Group Title: investigation of the turbulent mixing of parallel two-dimensional compressible dissimilar gas streams
Title: An Investigation of the turbulent mixing of parallel two-dimensional compressible dissimilar gas streams
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Title: An Investigation of the turbulent mixing of parallel two-dimensional compressible dissimilar gas streams
Physical Description: xv, 200 leaves. : illus. ; 28 cm.
Language: English
Creator: Tanrikut, Seyfeddin, 1947-
Publication Date: 1973
Copyright Date: 1973
 Subjects
Subject: Jets -- Fluid dynamics   ( lcsh )
Turbulence   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 193-199.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00097589
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000580529
oclc - 14046242
notis - ADA8634

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AN INVESTIGATION OF THE TURBULENT MIXINSe
OF PARALLEL TWO-DIMENUSIONAL. 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. Glaith-er, 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

comitt-ee. 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. Pa-tterson 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 Semi-Infinite 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 Semi-Infinite 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 Close-Up of Test Section with Two M~ach 2.0 N'ozzle
Blocks....................................... 104

7 Close-Up 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 Close-Up of the Probe Drive Miechanism with the
Control System................................... 106

12 Close-Up of Static Pressure Plate................. 107

13 Gas Chromatograph with a M~olecular Siev-e 5-A
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
I-A Tests...................................... 109

17 Streamwise Static Pressure Variation for Series
I-B Tests... ................. .............. ....... 110









LIST OF FIGURES (Continued)


F~ipure ase

18 Streamw~ise Static Pressure Variation for Series
II-A Tests...................................... 111

19 Streamwise Static Pressure Variation for Series
II-B 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 I-A (Argon-Air) Tests.....,. 123

31 Dimensionless Excess Velocity Profile at
x/a = 4.06 for Series I-A (Argon-Air) Tests....... 124

32 Dimensionless Excess Velocity Profile at
x/a = 5.56 for Series I-A (Argon-Air) Tests....... 125









LIST OF FIGURES (Continued)

EM rage
33 Dimensionless Excess Velocity Profile at
x/a = 12.17 for Series i-A (Argon-Air) Tests...... 126

34 Dimensionless Excess Velocity Profile at
x/a = 17.7 for Series I-A (Argan-Air) Tests....... 127

35 Dimensionless Excess Velocity Profile at
x/a = 4.06 for Series I-B (Argon-Air) Tests....... 128

36 Dimensionless Excess Velocity Profile at
x/a = 5.56 for Series I-B (Airgon-Air) Tests....... 129

37 Dimensionless Excess Velocity Profile at
x/a = 12.17 for Series I-B (Argon-Air) Tests...... 130

38 Dimensionless Excess Velocity Profile at
x/a = 17.7 for Series I-B (Argon-Air) Tests..,..,. 131

39 Dimensionless Excess Velocity Profile at
x/a = 23.26 for Series I-B (Argon-Air) Tests...... 132

40 Dimensionless Excess Velocity Profile at
x/a = 2.54 for Series II-A (Helium.-Air) Tests..... 133

41 Dimensionless Excess Velocity Profile at
x/a = 4.06 for Series II-A (Helium-Air) Tests..... 134

42 Dimensionless Excess Velocity Profile at
x/a = 5.56 for Series II-A (Helium-Air) Tests..... 135

43 Dimensionless Excess Velocity Profile at
x/a = 8.44 for Series II-A (Helium~-Air) Tests..... 136

44 Dimensionless Excess Velocity Profile at
x/a = 12.17 for Series II-A (Helium-Aiir) Tests.... 137

45 Dimensionless Excess Velocity Profile at
x/a = 2.54 for Series II-B (Helium-Air) Tests..... 138

46 Dimensionless Excess Velocity Profile at
x/a = 4.06 for Series II-B (Helium-Air) Tests..... 139

47 Dimensionless Excess Velocity Profile at
x/a = 5.56 for Series II-B (Helium-Air) Tests..... 140

48 Dimensionless Excess Velocity Profile at
x/a = 8.44 for Series II-B (Helium;-Air) Tests..... 141









LIST OF FIGURES (Continued)

_Figur r

49 Dimensionless Excess Velocity Profile at
x/a = 12.17 for Series II-E: (Helium-Air) Tests.... 142

50 Argon Miass Fraction Profiler at x/a = 2.54 for
Series I-A Tests................................ .. 143

51 Argon Mass Fraction Profile: at x/a = 4.06 for
Series I-A Tests..................... ........... 141

52 Argon Mlass Fraciton Profile at x/a = 5.56 for
Series T-A Tests..................... ..... ... ... 145

53 Argon MIass Fraction Profile at x/a = 12.17 for
Series I-A Tests.................................. 146

54 Alrgon Mass Fraction Profile at x/a = 17.7 for
Series I-A Tests.......... ........ 147

55 Argon Miass Fraction Profile at x/a = 4.06 for
Series I-B Tests................... 148

56 Argon. Mass Fraction Profile at x/a = 5.56 for
Series I-B Tests................... 149

57 Argon Mass Fraction Profile at x/a = 12.17 for
SeisI-B Tests.............. ................... 15

58 Argon Mass Fraction Profile at x/a = 17.7 for
Series 1-B Tests................... 151

59 Argon Mass Fraction Profile at x/a = 23.26 for
Series I-B Tests.................... 152

60 Helium Mlass Fraction Profile at x/a = 2.54 for
Series II-A Tests................... .......... ... 153

61 Heliuma Mass Fraction Profile at x/a = 4.06 for
Series II-A Tests........ .................15

62 Helium Hass Fraction Profile at x/a = 5.56 for
SeisII-A Tests.... .............. ............... 15

63 Hfl~ium Mass Fraction Profile at x/a = 8.44 for
Series II-A Tests... ............. .............. 15

64 Helium Malss Fraction Profile at x/a = 12.17 for
Series II-A Tests..................... ............17











LIST OF FIGURES (Continued)



65 Helium 11ass Fraction Profile at x/a = 2.54 for
Series II-B Tests................................. 158

66 Helium M!ass Fraction Profile at x/a = 4.06 for
Series II-B Tests................................. 159

67 Helium Mass Fraction Profile at x/a = 5.56 for
Series II-B Tests................ 160

68 Helium :ass Fraction Profile at x/a = 8.44 for
Series II-B Tests................................. 161

69 Helium Mfass Fraction Profile at x/a = 12.17 for
Series II-B Tests................................. 162

70 Velocity Similarity Plot for Series II-B Tests.... 163

71 Mass Fraction Similarity Plot for Series II-B
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 TWiO-DIMIENSIONA. 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 half-jet composed of dissimilar gas streams has been made. An

iso-energetic, non-reacting 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 half-jet 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 all-explicit

for-ward 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 two-dimensional test

section was designed and built to be used in conjunction with the

existing blow-down wind tunnel facilities. Interchangeable. nozzle

blocks were designed with the two-dimensional 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 re-entry and high-speed 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 first-stage 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 two-dimensional or

axi-symmetric.

vii. Chemical reactions may take place in the mixing region

or the flow could be frozen.

viii. The mixing may be isoenergetic or non-isoenergetic.

ix. The mixing streams could be contained by walls or the

streams may be semi-infinite.

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 non-similar, and second the asymoptotic (or

fully developed) region where the velocity profiles are self-preserving.

Figure 1 shows a schematic of the mixing region resulting from the

"contact" of two uniform parallel semi-infinite 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 high-speed 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) step-by-step

numerical solution of the equations on high-speed 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 closed-form 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

axi-symmretric jets. Data on the two-dimensional mixing in the initial

region ar-epractically nonexistent.

Thus, an experimental investigation was done to provide data

for the two-dimensional 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 two-dimensional 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 Navier-Stokes equations to the boundary layer

form are well documented in references [1], [2) and 13], and will not

be reviewed here. The two-dimensional 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


X-M~orenrtum: pu + pv =-(2.2)


Y-Momentuta: 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 semi-infinite 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 ex-perimental aspect of the

present investigation.


Mixing of Seni-Infinite 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 semi-infinite 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 two-dimensional

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 presente-d 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 y-coordinate 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

time-dependent 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 M-ch 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 Navier-Stokes 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 6-8) 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 non-linear 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 recently-to 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 non-turbulent 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 jet-to-confining 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 jet-to-

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 twuo-dimensional 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 axi-symmetric 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 air-air ejectors and air-augmented rockets.

Emmons (51] also developed an analysis for predicting the flow

characteristics in the mixing region of a particle-laden 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 Sh-apiro [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 f-or hydrogen-, belium-, and argon-air 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

two-dimensional mixing problems are:

Global Continuity

+ + = 0 (3.1)


X-Miomentum

a(pu) + (pu)u e (py)u = SP aT (3.2)
at ax oy ax ay

Y-Miomentum


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 x-and y-momentum equations are small, the

species are inert chemically (i.e., non-reacting flow) and thermal

diffusion is negligible. From the first of these assumptions, the

only information available from the y-momentum 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 y-momentum 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., 5-8 percent of the

exit plane value) since the region of interest is the









initial mixing region (i.e., 5-10 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 time-mean 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 time-mean-averages 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 time-mean 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., quasi-steady. 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 time-rean properties are constant with respect to time.

Thus, a state of quasi-steady 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

X-Momenztum


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 right-hand 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 y-coordinate 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 all-expli~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 m-1 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,m-1/2 n,m



n,m-1/2Fn,m-1]+(.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 nonl-linear, 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,m-1/2)un,m


+ n,m-1/2un,m-1]


(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,m-1/( l~-] (3.42)

Energy


n+1,m = n,m Pr n,m+1/2Hn,m+1 (n,mc1/2+ n,ml-1/2)Hn,m


i n,m-1/2Hn,m-1] 2 [n,nd-1/2un,m+1

2 2
-(n,m+1/2+ n,m-1/2)un,m+ n,m-1/2un,m-1]


+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 quasi-linear 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 variable-coefficient 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

quasi-linear 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 quasi-linear 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,m-1/2)(1~ -cos kaiy) + lY((n,m+1/2

(n,m-1/2)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,m-1/2

,m+1/2 ,m-1/2 -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,m-1/2


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 well-established 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,m-1/2


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 axi--symmetric flows has prevented its utilization;

and for the case of two-dimensional 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 half-radius. This model has yielded predictions of

unreliable accuracy for axi-symmetric 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 axi-symmetric 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 center-line

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 axi-symmetric 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

half-radius. 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

half-jet which is the case under study. The similarity region of a

symmetric jet is that part of the flow field far doblestream of the

so-called potential care. The half-jet 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

half-jet 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 so-called "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 two-dimensional

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 b-1 n,mun,mun+ unm1(1)


In the Ferri and Alpinieri models, the eddy viscosity is

expressed in terms of the half-radius based on either the velocity or

the mass flux. The half-radius 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 half-jet.

Therefore, rather than a width such as the half-radius 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 two-dimensional 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 two-stage 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 on-off 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

two-dimensional 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 one-inch aluminum side plates which when bolted together would

give a test section width of one-half 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 "O-rings". 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 "odd-shaped"

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 two-dimensional method of characteristics.

To get the shortest possible test section, a sharp-edged 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 0-rings

was placed between the side plates and the nozzle blocks to seal the

system.

The primary stream nozzles had a throat half-height 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 half-height 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 (shock-free)

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 two-stage 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 151-BAA-1 pressure transducer. Power was supplied to the trans-

ducer by a CEC-3-140 DC power supply. The output was recorded on a

CEC-5124A 20 channel recording oscillograph. This strip chart recorder

made traces on light-sensitive 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 chromel-alumel 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 32-92 'F, which was the temperature range of interest.











Static Pressure Mleasurements

A thirty-tube 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 mid-point, pressure mleasure-

ments in the range of 0-30 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 stop-cock 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 6-foot length

of Varian 5-A 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 02-001126-00 thermal con-

ductivity cell using two pairs of 30 ohm tungsten-rhenium 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

Electronik-194 strip chart recorder.

In the Series I tests where argon-air mixtures were being analyzed,

helium was used as the carrier gas; argon was used as the carrier gas to

analyze helium-air 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 helium-air 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 oxygen-argon mixture at

room temperature, then immerse the column in the dry ice-acetone bath

and inject the sample again. For a single chromatogram of a mixture, the

turn-around 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, self-diffusion coefficients obtained by

kinetic theory considerations [78] were examined. It was found that in

the temperature and pressure range of interest, the self-diffusion






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 argon-oxygen 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" argon-oxygen 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 cross-stream 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 "blow-down" 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

"fine-trace" 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

"blow-down" 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., 3-6).











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 micro-volume 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 cross-stream variation was less than + 5 percent. Either

method could be used to reduce the data of Series I-B and II-B 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 I-A and II-A tests. Both methods were used to

reduce the data of Series I-A and II-A tests. It was observed that the

two methods yielded reduced velocity data within 2-3 percent of each

other, provided the streamwise static pressure variations were within

6-8 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 cross-stream 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 cross-stream 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/1b-Ri)

ui = mass fraction of species (lb i/1b mixture)

C = Cp of species (Btu/1b-R)

= 1/ (5.2)


where W = average molecular weight of mixture

Wi = molecular weight of species





where R = gas constant of mixture (Btu/lb-R)

a = universal gas constant (Btu/lb mole-R)

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 non-uniform even for streams of equal total temperatures,

this non-uniformity is small if the Prandt1 number is close to unity.

For example, in air-air 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
V-U 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










T-T 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 35-70 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, one-dimensional, 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 argon-air 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 "left-running" 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 non-visibility) 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 non-dimensionalized

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 one-half 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 I-B and

II-B tests (Figures 17 and 19), the static pressure variation is within









2-3 percent of the average exit plane value for about 12 slot heights.

In Series I-A and II-A 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 II-A tests. Similar

results were obtained with the Series I-A 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 I-B 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 I-A 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 II-B 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 I-B 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 well-established 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

non-similar 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 non-linear 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

U-U
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 half-jet. 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




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