NBSSR 88-3340

Effects of Global Density and

Reynolds Number Variations on

Mixing in Turbulent, Axisymmetnc

Jets

William M. Pitts

U.S. DEPARTMENT 0 C COMMERCE

National 3ureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

March 1 986

U.S. DEPARTMENT OF COMMERCE

' YTIONAl GUPEi»U OF S7ANOAWOS

QC

100

.U56

NO. 86-5540

1986

NBSIR 86-3340

EFFECTS OF GLOBAL DENSITY AND

REYNOLDS NUMBER VARIATIONS ON

MIXING IN TURBULENT, AXISYMMETRIC

JETS

William M. Pitts

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

March 1986

U.S. DEPARTMENT OF COMMERCE. Malcolm Baldrige. Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

LIST OF TABLES v

LIST OF FIGURES vi

Abstract 1

1. INTRODUCTION 2

2. REVIEW OF AXISYMMETRIC JETS 3

2.1 Constant Density Axisymmetric Turbulent Free Jets 4

2.2 Axisymmetric Turbulent Free Jets Having Global

Density Differences 12

2.3 Effects of Buoyancy and Coflowing Surrounding Gases

on Mixing in Axisymmetric Turbulent Jets 25

3. EXPERIMENTAL SYSTEMS AND DATA TREATMENT 34

3.1 Flow System 34

3.2 Experimental System for Shadowgraphy 36

3.3 System for Rayleigh Light Scattering Measurements of

Concentration 38

4. EXPERIMENTAL FINDINGS 41

4.1 Shadowgraphy 41

4.1.1 Helium Turbulent Jet into Air (R p = 0.14) 41

4.1.2 Methane Turbulent Jet into Air (Rp = 0.55) 43

4.1.3 Propane Turbulent Jet into Carbon Dioxide (R p = 1.02) ... 43

4.1.4 Propane Turbulent Jet into Air (R p = 1.55) 44

4.1.5 Sulfur Hexafluoride Turbulent Jet into Air (Rp = 5.11) .. 45

4.1.6 Sulfur Hexafluoride Turbulent Jet into

Helium (R p = 37.0) 47

4.1.7 Summary of Trends in Mixing Behavior with

Changes in Rp and Re Derived from Shadowgraphs 47

4.2 Concentration Measurements Along the Centerlines of

Variable Density Jets 49

4.2.1 R p Dependence of Centerline Average Concentration 51

4.2.2 Unmixedness Measurements Along the Centerlines of

Variable Density Jets 53

4.2.3 Density Effects on Skewness and Kurtosis Values Along

the Jet Centerline 55

4.2.4 Reynolds Number Effects on Centerline Concentration

Falloff 56

4.2.5 Reynolds Number Effects on Centerline Unmixedness

Behavior 58

4.2.6 Reynolds Number Effects on Centerline Skewness and

Kurtosis Values 59

-iii-
TABLE OF CONTENTS (continued)

Page

5. DISCUSSION 59

5.1 Summary of Experimental Findings 60

5.1.1 Effects of Global Density Differences on Turbulent

Mixing 60

5.1.2 Effects of Reynolds Number Variations on Turbulent

Mixing 66

5.2 Comparison of Average Centerline Behavior with Integral

Analysis Predictions 67

5.3 Comparison of Experimental Results with Past Work 71

5.3.1 Average Centerline Concentration Measurements 72

5.3.2 Unmixedness Centerline Measurements 75

5.3.3 Centerline Measurements of Skewness and Kurtosis 82

5.4 Centerline Falloff of Average Concentration and Growth of

Unmixedness—Theortical Considerations 83

5.5 Validity of Generalizing Results of This Study to

Momentum-Driven Free Jets 90

6. FINAL REMARKS 96

7. ACKNOWLEDGMENTS 101

8. REFERENCES 101

9. NOMENCLATURE 106

-iv-
LIST OF TABLES

Page

Table 1. Suppliers and Purities of Gases Used in This Work 110

Table 2. Properties of Flows Studied Ill

Table 3. Estimated Values from Shadowgraphs 112

Table 4. Predicted and Observed Rayleigh Scattering Cross Section

Ratios 113

Table 5. Results for Average Concentration Measurements 114

Table 6. K Values which Result when Exponent in Equation (22) is

Set to 0.6 115

Table 7. Values of z maxCalculated Using Equations (58) and (59) 116

Table 8. Summary of Average Centerline Concentration Measurements for

Axisymmetric Turbulent Jets 117

Table 9. Literature Measurements of Centerline Unmixedness in

Axisymmetric Turbulent Jets 118

Table 10. Centerline Unmixedness Results for this Study 119

-v-
LIST OF FIGURES

Figure 1. A schematic representation of the three flow regions of an axisymmetric , constant-density, turbulent jet is shown.

Representative radial profiles of velocity and jet fluid

concentration are shown at one location in the potential

core portion of the flow and at three positions in the

self-similar regime

Figure 2. Calculated values for the ratio of the integrals I

3 (eq. (44)) and I ? (eq. (43)) are plotted as a function of

C ((P/P ) - 1) m 00 o

Figure 3. The experimental arrangement employed for time-resolved

shadowgraphy of variable density flows is shown

Figure 4. This figure shows the experimental system for making

real-time, spatially-resolved, concentration measurements

in turbulent flows of binary gas mixtures

Figure 5. Three shadowgraphs of a turbulent helium jet flowing into

a slow coflow of air are superimposed. The Reynolds number

for the jet is 3950

Figure 6 . Superimposed shadowgraphs of the same helium jet visualized

in fig. 5 are shown, but in this case, there is no coflow

and the surroundings are the ambient laboratory air

Figure 7. Four superimposed shadowgraphs are reproduced for a turbulent methane jet at a Reynolds number of 3950

entering a slow coflow of air

Figure 8 . The superposition of four shadowgraphs is shown for a turbulent jet of propane flowing into a slow coflow of

carbon dioxide. The Reynolds number for the jet is 39oo ....
LIST OF FIGURES (continued)

Page

Figure 9. Shadowgraphs of the same propane jet visualized in

fig. 8, but flowing into a slow coflow of air, are shown .... 128

Figure 10. The Reynolds number of the propane jet shown in fig. 9

has been increased to 7930 for the four shadowgraphs which

have been superimposed. The coflow gas is again air 129

Figure 11. Four superimposed shadowgraphs are shown for a turbulent jet of propane (Re = 11,880) entering a slow

coflow of air 130

Figure 12. The near-field region of a sulfur hexafluoride turbulent

jet (Re = 3950) is visualized by shadowgraphy . Note the

presence of jet fluid in regions well outside of the jet .... 131

Figure 13. A shadowgraph for the same flow described in fig. 12 is

shown at larger downstream distances. Again, jet gas is

observed in locations well outside of the jet 132

Figure 14. The formation of a fountain by the strongly negatively

buoyant jet of SF^ (Re = 3950) flowing into a coflow of

air is visualized by shadowgraphy. The range of downstream

distances extends from z/r

Q = 25 to 47 133

Figure 15. Four superimposed shadowgraphs of a turbulent jet of

sulfur hexafluoride (Re = 7890) entering a slow coflow

of air are shown 134

Figure 16. The Reynolds number has been increased to 11,860 for the

sulfur hexafluoride turbulent jet entering a slow coflow

of air 135

-vii-
LIST OF FIGURES (continued)

Figure 17. The near field region of a SF^ jet (Re - 3950) flowing

into a slow coflow of helium is visualized by shadowgraphy

Page

136

Figure 18. A shadowgraph centered at z/rQ

35 24 is shown for the SF 5 jet of Re = 3950 flowing into a slow coflow of helium ... 137

Figure 19. The fountain formed by the SF^ jet (Re = 3950) entering

into helium can be clearly seen in this shadowgraph

recorded for downstream distances extending from

z/rQ = 25 to 47 138

Figure 20. A portion of the time-resolved methane concentration

fluctuations within a turbulent jet of methane (Re = 3950)

flowing into a slow coflow of air is shown. The measurements were made on the centerline at z/rQ - 31.5 139

Figure 21. Values of Y /Y as functions of z/r are shown for the m

SF^ jet (Re = 3950) flowing into coflows of air and helium 140

Figure 22. Inverse centerline values of jet gas mole fraction,

normalized by multiplying by the jet gas mole fraction

(XQ = 1), are plotted as functions of nondimensionalized

downstream distance for six different jet/coflow gas pair

combinations. Straight lines are linear least squares

curve fits of the data 141

Figure 23. Values of the normalized centerline inverse mass fraction

(Y /Y ) are plotted as functions of z/r„ for the six jet/

o m u coflow pairs listed. Straight lines are linear least squares

curve fits of the data 1^2

-viii-
LIST OF FIGURES (continued)

Figure 24.

Page

Values of Y /Y versus z/r_ are plotted for the

o m ° helium jet (Re = 3950) flowing into both a slow coflow

of air and the laboratory. Lines are linear least squares

curve fits to the data 143

Figure 25. Experimental values of Y^/Y^ for the six jet/ coflow

combinations listed are plotted as functions of

downstream distance normalized by the effective radius,

r

£

. Results have been offset by the appropriate values

of virtual origin. Comparison with fig. 23 indicates that

the use of the effective radius concept gives a partial

correlation of the data 144

Figure 26. Experimental results of centerline unmixedness measurements

in mole fraction terms (X'/X mm) are plotted as functions of

z/rQ

for the six jet/coflow pairs listed 145

Figure 27. The results of Y'/Y measurements for the jet/coflow pairs

m m

listed are plotted as functions of z/r

Q

. Note that the

values for each flow approach a common asymptote of

(Y’/Y ) » 0.23 146 m m asym

Figure 28. Values of Y'/Y are plotted as functions of z/r for the m m o helium jet (Re = 3950) flowing into a coflow of air and into

the laboratory (free jet) 147

Figure 29. Mass fraction unmixedness values for the six jet/coflow

gas combinations are plotted as functions of the downstream

distance nondimensionalized by dividing by the effective

radius. An excellent correlation of the data results 148

-ix-
LIST OF FIGURES (continued)

Page

Figure 30. Skewness values for centerline concentration fluctuations

are plotted as functions of z/r

Q for the jet/coflow gas

pairs listed 149

Figure 31. Kurtosis values for centerline concentration fluctuations

are plotted as functions of z /r for the jet/coflow gas

pairs listed 150

Figure 32. Y^/Y^ values are plotted against z/r

Q

for propane jets

having the three different Reynolds numbers listed. The

coflow gas is air. Straight lines represent linear least

squares curve fits of the data 151

Figure 33. Y /Y values for two different Re flows of carbon tetrao m

fluoride into air coflows are plotted as functions of

z/rQ

. Linear least squares curve fits to the data are represented by the lines 152

Figure 34. Values of Y^/Y^ for SFg jets at Reynolds numbers of

7890 and 11,860 are shown as functions of z/r

o> Straight lines represent the results of linear least

squares curve fits of the data 153

Figure 35. Virtual origins which are obtained from centerline

concentration measurements in the CgHg/air and CF^/air

flows are plotted as functions of jet Reynolds numbers 134

Figure 36. Mass fraction unmixedness values are plotted as functions

of z/r

Q

for three different Re jets of propane entering

slow coflows of air 135

-x-
LIST OF FIGURES (continued)

Page

Figure 37. Mass fraction unmixedness values are plotted as functions

of z /rQ for two different Re jets of CF^ flowing into

slow air coflows 156

Figure 38. Unmixedness values for jets of SF^ at Reynolds numbers of

7890 and 11,860 are plotted as functions of z/r Q

. The

coflow gas is air 157

Figure 39. Values of 1/C calculated using eq . (34) are plotted as m functions of z/r

Q for the six gas pairs listed. These

results are based on a method suggested by Thring and

Newby [18] 158

Figure 40. Values of calculated in the manner suggested by

Rodi [5,23] are plotted as functions of z/r . Values of

jet to surrounding gas density ratios correspond to those

used in this study 159

Figure 41. The calculations shown in fig. 40 have been redrawn as

l/'C m

versus z/r

Q

. Comparison of this figure with

fig. 39 shows that the approximate treatments of Thring

and Newby [18] and Rodi [5,23] give similar predictions for

the dependence of average centerline concentration falloff

on the jet/coflow gas density ratio 160

-xi-

EFFECTS OF GLOBAL DENSITY AND REYNOLDS NUMBER VARIATIONS

ON MIXING IN TURBULENT, AXISYMMETRIC JETS

William M. Pitts

Center for Fire Research

National Bureau of Standards

Gaithersburg, MD 20899

Abstract

Shadowgraphy and laser-induced Rayleigh light scattering measurements of

centerline concentration have been utilized to investigate the effects of

global density and Reynolds number variations on the mixing behavior of a

turbulent, axisymmetric jet. Density variations are introduced by using

different gases for the jet and the surrounding coflow. Values of the

jet/coflow density ratio have been varied from 0.14 to 37. The shadowgraph

measurements give a qualitative indication of the variations in average mixing

behavior and turbulent structure which occur when the jet/coflow density ratio

and the Re are varied. These trends are quantified by the Rayleigh scattering

concentration measurements which give measurements of average centerline

concentration as well as the second through fourth moments of the concentration fluctuations. Extensive comparisons of experimental results are made

with previous literature findings. Integral analysis has been employed to

provide a framework for analysis of the data. The results of this study have

led to the proposal of a simple, qualitative theory based on two reasonable

assumptions concerning the flow behavior which gives predictions which are in

agreement with the experimental findings.

- 1 -
1 . INTRODUCTION

There has been a great deal of recent interest in improving the understanding of chemically reacting turbulent flow. The impetus for this research

has been the realization that the complicated coupling of heat release and

aerodynamics which occurs in such flows is crucial to control and improvement

in such important areas as combustion and engine efficiencies, commercial

chemical reactors, and the formation of pollutants in these systems.

With these considerations in mind, a long term experimental investigation

of chemically reacting turbulent flow has been initiated which has as its

primary goal an improved understanding of this important process. It is

anticipated that the experimental findings will yield an improved physical

understanding of turbulent flow behavior and at the same time provide the

input for workers who are trying to model this complicated process. As a

first step toward these goals, turbulent flows of two gases are being investigated in which there are global density differences based on molecular weight.

This is appropriate, since the major coupling of chemical reactions and aerodynamics in a turbulent flow occurs through the localized density fluctuations

introduced by chemical heat release. Despite the importance of density variations to understanding chemically reacting turbulent flow, the effects of

global density differences remain poorly characterized.

In this work the mixing behavior of an axisymmetric turbulent jet with i

slow coflow of a second gas has been considered. Gas pairs having jet carl >w

density ratios from 0.14 to 37 were investigated. Flow visualization using

shadowgraphy was employed to obtain an overall picture of the effects of

- 2 -
global density differences on turbulent mixing. These effects were quantified

by performing a series of measurements of jet gas concentration along the

centerlines of the axisymmetric jets. Laser-induced Rayleigh light scattering

[1-4] was employed as the real-time, spatially-resolved concentration probe

for these binary gas mixtures.

The following section (2) summarizes past experimental work on mixing in

constant and variable density axisymmetric jets and discusses integral analysis

approaches to the solution of the equations for momentum and jet fluid conservation which have been used to gain insight into the time-averaged behavior of

these flows. This is followed by a description of some considerations of special

importance to this study. In particular, the effects of the presence of a coflow

within an enclosure and the effects of buoyancy are emphasized. Section 3

describes the experimental systems for shadowgraphy and laser-induced Rayleigh

light scattering. The experimental findings are summarized in section 4 and a

discussion of the findings and comparison with past research is included in

section 5. A simple theoretical framework based on the experimental results is

proposed which is in qualitative agreement with the experimental findings. The

final section (6) summarizes the major conclusions of this work.

2. REVIEW OF AXISYMMETRIC JETS

Axisymmetric turbulent jets have been the subject of intense investigation (both theoretical and experimental) for several decades. Such flows are

of interest for several reasons. Among these are: (1) the axisymmetric

symmetry simplifies some analyses, (2) these flows are easily generated in a

laboratory environment, (3) many practical systems employ such flows, and

(4) these flows can serve as simple analogs for more complicated flows.

-3 -
The primary emphasis of this work is the effects of global density

differences on turbulent mixing. However, it is important to review some

general findings concerning the behavior of axisymmetric turbulent jets in

order to develop a foundation and consistent body of nomenclature for the

detailed discussion which follows. In the next subsection some general

findings of experimental studies as well as predictions and conclusions from

simple integral treatments of the equations of mass conservation and motion

for isothermal, constant-density, free jets (sometimes called submerged turbulent jets) are discussed. This is followed by a subsection describing similar

findings for variable density jets. Again, integral analysis of the equations

of motion and mass conservation are utilized to give insight into the mixing

behavior of these flows. The final subsection discusses specific aspects of

the current study which might be expected to modify the turbulent mixing from

that found for a purely momentum-driven axisymmetric free jet. In particular,

modification of flow behavior due to the presence of a surrounding coflow and

buoyancy effects are emphasized.

2.1 Constant Density Axisymmetric Turbulent Free Jets

There is a great deal of information available concerning the behavior of

this type of turbulent flow. Figure 1 shows an idealized picture of the

velocity and jet fluid concentration profiles of a constant density, axisymmetric, turbulent jet at high Reynolds number (Re). This figure represents a

synopsis of results from many different studies. It shows the time-averaged

behavior of the flow field which is found to maintain the axisymmetric

symmetry imposed by the circular nozzle.

- 4 -
The flow development of constant density jets can be broken into three

distinct flow regimes as a function of downstream distance. After the jet gas

exits the nozzle, a shear layer forms on the outer edge of the jet which grows

both inwards toward the centerline of the flow as well as outwards into the

surrounding gas as the flow moves downstream. There is a region in the center

of the jet, called the potential core, where the velocity and jet fluid

concentration have the same value as at z = 0. As the jet fluid moves downstream, the radius of the potential core decreases until it disappears. The

region of the flow field extending from the nozzle to the end of the potential

core defines the first flow regime. Following the potential core region, the

jet undergoes a development phase during which it assumes the velocity and

concentration behavior characteristic of its final regime. For this reason,

we will denote this second flow region as the development regime. The final

regime is called the self-similarity regime. The characteristics of this flow

region are discussed further below.

A flow field is said to be self-similar when only one geometrical

variable is required to characterize the nondimensionali zed , time-averaged

behavior of the concentration or velocity throughout the region. In an

analogous way, local similarity is defined as occurring when concentration or

velocity profiles can be collapsed in terms of a local geometrical variable.

These definitions are the same as suggested by Chen and Rodi [5].

The integral equations for momentum and jet fluid conservation can be

used to derive the requirements necessary for an axisymmetric , constant

density, turbulent jet to have a self-similar behavior. These conservation

equations can be written as [6],

-5 -
( 1 ) 2irp / U

2

rdr

o M

o

and

2ttp / CUrdr - N , (2)

o

o

where M

Q and N

Q are constants representing the momentum flux and jet fluid

concentration flux, respectively. For now, the exact nature of C has not been

specified since this is not critical for constant density flows. In section

2.2 a more precise description of C will be given. If the flow field is selfsimilar, the local values of U and C can be expressed as functions of the

similarity parameter n (where h =» r/z) as,

U = U f (n) (3) m 2

and

C - C f , (n)

,

(4)

m 1

where the subscript m indicates values on the jet centerline. The following

equations are obtained by substituting eqs. (3) and (4) into eqs. (1) and (2)

and assuming uniform exit profiles at the nozzle for both concentration and

velocity,

CO 2TTPU

2

z 2

/

f

2

ndn - TTPr

2 U2 (5) m i. oo o

and

- 6 -
2 „ irpr U C . ooo 2ttpC U z ( 6 ) 2

m m

f^hdh =

The assumption of uniform jet exit distributions for concentration and

velocity simplifies calculations, but is not necessary for the final

conclusions

Eq. (5) yields directly

U

m

where K

22 is a constant defined as

(7)

K

22

=

^ 2f

2 ndri o

( 8 )

Substituting eq. (7) into eq. (6) gives a similar equation for the centerline

concentration,

C m

zr

o

K

12

/K

22

(9)

with

K

12 =

^ 2f

l f2 ndn (10)

o

Equations (7) and (9) require that in the self-similarity region of the

flow field that both jet fluid concentration and velocity have a hyperbolic

dependence on downstream distance. Experimentally, it is often found that the

origins of plots of C /C and U /U as functions of z/r„ are not located at

o m o m o
the jet exit and a virtual origin ( z Q

) is introduced. In terms of z Q

, eqs.

(7) and (9) are rewritten as

U z - (z ) —— = K

Um

and

( 11 )

K

c

Z ~ < Z o>

ro

c

( 12 )

where K

22 and K^/

K

22 are rePl acec* by K

u

an<l K

c

» respectively, to reflect the

fact that these parameters are usually determined empirically from experiment

instead of directly integrating the radial profiles to obtain values for

and K

^ 2

•

In general, experiments by a large number of workers have yielded results

which are in agreement with the qualitative predictions based on the above

discussion. Both average velocity and concentration centerline decays are

found to have a hyperbolic falloff with downstream distance when measured from

a suitably chosen virtual origin. Radial profiles of these parameters are

found to collapse to single curves when plotted In terms of U/U^ or C/C^ as

functions of n. In almost all cases the resulting dependence of velocity and

concentration on n has been found to be well fit by Gaussian curves. The

halfwidth for the velocity (defined as the value of n where U/U = 1/2) is

ra found to be less than the corresponding value for concentration. This finding

indicates that momentum is transferred less efficiently than mass.

-8 -
Unfortunately, quantitative agreement among different studies is not

nearly as complete. Values of both and K

c are found to vary widely from

experiment to experiment. In a recent review, List [7] recommended a value of

0.081 for based on the results of several different studies. To give an

idea of the variations which are observed, this value can be compared with

that of K

u

= 0.093 reported by Wygnanski and Fiedler [8] in their widely

quoted investigation. Becker et al [9] have listed values of K

c measured by

several different authors using either temperature, concentration, or

particle-marked constant-density flows. A range of values from 0.088 to 0.139

are tabulated. Chen and Rodi [5] recommend a value of K =0.10 for the

c axisymmetric jet based on their review of literature results. These workers

conclude that the observed "differences merely represent experimental

scatter".

Some authors have concluded that variations observed in values of K , K , u c and z q are due to slow development of the jets to self-similar behavior. For

instance, Wygnanski and Fiedler [8] found that self-similar behavior of their

jet only occurred for values of z/r > 140. Measurements of U made at

o m considerably shorter downstream distances seemed to obey a self-similarity

relationship, but linear fits to the data gave different values of K

u and

(z

q

) u than found when data from further downstream were utilized. Birch et al

[10] have reached similar conclusions concerning the approach to selfsimilarity for jet fluid concentration. However, it should be noted that this

study was done for a flow of methane into air and density effects can not be

ruled out. In contrast to the results discussed above, most workers have

concluded that self-similarity for this type of flow occurs within 20 to 40

radii of the nozzle exit.

- 9-
In quantifying centerline decays of these jets the position of the

virtual origin may play a crucial role. Some workers utilize the value of z Q

as an experimental parameter while others have forced their fits of data to

pass through z = 0. Further complications arise due to the wide range of z Q

values which have been measured. Bradshaw [11] has argued that the position

of the virtual origin depends on initial conditions at the jet exit. For

instance, the nature of the boundary layer at the nozzle (laminar or

turbulent) and the level of turbulence of the fluid inside the nozzle are both

very important in determining the location of z Q

Despite the complexity of the factors which are expected to determine the

location of z Qt some information is available in the literature concerning its

dependence on Re. In his critical evaluation of turbulent data, Harsha [12]

notes a tendency for the virtual origin (he refers to it as the length of the

potential core) to increase with increasing Re. Using data of Alexander et al

[13], Harsha has derived an equation for z Q which can be written

(z

o

} u , w 0.097 = 4.3 Re (13)

Ebrahimi [14] has reported similar shifts in the location of ( z ^ / o

)

u (axial

distance where U = U /2) with Re which obey the empirical relationship,

m o

v—l/2'u ' -4

= 1.3 x 10 Re + 23.2 (14)

ro

If one assumes the virtual origin shifts in a manner similar to ( z ;/)) u , this

relationship indicates a downstream shift of the virtual origin with

increasing Re. Ebrahimi and Kleine [15] found a similar dependence for the

jet fluid concentration half length (value of z where = C

Q /2), namelv

- 10-
(15)

(z l/2 } c

r

o

= 2.6 x 10 Re + 12

This finding is evidence that the virtual origins for concentration also shift

downstream with increasing Re.

Thus far, the discussion has centered on the behavior of average centerline values of velocity and concentration in axisymmetric jets. It is also

necessary to consider the behavior of the turbulent fluctuations for these two

parameters. Such fluctuation behavior is usually characterized by measuring

average values of rms for the velocity (U') and concentration (C ' ) fluctuations which occur in the flow field. Often such data are normalized by

dividing by the average concentration or velocity at the measurement point.

These latter measurements are known as fluctuation intensities. Often the

concentration intensity is referred to as the unmixedness.

It is generally believed that both the concentration and velocity

intensities approach asymptotic values in the self-similar region of the flow

field. In fact, the attainment of such asymptotic values is often taken as a

criteria for self-similarity.

Very recently, Dahm [16] has reported that an axisymmetric turbulent jet

of water does not appear to attain an asymtotic value of unmixedness even at

extremely large downstream distances. This result has led this researcher to

conclude that intensity is not a suitable criteria for self-similarity. A new

similarity parameter defined as C/(l-C) has been proposed by this author to

treat data of this type.

- 11 -
Many measurements of U'/U and C'/C can be found in the literature.

m m m m

These measurements are characterized by rapid growth in values of the

intensity as the measurement point is initially moved away from the nozzle.

This is followed by a region at larger values of z/r

Q where the intensities

appear to level off or rise very slowly. As is the case for the average

values, quantitative agreement among different experiments which have measured

velocity or concentration intensities is poor. The basis for such variations

is not known at the present time.

2.2 Axisymmetric Turbulent Free Jets Having Global Density Differences

Historically, there has been a great deal of interest in variable

density, axisymmetric, turbulent jets. The reviews by Harsha [12] and Chen

and Rodi [5] discuss several of these studies. In section 5 the experimental

findings of this study on mixing will be compared with many of these studies

so a detailed description of individual studies will be delayed until then.

Only a short discussion of general experimental findings for jets having

global density differences is presented here along with some analyses of these

flows using various integral approaches similar to those given above for the

constant density case.

Only flows in which the global density differences are due to temperature

or gas composition are considered. Constant pressure and incompressible

behavior are assumed. For these conditions, global density differences due to

temperature or composition are generally assumed to have equivalent effects on

turbulent mixing.

- 12 -
Several generalizations can be made concerning the behavior of these

axisynnnetric jets. Here focus is upon mixing behavior. As discussed below,

analysis indicates that variable density jets can never attain true selfsimilarity. However, at far downstream distances where the centerline density

approaches the ambient surrounding density, but for which the initial jet

momentum forces still dominate, these jets will approach self-similarity.

Experimentally, it is found that such jets have hyperbolic centerline

concentration dependencies on downstream distance, a characteristic of selfsimilar, constant-density jets. This is true even at locations where the

centerline density is far from that of the ambient surroundings. Furthermore,

it has been found that for moderate downstream distances where the jets do not

display hyperbolic centerline concentration decay, local similarity still

occurs. In particular, the radial profiles of average temperature or jet

fluid concentration still have a Gaussian shape and such profiles, when taken

at various downstream distances, collapse when values of C/C are plotted as m

functions of r/(r. /0 ) . (r. ,„) is the radius for which C/C or 1/zc 1/2 c m

(T-T<jo )/(T -T^) = 0.5. A striking example of this behavior is found in the

study of O’Connor et al [17]. These authors investigated the turbulent mixing

of an axisymmetric jet of partially dissociated nitrogen with ambient air.

Jet exit temperatures as high as 5700 K were utilized. For axial locations as

near as z/r

Q = 8 from the nozzle, the radial profiles of mass fraction of jet

gas were found to be locally similar and to obey a Gaussian falloff quite

well.

for

All experimental results seem to agree that a decreasing density ratio

the jet to ambient surrounding gases (R = p /p ) leads to a faster fallp o 00

- 13 -
off of jet gas centerline concentration with downstream distance. Here

"concentration" refers specifically to either (T^-T^/CT -T^) in the case of

heated or cooled flows or to the mass fraction of jet gas in cases where

isothermal gas density differences are important. A justification for these

forms of "concentration" will be given shortly.

In his review, Harsha [12] concludes that the velocity potential core for

variable density jets lengthens as the ratio of jet to surrounding gas density

is increased. In contrast to this result, he reports that the potential core

for concentration (in terms of mole fraction) shortens with increasing R^.

This was considered to be an unexplained anomalous behavior. In section 5 it

will become clear that this observation is an artifact due to the manner in

which Harsha analyzed the data.

Several sets of experimental data for the decay of average centerline

concentration with downstream distance are available in the literature for

variable-density, axisymmetric jets. Specific results are to be discussed in

section 5. However, it is worthwhile to note that the quantitative results

are scattered in the same way as those found for the constant density flows.

There have been relatively few measurements of turbulent fluctuation

behavior in variable density jets. At the present time, it seems fair to

state that no consistent picture has emerged and very little is understood

concerning the predicted or experimental behavior of centerline velocity or

concentration intensities in these flows.

- 14 -
Some insight into the effects of global density differences on mixing

behavior can be obtained by considering the integral equations for conservation of momentum and jet fluid. In cases where the density is not constant,

eqs. (1) and (2) become

00

2t: / pU

2

rdr = M (16) 1 o

o

and

2tt / pCUrdr = N . (17) J a o

Since MQ

and N

q are constants, the integrals on the left hand side of these

equations must also be constant. However, the density terms inside the

integrals are not constant. It is clear that there are no physically meaningful functions having the forms of eqs. (3) and (4) which can be independent

of h. This conclusion requires that no variable density jet can have a truly

self-similar behavior.

Despite the above conclusion, eqs. (16) and (17) can be used to make

predictions concerning the behavior of certain types of flows having density

variations. Thring and Newby [18] were apparently the first to treat these

equations in such a manner. These workers first considered the case of a

burning free jet for which it was assumed that the only effect of chemical

heat release was to increase the temperature and hence decrease the density,

and that in the region of interest the density was essentially uniform. Of

course, the Re was assumed to be sufficient to insure that the flow was

dominated by the initial jet momentum and that buoyancy effects could be

- 15 -
neglected. These assumptions allowed the p term in eqs. (16) and (17) to be

placed outside of the integrals. By assuming initially uniform jet profiles

for velocity and jet fluid concentration, the equations can be rewritten as

00 2irp- / U^rdr = irr^P (18)

f

J o o o o

and

w o 2irp. / CUrdr = ir P U C (19)

f o o o o

o

The values of the integrals in eqs. (18) and (19) should assume the same form

as those for constant density flows. Making this assumption, the equations

become

U

o z

and

12

22

^P,/P

f o

( 21 )

where P^ is the density of the jet fluid at the exit, is the density at the

measurement point, and K92 and ^12 ^ave same forms as eqs. ( 8 ) and (10),

respectively. Note that since the density is assumed to be constant, can

be either in terms of mass or mole units.

Thring and Newby [18] incorporated the square root of the initial to

final jet density into an effective radius defined as

- 16-
r ( 22 ) =

£ r /p /p

o of

Physically, r

£ corresponds to the radius that a nozzle would be required to

have in order to result in a flow having the same momentum and mass flux as

actually exists, but with a density instead of P q

.

By using this definition and making similar modifications to the formulas

as made earlier for the constant density case, the expressions

U

o Um

- ( z ) o u

r

£ K

u

and

(23)

C

o

C m

2 - ( 2

0

> r

£

C K

c

are obtained.

(24)

These expressions indicate that for cases where the downstream density of

a variable density axisymmetric jet can be assumed to be constant, the flow

will behave in an analogous manner to a constant density flow if r Q

in eqs.

(11) and (12) is replaced by r^_. Since the centerline concentration of jet

fluid approaches that of the surroundings as the downstream distance

increases, it is clear that the behavior of a jet flowing into ambient

surrounding gas at a different density will approach asymptotically the selfsimilar behavior indicated by eqs. (23) and (24) when p^ in eq . (22) is

replaced by the surrounding gas density, p .

- 17 -
Thring and Newby [18] have also provided an approximate analysis of eqs.

(16) and (17) which allows the behavior of centerline temperature to be

predicted in cases where the difference between p and p can not be

m 00 considered as negligible. A similar analysis will be given here, but the

derivation will be generalized to include cases where density differences are

due to concentration or temperature. The following relationships for generalized concentration are obtained by assuming the ideal gas law holds everywhere

in the flow field,

Substitution of this definition of C into eq . (17) gives an expression for the

conservation of mass flow defect as originally defined by Corrsin and Uberoi

[6]. Note that eq. (25) can be rewritten as

Substituting eqs. (3), (4), and (26) into eqs. (16) and (17) and assuming

uniform initial jet profiles yields the following equations for conservation

of momentum,

Poo p (26)

z

r (27)

and jet fluid concentration

z

r

oo

1

2f f 0 ndn

(28)

o

- 18 -
Assuming C « 1 , it is possible to generate an approximate solution

m

-2

for C which is accurate to the order of C . This is done by using the

m m

binomial expression to expand 1 /

{ 1 +C [(p / p ) - l]f.} and neglecting terms

m 00 o I in C of higher order than one. Using this procedure, eqs. (27) and (28) m

become

2 z o

and

2

z Kn 2

]

(29)

(30)

where K

22 and Kj 2 are given by eqs. (8) and (10), respectively. ^122 anc* ^112

are defined analogously as

V

122 / 2f ^ndn

o (31)

and

K

1

2 * / 2f^f

2 ndn (32)

o

Dividing eq. (30) by the square root of eq. (29), again using a series

expansion, and neglecting higher order terms results in the following

expression,

z

1 + Cm

/ K122 K1 12

\

2K22

“ K

12

(33)

- 19 -
Equation (33) can be rearranged and solved for l/C . Again, series expansions m

are utilized and higher order terms are omitted. The final result is

1

C

K.

'12 /K

22

/ p \ I"k K „“]

1 — 1 ] 122 112

V’o V L

2K

22

K

12 J

(34)

For small z, the term in brackets is expected to vary with z. At larger

z this term should become nearly constant. Thring and Newby [18] noted that

the value of ^\ 22^22 be nearly equal to Ki22^12 anc* hence the term in

brackets is negative. This implies that the location of the virtual origin

for experimental data plotted as l/C versus z/r will be dependent on the m £ density ratio p /p , with (z ) (as defined by eq . (24)) becoming negative

Q CO oc

for Rp > 1 (negatively buoyant jet) and positive for R^ < 1 (positively

buoyant jet). Note also that the slope of the plot is predicted to be

independent of R

q and to be the same as that given in eq . (21).

There are several studies in the literature [e.g., 19, 20], mostly for

heated flows, which agree well with the predictions of Thring and Newby [18]

with respect to variations in the slopes of l/C versus z/r

c plots with

density ratio. There appears to have been no verification of their predictions for changes in (z ) c with the same parameter.

Ricou and Spalding [21] have described a study which also supports the

concept of the effective radius. These authors have experimentally determined

the entrainment behaviors of variable density, axisymmetric, turbulent jets.

For the constant density case, dimensional analysis can be used to show that

the average mass flow rate (m) through a cross section of the jet, defined as

- 20-
m = 2n ( 35 ) / pUrdr,

o

can be expressed in terms of other flow parameters as

m

zM

1/2 1/2

o

K

e

where Ke is a constant to be determined experimentally.

(36)

The form of eq. (36) was verified for an air jet flowing into surrounding

air. Interestingly, the same equation was found to be valid for jet flows of

CO

2

, CgHg, and H2 into air indicating the result is independent of density. A

value of K

e = 0.282 was determined.

By assuming that the jet profiles at the nozzle are uniform, it is

possible to rewrite eq. (36) as — = 0.16 ~ • L m o

o

This equation agrees with the hypothesis of Thring and Newby [18] that the

characteristic length for variable density jets is not r

Q

, but rather r^.

Equation (37) was found to be valid for Re > 2.5 x 10

4 by Ricou and

Spalding [21]. Measurements of the entrainment taken at z/r Q = 27.4 indicated

that for lower Re the amount of entrained fluid increased as the Re was

decreased. The value of K

0 and its independence of Re in the range 6-29 x 10

4

was verified in an independent study by Hill [22]. This study differed from

that of Ricou and Spalding in that the local entrainment coefficient was

(37)
measured at a given downstream location instead of the integrated entrainment

to that point. Hill’s study showed that there is an initial region of the Jet

near the nozzle where the entrainment coefficient is very small, and that it

rapidly increases with downstream distance until a constant value is reached

for z/rQ > 20.

The experimental observations that eq. (36) can be used to describe jet

mass flow rate through a cross section and that radial profiles of concentration and velocity obey local similarity even when the flow is not selfsimilar, can be utilized to predict the behavior of the centerline concentration falloff for variable density flows. The only additional assumption which

is required is that the turbulent Schmidt number ((Sc) tu ) be a constant. This

development should be valid for downstream locations where the above conditions are fulfilled, even if self-similarity behavior has not been approached.

The findings of such a study have been summarized by Chen and Rodi [5]. The

following analysis of this problem is based on the work of Rodi [23].

Expressions similar to eqs. (3) and (4) can be written in terms of a

local similarity parameter to describe the radial dependence of concentration

and velocity for a given value of z/rQ

. These are

C = C f.’ (n’> m 1 (38)

and

u = u f:(n'),

in Z

( 39 )

-22 -
where the primes indicate expressions written in terms of the local similarity

parameter, n' = r/(r, ..) . 1 /2 u

Utilizing the experimental observation that the radial falloffs of C and

U have Gaussian forms and assuming a constant value for (Sc) tu , f| and f^ can

be written as

- 0.693 .2

R

2 fj(n') = e (40)

and

fjcn’)

-0.693 n'

e (41)

where (r^y^c = ^( r l/2)u* With R prescribed, there are three unknowns,

namely C , U^, and (r^y^u e 9 s * (38-41).

Three independent equations are required to solve for these three

unknowns. Rodi [5, 23] utilized the integral momentum equation (eq. (16)),

the weight deficit equation (eq. (17) with C defined as in eq. (25)), and the

entrainment law of Ricou and Spalding [21] (eq. (37)). 3v making substitutions similar to those used above, these three equations can be written as

2,fl-<r l/2>»

o

f^

2 h'dn'

= M

1 + C — - 1 m . 0 q

c » x

i

2*p ( r i /o)

2 " U C /

1 /2 u mm

f Jf^n’dn’

1 + C l If]

m ' p 0

1

= N

(42)

(43)

- 23 -
and

2"P» <r 1/2 ) u 5m /

2 _ | j

f^n'dn’

l + c l Ilf;

m \ P Q

/

1

= KzM1 / 2

pi

/2

e o

00 (44)

Denoting the integrals in eqs. (42-44) as Ij, I 2

, and I

3

, respectively, and

assuming uniform profiles at the jet exit, these three equations become

U

2 — = 2 u

2m

(45)

= 2

^ r l/2^u

U C

m m

(46)

U 0 1 /2 (r, /0 ) 2 o _

2 tt v l/ 2

2 u - ~ K z r 3

U e e m

(47)

These three equations are easily solved simultaneously to give

l V X 2

C ”7*7 h m £

(48)

K

e

2

Um

1 1/2 I -5 r 3

e

(49)

and

( r l/2^u

=

1/2

K z 1/

(50)

- 24 -
The behaviors of 1/C , U /U , and (r,/ 9 ) assuming a value of R = 1.17

are shown in fig. 7 of Chen and Rodi's monograph [5] for values of Pq/p®

ranging from 3.3 to 0.07. Our primary interest here is the behavior of C^ as

a function of z/ r and P /p . In order to make this calculation, it is

o o " necessary to solve for values of I

2 /I 3 as a function of KP00 /P

o

) “ 1 ]«

Solutions have been obtained by numerical integration on a minicomputer.

Figure 2 shows a plot of the results. Using this curve, the desired results

for C as functions of z/r and R can be calculated using eq. (48). The

m op

results of calculations for the flows studied here are described in section 5.

2.3 Effects of Buoyancy and Coflowing Surrounding Gases on Mixing

in Axisymmetric Turbulent Jets

For the experiments to be reported here, it has been necessary to utilize

a coflow instead of quiet ambient surroundings in order to insure that no dust

particles enter the measurement volume. Rayleigh light scattering intensity

measurements, which are used as the concentration diagnostic, are extremely

sensitive to the strong scattering which occurs from small particles (Mie

scattering). In order to limit gas volume flow rates, a relatively small flow

system Is utilized. Unfortunately, this places a relatively low limit on the

volume flow rate of coflow gas which can be used; and, as will be shown, this,

in turn, limits the Re which can be utilized for the jets to relatively low

values. Since the initial flow velocities are limited, it is possible that

buoyancy effects can start to dominate these originally momentum-driven flows

as the jets move further downstream. For these reasons, it is necessary to

consider the effects of buoyancy and the presence of a surrounding coflow on

the jet measurements.

-25 -
A crude estimate of buoyancy effects can be obtained using an integral

analysis approach. The integral momentum equation (eq. (16)) can be modified

to include buoyancy effects [12] by writing the jet momentum at z as

2tt / pu rdr =* M + 2irg / / (p - p)rdrdz' = M + B(z), (51)

o

00 o o o o

where the flow momentum will increase for a positively buoyant jet (R < 1)

P and decrease for a negatively buoyant jet (R^ > 1). The several different

types of flow which can result are described by Chen and Rodi [5].

Equation (51) can be solved approximately in order to obtain an

expression for the ratio of momentum imparted to the flow at the nozzle to

that for a given z location due to buoyancy (Rm

= M

Q /B(z)). In order to

generate this solution, it is assumed that there is no coflow, self-similarity

holds throughout the flow field, and that the centerline concentration Is

described by eq . (24) with (z

Q

) c = 0. The last term of eq. (51) can be

rewritten as

z

00 r

B(z) = 2irg / / o o P„ "

1 + C — - If

m \ P

z

' ^ndndz

(52)

Expanding in C , ignoring higher order terms, and substituting eq . (24) gives

m

3(z) = Tr gPoo K

c

(53)

The half angle (9) of variable density jets which obey eq . (24) has been found

experimentally [5, 19] to be independent of R^. Using the experimentally

determined value and assuming Gaussian radial profiles, the value of the

remaining integral in eq. (53) can be expressed as

- 26-
-0.693

/ e

2

n

2 tan 0 ndn

tan^9

2 x 0.693 ( 54 )

The final form of B(z) then becomes

r

e /

p «

B(z) - 2.27 g P. ( p

2 2

z tan 0 ( 55 )

Assuming uniform profiles at the jet exit, M0 is given by TTr Q

p 0 U

0 and Rm( 2 )

becomes

v*>

1.39reU^e /P0 \

gz^tan^O \ P®

p o

/

( 56 )

The form of R^z) is reminiscent of that for the jet Froude number as defined

by Chen and Rodi [5],

F ( 57 )

The sign and magnitude of R^z) are indicators of the type of flow

behavior expected at various downstream positions for variable density jets.

If the absolute magnitude of R^z) » 1 the flow is dominated by the initial

momentum at the nozzle and the behavior is that of a forced jet. As the

absolute magnitude of Rm(z) approaches 1 this is no longer true. For positive

values of R^z) (positively buoyant jets), the momentum of the flow increases

as R

m(z) decreases and at downstream distances where Rm(z) "la mixed flow

behavior between that for purely forced jets and buoyant plumes occurs. As

Rm

(z) becomes still smaller the flow behavior becomes that for a buoyant

plume. On the other hand, for Rm

(z) < -1 (negatively buoyant jet), the total

flow momentum is decreasing as the downstream distance increases and the

- 27 -
absolute magnitude of Rm

(z) decreases. For R^z) = -1, the downward momentum

due to buoyancy is exactly balanced by the upward momentum imparted to the

fluid at the nozzle. As this point (zmax ) the jet will stop rising and jet

fluid will begin to fall. This type of flow behaves as a fountain.

Equation (56) can be used to generate an estimate for this height as

The problem of jets having negative buoyancy has been investigated by Turner

[24]. Dimensional arguments were used to show that zmax

can be written as

where is a constant to be experimentally determined. Equations (58) and

(59) are very similar except for a slightly different dependence on density

and the values of constants. From studies on mixing of salt water jets Into

pure water, Turner measured a value of = 2.46.

In section 5.5, eq. (56) is used to estimate the effects of buoyancy on

the variable density jets investigated in this study. The fountain-Like

behavior predicted by eqs. (58) and (59) has been observed and is discussed in

section 5.2.

It should be cautioned that eqs. (56) and (58) are based on assumptions

which cannot be correct throughout the flow field. For this reason, they

should be considered only as estimates for Rm

(z) and z max . These equations

(58)

(59)

- 28-
only apply to the behavior of the average concentration. The effects of

buoyancy on turbulent concentration fluctuations are not addressed and must be

considered as unknown.

The use of a secondary flow within an enclosure as the surrounding gas

for the primary turbulent jet can strongly perturb the mixing behavior from

that characteristic of free jets. Such modifications can be conveniently

categorized as those which arise only from the presence of an infinite

surrounding flow and those which occur due to the presence of a wall which

serves to contain the secondary flow. The latter modifications can be due to

several different effects including lack of enough mass volume flow in the

secondary stream to meet the entrainment capabilities of the jet [18], expansion of the primary jet until it strikes the enclosing walls [18], and the

development of pressure gradients within the enclosure [12].

The behavior of turbulent jets flowing collaterally into a secondary air

flow of infinite extent has been discussed extensively in the monographs of

Harsha [12] and Abramovich [25]. The nondimensionalized parameter which is

utilized to characterize this type of flow is m = U /U . Our primary interest

oo o here are flows for which 0 < m << 1. Experiments summarized in the two monographs lead to the following conclusions concerning the mixing behavior of

this type of flow:

1. Self-similarity is not possible, but is approached for constant

density flows when m « 1;

- 29 -
2. Radial profiles of concentration and velocity at moderate downstream

distances are locally similar even when the flow is not self-similar;

3. As the value of m is increased, the spreading rate of the jet

decreases and has a nonlinear dependence on downstream distance;

4. The potential core lengths of the turbulent jets increase as m is

increased; and

5. The potential core length increases with Re (as is the case for free

jets) for constant values of m.

Abramovich [25] has given approximate formulas for the variation of these

various properties of the flow with m.

The effects of limiting the mass flow rate of the secondary gas have been

considered by several workers. The qualitative discussion given by Thring and

Newby [18] is very instructive. For cases where the mass flow rate of the

secondary gas is large enough to meet the entrainment requirements of the jet

as far downstream as the point where the outer boundary of the jet strikes the

walls of the enclosure, it is expected that the jet will behave much like a

jet into a coflow until this point is reached. However, if the secondary mass

flow rate is not sufficient to meet the entrainment capability of the jet,

fluid will be drawn from downstream regions and recirculation eddies will

develop in the secondary flow. An estimate of the downstream location (z^)

where recirculation will develop can be obtained using Ricou and Spaldings'

entrainment law [21] (eq. (37)). After making the appropriate substitutions,

eq. (37) becomes

- 30 -
z^ =» 6.25 ( 60 )

where is the mass flow rate for the secondary gas. This relation is

similar to one given by Thring and Newby [18] using an earlier experimentally

determined expression for entrainment.

Later work by Becker, Hottel, and Williams [26] has provided a

quantitative means for dealing with enclosed primary jets entering secondary

flows of the same density. These workers utilized a similarity parameter

which they named the Craya-Curtet number (Ct) in recognition of Craya and

Curtet [27,28] who were the first researchers to utilize a rigorous similarity

analysis to treat this problem. Becker et al [26] introduced two velocities

known as the kinematic velocity, U^, and the dynamic mean velocity, U^, which

for uniform primary and secondary flows are defined as

( 61 )

and

2 2 Ur +

o o

TT2( 2 U® l r*

r J

1/2

( 62 )

where r

g

is the radius of the cylindrical enclosure. is the average

velocity for the total flow within the enclosure and the square of is a

measure of nonuniformity in the momentum flux distribution at z = 0 across

both the primary and secondary flows. In terms of and U^,

( 63 )

- 31 -
Experimentally, it has been found that recirculation only occurs for

Ct < 0.67 At higher values of Ct the jet strikes the walls of the enclosure

before recirculation eddies begin to develop. Becker et al [26] discovered

that measurements of (r, /0 ) , (r, /0 ) , 1/U , and 1/C as functions of z/r i/zu i/zc m m o were indistinguishable from those for a free jet when Ct was equal to 0.67.

These workers reported extensive measurements of the flow fields for velocity

and jet fluid in an axisymmetric turbulent jet flowing into a secondary flow

enclosed within a cylindrical enclosure. Their findings for downstream flow

regions can be summarized as follows:

1. Radial profiles of velocity for Ct < 0.67 are not locally similar.

Interestingly, for values of U/CU^ /2 } u there is a universal curve for

measurements made in the flow region extending from the jet axis to

r 3 ( r i/2)u* Local similarity only fails in the outer radial regions

of the jet;

2. The values of ^ or a §iven va l ue of z/r Q increase with

decreasing Ct;

3. Centerline velocity falloff as a function of z/r Q is faster for

smaller values of Ct;

4. As expected, the location of recirculation eddies in the secondary

stream move upstream as Ct is decreased; and

5. For values of Ct > 0.67 the jet fluid concentration is found to

display local similarity in the radial direction.

- 32 -
These conclusions are consistent with a picture in which mixing behavior

is dominated by recirculation effects for Ct < 0.67, behaves much like that

for a free jet for Ct = 0.67, and is dominated by the value of U^/U for

Ct > 0.67. With regard to flows where Ct > 0.67, the behavior of the experimental data obtained by Becker et al [26] is in qualitative agreement with the

conclusions given above for the dependence of flow behavior on U^/U .

Steward and Guruz [29] have extended the work of Becker et al [26] to

variable density flows. Their analysis indicates that the similarity

parameter Ct defined by eq. (63) is valid for cases where initially uniform

flows of primary and secondary gases have different densities if the

parameters and defined by eqs. (61) and (62) are replaced by the density

weighted variables

(64)

and

U

d

(65)

where

2 r U

p' =

_ p o o o ('! * ']

i J Uoo Kpoo 2tt , 2 „

r IJ + fr - r J U

o o K c o J °

( 66 )

These authors reported experimental measurements for heated and particlemarked air jets flowing into a secondary flow of air which confirmed that

treating their data in terms of the variable density Ct gave results which

were in good quantitative agreement with those reported for constant density

flows by Becker et al [26]

- 33 -
3. EXPERIMENTAL SYSTEMS AND DATA TREATMENT

Two different optical experiments have been performed during the course

of this study. Shadowgraphy was used to visualize the globt 1 flow field.

Quantitative measurements of the real-time concentration behavior for the jet

fluid were made along the jet centerline using the recently developed technique of laser-induced Rayleigh light scattering [1,2]. For both experiments,

the same flow system and conditions were utilized. The following sections

describe the flow system as well as the experimental systems utilized for

shadowgraph flow visualization and for laser-induced Rayleigh light scattering

measurements of concentration.

3.1 Flow System

The flow system employed for this study is the same as used in our

earlier study on the feasibility of Rayleigh light scattering as a concentration probe in variable density flows [1,2]. An axisymmetric turbulent Jet was

formed in the center of a square cylinder through which a slow coflow of a

second gas was passed. The enclosure served to prevent dust particles in the

laboratory from reaching the observation volume for the Rayleigh light

scattering measurement.

A 61 cm length of 6.35 mm inside diameter brass tubing formed the jet

flow. The tubing is tapered to a fine edge at the jet exit. A ballast

chamber was attached to the upstream end of the pipe in order to dampen

pressure fluctuations. The pipe was long enough to ensure that fullydeveloped turbulent pipe flow occurs before the gas exits the nozzle. The

- 34 -
axis of the jet was aligned vertically and the flow direction was upward so

that fluid flow was against the force of gravity.

A 10.4 x 10.4 x 61 cm^ square enclosure formed from 5 mm thick optical

crown glass served to contain the secondary flow. The coflow gas passed

through a bed of polystyrene balls and a wire gauze in order to create a

homogeneous flow before entering the enclosure well upstream of the jet nozzle

exit.

The primary and secondary gas flows were prefiltered using 0.3 micron

filters. Air was supplied by a regulated house airline while the other gases

utilized were obtained from cylinders. Table 1 lists the suppliers and stated

purities for the gases utilized in this study. Gas volume flow rates were

controlled and measured by Fischer and Porter Flowrator meters. Pressure

gauges monitored the pressure drop through the flow system and, where necessary, the volume flow rates were corrected to one atmosphere. Pressure drops

within the enclosure were assumed to be negligible.

Values of Re for the jet flows were calculated using the expression

Re

PUD o o o (67)

U

Q

is the velocity of the primary gas at the nozzle assuming a uniform flow

profile and is the diameter of the nozzle. Values of p and u were taken

° oo

from standard references as described in [30]

Certain commercial equipment, instruments, or materials are identified in

this paper in order to adequately specify the experimental procedure. Such

identification does not imply recommendation or endorsement by the National

Bureau of Standards nor does it imply the instruments or equipment are necessarily the best available for the purpose.

- 35 -
Since only the center region of the expanding Gaussian beam was collimated,

the 76 mm beam which resulted had a relatively uniform light intensity distribution.

The collimated laser light pulse next passed through the glass enclosure

where it interacted with the flow field formed by the turbulent jet and the

slowly flowing secondary gas. After passing through the far wall of the glass

enclosure, the beam traveled a 2.2 m before striking a thin diffuser which was

placed within the focal plane of a HP 197A oscilloscope camera. Images of the

shadowgraph were recorded using Type 107 Polaroid film. Camera shutter times

of 0.033 s were utilized to insure that only one image of the flow field was

recorded on each photograph. F-stop values were chosen to produce adequate

exposure of the film and generally fell in the 8-16 range.

3.3 System for Rayleigh Light Scattering Measurements of Concentration

The use of Rayleigh light scattering for concentration measurements in

binary gas mixtures has been discussed in detail in our earlier works [1,2,.

The basis of the technique is the dependence of the intensity of Rayleigh

light scattering on the index of refraction of a gas. In general, the indices

of refraction for different gases vary. For mixtures of two g3ses, the

scattering intensity is found to be linearly proportional to the mole

fractions of the gases which form the mixture. This observation along with

the relation

x

x + x

2 = 1,

- 38 -
where and X2 are the mole fractions for the two gases, allows the mole

fraction of a gas to be determined following calibration of the scattering

intensity by making measurements in pure gases 1 and 2.

A full description of the experimental system employed for laser-induced

Rayleigh light scattering measurements of real-time, spatially-resolved

concentration fluctuations in turbulent flows of binary gas mixtures is available [1,2]. Only a brief description of the experiment is repeated here.

Figure 4 shows a schematic for the experimental system. Rayleigh light

scattering was induced by the 8 W, 488 nm output of the Ar ion laser. The

laser beam was focused by a lens to a narrow waist ( *» 0.05 mm) at the desired

observation location within the flow enclosure. Scattered light was collected

at 90° to the laser beam by a f/2 optical system which imaged the scattered

light onto a 0.4 mm pinhole (the pinhole replaced the slits which were used in

the earlier work [1,2]). Since the laser beam was focused onto the pinhole as

a 1:1 image, the observation volume was a cylinder having an approximate

diameter of 0.05 mm and length of 0.4 mm. The resulting observation volume is

0.0008 mm^

After passing through the pinhole, the light was once again collimated

and then focused onto the photocathode of an EMI 9781B photomultiplier tube

(PMT). A narrow bandpass optical filter centered at 488 nm which was used in

the original experiments [1,2] was not employed in this study. The current

output of the PMT was converted to a voltage by shunting it to ground across a

load resistor. The voltage output of the amplifier was then digitized and

stored in the memory of a minicomputer. In general, the data collection rate

was 10 kHz and 32,768 individual measurements were taken. After collection,

the data was stored on a hard disk for later analysis.

- 39 -
Recorded time series of concentration behavior were used to calculate

time-averaged values of average, rms, skewness, and kurtosis (flatness) at

various downstream positions along the jet centerline. Note that for a

Gaussian distribution the skewness is zero and the kurtosis is 3. Results are

reported in terms of both mole (X) and mass fraction (Y) of jet gas. The

average and rms mole fractions were converted to mass fraction results using

the procedure described by Birch et al [10] in exactly the same way as

previously described [1,2]. Mathematical techniques exist for correcting the

raw data for the effects of electronic shot noise when calculating the moments

of the concentration distribution [1,2,10,33]. These procedures were employed

in this work even though such corrections were generally small.

One set of centerline measurements were recorded for a He/air free jet.

Light scattering measurements for this flow were heavily contaminated by the

presence of particles in the laboratory air which were entrained into the jet

flow. The resulting time-resolved light intensity measurements showed the jet

fluid concentration fluctuation behavior from the Rayleigh light scattering

along with numerous superimposed sharp spikes due to Mie scattering.

Some methods have been developed using computer algorithms to treat

Rayleigh light scattering data records which are contaminated by Mie

scattering [34], However, for the limited free jet data recorded during this

study a brute force approach was chosen. A stored data record of 32,7b8

points was visually examined and one hundred random concentration measurement

s

were chosen for which no particle interferences were present. The measurements were then averaged and the rms calculated to give X and X' . Values

m m of Y and Y' were then calculated as described above, m m

- 40-
4. EXPERIMENTAL FINDINGS

4 . 1 Shadowgraphy

Shadowgraphy has been used to visualize all of the flows listed in table

2 except those for the CF^ jets. Figures 5-19 show examples of the photographs which have been recorded. Shadowgraphs for each jet/coflow gas pair

combination are discussed below. The discussion is ordered from low to high

values of R . P

4.1.1 Helium Turbulent Jet into Air (R = 0.14)

P

Figure 5 shows a composite of three shadowgraphs recorded at different

times and downstream distances for the helium turbulent jet at Re = 3950

flowing into a secondary flow of air. An axis system has been included which

indicates the radial distance from the centerline in terms of r/rQ and the

nondimensionalized downstream distance, z/rQ

. The diffuse lines which are

evident in regions outside of the turbulent jet are due to index of refraction

variations within the glass enclosure which surrounded the entire flow field.

Since the light source was a laser, coherent effects are possible. Close

inspection of fig. 5 reveals that interference effects are present in the

photograph. Fortunately, these effects are not large enough to negate conclusions which are drawn concerning the flow behavior.

Several observations can be made concerning the flow behavior visualized

in fig. 5. Even though axisymraetric turbulent jets are known to have large

scale structure and to display a great deal of intermittency in the outer

-41 -
regions of the jet, the outline of the jet observed in this integrated view of

the flow field is relatively smooth. This is especially evident from the fact

that three independent shadowgraphs can be superimposed to give an apparently

continuous view of the flow. This observation is somewhat surprising since

the flow field has been "frozen" by the short light pulse and large scale

structures should be easily observed.

The spreading behavior of the outer boundary of the helium jet is evident

from fig. 5. As the flow moves downstream from the nozzle there is a

continuous increase in the diameter (D) of the jet. After a short distance

downstream from the nozzle, the spreading rate appears to be linearly

dependent on downstream distance. At a downstream distance of z/r Q = 40 the

apparent value of D is roughly 20 r .

Turbulent fluctuations of density in the jet are evident from the

cellular appearance of the shadowgraphs throughout the jet region. At certain

locations on the outer boundary of the jet, what appear to be localized

vortices can be seen. The fluctuations of density in this flow field extend

upstream to a point just above the nozzle exit and the "roll up" of the jet

fluid into vortices is evident within one jet radius of the nozzle exit.

As will be discussed later, there were experimental indications chat the

enclosed helium jet interacted with the surrounding glass wails. F )r this

reason, shadowgraphs were recorded for the helium jet flowing free:-’ into the

laboratory. Figure 6 shows the results. Once again, three independent photographs have been superimposed. The image appears clearer than tone in rig. 1

since the variations in the index of refraction due to the glass walls are

- 42 -
absent. Comparison of figs. 5 and 6 reveals no obvious differences between

the shadowgraphs for the free and enclosed helium jets.

4.1.2 Methane Turbulent Jet into Air (R = 0.55)

P

Figure 7 shows four independent shadowgraph visualizations of the CH^

turbulent jet flowing into a secondary flow of air. There are obvious differences in the appearance of this flow field as compared to those for the He/air

flows (figs. 5 and 6). The CH^ jet seems to spread radially much more slowly

than the helium jet. This is reflected in the observation that this jet has

not grown to a width of 7.5 cm (the observation diameter) by 55 r Q downstream

as compared to the He/air flow which reaches this width at **45 r^. At

z/r Q = 40, the visualized diameter of the turbulent jet is roughly 16 r

Q

.

The turbulent structure of the CH^ air shadowgraph also has a different

appearance than those seen in the He/air flows. Contrast the "cellular"

structure for the lighter jet with the "stringy" appearance of the turbulent

structure in fig. 7. Large scale structures in the outer radial regions of

the CH^/air turbulent jet also are more pronounced. A potential core region

is clearly evident near the nozzle. Indications of vortex roll up on the

outer edge of the flow field are not seen until a downstream distance of

** 1.6 r . o

4.1.3 Propane Turbulent Jet into Carbon Dioxide (R = 1.02)

P

The primary and secondary flow gases for this combination have nearly the

same densities and the jet is non-buoyant. In this regard, it should be

- 43 -
identical to jets of air which have been widely used in earlier turbulent flow

studies.

Many of the trends already noted on going from the small value of R^ of

the He/air flow to the higher value of R

p for the CH^/air flow continue to

develop as R

p is increased to that for this gas pair. Figure 8 shows the

superposition of four independent shadowgraph visualizations of the flow

field. The observed diameter for a given downstream distance is reduced from

that found for the jets having smaller values of R^. The value of D at

z/r

Q = 40 is 14 r

Q

. The turbulent structure is more pronounced and appears to

be very "stringy". Large scale structures are clearly seen near the edge of

the jet. A long potential core can be discerned and the first indication of

vortex roll up on the outer edge of the jet occurs at roughly 1.9 r G

.

4.1.4 Propane ’Turbulent Jet into Air (R = 1.55)

P

Figure 9 shows four superimposed shadowgraph views of the CgHg/air

jet/coflow combination for nearly the same Re (Re = 3960) as used for the

He/air, CH^/air, and CgHg/C^ flows. This jet has an appearance which is very

similar to that shown in fig. 8 for the CgHg/C^ case. However, the turbulent

structure seems to be less developed and to not have the same degree of small

scale structure as the combination with the smaller value of R . For

P z/r

Q = 40, the value of D is estimated to be 14 r

Q , the same value as found

for the CgHg/C02 flow. The distance from the nozzle to the first vortex roll

up is a 2.5 r .

- 44 -
As indicated in table 2, higher velocity propane flows have also been

studied. Figures 10 and 11 show superimposed shadowgraphs for Re = 7930 and

11880, respectively. The downstream spreading behavior for the different Re

flows (figs. 9-11) appear to be nearly the same (e.g., a value of D 3 14 r

Q

is

estimated for the three jets at z/rQ = 40). However, the turbulent structure

of these jets in the higher Re flows appears more developed and to extend to

smaller spatial scales.

The development of the C-^Hg/air jets in the potential core regions has a

strong dependence on Re. As the Re is increased, initial vortex roll up of

the outer portion of the jet fluid moves closer to the nozzle (values of

1.3 r Q and 1.1 r

Q are estimated for Re = 7930 and Re = 11880, respectively).

A well-developed potential core, which is very evident as a bright region near

the nozzle in fig. 9, is found for the lowest Re jet. The length of the

potential core decreases with increasing Re and has nearly disappeared in the

Re = 11880 flow (fig. 11).

4.1.5 Sulfur Hexafluoride Turbulent Jet into Air (R = 5.11)

P

The SFg/air combination has a jet/coflow density ratio of R

p = 5.11. For

Re = 3950, table 2 indicates that this jet should form a fountain. Figures

12-14 are individual shadowgraphs of this flow taken at progressively higher

values of z/rQ

. It was impossible to superimpose shadowgraphs since there was

a great variability in the appearance of the flow field for photographs taken

at the same downstream position. Figure 12 shows that this flow has an

extremely long potential core region and distance between the nozzle and the

onset of vortex roll up (z/r^ a 3.4). Once turbulence is initiated, it has a

very similar appearance to that observed in the CgHg turbulent jets.

-45 -
Jet fluid (SF^ and air mixture) can be seen in regions well outside of

the jet in fig. 12. Indications for the presence of SF 5 can even be seen in

the secondary flow at locations well below the nozzle. The presence of SF^ in

the coflow is even more evident in fig. 13 which is taken further downstream.

The turbulent jet is still visible in the center, but the concentration variations in the secondary flow region are now quite intense. The source of jet

fluid in the secondary flow is shown to be due to a fountain effect in fig.

14. This shadowgraph clearly shows the jet fluid reaching a maximum height

and then falling back down through the secondary gas. The maximum downstream

distance attained by the turbulent jet is estimated to be z max/r

Q = 38.

Measurements have also been made for SF^/air flows having higher Re.

Figures 15 and 16 display shadowgraphs for Re = 7890 and 11860. For these two

flows, it has been possible to once again superimpose independent shadowgraphs

in order to get an overall view of the flow field. In contrast to the jet

with Re = 3950, no jet fluid is observed in the secondary flow and the turbulence structure has an appearance similar to that already described for other

jet/coflow gas pair combinations.

Modifications in the appearance of the shadowgraphs for the two higher Re

flows are similar to those already discussed in section 4.1.4 for the C^Hg/air

flows. Downstream turbulent mixing apparently becomes more intense with

smaller scales, but the radial growth behavior of the jet is independent of

Re (D * 10 r for z/r = 40). The visible lengths of the potential cores for

o

0

the variable Re jets, which can be seen in figures 15 and 16, decrease on

going from Re = 7890 to Re = 11860. The distance from the nozzle to the first

appearance of vortex roll up is estimated to be 2.8 r Q for Re » 7860 and 7.7

r Q

for Re = 11860.

-46-
4.1.6 Sulfur Hexafluoride Turbulent Jet into Helium (R = 37.0)

P

This flow has only been investigated for Re = 3950. As might be

expected, this jet behaves much like the low Re SF^/air flow in that it has a

fountain behavior in which the dense SFg gas falls through the very low

density secondary He flow. Figures 17-19 show shadowgraphs for the SF^/He

flow field taken at three different downstream positions. The top of the

fountain at z /r Q

=* 44 is obvious in fig. 19 and the presence of SF^ in the He

coflow can be seen in figs. 17 and 18.

This jet spreads very slowly in comparison with the other jet/coflow gas

pairs already discussed. The shadowgraphs indicate that while concentration

fluctuations are present in the flow field, the flow does not have the fine

structure characteristic of turbulent flow.

Experimentally, it has been found that the appearance of shadowgraphs

taken near the nozzle for this flow vary from photograph to photograph. For

this reason, only an estimate for the distance from the nozzle to the first

vortex roll up is possible. It is clear, however, that this distance is

considerably further downstream than those found for the other gas pairs

investigated.

4.1.7 Summary of Trends in Mixing 3ehavior with Changes in R and Re

Derived from Shadowgraphs

In drawing conclusions concerning mixing behavior from the shadowgraphs

shown in figs. 5-19, it must be kept in mind that the technique inherently

gives an integrated view of the flow field. The observed shadowgraph is not

- 47 -
necessarily correlated with the local structure of the jet flow field.

Variations in indices of refraction among different gases can also lead to

misinterpretation when comparing shadowgraphs for flows of different gas

pairs.

Estimation of jet spreading rates from visualization studies should be

especially susceptible to such effects since it has not yet been shown that

the apparent diameter of the flow field is correlated with the true diameter

of the jet. Even though full radial concentration profiles have not been made

for all of the jet/coflow combinations, such measurements for z/r G = 31.5 were

previously described for the CH^/air case [1,2] and were made for the SF

& air

flow at Re = 7930 during the course of this study. For both of these flows it

was found that the apparent widths of the jets from the time-resolved shadowgraphs were less than estimates of the full widths derived from the radial

concentration measurements. This is not surprising since the average concentration of jet gas in the outer radial regions of the flow is due to the

passage of relatively infrequent large scale structures having much higher

average concentrations [1,2]. An instantaneous photograph is unlikely to

capture such structures. Comparison of the photographs and radial concentration measurements do show that the visualized jet widths are roughly constant

fractions of the true widths of the jets based on concentration. These

results suggest that visualization can be used to make relative comparisons of

flows having different values of R . P

With the above considerations in mind, cautious generalizations based on

shadowgraphy may be made concerning the effects of jet/coflow density ratio

and jet Re on the mixing behavior of the axisymmetric turbulent jets invest 1-

-48-
gated in this study. For a given value of z/rQ and Re, the radial extent of

the jet is larger for lower values of R^. As an example of this behavior,

table 3 lists values of D obtained visually for z /

Q - 40 which do indeed

decrease as R increases. Details of the turbulent structures observed within

P the different jet flow fields indicate that, for a given Re, turbulent mixing

is more intense for jets with smaller values of R^. The potential core

lengths as well as the distance from the nozzle to the first vortex roll up

are longer for higher values of R

p

. Values of the distances for vortex roll

up are included in table 3 for comparison purposes. These conclusions

concerning the potential core regions of constant Re jets are consistent with

the notion that mixing is more efficient when R^ is smaller.

The shadowgraphs of the C 3 Hg/air and SF6 /air flows at variable Re

indicate that there is very little dependence on Re for the radial spreading

rates of these jets. Values of D at z/r Q = 40 listed in table 3 support this

conclusion. On the other hand, the jets are found to have more intense

turbulent structures and smaller scales at higher Re. Similarly, the

potential core lengths as well as the distance from the nozzle to the point of

initial vortex roll up decrease as the Re is increased (table 3).

4.2 Concentration Measurements Along the Centerlines of Variable Density Jets

The laser-induced Rayleigh light scattering technique described in

section 3.3 was utilized to record time-resolved jet fluid concentration

measurements along the centerlines of the turbulent flows listed in table 2.

As part of the calibration procedure used in this technique, it was necessary

to record the scattering intensities from the two pure gases under investiga-

-49-
tion. These measurements allow the Rayleigh scattering cross section ratio to

be determined for the gas pair. The observed ratios serve as internal checks

of the measurement technique. Table 4 lists observed average values of the

cross section ratios for the gas pairs investigated along with average deviations based on a large number of independent measurements. Also included are

the predicted cross section ratios, which were calculated using known values

of gas indices of refraction in the same manner as described in our earlier

work [1,2]. The variations in the observed value of the scattering intensity

ratio for a given gas pair is small and good agreement is found between the

experimentally determined values and those predicted.

Figure 20 shows an example of the type of time records which were

recorded. These data were recorded for the CH^/air flow at a downstream

distance of z/rQ = 31.5. The raw data in this figure have been scaled using

the scattering intensity calibrations for CH^ and air to give CH^ mole

fraction measurements. It is clear from this figure, that the effects of shot

noise on the measurement are minimal.

The calculational procedures described in section 3.3 were used to

analyze data records such as that shown in fig. 20 in order to calculate

values of average and rms concentration in terms of mole and mass tractions as

well as the third and fourth moments normalized to give skewness and kurtosis.

In the following sections, measurements of the centerline behaviors of

X , Y , X', Y', Sm and K~ are discussed. In order to isolate the effects m m m m m ^

of R on mixing behavior; wherever possible, comparisons are made tor jets

P having different values of R^, but for which the Re are the same. he et* ect

of Re on the centerline mixing behavior is then discussed by comparing the

variable Re jet results for the CgHg/air, CF^/air, and SFg./air t low 1-

-50-
4.2.1 11^ Dependence of Centerline Average Concentration

In this section, data are discussed for the He/air, CH^/air, C3 Hq/C02,

C 3H8 /air, CF4 /air, SF 6 /air, and SF 6

/He jet/coflow combinations at

Re = 3950-3960. The SF^/air and SFg/He flows at these Re can not be utilized

for direct comparisons due to the fountain-like behaviors which can be clearly

seen in figs. 14 and 19. Since the SF^/air jet for Re = 7890 has a "normal"

turbulent jet appearance (see fig. 15 and the discussion in section 4.1.5),

data for this jet/coflow combination are included for comparison purposes with

the lower Re data of the other gas pairs.

It is interesting to compare the mixing behaviors of the SF^/air and

SF^/He turbulent jets at low Re. Centerline concentration measurements for

these two jets are shown in fig. 21 in terms of Y /Y . Concentration data for

o m

turbulent jets when plotted in this manner usually have an approximately

linear dependence on z/r

Q

. These two jets definitely do not have such a

dependence. At z/r 0 “35, an abrupt change in the dependence of Y^/Y^ on z/rQ

occurs for both. One data point at z/rQ

> 35 is included in fig. 21 for the

SF^/air flow. Comparison with figs. 14 and 19 shows that the abrupt changes

in centerline concentration behavior for these jets are at roughly the same

downstream locations as the tops of the fountains visualized by shadowgraphy

Experimental data are plotted in fig. 22 as inverse mole fraction versus

nondimensionalized downstream distance for the six jet/coflow gas combinations

covering a range of R values from 0.14 to 5.1. The values of 1/X are

P m normalized by multiplying by the mole fraction at z = 0 (XQ

) equal to

one for all the cases investigated in this study. For values of z/r Q > 20,

- 51 -
each of the different flows appears to have a linear dependency on downstream

distance. The mole fraction data for different gas pairs have been fit to a

relation of the form of eq. (12) using a linear least squares fitting

procedure. The results of these fits are indicated in fig. 22 by straight

lines and listed in table 5 by reporting the calculated slopes (K

x ) and

virtual origins (z Q

) x which result. A clear dependence on R

p is evident from

these results. As R

p increases, values of increase raonotonically . Values

of ( z0

) x also increase with R

p

, going from negative values for R

p < 1 to

positive values for R > 1.

The same experimental data have also been utilized to calculate the

centerline concentration behavior of the jets in mass fraction terms. Figure

23 shows the results for the experimental data taken at z/r Q > 20. All of the

data, except that for the He/air case, have an apparent linear falloff with

downstream distance. There is a slight curvature in the He/air data which, as

will be discussed in section 5.1.1, has been attributed to the jet flow interacting with the walls of the enclosure. In order to test this hypothesis,

average concentration measurements along the centerline of a helium free jet

at the same Re have been made in the manner described in section 3.3. Figure

24 compares the results of these measurements with those for the enclosed

He/air jet in mass fraction terms. Note that the free jet measurements extend

further downstream than those for the enclosed jet. It is clear that the two

sets of data agree well in near field regions, but begin to diverge as z/r

o increases. Unlike the enclosed jet, the data for the free jet lie on a

straight line.

-52 -
Linear least squares fits have been calculated for the different

jet/coflow combinations shown in figs. 23 and 24. Data are reported in the

form of eq. (12) by listing values of slopes (K

y

) and virtual origins (z 0

) y

in

table 5. The dependencies of Ky and (z Q )y on R

p are reversed from those found

for data plotted in terms of X /X . r o m

The discussion in section 2.2 has indicated that z/r

£

, where r

£ is

defined by eq. (22) with p = p , should be a more suitable length parameter

I

00

for treating variable density flows than z/rQ

. Data for the six jets are

shown plotted as functions of z/r

£

in fig. 25. Data taken from fig. 23 have

been offset by -(z Q )y in order to force each least squares fit to pass

through z/r^ = 0. This treatment does not give a complete correlation of the

experimental results, but it does collapse them into a narrower range of

values. A dependence on R

p still remains. This point is addressed further in

the discussion section. Values of Kc which result from linear least squares

fits of Y /Y as a function of z/r are included in table 5.

o m e

4.2.2 Unmixedness Measurements Along the Centerlines of Variable Density Jets

The rms values for the concentration fluctuations along the centerlines

of the six jets discussed in section 4.2.1 were calculated in terms of mole

and mass fractions. The results of these calculations are reported in terms

of unmixedness values defined as X’/X and Y'/Y . m m mm

Values of X'/X mm are plotted as functions of z/r^ in fig. 26. All of the

o variable density flows have an initially rapid rise in unmixedness with downstream distance and then tend to change rather slowly as z/r

Q

is increased

- 53 -
further. There is a definite ordering of values of X’/X at a given value of m m

z/r^opwith R value. Flows with larger R values have higher X’/X values than

p m m

those with smaller jet to coflow density ratio. Another trend in the results

is evident from fig. 26. After the initial rapid increase in the unmixedness,

jets J with R < 1 have values of X’/X which continue to increase slowly as the

p m m

J measurement volume moves further downstream. The opposite is true for jets

with Rp > 1. Interestingly, the data for R

p = 1 (C3Hg/C02 flow) reach a value

of X’/X " 0.23 between z/r„ “ 10 and 20 and then remain essentially constant mm o

to the furthermost downstream position measured.

The appearance of the plot for unmixedness as a function of z/r Q changes

dramatically when mass fraction results are used. Figure 27 shows experimental data for the six jets plotted in terras of X^/Y^. All of the experimental data now seem to be approaching a common asymptote of 0.23. The

downstream distance required for a given jet to reach this asymptote depends

strongly on R

p

. The He/air jet has reached an asymptotic value at z/r

Q

=» 20

while the SF^/air jet has not attained such a value at z/r Q = 63. Jets having

intermediate values of R

p lie between these two extremes.

A close examination of the data for the He/air jet shown in fig. 27

reveals that values of X^/Y^ first appear to reach an asymptote and then

decrease slightly downstream. This behavior is only observed for the He/air

jet and is taken as further evidence for a wall effect on this flow. Tht9

conclusion is supported by the comparison of unmixedness measurements for the

free and enclosed He/air jets shown in fig. 28. At lower values of z/r 0

, the

results for both jets lie close together. However, as z/r

Q increases, they

diverge somewhat. The free jet has an asymptotic value of Y^/Y^ which is

similar to those for the other jets shown in fig. 27.

-54 -
It has been shown using approximate theoretical developments and by the

experimental results in section 4.2.1 that the similarity parameter z/r

£

is

more suitable for plotting variable density data of jet average centerline

concentration than is z/r

Q

. We are aware of no such treatment which gives the

proper similarity parameter for the behavior of centerline concentration

intensity. Despite this, it might be assumed that z/r

£ is also more suitable

for treating intensity data. Figure 29 shows the unmixedness results for the

six jets replotted as functions of z/r

£

. The resulting figure has some spread

in the data, but there is a remarkable collapse of experimental results onto a

common curve. Note that no virtual origins are included in this plot. If the

virtual origins for the concentration intensity behavior have a similar dependence on Rp as values of the average concentration, the collapse of the data

in fig. 29 would be improved markedly.

4.2.3 Density Effects on Skewness and Kurtosis Values Along the Jet Centerline

Figures 30 and 31 show the results of measurements of S and K,

respectively, as a function of z/r

Q for the six gas pairs. At positions close

to the nozzle (z/rQ

< 20) there are large variations in the values of S and

K. For this reason, only values for z/

Q > 20 are plotted. S and K results

in terms of either mole or mass fractions are found to have very nearly the

same values. This equivalence occurs because the third and fourth moments of

the concentration fluctuations are normalized by the appropriate values of the

rms to give S and K. Only the values in mole fraction terras are shown.

There are some clear trends evident in the results for the centerline

dependence of S on r For the jet with p q = Poo (C

3 Hg/C02 ) the values of S

-55-
appear nearly constant (S 13 -0.26) over the range 20 < z/r

Q < 65. All of the

different jet/coflow combinations seem to approach this same value of S with

increasing z/r Q

. However, jets with < 1 have values of S which increase

with downstream distance while jets with R > 1 have values which fall. For

P all downstream distances there is a clear ordering of S values with R^, the

He/air flow has the most negative values of S and the SF^/air jet the most

positive. Note that the values of S for the CF^/air and SF^/air flows are

initially positive, but decrease to negative values as z/r Q

is increased.

A similar, but not nearly as pronounced behavior, is found for values of

kurtosis as a function of downstream distance. Figure 31 shows the results of

the experimental measurements. By the time the flow fields reach z/r

Q = 63,

values of K for all of the gas combinations are approaching S = 3, the value

for a Gaussian distribution. There is a dependence on R^ such that jets

having smaller values of R

p have larger values of K. The two jets with R^

less than one have the largest deviations from K = 3. K values for these jets

decrease with increasing z/r

Q

. The negatively buoyant jets have only minor

departures from K = 3 throughout the flow region shown. There are apparently

only minor increases in K for these jets as the observation volume is translated from z/r

Q = 20 to 65.

4.2.4 Reynolds Number Effects on Centerline Concentration Falloff

Table 2 lists the jets for which variable Re measurements have been

made. These include three different Re measurements for the C^Hy/air and

SF^/air flows and two for the CF^/air case. Only the two higher Re cases will

be considered here for the SF^/air jet since, as already discussed, the mixing

- 56-
behavior for Re = 3950 is strongly modified by the strong negative buoyancy

present for this jet.

Figures 32-34 show the centerline concentration behaviors for each of the

variable Re flows. As before, the data is plotted as Y /Y versus z/r_. The

solid lines are results for linear least squares fits of the various sets of

experimental data. Results of these fits are summarized in table 5 where

values of the slopes (Ky) and virtual origins (z Q )y are listed for each flow.

For the C^Hg/air flow at Re = 11,860, the three data points at the lowest z/r0

values shown have not been used in the least squares fit.

For the C^Hg/air and CF^air jets the principal effect of increasing the

Re is a downstream shift in the location of the virtual origin. There are

only relatively minor variations in the slopes. Figure 35 shows the values of

the virtual origins plotted as a function of Re for these two jets. The data

are too limited to allow definite conclusions, but it appears that both flows

have the same dependence of (z

Q )y on

The SF^/air data for Re = 7890 and 11860 shown in fig. 34 lie very close

together. Linear least squares fits of the data give lines having slightly

different values of (Ky) and (z Q )y. It is not surprising that the observed Re

dependence for this jet/coflow combination differs from those found for the

two gas pairs with lower values of R

p

. As shown in section 4.2.2, the SF^/air

flow approaches an asymptotic value of Y^/?

m very slowly which indicates this

flow is still developing over the range of z/r Q

values where measurements have

been made.

-57 -
4.2.5 Reynolds Number Effects on Centerline Unmixedness Behavior

Values of Y'/Y mm as a function of z/r^ for the jet/coflow combinations o discussed in the last section are shown in figures 36-38. Once again, the

C^Hg/air and CF^/air jets seem to have similar dependencies on Re. For the

lowest Re studied, both jets have unmixedness values which attain an

asymptotic value well before the end of the flow region over which measurements are made. In each case the value is 3 0.23. As the Re is increased, the

growth of the concentration intensity along the centerline is altered

markedly. After initial rapid rises in unmixedness values with downstream

distance, the growth slows down and slow increases continue until the

asymptotic value is attained considerably further downstream. Figure 36 shows

that the variations in the centerline development behavior of unmixedness with

Re are largest when the Re are small, but that variations are observed over

the entire range of Re investigated.

The Re dependence of the centerline development of concentration

intensity for the SF^/air flows is shown in fig. 38. As was the case for the

centerline average concentration, there are only small differences between the

measurements for Re = 7890 and Re = 11860. As already noted in section 4.2.2,

the values of unmixedness of this jet/coflow gas combination grow slowly along

the jet centerline. For z/r

Q = 63, the values of Y^/Y

m have only increased

to 3 0.19. The experimental results for the two Re lie very close together.

If a trend is present, it is that the lower Re jet develops slightly more

rapidly with downstream distance than that for Re 3 11860. This conclusion is

the same as that for the two flows with smaller R values.

P

- 58 -
4.2.6 Reynolds Number Effects on Centerline Skewness and Kurtosis Values

The higher moments for the concentration fluctuations of the variable Re

flows have been utilized to generate values of S and K along the centerlines

of these flows. Increasing the Re results in no measureable variations of S

or K from those recorded at lower Re and shown in figs. 30 and 31,

respectively.

5. DISCUSSION

The two measurement techniques utilized in this work provide very

different types of information concerning the behavior of turbulent flows.

The shadowgraph experiment is sensitive to integrated variations of the index

of refraction and, at best, gives a qualitative view of the flow field. On

the other hand, the Rayleigh light scattering technique provides quantitative

measurements of concentration, but results are only reported for the concentration behavior along the centerlines of the jets studied. Despite the very

different nature of the two experiments, the measurements are found to complement each other and, when taken together, provide a clear indication of the

effects of global density differences on turbulent mixing.

In this section the experimental findings of this study are first

summarized for both the shadowgraph and concentration results. Centerline

concentration measurements are then compared with predictions based on

integral treatments of the equations for conservation of momentum and jet

fluid given in section 2.2. Experimental findings of similar studies available in the literature are then discussed. A simple theoretical framework is

-59-
(

proposed which provides qualitative predictions concerning the effects of

jet/coflow density ratio and Re on centerline average concentration fall off

and unmixedness behavior. The last part of this section considers the question of whether or not the flows investigated here can be considered to be

representative of free, purely momentum-driven, turbulent, axisymmetric jets.

5.1 Summary of Experimental Findings

5.1.1 Effects of Global Density Differences on Turbulent Mixing

That global density variations have an effect on turbulent mixing is

evident from an examination of the shadowgraphs shown in figs. 5-19. These

photographs indicate quite clearly that, for jets having the same Re, as the

value of Rp increases the initial radial growth rate of the jet as a function

of z/r

Q is slowed. This is also evident from the variations in visually

estimated values of D for the flow at z/r

Q = 40 listed in table 3.

Perhaps the most dramatic manifestation of density effects on the mixing

behavior of jet flows are the fountains observed for the slow Re flows of

SF^/air and SF^/He. Figures 12-14 and 17-19 show examples of shadowgraphs

recorded for these two combinations of jet and coflow gases. The concentration measurements shown in fig. 21 indicate that the tops of the fountains for

both of these jets occur at z/rQ* 35. These downstream distances agree very

well with the location of the tops of the fountains indicated by the shadowgraphs shown in figs. 14 and 19.

-60-
As suggested by the shadowgraphs of figs. 12 and 17, concentration

measurements made near the nozzle, but well outside of the jet, indicate the

presence of SF^ in the coflow. It is clear that the general characteristics

of these fountain flows, which can be inferred from their shadowgraphs, are

consistent with the more exact concentration measurements.

In section 4.2.1 it was noted that plots of Y

Q

/Ym

for different gas pairs

as functions of z/

£ are expected to give straight lines having a common

slope. Figure 25 shows the experimental data for six jet/coflow combinations

plotted in this manner. It is clear that this data treatment gives a partial

collapse of the experimental data. Values of the slopes, K which result

from least squares fits of the data are included in table 5. Both fig. 25 and

the K

c values show that there is a systematic variation in Kc values with .

The practical effect of plotting the experimental data in terms of z/r

£ instead of z/r0 is to multiply the slopes of the fits in terms of z/r 0 (Ky)

1 /2 by Rp in order to give K

c# Comparison of Kc values which result from

measurements on the different jets indicates that improved fits are obtained

if a higher power for R

p is used than 0.5. Table 6 compares values of slopes

which are found for plots of Y /Y versus z/(r ri R^’^). There is obviously a

o m u P much improved correlation of the data compared to that found for the values of

K

c listed in table 5. It should be emphasized that an exponent of 0.6 is

based purely on empirical observation. We are unaware of any theoretical

development which would predict such a dependence on R . P

Comparisons of the shadowgraphs and centerline measurements also yield

consistent conclusions regarding the development and growth of turbulent

-61 -
fluctuations in these variable density jets. The apparent lengthening of

potential cores as visualized by shadowgraphy are well correlated with the

downstream distance required for the rapid rise of centerline unmixedness in

mole fraction terms (see fig. 26). The distances required f ->r values of X'/X m m

to attain a relatively slowly changing value increase as R

p is increased.

Similarly, the lengths of the potential cores in the shadowgraphs are observed

to increase as the value of R

p becomes larger. This tendency for jets having

larger R

p values to require a longer downstream distance for the development

of turbulence is also evident from a comparison of the downstream distances

required by the various flows for initial vortex roll up which are listed in

table 3.

When the centerline unmixedness values are plotted in mass fraction

terms, the importance of R^ values to jet development becomes even more

evident. All of the different jet/coflow combinations studied approach an

asymptotic value of Y'/Y * 0.23 (see fig. 27). However, the downstream

m m distances required for the various flows to approach this value are strongly

dependent on the value of R . Jet/coflow combinations having R < 1 are found

P P to develop very rapidly and to only require a short downstream distance to

attain an asymptotic value of Y'/Y . On the other hand, gas combinations

m m which have R

p much greater than 1 develop very slowly with downstream

distance. In fact, at the largest downstream distance for which measurements

have been made (z/rQ = 63), the SF^/air jet has not yet attained its

asymptotic value of Y^/Y^. Note that the results quoted here for the SF^/air

flow are for Re = 7890 while the other jet/coflow combinations shown in fig.

27 are for Re 3960. As will be discussed shortly, higher Re are also found

to be associated with longer flow development lengths. Despite this, it is

-62 -
clear that the major source of the extremely slow development of this

turbulent jet is its large value of R^.

Further evidence for the importance of density to development in

turbulent flows appears in the plots of for the different flows as

functions of z /r

£ shown in fig. 29. The use of the effective radius concept

has been shown to allow a partial correlation of the average centerline

concentration behavior for jets having global density differences from their

surroundings. The same concept also appears to collapse data for the unmixedness behavior. It is interesting to note that if R were taken to the 0.6

P power as suggested by the average concentration results discussed above, there

would also be an improved collapse of the unmixedness data.

The behaviors of S and K values which have been measured for the various

jet/coflow combinations have clear dependencies on R^ as described in section

4.2.3. However, it is difficult to correlate these measurements with those of

average concentration or unmixedness behavior. The unmixedness measurements

indicate that the He/air flow develops asymptotic behavior at the shortest

downstream distance of any of the flows studied here. It might be expected

that this flow would also have small variations in S and K values. This is

not the case. Figures 30 and 31 show that this particular flow has the

largest variations in the values of S and K. It may well be that the variations in the values of S and K are not only due to turbulent fluctuations , but

may also be due to other processes such as molecular diffusion.

In section 4 it was noted that the mixing behavior for the He jet flowing

into a coflow of air is believed to have been influenced by the presence of

-63 -
the enclosure to a much greater degree than the other gas combinations

studied. There are no clear indications of such variations indicated by

comparison of the shadowgraphs of the enclosed and free helium jets shown in

figs. 5 and 6, respectively. Spreading rates with downstream distance and

turbulent structure appear to be the same in both cases. Nevertheless,

comparison of these two shadowgraphs with those for the other jet/coflow gas

combinations shows clearly that the He turbulent jet into air spreads radially

much more rapidly as a function of downstream distance than do the other

flows. For this reason, the He/air jet will interact with the enclosure walls

at a much shorter downstream distance than the other gas flows.

To further complicate the problem, the flow volume of air for the

enclosed He jet is limited and is not high enough to insure that recirculation

eddies will not develop. In fact, the calculated downstream distance for

complete depletion of the coflow listed in table 2 is z^/r Q= 56, which lies

within the range of downstream distances observed during the study. Such

recirculation eddies will surely modify the mixing behavior along the jet

centerline. This situation is exacerbated by the extremely rapid flow

development observed for the He turbulent jet. This rapid development assures

that the turbulent flow reaches its maximum entrainment capability after only

a short distance downstream.

The behavior of the centerline mass fraction unmixedness provides a clear

indication that modifications of mixing behavior for the He/air flow occur due

to wall and/or recirculation effects. Figure 27 shows that the He/air flow is

the only gas pair studied for which values of Y^/Y^ reach a maximum and then

decrease. Further evidence for the importance of these effects is found by

-64-
comparing the measurements of average concentration and mass fraction unmixedness made for the enclosed He/air jet with similar results for a free He jet

at an equivalent Re (see figs. 24 and 28). These results show that values

of Y /Y and Y'/Y for these two flows lie very close together at z/rn = 31.5,

but then have an increasing divergence as the flow fields move further downstream. For both average concentration and unmixedness, the values for the

enclosed jet fall below those of the free jet at equivalent downstream

distances.

It is significant that the values of Y

Q

/Ym

for the He free jet form a

straight line for values of z/r Q extending to 110. Values of Y^/Y^ for the

free jet (fig. 28) show some variation, which might be expected due to the

small number of concentration measurements (100) used in their calculation.

The observed values of Y'/Y indicate that the downstream unmixedness for the m m

free He jet is roughly constant at Y'/Y ® 0.26. This value is only slightly

m m higher than that of 0.23 which is estimated for the jets flowing within the

enclosure.

On the basis of the above discussion, we have chosen to use the free jet

measurements of the average concentration when making comparisons with similar

results for other jets. It has also been assumed that the drop in unmixedness

values observed for the enclosed He/air flow at downstream distances is an

artifact and that if recirculation and wall effects were absent, observed

values of Y'/Y mmm would remain constant as z/x n is increased,

o

- 65 -
5.1.2 Effects of Reynolds Number Variations on Turbulent Mixing

The majority of measurements which have been performed during the course

of this study were made for Re a 3960. The use of such low Re serves to limit

the volumes of gas flows and at the same time are more compatible with the

simple flow system used. Unfortunately, turbulence researchers often question

whether axisymmetric turbulent jets operating at such low Re are fully

turbulent. For this reason, a limited study of Re effects on the turbulent

mixing behavior of the C

3 Hg/air, CF4 /air and SFg/air flows has been made.

Shadowgraphs of the three CgHg/air (figs. 9-11) and the three SFg/air

(figs. 12-16) variable Re flows are described in sections 4.1.4 and 4.1.5,

respectively. Similar variations in the appearance of the shadowgraphs are

observed for each gas pair as the Re is increased. With the exception of the

lowest Re SFg/air case, the spreading rate of the jet for a particular gas

combination seems to be independent of Re. On the other hand, the turbulent

structures observed within the turbulent flow fields of the jets become more

intense and have smaller scales as the Re is increased. The development of

turbulence near the nozzle is also modified by changes in the flow Re.

Decreasing potential core lengths occur for increasing Re. Similar reductions

in the distance from the nozzle to initial vortex roll up with increasing Re

can be seen in table 3.

The concentration measurements along the centerline produce a different

view of Re effects on the turbulent mixing behavior. These experiments show

that increasing Re result in downstream shifts in the positions of tne virtual

origins which are derived from plots of Y^/Y^ versus z/r Q (see rig. 3 5 in*:

-66 -
table 5), but that the values of remain essentially constant. Values of

unmixedness as a function of downstream distance still approach a common

asymptote, but as the Re becomes larger the flow distance required to achieve

asymptotic behavior is increased dramatically.

Measurements made for constant Re have indicated that increasing the

value of Rp leads to a slower development of the jet which is reflected in

slower initial development of the turbulence near the jet nozzle as well as a

slower approach to an asymptotic value of Y^/Y^. Holding R^ constant and

increasing the Re leads to a different turbulence behavior. Increasing the Re

results in a more rapid initial development of the jet near the nozzle as

reflected in the observed shorter potential core lengths. At the same time,

the growth of the jet towards an asymptotic unmixedness value is slowed and

the locations of the virtual origins for plots of the centerline concentration

data are shifted downstream.

5.2 Comparison of Average Centerline Behavior with

Integral Analysis Predictions

The results of several approximate solutions of the integral equations

for variable density turbulent flows are given in sections 2.2 and 2.3. In

this section the experimental findings of this study will be compared with

predictions of mixing behavior based on these solutions.

Perhaps the most important conclusion that can be derived from this type

of analysis is that plots of Y /Y versus z/r for data recorded at suffio in £

ciently large downstream distances should have linear behaviors with slopes

which are independent of R^. It has already been shown that the use of the

-67 -
effective radius concept, where r_ is defined by eq. (22) with p = P^, gives

c f a partial correlation of the centerline concentration falloff data for the

variable density jet/coflow combinations investigated. However, it has also

been shown that such a correlation is improved markedly if an exponent of 0.6

is employed for R

p instead of that of 0.5 derived from integral analysis.

Deviations from the predicted dependence of K

c on R

p may be a characteristic of free jets or it may be due to one or all of several differences which

exist between the flows studied here and truly free, momentum-driven variable

density jets at far downstream distances. It must be remembered that the

integral analysis used to derive the dependence of the centerline concentration falloff on R

p assumes that the centerline density and the density of the

surrounding gas are very nearly the same. The measurements reported in this

study have been made at relatively short downstream distances, 20 < z/rQ

< 63,

for which this condition is not met. The ordering and relative magnitudes of

the variations in the slopes of the linear least squares fits of the data to

eq. (24) are consistent with such an interpretation.

The use of a coflowing surrounding gas may also modify the observed

slopes. Table 2 shows that as the primary flow to secondary flow density

ratio increases from that for He/air to that for SF^/He, the relative velocity

ratio of the coflow and jet, m = U /U , increases. In section 2.3 it was

oo o pointed out that increasing values of m result in less rapid spreading rates

for a turbulent jet with increasing downstream distance and to a less rapid

falloff of centerline concentration. Such a modification from free jet

behavior Is consistent with the observed variations of K

c with changes in .

However, it must be remembered that m values are relatively small even in case

of the SF^/air flow.

-68 -
The predicted dependence of K

c on R

p was derived assuming uniform

profiles at the exit of the jet nozzle. It is possible that the exit profiles

actually vary somewhat for the different gas flows. If this is the case, the

values of M

q

will also vary and the slopes of the least squares fits will not

have the expected dependence on R^.

Equation (34) has been derived in section 2.2 using an analysis similar

to one given by Thring and Newby [18]. This equation predicts a variation in

the locations of virtual origins with R

p for mass fraction data plotted in the

form of eq. (24). By choosing an appropriate constant value for the term

^122^^22 ” *422^12 (here — 0.5) and setting = 0.114, eq. (34) leads

to the predictions of 1/C falloff as a function of z/rQ

shown in fig. 39 for

the values of R

p used in the experimental study. Comparison of figs. 23 and

39 shows that there is good qualitative agreement between the experimental

findings and the calculated variations of Y^/Y^ with z/rQ

. Of course, quantitative agreement is not found, since, as we have already noted, the use of the

effective radius concept does not yield a complete collapse of slopes for the

experimental data. On the other hand, the magnitudes and signs of the virtual

origins are predicted quite well by this approximate treatment of the integral

equations.

The analysis suggested by Rodi [5,23], which is discussed in section 2.2,

can also be used to generate predicted behaviors for C^ as a function of z/r0

for variable density flows. The results of such calculations for the values

of Rp employed are shown in fig. 40 where values of are plotted as

functions of z/r Q

. These calculations require the input of an empirical value

for the ratio of the concentration to velocity radial half widths, R (see eq

-69 -
40). The value recommended by Chen and Rodi [5], R = 1.17, has been used.

Note that the analysis assumes that R is constant throughout the flow field.

Figure 41 shows the results of linear least squares fits of 1/C versus

m

z/rQ for the results of fig. 40. The fit for R^ » 1 gives a slope of

Kc » 0.92. This value is smaller than that of Kc " 0.114 which has been

measured experimentally in this study. For this reason, the calculated curves

for the different flows are in poor quantitative agreement with experimental

results. Increasing the value of R leads to a higher slope, but a systematic

variation has not been attempted and only the results for R - 1.17 are

reported.

The lines shown in fig. 41 are found to collapse to a single line if the

calculated values of 1/C^ are plotted as functions of z/r^. The analysis of

Rodi therefore gives the same dependence of Kc on as that deduced from

Thring and Newby [18]. This is not too surprising since both analyses use the

conservation equations for momentum and jet fluid.

Rodi's formulation seems promising as a useful means for predicting Jet

centerline concentration behavior. It will be interesting to apply it to a

set of experimental data where velocity and concentration measurements have

both been made throughout the flow field. Such measurements will allow a much

better evaluation of whether or not the assumption of a constant value for R

is correct and also allow an experimental determination of its value.

An integral analysis is given in section 2.3 which generates an approximate expression for the height of the fountain ( z max/r Q ) formed by a negative-

- 70-
ly buoyant jet. These heights have been calculated using eq. (58) for the

flows studied which have values of R

p greater than one. The value of the jet

spreading angle required for use in eq. (58) was chosen to be that recommended

by Chen and Rodi [5] (0 = 6.3°). The empirical expression of Turner [24], eq.

(59), has also been used to calculate zmax . Results for both expressions are

summarized in table 7.

Both equations predict values of z max which are in good agreement with

the estimates of zmax/ ro = 38 for the low Re SF^/air and zmax/rQ = 44 for the

SF^/He flows from the shadowgraphs. As expected, the other flows have values

of zmax which lie outside of the range of experimentally observed values. The

experimental results are too limited to allow a determination of which

equation ((58) or (59)) describes the density dependence of zmax

most

accurately

5.3 Comparison of Experimental Results with Past Work

Experimental analysis of the concentration fields of axisymmetric

turbulent jets has historically been and remains the focus of a great deal of

interest in the literature. In this section the results of the current study

will be compared with and discussed in relation to this earlier work.

Concentration measurements in turbulent axisymmetric jets fall into two

general classes—cases where the jet and the surrounding gas have the same

density and cases where these densities vary (i.e., p # p ). Constant

o °° density flows (usually air jets into air) are marked in some manner in order

to allow concentration variations to be followed. Markers have included small

- 71 -
particles, minor temperature variations, and trace concentrations of a second

gas. In cases where temperature or chemical markers are utilized, the variations in density are small enough to assume constant density mixing. Variable

density cases have been studied for flows having variable temperature,

velocity (compressibility effects), and chemical composition. Density variations due to compressibility will not be included in this discussion.

5.3.1 Average Centerline Concentration Measurements

There are a large number of experiments which have measured average jet

fluid concentration along the centerlines of axisymmetric jets. Table 8

summarizes the results of many of these studies along with parameters for the

various flows. The centerline concentration falloff is characterized in terms

of eq. (24) where C , as defined by eq . (25), can represent either chemical m composition or temperature. Values of K

c and (z Q ) c are listed where available. In some studies, marked with an *, these values have been generated

using results taken from figures in the original papers. It has also been

necessary for a few of these data sets to convert from mole fraction measurements to mass fraction results. No corrections for turbulent fluctuations

were utilized in these conversions. Resulting values of are expected to be

accurate to within a few percent [10].

There are several experimental investigations which have not been

included in table 8 because their results are not In formats which allow

values of Kc and (z 0 ) c to be readily calculated. Among these are studies by

Keagy and Ellis [43], Uberoi and Gariy [44], and Shaugnessy and Morton [45].

Keagy and Ellis [43] utilized a quantitative Schlieren method to measure the

- 72 -
falloff of centerline concentration for a He jet into air. The Re for the

flow was not reported, but was apparently quite high. For results plotted

as 1/X versus z/r_ they find a straight line having a slope of K„ = 0.053. m ° x This value is in good agreement with that found for the same gas pair in this

study, K

x

= 0.048 (see table 5). Uberoi and Garby [44] have investigated

flows of cooled air into room air. Jets having initial temperatures as low as

100K were used. The experimental results are sparse, but they do indicate

that the normalized average temperature, expressed in the form of C , falls m

off much more slowly along the centerline for the cooler (more dense) jets.

This finding is consistent with the results of this work. Shaughnessy and

Morton [45] have investigated mixing in high Re flows of air into air using

nephelometry . Radial profiles are presented at several downstream positions.

Unfortunately, the centerline behavior is never explicitly given.

The measurements summarized in table 8 have been made using a wide range

of experimental techniques and for various flow velocities, nozzle diameters,

and nozzle shapes. For this reason, the initial jet velocity profiles and

turbulence levels are expected to vary markedly from study to study. Given

these variations in measurement techniques and initial flow conditions, values

of K

c for the different studies summarized in table 8 agree to a remarkable

degree. An average of all of the K

c values listed in the table yields

K

c = 0.105 with an average deviation of ± 10%. K

c values for the variable

density flows investigated in this study (table 5) lie well within this range

of values.

It is very difficult to discern any dependence of K on R or Re from the

^ P results listed in table 8. Even though the variations in the values of K„ are

- 73 -
relatively small from study to study, they are large enough to mask small

dependencies on these parameters. If any trend is evident in the data, it is

that the jets having the largest Re seem to have lower values of K

c

. it must

be emphasized that this is a highly tentative conclusion based on the results

summarized in table 8. Such a dependence has not been identified in the

limited investigation of Re effects in the current study.

There are very few studies where variable density flows have been

systematically investigated. From the results listed in table 8 it is impossible to verify whether K has the dependence on R which has been deduced c p from this study. The value of K

c listed for the results of Wilson and

Danckwerts [20] is actually a fit of several sets of data covering a range

of R values from 1.04 to 1.68. The concept of the effective radius [18] was

} used for the fit, but no allowance was made for shifts in virtual origins and

a value of 0 was assumed for each set of results. This study [20] supports

the validity of Thring and Newby's concept, but the data are not recorded over

a wide enough range of R^ to determine an accurate value of the exponent for

r Because of the treatment of the data, any dependence of ( Z Q

) C on R^ is

masked.

Scheffer and Dibble [42] have reported a detailed study of a C>Hg/air

flow. In this work some measurements of a CH^/air flow were also made. When

both sets of data are plotted in the form of eq . (24), the CH^/air measurements lie above those for the CgHg/air flow. This experimental result for

their high Re flows is consistent with a stronger dependence of the effective

radius on R^ than indicated by the half power included in eq . (22).

- 74 -
It is impossible to reach any conclusions based on the tabulated results

concerning the dependence of (z

Q

) c on Rp or Re* The clear dependence of (z

Q

) c

on R

p identified in this study will be masked by Re variations and the different initial conditions among the summarized investigations. Similarly, variations in (z ) c with Re seen in this investigation (see table 5 and fig. 35)

cannot be discerned from results listed in table 8. Ebrahimi and Kleine [15]

do indicate that values of the axial concentration half lengths shift downstream with increasing Re (see eq. (15) and discussion in section 2.1). This

is the only evidence of which we are aware that values of (z Q

) c increase with

increasing Re. However, as discussed in section 2.1, similar shifts with Re

have been identified in velocity measurements.

It must be concluded that the data available for average jet fluid

concentration along the centerlines of axisymmetric jets all yield roughly the

same value for slopes of data plotted as 1/C versus z/r and that there may

in £

be a very weak dependence of K

c values on R^. However, the data are not

sufficiently reproducible from experiment to experiment to confirm the relatively small effects of R

p on K, values or the shifts in values of (z

Q

) c with

changes in R

p or Re which have been characterized in this study. The use of a

single flow system and flow diagnostic for a number of jet/coflow gas pairs

covering a wide range of density ratios is the key to the identification of

these trends in the current work.

5.3.2 Unmixedness Centerline Measurements

The measurement of unmixedness in turbulent flows is considerably more

difficult than the determination of average concentration values. For this

- 75 -
reason, until recently the number of unmixedness measurements has been very

limited. However, the development of new diagnostics in the past few years

has led to many such measurements. As will be shown, findings among different

studies have not agreed well.

Table 9 lists various literature sources for centerline unmixedness

measurements of jet fluid in turbulent jets. Included are tracer measurements

in constant density flows as well as variable density flows having varying

temperature or chemical composition.

In the past, many authors [10,15,16,39,41,48,50] have shown plots which

compare their centerline unmixedness measurements with those of several

different workers. Such plots are extremely useful for comparing different

sets of data and do give a good indication of the wide variation in concentration intensity behavior which has been observed. Unfortunately, the number of

studies available is becoming so large that several plots would be required to

compare our results for centerline unmixedness shown in figs. 27-29 and 36-38

with available literature findings. Instead, the various studies are compared

in table 9 by listing some general characteristics for each flow and some

parameters for the behavior of centerline unmixedness.

All of the studies discussed have found that there is a rapid increase in

unmixedness with downstream distance beyond the potential core region. After

this rapid increase, the unmixedness begins to level off. At downstream

distances beyond the region of rapid rise, two different behaviors for

unmixedness growth have been reported. Some workers find that the values

of C'/C mm attain an asymptotic value while others claim that no such asymptote

- 76 -
is reached and unmixedness values continue to increase slowly over the entire

range of z/rQ

values investigated. Table 9 lists whether or not asymptotic

values are achieved and, if so, the approximate downstream location where

asymptotic behavior is first reached. Values of the maximum unmixedness

observed along the centerline are also included in the table. For those jets

which have an asymptotic behavior this value is the observed asymptote while

it is simply the largest value of C’/C observed for jets which do not. The mm

final column of table 9 lists the maximum downstream distance in jet radii for

which measurements have been reported.

A few notes concerning the studies listed in table 9 are appropriate.

Corrsin and Uberoi [6] report unmixedness values for a heated jet which reach

a maximum near the nozzle and then fall off with downstream distance. This is

the only study listed in table 9 which has such a behavior and is most likely

due to an experimental artifact. Some of these studies [9,15,37] have shown

maxima near the potential core region of the flow field followed by development of unmixedness much like that observed in this and other studies. We

feel that these observations may also be due to experimental artifacts. The

most likely source is a lack of sufficient spatial resolution for measurements

taken near the jet nozzle where the unmixedness is changing rapidly with

downstream distance.

Values for the maximum C^/C

m observed generally fall in the range 0.18 to

0.25. Measurements using hot-wire anemometry techniques by Way and Libby [47]

and McQuaid and Wright [37] lie well outside of this range. It may be that

the hot-wire measurement technique is not able to compensate properly for the

high turbulence levels found in this type of jet and the high readings

- 77 -
observed are the result. Excluding these two studies, there is still a wide

variation in measured results. It would be highly speculative on our part to

comment on possible sources for these variations. There is no reason to

suspect that one experimental technique is superior to another. On the other

hand, it is difficult to accept that such large variations in C^/Cmare

indicative of the actual behavior of these flows. Clearly, more extensive

studies will be necessary to unravel the true source of these variations.

It is also perplexing that some jets seem to attain asymptotic values of

unmixedness while others do not. Some of these studies have extended measurements to quite large downstream distances. The data are not extensive enough

to make definite statements, but as will be discussed shortly, most of the

data are not inconsistent with the variations in centerline unmixedness with

Rp and Re which have been identified in this study.

Table 10 summarizes the centerline unmixedness results of this study in

the same manner as the literature results listed in table 9. As already noted

in section 4.2.2 based on plots of the results, the measurements show clearly

that asymptotic values of unmixedness are attained for the low Re (* 4000)

jets. The downstream distance required for a given jet/coflow pair to reach

an asymptotic value increases rapidly with R . Higher Re also lead to slower

P centerline growth of C’/C values.

m m

Table 9 includes the results of an experiment on a CH^/air flow which we

have previously described [1,2]. Comparison with table 10 shows that different conclusions have been reached concerning the existence of an unmixedness

asymptote despite the fact that the two sets of data are in close quantitative

-78 -
agreement. The signal to noise levels in the earlier experiment were lower

and the resulting measurements of unmixedness are more scattered than are the

current results shown in fig. 27. Furthermore, a small misalignment of the

jet centerline and the light scattering observation volume has been corrected

since the original work was completed. These two differences are believed to

be sufficient to explain the conflicting conclusions for studies made in the

same flow.

Most of the studies in table 9 which show an asymptotic behavior for

unmixedness either have relatively low Re or have values of < 1. This

observation is consistent with the conclusions of this study concerning the

dependence of jet centerline concentration fluctuation behavior on R

p and

Re. Some experiments clearly do not conform to these criteria. For instance,

the flow conditions of the free CH^/air flow of Birch et al. [10] indicate

that this jet should attain an asymptotic value of unmixedness. Their data

indicate quite clearly a continual increase along the entire centerline length

for which measurements were made. The data of Niwa et al. [41] for the He/air

flow were taken for a jet having an extremely low Re and it is not clear that

this jet will be turbulent. Two recent experiments on water jets [16,48] at

Re = 5000 are included in table 9 which give different conclusions concerning

the existence of an unmixedness asymptote. Both of these studies utilized

absorption measurements of dye solution for jet fluid determination. Dahm

[16] used an innovative external line measurement technique while Nakamura et

al. [49] employed an internal fiber optic probe. Given the similarity of the

two flows it is difficult to understand the source of the different results of

the two studies.

- 79 -
Indirect evidence for the dependence of centerline unmixedness

development on Re and the existence of an asymptotic value of unmixedness is

available in studies of jet entrainment. It is expected that centerline

unmixedness will be correlated in some manner with the entrainment behavior of

the jet. On this basis, one would predict that more surrounding fluid would

be entrained into these flows near the nozzle for lower Re flows. Evidence

for this can be found in the studies of Ricou and Spalding [21] and Hill

[22]. Hill's local entrainment measurements show clearly that a finite downstream distance is required for a jet to attain an asymptotic entrainment

rate. At positions near the nozzle, the jet was found to have a lower

entrainment rate which grew with downstream distance. These findings suggest

that there is indeed a correlation between the entrainment behavior and

centerline unmixedness. Ricou and Spalding have shown that the entrainment of

ambient fluid is dependent on Re for positions near the nozzle. Flows having

Re on the order of 10^ entrained fluid much more efficiently than jets at

higher Re. This observation is In agreement with the hypothesis, based on

centerline unmixedness observations, that the entrainment capability for lower

Re flows will increase more rapidly as a function of downstream distance.

It has been pointed out by several groups [1,2,9,10,15] in the past that

when inverse values of C' are plotted as a function of z/r^ a linear growth Is m ° found after the initial jet development region. This growth behavior can be

expressed mathematically with an expression similar to eq . (12),

z - <«;>e

ro

( 69 )

-80-
where the primes indicate a slope (K^) and virtual origin (z^)^ f°r plots of

inverse C’. Dividing eq. (12) by eq. (69) yields an expression for unmixedm

ness,

C'm

Cm

2 ~ (z

o

)

z - (z

o (70)

This equation predicts that at sufficiently large downstream distances the

unmixedness will attain an asymptotic value of K /K^. The downstream distance

required for an approach to this value is determined by the difference between

the locations of the virtual origins for the average and rms plots. If (z Q

) c

is nearly the same as (z') , the jet will attain an asymptotic value of

o c unmixedness at relatively short downstream distances. On the other hand,

large differences in the values of virtual origins will insure that an

unmixedness asymptote will occur only at large z/r

Q

. Equation (70) is useful

because it treats the data in a manner which allows easy prediction of average

and rms concentration values for a given flow at any value of z/r

Q beyond the

potential core and also allows the approach of the jet toward an unmixedness

asymptote to be predicted.

The existence of different virtual origins for plots of C /C and C /

C

f o m o m versus z/r

Q implies that these jets can not be truly fully developed over the

axial downstream distances for which the least squares fits have been made.

This requires further that K

c and K/ cannot be true constants. However, these

values are found to vary so slowly with downstream distance that linear fits

of the data remain quite accurate. For this reason, their use is appropriate.

-81 -
5.3.3 Centerline Measurements of Skewness and Kurtosis

Very few measurements of the higher moments of jet fluid concentration

fluctuations along the centerlines of axisymmetric turbulent jets are available in the literature. Ebrahimi et al [50] have measured both skewness and

flatness values for a constant density air/air jet. Values of variable

density flows have been previously described for CH^/air [1,2], natural

gas/air [10], and heated air/air [39] flows. The heated air jet had a density

ratio of R = 0.51 which is very close to the value of 0.55 for the methane

P jet and 0.60 for the natural gas jet.

Figure 30 shows that the constant density C^Hg/CO-, gas pair gives skewness values which are nearly constant over the range of z/r

Q shown. There is

a gradual drop from S

°* -0.25 at z/r

Q = 20 to S

a -0.30 at z/r

Q = 62. Values

of K (fig. 31) over the same range of downstream distances remain constant at

the Gaussian value of 3. The S and K results reported by Ebrahimi et al [50]

show a very different behavior. Values of S increase from * -0.3 at z/r Q

* 20

to * 0.6 at z /r = 100. On the other hand, values of K remain nearly constant

at roughly 4.3 over the same range of z/r

Q

.

Our previous measurements of S and K [1,2] along the centerline of the

CH^/air jet are in excellent agreement with results shown in figs. 30 and

31. The value of S for this turbulent jet is a -0.6 at z/r = 20 and

increases with downstream position to " -0.3 at z/r

Q

= 62. Values of K ire

found to be * 3.4 at z/r

Q = 20 and to fall to K > 3.0 at z/r

o 62. Birch et

al [10] also find negative values of S which remain relatively constant it

S * -0.3 over the range of downstream distances investigated while Lockwood

-82 -
and Moneib [39] report a fall in S values from ** -0.05 to w -0.3 on going from

z/r

Q = 20 to 100. The centerline behavior of K values measured by Birch et al

are in excellent agreement with those shown in fig. 31. Again the results of

Lockwood and Moneib differ slightly, showing a relatively constant value

of K " 2.9 from z/r

Q = 10 to 100.

Based on these limited comparisons, it is concluded that the results of

Ebrahimi et al [50] are inconsistent with other published measurements and

that our measurements of S and K are in good agreement with the other two

experiments which have reported axial measurements of the higher moments for

concentration fluctuations. There is no independent experimental verification

for the dependencies of S and K on R

p shown in figs. 30 and 31, but, based on

the agreement between experiments having a 0.55, it is felt that the

measurements for the other jet/coflow combinations provide a true indication

of variations in S and K with R . P

5.4 Centerline Falloff of Average Concentration and Growth of Unmixedness

Theoretical Considerations

It has been shown for the low Re jets investigated in this study that the

average centerline concentrations and intensities are dependent on both R

p and

Re. Concentration data for constant Re jets plotted as Y /Y versus z/r„ give

straight lines with slopes which decrease and values of (z Q )y which move

upstream as R increases. If R is fixed, increasing the Re gives results

P P which lie on lines having the same slope as at lower Re , but for which the

virtual origins have moved downstream. Values of Y’/Y initially increase

with downstream distance and then attain an asymptotic value. The value

of (Y'/Y ) is invariant with R or Re . Increasing values of either R or m m asym p &

p

-83 -
Re increase the downstream distance required for unmixedness to reach this

value. Comparisons with available literature results are inconclusive, but

suggest that similar dependencies exist for axisymmetric jets of considerably

higher Re. The obvious question is: Is there a theoretical framework with

which to explain these empirical observations? In this section, a very simple

analysis will be proposed which seems to provide such a qualitative framework.

Further experimental and theoretical effort will be necessary to demonstrate

whether this interpretation is correct and, if so, to quantify the

predictions.

The "classical” picture of turbulence [51-54] is often discussed in terms

of the eddies which are characteristic of turbulent flow. For sufficiently

high Re, theoretical analysis indicates that eddy behavior falls into three

distinct size ranges—large eddies which contain most of the turbulent energy

of the flow and whose size is determined by the physical dimensions of the

flow, an intermediate range of eddy sizes which are isotropic in space and for

which the dynamic behavior is determined solely by the energy transfer rate

(e) from larger to smaller eddies, and the smallest eddies of the flow which

are isotropic and energy dissipating. These three types of eddies are often

described as being in the energy containing, the inertial, or the dissipating

ranges. Here the primary interest is the largest and smallest eddies. The

sizes of these two types of eddies are characterized by length scales which

are referred to as the integral (L), and the Kolmogorov (n) length scales,

respectively.

Values of L are extremely difficult to measure experimentally. Estimates

are often obtained by recording the autocorrelation function, R(x), of the

- 84 -
velocity fluctuations at a point in the flow. The Eulerian integral time

scale (T) can be obtained from R(t). For instance, if R(t) has an exponential

dependence on t, it can be expressed as

Utilizing Taylor's approximation, the value of L can be obtained from T as

For a coflowing jet with m = 0.33, Antonia and Bilger [55] measured L values

which fell in the 0.73 ( r i/2^u to 0.65 ( r i/2^u ranSe over the downstream

distances from z/r

Q = 80 to 260. These authors noted that for free jets

Wyganski and Feildler [8] reported a value of L = 0.45 while Corrsin and

Uberoi [56] measured a value of 0.82. On the basis of these results, it is

concluded that values of L for free and coflowing jets are similar and lie in

the range from 0.4 (r

1//2 ) u to 0.8 (r 1 / 2 ) u

.

The Kolmogorov scale can be obtained from dimensional analysis as

(71)

L = UT (72)

(73)

where

c .m

L

3

(74)

-85 -
Flow fields are generally considered to be fully turbulent when local Re

are sufficiently high to insure the existence of all three eddy size ranges.

Bradshaw [51] reports that this condition is satisfied when L/n > 100.

In general, the flow velocities utilized in this study are not expected

to be high enough to result in fully developed turbulent flow behavior. This

point has been discussed in detail by Drake et al [58]. In their report,

these authors have made detailed comparisons of their results for a hydrogen

turbulent jet diffusion flame with our earlier measurements of an isothermal

CH^/air flow [1,2]. As part of this comparison, they estimated values of

n = 0.01 cm and L = 0.71 cm for the length scales of our jet at z/r Q

=* 35 and

r/

(

r

i

/

2 ) c = Similar estimates would be expected for the other gas flows

investigated in the present work. Based on the criterion of Bradshaw [51],

these jets are not fully turbulent. Drake et al reached the same conclusion

based on a similar criterion given by Saffman [57].

It can be shown that one way to meet the criteria of fully developed flow

for a given gas pair is to increase the Re of the axisymmetric jets [54]. We

have already discussed how values of C^/C

ffl are either constant or slowly

varying after an initial development region near the jet nozzle. A similar

behavior is also expected and has been experimentally observed for the centerline velocity fluctuation intensity, Using eq . (23), it Is easy to see

that is proportional to which is in turn proportional to U

Q

. These

proportionalities along with eqs. (73) and (74) lead to the conclusion that n

is proportional to (1/U )^^ since values of L and local v are expected to be

independent of Re. Since the value of Re is proportional to U

Q

(see eq. 67),

the local value of n will depend on (Re)

- ^4

. Thus increasing Re leads to

smaller values of n and increasing values for L/n.

-86 -
There is an apparent contradiction in the experimental findings of this

study and the discussion just given concerning fully developed turbulent

flow. It has been indicated that the turbulent structure of the jets investigated in this study are not fully developed in the classical sense. Yet these

jets have a centerline average concentration falloff which approximates that

of a flow having a self-similar behavior and are in quantitative agreement

with experimental results for jets having much higher Re. Perhaps more

disconcerting, at least at first glance, is the observation that the low Re

jets investigated in this study achieve an asymptotic value of unmixedness at

much shorter downstream distances than higher Re flows. Achievement of such

an asymptote may be considered as a criterion for a fully developed flow

field. Using this criterion, the lower Re flows apparently become fully

turbulent at shorter downstream distances than flows at higher Re.

This contradiction between experiment and theory disappears if two simple

and reasonable assumptions are made:

1. The time-averaged mixing behavior as well as the values of unmixedness

are determined primarily by large scale structures; and

2. A finite time and downstream distance are required for the entire

length scale structure of the turbulent flow to come into a pseudoequilibrium. The flow length required increases as the turbulent

structure extends to smaller scales.

These two assumptions not only remove the contradiction between theory and

experiment, but also lead to predictions for the dependencies of unmixedness

-87 -
centerline behavior on and Re which are in qualitative agreement with

experimental observations.

Consider initially the effects of varying R^ on the values of

and Y^/Y^ which are observed. Integral analysis has already been employed

to show that measurements of centerline concentration for jets having different R„ should fall on a common curve when 1/C values are plotted as functions

P m

r

of z/r^. The use of the effective radius concept allows the comparison of

jets having different initial momentum values due to molecular weight differences. Sunvala et al [19] and Chen and Rodi [5] have argued that the jet

angle will be constant. Since the large scale structure should be nearly

independent of v and the Re is constant for each jet, it is required that the

large scale structure be dependent on R^. It is expected that in real space,

the eddies will be shortened in the axial direction relative to the constant

density case for positively buoyant jets while the opposite will be true for

jets J with R >1.

P

The shadowgraphs of the variable density jets give qualitative support

for these conclusions. Increases in the lengths of turbulent structures with

increasing R^ are apparent when jets having the same Re are compared. The

structures of flows having higher jet/coflow density ratios are longer axially

than flows having the same Re but lower density ratios simply because the

range of z/r

£ viewed is much shorter.

It is impossible to reach conclusions concerning the spreading rate

behavior of the variable density flows from the shadowgraphs. The jets having

lower values of R^ appear to spread faster. This observation may be an ill u

-88-
sion due to shifts in the virtual origins for the spreading angle. A determination of the actual spreading rate behavior must await quantitative

measurements of concentration in the radial directions of the various flows.

The conclusion that the axial turbulent structure for constant Re jets is

determined by a length scale of r

£ along with assumption 2. above concerning

the finite length required for growth of the unmixedness to an asymptotic

value requires that the growth curves for centerline unmixedness for variable

density cases fall on the same curve when plotted in terms of z/r

£

. Figure 29

shows that this prediction is closely matched by experimental measurements

covering a wide range of R

p values. The assumption of a common turbulence

structure also predicts a common value of (Y^/Y

m

) ag

for jets having varying Rp. This conclusion also agrees with experimental observation.

The variation in the downstream distance required for a turbulent jet to

attain (Y'/Y ) with Re also follows naturally from assumptions 1. and 2. m m asym y r

The large scale structure of the flow is expected to be nearly invariant with

Re as long as the Re is high enough to insure the development of the large

scale turbulent eddies. It has already been shown that as the Re is

increased, values of 9 are expected to decrease as Re^^ with the result that

the turbulent eddies extend to smaller sizes. The effect can be clearly seen

in the shadowgraphs of the C^Hg/air and SFg/air flows at different Re where

the apparent observed turbulent structure becomes finer as the Re is

increased. These observations are almost certainly due to the reduction

of n at higher Re.

-89-
The centerline concentration falloff has been shown to have a hyperbolic

dependence on z/r

Q which, except for shifts in the virtual origin, is nearly

independent of Re for a given jet/coflow gas pair. This observation is

consistent with the assumption that the mixing behavior is determined

primarily by the large scale structure of the flow field which is independent

of Re. Assumption 2. leads to the prediction that the rate of growth to and

the downstream distance required for the attainment of f Y ’ /

1 will ^ m m'asym

increase with Re. Figures 36-38 show that experimental measurements for

C^Hg/air, CF^/air, and SFg/air flows all conform to these predictions. The

observed downstream shift in the growth of unmixedness is also sufficient to

explain the observed downstream shift in virtual origins for a given flow with

increasing values of Re.

The use of assumptions 1. and 2. have allowed the generation of a number

of qualitative predictions concerning the centerline concentration behavior of

axisymmetric jets which are in agreement with experimental observations.

Perhaps the most important conclusion of this analysis is that the criteria

normally employed for determining whether or not a turbulent flow is fulLy

developed are too stringent for the role in which they are sometimes used,

namely, that of determining whether or not comparison of jet measurements at

different Re are valid. Clearly, the nature of the measurement must be

considered.

5.5 Validity of Generalizing Results of This Study to

Momentum-Driven Free Jets

Much of the discussion above has dealt with the comparison of the results

from this study with similar measurements in free jet flows as well is .»

- 90 -
theoretical development which assumes the lack of wall, coflow, or buoyancyinduced momentum effects on the turbulent mixing. The implicit assumption

which has been made throughout the discussion is that the flows which have

been investigated ar6 close approximations to those which would be found for

free jets. Since examples have already been discussed where wall effects are

present (He/air) and buoyancy strongly modifies the flow behavior (the

fountains observed in the SF^/air and SF^/He flows), it is important to

question the validity of this assumption. It will be shown that for most of

the flows studied the concentration behavior should indeed be that expected

for the corresponding free jets. In section 2.3 possible modifications due to

buoyancy and the utilization of a coflow within an enclosure have been

discussed. This discussion forms the foundation for the remarks which follow.

The confined He/air flow apparently interacts with walls of the enclosure

and/or is effected by recirculation eddies in the surrounding coflow. This

conclusion is based primarily on the observation that the unmixedness values

for this flow falloff as a function of downstream distance while the comparable measurements for the free jet have an unmixedness behavior which is

similar to that found for the other jet/coflow gas pairs investigated. Flow

visualization results show that the other jets have grown to much smaller

radial diameters at the maximum downstream distance for which measurements

have been made. Furthermore, as can be seen from table 2, values of z^/r0 and

Gt for these other flows are such that recirculation effects should be absent.

It is therefore concluded that direct wall interaction of the jet or recirculation in the coflow is important only for the He/air jet at the downstream

distances observed in this study.

- 91 -
No pressure gradient measurements have been made in this study. Strong

pressure gradients can modify the turbulent mixing behavior of axisymmetric

jets within enclosures [12]. Based solely on comparisons of the findings of

this study with those for free jets, such effects have been assumed to be

small.

Surrounding a jet with a coflow instead of stagnant surroundings modifies

the turbulent behavior of the jet (see the discussion in section 2.3). Values

of Ct for the various flows studied here are listed in table 2. Most of the

values fall in the 0.8 to 0.9 range. These Ct values have been calculated

assuming a cylindrical enclosure of the same cross sectional area as the

square enclosure actually employed. Based on the constant density measurements of Becker et al [26], a value of Ct = 0.67 corresponds to a jet into a

confined coflow which is indistinguishable from a free jet into stagnant

surroundings. The variable density Ct derived by Steward and Guruz [29] has

been shown to provide an equivalent description for variable density flows.

Most of the Ct for the flows investigated in the current study are only

slightly larger than 0.67. The findings of Becker et al [26] indicate that

very minor variations in centerline mixing behavior will be found for these

flows compared to those with Ct =* 0.67. On this basis, we conclude that, with

the exception of the He/air flow and the low Re flows of SF^/air and SF^/He

discussed below, the jets investigated in this study will have a centerline

concentration behavior which is very similar to the corresponding free jets.

In section 2.3 an approximate analysis assuming stagnant surroundings was

used to obtain an expression for the downstream location (z max ) of the top of

the fountain formed by a negatively buoyant jet. The presence of a coflow

- 92 -
would be expected to substantially modify the formation of such a fountain.

Discussions in Turner [24] and Chen and Rodi [5], as well as common experience, indicates that such fountains are formed when jet fluid on the centerline comes to a standstill and then falls through the less dense fluid

surrounding the jet. If the surrounding gas is flowing, the resulting

momentum counteracts the negative buoyancy of the jet fluid and the value of

zmax exPected to increase. We are not aware of any theoretical analysis

which deals with this complex flow phenomenon. Such an analysis is beyond the

scope of this work. It is worthwhile to note that both the approximate

analysis given in section 2.3 and the experimental correlation of Turner [24]

(eq. (59)) do give rough agreement with experimental results (see table 7)

despite the fact that both assume stagnant surroundings.

The importance of deviations from purely momentum-driven jets due to

buoyancy are difficult to quantify. An approximate expression (eq. 56) was

given in section 2.3 which allows the ratio (Rm

) of the initial momentum of

the flow to that imparted by buoyancy to be calculated as a function of z/r

Q

.

Values of Rm

(z) at the most distance z/rQ for which a given set of measurements has been made are listed in table 2. From these calculations, it is

clear that buoyancy effects should be very small for the He/air, CH^/air, and

C^Hg/CC^ flows. Such a conclusion cannot be definitely made for the other

jets which have larger values of R . The results given in table 2 indicate

that for jets with R

p greater than 1 the momentum of the flow will have fallen

substantially by the time the flow has reached the end of the observation

region. If variations in flow momentum are affecting the experimental

measurements, the modifications should be most evident for these flows.

- 93 -
Fortunately, the flows having larger values of R have been studied at

P variable Re. As expected, increasing Re led to a rapid reduction in the

importance of buoyancy. This can be seen clearly by comparing the values of

Rm

at z/rQ » 63 for the three different C

3Hg/air jets listed in table 2. For

the Re - 11,880 case, the value of has decreased to 24.0 and this flow

should behave very much like a purely momentum-driven jet. Comparison of the

centerline measurements of Y and Y'/Y for this gas pair at different flow Re m mm

therefore provides a means for assessing the effects of buoyancy on the

measurement results.

Figures 32 and 36 show plots of Y /Y and Y ' /? versus z/v n for the three

o m m m o different CgHg/air flows. The results of the measurements are summarized in

tables 5 and 10. Comparison of values of K

c for the three jets shows that Che

slopes of these plots are identical within experimental accuracy. However,

downstream shifts are found for the virtual origins (z

Q )y as the Re is

increased. It has been shown that similar shifts in virtual origin have been

found for constant density turbulent flows. Thus, while it is possible that

there is a buoyancy effect on virtual origin position, it is believed that the

primary source of these variations is a Re dependence. Using similar reasoning, the behavior of the centerline unmixedness is also found to be unaffected

by buoyancy effects. Both the asymptotic value and unmixedness development

along the jet centerline for the Re » 3960 flow are consistent with Jets

having lower values of R

p in which buoyancy effects should be absent. Values

of S and K are also found to be invariant with Re and to have Jet centerline

variations which are determined primarily by the jet to coflow density ratio.

It must be concluded that there are no indications of buoyancy effects in the

experimental measurements for these three C

3 Hg/air flows.

- 94 -
The absence of a measurable buoyancy effect on the centerline

concentration behavior of variable Re CgHg/air flows requires further explanation. The values of R^z) listed in table 2 for the three flows suggest that

the total momentum of the flow at the lower Re is reduced considerably from

its initial level at z = 0 due to a substantial buoyancy-induced component.

Equation (55) indicates that the negative momentum contribution due to

buoyancy has a squared dependence on z. This requires that the importance of

buoyancy-induced momentum changes decreases rapidly upstream. Apparently, the

measurement results for the CgHg/air flows at a given downstream distance are

determined primarily by the flow conditions upstream, (i.e., during passage

through regions where buoyancy contributions to the flow momentum are small).

Presumably, measurements made at locations still further downstream would be

strongly modified by buoyancy.

Some indication of the modifications in turbulent behavior which are to

be expected for these negatively buoyant flows can be found in the results for

the SF^ turbulent jets at Re = 3950 flowing into coflows of He and air.

Figure 21 shows the results of experimental measurements of average centerline

concentration for the two flows plotted as Y /Y versus z/r_. Both plots show

o m or an initial region at upstream positions for which the data have a roughly

linear dependence on z/r

Q

. This is the same dependence which has been

observed for other jet/coflow gas pairs in which buoyancy effects are absent.

However, as z/r

Q is increased, the jet begins to slow down significantly due

to buoyancy and values of Y

Q

/Y m increase dramatically until the downstream

position where the tops of the fountains form. These findings suggest that as

the momentum of the flow decreases significantly, the centerline mixing rate

increases

- 95 -
The results of calculations of ^ at z/r Q = 63 for the CF^air flow at

Re = 3960 and the SFg/air flow at Re = 7890 are included in table 2. For

these two jets the effects of buoyancy on the flow momentum are quite large at

this downstream position. However, comparisons of experimental results for

these flows with the corresponding flows at higher Re show no indication of

variations in centerline concentration behavior due to buoyancy. As is the

case for the C^Hg/air flows, the integrated effects of buoyancy are too small

to be detected as variations in the average concentration or unmixedness

values over the range of downstream distances investigated. The experimental

results for these flows must correspond closely to those which would be found

for purely momentum driven flows.

The above discussion indicates that most of the flows which have been

investigated during this study will be good approximations of the corresponding free jet flows and that buoyancy effects on the measurements at the relatively short downstream distances investigated are minimal.

6. FINAL REMARKS

This work has reported flow visualization studies and centerline

concentration measurements for axisymmetric jets of gases into slow coflows of

a second gas. In this manner it has been possible to investigate turbulent

flows where values of the density ratios of the jet to surrounding gases vary

from 0.14 to 37. It is interesting to note that if this range of values of

density ratio were due to gas temperature variations of a single ideal gas,

the corresponding temperature range would extend from 8 to 2100 K. To our

knowledge, this is the most extensive range of global gas density variations

ever systematically investigated within a single experimental flow system.

-96-
There are a number of experimental parameters which may influence the

flow behavior. Potential effects due to the experimental requirement for the

use of an enclosure and coflowing gas as well as flow momentum variations due

to buoyancy have been discussed. In many cases, approximate treatments have

been utilized in order to estimate the variations in experimental results

expected from those which would be found for purely momentum-driven, turbulent

free jets. In general, these treatments and comparisons with experimental

results have led to the conclusion that such effects are small. However, it

cannot be claimed that this study has fully characterized such effects. Their

possible influence on the experimental findings as well as the validity of the

conclusions which have been drawn from the results must be evaluated in terms

of this incomplete characterization.

The findings of this study have been compared whenever possible to other

literature measurements. Unfortunately, despite the intense experimental and

theoretical interest in axisymmetric jets and the availability of a large

number of studies, it is extremely difficult to reach definite conclusions

concerning the effects of such important flow parameters as initial flow

conditions, Re, and jet to surrounding gas density ratio on the turbulent

mixing behavior.

The lack of a concensus on the existence of an asymptotic centerline

unmixedness is an example of the difficulty. For instance, in a very recent

study, Dahm [16] has concluded that such an asymptote does not exist. This

led this author to suggest a new type of concentration similarity parameter

for treating the concentration fluctuation behavior of these jets. On the

other hand, the findings of the current study do indicate the existence of an

-97 -
asymptotic value for centerline unmixedness and a very simple, qualitative

theory based on variations in unmixedness development with Re and R

p has been

suggested as an explanation for the variations of results among different

studies. The answer to the question of whether or not such an asymptote

exists is very important to both experimentalists and theoreticians who are

trying to characterize turbulent flows of this type. Clearly, the question

has not been answered satisfactorily and more experimental and theoretical

study is necessary to solve this important puzzle.

Similarity analysis has been employed in this study as a means for

comparing jets having different global density ratios and Re. For example,

the fitting of the centerline concentration data plotted as ^

Q

'^ m versus z/rQ

to a straight line is based on the assumption of self-similarity. Selfsimilarity is also the basis of the effective radius concept as originally

developed by Thring and Newby [18]. These conclusions have been utilized

freely despite the fact, as discussed in sections 2.2 and 2.3, that true seLf

similarity is not possible for either jets into a coflow or for jets Into

surroundings at a different density. Future work will be necessary to determine whether such observations as the incomplete correlation of the average

centerline concentration data in terms of eq . (24) for variable density flows

has its basis in a breakdown of self-similarity or in some other effect.

The use of both shadowgraphy for flow visualization and Rayleigh light

scattering concentration measurements has allowed much to be learned

concerning centerline mixing in these turbulent flows. It is clear, however,

that a more complete understanding can be achieved by extending these measure

ments to radial locations in the jets and by obtaining velocity information

- 98 -
for the flow field. A method for making simultaneous concentration and

velocity measurements has been previously described [59]. In the near future,

we plan to utilize this technique to map out the axial and radial concentration and velocity fields for the variable density jet/coflow gas pairs investigated in this study.

As for most investigations of this type, the findings of this study have

proven to be somewhat inconclusive and have perhaps raised more questions than

have been answered. However, the detailed nature of the quantitative measurements is such that very clear dependencies of flow behavior on R

p and Re have

been elucidated. Many of these findings have not been reported or are not

widely quoted in the literature. For this reason, and at the same time

keeping in mind the caveats noted above, this report concludes by listing the

most important findings concerning density and Re effects on mixing for an

axisymmetric turbulent gas jet flowing into a second gas.

(1) General conclusions drawn from integrated flow visualization studies

agree with those based on the quantitative centerline concentration

results

(2) The average centerline concentration behavior for jets and coflows

having different density ratios all have linear dependencies when

values recorded for z/r

Q > 20 are plotted as Y

Q

/Y m versus z/r 0

.

(3) The use of the effective radius concept of Thring and Newby [18]

gives a partial correlation of the concentration results for variable density flows. An improved correlation is found if the square

- 99-
root dependence on density ratio in eq. (22) is changed to

(P /Pj o 00 0.6

(4)

Virtual origins for the concentration plots are highly dependent on

density ratio, going from negative to positive values as is

decreased from values greater than to values less than 1. This

behavior is predicted qualitatively by eq. (34) which is based on an

approximate solution of the integral equations.

(5)

The centerline unmixedness values in terms of Y’/Y have a common

m m asymptote of 0.23 for all of the flows which have been investigated.

(6) The effective radius concept provides a good collapse of the

centerline unmixedness growth curves for flows at the same Re, but

having different values of R . P

(7) Values of S and K for the centerline concentration fluctuations are

close to their Gaussian values of S = 0 and K = 3, but variations

with R are found.

P

(8) Increasing Re shift the positions of the virtual origins downstream

for plots of Y /Y versus z/r„, but has no measurable effect on the r o m o slope

(9)

The asymptotic value of centerline unmixedness is independent ot Re,

but the downstream distance required to attain (Y'/Y ) increases

^ m asym

rapidly with Re.

- 100-
(10) Values of S and K as functions of z/r

Q are insensitive to Re.

7 . ACKNOWLEDGMENTS

This research has been partially sponsored by the Air Force Office of

Scientific Research, Air Force Systems Command, USAF, under Grant Number

AF0SR-ISSA-85-00012 . TheU.S. Government is authorized to reproduce and

distribute reprints for Governmental purposes not withstanding any copyright

notation thereon.

The author would like to thank Drs. Bernie McCaffrey and Howard Baum of

the Center for Fire Research for thorough reviews of this manuscript and their

many helpful comments. Thanks are also due to Professor Wolfgang Rodi of

the Universitat Karlsruhe for providing a description of unpublished research

and to Dr. Robert Schefer of CFR, Sandia National Laboratories for providing a

preprint of work prior to publication.

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- 101 -
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[20] Wilson, R.A.M. and Danckwerts, P.V., Studies in turbulent mixing—II a hot jet, Chem. Eng. Science 19:885-895, 1964.

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[24] Turner, J.S., Jets and plumes with negative or reversing buoyancy, J.

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[25] Abramovich, G.N., The theory of turbulent jets, The MIT Press,

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ducted turbulent jets, Ninth Symposium (International) on Combustion,

Pittsburgh, PA, The Combustion Institute, 1962, 7-20.

[27] Craya, A. and Curtet, R., Sur l’evolution d'un jet in espace confine,

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variable density, Com. Science Tech. 1 6( 1-2 ): 29-45 , July 1977.

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films to flows of different gases, Nat. Bur. Stand. (U.S.), NBSIR

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[31] Goldstein, R.J., Optical systems for flow measurement: shadowgraph,

schlieren, and interferometric techniques, Chapter 8 in Fluid Mechanics

Measurements, Richard J. Goldstein, ed. , Hemisphere Publishing,

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[32] Weinberg, F.J., Optics of flames, Butterworths , Washington, DC, 1963,

251 p.

[33] Pike, E.R., Photon statistics, Riv. Nuovo Cimento 1 (Numero Speciale)

277-314, 1969.

[34] Arcoumanis, C., A laser Rayleigh scattering system for scalar transport

studies, Exp. Fluids 3(2 ): 103-108 , 1985.

[35] Hinze, J.O. and van der Hegge Zijnen, B.J., Transfer of heat and matter

in the turbulent mixing zone of an axially symmetrical jet, Appl. Sci.

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- 103 -
[36] Keagy, W.R. and Weller, A.E., A study of freely expanding inhomogeneous

jets, Proceedings 1949 Heat Transfer and Fluid Mechanics Institute,

Berkeley, CA, May 1949, 89-98.

[37] McQuaid, J. and Wright, W., Turbulence measurements with hot-wire

anemometry in non-homogeneous jets, Int. J. Heat Mass Trans.

17(2) :34l-349 , February 1974.

[38] Chassaing, P. and Claria, A., Transfer de masse dans des jets turbulents

de revolution en milieu non homogene, Int. J. Heat Mass Trans.

19(3) :249-258 , March 1976.

[39] Lockwood, F.G. and Moneib, H.A., Fluctuating temperature measurements in

a heated round free jet, Com. Science Tech. 22(1-2) :63-81 , April 1980.

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and concentration decay of high pressure jets of natural gas, Com.

Science Tech. 36(5-6) :249-261 , April 1984.

[41] Niwa, C., Ichizawa, J., Yoshikawa, N., and Ohtake, K. , Time-resolved

concentration measurements of jets by laser Rayleigh method, Proceedings

of the Fourteenth International Symposium on Space Technology and

Science, Tokyo, 1984.

[42] Schefer, R.W. and Dibble, R.W. , Rayleigh scattering measurements of

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published

[43] Keagy, W.R. and Ellis, H.H., The application of the schlieren method to

the quantitative measurement of mixing gases in jets, Third Symposium on Combustion, Flame, and Explosion Phenomena, Williams and Wilkins,

Baltimore, MD, 1949, 667-674.

[44] Uberoi, M.S. and Garby, L.C., Effect of density gradients on an air jet,

Phys. Fluids Supp. 10: S200-S202 , 1967.

[45] Shaughnessy, E.J. and Morton, J.B., Laser light-scattering measurements

of particle concentration in a turbulent jet, J. Fluid Mech. 80(1):

129-148, April 1977.

[46] Rosenweig, R.E., Hottel, H.C., and Williams, G.C., Smoke-scattered light

measurement of turbulent concentration fluctuations, Chera. Eng. Science

15:111-129, 1961.

[47] Way, J. and Libby, P.A., Application of hot-wire anemometry and digital

techniques to measurements in a turbulent helium jet, AIAA J. 9(8):

1567-1573, August 1971.

[48] Nakamura, I., Sakai, Y., and Miyata, M. , A study of the fluctuation

concentration field in a turbulent jet, Mem. Fac. Eng., Nagoya Univer.

34 ( 1 ) : 1 13-124 , 1982.

- 104 -
[49] Ebrahimi, I. and Kleine , R. , The nozzle fluid concentration fluctuation

field in round turbulent free jets and jet diffusion flames, Sixteenth

Symposium (International) on Combustion, The Combustion Institute,

Pittsburgh, PA, 1977, 1711-1723.

[50] Ebrahimi, I., Gunther, R. , and Haberda, F.,

Wahrscheinlichkeitsdichteverteilungen der Konzentration in isothermen

Luf t-Freistrahlen, Forsch. Ing.-Wes. 43(2):47-52, 1977.

[51] Bradshaw, P. , An introduction to turbulence and its measurement,

Pergamon Press, New York, 1971, 218 p.

[52] Hinze, J.O., Uirbulence , 2nd ed., McGraw-Hill, New York, 1975, 790 p.

[53] Tennekes, H. and Lumley, J.L. , A first course in turbulence, The MIT

Press, Cambridge, MA, 1972, 300 p.

[54] Landau, L.D. and Lifshitz, E.M., Fluid Mechanics, Pergamon Press,

London, 1959, 536 p.

[55] Antonia, R.A. and Bilger, R.W. , An experimental investigation of an axisymmetric jet in a co-flowing air stream, J. Fluid Mech. 61(4):

805-822, December 1973.

[56] Corrisin, S. and Uberoi, M. , Spectra and diffusion in a round turbulent

jet, N.A.C.A. Report 1040:915-935, 1951.

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conserved scalar pdfs in a H2 turbulent jet diffusion flame, Proceedings

of the Fifth Symposium on Turbulent Shear Flows, August 7-9, 1985,

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[59] Pitts, W.M., McCaffrey, B.J. , and Kashiwagi , T. , A new diagnostic

technique for simultaneous, time-resolved measurements of concentration

and velocity in simple turbulent flow systems, Proceedings of the Fourth

Symposium on Turbulent Shear Flows, September 12-14, 1983, Karlsruhe,

W. Germany.

- 105 -
9. NOMENCLATURE

B(z) flow momentum at z due to buoyancy, see eq . (51)

C generalized jet fluid concentration, see eq. (25)

Ct Craya-Curtet number

D diameter of axi symmetric jet

*1 radial function of C in terms of n, see eq. (4)

f 2 radial function of U in terms of n, see eq . (3)

radial function of C in terms of n' h radial function of U in terms of n'

F Froude number

g gravitational constant

h> l 2> 1

3

integrals given by eqs. (42), (43), and (44)

K kurtosis

Kc slope for plot of 1/C^ versus r

£

, see eq. (24)

Ke numerical constant in eq. (36)

K

m numerical constant in eq. (59)

Ku slope for plot of U

Q

/Um versus z/rQ

slope for plot of X

q

/X^ versus z/rQ

K

y slope for plot of Y0

/Y

m versus z/r0

*12 integral defined by eq. (10)

K

22 integral defined by eq. (8)

*112 integral defined by eq. (32)

K

122 integral defined by eq . (31)

L integral length scale for turbulent eddy size

m ratio of U /U * 0

A mass flow rate

- 106 -
M

N

r

re

r

e

r l/2

R(t)

R

R

P Re

S

Sc

T

U

Uk

ud

X

Y

z

z

z d

z max

momentum flux

jet fluid concentration flux

radial coordinate

radius of cylindrical enclosure

effective radius, see eq. (22)

radial distance for which measured value of scalar or velocity

is equal to one half of the jet centerline value

autocorrelation of velocity fluctuations

< r l/2)c/(ri/2)u

ratio of flow momentum due to initial momentum of jet to that

due to buoyancy

ratio of initial jet gas density to the surrounding gas density

Reynolds number for jet, see eq. (67)

skewness

Schmidt number

Eulerian integral time scale; temperature

flow velocity in axial direction of jet

kinematic velocity, see eq. (61)

dynamic velocity, see eq. (62)

mole fraction of jet gas

mass fraction of jet gas

axial coordinate

dummy variable for integration, see eq. (51)

axial location where complete entrainment of a coflow by a jet occurs

axial location of the top of the fountain formed by a negatively

buoyant jet

axial location of virtual origin for plots of inverse average

concentration or inverse velocity versus downstream distance

- 107 -
axial location of virtual origin for the inverse rms of

concentration versus downstream distance

z l/2 axial distance for which a centerline value of a scalar is one half of its value at z = 0

Greek letters

P gas density

p' term defied by eq. (66)

£ rate of energy cascade through turbulent eddies

u dynamic viscosity

V kinematic viscosity

n nondimensionalized similarity parameter, r/z

Kolmogorov length scale

n' nondimensionalized local similarity parameter, r/r ^2

9

Subscripts

half angle for jet

asym value at asymptote

c value for generalized concentration

f scalar value in downstream region of jet where it can be treated

as a constant

m value of velocity or scalar along the jet centerline

0 value at the jet exit, or with z, a virtual origin

tu value for a turbulent flow

u value for axial velocity

X value in terms of mole fraction of jet gas

y value in terms of mass fraction of jet gas

00 value in gas surrounding jet

1 gas 1

2 gas 2

- 108-
Superscripts

overbar time-averaged value

rms of fluctuating value

- 109-
Table 1. Suppliers and Purities of Gases Used in This Work

Gas Supplier Purity

He Air Products > 99.995%

ch4 Matheson > 98%

air house line —

CO2 Roberts Oxygen > 99.9%

c

3 h8 Matheson > 99%

cf 4 Dupont > 99.7%

SF

6 Matheson > 98.8%

- 110 -
Table

2.

Properties

of

Flows

Studied * X on PM i-* o mo- o N • • • • • • • • • • • • • co mt o 00 CM O 'O- CM ON cvi CM X X

of

co -a-1 1 1

CM1 1 1 1 ' l 1

r+~ ON

m 00 CM CM

X

Xm 00 oo 00XXXXX

X

XX

u • l • • • • • • • • • • • o i o o o o o o o o o o o ox X l fM. 'O- ON Mf I—

<r ms- Mf MO- MO- -3-

m l ON ON ON ON ON on ON O' ON ON O'

-a N o o <r CO

m ON ON On o —

o on 'T m m m r-» X X X

B o o o o o o o O o o o O —l • • • • • • • • • • • • • o o o o o o o o o o o o o o o o o o o o o o o o o o X m in m NO NO CO X NO CM m ON X X

06 ond ON ON On ON

X ON On ON

X

X O'

co co co co CO pm co

X r-~ •H co

X

0)

X > m in CM ON ON X NO mo- x CM X X a CM CM CM NO NO co o ON ON in o m m 0) o CO co o CM CM m X cm m Mt X w n* i-m X O o 3 c x

"v.N

>3-

m CM

m

mm

H

__ ON

X

Q.

H

m o in

mm

OO

—

-4 O'

X

OS • • • • • • • • • • • • • X o o o —* x CO CO m X X X co xX3

IMXXx 3O X

r-4 CM

u

x

x

X

4-1 (-1

O •H

X •H

X

uX

X

u o M u •H o CO X CO •H iM •H X •H 2 X u •H H 03 '-«» —

x a CO <0 3

03 03 '

v. 00 00 00 00 -

s X 4-1 —^ mo- X X X X mo- mj- nO X X X X

01

X 0) pe CO c*0 c^) ro Ct, (i, t*-4 Cb > n as as o o o CJ o C/3 cn CO X *
Table 3. Estimated Values from Shadowgraphs

Jet/Coflow Re r

p

z/rQ for Initial

Vortex Roll Up

D/r0

* z/rQ -

He/ air 3950 0.14 0.8 20

free He/air 3950 0.14 1.4 20

CH^/air 3950 0.55 1.6 16

C

3 H8 /CO2 3960 1.02 1.9 14

C^Hg/air 3960 1.55 2.5 14

CgHg/air 3960 1.55 1.3 14

CgHg/air 3960 1.55 1.1 14

SF 6 /air 3950 5.11 3.4 —

SFg/air 7890 5.11 2.8 10

SF6 /air 11860 5.11 2.2 10

SFg/He 3950 37.0 5.6 —

- 112 -
Table 4. Predicted and Observed Rayleigh Scattering Cross Section Ratios

Gas Pair Predicted Ratio Observed Ratio

CH^ :air 2.34 2.29 ± 0.02

C

3 H8 :air 14.0 14.6 ± 0.2

CO2 :air 2.40 2.48 ± 0.03

CF 4 :air 2.68 2.66 ± 0.04

SF^ :air 6.74 6.88 ± 0.05

- 113 -
Table 5. Results for Average Concentration Measurements

Jet/ Co flow Re Kx

^ z o^x^ r

o Ky ^ z o

ro K

c

He/ air 3950 0.038 - 17.8 0.281 5.5 0.104

free He/air 3950 0.048 - 9.3 0.344 8.9 0.128

CH4 /air 3950 0.085 - 3.4 0.154 1.9 0.115

CgHg/ CO 2

1 3960 0.113 - 0.7 0.113 - 0.7 0.114

CgHg/air 3960 0.130 0.1 0.084 - 4.1 0.104

C 3 Hg/air 7930 0.131 4.7 0.084 0.4 0.105

CgHg/air 11880 0.128 6.7 0.082 2.1 0.103

CF^/air 3960 0.164 5.5 0.055 - 7.3 0.095

CF4 /air 7920 0.164 10.2 0.055 - 2.7 0.095

SFg/air 7890 0.166 9.8 0.032 - 17.5 0.072

SF6 /air 11860 0.170 12.0 0.034 - 12.8 0.076

- 114 -
Table 6. K

c Values which Result when the Exponent in Equation (22)

is Set to 0.6

Jet/ Coflow Re Kc

He/air 3950 0.086

free He/air 3950 0.105

CH4

/ air 3950 0.108

C^Hg/ CO2 3960 0.114

C3Hg/air 3960 0.109

C

3 Hg/air 7930 0.110

CgHg/ air 11880 0.107

CF 4 /air 3960 0. 106

CF4 /air 7920 0.106

SFg/air 7890 0.084

SF

6 /air 11860 0.089

- 115 -
Table 7. Values of zmax Calculated Using Equations (58) and (59)

Flow Re Eq. (58) Eq

(59) Experimental

C

3 H8

/ CC>2 3960 426 rQ 296 ro C^Hg/air 3960 100 rQ 63 r

o CgHg/air 7930 201 r

Q 126 r

o CsHs/air 11880 301 r

0 188 r

o CF^/air 3960 96 r

o 51 r o CF^/ air 7920 191 r

Q 101 r o SFg/air 3950 51 r o 24 r

o 38 r o SF

6 /air 7890 101 r

Q 47 ro SFg/air 11860 152 r

Q

71 r o SFg/He 3950 75 r 0 21 r

o 44 rQ

- 116 -
Table 8. Summary of Average Centerline Concentration Measurements

for Axisymmetric Turbulent Jets

First Author,

Ref. , Date Jet/Coflow Po /p»

D

0 (cm) Re Kc

^ z o^c^ r

o Method

Hinze, (35], 1948* air/air 1.00 2.54 67,000 0.114 4.0 tracer TC

Corrsin, [6] air/air 0.95 2.54 > 33,000 0.103 2.8 resistance

1949* air/air 0.50 2.54 > 33,000 0.093 7.6 thermometry

Keagy, [36], He/air 0.14 0.33 3,360 0.107 2.3 Orsat probe

1949* No/air 0.97 0.33 26,300 0.092 - 0.9 C0

2 /air 1.53 0.33 49,800 0.117 5.9 Sunavala, air/air 1.00 0.48- 28,700- 0.110 13.6 tracer NO IR

[19], 1957 air/air 0.50 0.95 56,900 0.108 14.0 thermocouple

Wilson, air/air 0.94 1.27 20,000- 0.088 0.0 resistance

[20], 1964 air/air 0.60 1.27 40,000 0.088 0.0 thermometry

Becker,

[9], 1967 air/air 1.00 0.64 54,000 0.093 4.8 nephelometry

McQuaid

,

Ar/air 1.38 1.27 10,000 0.115 - 0.1 hot-wire

[37], 1974 CO2/ air 1.53 1.27 10,000 0.121 - 0.1 ChaBsalng,

[38], 1976* C02/air 1.53 4.00 26,500 0.086 0.8 infrared

Ebrahlmi

,

[15], 1977 air/air 1.00 0.80 15,400 0.115 0.0 nephelometry

Birch, [10], natural 0.56 1.27 16,000 0.125 11.6 Raman

1978 gas/alr

Lockwood,

[39], 1980* air/air 0.54 1.93 50,400 0.092 5.5 thermocouple

Birch, [40], natural 0.56 0.27 - 0.102 - gas

1984 gas/air

C

2 H 4 /air 0.97 0.27 - 0.102 - chromatography

Niva, [41], He/ air 0.14 0.07 1 ,300 0.156 - Rayleigh

1984 f^/air 1.53 0.07 5,000 0.114 - CF 2 Cl 2 /air 4.30 0.07 10,000 0.104 Pitts, [1,2], 1984 CH4 /air 0.55 0.64 4,000 0.112 - 2.0 Rayleigh

Dahm, [16], 1985h

2o/h 2o 1.00 0.25 5,000 0.093 - Absorption

Schefer

[42], 1985 CjHg/air 1.53 0.50 68,000 0.093 6.0 Rayleigh

Values of K

c and (z 0

) c /r

o calculated for data taken from figures in paper.

- 117 -
Table 9. Literature Measurements of Centerline Unmixedness

in Axisymmetric Turbulent Jets

First Author

Ref., Date Jet/Coflow a 1 o

a. Re Asymptote

z/r0

Maximum

C’/C

m a

Maximum

*/ro Corrsin1

, air/ air 0.66 - none 0.19 54 [6], 1949 Rosenwelg

[46], 1961 air/air 1.00

Wilson,

[20], 1964 air/air 0.62

Becker,

[9], 1967 air/air 1.00

Way2

, [47],

1971 10: He/airHe/ air

0.91 0.14 McQuaid

,

[37], 1974 Ar/air 1.38

Ebrahlmi,

[15], 1977 air/air 1.00

Birch, [10], 1978 natural gas/air

0.56

Lockwood

,

[39], 1980 air/air 0.53

Nakamura

[48], 1982

h

2 o/h2 o 1.00

Niwa, [41], 1984 He/air

C0

2 /air

CF

2 Cl 2 /air

0.14

1.53 4.30

Piets, [1,2] 1984 CH4 /air 0.55

Dahm, [16]

,

1985

h

2o/h2o 1.00

Schefer

,

[42], 1985

C

3 Hg/air 1.53

26,200 none 0.21 67

20,000 90 0.18 200

54,000 none 0.21 128

7,170 30 0.24 40 3,290 none 0.37 40 10,000 34 0.36 40

15,400 80 0.25 100

16,000 none 0.28 140

50,400 73 0.21 92

5,000 200 0.21 600

1 ,300 none 0.20 160 5,000 90 0.20 160 10,000 none 0.20 160 4,000 none 0.23 63

5,000 none 0.30 600

68,000 none 0.25 123

^Experimental data for unmixedness have a maximum and then falloff as a function of downstream distance.

^Measurements are reported for only two downstream positions.

- 118 -
Table 10. Centerline Unmixedness Results for This Study

Jet/ Coflow p o /p~ Re

Asymptote

z/r0

Maximum

C’/C

m m

Maximum

z/r o

He/air 1 0. lu 3,950 < 23 0.23 63

free He/air 0.14 3,950 40 0.26 110

CH4 /air 0.55 3,950 20 0.23 63

c 3h8 /co2 1.02 3,960 23 0.23 63

C

3 H3

/ air 1.55 3,960 28 0.23 63

C 3 H8 /air 1.55 7,930 50 0.23 63

C

3 H 3

/ air 1.55 11,880 none 0.22 63

CF4 /air 3 . 00 3,950 50 0.23 63

CF 4

/ air 3.00 7,390 none 0.22 63

SF^/air 5.11 7,890 none 0.19 63

SF 6 /air 5 . 1 i 11,960 none 0.19 63

^The unmixedness values for this flow fall with increasing downstream distance

(see fig. 28 and discussion in text).
Figure 1. A schematic representation of the three flow regimes of an axisymmetric

,

constant-density, turbulent jet is shown. Representative radial profiles

of velocity and jet fluid concentration are shown at one location in the

potential core portion of the flow and at three positions in the sel^-

similar regime.

z

- 120-
Figure 2. Calculated values for the ratio of the integrals 1^ (eq. 44) and I 0 (eq. (43)) are plotted as a function of C^( (p^/o ) -1)

- 121 -
Figure 3. The experimental arrangement employed for time-resolved shadowgraphv

of variable density flows is shown.

- 122 -
BEAM

EXPANDER

Figure 4. This figure shows the experimental system for making real-time,

spatially-resolved, concentration measurements in turbulent flows of

binary gas mixtures.

-123 -
He/air, Re = 3950

Figure 5. Three shadowgraphs of a turbulent helium jet flowing into a slow coflow

of air are superimposed. The Reynolds number for the jet is 2950.

-124 -
Free He/air, Re = 3950

Figure 5. Superimposed shadowgraphs of the same helium jet visualized in fig. 5 are shown, but in this case, there is no coflow and the surroundings are

the ambient air.

- 125 -
Figure 7 . Four superimposed shadowgraphs are reproduced for a turbulent methane

jet at a Reynolds number of 3950 entering a slow coflow of air.

CH4/air, Re = 3950

- 126 -
Figure 8 superposition of four shadowgraphs is shown ior a turbulent jst

of propane flowing into a slow coflow of carbon dioxide. The Reynolds

number for the jet is 3960.

40 -

30 -

CO2, Re —1

1 1

= 3960
Figure 9. Shadowgraphs of the same propane jet visualized in fig. 8, but flowing

into a slow coflow of air, are shown.

C3 H8/air, Re = 3960

- 128 -
Figure 10. The Reynolds number of the propane jet shown in fig. 9 has been

increased to 7930 for the four shadowgraphs which have been superimposed. The coflow gas is again air.

C3 H8/air, Re = 7930

- 129 -
Figure 11. Four superimposed shadowgraphs are shown for a turbulent jet of

propane (Re = 11,880) entering a slow coflow of air.

C3 H8/air, Re =

1 1 1 r

j L -15 -10 -5 0

r/r0

1 1 330

T

'4

- --- .

10 15

-13C-
SF6/air, Re = 3950

Figure 12. The near-field region of a sulfur hexafluoride turbulent jet (Re = 3950)

is visualized by shadowgraphy . Note the presence of jet fluid in

regions well outside of the jet.

-131 -
Figure 13. A shadowgraph for the same flow described in fig. 12 is shown it

larger downstream distances. Again, jet gas is observed in locations

well outside of the jet.

- 132 -
SF6/air, Re = 3950

45

z/r0 35

25

-15 -10 -50 5 10 15

r/r0

Figure 14. The formation of a fountain by the strongly negatively buoyant jet of

SF (Re = 3950) flowing into a coflow of air is visualized by shadow6

graphy. The range of downstream distances extends from z/r^ = 25 to 47.

- 133 -
Figure 15. Four superimposed shadowgraphs of a turbulent jet of sulfur hexafluoride

(Re = 7890) entering a slow coflow of air are shown.

SF6/air, Re = 7890

-134-
Figure 16. The Reynolds number has been increased to 11,860 for the sulfur hexafluoride turbulent jet entering a slow coflow of air.

SF6/air, Re = 1 1 860

i i 1 1 r

l i i i

i i i -15 -10 -50 510 15

r/r0

- 135 -
SF6/He, Re = 3950

20

z/r0

10

o

-15 -10 -5 0 5 10 15

r/r0

Figure 17. The near field region of a SF jet (Re = 3950) flowing into a slow

D coflow of helium is visualized by shadowgraphy

- 136 -
SF6/He, Re = 3950

35

25

z/ro

15

-15 -10 -5 0 5 10 15

Figure 18. A shadowgraph centered at z/r^ ~ 24 is shown for the SF^ jet of Re

3950 flowing into a slow coflow of helium.

- 137 -
SF6/He, Re = 3950

45

z/r0 35

25

-15 -10 -50 5 10 15

r/r0

Figure 19. The fountain formed by the SF^ jet (Re = 3950) entering into helium

can be clearly seen in this shadowgraph recorded for downstream

distances extending from z/r =25 to 47.

o

- 138 -
Figure 20. A portion of the time-resolved methane concentration fluctuations within

a turbulent jet of methane (Re = 3950) flowing into a slow coflow of air

is shown. The measurements were made on the centerline at z/r^ = 31.5.

-139 -
z/r0

Figure 21. Values of Y /Y as functions of z/r are shown for the SF. Jet (Re - o m o b 3950) flowing into coflows of air and helium.

- 140-
zAo

Figure 22. Inverse centerline values of jet gas mole fraction, normalized by

multiplying by the jet gas mole fraction (X = 1), are plotted as functions of nondimensionalized downstream distance for six different

jet/coflow gas pair combinations. Straight lines are linear least

squares curve fits of the data.

-141-
z/r0

Figure 23. Values of the normalized centerline inverse mass rraction ( V* Q

' : m

'

)

are plotted as functions of z/r^ f° r s ^- x jct/coflow pairs listed.

Straight lines are linear least squares curve fits of the data.

- 142 -
Y

0

/Y

Figure 24. Values of Y /Y versus z/r

Q are plotted for the helium jet (Re -

3950) flowing into both a slow coflow of air and the laboratory.

Lines are linear least squares curve fits to the data.

- 143 -
r£

Figure 25. Experimental values of Y^/Y^ for the six jet/coflow combinations

listed are plotted as functions of downstream distance normalized

by the effective radius, r . Results have been offset by the

appropriate values of virtual origin. Comparison with fig. 23

indicates that the use of the effective radius concept gives a partial

correlation of the data.

-144 -
0.35

0.30-

0.25

£ 0.20

IX\. E X 0.15

0.10

0.05

f •

a *

& a

x

X

x

A *

A

A

X X

•

• • .

A A *

A A

a

XX * X X

O O

o O

A A •

A AA • & x X x S

<>S

o «o

o o

o

OV

o OOO

00

O'

A

°„0 oo

O

0 oOO

o

O

OO 0° C oO^C

0

W

o

00

$

g

o He/air a C3 H 8/air

o CH 4/air a CF4/air

x C3 H a /C02 • SF6 /air

'0 10 20 30 40

z/ ro

50 60

Figure 26. Experimental results of centerline unmixedness measurements

fraction terms (X'/X ) are plotted as functions of z/r for mm ° jet/coflow pairs listed.

in mole

the six

-145 -
Figure 27. The results of Y ' /Y measurements tor the jet/coflow pairs listed

° mm

are plotted as functions of z/r^. Note that the values for each

flow approach a common asvmptote of (Y'/Y ) =0.23. rr • mm asvm

- 146 -
i r

0 . 30 -

0 . 25 -

0 . 20 -

l>-

£ 0.15 >-

0 . 10 -

0 . 05 -

o^o

xo

00

o

cboco

o cP°°

0

o

0 o

00

oi 1—

He/air

X FREE JET

o ENCLOSED JET

'0 20 40 60 80 100 120

z/r\

Fieure 28. Values of Y'/Y are plotted as functions of z/r for the helium jet

6 mm o (Re = 3950) flowing into a coflow of air and into the laboratory (free

jet)

- 147 -
i

r

1

••

% *

" “1 1 1

XA ^8

x <> — O O O O- x*»0°%0 ° ° °

0>9* o 0

O O O

O O

O O

o

o o

o

o o

o

o

o

o

]

1

1

%

om

•

O He/air A C3 H 8/air “ •

X O CH 4/air a CF4/air

9

X CjH,/C02 • SF6/air

o

0

?

n ± 1I 1I I 1I 1I 1I 11 1L_

0 20 40 60 80 100 120 140 160

z/rf

Figure 29. Mass fraction unmixedness values for the six jet/coflow gas combinations are plotted as functions of the downstream distance nondimensionalized by dividing by the effective radius. An excellent corre- lation of the data results.

-148 -
Skewness

Figure 30. Skewness values for centerline concentration fluctuations are plotted

as functions of z/r for the jet/coflow gas pairs listed.

o

- 149-
T

5-

O He/alr

O CH4/alr

* CjHg/COj

a CjH,/air

a CF*/air

• SF,/alr

.«2 4

(/) O

"C

3 - 2S a a A

o o

o o

o

o

X A

o •

G• x

o

o

x A •

a

O

o 0

9 o 3 o •a *i

a H

20 30 40

z/rc

50 60

Figure 31. Kurtosis values for centerline concentration fluctuations are plotted

as functions of z/r for the jet/coflow gas pairs listed.

o

-150 -
Y

0

/Y

z/r0

Figure 32. Y /Y values are plotted against z/r for propane jets having the

o m o three different Reynolds numbers listed. The coflow gas is air.

Straight lines represent linear least squares curve fits of the data.

-151 -
Figure 33. Y /Y values for two different Re flows of carbon tetraf luoride into

o m

air coflows are plotted as functions of z/r^. Linear least squares

curve fits to the data are represented by the lines.

-152 -
zAo

Figure 34. Values of Y /Y for SF, jets ar Reynolds numbers of 7890 and 11,860

o m b are shown as functions of z/r . Straight lines represent the results

o of linear least squares curve fits of the data.

-153-
Figure 35. Virtual origins which are obtained from centerline

measurements in the C^Hg/air and CF^/air flows are

of jet Reynolds number.

concentration

plotted as functions

- 154 -
Figure 36. Mass fraction unmixedness values are plotted as functions of z/r^ f° r three different Re jets of propane entering slow coflows of air.

-155-
Figure 37. Mass fraction unmixedness values are plotted as functions of z/r for

o two different Re jets of CF^ flowing into slow air coflows.

j

-156-
Figure 38. Unmixedness values for jets of SF^ at Reynolds numbers of 7890 and

11,860 are plotted as functions of z/r^. The coflow gas is air.

- 157 -
Figure 39. Values of 1/C calculated using eq. (34) are plotted as functions of

z/r for the six gas pairs listed. These results are based on a o method suggested by Thring and Newby [18].

- 158 -
z/ r0

Figure 40. Values of C calculated in the manner suggested by Rodi [5,23] are

m plotted as functions of z/ r . Values of jet to surrounding gas

density ratios correspond to those used in this study.

- 159 -
z/ ro

Figure 41. The calculations shown in fig. 40 have been redrawn as 1/C

m versus z/r

Q

.

Comparison of this figure with fig. 39 shows that the approximate

treatments of Thring and Newby [18] and Rodi [5,23] give similar

predictions for the dependence of average centerline concentration

falloff on the jet/coflow gas density ratio.

p-U.S. GOVERNMENT PRINTING OFFICE; 1986 491 070 40048

- 160 -
HBS-114A (rev. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR-86/33^0

2. Performing Organ. Report No. 3. Publication Date March 1986

4. TITLE AND SUBTITLE

Effects of Global Density and Reynolds Number Variations on Mixing in Turbulent,

Axisymmetric Jets

5. AUTHOR(S)

William M. Pitts

6. PERFORMING ORGANIZATION (If joint or other than N BS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. AFOSR-ISSA-00012

t. Type of Report & Period Covered

Internal Report & Archival Publication

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Air Force Office of Scientific Research

Bolling AFB

Building 410

Washington, DC 20332

10.

SUPPLEMENTARY NOTES

| | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gnificant

bi bl iography or literature survey, mention it here)

Shadowgraphy and laser-induced Rayleigh light scattering measurements of

centerline concentration have been utilized to investigate the effects of

global density and Reynolds number variations on the mixing behavior of a turbulent, axisymmetric jet. Density variations are introduced by using

different gases for the jet and the surrounding coflow. Values of the

jet /coflow density ratio have been varied from 0.14 to 37. The shadowgraph

measurements give a qualitative indication of the variations in average

mixing behavior and turbulent structure which occur when the jet/coflow

density ratio and the Re are varied. These trends are quantified by the

Rayleigh scattering concentration measurements which give measurements of

average centerline concentration as well as the second through fourth

moments of the concentration fluctuations. Extensive comparisons of

experimental results are made with previous literature findings. Integral

analysis has been employed to provide- a framework for analysis of the data.

The results of this study have led to the proposal of a simple, qualitative

theory based on two reasonable assumptions concerning the flow behavior

which gives predictions which are in agreement with the experimental findings,

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) concentration fluctuations; concentration measurement; density eff ects ; flow visualization

Rayleigh light scattering; Reynolds number; shadowgraphy; turbulent flow

13. AVAILABILITY

Pn*| Unlimited

| | For Official Distribution. Do Not Release to NTIS

| | Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402. 22 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

172

15. Price

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