in the two streams, the mixing layer is
dominated by large coherent structures. High-speed movies showed that these
convect
at
nearly constant speed, and increase their size and spacing discon-
tinuously by amalgamation with neighbouring ones. The pictures and measure-
ments of density fluctuations suggest that turbulent mixing and entrainment
is
a
process of entanglement on the scale of the large structures; some statistical
properties of the latter are used to obtain an estimate of entrainment rates.
Large changes of the density ratio across the mixing layer were found to have
a
relatively small effect on the spreading angle;
it
is concluded that the strong
effects, which are observed when one stream is supersonic, are due to com-
pressibility effects, not density effects, as has been generally supposed.
1.
Introduction
Several years ago we undertook to build an experimental facility in which to
study plane turbulent mixing layers between gases of different molecular
weights. By that time there was a great deal of data in the literature from many
experiments on the mixing of dissimilar gases in coaxial flows, and almost equally
many proposals for taking into account the effects of the dissimilarities on
turbulent mixing. The difficulty in achieving any satisfactory description
of
the
effects was due, it now seems quite clear, to the limitations of the axially sym-
metric configuration. As the jet entrains the surrounding fluid
it
is rapidly diluted
and, by the time it has achieved a similarity state far downstream, is practically
at the same density as the surrounding fluid. At that point the only problem
connected with the non-uniformity concerns the spreading of
a
passive con-
taminant by the otherwise uniform, turbulent flow; this problem has, in fact,
been fruitfully studied in the past.
On
the other hand, to study the
dynamic
effects of density non-uniformity on turbulent structure it is essential to maintain
a
large density difference and, to do it in a scientifically simple context, it is
desirable to find a flow which will have similarity properties under these condi-
tions. Such a flow is the plane mixing layer.
776
G.
L.
Brown
and
A.
Roshko
It
has been found in the past that the plane mixing layer seems to be fairly
well approximated by the initial mixing region at the boundary of an axisym-
metric jet, therefore the latter might be
a
suitable configuration for a study of the
problem
at
hand.
It
has the advantage of being free of ‘end effect
’
problems and
of being fairly simple to realize experimentally. On the other hand, because of the
finite thickness of the mixing layer, it cannot be strictly self-similar in this con-
figuration. Although
it
has been adopted by some investigators (see e.g.
Abramovich
et
al.
1969),
we came to the conclusion that the plane-flow con-
figuration allows more flexibility in choice of parameters, especially a second
velocity, and has advantages for flow visualization and other purposes which
outweigh its disadvantages, and came to design and build the apparatus described
in what follows.
Another motivation
for
studying the plane mixing layer between gases
of
different densities came from the problem of a supersonic turbulent mixing layer.
It
was known that increasing the Mach number of a supersonic jet results in
a
decrease in the spreading angle of the mixing region at the boundary of the initial
portion of the jet. In most such experiments, increasing Mach number is accom-
panied by decreasing temperature and thus increasing density of the jet, and the
observed effects were attributed by many investigators to this increasing density
ratio between the jet and the external gas. One result of this point of view was
attempts to relate the supersonic mixing layer to its low-speed, uniform-fluid
counterpart through transformations of the Howarth-Dorodnitsyn type that had
been developed for laminar shear layer theory. Implicit in this idea is the conse-
quence that
a
supersonic mixing layer would have the same growth rate as
a
low-speed layer with the same density ratio across
it,
and
this
in fact seems to be
the assumption made. We thought, therefore, that our experiments using different
gases could help throw some light on this aspect of the problem. With helium
and nitrogen for example, it would be possible to have the same density ratio
across the mixing layer as in
a
supersonic air jet at
M
=
5.5.
If the density ratio
played the same role
as
in the supersonic case,
a
very marked thinning of the
mixing layer should be observed.
Using dissimilar gases in these experiments, it was reasoned, would not only
give some information on questions about the effects of density difference out-
lined above, but would also provide dissimilarities in index of refraction and other
physical properties that could be used to advantage in various optical and
sampling techniques to obtain information about the turbulent structure, details
of the mixing, etc.
To
enhance the sensitivity of such techniques,
it
was decided
to operate at elevated pressures, up to
10
atm. A further compelling reason for
the high pressure was one of economy: for given Reynolds number the mass flow
rates of the gases scale linearly with pressure but as the square of a linear dimen-
sion; thus it
is
more economical to obtain large values of Reynolds number by
increasing pressure
(or
velocity) rather than size.
The possibility
of
obtaining density differences by heating one of the streams
(and possibly cooling the other) was considered but not adopted, because
it
appeared to be easier to design a system for density ratios of the order of
10
by
using different gases.
Density effects and large structure
in
turbulent mixing layers
777
While the original motivation for these experiments came from questions as
to density effects on turbulent mixing, our attention was soon turned to even
more fundamental questions about the flow structure, of which various facets
were revealed in the shadowgraphs and in the density fluctuation measurements
obtained. Thus, two themes run through this paper: one concerning the effects
of density difference on the mean flow, the other concerning the turbulent flow
structure.
2.
The plane turbulent mixing layer
Figure
1
illustrates the basic elements of the family of flows which we wish to
establish in our experiments. Two plane flows with velocity
Ul
and
U2
and
densities
p1
and
p2,
respectively, are initially separated by
a
partition which ends
at
x
=
0,
where the flows begin to mix.
As
is well known,
at
sufficiently high
Reynolds number based on
x
the mean flow becomes independent of molecular
diffusion rates and approaches similarity in the variable
ylx.
The profiles of the
x
component of velocity and of the density have the similarity forms
UlUl
=
mr;
r,
81,
PIP1
=
f4r;
r,
a),
(2.la,
b)
where
r
=
Y/(X-Xo),
r
=
U2lU1,
s
=
P2IP1.
(2.2
a-c)
A
shift of origin to
xo
is introduced in
(2.2)
to correct for the effect of finite
thickness, non-similarity, etc. of the initial part of the mixing layer near
x
=
0.
Because of these initial effects, the flow strictly speaking approaches asymptoti-
cally to the similarity state only at values of
x
so
large that
xo/x
-+
0.
Practically,
it
is
usually necessary to determine and include
x,,
as described below.
The ratios
U2/U,
and
p2/p1
appear in
(2.1)
as
parameters on which the mixing
layer structure and its various statistical mean properties depend. The effect of
U,/Ul
in homogeneous flow
(s
=
1)
has been fairly extensively investigated in the
past and we shall review these results in
$
5.
To determine the effect
of
p2/p1
was
the initial objective of the present work. Two more parameters, namely the Mach
numbers
Ml
and
M2,
could be added to the functional dependence in
(2.1).
In our
experiments these are both zero, and
so
are not explicitly exhibited. But,
as
explained in
$
1,
another purpose of our work was to compare mixing
rates at
M
=
0
with those where at least one of the streams is supersonic
(MI
>
0);
this
is
done in
$7.1.
778
G.
L.
Brown
and
A.
Roshko
Each mixing layer in the family of flows described by
(2.1)
will spread linearly,
(2.3)
i.e.
where
6(x)
is some measure of the local scale of the flow,
a
thickness defined in
some particular way. The proportionality factor
C
depends on the parameters
aspx
=
S’
=
6/(x-x,)
=
c,
U2lUI
and
P2IP1:
c
=
C(r,s).
(2.4)
Equation
(3.4)
states
a
functional dependence of spreading rate on velocity ratio
and density ratio; the objective is to determine this experimentally.
It
is
this
clear statement that attracted us to the study of the plane mixing layer as the
best if not the only prospect for beginning to untangle the confusion about the
effects of density on turbulent mixing.
Equations
(2.3)
show how
xo
is usually determined operationally. If an
asymptotic value of
6’
=
C
can be determined from the experiment then
xo
is
determined from the tangent,
6
=
C(x
-
xo).
For consistency, different thick-
nesses (e.g. momentum thickness, energy thickness, vorticity thickness) should
lead to the same value of
xo.
Furthermore, the attainment of constancy in
6’
should be accompanied by the attainment of constancy in various turbulent
correlations such as the dimensionless profile of Reynolds stress. This
is
the more
sensitive indication of the attainment of similarity
or
self-preservation (Townsend
1956).
While there are many quantities that are of interest, such as entrainment rate,
dissipation rate,
or
maximum shear stress, some measure of the spreading rate
is
indispensible for discussion and comparison of mixing layers.
A
frequently
used measure is the parameter
(r,
determined by plotting the velocity profiles
onto
a
standard curvef(C) with
[
=
(ry/x
and choosing
(T
for best overall
fit.
Thus
(r
is actually an inverse measure of the spreading rate. Its superiority over other
definitions
is
that all parts of the profile are involved in the fitting. Shortcomings
are that the definition of thickness is tied to a shape function or
a
computational
model, that low-speed parts of the profile may not be accurate, that velocity
profiles vary in shape (and that the fitting procedure is tedious). Simpler measures
based on the definition of some
6
have therefore been used by various authors.
A
useful discussion and compilation of values of
(r
from the literature
is
given
by Birch
&
Eggers
(1972).
Whatever definition is used for the thickness of the
velocity profile,
it
will not necessarily be adequate for the density profile.
Here we use mainly the velocity-profile maximum-slope thickness
UI
-
u2
6,
=
(8
U/ay)max
’
and
its
x
derivative
as
a measure of spreading rate, but other measures are introduced where appro-
priate. This thickness was used by Spencer
&
Jones
(1971),
who note that it may
be related to
(r
by
(rs:
=
7rrg
Density effects and large structure
in
turbulent mixing layers
779
when the profile shape
is
fitted by an error function. We choose for this thickness
the notation
S,,
because it can also be interpreted
as
the
vorticity th,ickness,
i.e.
where
--w
=
aU/ay.
In addition to being convenient, the vorticity thickness is
appropriate, the problem of the growth of the turbulent mixing layer being
basically the kinematic problem of the unstable motion induced by the vorticity.
In appraising the possible combinations in the
r,
s
plane
(2.4),
various parti-
cularly interesting ones can be found, We have been particularly interested in the
two cases
TS
=
1
and
rs2
=
I.
For the
first
one,
pz
U2
=
p1
U,,
the mass flux rate is
the same in both streams. If an eddy-viscosity model is used to solve the Reynolds
and turbulent diffusion equation and the turbulent Schmidt number is assumed
to be unity, it is found that
pU
=
const.
=
p1
Ul
=
p2
U,
across the layer,
for
any
dependence of the eddy viscosity on
ylx.
The deviation of
a
measured
p
U
profile
from this constant value will be related to the relative magnitudes
of
the
momentum and mass diffusivities. Thus we were very interested in measuring the
actual profile of
pU
for this case.
For
the second case,
p2
Ui
=
p1
U;,
the dynamic
pressure is the same in both streams.
It
is a special case of a set of similarity flows
with streamwise pressure gradients. These have been studied by Rebollo
(1973),
and will be discussed in a later paper.
3.
Apparatus and measuring
techniques
The principal requirements for designing the apparatus were two: (i) high
density ratio between the two streams; (ii) high Reynolds number.
A
value of
at least
2
for the density ratio
p2/p1
was required
so
that
dynamic
effects of density
non-uniformity could be studied; even higher values, comparable to those in
supersonic flow at high Mach number, were desired. For Reynolds numbers, the
aim was to reach values comparable to those in the experiment of Liepmann
&
Laufer
(1947),
in which values of
Uxlv
up to
lo6
were achieved.
The possibility
of
achieving a large density ratio by heating one stream (and/or
cooling the other) was considered but,
at
high flow rates, this
is
rather impractical.
We therefore decided to provide density differences by using different gases, in
particular the combination of nitrogen and helium, which gives a density ratio of
7.
No
other combination is less expensive for a density ratio of at least two. The
consumption of gases is kept down to economical values by operating for short
%ow
times at high pressure. These considerations led to a new kind of high-
pressure, short-running-time wind tunnel designed particularly for the mixing
layer experiment. Details of its design and construction will be presented else-
where. Basically, two gas streams supplied from two banks
of
2OOOpsi bottles
are brought together at the exit of two
4
x
1
in. nozzles in the working section
shown in figure
2
(plate
I).
Upstream of the nozzle in each stream there is
a
pressure regulator,
a
flow metering valve, and noise- and turbulence-reducing
sections. Downstream of the test section both streams flow through
a
pressure-