AN INVESTIGATION OF THE TURBULENT MIXINSe
OF PARALLEL TWODIMENUSIONAL. COMPRPESSIBLE
DISSIMILAR GAS STREAMS
By
SEYFEDDIN TAN~RIKUT
A DISSERTATIONI PRESENiTED TO THE GRADUATE
COUNCIL OF THE UNIIVERSITY OF FLORIDA IN PARTIAL
FULFILDIENT OF THE REQUIREMENTS FOR TH1E DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1973
ACK'OkLEDGEYENT~S
A research program, especially one involvingp an experimental
investigation, is rarely completed wuith the sole efforts of one individual.
Thec author wishes t:o express his gratitude for the guidance and
assis~ta~ce of his mnajor zdvisor, Dr. V. P. Roan. Special thanks are due
to Dr. R. B. Glaither, Dr. R. A. Gater, Dr. C. K. Uisieh, Dr. U. H. K~urzweg
and Dr. J. F. Reading for serving as members of the special supervisory
comittee. Th~e financial support of the department of M~echanical Engineerinl
throughout thle doctoral programs is gratefully acknow~ledged.
The construction and erection, of the test apparatus wias done by
the personnel of the Mech'anical Research Laboratories, headed by Professor
L. F. Patterson and Mr. R. T. Tomlinson. Their suggestions and contributions
to the effort are also gratefully ack~nowledged.
Finally, the author ir indebted to his w~ife, Yasar, for her patience
and~ confidence, in addition to lir assistance in preparing this manuscript.
TABLE OF CONTENDS
ACKNOULEDG:ENT, .......... ...............
LIST OF TABLES. ........... .............
LIST OF FIGURES. .......... ........ ....
LIST OF SYMBOLS. .......... ......... ...
ABSTRAICT .............................
CHAPTERS:
I. INTRODUCTION......................
Approach. ..........,........
II, LITIERATURE REVIEW...
Mixing of SemiInfinite Streams.. ......
Mixing of Confined (Ducted) Streams.......
Mixing of Dissimilar Gases. ..........
Major Experimental Investigations... ..
III. ANALYSIS FOR THE MIXING OF DISSIMILAR STREAlS....
Introduction .................
Boundary Layer Equations. ...........
Turbulent Equacions ...............
The von MIises Transforneation ..........
Boundary Conditions.. .......... ..
Finite Difference Equations..........
Boundary Conditions for the Difference
Equations. .......... ........
Analysis of the Difference Scheme........
IV. TURBULENT NOMENTUML AD MAYSS TRANSPORT ..... .,
General Considerations.. ......... .
TABLE OF CONTENTS (Continued)
rag
Classical Eddy Viscosity M~odels.............. 41
Choice of Mlodels for the Present Study....... 46
Modification and Correlation of Models....... 48
Remiarks on the Turbulent Transport of Mlass
and Energy................................... 50
V. EXPERIMENiTAL INVESTIGATION........................ 52
General Considerations....................... 52
Experimental Apparatus....................... 53
Test Section............................ 54
Nozzle Blocks........................... 55
Gas Supply and Control System........... 57
The Schlieren System.................... 57
Probe Drive Mechanism................... 58
Static Pressure Plate... ................ 59
Gas Analysis System. ................... 59
Measuring Devices............................ 60
Total Pressure Measurements............. 60
Total Temperature Mleasurements.......... 61
Static Pressure Measurements. ........... 62
Concentration Measurements.. .........., 62
Probe Design....................... 62
Chromatograms...................... 63
Testing Procedure............................ 66
Data Reduction.................. ............ 69
Accuracy of Results.......................... 74
TABLE OF CONfTENTS (Continued)
rage_
VI. DISCUSSION OF THEORETICAL ANYD EXPERIMENTAL RESULTS 77
Schlieren Photographs........................ 77
Static Pressure Variations................... 78
Growth of the Mixing Region. ..............., 80
Correlation of Velocity Data................. 82
Correlation of Mass Fraction Data............ 85
Remarks on the Similarity of Profiles....,... 88
VII. SUMMARY OF RESULTS AND CONCLUSIONS ................ 89
TABLES............................ ............ 93
FIGUIRES...................... .. ............. 98
APPiEDIXii .................. ........... ............. 165
EISLIOGRAPHY .................. .......... .......... 193
BIOGRAPZHICAL SKETCH .................. .. .. ........... ... 200
LIST OF TABLES
labl Pg
I M~ost Promlinent Moadels Proposed for Eddy Viscosity. 94
II Test Conditions and Configurations................ 96
III Comparison of Calculated and M:easured Mass Flow
Rates................ ... ........ ........... 97
IV Enthalpy T~t Coefficients................ ......... 171
LIST OF FIGURES
ngr rae
1 Schematic of the Mixing Region Resulting from
the Contact of Two Parallel SemiInfinite Streams. 99
2 A Typical Geometry of the Problem Under
Consideration................................ 100
3 The Grid Network Utilized in the Finite
Difference Scheme................................. 101
4 Test Section Installed in Blowdown Wind Tunnel
Faility...................................... 102
5 Schematic Diagram of the Gas Dynamics Facilities
Used in the Experimental Phase of the
Investigation................................ 103
6 CloseUp of Test Section with Two M~ach 2.0 N'ozzle
Blocks....................................... 104
7 CloseUp of the Air Ma~ch 2.0 N~ozzle Block......... 104
8 Schlieren Photograph of Mlixing Flow with Mach 2.0
Air and Mlach 1.3 Argon. ,.................. ........ 105
9 Schlieren Photograph of Hixring Flow with Mlach 1.3
Air and Mlach 1.3 Argon.............,...... ........ 105
10O Probe Drive Mlechanism Miounlted on the Test Section. 106
11 CloseUp of the Probe Drive Miechanism with the
Control System................................... 106
12 CloseUp of Static Pressure Plate................. 107
13 Gas Chromatograph with a M~olecular Sieve 5A
Column in Front................................... 107
14 Total Pressure Probes............................. 108
15 A Sample Bottle with Species Sampling Rakes........ 108
16 Streamw~ise Static Pressure. Variation for Series
IA Tests...................................... 109
17 Streamwise Static Pressure Variation for Series
IB Tests... ................. .............. ....... 110
LIST OF FIGURES (Continued)
F~ipure ase
18 Streamw~ise Static Pressure Variation for Series
IIA Tests...................................... 111
19 Streamwise Static Pressure Variation for Series
IIB Tests...................................... 112
20 Transverse Static Pressure Variation for Series I
Tests. x/a = 2.54................................ 113
21 Transverse Static Pressure Variation for Series I
Tests. x/a = 4.06................................ 114
22 Transverse Static Pressure Variation for Series I
Tests. x/a = 5.56................................ 115
23 Transverse Static Pressure Variation for Series I
Tests. x/a = 12.17............................... 116
24 Transverse Static Pressure Variation for Series II
Tests. x/a = 2.54................... ............. 117
25 Transverse Static Pressure variation for Series II
Tests. x/a 4.06................................ 118
26 Transverse Static Pressure Variation for Series II
Tests. x/a = 5.56................................ 119
27 Transverse Static Pressure Variation for Series II
Tests. x/a = 8.44,............................... 120
28 Growth of the Mixing Zone for Each of the Four
Series of Tests.................................., 121
29 Empirical Coefficients for the Turbulent
Viscosity Models as a Function of the Mass Flux
Ratios of the Two Streams......................... 122
30 Dimensionless Excess Velocity Profile at
x/a = 2.54 for Series IA (ArgonAir) Tests.....,. 123
31 Dimensionless Excess Velocity Profile at
x/a = 4.06 for Series IA (ArgonAir) Tests....... 124
32 Dimensionless Excess Velocity Profile at
x/a = 5.56 for Series IA (ArgonAir) Tests....... 125
LIST OF FIGURES (Continued)
EM rage
33 Dimensionless Excess Velocity Profile at
x/a = 12.17 for Series iA (ArgonAir) Tests...... 126
34 Dimensionless Excess Velocity Profile at
x/a = 17.7 for Series IA (ArganAir) Tests....... 127
35 Dimensionless Excess Velocity Profile at
x/a = 4.06 for Series IB (ArgonAir) Tests....... 128
36 Dimensionless Excess Velocity Profile at
x/a = 5.56 for Series IB (AirgonAir) Tests....... 129
37 Dimensionless Excess Velocity Profile at
x/a = 12.17 for Series IB (ArgonAir) Tests...... 130
38 Dimensionless Excess Velocity Profile at
x/a = 17.7 for Series IB (ArgonAir) Tests..,..,. 131
39 Dimensionless Excess Velocity Profile at
x/a = 23.26 for Series IB (ArgonAir) Tests...... 132
40 Dimensionless Excess Velocity Profile at
x/a = 2.54 for Series IIA (Helium.Air) Tests..... 133
41 Dimensionless Excess Velocity Profile at
x/a = 4.06 for Series IIA (HeliumAir) Tests..... 134
42 Dimensionless Excess Velocity Profile at
x/a = 5.56 for Series IIA (HeliumAir) Tests..... 135
43 Dimensionless Excess Velocity Profile at
x/a = 8.44 for Series IIA (Helium~Air) Tests..... 136
44 Dimensionless Excess Velocity Profile at
x/a = 12.17 for Series IIA (HeliumAiir) Tests.... 137
45 Dimensionless Excess Velocity Profile at
x/a = 2.54 for Series IIB (HeliumAir) Tests..... 138
46 Dimensionless Excess Velocity Profile at
x/a = 4.06 for Series IIB (HeliumAir) Tests..... 139
47 Dimensionless Excess Velocity Profile at
x/a = 5.56 for Series IIB (HeliumAir) Tests..... 140
48 Dimensionless Excess Velocity Profile at
x/a = 8.44 for Series IIB (Helium;Air) Tests..... 141
LIST OF FIGURES (Continued)
_Figur r
49 Dimensionless Excess Velocity Profile at
x/a = 12.17 for Series IIE: (HeliumAir) Tests.... 142
50 Argon Miass Fraction Profiler at x/a = 2.54 for
Series IA Tests................................ .. 143
51 Argon Mass Fraction Profile: at x/a = 4.06 for
Series IA Tests..................... ........... 141
52 Argon Mlass Fraciton Profile at x/a = 5.56 for
Series TA Tests..................... ..... ... ... 145
53 Argon MIass Fraction Profile at x/a = 12.17 for
Series IA Tests.................................. 146
54 Alrgon Mass Fraction Profile at x/a = 17.7 for
Series IA Tests.......... ........ 147
55 Argon Miass Fraction Profile at x/a = 4.06 for
Series IB Tests................... 148
56 Argon. Mass Fraction Profile at x/a = 5.56 for
Series IB Tests................... 149
57 Argon Mass Fraction Profile at x/a = 12.17 for
SeisIB Tests.............. ................... 15
58 Argon Mass Fraction Profile at x/a = 17.7 for
Series 1B Tests................... 151
59 Argon Mass Fraction Profile at x/a = 23.26 for
Series IB Tests.................... 152
60 Helium Mlass Fraction Profile at x/a = 2.54 for
Series IIA Tests................... .......... ... 153
61 Heliuma Mass Fraction Profile at x/a = 4.06 for
Series IIA Tests........ .................15
62 Helium Hass Fraction Profile at x/a = 5.56 for
SeisIIA Tests.... .............. ............... 15
63 Hfl~ium Mass Fraction Profile at x/a = 8.44 for
Series IIA Tests... ............. .............. 15
64 Helium Malss Fraction Profile at x/a = 12.17 for
Series IIA Tests..................... ............17
LIST OF FIGURES (Continued)
65 Helium 11ass Fraction Profile at x/a = 2.54 for
Series IIB Tests................................. 158
66 Helium M!ass Fraction Profile at x/a = 4.06 for
Series IIB Tests................................. 159
67 Helium Mass Fraction Profile at x/a = 5.56 for
Series IIB Tests................ 160
68 Helium :ass Fraction Profile at x/a = 8.44 for
Series IIB Tests................................. 161
69 Helium Mfass Fraction Profile at x/a = 12.17 for
Series IIB Tests................................. 162
70 Velocity Similarity Plot for Series IIB Tests.... 163
71 Mass Fraction Similarity Plot for Series IIB
Tests..... .......... ......... .......... .. n 16
72 Flow Diagram for YAIN~ Routine...................o. 172
73 Flow Diagram for Routine PREL~TIM................... 173
74 Flow Diagram for Routine SHEAR.................... 174
75 Flow Diagram for Routine SOL~VE.................... 175
76 Flow Diagram for Routine SUPP..................... 176
LIST OF SYMBOLS
slot height
mixing zone width
specific heat at constant pressure
Crocco number
total enthalpy
static enthalpy
mass diffusion
specific heat ratio
Lewis number
MI Mach number
m mass flow8 rate
P pressure
Pr Prandt1 number
R gas constant
Sc Schmcidt number
T temperature
u velocity component in streanwise direction
v velocity component in lateral direction
W molecular weight
x,y physical coordinate system
Greek Letters
ai mass fraction of specie i
B mass flux ratio
v x/(Ag)22
e error term in finite difference equation
Ed eddy diffusivity
Lh eddy conductivity
Em eddy viscosity
h eigenvalue of difference equation velocity ratio
a density
excess velocity (U Us)/(Up Us
stream function
u dynamic viscosity
a spread rate parameter
shearing stress
subuscripts
e nozzle exit plane
i specie (021 2, A, He)
m lateral grid incremlent in difference equation
n streamrwise grid increment in difference equation
o stagnation conditions
p primary stream
s secondary stream
Sturbulent quantities
Suoe~rscrip
turbulent fluctuation
incomlpressible initial condition
xili
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy
AT INVESTIGATION OF THE TURBULENT MIX:ING
OF PARALLEL TWiODIMIENSIONA. L COP}TRESSIBL~E
D)ISSIMILARY GAS STREAMS
By
Seyfeddin Tanrikut
December, 1973
Chairman: Dr. R. B. Gaither
Cochairm~an: Dr. V. P. Roan
Major Departnent: Mlechanical Engineering
An investigation of the turbulent mixing in the initial region
of a halfjet composed of dissimilar gas streams has been made. An
isoenergetic, nonreacting and isobaric system with various velocity
and mass flux ratios was studied both experimentally and theoretically.
The flow problem was formulated as an initial value problem
using the turbulent boundary layer equations in conjunction with
phenomenological models for the turbulent eddy viscosity. These
models consisted of Prandtl's mixing length hypothesis, Ferri's
differential mass flux formulation, Schetz's extension of th1 Clauser
integral m~odel and Alpin~ieri's momencum and mass flux model. The
validity of the above models for the turbulent momentum transport
mechanisms in the initial region of a halfjet was investigated on
a consistent basis. The eddy diffusivity was obtained from the addy
viscosity model by considering the turbulent Schmnidt number as a
parameter. The analytical solution was obtained using an allexplicit
forward marchi~ng finite difference scheme with the stability of: this
scheme being ensured by satisfying the von Neumann stability
criterion.
In the experimental phase of the study, a twodimensional test
section was designed and built to be used in conjunction with the
existing blowdown wind tunnel facilities. Interchangeable. nozzle
blocks were designed with the twodimensional method of characteristics
to provide the supersonic flow. A primary stream of air at Mach
numbers of 1.3 and 2.0 was allowed to mix: with a secondary stream of
either helium or argon at a Maech number of 1.3. Quantitative data in
the form of total and static pressure measurements, total temperature
and mass fraction measurements were collected. Qualitative data in
the form of schlieren photographs were also obtained. These data
were used to correlate and complement the analytical study.
Results of experimental static pressure measurements, both
experimental and theoretical excess velocity and mass fraction profiles,
are presented in graphical form for all configurations tested.
XV
CHAPTER I
INTRODDUCTION
The occurrence of free viscous layers and their consequent
effect on the performance of many contemporary devices has stimulated
considerable study of these flow processes. Some examples are slot
cooling, supersonic diffusion flames, dumping of fuel at high velocities
and thrust augmentation in jet and rocket engines. It is deemed
worthwhile to briefly describe how the free viscous layers, resulting
from the mixing of coflowing streams, are encountered in some of the
physical applications cited above.
1) Slot (or film) cooling: The possible structural failure
of modern aircraft due to excessive heating has been a major problem.
This problem arises in rotating machinery components of aircraft
engines as well as in all types of reentry and highspeed vehicles.
Some examples are turbine blades, rocket nozzles and leading edges of
hypersonic aircraft. One method of solution to this problem lies in
slot cooling. Slot cooling is a method whereby a coolant gas (or
lIquid) is injected into the boundary layer of the surface to be cooled.
The purpose is to create a cold film layer of gas between the surface
and the hot mainstream. If the film is maintained over a large portion
of the surface to be cooled, it acts as a partial heat shield and
therefore reduces the heating process sufficiently so as to ensulre
safe operation of the vehicle.
2) Diffusion flames: A mixing controlled combustion has
several features. The heat release is distributed over a finite length
in contrast to a premixed configuration wherein the heat may be released
abruptly through a detonation process. The inherent distribution of
heat release in a mixing controlled system provides a mechanism for
obtaining a controlled pressure variation enhancing the possible use
of a fixed geometry system. Furthermore, the mixing controlled
combustion process can take place in supersonic flow eliminating flow
losses (i.e., total pressure loss due to shock waves) and critical
design problems required for subsonic burning.
3) DuPin~ff : Mixing of an injected gaseous fuel in
combustible proportions, although desirable in combustion chambers,
is usually undesirable when fuel is vented from a flight vehicle. In
the case of the multistage chemical rocket using cryogenic propellant.
such as hydrogen, large quantities of waste gaseous fuel must be
vented overboard. After the waste gas is dumped overboard, it may be
exposed to regions of high temperature such as the surface boundary
layer and the nczzle base area of the operating firststage engine. A\
solution to this problem is the venting of the gas in such a manner
that the mixture is diluted below the lower limit of flamcmability in
a reasonably short distance from the point of injection.
4) Thrut u~igmenaton In addition to combustion and heat
transfer problems, secondary injection is of importance in ejector
systems for jet and rocket engines. High velocity secondary gases are
ducted into the nozzles to increase the total exit momentum flux of
the flow, thus obtaining higher thrust vectors.
For any kind of analytical study to be conducted on the above
physical problems there is a need to know the rate of growth of the
shear layer, subsequently referred to as the mixing region, the
variables affecting this growth and the velocity, concentration and
temperature profiles inside the mixing region.
The mixing process may take place under a variety of conditions
which determine the method of solution to the problem. Some of these
conditions are summarized below:
i. The transport mechanism governing the mixing process may
be laminar or turbulent.
ii. The mixing process may be steady or unsteady.
iii. Mixing may be isobaric or take place in the presence of
pressure gradients.
iv. The compositions of the mixing streams may be similar
or dissimilar.
v. The flow~ may be compressible or incompressible.
vi. The geometry of the system may be twodimensional or
axisymmetric.
vii. Chemical reactions may take place in the mixing region
or the flow could be frozen.
viii. The mixing may be isoenergetic or nonisoenergetic.
ix. The mixing streams could be contained by walls or the
streams may be semiinfinite.
In m~ost physical problems many of the complicating factors are present
but some have to be neglected or simplified in order to obtain
analytical solutions.
A brief investigation of the mixing problem first reveals
that the mixing processes are almost unanimously turbulent. Second,
one finds that at least one of the trixing streams will be compressible.
Thus, the mixing process of interest must account for corapressibility
of the streams. A third observation one might note is that the
two streams are very likely to be at different thermal levels, requi~ring
an assessment of these effects as well.
For all cases of mixing previous investigators [l)1 bave
concluded that for the purpose of analysis the mixing process may be
divided into two regions. The first is the developing region where
the velocity profiles are nonsimilar, and second the asymoptotic (or
fully developed) region where the velocity profiles are selfpreserving.
Figure 1 shows a schematic of the mixing region resulting from the
"contact" of two uniform parallel semiinfinite streams which are
initially separated by a thin splitter plate. The rate of growth
of the mixing region is determined by the turbulent transport r;echanismrs
of mass, momentum and energy.
Approach
There are essentially three methods in use today by which the
mixing of parallel coflowing streams are studied. These consist of
two methods that have been in use for some time, and a method that has
become available within the last several years as a result of the rapid
development and use of highspeed computers. The three methods are:
1) the simple momentum integral method, 2) solution of the equations
of motion where certain assumptions are made which render these
equations tractable to existing analytical techniques, and 3) stepbystep
numerical solution of the equations on highspeed digital computers.
All of these methods depend ultimately on experimental turbulent mixing
1Numbers in brackets designate references.
data since an ssential part of all these techniques is the specifica
tion of the local turbulent transport coefficients (eddy viscosity,
diffusivity, conductivity) of the particular turbulent flow in question
at, at least, one general location in the mixing region.
In practical applications one is sometimes interested in the
initial stage of mixing, where the upstream velocity has an important
effect. The classical similarity solutions for the shear layer cannot
account for the effect of the upstream boundary layer on the mixing
process. The profile similarity assumption limits validity of the
solution to the region past the developing region [2,3]. Solutions
for the nonsi~nilar problems may be derived using the integral technique;
however, this application is limited in practice to reasonably smooth
profiles. Profiles that cannot be approximated by analytical
expressions such as the step function, or profiles that exhibit large
boundary layer deficits lie outside the scope of the integral method.
In some of the analyses, transformations are made to obtain closed
form solutions that severely limit the variation in flowr variables. In
reference [4] a closedform solution for this problem is obtained, but
the linearization of transformed equations that is employed again limits
the variation in flow variables.
Therefore, the analytical approach taken in this investigation
wlas To develop a solution that would be numerical in nature and permit
initial variations of density, velocity and temperature profiles, and
would be valid close to the flow inlet as well as far downstream. The
flowu field was treated by employing the von Mises transformation to
the boundary layer equations and utilizing several different hypothesized
models for the eddy viscosity. The resulting conservation equations
were solved numecrically with~ an explicit type "narchiing" technique.
As stated earlier, experimental dataareessential to evaluate
the validity of any formulated turbulent transport coefficient. Such
data on the miixing of compressible streams of different compositions
are available in literature only in scarce quantity. The main body
of available material deals with the fully developed flow region of
axisymmretric jets. Data on the twodimensional mixing in the initial
region arepractically nonexistent.
Thus, an experimental investigation was done to provide data
for the twodimensional mixing of dissimilar streams. The data were
used to complement and verify the analytical approach taken in this
srudy.
The gas dynamics facilities in the mechanical engineering
department were adapted to utilize different species of gas for the
mixing analysis. The high pressure air supply system was utilized in
the primary stream and a bank of commercial bottled gases, i.e., argon
and helium, w~as used to supply the secondary stream. A test section
was designed which permitted the use of interchangeable nozzle blocks.
The isoenergetic mixing took place in this twodimensional test section
where both1 the primary and secondary streams were supersonic, Static
and total pressure measurements were made in the test section. Gas
samples were withdrawn from the mixing region and collected in a series
of vacuum bottles. These samples were subsequently analyzed with a
gas chromatograph. A schlieren system was also employed to obtain
qualitative results.
CHAPTER II
LITERATURE REVIEW
The analyses of the mixing of turbulent flow fields have been
performed by employing the laminar flowl boundary layer equations modified
by replacing the lam~inar viscosity with the eddy viscosity and by
replacing the laminar Prandt1, Lewis and Schmridr numbers by their
turbulent counterparts. The use of th~e boundary layer equations is
justified by the fact that the region of space in which a solution is
being sought does not extend far in the transverse direction, as
compared with the main direction of flow, and th~at the transverse
gradients are large. The assumptions and details involved in the
reduction of the general NavierStokes equations to the boundary layer
form are well documented in references [1], [2) and 13], and will not
be reviewed here. The twodimensional continuity and momentum equations
for steady flow are preSEnted below to aid i~n understanding some of
the assumptions and techniques used by various investigators in
obtaining solutions. These equationls are discussed in detail in!
Chapter III.
ConyIiitinury ao+oP = 21
XM~orenrtum: pu + pv =(2.2)
YMomentuta: 0 (2.3)
(Here ? denotes the turbulent shearing stress or the Reynolds stress
and ic usually expressed in terms of the time mean average of the
velocity perturbations, i.e., T = (py)'u' .)
The works of previous investigators will be reviewed in three
sections; 1) ni:;ing of semiinfinite streams, 2) mixing of contained
(ducted) streams, and 3) some experimental investigations on the
mixing of dissimnilar gases. The first two sections will primarily be
involved in the analytical. approaches. The last section is included
because it has a definite bearing on the experimental aspect of the
present investigation.
Mixing of SeniInfinite Streams
An excellent bibliography of both experimental and theoretical
worki on turbulent mixing prior to 1950 is given in a paper by Forstall
and Shapiro [5). Tw~o of the works in this paper should be mentioned
here since they formed the starting point and basis for some of the
more recent in,etigations.
The mixing of semiinfinite incompressible streams was considered
analytically as early as 1926 by Tollkien [6]. Using a similarity
transformation of the type n = y/x and Prandtl's [7] mixing length
bypothes~il for the turbulent transport mechanism, he obtained a
numerical solution for the fully developed region of a twodimensional
turbulent jet exhausting into a quiescent atmosphere. This solution
was later extended by Kuethe [8] for various boundary conditions.
Cortler (9] utilized Prandtl's (10) second hypothesis for the
eddy viscosity: to obtain a new analytical solution for the incompressible
mixing of two parallel streams. The result was a series solution in
contrast to Tc11mien's numerical solution and offered the further
A detailed discussion on the turbulent transport mechanisms
is presented in Chlapter IV.
advantage that for sufficiently large secondary velocities, the velocity
profile could be approximated by the error function.
Both Tollmiein's and Gortler's solutions involved the utilization
of a stream function which was proportional to a function, F, of the
similarity variable n = cy/x. Thus, the partial differential
equations of motion were reduced to a single ordinary differential
equation:
Tiollmien's problem: F"' + F = 0
Gortler 's problem: F' '' + 2oFF' = 0
Tollrrien experimentally determined a value of 12 for a from low subsonic
turbulent mixing experiments.
it was desired to extend this type of solution to compressible
flows. Thus, mixing analyses have primarily been concerned with the
development of the theoretical expressions for the mixing similarity
parameters: specifically, the evaluation of the "spread rate parameter",
o. One widely used method is to apply a coordinate transformation to
the compressible flow equations such that the transformed equations are
in the form of the incompressible ones. Then, data from both the
compressible and incompressible domain may be used to predict o. One
such mlthocd is the extension of Howarth's [ll] transformation which
was developed for the analysis of compressible laminar boundary layers.
The transformation is essentially stretchiing of the ycoordinate and may
be defined in one form by
+ ~ (2.4)
ax
(2.5)
These relations wrill reduce the compressible flow equations to the
incompressible flow equations in the absence of external pressure
gradients. Meager [12) removed the restriction on the pressure gradient
by postulating that the shear stress was invariant under the transfor
mation.
There are a number of theoretical estimates of the effects of
compressibility on the spread rate parameter. These estimates relate
the ratio a/a to the free stream Mach or Croccol numbers; a is the
similarity parameter for incompressible, isoenergetic flow. Abranovich
[1] estimated lateral turbulent transport using the classical Prandt1
bypothesis for eddy viscosity in conjunction with a hypothetical
characteristic longitudinal velocity in the shear layer in order to
predict the growth rate of the mixing region. Bauer [13] based his
model on a compressible analog of the mixing length concept; together
with the error function approximation of the velocity profile, he
was able to estimate the spread rate.
Channapragada and ICoolley[14], using the Howarth transformation
in conjunction with Mager's postulate, reduced the governing equations
to the form of Tollmien's or Gortler's problem depending on the model
for the eddy viscosity. From the transformation they concluded that
the parameter a varied across the mixing region for compressible flow
fields. For the two stream mixing problem, they related a to the total
temperature andi velocity ratios of theztwo streams in addition to the
primary stream Crocco number. This model for a agreed well with the
IThe Mach and Crocco numbers are related through the equation
M =cl C 1 ( 1 Cb
emipirical relation of Korst and Tripp [15] for total temperature ratios
of unity, and exhibited simiilar trends to the predictions of
Channapralgada [16]. This work was later extended by IWooLley [17] to
the case of two dissimilar streams (with same specific heat ratios) in
the presence of small pressure gradients.
Laufer [18) applied a Howuarth type of transfonrmation to the
timedependent rather than the mean equations of motion, as had been
done in the past. Wjhen the transformed equations were averaged and
the correlation between the fluctuations of the temperature and velocity
gradients neglected, the incompressible turbulent equations for the
mean flow were obtained; and these could be solved with conventional
methods.
Utilizing Prandtl's second hypothesis and an extension of
IWarren's (19] momentum integral method, Donaldson and Gray [20]
analyzed the turbulent mixing and decay of axially syamu~etric, compressible
free jets of dissimilar gases. They concluded from a comparison of
data with theoretical results that a general relationship existed, at
each axial position in the jet, between a local mixing rate parameter
and thle local Mch number. Furthermore, this relationship was
independent of the physical properties or the thermodynamic state of
the m~ixing gases (i.e., independent of molecular weight and enthalpy).
This method of analysis was later extended to the case of coflowing
stream~s by Smoot and Purcell [21].
Peters [22] presented a detailed discussion on various eddy
viscosity theories and compressibility transformations during the
develo~pient of , transport model incorporating a dual scale of eddy
sizes which was ulsed to predict o in the compressible regime. Lamb [23]
developed a theory which permitted the estimation of the effect of
compressibility and heat transfer on the spread rate parameter for
fully developed mixing zones. Integral forms of the conservation
equations were used to specify the flow characteristics along the
"dividing streamline" between the two streams. By the application of
the NavierStokes equations to this streamline he was able to calculate
a position parameter which in turn yielded an expression for o.
In a later work by Lamb and Bass [24] an analysis of the methods
involved in the correlation of the parameter a was made. It was
observed that differences in the various predictions at high Mach
numbers (on the order to 68) were as much as 50 peteent. It was
also seen that the trends of the Channapragada and W'oolley theory
appeared to be opposite to those of the other analyses, showing a large
effect of compressibility at low Mach numbers and very little
influence at higher values of Mm
In an effort to compare and consolidate different theoretical
velocity profiles and expressions for o, Korst and Chow [25] pointed
out that the spread rate parameter depended on 1) the selected eddy
viscosity model, 2) the methods of theoretical analysis and 3) the
definition of profile matching. On this basis they established
theoretical relations which attempted to reconcile discrepancies
between different analytical solutions so that all available information
on o could be utilized.
It can be concluded that many of the compressible mixing investi
gations have placed an emph~asis on obtaining similarity solutions
which are valid for regions far removed from the initial point of
contact of the two streams. Major effort has gone into methods which
attempted to predict the behavior of the spread rate parameter in the
compressible domain. Unfortunately there is no widely accepted relation
for o. This may be coupled to the fact that there is no universal
model for the turbulent transport mechanism due to the lack of
understanding of the physics of the phenomena. Another drawback; is
that no information can be extracted about the initial and transitional
regions of mixing from the spread rate parameter or similarity solutions.
Little work has been done on the effect of nonuniform initial
velocity profiles on the mixing process. Wygnanski and Hawalfehka
[26] attempted a series solution to the case of the incompressible
asymmetric jet (i.e., created by the mixing of a wall jet with quliescen~t
surrounding fluid downstream of the trailing edge). Powrever, the region
of validity of this method as well as its accuracy depended critically
on an accurate knowledge of the initial velocity profile and its
derivatives with respect to y and the number of terms retained in the
series expansion. Korst and Chow [27} used integral methods to account
for the effects of initially disturbed profiles on the mixing region.
Lamb's (23,28] "dividing streamline" solution using the momentuml
integral method yielded some information on the transition region.
Most of the investigators have assumed a value of unity for
the turbulent Frandt1 and Lewis numbers. This assumption enabled
them to obtain a solution to the momentum equation and use this
solution in conjunction with the Crocco integral relation to automatically
satisfy the speciesand energy equations. Although a value of unity
for the turbulent Frandt1 number may be justified in most cases, the
turbulent Schmidt number (rr/Le) has been shown ex~perimentally to vary
between 0.5 and 2.0 [29,30,31].
Another widely used method in obtaining solutions to the
mixing problem is one which is based on the linearization of the
conservation equations in the plane of the von Mises variables while
retaining the essential nonlinear nature of the equations in the physical
plane. This results in a linear partial differential equation in the
form of the unsteady heat equation which can be treated with conventional
methods [32] subject to initial and boundary conditions. However,
relating the intrinsic coordinate system to the physical plane by
means of integral forms of the conservation equations is rather involved
and requires the specification of a compressible eddy viscosity model.
The works of Libby [33], Ferri et al. [34], Alpinieri [35], Kleinstein
14), and Schetz [36] were all based on this type of a solution with
each solution deviating from the other in the formulation of the
turbulent transport mechanisms. These formulations will be analyzed
in Chapter IV.
Numnerical solutions have been utilized recentlyto obtain
mixing characteristics throughout the flow field [37,38,39]. These,
however, required appropriate models for the Reynolds transport terms
for each region of mixing; that is, the initial, transition and
fully developed regions.
Mixing of Confined (Ducted) Streams
Integral techniques have been used in general to obtain
solutions to confined turbulent jet mixing problems. In most of the
analyses the axial velocity and the shear stress were assumed to obey
similarity laws, so that expressions for the turbulent transport
coefficients were not needed. In 1955, Craya and Curtet [40]
established an approximate theory for confined jet mixing of streams
of identical composition. This theory w~as further developed by Curtet
[41,42] and was followed by additional theoretical and experimental
studies by Curtet and Ricou (43] and Curtet and Barchilon (44]. The
theoretical analysis was based on assumptions of zero radial pressure
gradient, uniform and nonturbulent axial velocity outside the mixing
region, and similarity of the axial velocity profiles inside the mixing
region. Experimental observations led to the assumption of similar
Gaussian velocity profiles in the developing region.
In 1965, Hill [45,46] carried out analytical studies of an
isothermal homogeneous confined jet raixing in order to predict the
mean velocity field in the flow. In this analysis, an integral
technique was also used and the shear integrals were evaluated using
free jet data. However, the assumptions made were such that the
effects of a confining wall were not significantly taken into
consideration.
Dealy [47] studied the effects of conditions at the inlet on
the flow phenomena in a confined jet mixing system. Dealy concluded
that, for systems with low jettoconfining tube radius ratio, the flow
in the near regime wuas indeed independent of the nature of the jet
source; had similar velocity profiles and was amenable to analysis
by the common momentum integral technique. But for large jetto
confining Lube radius ratios, thle mixing mechanism was found to be
dependent strongly on the flow conditions in thle Jet exit. Further
experiments by Dealy~ also showed that, for fully developed turbulent
flow at the jet source, mixing took place mlore rapidly (because of
larger turbulent stresses) than for the case corresponding to a uniform
flow at the jet source.
Trapani (48] carried out en experimental study of turbulent
jets with solid boundaries in the transverse direction inl order to
investigate their application in certain fluidic devices. Comparison
with the flow characteristics of a twuodimensional turbulent free jet
showed that the presence of solid transverse boundaries definitely
alters the behavior of the flow. 'The bounded jet (i.e., the jet
bounded by plates above and below) was seen to spread less rapidly
than the free jet. On the other hand, the confined jet (i.e., the jet
enclosed on all sides) was observed to spread more rapidly than the
free jet; this effect was attributed to the development of an adverse
axial pressure gradient in the confined flow.
An extensive study of the ducted turbulent mixing process
for supersonic flows was carried out by Peters et al. [39,49,50]
experimentally as well as analytically. An integral theory for the
ducted flow was presented for arbitrary axisymmetric duct geometry.
Cases of both frozen~ and equilibrium chemistry were considered as the
mode of chemical reaction in the mixing zone. At initiation of mixing,
the boundary layer was considered negligible as were the viscous effects
at the duct wall. The velocity profiles in the turbulent mixing zone
were assumed to be similar and were represented by a cosine function.
The turbulent shear stress in this variable density mixing layer was
treated by the use of a modified Prandt1 eddy viscosity model. The
free mixing concept of shear and velocity profile similarity were
assumed to be applica'ole in the main region. The turbulent Frandt1
and Lewis numlbers of unity were used in the analysis. From the resullts
of analysis and experiment, it was concluded that the integral method
developed permitted reasonably accurate computations of the flow in
complex mixing systems such as airair ejectors and airaugmented rockets.
Emmons (51] also developed an analysis for predicting the flow
characteristics in the mixing region of a particleladen turbulent
rocket exhaust and the surrounding air stream, N~eglecting the boundary
layer at the confining wall, the turbulent boundary layer equations
were used to describe the flow in the mixing zone. The eddy viscosity
model was assumed to vary with the streamwise coordinate. The
system of partial differential equations governing the flow was
transformed using the von M~ises transformation and then solved by
finite difference methods. Similar approaches were taken by Cohen
[52], Edelmean and Fortune [383 and Chia et al. [53].
Mixing of Dissimilar Gases
Major Experimental Investigations
Perhaps the earliest experimental study on the relative rates
of diffusion of momentum and energy in turbulent jets was done by
Ruden [54), who found that energy diffused more rapidly than momentum
in isobaric incompressible jets. Forstall and Shapiro [5] studied the
diffusion of mass and momentum between coaxial jets of air where the
central jet was composed of approximately 10 percent of helium for use
as a tracer gas. The velocities considered in their experiments were
in the low subsonic range. They concluded that mass diffusion was
more rapid than momentum diffusion. Furthermore, the Schmidt number,
which measures the relative rates of transfer of mass anid momentum,
was found to be independent of the velocity ratio. A similar result
was obtained by Keagy and lieller (55] from their experiments on helium
and carbon dioxide jets exhausting into quiescent air.
Corrsin and Uberoi (56], using a heated jet exhausting into
a quiescent region of different density, investigated the effect of
large density differences in the mixing zone. They found that a decrease
of jet density with respect to that of the receiving medium caused an
increase in the rate of spread of the jet.
Isoenergetic mixing between carbon dioxide and hydrogen central
jets exhausting into a moving concentric stream of air was investigated
by Alpinieri [35]. Using velocities in the low to high subsonic range,
he concluded that the Schmidt number was essentially constant and, in
agreement with previous results, that mass appeared to diffuse more
readily than momentum. It was also observed that no tendency toward
segregation of the two jets was evident when either the velocity ratio,
mass flow ratio or momentum flux ratio was made equal to unity. Yates
[57] studied the supersonic slot injection of hydrogen into a
supersonic air stream. He found that the concentration profiles exhibited
similarity before the velocity profiles did. The rate of growth of the
energy and concentration layers w~as observed to be about the same and
exceeded the growth of the momentum layer.
Zakkay et al. [58] undertook an extensive experimental inves
tigation of the turbulent mixing of two dissimilar gases. The axi
symmetric mixing analysis was carried out to detennin~e the turbulent
transport coefficients for hydrogen, belium, and argonair mixtures.
The external stream of air was maintained at a constant Mach number of
1.6 and the inner jet was either subsonic or supersonic, Their
conclusions mlay be summarized as, 1) the centerline decay was not
influenced by the molecular weight or the initial boundary layer of
the jet; 2) the radial velocity profiles exhibited similarity past the
potential core and could be correlated by the mass flux ratio; 3) no
dependence of Schm~idt number on molecular weight could be observed;
4) the deviation of the turbulent Schmidt number from unity was
considerable in several cases.
However, from their experimental investigation of the axi
symmentric turbulent jets of air, helium and F~reon 12, Abramovich et al.
[59) concluded that it was ineffective to attempt to describe the
characteristics of the jet by any cojoint complex of the variables
(p,u) such as the mass flux ratio or the momentum flux ratio.
CHAPTER III
ANALYSIS FOR THE MIXING OF DISSIMILAR STREAS~h
Introduction
The literature survey presented shows that several simplifying
assumptions were made to render the analytical model tractable from
the mathematical. point of view. The "boundary layer" forms of the
conservation equations have been shown to be applicable to the cases
of both confined and unconfined mixing problems. In a great majority
of the investigations, the correlation of the analysis with data was
restricted to the main region of mixing, where similarity of the
velocity profiles was assumed.
In the present study, the mixing problem is formulated as an
initial value problem in the von M~ises plane using the turbulent
boundary layer equations. Unlike the laminar problem, the transport
mechanisms in turbulent mixing do not depend on the fluid properties
alone, but also on geometric and dynamic factors of the flow system.
Determination of the necessary transfer coefficients by an exact
theoretical analysis is presently not possible due to the lack of
understanding of the turbulence phenomena; therefore, a semi
empirical approach is employed in order to correlate the data obtained
in the experimental phase of the investigation. The governing
conservation equations of the flow~ are approximated by their finite
difference forms and the solution is obtained by an explicit numerical
scheme. Numerical stability is ensured by satisfying the von Neumann
stability criterion. A typical geometry of the problem is shown in
Figure 2.
Boundary Layer Equations
The Frandt1 boundary layer equations used in the analysis of
twodimensional mixing problems are:
Global Continuity
+ + = 0 (3.1)
XMiomentum
a(pu) + (pu)u e (py)u = SP aT (3.2)
at ax oy ax ay
YMiomentum
0 (3.3)
Specie Continuity
apai a(pu)ai a(py)ai 3 i
at ax ay ay(3)
Total~
apH a(pu)H a(py)H _By But SP(35
at ax ay ay ay at
where I = (molecular shear) (3.6a)
Ji = Di ay(molecular diffusion) (3.6b)
q9q 9 d = k~ + hi
ah
=~ ai i+ hi i (molecular heat flux) (3.6c)
Ci i
If the differentiations in Equations (3.2), (3.4) and (3.5) are carried
out, the global continuity equation can be extracted and the equations
reduce to their more conventional form. The equations are written
in the above forms for the sake of mathematical, expediency in the time
averaging technique necessary to obtain the turbulent forms.
Inherent in the above equations are the assumptions that the
body force terms in the xand ymomentum equations are small, the
species are inert chemically (i.e., nonreacting flow) and thermal
diffusion is negligible. From the first of these assumptions, the
only information available from the ymomentum equation is the result
that pressure is invariant in the lateral direction. Pressure gradients
(if any) are restricted to the axial direction and no further use
of the ymomentum equatica can be made.
Further assumptions necessary for the present analysis are
summarized below:
i. Flow outside the mixing zone is inviscid, non
conducting and uniform.
ii. Boundary layers on the confining walls will be neglected.
No correlation will be attempted for regions darmstream
of the point of interaction of the wall with the mixing
zone.
iii. The mixing region of interest is under isobaric conditions;
that is, in addition to aP/8y = 0, it is assumed that
aP/ax = 0. Although some streamwise pressure changes are
expected since the experimental configuration under considera
tion is one of confined supersonic flow, these pressure
changes are expected to be small (i.e., 58 percent of the
exit plane value) since the region of interest is the
initial mixing region (i.e., 510 inches downstream of the
nozzle exit) where there is no interaction of the confining
walls with the mixing zone. Data from the experimental
investigation justify this assumption as does other
literature [1] .
Turbu~lenat Eqution
In the analysis of turbulent flows, it is customary to assume
that the instantaneous value of each property is the sum of the mean
value wh7~ich varies with a timemean average value and a fluctuating
component which is a function of time [3]. In addition to fluctuations
of velocity, density, temperature and mass fractions, Van Driest [60]
hypothesized that there are fluctuations of mass flow (pu), (pv)
regarded as a single property. Hence, the following relation is
defined for any property O:
4 = + (3.7)
where "bar" quantities are timemeanaverages defined by
to+t
O=, dt (3.8)
tt
and the "p~rimie" quantities are fluctuating components. It is readily
seen from Equation (3.8) that the timemean value of any linear
fluctuating quantity and its derivatives vanisles;L.e., ', v', Bu'/ay,
etc. are all zero.
Since turbulent flow is unsteady in nature, it is somewhat of
a contradiction in terms to speak of a "steady turbulent flow", but
the term has usually been used to denote a turbulent flow which is
steady in the mean, i.e., quasisteady. Moreover, at each point the
fluid properties and velocity may be observed to fluctuate wildly, but,
when averaged over time periods comprising many cyclic fluctuations,
the timerean properties are constant with respect to time.
Thus, a state of quasisteady turbulent flow is assumed; the
instantaneous values of the quantities are replaced by the relation
in Equation (3.7) and the flow is considered in the mean by time
averaging as defined in Equation (3.8).
Global Contin~uityi
apu 8cy = 0 (3.9)
3x ay
XMomenztum
ou +3 = (py)'u' (pu)'u'
This equation differs from the lamninar counterpart by the last two
terms. The last term is the derivative of the turbulent normal stress
and is usually neglected by assuming that boundary layer approximations
are valid for cojoint complex perturbation quantities, i.e.,
Ci 14'" (3.10)
The second term on the righthand side is the Reynolds (or apparent)
stress which cannot be neglected. Thus, a turbulent stress is defined
as
It = (py)'u'
and the miomentum equation takes the form:
pu +tp = i [r + Tt] (3.11)
Species
pu bx p y Byv (py)'ai x (pu)'ai
Withl the ap~proximation of Equation (3.10) and defining turbulent mass
diffusion as
Jit = (py)'a'
the specifsequation takes the same form as the momentum equation:
pua + PVa = [Ji Jit] (3.12)
Total I nergv
H 8H 3c d a
pu x ay ay ay 3y (uT + '' p)H x(p)H
Although the last term can be neglected with the aid of Equation (3.10),
the present form of the equation does not permit a comparison between
the molecular and turbulent transport properties. To overcome this
difficulty, some assumptions have to be made. From Equation (3.6a),
the third term on the right may be written as
2 y"a u2 + u2
Along the lines of Van Driest's [60) workl, it is expected that u'2 is
small compared to u2 ; since u2 is assumed to be of the same order as h,
u'2 can be dropped from the equation.
Now an equivalent expression for the perturbation term, 11',
will be formulated. From the definition of total enthalpy and assuming
v2 << u2 which is a valid boundary layer assumption:
2 2
H h +2 u= ihi F
2 ,
+i H'. = {i hi + ihi + Ui i + aihi + + uu
2 2 U,2
Noting that a "hi = h a hi and = u
+i H' = (FL + ) + {i hi + aii ih i a aih'} + uu
Thus,
,2 ,2
H' = a ihi u + iai + hiai +L aii+u'+ (.3
It is also noted that Equation (3.13) satisfies the condition H' 0.
W'hen Equation (3.13) is multiplied by (pv)', time averaged and the
third order correlations neglected, the result is
(ou)'H' = b (py)'ol (py)'u' t i (py)'hi (3.14)
i i
The first and second terms on the right may be recognized as containing
the turbulent diffusion and shear terms, respectively. The third term
is similar to q; in Equation (3.6c) and is defined as the turbulent
"conduction" term:
Act = at" (py)'h'
Equation (3.14) no* has terms analogous to the molecular terms and the
total energy equation is written as
PY~ + PY ay ay (qC )ly! 44) i ayd d (T t
(3.15)
It is now assumed that the boundary layer is fully turbulent
and, thus, molecular transports are negligible, i.e.,
get a)9
qdt > d
The turbulent shear is related to the mean flow variables
following Boussinesq [61)
S= (py)'u' = Em a 3.16)
where E is the turbulent momentum transfer coefficient and has the
same units as the molecular dynamic viscosity. Similarly, the
turbulent diffusion and conduction terms are related to the mean flow
variables:
it = ~ (y'i = d (3.17)
qc = l Ci (py)'hi I: ai h (3.18)
i
The relation indicated by Equation (3.17) implies that all mass
transfer coefficients, Edi, of the various species are equal. Woolley
[17) argued that if two streams, each composed of multiple chemical
species in homogeneous mixtures were considered, then, upon exposing
the two streams to each other, all gradients of concentration differences
for species between the streams would be identical. If their mass
transport coefficients were also equal, they would diffuse through the
mixing zone at the same rate. The relative concentrations of the species
from a given stream would, then, remain unchanged at any position in
the mixing zone. However, this was equivalent to each stream behaving
as a single species. Thus, under the present assumption, each homogeneous
stream, no matter what its detailed composition, may be treated as a
single speciesin the mixing study. Therefore, it is only necessary to
treat the mixing of two dissimilar gases, each having the average
chemical and thermodynamic properties of their respective mixtures.
This is particularly true for the primary stream of air which is
usually considered as a single species[62].
Also of interest are some dimensionless turbulent quantities
which measure the relative rates of different transport mechanisms.
The turbulent Prandt1 number is a measure of the relative rates of
transport of momentum and energy; the turbulent Lewis number is a
measure of the relative rates of transport of mass and energy.
These quantities together with the turbulent Schmidt number
are defined as
Pr E
t ch
Le (3.19)
t Eh
Prt Em
Sc 
t Let Ed
Since all equations are time averaged and all transports are
turbulent, the bar notation and the subscript "t" will be dropped
from here on. With thle definitions in Equations (3.16) through (3.19)
and after some rearrangement, the governing equations take the final
form:
Global Continuity
ap+ = 0 (3.20)
Morentum
pu au + vi y u (3.21)
ax ayBymby
Species
pu aa+ pv a = (3.22)
Total ~Eng_
pu + ov = E
+ e~ hi (3.23)
The von Miises Transformation
The solution of Equations (3.20) through (3.23) provides the
details of the flow field including the velocity, species and enthalpy
(thus temperature) fields. The global continuity equation, Equation
(3.20), can be eliminated from the system of differential equations by
introducing the von Mlises coordinates as the independent variables.
The transformation (x,y) ) (x,0(x,y)) is defined according to the
relations:
= v ; pu
The derivatives in the phlysical coordinates are mapped onto the
von MIises plane via
x x,~] x x,~!  = pu (3.24)
(3.25)
Substitution of the above relations into the system of equations
completes the formulation of the problem in the von Hises plane.
Momentum
usa a (3.26)
=xa pUE, (3.27)
Total Energy
asH ai pus 1 8H Pr 1 u2/2
ax B m r B Pr ao
+ Le 1~ hi BiiL (3.28)
Pr i
The physical ycoordinate is obtained by the inverse transformation:
yj = (3.29)
and the transvefrse component of velocity, v, is given by:
v = 3.30)
p ax
Boundary Conditions
The governing equations exhibit parabolic characteristics
and thus require initial conditions at some x x and boundary
conditions at 9 = 0 and $ = =
In any real experimental situation, there is an inevitable
accumulation of boundary layer on the jet dividing boundary. In
order to avoid making an error in initial conditions by assuming
either a step profile or a computed boundary layer, the calculation
of the mixing region is started at a position downstream of the mixing
interface where measured data are available. Hence, the initial
conditions may be expressed as:
@x=x ; 0i<$<=
u(x ,4) = U (9)
(3.31)
H(x ,9) = H ( )
aicx,9) = at ~)
Since the warll boundary layer is assumed to be negligible,
constant flow conditions equal. to the secondary stream conditions are
assumed at the wall, i.e., slip condition. It is observed that zero
axial velocities are not permissible at any location in the flow field
due to thle inverse transformation of Equation (3.29). Then, the
boundary conditions become:
@y=0+ =0 ;
u(x,0) = u = constant
s (3.32)
H(x,0) = Hs =constant
.O2(x,0) = 0 ; a 2(x,0) = 0 ; a ~H(x,0) = 1
07== =m~
u(x, ) = u =constant
P (3.33)
H(x, ) = H_ = constant
a02(x,,) = 0.23 ; a 2xw .7;Gexm
The condition of y = = (\p = m~) are all regions beyond the upper
boundary of the mixing zonle.
Finite Difference EquationR
A forward m:arching allexpli~cit numerical method was used in
this analysis. Accordingly, for the transverse derivatives, central
differences are used in the interior grid points. Forward differences
are used for the longitudinal derivatives everywhere. Figure 3 shows
a generic point (n+1,m) in the (x,tb) grid network, for which the solution
is obtained by using the following explicit difference relations [63]
where F is any one of the three pertinent variables u, ai, or H:
8.2 F+1,m n,m (3.34)
Sx dx
=F n ml n m1 3.5
1 (3.36)
n;,m+/ (F n nEl (3.38)m nm
by her finite difernc euaio
n+1,m = n,n n,m+1/2Fn,mi+1 n,m+1/2 n,m1/2 n,m
n,m1/2Fn,m1]+(.9
where y = ax/(60)2 and t is the error introduced by the finite difference
approximation of the differential equation and may be defined as [63):
t = kl[6(bx)] + k2[8(6;)2] (3.40)
Equation (3.40) implies that Ax and At have to be sufficiently small
for the difference scheme to be accurate. This point will be
discussed further with respect to consistency and stability of the
scheme.
It should be recognized that, although the partial differential
equations are nonllinear, the present explicit difference formulation
results in a locally linear system. Inherent in this result is the
assumption that the solution is "fairly smooth" and the quantity, 5,
is a "slowly varying" function. These two assumptions imply that
discontinuities such as shock waves cannot be present in the flow
field, and thus pose no major restriction on the solution since no
such phenomena are considered.
The conservation equations for the interior grid points
(i.e., m # 0) in difference form are:
Momefntum
un+1,m = un,m+ Yn,m+1/2nm+1~ (n,mt/+1/ n,m1/2)un,m
+ n,m1/2un,m1]
(3.41)
species
(sin+,m= a n,m Sc n,m+1/2(ai)n,m+1 n,m+1/2
n~m1/2 (s 5 n,m1/( l~] (3.42)
Energy
n+1,m = n,m Pr n,m+1/2Hn,m+1 (n,mc1/2+ n,ml1/2)Hn,m
i n,m1/2Hn,m1] 2 [n,nd1/2un,m+1
2 2
(n,m+1/2+ n,m1/2)un,m+ n,m1/2un,m1]
+r [(Shi n,m+1/2(Cti n,m+1 [(5hi nn~m+12
+ ( i n m 1 /2 ] ( )n m + i n m 1 / 2 ( ai n m 
(3.43)
Boundary Conditions for the Difference Equation
The initial and boundary conditions for the difference
equations are similar to their counterparts for the differential
equation and are input in equal intervals of A$.
@ x = x" ; 0 I 4 NCJI
uo,m = Um(Pf)
(3.44)
H = H (d)
o,m m
(ai)o~m = [ai ml)l
An interesting characteristic of the general difference equation,
Equation (3.39), is observed when the boundary condition at Jl= 0, i.e.,
m = 0, is applied; that is, the terms Fn and 5n/ are undefined.
The conventional method Employed to circumvent this difficulty is to
define an "artificial" boundary condition such that the value of
n,1ieqlton1.This implies that the gradient of the quantity
F is zero at the boundary and, in the physical sense, is equivalent to
a condition of symmnetry at the axis of a jet. Since an assumption of
constant velocity at the wall has already been made for the physical
boundary condition, the application of the above principle will not
introduce any new assumptions into the analysis. With the application
of the condition of symmetry, the difference equations at the wall
become:
Momentum:
un+1,o= u,o + 2YSn,o(un,l un) (3,45)
Species
(ai~n+1,o = [ (onoai n,l (Oi n,0] (3.46)
Energy
H =H + ,,R nl H )+ (
n+1l,o n,o Pr n~(n1 n,o Pr n,o un,l
u2,o n,o i n,olan,( )o
(3.47)
Together with the boundary condition:
nu = us = constant
Hn~ = Hs = constant (3.48)
(a0 n~ = ; aN2n~o= 0; (A,He n,o
The boundary condition at $ = = is one of "floating" type; that is,
calculations proceed until the condition Enm+ n m is satisfied.
Analysis of the Difference Scheme
Once a finite difference scheme is set up, it must satisfy
three conditions [64]:
i. The difference equation must be "consistent" with
the differential equation. That is, the error involved
in approximating a differential equation by a difference
equation must vanish in the limit nx, 69 +t 0.
ii. The difference scheme must be "con~vergent". That is, the
solution to the difference equation must approach the
solution of the differential equation in the limit.
iii. The marching rate must be "stable". That is, the grid
size should be such that the solution does not become
unbounded anywjhere. A relation between Ax and Ath must
be found to insure this condition.
The consistency of the present method may be shown by
observing that the quantity E*, defined in Equation (3.40), approaches
zero unconditionally as (Ax, AS) approach zero. The proof of conver
gence for quasilinear difference equations is lengthy and is not
presented here; however, the numerical scheme employed can be shown
to be convergent with the method of Strang (65].
One of the important characteristics of explicit finite
difference schemes is the stability of the solution. For a system of
linear constant coefficient equations, it can be shown that the
von N'eumann stability critericnis the necessary and su~ffcient condition
for stability and convergence [63]. In variablecoefficient cases it
is necessary for stability that the von Neumann conditions (derived as
though the coefficients were constant) be satisfied at every point
in the grid network. This is based on the observation that when
instability occurs in practice, it often appears as a local disturbance
in a region where the von Neumann condition is violated. John [in 63] has
shown that for a general class of explicit difference equations for
quasilinear parabolic partial differential equations, the von N'eumann
condition is necessary for stability and that a slightly modified form
of the von Neumann condition is sufficient.
On this basis the von Neumann stability criterimwas derived,
and this criterionwas checked at each point in the grid network since
the system of governing equations are quasilinear An outline of the
von Neumann analysis for the difference equations used is given below.
A linear, constant coefficient parabolic partial differential
equation with periodic boundary conditions has a solution of the form
u(x,y) = A(x)eiky
Assuming that in the limit the difference equation has the same type
of solution, let
En~m nikmby(3.49)
F, Ikin~~d
Equation (3.49) is substituted into the general difference equation,
Equation (3.39), and since it is also assumed that all Fourier
coefficients decay exponentially, the kth term is examined:
lk 1 Y(i(n,m+1/2 n,m1/2)(1~ cos kaiy) + lY((n,m+1/2
(n,m1/2)sin kfiy (3.50)
For Fn~ to be bounded, the von N\eumann criterion is
After some manipulation, the restriction on the grid size relation,
y, is obtained
n,m+1/2 n,m1/2
,m+1/2 ,m1/2 lj2 2(n,m+1/Zn,m m1/2 cos k~y
and the most restrictive condition occurs when the quantity on the
right is a minimum, which is the case when cos(k~y) = 1. Then, the
final form of the stability criterion which governs the interior
grid size is
2 5 +1 (5 = pusm) (3.51)
(&) n,m+1/2 n,m1/2
It is further observed that if 5 is a constant, C, throughout
the domain of interest, Equation (3.51) reduces to the form C~t/(ox)2
1/2 which is the wellestablished stability condition for the linear
"heat Equation" with the difference scheme of Equation (3.39).
Since Equation (3.51) is valid only at the interior grid
points, a stability condition for the boundary is derived in the
same manner with the result
(3.52)
The above stability conditions were derived for the general
difference equation. W~hen applied to the governing system of equations,
they are seen to be different by a constant K where
K = 1 for the momentum equation
K = 1/Sc for the speciesequation
K = 1/Pr for the energy equation
It was found, during the analysis of truncation error of
linear equations, that when the stability criterion is multiplied by
a factor of 1/3, a higher order of accuracy could be obtained, i.e.,
=. K [6(ix)2] + K2[ (b ) ]. This is not strictly correct for quasi
linear equations, but if 5 is a "slowly varying" function, considerable
improvement in accuracy can still be obtained by making use of the
above result. Therefore, with the above modifications the final form
of the stability conditions are
> Ax < ( (3.53)
n,o n,m+1/2 n,m1/2
This results in six conditions, the most restrictive of which is
utilized, depending on the magnitude of thle dimensionless quantities
Pr, Sc and Le.
Thus, Equations (3.41) through (3.48) together with the stability
condition, Equation (3.53), constitute the numerical solution to the
flowy field. The only remaining point is the formulation of the
quantity em, and this will be discussed in Chapter IV.
CHAPTER IV
TURBULENT MIOLENTIU AND MASS TRANSPORT
General Considerations
Although the first portion of this chapter could have been
included in the literature survey, it is felt that the presentation
of the analysis would be more continuous if it were included in this
section.
To this point the consideration of the analytical treatment
of the turbulent mixing problem has dealt primarily with the question
of solving the questions of motion. While being necessary, this is
not the area of greatest difficulty. The major difficulty associated
with these problems is the mathematical representation of turbulent
transport processes. The present understanding of the turbulence
phenomenon is such that turbulent processes within a shear flow cannot
be treated locally. Rather, the most that can be expected is some
prediction of the "mean" flow properties.
There are essentially two distinct approaches previously taken
in the analysis of mixing problems. The more recent approach (66,67,68]
is to introduce additional conservation equations which describe the
Reynolds stress. This approach is appealing since it guarantees
conservation of the turbulent quantities. Unfortunately, the resulting
equations contain second and higher order correlations and, thus, to
apply this approach, empirical relations are required for the third
and higher order correlations. Furthermore, due to its complexity,
this approach has not been applied to compressible free shear layer
flows. Therefore, solutions of practical problems of current interest
have been attained only by employing the more commonly used approach
of utilizing an eddy viscosity model along with assumed constant Sc,
Pr and hence LE. The eddy viscosity models that have been proposed
for free mixing flows are summarized in Table I.
Classical Eddy Viscosity Mlodels
The famous mixing length theory for turbulent shear was
formulated by Prandrl [7] who hypothesized that the mean value of the
fluctuating velocity component in a turbulent flow field is equal
to the product of the local mean velocity gradient and a characteristic
mixing length, 2. The quantity e is defined as a distance in the flow
field such that a fluid element conserves its longitudinal velocity
as it moves across this distance. In the case of free mixing, the
mixing length is assumed to be constant across the mixing layer and
also assumed to be proportional to the local width of the mixing zone.
Thus, Prandtl's mixing length theory gives the following relation
for the Reynolds stress in the longitudinal direction
I = py)u : c2b2 au au (4.1)
t ay ay
where c is an experimentally determined constant and b is the width
of the mixing layer. Using the concept of eddy viscosity in Equation
(3.16)
q = cb2 Su(4.2)
Based on a similar mixing length concept, Taylor [in 1] derived a
vorticity transport theory where the vorticity of the fluid element
is assumed to be conserved across the mixing length. In both of these
mixing length concepts, the eddy length scale was assumed to be much
smaller than the local width of the mixing layer. The complexity of
Taylor's model for axisymmetric flows has prevented its utilization;
and for the case of twodimensional flows, except for a numerical
constant, the vorticity theory results in the same expression for
turbulent shear as obtained from Prandtl's mixing length theory.
Prandt1 [10] later proposed another model for the turbulent
eddy viscosity based on the hypothesis that the eddy scale was of the
same order as the width of the mixing layer. This model was based on
the assumption that the eddy viscosity is related to the local mean
velocity gradient and was expressed as
Eol = Cb(Umax Umin) (4.3)
where c is an empirical constant and Umax and Umin are the maximum and
minimum longitudinal velocities, respectively. Prandtl's second model
predicts a constant eddy viscosity across the mixing layer since the
width, b, is not a function of the lateral coordinate.
Equation (4.3) has been widely applied to a variety of free
mixing problems due to its mathematical simplicity and the results it
yields agree satisfactorily with experimental data for several flow
configurations. It, therefore, forms the basis for most eddy
viscosity models existing in the literature, and is used for a
particular flow field by appropriately including the effects that may
be of significant interest in that case. However, it is noted that
the model fails completely when the velocities of the two streams are
equal. Hence, it predicts that two streams of equal velocities flow
along as segregated without turbulent mixing. This implication has
been shown to b incorrect by the experimental results of references
[5] and [35].
The various eddy viscosity models available today are the
result of attempts by several investigators to include the cases of
equal velocities within the framework of Prandtl's original hypothesis
for free turbulent flow. Ferri et al. [34] suggested the following
model by simply extending the second Prandt1 model to describe flows
with density gradients
Pm = C rl/2[(pu)max (pu)min] (4.
where rl/2 is the halfradius. This model has yielded predictions of
unreliable accuracy for axisymmetric flows [35,58] but when applied
to the planar case, good predictions were achieved [36]. However, the
Ferri model fails when the mass flux of each stream is equal.
In order to circumvent such irregularities, Alpinieri (35]
considered the eddy viscosity to be proportional to the sum of the
mass flux and the momentum flux and for an axisymmetric jet, proposed
the relation
(PE )C.L.u pu p)
(pu)j c rl/2 "+(45
where (PE ) is the "dynamic" turbulent viscosity at the centerline
of the coaxial jet with the subscripts e and j referring to the
properties of the external stream and the inner jet, respectively.
The Alpinieri model is contrary in form to any other model and is
viewed as essentially empirical, qualitatively as well as quantitatively
[69].
Density differences may arise within the mixing region either
due to compressibility effects, as in the case of heated jets and
supersonic flows, or due to streams of different composition. Ting
and Libby [70], employing a Miager transformation, postulated the
following relation between the eddy viscosity for constant density
mixing and that for variable density axisymmetric flows
E = c*0 21 2P rdr (4.6)
m p 2) Po
where E is the eddy viscosity for incompressible flows and po is a
reference density. As can be seen, the above relation is essentially
a conversion of the incompressible eddy viscosity, E to one applicable
for flows with density variations either due to compressibility or
stratification. It should be recognized, however, that while this
transformation admits possible practical applications, no definite
form of a or po is suggested and the results vary depending on the
forms of E and po used. A planar form of this model was utilized
by Schetz [36] with the expression
p~~cm C.. i (4.7)
Donaldson and Gray [20] attempted to account for compressibility
via modifying the Prandt1 model, Equation (4.3), by employing an
empirically determined constant which varied with Mach number at the
halfradius. This resulted in the expression
cm = p(0.66 + 0.34 exp(3.42 M2)
where a_ denotes the Prandt1 eddy viscosity model of Equation
(4.3).
Schetz [69] proposed a model which was developed from an exten
sion of Clauser's model for the wake region of a turbulent boundary
layer to free shear layers. The specific functional expression for
a given flow problem was derived from the general statement: "the
turbulent viscosity is proportional to the mass flowJ defect (or
excess) in the mixing region." [69, page 1] Thie model was expressed as
pe, = ep u 1 p~ue dy (4.9)
Similar to the Ferri model, the above expression fails when the mass
defect (or excess) is zero. Also, the model has been shown [31] to
fail for the case of the quiescent jet since the mass entrainment,
and therefore, mass defect, increases in the downstream direction
thus predicting continually increasing eddy viscosity. It has been
demonstrated by Eggers [71] that for an accurate prediction of the
flow field in quiescent supersonic jets, the eddy viscosity must
remain very nearly constant.
There have been other models formulated for the turbulent
transport mechanism, but these are either too complicated to be
of practical use, i.e., von Karman model (Table I), or are not
applicable to the present study, i.e., Zakkay model. Each of the eddy
viscosity models presented in Equations (4.2) through (4.9) are
deficient in certain respects. Of course, some of the deficiencies are
relatively unimportant and any one of these models is acceptable
provided that they are used within the flow region for which they have
been verified.
Choice of Models for the Present Study
It should be mentioned that the eddy viscosity models cited
have been used for extensive correlations but only in the similarity
regions of symmetric jets, subject to specific flow conditions and
configurations. It is important to note the distinction between the
similarity region of a symmetric jet and the similarity region of the
halfjet which is the case under study. The similarity region of a
symmetric jet is that part of the flow field far doblestream of the
socalled potential care. The halfjet may be thought of as a symmetric
jet with an infinite jet radius or eight, resulting in a potential
core of infinite length. The important point is the fact that the
models in Equations (4,4) through (4.9) have been verified only in
regions past the potential core of a symmetric jet. Thus, no state
ment may be made regarding their application to other jet mixing
configurations. In fact, this is true for all available expressions
for em because of the lack of complete and accurate data used in
studying any eddy viscosity model. Hence, the results obtained in
this study apply to both the initial and fully developed regions of a
halfjet and the potential core of a symm~etric jet.
The fact that several models may be used to correlate the same
experimental data is shown by the results obtained by Ragsdale and
Edwards [72,73] in their analytical and experimental study with air
bromine system. In their analytical study, various expressions for
eddy viscosity were compared on a consistent basis. It was concluded
that modifications of Plandtl's second hypothesis, that introduce mass
flux or momentum flux or both, produce expressions whose differences
are more apparent than real. It was shown that these various expressions
predict essentially the same eddy viscosity as long as they are applied
only within the range of conditions for which they have been experimentally
verified. It was concluded that this was perhaps because the initial
turbulence present in the streams contributes significantly to the
mixing process and may domlinaite the situation for nearly equal stream
velocities.
The effect of free stream turbulence was also considered by
Hokenson and Schetz [74] in their study of turbulent mixing with
pressure gradients. The results of their investigation demonstrated
that the empirical constant in the modified Clauser model of
Equation (4.9) must reflect the turbulence intensity and therefore is
not a universal constant. It was also observed that this empirical
constant was approximately independent of the longitudinal distance
from the initial station, and that if adequate information (i.e.,
turbulence intensity) is known at the initial station, a numerical
evaluation of the flow field could reasonably be assumed within the
framework of the generalized Clauser eddy viscosity.
To correlate the data obtained in the experimental phase of the
present investigation, four models of the turbulent momentum transport
mechanism were chosen; namely, PrandtlB mixing length hypothesis
(Equation (4.2)), Ferri's differential mass flux model (Equation (4.4)),
Alpinieri's momentum flux model (Equation (4.5)) and Schetz's
extension of the Clauser model (Equation (4.9)). As stated earlier
these models have been shown, within certain limitations, to
correlate data in the "far field" or similarity region of symmetric
jets. Since the region of interest for the present analysis is the
initial mixing region, the primary objective was to examine the
validity of these models in the socalled "near field". It is also
of passing interest to note that, to the author's knowledge, the
Alpinieri model has never been applied to the case of twodimensional
mixing problems in any region of flow. Minor modifications necessary
to compare the models on a consistent basis within the framework
of the present analysis, is discussed in the next section.
Modification and Correlation of MLodels
The form of the governing equations, Equations (3.26) through
(3.30), require that the turbulent transport mechanism be specified in
the form (or units) of "dynamic" turbulent viscosity; i.e., as the
counterpart of u in the laminar case. To avoid confusion in terms of
notation, (yt P' (t F' (bt A and (llt S will be used to denote the
Prandtl, Ferri, Alpinieri and Schetz models, respectively.
To be consistent with the governing equations, the Prandt1
model is expressed in terms of the von Mises variables
(ut m c b 2u(4.10)
and with the aid of Equation (3.35) is put in finite difference form
(p b1 n,mun,mun+ unm1(1)
In the Ferri and Alpinieri models, the eddy viscosity is
expressed in terms of the halfradius based on either the velocity or
the mass flux. The halfradius is defined as the distance from the
axis of symmetry at which the axial velocity (or mass flux) is equal to
the average of the maximum and minimum velocities (or mass flux), i.e.,
r at which U = 0.5(Una +Umi
or r at which pU = 0.5((pU)ma + (pU)mi
Although this has some physical meaning in the similarity region of a
symmetric jet, it has no meaning in the initial region of a halfjet.
Therefore, rather than a width such as the halfradius being used, the
actual width of the mixing region will be utilized. Since there is
always an uncertainty as to the physical boundaries of the mixing
region in any experimental study, the width will be defined as the
distance in which the velocities are within 5 percent of the free
streak values, i.e., referring to Figure 2, if
b = Y1 Y2
then Y1 = Y at which IU Up /Up 0.05 (4.12)
Y2= Y at which IU Us /Us= 0.05
In the Alpinieri model the centerline velocity is replaced by the
secondary stream,which is analogous to the condition in the potential
core of the symmletric jet.
With the above modifications, the Ferri and Alpinieri models
take the form
("t F = c2b[(pu)max (Pu)min] (4.13)
(9 A =03bpuu
where the subscripts p and s refer to primary and secondary stream
conditions, respectively.
The Schetz model when expressed in terms of the von Mises
variables reduces to
(ut s = C4 m 1I dt (4.15)
It was necessary to have a common frame of reference to be
able to compare the different models on a consistent basis. The
mixing zone width, defined in Equation (4.12) was chosen to be this
common reference. The coefficients of each model were varied until
the predicted growth of the mixing zone matched the experimental data
for each of the four configurations of Table II. With each model
predicting the same growth rate, the theoretical and experimental
velocity profiles were compared as to the "goodness of fit" of each
model.
It was also expected that the coefficients of a given model
would vary from configuration to configuration and thus be a function
of the flow field as was concluded by Hokenson and Schetz [74].
However, rather than use the turbulence intensity at the initial
station, an attempt was made to correlate the coefficients (Cl through
C ) with the initial mixing conditions such as the velocity ratio or
the mass flux ratio. The results will be discussed in Chapter VI.
Remarks on the Turbulent Transport of Mlass and Energy
In early analysis of mixing problems, it was often assumed
that the turbulent Schmidt and Prandt1 numbers were unity; an
assumption which simplifies the governing equations considerably.
However, recent experiments indicate that the Schmidt number may differ
significantly from unity. Furthermore, the experimental data of Forstall
51
and Shapiro [5] show that the Schmidt number remains constant at
approximately 0.7 throughout the mixing region, so that the eddy
diffusivity Ed is merely a constant times the eddy viscosity Em.
For gaseous components in binary mixing, the values of Sc most
frequently cited vary between 0.5 and 1.2. In the present investigation,
the turbulent Schnidt and Prandt1 numbers are considered as parameters
and are retained constant in the entire mixing region, Using suitable
values for Sc and Pr yields the values for Ed and sh from calculated
values of Em
Hence, with the choice of the transport mechanism, the
formulation of the problem is completed. The equations are solved
numerically on. an IBMI 370/165 and the results are discussed in
Chapter VI. A discussion and printout of the computer program is
presented in the Appendix.
CHAPTER V
FXPERIMLENTAL INVESTIGATION
General Considerations
The main objective of the experimental investigation was to examine
the mixing of parallel twodimensional coflow~ine supersonic gas streams.
The primary emphasis was placed on obtaining pressure, velocity and
speciesconcentration profiles inside the mixing region.
The secondary stream design Miach number wras chosen to be 1.3
for the following reasons:
i. to minimize disturbances at the nozzle exit, the pressure in
the two streams was matched by presetting settling chamber
conditions in each. To avoid pressure communication back
to the repective chambers, both streams had to be supersonic,
i.e., M > 1. Waves due to supersonic flow at Mlach 1.3
(if any existed) would tend to be weak.
ii. an average run time of approximately thirty seconds was
necessary to obtain various measurements. Due to the large
number of runs needed to accurately define the specie
profiles, the total pressure in the secondary settling chamber
would have to be as low as possible to conserve the con
sumption of commercial bottled gas. High Mach numbers in
the secondary stream would have forced the use of high
total pressures to match exit conditions. Since flow rate
is directly proportional to the total pressure, a design
Mlach number of 1.3 was chosen to keep the secondary stream
mass flow rate relatively low.
iii. the contour of the supersonic portion on a Mach 1.3 nozzle
was explicit enough to ensure relatively error free machining.
The primary stream design Mach numbers of 1.3 and 2.0 were
chosen to obtain various velocity and mass flux ratios between the two
streams. A summary of runs with various configurations and test con
ditions is shown in Table II.
Experimental Apuaratus
The gas dynamics facilities in the mechanical engineering de
partment were modified to utilize different species of gas for the mixing
analysis (Figure 4). A block diagram of the system is also presented
in Figure 5.
The existing facility consists of a twostage positive displace
ment type compressor, feeding a series of high pressure air storage
tanks. Air from the tanks is brought to the laboratory in two separate
lines to a pair of onoff valves. The line pressures are stepped down
to the desired levels by a pressure regulator in each stream before the
air enters the settling chambers. Inside the two separate settling
chambers are a series of flow straighteners and dampening screens. Thus,
fairly uniform streams are introduced through converging sections into
the test section. The test section is followed by a diffuser section;
then the air is passed through a sound attenuator and exhausts into the
atmosphere.
This system was modified by the addition of a storage tank that
supplied the secondary stream with either argon or helium. The
existing dry air supply system was utilized for the primary stream. A
twodimensional test section was designed that permitted the use of
interchangeable nozzle blocks.
Test Section
A photograph of the test section with the splitter plate and a
set of nozzle blocks installed is shown in Figure 6. The designed test
section consisted of a section which reduced the existing dimensions of
the system to the desired dimensions of 8.5 by 0.5 inches for the primary
and 4.5 by 0.5 inches for the secondary stream. This section was
constructed out of steel and had a secondary function of supporting most
of the length of the splitter plate. Erected between the reducer and
the diffuser sections was the main frame also made from steel. The
upper and lower portions of the frame were used to position and secure
the primary and secondary stream nozzle blocks. Hard neoprene gasket
material was used to seal the flanges. The frame was "sandwiched"
between two oneinch aluminum side plates which when bolted together would
give a test section width of onehalf inch. Although a larger test
section might have been desirable, the flow area (thus the flow rate)
of the secondary stream was the governing factor in the test section
dimensions. The frame and the side plates were sealed from the environ
ment by linear "Orings". The side plates had sections cut out to accom
odate optical windows and the static pressure plate.
Since having the two streams an infinitesimally small distance
apart when they came into contact with each other was physically impos
sible, the splitter plate was machined down to 0.015 inches at the tip.
A thinner plate would have caused strength problems since the splitter
plate had to support the force due to the pressure difference between
the streams.
To support the protruding portion of the splitter plate, a 1.25
inch w~ide by 0.25 inch deep groove was machined in the side plates. W~ith
the splitter plate in place, epoxy resin w~as poured into the groove and
allowed to harden. Since the splitter plate was coated with silicone
grease before this operation, it was easily removed when the resin had
hardened. The excess resin was then sanded down smooth with thle side
plate surface. This method permitted the support of the "oddshaped"
portion of the splitter plate. The snug fit also served as a seal
between the primary and secondary streams.
Nozzle Blocks
The contour of the nozzle blocks guiding the subsonic flow was
an arbitrary shape which permitted smooch transition to sonic condi
tions at the minimum area. The contour providing the supersonic flow
was determined from the twodimensional method of characteristics.
To get the shortest possible test section, a sharpedged throat with
a single wave reflection design was used. No allowance for the boundary
layer was made in the nozzle design; however, the splitter plate had a
taper of 0.007 inches per inch at the straight section. This made up for
some of the boundary layer accumulation which is small in accelerated
flows.
The nozzle blocks were cut out of 0.5 inch aluminum plates,
and machined to the desired contour. The final polishing was done by
hand using fine grained emery cloth. Cushioned tape instead of 0rings
was placed between the side plates and the nozzle blocks to seal the
system.
The primary stream nozzles had a throat halfheight of 2 inches.
Thus, except for the reflected waves, the effect of the wall bounding
the primary stream could be neglected. The secondary stream nozzle had
a throat halfheight of 0.5 inches. For this case the wall effects
could not be ignored and limited the collection of data to the downstream
Iccation where the mixing region and the wall boundary layer interacted
with each other. It should be recognized that the secondary stream
dimensions were determined by the maximum feasible flow rate of the
stream.
It was observed during calibration runs that the primary and
secondary stream Mach 1.3 nozzles gave surprisingly clean (shockfree)
flows. The primary stream Mach 2.0 nozzle, however, did display some
wave patterns, but pressure measurements showed these waves to be weak.
The actual Mach numbers of the two streams were checked by
three methods: 1) ratio of the settling chamber total pressure to
probe total pressure, 2) ratio of the local static pressure to probe
total pressure, and 3) measuring the wave angles on the schlieren
photographs. With the air Mlach 2.0 and 1.3 nozzles, the result was an
average M~ach number of 1.97 and 1.28 respectively. The secondary stream
nozzle (designed for MI=1.3) yielded an average Mach number of 1.27.
A photograph of the air Mlach 2.0 nozzle block is presented in Figure 7.
Gas Supply and Control Svstem
The primary stream utilized air from the existing air storage
system of the gas dynamics laboratory. This system consists of a
TWorthington twostage compressor feeding a series of tanks capable of
holding approximately 420 cubic feet of air at 300 psi. The air is
passed through several oil and water traps and a regenerative type gas
dryer before it goes into the storage tanks.
The secondary stream utilized argon (or helium) from a separate
tank capable of holding approximately 50 cubic feet of gas at 150 psi
pressure. This tank itself was supplied argon (or helium) from a series of
commercial bottled gas manifolded together. A regulator was necessary
to reduce the commercial pressure from a maximum of approximately 2500
psi to the desired 150 psi.
The pressures in the two settling chambers were controlled by
Fisher pressure regulators. These regulators were activated by remote
control with a single switch, and if necessary, the sequence of opera
tion could be staggered with the bleed valve on each regulator.
The Schlieren System
A schlieren system was available to observe the behavior of the
flow field. A xenon lamp was used as the monochromatic light source
which converged on a 16 inch diameter parabolic mirror through a
condensing lens and a knife edge. The parallel beam of light reflected
from the Irirror passed through the test section and was reflected off of
another mirror and focused on a knife edge. The image was projected on a
ground glass plate and by the use of a Graflex camera polaroid pictures
of this image were taken. Figures 8 and 9 show two typical results
obtained. The shock free nature of the flow in the case where both
streams are at M~ach 1.3 should be noted.
Probe Drive Mlechanism
Due to the small dimensions of the test section, it was necessary
to have the total pressure probe very close to the primary stream wall
during the start of the run. After steady flow conditions were es
tablished in the test section, the probe had to be passed through the
free stream quickly in order to conserve run time. It then had to be
slowed down in the neighborhood of the mixing region and traverse the
mixing region at a relatively slow rate to obtain an accurate pressure
profile. This was accomplished by the use of a variable speed reversible
electric motor to drive the probe.
Eight holes were drilled at various intervals on top of the
test section frame and the nozzle block to accommodate the probe shaft.
The probe shaft was attached to a threaded rod which in turn was rotated
by the electric motor. This mechanism was mounted on top of the test
section and controlled from the instrument table. Provisions were made
to accommodate a linear potentiometer and a pressure transducer on the
mechanism frame. Limit switches were installed to automatically stop the
probe drive once it reached either the upper or lower wall. The probe
holes not in use were closed with a threaded brass plug which was
screwed in until the tip was flush with the nozzle block surface.
Photographs of the probe drive mechanism and the mounting on the test
section are shown in Figures 10 and 11.
Static Pressure Plate
It was undesirable to obtain static pressure measurements with
probes in such a small test section since the presence of the probe
would influence the results. This could have been accomplished by
installing pressure taps on the side walls if it were not for the schlieren
window~s. An alternative was to machine an aluminum plate, the exact
size of the windows, which would be interchangeable with one of the
windows. The pressure taps could then be drilled in this plate. This
method was chosen since it allowed the determination of static pressure
with the least amount of external disturbance in the flow field.
A total of ninety holes were drilled on the flow field side with
each hole having a diameter of 0.030 inches. These holes were distributed
among 13 axial stations ranging from 0.25 inches downstream of the nozzle
exit plane to approximately 15 inches downstream. On the outside of the
plate, these holes were enlarged to a diameter of 0.080 inches so that
short pieces of stainless steel tubing could be pressed in. These tubes
were sealed at the bases with epoxy resin. Vinyl tubing was used to
connect the stainless steel tubing in the taps to the monometer board. A
photograph of the static pressure plate is shown in Figure 12.
Gas Analysis System
Since one of the main objectives of the experimental program
was to obtain species concentration profiles, gas samples were withdrawn
from the flow field and collected in a series of vacuum bottles. These
samples were subsequently analyzed on a Victoreen M~odel 4000 Gas chroma
tograph. A photograph of the gas chromatograph together with the columns
used is shown in Figure 13.
The fundamentals of gas chromatography can be found in reference
[75]. The chromatograph measures the volumetric concentration of each
constituent of the sample. The components are separated when passed
through a column consisting of a length of stainless steel or copper
tubing packed with a solid phase such as charcoal. Since each component
progresses through the column at different rates, the travel time
(or elution time) identifies each component qualitatively. Thermal
conductivity detectors measure the quantity of each of the separated
gases relative to the carrier gas and concentrations are printed out
on a strip chart recorder.
Measuring Devices
Total Pressure M~easurements
Total pressure profiles in the mixing region were obtained by
introducing a probe into the stream. The pressure registered via the
probe was transformed into an electrical output using a MIB Electronics
Model 151BAA1 pressure transducer. Power was supplied to the trans
ducer by a CEC3140 DC power supply. The output was recorded on a
CEC5124A 20 channel recording oscillograph. This strip chart recorder
made traces on lightsensitive tape which was 6 inches wide. Thus, the
voltage output from the transducer had to be scaled down by means of
an external attenuation circuit so that a full scale deflection registered
180 psia pressure. With this calibration, probe pressures could be
read to within + 0.75 psia.
The probes had to be small enough so that the least amount of
disturbance would be introduced into the flow field and thus give
accurate pressure readings. Yet they had to be strong enough to withstand
the bending moments due to high speed flow. For this purpose stain
less stedl tubing of 0.060 inches OD (0.036 inch ID) was used. A
length of this tubing was bent at a 90 degree angle and welded into a
short piece of 0.25 inch diameter stainless tubing which in turn screwed
into the probe shaft on the drive mechanism. The length of the
"sting" wras determined by the location of the probe holes in the test
section frame relative to the static pressure taps since total pressure
profiles were needed at the point where static pressure data were taken.
It was found that three different "sting" lengths of 0.2, 0.5 and 0.75
inches were needed. After the desired lengths were cut, a 0.005 inch
thick shim stock was inserted into the end of the sting and compressed
to form a slit 0.005 by 0.040 inches. This design yielded satisfactory
results in terms of accuracy and strength. A photograph of the probes
is Shoeml in Figure 14.
The pressures of the two settling chambers were monitored both
visually on pressure gauges and also on the recording tape by the use
of two Giannini Hlodel 46139 pressure transducers.
Total~ '~~~ TemerturMasur'"emets
Total temperatures were monitored only in the two settling cham
bers using chromelalumel alloy themocouples, the output of whiich was
recorded on the recording tape. The reference junction was held at
32DF by immersing it in an ice bath. A full scale deflection of six
inches on the recording tape corresponded to temperature readings
between 3292 'F, which was the temperature range of interest.
Static Pressure Mleasurements
A thirtytube illuminated mercury manometer was used to measure
static pressures. A pressure differential of approximately 60 inches of
mercury could be measured on this manometer. By using atmospheric pressure
as reference and setting the zero at the midpoint, pressure mleasure
ments in the range of 030 psia were obtained.
Since there were ninety pressure taps available and only thirty
manometer tubes, a manifolding system had to be devised. This was accom
plished by connecting each manometer tube to a six inch length of brass
pipe at one end and welding shut the other. Three brass stopcock valves
were mounted in each piece of pipe and a pressure tap connected to each
valve. Thirty of these manifolds wrere mounted on the mianometer board.
Vinyl tubing was used to make the connections and each connection was
sealed with enamel paint (Glyptal). No leakage problems were encountered.
Thus, each manometer tube was capable of reading one of three pressure
taps depending on which valve was turned on.
A Graflex camera using 4 by 5 Polaroid plate film was used
to record the pressure measurements. From the photographs, the pressures
could be read to within 0.1 inches of mercury allowing pressure measure
ments to within + 0.05 psia.
Concentration M~easurements
1) Probe design
In order to reduce the number of runs necessary for the accurate
determination of concentration profiles, gas sample rakes made up of
three probes each were designed. The tips of the probes were made from
0.040 inch OD by 0.009 inch wall stainless steel tubing. The tips were
then immediately expanded to 0.040 inch TD to prevent the flow inside
the probe from choking. As was done in the case of the pressure probes,
the tubing was bent at a 90 degree angle and mounted inside a 0.25
inch diameter by 1.0 inch long stainless steel tubing which in turn
screwed into the. probe shaft on the drive mechanism. The nominal
distance between the probe tips was 0.1 inches.1 Each of the three probes
was connected to a 2.5 cubic inch volume evacuated sample bottles by
means of vinyl tubing. A photograph of the gas sampling rakes and a
sample collection bottle is shown in Figure 15.
2) Chromatograms
The column used in the gas chromatograph was a 6foot length
of Varian 5A molecular sieve. The column was conditioned by drying
it in the chromatograph oven set at 750 OF for 24 hours. During the
conditioning time the carrier gas was allowed to flow through the column
at a rate of 40 ml/minute. A Varian Mlodel 0200112600 thermal con
ductivity cell using two pairs of 30 ohm tungstenrhenium filaments was
used to measure the amount of each constituent in the mixture. The
output from the conductivity cell was recorded by a Honeywell Model
Electronik194 strip chart recorder.
In the Series I tests where argonair mixtures were being analyzed,
helium was used as the carrier gas; argon was used as the carrier gas to
analyze heliumair mixtures. For both series of tests, the injection
ports, the column and the thermal conductivity cell were maintained at 86 OF
1The spacing between. the sample probes was determined from the
recommendations of reference [17). With the above design no inter
ference problems were encountered.
The Series II tests of heliumair mixtures presented no problems
since the molecular sieve column separated oxygen, nitrogen and helium into
distinct bands and the mass fractions could be calculated from peak
areas. However, argon cannot be separated from oxygen when column
temperatures are above approximately 95 OF [76, 77]. If an acetone
dry ice bath was used to attain this temperature then nitrogen would not
be eluted and thus be irreversibly adsorbed in the column. Another
alternative was to separate nitrogen from the oxygenargon mixture at
room temperature, then immerse the column in the dry iceacetone bath
and inject the sample again. For a single chromatogram of a mixture, the
turnaround time using this technique was estimated to be over 30 minutes.
It was necessary to have three or more chromatograms of the same mixture
to obtain a statistical average of the mass fractions of the constituents.
This, together with the large number of samples collected, made the above
method impractical.
Ant indirect method of calibration was devised to avoid this problem.
It was assumed that air behaved as a single species.1 As a measure on
the validity of this assumption, selfdiffusion coefficients obtained by
kinetic theory considerations [78] were examined. It was found that in
the temperature and pressure range of interest, the selfdiffusion
1Air is conventionally treated as a single component in evalu
ation of transport properties for low density systems (i.e., pressures
on the order of one atmosphere) [62]. Example calculations of binary
and ternary diffusion in air (i.e., considering it as a single specie
and as a mixture) may be found in reference [62]. As is usually expected,
the two methods are in good agreement.
coefficients were within 5 percent of each other. Thus, for diffusion
purposes, the nitrogen and oxygen molecules were practically indistingu
ishable. Then, if the mass fraction of nitrogen in the mixture was known,
the mass fraction of oxygen could be computed since for every 0.79
moles of nitrogen there are 0.21 moles of oxygen. Knowing the mass
fractions of two of the three constituents, the third could be deduced.
Exact amounts of pure argon, oxygen and nitrogen were injected
into the column. The values of the peak areas, calculated by the method
of triangulation, were then plotted against the known weight injected.
As expected the curves were linear and passed through the origin (no
sample, no response) .
When an exact amount of unknown sample was injected into the
column at room temperature, tw~o peaks would appear on the chromatogram;
one of pure nitrogen and one of the argonoxygen mixture. Using the
calibration curves the weight of nitrogen could be determined from its
peak area. Then a simple ratio would yield the weight of oxygen present
in the mixture. The calibration curves would again be used, this time
somewhat in reverse, to obtain the peak area corresponding to this weight.
This area would then be subtracted from the "compound" argonoxygen peak
area to get the argon peak area and therefore its weight. Knowing the
weights of each of the constituents of the mixture, the mass fractions
could be calculated.
It was realized from the start that errors would be magnified
when the concentration of argon in the mixture fell below approximately
10 percent, i.e., samples collected from crossstream locations close
to the primary stream. The accuracy of the overall data, discussed in
the next section, showed this error to be at an acceptable level.
To keep all other sources of error at a minimum, the following
precautions were taken:
i. calibrations were frequently checked.
ii. injection port, column and detector cell temperatures
together with the carrier gas flow rate were maintained
at the same levels as used in calibration runs,
iii. fast recorder chart speed was used to make peak width
measurements more accurate.
iv. signal attenuations were adjusted to obtain full scale peaks
so that peak height measurements were accurate.
v. the mximum allowable filament current was used in the detector
cell to increase the overall accuracy of the chromatograms.
vi. number of chromatograms per mixture were increased to get
better statistical values.
Since all of the measuremoents, namely static and total pressure,
concentration and schlieren photographs, could not be made during the
course of one run, it was necessary to make different sets of runs. The
data collection technique, repeated for each set of nozzle blocks, is
summarized below:
1) The settling chamber total pressures necessary to match
static pressures in the exit plane of the nozzles were calculated.
Using air in both streams the regulators were adjusted to yield these
pressures. The secondary stream air line was then shut off and the static
pressure plate was installed in the test section, With the secondary
stream now utilizing argon (or helium), further adjustments were made
in the chamber total pressures by observing the exit plane static pressures.
The static pressure plate was removed and the schlieren window was in
stalled, after which the flow field was observed for "cleanliness",
i.e., the presence of unwanted expansion or compression waves. The
repeatability of chamber pressures from run to run was within 3.5
percent for the primary and 1.5 percent for the secondary. Better
repeatability (i.e., 1 percent) was obtained with the primary stream
when M~ach 1.3 nozzle block was used.
2) After a satisfactory flow field was established, a schlieren
photograph was made. The static pressure plate was replaced and three
runs made to determine the static pressure distribution. In between the
runs the appropriate manifold valves were turned on,
3) The total pressure runs were made with the windows back in
place so that the shock pattern due to the presence of the probe in the
flow field could be viewed on the schlieren screen. The number of
runs at each axial location depended on repeatability and the quality of
the traces. As the pressure probe was traversing the flow field, the
exact location of the probe tip had to be known. To accomplish this a
linear potentiometer was connected to the travelling probe mechanism.
Before each run, position calibration was done by getting a trace on
the recorder tape while the probe tip was located at the lower wall.
Thus, knowing a reference position and measuring the displacement of the
potentiometer output trace obtained during a run, the position of the
probe and the corresponding pressure at that point could be obtained.
At the beginning of each run the probe was located next to the primary
stream wall. The "blowdown" was started and allowed to reach a steady
state after which the probe mechanism was activated. The probe was allowed
to "sweep" the entire flow field, but as it approached the lower wall,
the direction was reversed and the speed reduced so as to obtain a
"finetrace" through the mixing region.
4r) A vacuum pump was connected to a manifoldwith three outlets;
one outlet wras connected to a 30 inch vertical mercury mlanometer, the
second to the injection port of the gas chromatograph and the third to
the sample bottle. Before each sample collection run, the sample bottles
were evacuated, after which the system was purged with carrier gas so as
to minimize the concentration of any possible residual sample from the
previous run. Position calibration was done in the same manner as was
done in total pressure measurements, by obtaining a trace with the
outermost probe on the lower wall. The probe rake was then moved to the
location where sample collection was desired by activating the drive
mechanism. This location was determined from schlieren photographs;
that is, most of the samples were collected from inside the mixing
region with only a couple of measurements in the free streams. The
"blowdown" was started and the flow of sample was established through the
probes. After steady flow conditions prevailed, the probes were connected
to the sample bottles and the bottle valves were opened. It was observed
that approximately 20 seconds of run time was required to obtain an
adequatee" quantity of sample. After each run the samples were normalized
"adequate" quantity was determined by trial and error to be a
sample at approximately 1/3 to 1/2 atmosphere pressure. This yielded a
sample of high enough concentration after normalization to one atmosphere
pressure for the required number of chromatograms (i.e., 36).
to one atmosphere pressure by the addition of carrier gas into the bottles.
Then 1 ml. of each sample was injected into the gas chromatograph by
means of a microvolume gas sampling valve which is an integral part of the
gas chromiatograph. A slight vacuum had to be applied to the exhaust port
of the sampling valve to "suck" in the sample which was at one atmosphere
pressure.
Data Reduction
Two methods of reducing the data were considered. The first
method was to assume uniform static pressure equal to an average test
section pressure throughout the flow field. The second method was to use
the average static pressure at each downstream station after making sure
that thie crossstream variation was less than + 5 percent. Either
method could be used to reduce the data of Series IB and IIB tests
with practically the same results since the streamwise static pressure
variation is very small (Figures 17 and 19). On the other hand, as may
be seen in Figures 16 and 18, some static pressure variations were
observed in the Series IA and IIA tests. Both methods were used to
reduce the data of Series IA and IIA tests. It was observed that the
two methods yielded reduced velocity data within 23 percent of each
other, provided the streamwise static pressure variations were within
68 percent of the exit plane value. Hence, the results of the second
method of data reduction are presented because retention of a true
representation of conditions in the mixing region is desirable. It is
noted, however, that an average test section static pressure was used in
the mixing analysis of Chapter III.
Since the two streams had different specific heat ratios, con
centration values were needed in conjunction with the pressure measure
ments. The total pressure data obtained wefrein the form of a smooth
trace from the recorder, whereas concentration measurements were points
spread throughout the mixing region. Total pressure and concentration
measurements corresponding to a specific crossstream location were
obtained by plotting the total pressure and concentration profiles on
the same graph, then drawing a smooth curve through the concentration
values; thus, the speciesconcentration and the total pressure corres
ponding to a specific crossstream location were available.
The following equations were used to obtain the average
specific beat ratios of the mixtures:
Cp= a YCpi (5.1)
whr = average Cp of the mixture (btu/1bRi)
ui = mass fraction of species (lb i/1b mixture)
C = Cp of species (Btu/1bR)
= 1/ (5.2)
where W = average molecular weight of mixture
Wi = molecular weight of species
where R = gas constant of mixture (Btu/lbR)
a = universal gas constant (Btu/lb moleR)
c; 54
C R
where k = average specific heat ratio of mixture.
Velocities were computed by first determining the Mach numbers
through the Rayleigh pitot formula [79]:
4= ( M2kk1 / (5.5)
where Py = the measured probe pressure at (x,y)
Px~ = the measured average static pressure at (x)
k = the average specific heat ratio a (x,y)
M = the local. Mach number.
Since local values of static and total pressures in conjunction with
mass fractions are used, the utilization of Equation (5.5) for mixtures
is justified. In other words, no attempt is being made to relate any
of the quantities along streamlines to the undisturbed portion of the
flow.
Although the distribution of total temperature across the
mixing zone is nonuniform even for streams of equal total temperatures,
this nonuniformity is small if the Prandt1 number is close to unity.
For example, in airair mixing, the variation of the total temperature
is only about 0.1 percent for a Mach number of unity [1]. In his
investigation of the supersonic mixing of hydrogen and air, Morgenthaler
[80] observed that typical experimental profiles at Mach 2.0 indicated
a 3 percent variation in total temperature.
On this basis the total temperature was assumed to be constant
through the mixing region and the average of the total temperatures of
the two streams was used for To. The total temperatures of the two
streams were never more than 100 apart with the average value being
approximately 535 R. Thus, static temperature profiles were determined
through the relation
T = 7/(1 + M2) (5.6)
For a mixture of n species, the local mass average velocity U
is defined as [62]
~ iUi n
U i= aiU (5.7)
i=1
It is noted that U is the velocity one would measure by means of a
pitat tube (i.e., incompressible flow) and corresponds to the velocity
as used for pure fluids. The local velocity of each specieswas
calculated using the adiabatic flow equation
Ui = z~f 2C( T5. 8)
Finally, densities were computed from the perfect gas law
P = (5.9)
RT
The assumption of argon, helium, oxygen and nitrogen being perfect
gases was valid since the pressures were much less than the critical
pressure and the temperatures much greater than the critical
temperature for all species during all test conditions.
It was desired to relate the reduced velocity and temperature
data to a set of average initial conditions for each test configuration
(i.e., an average primary stream velocity and temperature together
with an average secondary stream velocity and temperature). This was
necessary because the theoretical analysis in Chapter III required
initial profiles of velocity and temperature and with these profiles
the mixing program "marched" downstream. Since the experimental
system was not perfectly repeatable, each set of initial conditions
varied somewhat from run to run, In addition to the above, in some
of the experimental configurations as discussed earlier, there were
slight variations in the static pressure which affected the velocity
profiles. Eggers and Torrence [81], in their experimental investigation
of compressible air jets encountered similar problems. They compensated
for the above stated variations by suggesting a velocity modification
of the type
VU V U
~=4~i_(5.10)
p s px sx
where V = the new local velocity modified for pressure and free
stream deviations
U_ = the average primary stream velocity at the nozzle exit
plane
Us = the average secondary stream velocity at the nozzle exit
plane
V_ = the local velocity modified for static pressure changes
Upx = the primary stream velocity modified for static pressure
changes
Us = the secondary stream velocity modified for static pressure
changes
The above modification was adopted for the present analysis together
with a temperature modification of the same type
TT T'T
T T T T(5.11)
p s px sx
where T = the new local temperature (modified)
T_ = the average primary stream temperature at the nozzle
exit plane
Ts = the average secondary stream temperature at the nozzle
exit plane
T' P the local temperature
Tx = the local primary stream temperature
Ts = the local secondary stream temperature,
It should be recognized that these modifications were adopted
for the sake of consistency between the experimental data and the
theoretically predicted profiles. Since both the experimental and
theoretical results are presented in the form of excess velocity
profiles (discussed in the next chapter) there are no consequences due
to these modifications.
Accuracy of Results
Based on the chart and photograph resolutions of recorded data,
repeatability of runs and calibration, the estimated accuracy of the
measurements are:
Static pressures ........... 0.1 psia
Probe pressures ................ ..... f 0.75 psia
Total temperatures .................. f 100
Probe position ...................... + 0.025 inches.
The test section static pressures are on the order of 10 psia;
the probe pressures range from 3570 psia; the temperature range of
interest is on the order of 535 R and measurements are made in a region
of approximately 2 inches. Thus, in terms of percentage errors:
Static pressures .................. .. f 1%
Probe pressures ................... .. f 2.5%
Totl tmpeatues.................. f 2%
Probe position ................. .... f 1.5%
k'ith the above values and the equations used to reduce the
data, it is estimated that the velocity data are accurate to within
3 percent. This estimation does not include any uncertainty due to
concentration measurements.
One means of assessing the overall accuracy of the data is
to apply the principle of conservation of mass to the secondary
stream. This would also indicate the degree of accuracy of the
concentration measurements. The following equation was evaluated
numerically for each axial station at which data were taken:
me = ipudA (5.12)
where pu = local mass flow per unit area evaluated from experimental
data
ai = local mass fraction of the secondary stream constituent
A = area of the flow field over which ai is nonzero.
If the data were correct, me would be equal to the secondary
stream flow rate which can be approximated from settling chamber
conditions (for uniform, onedimensional, isentropic flow)
=const. (5.13)
t o
where the constant depends on whether argon or helium is being used.
Differences between m and me are due to experimental error. A
comparison between these two mass flowu quantities is an essential
criterion in assessing the accuracy of concentration measurements
because of uncertainties in obtaining representative samples from
flowing streams.
The application of this criterion to the data presented herein
is reported in Table III. Large errors in concentration measurements
taken from binary streams may occur; the probe design, sampling technique
and the local turbulence level in the flow field have a significant
effect upon the results [35,81]. The actual physical mechanism which
causes unrepresentative sample collection is not known, but satisfactory
results were obtained with the probe and sampling technique used in
this investigation. As can be seen in Table III, the overall error
is less than 10 percent for all cases, and 75 percent of the cases
have an error of 6 percent or less. Similar sampling problems were
found in references [35,81,82,83,84,85] where errors of up to f 25
percent were encountered. Therefore, it is concluded that the accuracy
of the data is well within acceptable limits. Furthermore, it is
deemed that the error involved in the method of determining argon mass
fractions in argonair mixtures is negligible as compared to the
uncertainties of the samples themselves.
CHAPTER VI
DISCUSSION OF THEORETICAL ANiD EXPERIMENTAL RESULTS
Schlieren Photographs
Schlieren photographs were made to observe the quality of the
supersonic flow for each of the four test configurations. Figure 8
shows the flow field with primary stream of air at a Mach number of
approximately 2.0 and the secondary stream of argon at a MIach number
of about 1.3. Here, the wave patterns are distinct and although the
mixing zone is not too clearly visible, the waves may be observed
to bend as they pass through the mixing zone. The "leftrunning" wave
emanating from the left middle of the photograph is the typical
"lip shock" resulting from two supersonic streams coming into contact
with each other. Static and total pressure measurements confirmed
these waves as being weak and the local wave angles in the primary
stream indicated a Mach number on the order of 1.97.
There is considerable difference between the flow fields
depicted in Figures 8 and 9. In Figure 9 both streams are at a Mlach
number of 1.3 again with air in the primary and argon in the secondary
streams. The possibility of the flow being subsonic due to the
absence (or nonvisibility) of the "lip wave" and other waves was
discarded with total pressure measurements. When a total pressure
probe was injected into the flow field, weak oblique shocked waves
were also observed around the probe tip. Due to the cleanness of the
flow, the mixing zone is more distinct than in Figure 8.
Similar qualitative results were obtained when helium was
utilized in the secondary stream. The small width of the test section
was a factor in the quality of the schlieren photographs since the
quality of the image is a function of the width over which the
initially parallel light beams are diffracted.
Static Pressure Variations
The mixing analysis of this study involved the use of the
boundary layer form of the conservation equations, from which it was
deduced that the transverse pressure gradient (aP/ay) was negligible.
It was further assumed that the streamwise pressure gradient (SP/ax)
could also be neglected. The validity of these assumptions are now
analyzed in light of the experimental data obtained. The static
pressure measurements are plotted in Figures 16 through 27 with the
average static pressure at the exit plane of the nozzles used as a
reference pressure and with the physical coordinates nondimensionalized
with respect to the exit height of the secondary stream, i.e., slot
height.
For all four test conditions, the streamwise pressure distri
bution is plotted along three lateral locations; along the plane of
the splitter plate (y/a = 1.0), and onehalf slot height above and
below the plane of the splitter plate (y/a = 1.5 and y/a = 0.5). It may
be observed in Figures 16 through 19 that the static pressure increases
monotonically after a certain axial location. This is a typical
characteristic of confined (ducted) flows. The axial location at which
this steady increase is observed usually corresponded to approximately
the downstream location where the mixing zone interacted with the wall
boundary layer. When both streams are at Mach 1.3, Series IB and
IIB tests (Figures 17 and 19), the static pressure variation is within
23 percent of the average exit plane value for about 12 slot heights.
In Series IA and IIA tests, where the primary stream of air is at
Mach 2.0 and the secondary stream of argon or helium is at Mlach 1.3
(Figures 16 and 18), higher variations of up to 10 percent for regions
within 12 slot heights are observed. When the stream~wise static
pressure variations and the schlieren results are considered together,
it is concluded that, for the configurations involving the Mach 2.0
nozzle block, the variation of the static pressure can be attributed
to the weak waves present in the flow field.
A further check on the magnitude of the streamwise pressure
gradient was also made by comparing it to one of the convective terms
(i.e. pu ) in the momentum equation.1 The results are presented
below in terms of the parameter 6, where 6 is defined as
*
y y
a =x pu'aldy/ Pdy
x/a a
1.0 55
3.0 42
5.0 39
7.0 36
9.0 32
The above values are from the data of Series IIA tests. Similar
results were obtained with the Series IA test data.
Therefore, if only the region upstream of the point of
Th~lese values were obtained by getting intermediate printed out
put from the computer program.
interaction of the mixing region and the confining walls is considered,
the assumption of constant axial pressure distribution is well justified
for one configuration and at least acceptable for the other.
The transverse pressure variations for the Series I and II
test at various axial locations are presented in Figures 20 through
27, For Series IB tests involving both streams at the same Mlach
number of 1.3, the variation is within 5 percent for all stations up
to x/a = 12. The same is essentially true for Series IA tests except
that at about x/a = 12 (Figure 23) the variation increases up to about
8 percent. This again is attributed to the waves present in the flow
field when the Mach 2.0 nozzle block is utilized. The same trend may
be observed in the helium tests (Series II). With a Mach 1.3 primary
stream, static pressure variations are on the order of + 4 percent.
With a Mach 2.0 primary stream, pressure variations of up to 9 percent
at about x/a = 5.5 (Figure 26) may be seen.
Even though there is some variation of the static pressure in
the lateral direction, this variation is due to the presence of weak
waves in the supersonic flow field, and the assumption of negligible
transverse pressure gradient seems to be justified.
Growth of the Mlixinn Region
The growth of the shear layers for each of the four test
conditions is presented in Figure 28. There is always some uncertainty
in locating the edges of the mixing zone. The range of uncertainty
for each case is shown on the curves in Figure 28.
1The range of uncertainty is the maximum lateral distance in
which the velocities change from 5 percent of their free stream values
to the free stream values.
The largest growth is observed in the Series IIB tests where
helium with a velocity of about 3500 ft/sec is mixing with air flowing
at 1300 ft/sec. The least growth rate is observed in the case of
argon (U = 1150 ft/sec) mixing with air (U = 1350 ft/sec), i.e.,
Series IB tests. Thus, with the two other configurations showing
the samre trend of increased mixing zone growth with an increase in the
velocity difference, it is concluded that as the velocity difference
between the two streams increases so does the growth rate of the mixing
region. This is consistent with the wellestablished fact [1] that the
growth rate is a maximum when one stream exhausts into a quiescent
medium.
Although curve (a) in Figure 28 corresponds to the test
condition with the largest mass flux difference, curve (d) does not
correspond to the case of the smallest mass flux difference, Therefore,
the same reasoning that holds for velocity differences does not hold
for mass flux differences.
The curves have been started at approximately 2 slot heights
downstream of the exit plane due to thle lack and uncertainty of data
at previous locations. It is also noted that if the curves are
extrapolated to determine the intercept, none of the curves pass
through the origin. This is attributed to twoe possible reasons; the
first is the fact that the growth of the mixing zone might be non
linear in this region. The second and more probable reason might be
that this initial thickness is due to the accumulation of boundary
layers on both sides of the splitter plate. It is the opinion of the
author that thle above phenomencnis due to a combination of the two
possibilities rather than due to solely one. Of course, experimental
error could also have a significant effect.
Correlation of VTelocity Data
A step function velocity profile as the initial profile input
to the mixing program could not be expected to satisfactorily predict
the mixing in the near field since the profiles are expected to be
nonsimilar and there is the possibility of a "wakelike" profile
stemming from the boundary layer accumulation on the splitter plate.
Thus, it was necessary to input measured profiles rather than assumed
or calculated profiles; and this was accomplished by starting the
mixing program with experimental profiles at x/a = 0.57. At this
axial location, no "wakelike" profiles were observed in any of the test
conditions. As described in the Appendix, the input consisted of velocity,
static temperature and mass fraction profiles in equal intervals of kS.
The constants in Equations (4.10), (4.13), (4.14) and (4.15)
were varied until the theoretical and experimental mixing zone growth
rates matched. However, difficulties were encountered in Prandtl's
mixing length model as it predicted a highly nonlinear growth rate for
the mixing zone in the region of interest, i.e., 2 < x/a < 10. No
evident reason can be given for this behavior. It is speculated
that the dynamic eddy viscosity (pe) is a very weak function of the
transverse coordinate. References [35] and [57} tend to support this
speculation. The Prandtl model is a strong function of the transverse
coordinate since it involves the gradient of the longitudinal velocity.
Hence, the behavior may be related somewhat to the above speculation.
The mixing length model was thus eliminated from further analysis.
The remaining three momentum transport models (Schetz, Ferri
and Alpinieri) were correlated, as stated earlier, in the "very"
near field, 2 1 x/a < 10. The reason for using only a portion of the
data was to see how well the models could predict available mixing
data in the region x/a > 10.
Four different empirical constants were determined for each of
the three models, i.e., one for each test condition. Since the eddy
viscosity is semiempirical in nature, it is too much to expect a single
correlation to be valid for all conditions encountered. These
constants were then examlined to see if any trend could be observed.
In other words, it was desired to relate the coefficients to initial
mixing conditions. The only flow property that depicted any trend in
values of the empirical constants was found to be the ratio of the mass
flux per unit area of the secondary stream to the primary stream.
Figure 29 shows the values of the coefficients for the turbulent eddy
viscosity models as a function of the mass flux ratios of the two
streams. The "asymptotic" characteristic of coefficients for the Ferri
and Schetz models as the mass flux ratio approaches unity may be
related to the fact that the two models fail when the mass fluxes of
the two streams are equal. N'o reason can be given for the same trend
shown by the Alpinieri model. It is also recognized that more test
data areneeded in the region of unity mass flux ratio (both less
than and greater than) as well as large values of mass flux ratio to
validate the proposed relation between the empirical coefficients
and the mass flux ratios.
The velocity profiles predicted by each model as well as the
experimental velocity profiles for each of the four test conditions
are presented in Figures 30 through 49. The profiles are presented
in the form of the dimensionless excess velocity
UU
U = U (6.1)
p s
and the dimensionless transverse coordinate y/a at five axial stations.
Only sample experimental points are plotted showing the trend of the
data to avoid a cluster of points in obscuring the plots.
With all three of the eddy viscosity models predicting nearly
the same growth rate for the mixing zone, the Schetz model is
observed to be superior in predicting the velocity profile compared to
the Ferri and Alpinieri models as may be seen in Figures 32, 39, 43,
47 and 49. The Ferri model tends to under predict the velocity profiles
in the region x/a y 8, i.e., past the region of correlation
(Figures 34, 38, 44 and 49). The Alpinieri model falls in between the
predictions of the Ferri and Schetz models. The better correlation
obtained with the Schetz model may be explained by the fact that this
model takes into account the velocity and density profiles in the shear
layer. By integrating these profiles at each axial station an
"average" value for the eddy viscosity is obtained. On the other hand,
the Ferri and Alpinieri models predict a turbulent eddy viscosity
by means of free stream properties and a representative width for the
mixing zone.
Although the Alpinferi and Ferri models correlate the data
presented somewhat satisfactorily, certain limitations are inherent
in their formulations. For example, in both cases, the mixing zone
width b is based on velocity difference; thus as the velocity
difference between two streams becomesvery small, the mixing zone
width becomes undefined. As discussed earlier, the Ferri model also
fails when the mass flux difference between the two streams approaches
zero. Although the Schetz model also fails when the mass flux gradients
in the flow field disappear, the former problem is circumvented since
this model is not based on a mixing zone width.
Thus, it is concluded that on the basis of formulation
characteristics and the satisfactory correlation of the experimental data,
the Schetz extension of the Clauser integral model for the turbulent
momentum transport mechanism is superior to Prandtl's mixing length,
Ferri's differential mass flux and Alpinieri's momentum flux models
in terms of predicting the velocity profiles in the initial region
of a confined halfjet. It should be added, however, that the Ferri
and Alpinieri models may also be utilized in the initial region with
fairly good results. Although the empirical constant in all three
of the models requires adjusting for different flow configurations,
it is anticipated that the application of a given eddy viscosity model
to a sufficient quantity of data will produce a relationship in terms
of the mass flux ratios of the two streams, which will enable calculation
of the constant for initial mixing conditions.
Correlation of Mass Fraction Data
As may be recalled from the discussion of Section 4.5
involving the turbulent transport of mass, the approach taken in this
study is to formulate and correlate a turbulent momentum transport
mechanism and then to use a suitable value of the turbulent Schmidt
number to determine the speciesmass fraction profiles. Thus, the
