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NUMERICAL SIMULATIONS OF MULTIPHASE TURBULENT
JETS

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THOMAS GABRIEL ALMEIDA

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NUMERICAL SIMULATIONS OF MULTIPHASE TURBULENT JETS
By

Thomas Gabriel Almeida

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2004

ABSTRACT
NUMERICAL SIMULATIONS 0F MULTIPHASE TURBULENT JETS
By

Thomas Gabriel Almeida

The methods of direct numerical simulation (DNS) and large-eddy simulation (LES) are
used to investigate the effects of heavy solid particles and evaporating droplets on planar
and axisymmetric jets. The carrier gas is solved in an Eulerian frame, while the dispersed
phase is computed using a Lagrangian method of tracking the droplets/particles. The
DNS are carried out on a two—dimensional, droplet-laden harmonically forced jet. The
efl‘ects of particle time constant, mass-loading ratio, inlet carrier gas temperature, and
phase-coupling are studied and those of body forces and droplet-droplet interactions are
neglected. The LES are performed on a three-dimensional, particle-laden turbulent
axisymmetric jet. An a posteriori analysis of the subgrid-scale (SGS) closures used in the
LES is conducted via comparison with experimental data. The effects of a stochastic SGS
particle closure and inlet particle conditions are investigated by effectively taming the
SGS particle closure on and off and by examining the statistics for simulations with an
assumed uniform particle distribution at the inlet and those for simulations utilizing a
random (Gaussian) particle size distribution. The results indicate that the proposed SGS
closures implemented herein accurately capture both the effects of the particles on the
carrier gas and those of the carrier gas on the particles. It is also shown that the
implementation of certain physical realities, such as a non-uniform particle size

distribution, can significantly increase the accuracy of the results.

DEDICATION

For my wife, Staci and my son, Gabriel.

iii

ACKNOWLEDGEMENTS

I would like to thank those responsible for the funding of my work at Michigan State
University. This research is sponsored by the US. Office of Naval Research under Grant
NOOOl4-Ol-l-O843. I am indebted to Dr. Gabriel D. Roy for his continuous support and
encouragement through the course of this research. Computational resources are provided
by the National Center for Supercomputer Applications (N CSA) at the University of

Illinois.

I cannot express how much appreciation I have for the guidance of my advisor, Dr. F .A.
Jaberi. He has been unnecessarily supportive and helpful to me from the first day I
stepped into his undergraduate thermodynamics class. I would also like to thank my
committee members, Dr. 2.]. Wang and Dr. NT Wright. They have been patient with
me and vital to my grth as an engineer and researcher. Additionally, I would like to
thank all of the faculty and staff at Michigan State University for producing an excellent

environment for learning and research. I will be forever indebted to all of them.
Finally, I would like to thank my family and friends for always being there for me and

pushing me to always do my best in everything that I encounter in life. Without the love

of my wife and son, I would be lost.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................................... vi
LIST OF FIGURES ................................................................................. vii
NOMENCLATURE .................................................................................. x
INTRODUCTION .................................................................................... 1
CHAPTER 1
BACKGROUND ..................................................................................... 3
1.1 Particles and Isotropic Turbulence ..................................................... 3
1.2 Single-phase Free Jets ................................................................... 5
1.3 Multiphase Free Shear Flows ........................................................... 7
CHAPTER 2
DIRECT NUMERICAL SIMULATIONS OF A PLANAR JET .............................. 15
2.1 Governing equations and computational methodology ........................... 15
2.2 Results and Discussion ............................................................... 19
2.2.1 Nonevaporating droplets ................................................... 19
2.2.2 Evaporating Droplets ...................................................... 34
2.2.3 Interpolation ................................................................. 44
CHAPTER 3
LARGE-EDDY SIMULATIONS OF A ROUND JET ......................................... 46
3.1 Governing equations and computational methodology ........................... 46
3.2 Results and Discussion ............................................................... 50
3.2.1 Comparison with experiment ............................................. 50
3.2.2 Additional physical Observations ........................................ 60
CHAPTER 4
CONCLUSIONS .................. , .................................................................. 72
REFERENCES ...................................................................................... 77

LIST OF TABLES

Table 1. Physical parameters ....................................................................... 19
Table 2. Two-way coupling cases ................................................................. 26
Table 3. Evaporative case descriptions ........................................................... 35
Table 4. Physical parameters ...................................................................... 50
Table 5. Case summary ............................................................................. 51
Table 6. Sample percent error for mean axial velocity at jet centerline, ucl ................. 53
Table 7. Sample percent error for mean axial velocity, um .................................... 54
Table 8. Sample percent error calculations for um ........................................................... 57
Table 9. Sample percent error calculations for Tu .............................................. 58

vi

LIST OF FIGURES
Figure 1. Vorticity magnitude and particle locations, one-way coupling.
(a) rp=0.1,(b) rp= 1.0 and (c) rp= 10.0 ............................................... 21
Figure 2. Particle inertia effects on integrated particle number density ...................... 22

Figure 3. Particle number density transverse profiles at x/h = 8. Note the

vertical shift applied to the IP = 1.0 and the rp = 100.0 cases ........................ 23
Figure 4. Particle dispersion as a function of injection. (a) uniform over 4h,

(b) uniform over 2h and (c) at shear layer over 2Ay ................................... 24
Figure 5. Particle injection effects on particle dispersion. The x/h = 3 data is

shifted up by 300 ........................................................................... 25
Figure 6. Effect of particle time constant on jet growth rate. (a) whole

field and (b) regression analysis of self-preserving region ............................ 27
Figure 7. Particle centerline axial velocity versus particle size ............................... 28
Figure 8. Effect of mass loading on jet growth rate. (a) whole-field and

(b) regression analysis of self-preserving region ....................................... 29
Figure 9. Centerline axial velocity profiles as a function of mass-loading .................. 30

Figure 10. Reynolds stress profiles versus mass-loading. (a) axial variation
at the shear layer and (b) transverse variation at x/h=6 ............................... 31

Figure 11. Production of TKE profiles versus mass loading. (a) axial

variation at the shear layer and (b) transverse variation at x/h = 6 .................. 32
Figure 12. Temperature effects for two-way coupling at centerline .......................... 34
Figure 13. Two-way coupling temperature effects on velocity profiles at x/h=6 ........... 34
Figure 14. Jet half-width versus coupling (T =600) (a) actual growth

and (b) linear regression of the linear portion of the growth rate .................... 36
Figure 15. Centerline axial velocity profiles as a fimction of coupling (T=600) ........... 37

Figure 16. Reynolds stress profiles for different coupling (T=600). (a) axial
variation at shear layer and (b) transverse variation at x/h=6 ........................ 38

vii

Figure 17. Production of TKE profiles for different coupling (T=600). (a) axial
variation at the shear layer and (b) transverse variation at x/h = 6 .................. 39

Figure 18. Temperature variation in the axial direction for different
coupling cases .............................................................................. 40

Figure 19. Density and mass fraction of vapor variation in the axial direction

for different coupling cases ............................................................... 41
Figure 20. Transverse variation of temperature and density ................................... 42
Figure 21. Development of the probability distribution function of particle mass ......... 43
Figure 22. Eulerian averaged particle mass and Reynolds stress profiles

at various downstream locations ......................................................... 43
Figure 23. Temperature profiles versus interpolation schemes ............................... 44
Figure 24. Jet growth rate as a function of interpolation scheme ............................. 45
Figure 25. Vorticity magnitude and particle distribution ....................................... 51
Figure 26. Centerline mean axial velocity versus downstream location ..................... 53
Figure 27. Axial velocity profiles at (a) x/D = 5 and (b) x/D = 12.5 ......................... 54
Figure 28. Spatial Development of Axial Velocity Profiles at

x/D = 2.5, 5 and 12.5 ...................................................................... 55
Figure 29. Radial profiles of RMS of axial velocity at (a) x/D = 5 and

(b) x/D = 12.5 ............................................................................... 5 7
Figure 30. Radial profiles of turbulence intensity at x/D = 5 and 12.5 ...................... 58
Figure 31. Mean centerline axial velocity of Single- and two-phase flow ................... 59
Figure 32. Radial profiles of mean axial velocity at x/D = 7.5 ................................ 60
Figure 33. Particle number density versus downstream distance ............................. 61

Figure 34. Carrier gas and particle velocity versus radial position at
(a) x/D=2.5, (b) x/D=5.0 and (c) x/D=12.5 ............................................. 61

Figure 35. Correlation between particle axial velocity and particle mass
at different downstream locations ........................................................ 62

viii

Figure 36. Particle mass versus radial position at different downstream locations. . . . . ....63

Figure 37. Particle mass and number density radial variation at different
axial locations .............................................................................. 64

Figure 38. Probability distribution function of particle mass ................................. 65

Figure 39. Axial variation of correlation between particle and
carrier gas velocities ....................................................................... 66

Figure 40. Radial variation of correlation between particle and
carrier gas velocities ...................................................................... 68

Figure 41. Particle temperature as a function of particle mass .......... . ...................... 6 9

Figure 42. Mean axial velocity versus radial position for varying average
particle inertia at (a) W = 5 and (b) x/D = 12.5 ....................................... 70

Figure 43. Mean axial velocity (a) versus axial position at the centerline
and (b) versus radial position at x/D = 7.5 .............................................. 71

ix

bmsppg

NOMENCLATURE

mass transfer number

specific heat at constant pressure of fluid

jet diameter

particle diameter

total energy

coefficient related to particle drag

coefficient related to particle heat and mass transfer
coefficient related to particle heat and mass transfer
mass flux of species or in ith direction

thermal conductivity

Mach number

mass of particle

number of species

Prandtl number

pressure

heat transfer in ith direction

universal gas constant

molecular weight gas constant

Reynolds number

radial position

jet radius

energy source term

momentum source term in ith direction

mass source term

fluid temperature

particle temperature

turbulence intensity

time

ith component of fluid velocity vector, U
deviation from the mean velocity in ith direction
centerline axial velocity

mean axial velocity

root-mean-square of axial velocity

Reynolds stress

ith component of particle velocity vector, V
molecular weight of species or

ith component of Lagrangian coordinate system
i[11 component of Eulerian coordinate system
mass fi'action of species or

Greek Symbols

@15th 9%“qu

.9

1'

ratio of specific heat of the particle to that of the fluid
Kronecker delta

Eulerian differential volume

grid spacing

ratio of specific heats of the fluid
coefficient related to droplet evaporation
fluid viscosity

fluid density

particle density

Newtonian fluid stress tensor

particle time constant

xi

INTRODUCTION

Multiphase flows occur in a wide range of engineering applications. Ink jet printers,
internal combustion engines and fire prevention systems are obvious examples of
physical situations for which the understanding of multiphase transport phenomena are
very important. Due to the range of applicability of these flows, scientists and engineers
continue to work on these problems with fervor. This work is focused on a specific class
of multiphase flows, that of dilute turbulent free shear flows laden with a dispersed
medium, either solid particles or evaporating droplets. There exists an underlying
difficulty in dealing with such flows for various reasons. For experimentalists, one such
difficulty is resolving the properties of both (or all) phases involved. For example, to use
phase-Doppler anemometry (PDA) or particle imaging velocimetry (PIV) to capture the
velocity field of both the carrier gas and the dispersed phase would require some method
of differentiating between the seeding particles and the actual dispersed particles, such as
a separation of scales. Analytically, the amount of simplifications required to make the
governing equations tractable would greatly reduce the range of applicability of said
equations. One of the more promising methods of predicting turbulent multiphase flows
is the use of computational methods. While there are limitations to the use of numerical
methods, the benefits seem to outweigh the costs. The long—range impact of this work is
to aid in the development of subgrid~scale (SGS) closures for the large-eddy simulation
(LES) simulation methods of computing multiphase flows. The calculations performed
herein are direct numerical simulations (DNS), utilizing a particle-source-in-cell (PSIC)
methodology. While some critics argue that this is not true DNS because the smallest

scales of the flow around each individual particle or droplet are not resolved, the careful

analysis of the results indicates that the assumptions and correlations used are indeed

valid.

In order to develop accurate closure models for LES, one must establish an effective way
of validating the proposed models. This work is an effort both to understand the complex
physics associated with multiphase turbulent flows and to develop a usable database of
DNS results. Once the database is established, one can use those results to develop
closures for various turbulence modeling techniques that are robust enough to accurately
predict the effects of adding solid particles or evaporating droplets to various turbulent

flows.

To this end, the author has conducted numerous numerical experiments of two-
dirnensional, harmonically—forced planar jets laden with heavy solid particles or
evaporating droplets. The code utilized is a fully-compressible, three-dimensional,
density-based research code. The simulations discussed herein are ‘pseudo-two-
dimensional’, for the longitudinal depth of the planar jet is small enough to force the
variations in the z-direction to zero. Eventually, this same code will be utilized to run
large-scale, three-dimensional numerical experiments of fully turbulent planar jets laden
with particles or droplets. The details of the simulation parameters will be discussed in

more detail in sections to follow.

CHAPTER 1
BACKGROUND
To better understand the underlying physics of multiphase free shear flows, one must first
understand the ‘building blocks’ of multiphase turbulent flows, i.e., first consider single-
phase free shear flows and modulation of isotropic turbulence by particles separately,
then combine the two. Another important aspect of this work is the inclusion of
evaporative effects. While understanding evaporation may be fairly straightforward, the
problem of understanding how evaporation will affect transitional and turbulent flows has

more intrinsic difficulties associated with it.

1.1 Particles and Isotropic Turbulence

Elghobashi and Truesdell (1993) investigated the effects of dispersed, small, solid,
spherical particles on decaying homogeneous turbulence, noting a “selective spectral
redistribution” of turbulent kinetic energy and an increase in the decay rate of turbulence.
The particles increase the turbulent kinetic energy for lower wave numbers and increase
it for higher ones. Yang and Lei (1998) studied the effects of turbulence scales on particle
settling velocity by considering homogeneous isotropic turbulence laden with particles.
They noted that low wave number components affect the particle accumulation, while the
small wave number components do not. However, they also found that the settling rate
depends on the large scales of turbulent motion. Boivin, Simonin and Squires (1998) used
direct numerical simulation (DNS) to study turbulence modulation due to particles, and
found that they tend to dissipate the kinetic energy. The degree to which they affect the

flow was found to scale with the mass loading. They also noted that the particles distort

the energy spectra (as opposed to attenuating or amplifying uniformly). Upon
investigation of particle size effects, they found that the effects on low wave number
energy region were independent of particle Size, but that at high wave numbers, the larger
particles damp the turbulence and the smaller ones amplify it. Ooms and Jansen (2000)
also studied the effects of particles on homogeneous, isotropic turbulence. They give a
reasonable physical argument as to why the size of the particle can lead to different
turbulence modulation. They state that, if the volume fraction is held constant, smaller
particles have more surface area than large ones and thus cause more friction, which
leads to a larger suppression of fluid turbulence. They also give this as a reason to use

caution when using point-source particle approximations.

J aberi (1998) conducted a study of fluid-particle thermal interactions in a particle-laden
homogeneous turbulent flow. His findings indicate that the temperature of both the
carrier gas and the dispersed phase are dependent upon various properties, such as the
particle time constant and the mass-loading ratio. The specific intent was to determine the
effects on the temperature fields. He found that increasing the mass loading causes the
p.d.f. of particle temperature to deviate from a Gaussian distribution. Mashayek and
Jaberi (1999) collaborated to study isotropic turbulence and particle dispersion, noting
that the particle velocity auto-correlation coefficient drops off more sharply with lower
mass loading ratios. They also noted that the pressure-dilatation correlation is

significantly damped by the presence of particles.

1.2 Single-phase Free Jets

One specific flow of particular significance to this work is that of free jets. Free jets, both
planar and round, have been the focus of much research. The general features of the
turbulence fields in such flows are well-established. Hinze (1975), Pope (2000), Bernard
and Wallace (2002) and others have discussed these flows in detail, noting the general
characteristics of the self-preserving portion of the flow. It has been shown that the jet
half-width grows linearly and the centerline mean velocity decays proportional to the
inverse of the square root of the axial position. The self-preserving nature of the jet leads
to the formulation of a non-dimensional velocity profile, which is unchanging. It has also
been shown that the root-mean-square (RMS) of velocity fluctuations and the ‘Reynolds-
stress’ profiles are self-similar as well. The development of free shear flows has been

attributed to instabilities which are controlled by the Kelvin-Helmholtz mechanism.

More recently, researchers have turned their focus to determining the specific causes and
physical explanations for the observed behavior. The instabilities which cause a jet to
develop have been investigated in detail by Hsiao and Huang (1990). Their work
involved a plane jet that they subjected to acoustic forcing. They investigated the growth
of instabilities and the velocity correlations within the jets. They found both the
strearnwise and transverse fluctuating components to be important to the development of
said instabilities. Their work is of particular relevance to this work because they used
experimental methods to force the jet harmonically, similar to the way in which the jets in
this work are numerically forced. They found that small perturbations caused the jet to

develop much more rapidly than unforced jets, and that the structure of the flow was

greatly modified. One finding of particular significance was that the fluctuating velocity
profile along the inside of one shear layer displayed a convenient way to see vortex
merging and saturation. This helped them conclude that the harmonic forcing adds energy
to the fundamental frequency to the point of saturation and then adds to the first
subhannonic, then the second, etc. This was also shown in the development of the half-

width of the jet, which showed plateaus near the harmonics.

Stanley and Sarkar (1997) studied two-dimensional shear layers and jets, noting the
impact that external forcing has on the jet development. Their work aided the author of
this work in choosing the harmonic forcing used (as described in Chapter 2). They found
that, although the down-stream grth was nearly unaffected by forcing at the inlet, the
near-field was modified. They reported some interesting result related to the symmetry of
‘weak’ and ‘strong’ jet flows due to forcing. The simulations performed herein would be
classified as strong. Stanley and Sarkar found that the sinuous (antisymmetric) forcing of

the jets lead to a more realistic velocity field.

Another work of interest was that of Kennedy and Chen (1998). They studied the effects
of temperature on the growth and stability of free jets and found that cold jets tended to
be significantly more stable than isothermal or hot jets. They attribute much of this
observation to increased compressibility. They also indicated a modification of mean
velocity profile, where the cold jets profile were ‘more narrow’ and had a “more gradual
taper of velocity” than their isothermal and hot counterparts. They also noted that the

transverse location of the vorticity maxima decreased with decreasing temperature and

the value of that maxima increased. These are explained by Colucci (1994) investigated
the linear stability of shear flows under density-stratified conditions. He found that
decreasing the density of the high-speed stream would increase instabilities, thereby
increasing turbulence and mixing. He also found that either increasing or decreasing the
density of the stream would stabilize the low-wave numbers. Another important finding
was that the convective wave speed is biased towards the higher density stream. He also
investigated the effects of ‘density spikes’ at the region of highest shear, noting that
decreasing the density of the shear zone darnps the instabilities, thus slowing the growth

rate of the jet.

1.3 Multiphase Free Shear Flows

Recent advances in experimental methods and computational power have given
researchers more access to multiphase flow analyses. Whole texts have been authored on
this subject (e.g., Fan and Zhu (1998), Sommerfeld, Tsuji and Crowe (1997)). Yet, there
is still room for advancement of the ever-broadening field. Crowe, Chung and Troutt
(1988) used numerical and experimental evidence to show that particle dispersion in free
shear flows is controlled by large-scale vertical structures, and not so much by diffusion
(particle concentration gradients). They also classified particles by their aerodynamic
response time (as compared to the fluid characteristic time scale) into three categories:
small, intermediate and large particles. They also mention the effects of forcing a jet on
the particle dispersion by regularizing the vertical structures, which increases the

dispersion of smaller particles.

Hansel], Kennedy and Kollrnann (1992) used numerical methods to study the effects of
individual forces on particle dynamics. They found that, under certain circumstances, the
omission of forces beyond those of the drag, such as Bassett history and virtual mass,
could lead to errors in droplet dispersion on the order of 25%. They discuss the
differences between the vaporizing and non-vaporizing droplet models. The findings are
useful but only pertain to droplet dispersion, not to the two-way coupling effects of the
particles on the carrier gas. The additional forces involved in particle dynamics may have
a significant effect on dispersion, but the effects on the carrier gas properties have yet to

be determined with great accuracy.

Eaton and F essler (1994) studied preferential concentration of particles due to turbulence.
They found that the coherent turbulent structures are still present in particle-laden flows,
but that they are modified. Once again the particle concentration was found to be
determined by the large-scale structures in the flow. They found that the size of the
particle has an important effect on the concentration; neither large nor small particles will
preferentially concentrate, while intennediate—sized particles will. The preferential
concentration of droplets in particular can have drastic effects on the evaporation and
bunting of fuels. For this reason, the size of the droplets should be carefirlly considered

before designing a system involving evaporation.

Mashayek (1998a, 1998b) used DNS to study the droplet size, vapor mass fraction and
temperature field in droplet-laden, forced, low-Mach number turbulence. An important

finding was that the addition of solid particles to homogeneous shear flows causes a

decrease in turbulent kinetic energy and an increase in the anisotropy of the flow, but that
evaporation can effectively decrease that anisotropy. He also found that increasing the
mass-loading ratio causes the temperature of the carrier gas to decrease more initially and
then increase more at longer times. What can be seen to occur in temporal variations here
may be seen in spatial development in anisotropic flows. This is attributed to the
increased temperature difference between phases, and the increased heat capacity of the
larger droplets. He also found the probability distribution fimctions of the particle
diameter to be skewed towards smaller droplets. The vapor mass fraction was found to
become saturated at long times. He also noted that the evaporation rate is higher in
regions of high shear, which then indicates that the Reynolds number is a very important

parameter governing droplet vaporization.

Experimental results for a particle laden round jet were reported and analyzed by
Longmire and Eaton (1992). The visualizations offered in their work are fascinating.
They too found that the large-scale vortices controlled the particle dispersion. The jet that
they studied was acoustically forced. They found that the total particle number density
decreased in the axial direction, and that the number of particles in low-concentration
areas increased with axial distance from the inlet. They also noticed that the slightly
smaller particles tended to accumulate in regions between the large vortices, lending to
periodic peaks and troughs in particle number density corresponding to the forcing
frequency. Round jets and particle dispersion were the focus of Kennedy and Chen

(1998). In an effort to simplify the theory of particle-laden shear flows, they suggested

that particles with Stokes numbers less than unity should behave like fluid particles and

therefore self-preserving laws should hold for these cases.

Several different models for evaporation models were compared by Miller, Harstad and
Bellan (1998). They fund that the Langmuir-Knudsen model was the most accurate (and
thus was chosen as the formulation for this author’s work). The work is a very thorough
and direct synthesis of various important aspects related to droplet-laden shear flows.
They also found that the constant property assumption was an accurate one for
temperatures calculated at the boiling temperature. Armenio and Fiorotto (2001) studied
the importance of the different forces acting on particles. The intent was to determine
which, if any, could be neglected in favor of computational efficiency. They found that
the most important force is that due to drag and that added mass effects are always
negligible, but that Bassett history forces may be important in certain flow regimes. They

are definitely not important to particle dispersion, though.

Armenio, Piomelli and F iorotto (1999) and Bellan (2000) have investigated some of the
important issues related to turbulence modeling and multiphase flows. Both discuss the
state of the art and give more suggestions for further work. Miller and Bellan (2000)
contribute even more to the underlying theme of computational modeling of multiphase
flows. All of these researchers found that the effects of the sub-grid scales on various
particle properties are significant. Specifically, Miller and Bellan (2000) mention that
there may be a need to model some of the subgrid scale fluctuations to accurately predict

the particle dispersion and droplet evaporation. Boivin, Simonin and Squires (2000) also

10

discuss some of the issues related to the LES of multiphase flows. They are careful to
mention the restrictions on a priori testing, but they also found their models to be fairly
accurate for gas-solid flows (note that there is still much room for studies involving

evaporating droplets).

One of the more promising methods of doing so is the subset of numerical methods
termed large eddy simulations (LES). As Bellan (2000) described, there remains a
seemingly inexhaustible amount of work to do on the development of LES closures to
accurately model the interaction between the small-scale turbulence of the carrier fluid
phase and the dispersed phase. While there are a steadily growing number of studies
involving the deve10pment and verification of SGS models using a priori studies applied
to direct numerical simulation (DNS) results, there is a noticeable deficit in the area of
validation of LES results via a posteriori analysis of numerical results by comparison
with experimental data. To ensure the accuracy of a given model, both verification and
validation studies should be conducted as suggested by Boivin, Simonin & Squires
(2000). Their work mostly concerns the spectral analysis of DNS and various LES
models. The importance of correlation with experimental data as well as with DNS data
cannot be overlooked. While there have been several experimental studies (e.g.,
Longmire and Eaton (1992) and Schreck and Kleis (1993)) on the subject of particle-
laden flows, many of these deal with the far-field of as yet prohibitively high Reynolds
number flows and/or do not report results for both the carrier and dispersed phases
concurrently, and thus are insufficient to determine the direct effect of the particles on the

carrier gas and vice versa. The goal of LES is, of course, to be able to predict the near and

11

far flow fields of increasingly high Reynolds numbers, but it seems to be more prudent to
focus first on accurately predicting the near field of lower Reynolds number flows before

moving on to the far field of high Reynolds number flows.

Armenio, Piomelli and Fiorotto (1999) investigated the effects of the SGS on particle
motion. Their work indicates that using a filtered velocity field to advance the particles
can lead to serious inaccuracies; thus the importance of the SGS closures is emphasized.
In Chapter 3, the importance that they refer to will be investigated. Miller and Bellan
(2000) conducted a thorough a priori analysis of the SGS using DNS results for a
transitional mixing layer, and they also concluded that neglecting the SGS velocity
fluctuations in LES might lead to gross errors in the prediction of the particle drag force.
This, in turn, will lead to errors in both the carrier-phase and the dispersed-phase. Miller
(2001) went on to investigate the effects of solid particles on an exothermic reacting
mixing layer. He found that the particles were preferentially concentrated in the high-
strain braid regions of the mixing layer, which can lead to local flame extinction.
Additionally, several researchers have used DNS methods to gain a better understanding
of multiphase flows. Mashayek (1998a, 1998b) and J aberi (1999) noted that the presence
of particles effectively decreases the turbulent kinetic energy while increasing the
anisotropy of homogeneous turbulent shear flows. These effects were shown to be
magnified by increasing either the mass-loading ratio or the droplet time constant. They
also found that the autocorrelation coefficient of the velocity of the carrier gas in an
isotropic two-phase flow increases with an increase in mass-loading ratio. Jaberi (1998)

and Jaberi and Mashayek (2000) studied particle temperature in homogeneous

12

turbulence. They noted that the temperature intensity decreases with increasing particle
time constant, thermal diffusivity and Prandtl number. Also of great significance was the
finding that velocity coupling is not sufficient to resolve the physics of non-isothermal
two-phase flows. It is necessary to include thermal coupling effects as well. Additionally,
their results indicate that increasing the mass-loading ratio causes the carrier fluid
temperature fluctuations to increase (further causing a decrease in the decay rate of fluid
temperature in decaying isotropic turbulence) and increasing the particle time constant

increases the temperature difference between the particles and the carrier fluid.

Chapter 3 will offer evidence that the SGS closures discussed and implemented herein
are both applicable and accurate. This is accomplished through correlation with the
experimental data obtained by Gillandt, et al. (2001). They have generated phase-
Doppler-anemometry (PDA) results for a moderately high Reynolds number round jet
laden with heavy particles. The desire to improve the applicability of LES to multiphase
flows is complemented by the current limitations of experimental methods of flow
measurement. The PDA system described can measure the velocities of both the carrier
gas and the particles, but the particles must be much larger than the tracers (to offer a
definitive separation of scales). This results in a description of a flow that involves
particles larger than those that may be seen in some industrial applications. In contrast,
the LES methods described herein may be used for various particle sizes and Reynolds
numbers at no more expense than the original studies. This work is somewhat similar to
the investigation of a slit—j et by Yuu, Ueno, and Umekage (2001). However, an additional

emphasis is placed on the comparison of the LES with and without the SGS models to the

13

experiment. Also, there are important physical differences between planar and

axisymmetric free jets.

l4

CHAPTER 2
DIRECT NUMERICAL SIMULATIONS OF A PLANAR JET

2.1 Governing equations and computational methodology

The formulation used for this study is similar to that of Miller and Bellan (1999). The
carrier gas phase is treated in an Eulerian frame, and a Lagrangian flame of reference is
used for the dispersed phase (either solid particles or liquid droplets). While there has
been some discussion and research in the area of Eulerian-Eulerian frames versus
Eulerian-Lagrangian flames, the latter has been chosen for simplicity of implementation
and lack of modeling requirements. The two-way coupling effects of the carrier gas on
the droplets and the droplets on the carrier gas are due to the inclusion of each particle as
a point source or sink of mass, momentum, and energy. This is commonly referred to as
the particle-source-in-cell (PSIC) or particle-in-cell method. The equations are derived
under the assumptions of calorically perfect species (carrier gas and evaporate), no body
forces (e.g., gravity) and the dispersed phase volume flaction much less than unity. The
non-dimensional Eulerian equations for continuity, momentum, energy and scalar

(evaporate) are

 

 

 

 

 

 

 

5,0 6m
at axi ,0 ( )
. a -u — 50"
6pm + P“! 1 :_5_P+__1_ '1 +51“. (2.2)
at ij 8x,- Re 6x}-
apE + apEu. = _6Pu. +_1_ J I} _ 1 2 5‘12 +515 (2.3)
at ax,- 5xi Re 6«Ii (7 —1)Ma Re Pr 6x,-
apY 6pY u- l aJ'a
a+ az:_ 1+é’a (2.4)
at 6x,- Re PrLea ax,-

15

The pressure, gas constant, Newtonian shear stress, heat flux, mass flux, total energy and

enthalpy are defined by

1
Ma

RzkuZEYfL
a a

 

P: 2pm"

 

I] 6xj 6x,- 3 i162%

 

61‘ h 61/
qi=-/1 cp—+z 6* 4
6x,-

a Lea ax,-

 

———Zha Ya —5+ “'“'

E:-(r 1W022

ha =12}; +cpaT

(2.5)

(2.6)

(2-7)

(2.8)

(2-9)

(2.10)

It should be noted that for a perfect binary mixture (of the carrier gas and the evaporate),

a =1, 0') = S p The Lagrangian equations for the droplets are defined by

 

:T—p—=Z-2—(T—T )+ L" dmp

_ Qconv 'I’ Lvmp

 

 

dt 1,, P "1ch dt
dm m f
——p=— p 31n(1+BM)= '1,
dt rp

mch

16

(2.11)

(2.12)

(2.13)

(2.14)

 

The subscript p indicates the droplet/particle property, and the fluid properties are
interpolated to the droplet position. The value of f, is empirically evaluated for the
Stoke’s drag, based on the particle slip and blowing Reynolds numbers, Res] and Reb.
Similarly, f2 and f3 are functions of the droplet properties; both involving heat and mass
transfer properties, such as the Nusselt number (Nu), Prandtl number (Pr), Sherwood
number (Sh) and Schmidt number (Sc). Lv, cL and BM are the droplet heat of
vaporization, heat capacity and mass transfer number, respectively, and the
particle/droplet time constant is defined as

2
7p = Eۤpo__ (2.15)
Evaporation is taken into account via the Langmuir-Knudsen evaporation model, which
takes into account both equilibrium and non-equilibrium effects, and the traditional ‘D2
law’ of Godsave and Spalding is also implemented. Finally, the two-way coupling effects

are taken under consideration through the use of the source terms which, for mass,

momentum and energy, are defined as

1 .
up
1 .
Sui :—3V'ZIFI +mpvi) (2.17)
np

 

Qconv hVS +Vivi
=—— + iFi+ (2.18)
EWZPIG— (y—lea v m va- 11m 2 II

These source terms are evaluated for each individual particle and then summed over a

finite Eulerian volume, 5V.

17

The carrier gas Eulerian equations are temporally solved via a second—order, modified
MacCormack technique (predictor-corrector), and the spatial derivatives are calculated
using a sixth-order compact finite differencing scheme. The dispersed phase equations
are calculated via traditional Eulerian time stepping. The carrier gas properties are
interpolated to the droplet position via two methods, first-order linear and fourth—order
Lagrange polynomial. The accuracy of the linear interpolation was determined to be
sufficient to resolve the interaction between phases, and thus was used in the great

majority of simulations (see Section 2.2).

The jet inlet velocity profile chosen was tangent hyperbolic and the jet was subjected to
random-phase harmonic forcing at the shear layer in the transverse (y) direction. The
forcing energy peak was set at 5% for each of five flequencies, the harmonic, one super-
hannonic and three sub-harmonics. Each harmonic had a randomly generated phase angle
between 0 and 1t/2, which was changed each time step. The phase angles applied to the
top shear region were different than those applied to the bottom region. The inlet
boundary condition was derived flom the work of Poinsot and Lele ( 1992). The y-
direction boundary conditions (BCS) were chosen to be zero-derivative flee-stream, the z-
direction BCS were periodic, and the non—reflecting outlet boundary condition used was

that of Rudy and Strikewerda (1980).

The statistical properties were averaged over three pass—over times, where the pass-over

time is given by

 

xmax
t = ' * 2.19
pass (“jet + “co )/2 ( I

18

Convergence was confirmed via comparison between the mean and RMS properties at
two different times, noting less than 1% difference. The jet was given one and a half
pass-over times to develop initially before the averaging was started. In different
simulations, various parameters were varied. Specifically, the particle time constant, the
carrier gas temperature, the mass loading ratio and the ‘coupling’ were treated as
independent variables. The mean and RMS of axial velocities, temperature, density and
mass-flaction of species, as well as the Reynolds stress and production term of turbulent

kinetic energy were all calculated as dependent variables for this study.

2.2 Results and discussion

Some of the physical parameters for the numerical simulations are given in Table 1. Note
that these simulations are chosen to emulate a planar jet of air laden with droplets of
decane. The parameters can easily be modified to more closely match any of several
physical configurations. In the following sections, more parameters are discussed as they

relate to the particular scenario.

Table 1. Physical parameters

 

Carrier Fluid Air

Dispersed Phase Decane Droplets
Jet Reynolds Number 3165

Jet Mach Number 0.291

Prandtl Number 0.75

 

2.2.1 Nonevaporating Droplets
The following section is primarily concerned with how the carrier gas affects the non-

evaporating droplets. For these simulations, all of the source terms were set to zero, but

19

 

the droplet equations were solved. The droplets (or particles) are therefore not allowed to
have any effect on the carrier gas. The gas ‘pushes’ the particles around without ‘feeling’

any reciprocal force.

The effects of particle inertia on particle dispersion are shown in Figure 1. Expectedly, it
is observed that the small particles tend to follow the flow of the carrier gas, while the
large ones tend to move at their own inlet/initial velocity (less affected by the carrier gas).
For a more quantitative understanding of particle dispersion, the particle number density

has been compared for different particle time constants in Figures 2 and 3.

20

 

      

, ((#‘3.

 

magnitude and particle locations, one-way coupling. (a) r, = 0.1,
(b) 1,, = 1.0, and (c) 1,, = 10.0

Figure 1. Vorticity

Figure 2 shows that the number of particles integrated over the y—direction is relatively
constant for high inertia particles, erratic for intermediate size particles, and
approximately harmonic (farther downstream of the jet inlet) for small particles. Again,
this is fortuitous in that the flow is harmonically forced, and the small particles behave

like fluid particles. Figure 3 shows the transverse variation of particle number density as

21

it relates to particle size. The striking characteristic of the flow observed in this plot is
what we call ‘local particle dispersion’. Both the small and the large particles have
regions of high particle dispersion, while the intermediate size particles (2;, = 1.0) clearly
shows minimal local particle dispersion in the narrow width and high peaks of the high
particle density regions. For the other particles, the high particle density regions are

broader and, on average, less dense. This has important implications when evaporative

and/or reactive particles are considered.

 

 

 

 

150 . . . v . . . . . T
—- «5100.0 (+100) ‘ .
. a f q ‘
J :I' ':
I ' h
100 ' :, :1 -
-_. _ r I
”’1;°‘*’°’ . -‘ : 5'.
z ‘1 \ ‘l | g I ll
' 3 II II
a so - I} L---} I-
“ — 5:001
0 3 6 9
Axial Position

Figure 2. Particle inertia effects on integrated particle number density. The curves
for 1;, = 1.0 and 100 are shifted upward for clarity.

22

 

 

 

 

 

w P d
i W
E O ,1
—>iI.— — 5:001
.8 |I: b --- :,=1.o (~95) i
t ' I —- t =100.0(-145)
_ _ I J a
E 50 I : 1’1” 1'.
'8 c t I t
g t, ‘\ 1' It "\
m ....... a -——. s ............ \......
-1(X)
F N AA‘ 1
l (r‘f’ */ 1...; i
e I
_150 ------ 1'", \L 1 l'_-T ——————
—2 -l 0 1 2
TransversePoslflon

Figure 3. Particle number density transverse profiles at xlh = 8. Note the vertical
shift applied to the 1;, = 1.0 and the r, = 100.0 cases
The effects of particle injection are shown in Figures 4 and 5. Notice that the downstream
behavior of the particles with 1;, = 1.0 is basically independent of particle injection
location. Specifically, in Figure 5, the particle number density plots Show that at
downstream locations of x/h 2 9, the local particle dispersion is approximately constant
for the given particle size. This statement only holds true for one-way coupling, as the
particle concentration in the shear layer affects the growth or decay of instabilities in two-

way coupling cases and the results are sensitive to inlet/initial particle distribution.

23

 

 

Figure 4. Particle dispersion as a function of injection. (a) uniform over 4h, (b)
uniform over 2b, and (c) at shear layer over 2Ay. r, = 1.0 in all cases.

24

 

I I V
‘5
"‘
——---~

I
-- shear layer. xlh=3 (+300)

  

 

 

 

I
-- uniform. xlh=3 (+300) |

a 200 . -- shear Iayer,xlh=9 I ‘

9" ' —- uniform, xih=9 I l

100 - .

. ll '\ A
0 [A 1 "‘ “All I;
—2 —1 o 1 2
Transverse Position
Figure 5. Particle injection effects on particle dispersion. The x/h = 3 data is shifted

up by 300.

For the two-way coupling cases considered in this section, the energy and momentum
source terms are affecting the canier gas, however the droplets are still non-evaporating.
This allows for realistic physical simulations with two-way coupling effects present, but
removing the complexities due to evaporation. The effects of particle size, mass loading
and carrier gas temperature were investigated in detail. The outline of the different cases
and their parameters is given in Table 2. Case 7 was conducted as a reference to verify
the modifications due to the temperature-dependency of density and viscosity, and to
separate flom those the effects of the particles as ‘temperature sinks’ on the carrier gas
field. The results of cases 1, 2 and 3 aid in the understanding of the effects of particle size
on various turbulent properties. Cases 2, 4, 5 and 6 were designed to Show the effects of
mass-loading (or initial particle number density). Cases 2, 7 and 8 show the effects of

varying the carrier gas temperature.

25

 

Table 2. Two-way coupling cases

 

 

Case 1;, q)", T
l 0.1 0.2184 293
2 1.0 0.2184 293
3 10.0 0.2184 293
4 1.0 0.2908 293
5 1.0 0.3637 293
6 1.0 0.4365 293
7 1.0 0.4472 600 (dTp/dt=0)
8 1.0 0.4472 600

 

Figure 6 shows the jet half-width growth rate for different particle sizes. The half-width
is defined as the transverse difference between the y-positions where the mean axial

velocity is equal to the average of the centerline velocity and the flee stream velocity

u +u u +u
Y1/2zy CI 00 _y c1 00
2 t 2
0P

The findings confirm the existence of “ghost particles”, which were hypothesized by

(3.1)

 

bottom

Ferrante and Elghobashi (2003). From the plot, it is clear that the larger particles damp
the Kelvin-Helmholtz instabilities which, in turn, decrease the growth rate of the jet; yet
it is also apparent that the addition of tiny particles slightly amplifies the instabilities
leading to a small increase in the growth rate. Therefore, it seems entirely probable that
there is a particular particle size that will have a minimal effect on the turbulence. A
poignant finding is that the particle with 1;, = 1.0 have the largest damping effect on the
carrier gas. Physically this could be explained as particles that have a particle response
time that is on the order of the time scale of the carrier gas turbulent kinetic energy
(TKE) will tend to dissipate the TKE in a way Similar to added viscous effects, damping

the instabilities. The non-linearity of the correlations for particle drag makes it very

26

difficult to correctly analyze (or verify) the effect of particle inertia on turbulence directly

 

 

 

 

 

 

    

 

 

 

flom theory.
2 , v I ‘ ' I
a
--- =10 r“
1.8 '- —- 1,510.0 // ’7
-— single—phase // I”
1.8 ' '
(b)
_ single—phaseslope:01302
— ty:0.l,slope=0.13l6 I
_ 1P:1.0,slope=0.1019
1.6 - --. 3:100. slope=0.1203
g .2
3. r 4 i
1.2 '
It", ‘
1 ‘ ' T
4 6 8

Axial Position

Figure 6. Effect of particle time constant on jet growth rate. (a) whole field and (b)
regression analysis of “self-preserving” region

Figure 7 shows the instantaneous values of the particle centerline axial velocity for
various rp. Note the similarity between the cases with 1,, = 0.1 and IP = 1.0. The
difference in peaks of these two cases indicates that the larger particles are not

accelerated up to the fluid’s local velocity, but rather have a finite drag. It is important to

27

emphasize that the plot represents the instantaneous velocity, not the mean (time-

averaged or ensemble).

1.2

 

AxialVdodty

0.8

 

 

 

 

0.6
0

Axial Position
Figure 7. Particle centerline axial velocity versus particle size

Figure 8 shows that an increase in mass-loading will amplify the effect of the particles on
the carrier gas as expected. There is a significant decrease in the slope of the linear region
as the mass-loading ratio is increased. This decrease follows a linear trend. Using the
mass loading as the independent variable (x) and the slope of the half-width in the self-
sirnilar region as the dependent variable (y), a linear regression analysis shows that
y = —0.1051x+0.1284,r = —O.989l; verifying the linear nature of the effects of mass-
loading on jet growth rate. Although not investigated directly, it would seem that if the
particles were small, they would increase the grth rate (as discussed previously). This
increase would be amplified by increasing the mass-loading ratio just as the growth rate

is attenuated by the addition of more large particles.

28

. "a \‘n'... .‘

1!».‘31- has

 

 

1.9 . . ,

 

 

 

 

I. a,"
N‘wl(a)
— 42.503134 ,jz; .
-- ¢.=0.2908 x / 1
- - - Que-0.3637 f ,’/
1.6 l' _..... 911504365 ,-"' ’I/ .4
g -—- single—phase [/34/ 1
g i/ 71/
' ,/ 3’1";
3 5/
1.3 _ ...._..,-' ’/;.:/’f _
,v" ..-" //
.-‘ 2 fl ’/
i- " ,v“, ’/ q
.o” ..-'” l
.i'!’
w" 1
1 A l A; l
3 6 9
Axial Position
1.8 ' r 1 f u I
L — single—phase, slope=0.l302 . (b)

— 63:022. slope=0.1019
-— 9,150.29. slope=0.0985
- - - ¢m=0.36, slope=0.0883
-- 6,5044. slope=0.0853

1.6 -

    

L

 

 

 

Axial Position
Figure 8. Effect of mass-loading on jet growth rate. (a) whole-field and (b)

regression analysis of self-preserving region
These effects can also be seen by looking at the axial velocity profiles. The centerline
mean axial velocity of single-phase free jets decays as x‘“2 in the self-similar'region. This
decay rate is expected to be lowered by the addition of large particles. Figure 9 shows the
centerline axial velocity profiles. Note the decrease in jet mean velocity decay as the
mass-loading ratio is increased. The root-mean-square (RMS) of axial velocity trend is

altered in a similar way. The RMS value reaches a higher peak earlier in the flow with

29

 

lower mass-loading. Then it decays faster than the higher mass-loading cases. There is a

‘cross-over’ point around x/h = 9.

 

 

 

 

 

 

Axial Position
Figure 9. Centerline axial velocity profiles as a function of mass-loading

Another important variable to consider when investigating planar jets is the Reynolds

stress term, u'v' , that appears in the production part of the TKE equation. For

convenience, the Reynold’s stress and the Production of TKE are averaged over the

length normal to the direction of interest. For example, the plot of W versus 2: is

actually the plot of TI—Ifidy versus x. Figure 10 shows the axial and transverse
°Ymax

variations of the Reynolds stress as functions of mass-loading. The Reynolds stress peak
magnitude is nearly halved with the addition of particles at a mass-loading ratio of 0.44,
and the integral of the velocity correlation term in the transverse direction is clearly less
than half of the single-phase case. When particles are added to the flow, the growth of the
Reynolds stress along the shear layer is severely hampered, indicating an increase in the

stability of the jet.

30

 

 

.3: a... .
— single-phase
8 -—— 9.50.22
g * --—- ¢n=o.29
a: —- ¢I=035

—0.004 - —- 0.50.44

 

 

 

 

fl.“ A A l A- ; l L a I
0 3 6 9
Axial Position

 

-0.01

Reynolds Stress

 

 

 

 

_0.m a I a l a I a
—2 —1 0 1 2

'h'ansverse Position
Figure 10. Reynolds stress profiles versus mass-loading. (a) axial variation at the
shear layer and (b) transverse variation at x/h=6
It is also easy to see the effects of the mass loading on the turbulence by considering the
production term (of the TKE equation) as a whole. The inclusion of the mean velocity
gradient helps to see the net production effects, not just the Reynolds stress. Figure 11
shows the production profiles for different mass loading ratios. The most significant

observation is that the production of TKE on the positive y-half of the jet swaps flom

negative to positive when moderately-sized particles are added to the flow. This shift

31

flom somewhat antisymmetric to symmetric production of TKE causes a change in the jet

development mechanism. There is also a noticeable difference in the magnitude of

production of TKE in the axial direction downstream of the jet inlet. The shear layer is

where the production should be close to its maximum. The cases with particles clearly

show a significant deviation flom the results for single-phase flow.

0.002

0.001

Production of TKE

 

 

   
 
     

— single—phase

 

 

 

  
 

0 —- 6,5022
--- ¢m=0.29
-- ¢m=0.36
-- ¢m=0.44
—0.001 L ‘ ‘ L
0 6 9
Axial Position
0.01 r . u u
— single-phase (b)

—— 6:022

-—- 6,5029

- - 6350.36

0.005 -- ¢m=o,44

Production of TX E

—0.005

 

 
   

 

 

—2

0 1 2
Transverse Position

Figure 11. Production of TKE profiles versus mass loading. (a) axial variation at the
shear layer and (b) transverse variation at x/h = 6

32

The temperature effects for the two-way coupling cases 7 and 8 without evaporation
show interesting aspects of the particles’ thermal inertia. The non-physical case (Case 7),
for which the thermal interactions between particles and canier gas is not allowed and the
particle temperature is artificially held constant, aids in the understanding of the
additional density effects due to the temperature difference. The more physical case
(Case 8) involves much more complicated particle-gas interactions. For one, the particles
act as temperature sinks, dropping the canier gas temperature in the core of the flow.
This causes a density stratification wherein the high-speed flow is cooler (and thus higher
density) than the low speed flow. This acts to stabilize the jet, decreasing the growth rate
(Colucci (1994)). It also causes the entrainment velocity to increase. The centerline mean
axial velocity profiles of Cases 2, 7 and 8 are shown in Figure 12, and the variation of the
mean axial velocity profiles are shown in Figure 13. When the gas temperature is
increased but the particles are not allowed to have thermal interaction, the mean axial
velocity is nearly unchanged, while the RMS of axial velocity is significantly damped,
especially in the region where the particles are more highly concentrated. When the
particles are allowed thermal interaction with the carrier gas, the centerline mean axial
velocity is attenuated, while the RMS of axial velocity is nearly unchanged. This would
seem to imply that the instabilities generated by the particles are nearly balanced by the

attenuation of the instabilities due to temperature differences.

33

w mfiqfl.“mmfi - any.”

 

   

 

  

 

 

 

1.1 - - u w - 0.2
— theold (Case 2)
--- inflict, dT/dt=0(Case 7)
—— ltflhot (Case 8)
1 .
s f
3 '3
> >
'6 0.9 0.13
a --- umrhot, dT/dt=0
__ “ha 3
I, \m‘ g
0.8 ,” \‘\\ ‘
I ‘1
0.7 ‘ ‘ ‘ ‘ L 0
3 6 9
Axial Position

 

.0
oo

.0
05

Mean Axial Velocity

 

 

 

 

Transverse Position
Figure 13. Two-way coupling temperature effects on velocity profiles at x/h=6

2.2.2 Evaporating Droplets

There are several interesting effects that evaporation has on the gas-droplet interaction in
a planar jet. These are best analyzed by isolating the evaporative effects flom the
momentum and heat interactions that are discussed above for the non-evaporative cases.

Table 3 explains the different cases relevant to evaporation, Figure 14 shows the growth

34

 

rate of the jet half-width for the different cases, and Figure 15 shows the centerline
velocity trends for the same cases. A special case was considered to discern the flow
modification due to density stratification and due to phase interactions. A case without
particles was conducted wherein the initial temperature profile was chosen to
approximate the profile observed in the evaporating cases. The temperature of the jet was
reduced flom nondimensional temperature T=2.04 to T=l.5; the co-flow was kept at
T=2.04. In Figure 14(b), the jet development is quantified by calculating the approximate
slope of the jet half-width at different locations flom the jet nozzle. This plot shows that
the addition of particles with no temperature or mass interaction does not significantly
change the jet growth rate. When the temperature effects of the particles are considered,
there is a very large change in the growth rate. When there are no particles and the initial
non-uniform jet density profile is introduced, the modification to the jet growth is slightly
less than the thermally coupled case. Also, when the effects of evaporation are
considered, the jet grth rate is damped as compared to the two-way coupled case with
no evaporation. The density stratification effects seem to contribute significantly, but not
exclusively to the modification of the jet growth rate.

Table 3. Evaporative case descriptions

 

 

Case Description

Case 1 Single-phase/One-way

Case 2 One-way with Density Stratification
Case 3 Two-way, with dTp/dt = 0

Case 4 Two-way

Case 5 Evaporation

 

35

 

 

 
    

 

 

 

 

 
 
   
 
   

 

 

 

2 ' I ' I ‘ F
--- singIe—phase (Case 1) ”en” (a)
— - single—phase, dens—strat. (Case 2) ./ ...--—
1-75 " — two-way, dTvldt=0 (Case 3) /I/ '
— — two-way (Case 4) '
--- evaporation (Case 5)
5 15 -
3
B .
i
:1: 1.25 -
1 "" /‘{$/ "‘
affl/ 1
0.75 ‘ ‘ ‘ P ‘ 1
2 4 6 8 10
Axial Position
2 ' T ' l ' F r )
--— slope=0.1372
-- slope=0.1625 ’,/'
1-75 r — slope=0.1371 f.»
L —- slope=0.1714 ','
--- slope=0.1686 ’
g 15 -
s .
’3
a: 1.25 s
x?”
1 :’;”/ J
4
0.75 L 4 ‘ J ‘ 1 ‘
5 6 7 8 9
Axial Position

Figure 14. Jet half-width versus coupling (T=600) (a) actual growth, and (b) linear
regression of the linear portion of the growth rate

What is striking in Figure 15 is the modification of the RMS of axial velocity. The
difference between the two-way coupling cases with and without evaporation seems to be
more pronounced in the velocity fluctuations. The tendency of the RMS of axial velocity

to decay rapidly after reaching its peak is reduced in the evaporative case. It would seem

36

as though the added mass due to evaporation clamps the peak value, but sustains the

turbulence levels further beyond that peak.

 

 

    
 

  
 
 
 
 
 
  

 
 

 

 

 

1 -------
‘ ‘ ~ - 0.4
in- M—mvag-z‘ ‘
_ un’ l-Way ~ QTX?‘\
5. —- um, 1-way, dcns.-strat. ‘- 3’
‘2 --- um, 2—way, dT ldt=0 \‘~‘§. ' 033
3 0'8 ' —- u 2—w P -=.. 0
> m’ 3y >
a ’- - - um, cvap. ' 3
< - 0.2:
g —- um, l-way ‘5
—- um, 1-way. dens.—-strat.
2 0-6 ' --- u“. 2—way, dTpldt=0 ..... 2
— - um, 2—way 0. 1
- - - um. cvap. ’,
- -"' ' ’ av ’ ’ I I
0.4 " "' '1 ' - - 1 - I o
2 4 6 8

Axial Positim
Figure 15. Centerline axial velocity profiles as a function of coupling (T=600)
The Reynolds stress profiles for the different one- and two-way coupling cases are shown
in Figure 16. Note the considerable modifications in production due to the evaporative
effects. Specifically, there is a considerable decrease in the Reynolds stress at the shear
layer for the two-way coupled case. The case with evaporation showed an increase in
Reynolds stress over the two-way coupled case except between x/h 2' 7 and 9. A decrease
in the Reynolds stress is generally associated with a decrease in the velocity fluctuations,

effectively increasing the stability of the jet. Hence, a decrease in the growth rate of the

jet is to be expected.

37

 

 

  

 

 

 

 

    

 

 

 

l a)
0 ‘5 5., - h -' - 5 ‘ d
x ' d. - ‘7
. / ' l
3 ' /
\\ ,z”
5’, —o.002 - \T" ..
é ‘ ~ ~ ‘ s
g. — one—way I
03 — one-way. dens—stint.
—0.004 - --' two—way, dTvldt=0 ~..
- - two—way T“
-- evaporation \
—0.006 ‘ ' ‘ L ‘ i
2 4 6 8
Axial Position
(“0)
0.005 P 4
I“ g \‘
8 /' /”>\‘ “‘3
3'3
fl —0.00S
i»
or
-— one-way, dens-mat.
—0.015 - --- two—way. dT/dt=0 '
—- - two—way
—- evaporation
_0.m5 a I A. l a l a
—2 —l 0 l 2
Transverse Position

Figure 16. Reynolds stress profiles for different coupling (T=600). (a) axial variation
at shear layer and (b) transverse variation at xlh=6

The modification of TKE production due to evaporative effects can be seen in Figure 17.

Of particular interest is the increase in the peak of the production of TKE flom the non-

evaporating to the evaporating case near x/h z 9, as seen in Figure 17(a). Farther

downstream, the two curves tend toward each other. There is also a definite increase in

the production of TKE in the evaporation case over the one-way case with density

38

stratification. Evidently, the density—stratification has increased production initially,

followed by a steady decline towards the one-way coupled case with uniform density.

 

 

 

 

 

 

  
  
   

0.002 v . u 1 f r - - u
(a)
g 0.001 -
‘5
i
2 o
-— one-way. dens-strat. \,
--- two-way. dT/dt=0
-- two—way
—- evaporation
_0.ml A A I A A l A A l
0 3 6 9
Axial Position
0.01 V l V ' v I u (b)
r- — one—way
--- one-way, dens—stat.
- -- two—way. dT,/dt=0
—- two-way
0_005 —- evaporation

ProductionofTKE
o

 

 

 

_0-m5 A l A L A l A
-2 — 1 0 l 2

Transverse Position

Figure 17. Production of TKE profiles for different coupling (T =600). (a) axial
variation at the shear layer and (b) transverse variation at x/h = 6

It is important to realize that the effects of evaporation are attributed to several competing
physical occurrences. The non-evaporative particles in the hot environment will act as

temperature sinks, but the addition of evaporation will decrease the temperature even

39

more. The axial temperature profiles for evaporating and non-evaporating cases are
shown in Figure 18, which clearly shows a nearly constant decrease in temperature
downstream of the jet inlet. The value of this decrease is about 0.2 non-dimensional
temperature units, or approximately 55K in standard units, give the parameters used for
this study. The one-way density-stratified case seems to agree more favorably with the
two-way coupled case than the evaporative case, indicating that the added mass effects in

the evaporative case uniquely modify the jet.

 

 

 

— one-way

-— one—way, dens—swat.
Otwo—way, dT 1,ldt=0

— - two—way

—- evaporation

i.

 

 

l L A l L n l n A I A

0 3 6 9
Axial Position
Figure 18. Temperature variation in the axial direction for different coupling cases.

 

In addition to its effects on temperature, the evaporate vapor will contribute mass, adding
to the density of the carrier gas. These combined effects may explain the observed
differences between the cases with and without evaporation considered. The density and
mass fiaction of vapor variation due to temperature and evaporate added mass effects are
shown in Figure 19. It is important to note that the nondimensionalization of the variables
leads to the ability to directly add or subtract the density and the mass fractions. The

difference in density between the two-way coupled cases with and without evaporation is

40

nearly mimicked by the mass fraction of evaporate plot. There is, however, a small
difference between these two that is due to the actual heat of vaporization of the droplets

modifying the temperature and density fields.

0.8

 

p
Q

 

 

—- p. one-way

— p, two-way

--- p. evaporation

- - Y, evaporation
— - p-Y, evaporation

Density, Mm Fraction
,0
h

 

 

 

Axial Position
Figure 19. Density and mass fraction of vapor variation in the axial direction for
different one- and two-way coupling cases
The transverse variation of temperature and density are shown in Figure 20. Not the clear
density stratification due to the temperature and added mass effects. The effects of
density stratification are explained in the introduction section and throughout this paper.
To summarize, if the higher speed stream is of lower density than the lower speed stream,

the Kelvin-Helmholtz instabilities will be attenuated. Add to that the effects of the

particle drag, etc. and there are interesting modifications to the jet structure.

41

 

 

 

 

 

 

 

1.2 ' f T T Y fir
r ‘79 r? '—
c“. f,’ - 2
‘Q .-'I
1 - <3... ,’
:“a y. I
\ "a I, I
\ ’0' ’
\\ ----- 'ro‘” é
‘ K I, l
‘ x _. , q 1
b \ \ V I .p _ 'I‘. onkway . ‘
I
0.8 - --- p. one—way "' ---- T. two—way g.
-- p. two—way ,’ a --- T. evaporation a
. \ _
--. p. wamration ’1’ “\ h
I, \
I 'm‘ \ .1
06 . _,..- .-_ . 1.2
. I, M}; .\ \\
I .-" \
" a. x
47 ' ,3‘
0.4 ‘ ‘ ‘ ‘ ‘ l ‘ 0-3
-2 -l 0 1 2
Transverse Position

Figure 20. Transverse variation of temperature and density

As discussed previously, different sizes of droplets or particles have different effects on
the carrier gas (and the carrier gas affects them differently as well). Clearly, if droplets
evaporate at different rates, there will then be a size distribution that could cause
interesting modifications to the statistical properties of the turbulence. A plot of the
development of the probability distribution function (pdf) of particle mass for the 1;, = 1.0
case is shown in Figure 21. The pdf of particle mass at the inlet would look like a delta
function, as the injection parameter was set to uniform particle size. As the flow develops
and droplets begin to evaporate, the pdf shifts towards a more Gaussian shape, indicating
that there are some particles that are not evaporating very much and that some are
vaporizing very rapidly. The plot does not allow for the discernment of local vaporization
rates, as it represents the pdfs of all of the particles at a particular x-location (integrated
over y). Figure 22 show a comparison of the local average particle mass and the Reynolds
stress. The transverse profiles show that at locations where the droplets have evaporated

(i.e. small dr0p1ets), there is an increase in the Reynolds stress. This seems to correlate

42

well with the notion that smaller particles enhance the TKE, while larger particles

attenuate the turbulence.

 

 

 

 

0.3 P a
— xih=3
"" xlh=6
b -- “:9
g 0.2 '- fi "
,t
, t
i I ' ‘.
t“ ’ it 'I ‘ i
I \ I ‘
g I t ,’\ f ‘.
G 0.1 - I i \ I i -'
m I I \' \
/ ,. \ i
/ ,’ \ k
I — ’--"- \ \ \\
(1"51’ \ ‘\\
0 g A l A \‘l'\ A‘s L
0 213—06 Ale-06 6e—06 8e—06 le—OS
ParticleMass

Figure 21. Development of the probability distribution function of particle mass

 

 

 

 

 

 

 

 

0.04 . r - r r I . 8e—06
- 443-06
3 0.02 " a
1
a z
i 0 g
a rat ,,
Q ’2 x ‘ —' _ 5 \ m
m o --—-—~.v-¢-~ /’ ,’ \M r
\ I I
\ E ‘ .. 4e 06
\ "-._ I
\\ V I
\ ~ /’
p.02 - 1 . . . . . ~8e—O6
—2 —1 0 l 2

Transverse Position
Figure 22. Eulerian averaged particle mass and Reynolds stress profiles at various
downstream locations

43

2.2.3 Interpolation

To verify the accuracy of the numerical schemes used in this study, various tests were
devised and implemented. One of the more crucial aspects of the Eulerian-Lagrangian
formulation requires the accurate calculation of the carrier gas properties at the particle or
droplet location, which requires interpolation. Two different interpolation schemes were
implemented for this work: first-order linear and fourth-order Lagrange polynomial. The
results for the two cases were compared, and the accuracy of the linear interpolation was
determined to be sufficient to resolve the physics of the jet. This was indeed fortuitous, as
the computational efficiency of the linear scheme was nearly twice that of the Lagrangian
scheme. The case used for comparison was the most complicated physically, including
evaporative effects at high temperatures. Figure 23 shows the temperature profiles of the

two interpolation schemes, and Figure 24 shows the jet half-width growth rate.

 

24 - . , . . . . . I . 0.3
-- - Tn, linear
-- - T...- Lagrange
— T linear
21 >- at?
L — T Lagrange
E i m” 0.25
a 2
“ 8.
8. ;
g 1.8
*' §
3 m
z " 0.15
1.5
1.2 0

 

 

 

 

Axial Position
Figure 23. Temperature profiles versus interpolation schemes

While there are some differences to be noted, the two are within any acceptable

experimental margin of accuracy. It is important to realize that some of the larger

44

variations at near the outlet could be attributed to numerical growth of physical

inaccuracies that are commonly associated with certain boundary conditions.

 

    

 

 

 

2 ' l I r I r
L — lst—order linear .m‘j
— - 4th—crder Lagrange
1.5
i
E
1
J
0.5 A A l A A L A A l A
0 3 6 9
Axial Position

Figure 24. Jet growth rate as a function of interpolation scheme

45

CHAPTER 3
LARGE-EDDY SIMULATIONS OF A ROUND JET
3.1 Governing equations and computational methodology
In large eddy simulation (LES) methods, the “resolved” carrier gas field is obtained by
solving the filtered form of the compressible continuity, Navier-Stokes, energy and scalar

equations, together with the equation of state for pressure

 

607) 5<P>I<Ui>L_
at + 0x.- "(SP>1 (3.1)

6(p)1(u,-)L +a<P)I<“i)L<uj>L :_5(p)1 +_1_5(Tij)1

 

 

 

 

 

 

 

 

 

 

at axj 6x,- RC 6x]-
1 5(1)]?!
’1‘; axj +<Suz)l (3’2)
6<p>1<E>L +6<p>1<ui>L<E>L =_ 1 6<qi>1_0Nz
at an (y—1)Maz Re Pr an 6X:
+<nSe>1 (3.3)
a< ><r> a< ><u~> <Y> 0W) ,M.
plaL+ ”11L aL:_ 1 1__ r
at axi RePrLea 6xi axi
+(psa), +<Sg>l,a =1,2,...,N, (3.4)
NS Y
<p>1 z<p>1RO<T>12< ;>L (3.5)
1 a
(r--)z(,u) a<ui>L+a<uj>L -3 ~a<uk>L (y) =Pr<k/C ) (36)
y L 6xj 6x,- 3 l] axk ’ L p L '

46

where (f(x, 0); and ( f (x,t)) L =( p f) I /( p) I represent the filtered and the Favre-
filtered values of the transport variable f (x, t) and p, u), p, T, E, and Ya are the fluid

density, velocity, pressure, temperature, energy and mass fraction of species a,
respectively. In equation (3.5) ,u, k, Cp and Pr are the viscosity coefficient, the thermal
conductivity coefficient, the specific heat and the Prandtl number, respectively. R0

denotes the universal gas constant and Wa is the molecular weight of species a. The
subgrid-scale (SGS) closures that appear in the above filtered equations include the SGS
stress (1‘3) 2 (p) L Kuiuj>L —<u,-) L (u j>L land the SGS energy and scalar fluxes. Jaberi

and Colucci (2003) describe the SGS models chosen for this particular body of work in
detail. Here all of these terms are modeled with “standard” diffusivity closures. The

effects of the droplets on the carrier fluid are expressed through the mass (S p),

momentum (Sui) and energy (S e) source/sink terms as described below.

The droplet field is solved via a Lagrangian method under the point-source

approximation. In this method, the evolution of the droplet displacement vector (X i ), the
droplet velocity vector (vi ), the droplet temperature (T p ), and the droplet mass (m p ), is

governed by the following equations

 

dX-

dtl =vl- (3.7)
d .

31— :11—(uf —v,-) (3.8)
dz rp

47

 

dTp :12—(1'*—T )+ Lv dmp

 

3.9
dt 7p P mch dt ( )
dm m
_P_=_ ”f3 1n(1+BM) (3.10)
dt 7p

where the asterisk refers to the local fluid variables, which are interpolated to the droplet
position, and r p is the normalized droplet time constant. The variables 77, f1,f2 and f3
are functions of the droplet and carrier gas parameters such as the droplet drag

coefficient, the drOplet Reynolds number, and the Prandtl number. BM is the mass transfer

number as calculated by the Langmuir-Knudsen non-equilibrium model. In the following

 

 

 

terms
3Pr2' ldm
n= fl’B . fl=-[ 1’] p (3.11)
e _1 2 mp dt
CDRC
P
= 3.12
f1 24 ( )
Nu C1
= , =_ 3.13
f2 3pm 0 Cp ( )
Sh
= 3.14
f3 3S6 ( )

where Rep is the Reynolds number based on the particle diameter and the slip velocity,
Reb is the blowing Reynolds number due to evaporative blowing velocity, Nu is the

Nusselt number, Sh is the Sherwood number and Sc is the Schmidt number.

The volumetric source terms appearing in the carrier gas equations are evaluated based

on the volumetric averaging of the Lagrangian field as

48

Sp =7}; {-mp[-f—3—ln(l+BM)]} (3.15)

 

7p
l * m Vd'
Sui S8:_-§—V-Z{_—m:p fig-(w -v,-)+ :p n _mp[z_i:_ln(l+BM )Vi:|} (3.16)
12 +f—1— Vdr1V1
=‘— T Tpu1 ”V1 1

 

f3 has ViVi
inp[—T;1nl(1+131:)][(7”Um2 + 2 J} (3.17)

where the summation is taken over all droplets in the volume 5V = 6x3 centered at each
Eulerian (grid) point. Equations (3.1)-(3.17) describe the general Eulerian-Lagrangian
formulation of a compressible two-phase turbulent system with full mass, momentum and
energy coupling between carrier and dispersed phases. For the purposes of this study, the

effects of gravity and evaporation are ignored. Hence, gi, vdn, 151p and S p are zero.

In LES, the above filtered carrier-gas equations are solved together with diffusivity-type
closures for the SGS stress and the SGS energy flux terms. For the droplet phase, a
stochastic model is considered in which the SGS diffusivity (evaluated from large scale
quantities) is used to construct the residual or subgrid velocity of the carrier fluid at the
droplet location. The combined large- and small-scale fluid velocity is used to move the
droplets and to calculate the droplet drag force. The discretization procedure of the carrier
fluid is based on the “compact parameter” finite difference scheme, which yields up to
sixth order spatial accuracies. The time differencing is based on a second order method.

Once the fluid velocity, density and temperature fields are known, the droplet transport

49

equations are integrated via the second order Adams-Bashforth scheme. The evaluation
of the fluid properties at the droplet locations is based upon a second and fourth order
accurate Lagrangian interpolation scheme. The location and size of droplets at the jet

inlet vary in different simulations.

3.2 Results and discussion

3.2.1 Comparison with experiment

An a posteriori analysis is conducted to assess an stochastic subgrid-scale (SGS) closure
used in the large-eddy simulation (LES) method for computing two-phase turbulent
flows. Specifically, LES and experimental results for a particle-laden axisymmetric
turbulent jet are compared. The experimental results are taken from U. Fritsching, et a1.
(2001), for which a phase-Doppler anemometry (PDA) system is used to measure the
velocities of the seeded carrier gas and particles. The physical parameters for the two-
phase flow configuration are given in Table 4. An analysis of the results for the single-
phase flow configuration is also included, although an emphasis is placed on the ability
of LES to accurately represent multiphase effects. A qualitative picture of the flow

configuration is shown in Figure 25.

Table 4. Physical parameters

 

Reynolds number 5700

Tube diameter 12mm
Carrier gas Air

Mean particle diameter 110mm
Mass loading 1

Dispersed phase Glass beads

 

50

 

Figure 25. Vorticity magnitude and particle distribution

The data for three different LES cases are compared to the experimental results: Case 1)
no SGS model for particle-carrier gas interaction with uniform inlet particle size
distribution; Case 2) an stochastic SGS model for particle dispersion with uniform
particle size distribution; and Case 3) an stochastic SGS model with Gaussian particle
size distribution. A summary of these cases is given in Table 5. Three variables in
particular are used to compare the results: mean axial velocity, um, root-mean square
(RMS) of axial velocity, um, and turbulence intensity, Tu. These variables are
measured/calculated at various locations in the flow field. The results are reported for

single— and two-phase flows, and for LES with various SGS closure/particle size

 

 

configurations.
Table 5. Case summary
Case SGS Particle Model Particle Size Distribution
1 No Uniform
2 Yes Uniform
3 Yes Gaussian

 

51

A plot of centerline mean axial velocity versus downstream location for the experimental
and computational results of two-phase cases is shown in Figure 26. Note that Case 3, the
case with SGS particle closure and initial Gaussian particle size distribution, best
reproduces the experimental centerline velocity. Case 2 is nearly as accurate as Case 3
and Case 1 is considerably inaccurate in comparison to the other cases. This is the first
indication that the new SGS closures are indeed viable. The slight increase in accuracy
from Case 2 to Case 3 indicates that some of the errors associated with the LES
calculations are due to actual physical variations in inlet particle size and locations, and
are not simply due to modeling errors. The difference between experimental and
numerical results is better shown in Table 6, where the percent error for centerline mean

axial velocity at several axial locations is considered.

Figure 27 shows the radial variation of axial velocity at two different axial locations and
Table 7 gives some sample percent error calculations for those locations. Note that the
difference between LES and experimental results at x/D = 7.5 increases with increasing
radius (i.e., the tails match less), but that farther downstream (at x/D = 12.5), they match
much better. The percent error calculations show that there are positions where each case
best matches the experimental data. However, on average, the results of Case 3 are

significantly better than the other two.

52

 

 

 

 

0.4 A A 1 A A l A A l A A l A A
0 3 6 9 12 15

Axial Position
Figure 26. Centerline mean axial velocity versus downstream location

 

Table 6. Sample percent error for mean axial velocity at jet centerline, ud

 

 

Axial Location (x/D) Case 1 Case 2 Case 3
1.25 4.24 3.70 3.69
6 3.93 1.63 0.09
7.5 5.49 3.96 1.40
9.25 10.46 2.34 1.35
12.5 10.40 2.29 0.27

 

53

 

0.8 . u . I . i . r
(a)

OEXP

.0
a

Mean Axial Velocity
.0
h

.0
N

 

 

 

 

 

 

 

 

 

0A0.5A1A1.5‘2A25 3‘35
Radial Position
Figure 27. Axial velocity profiles at (a) x/D = 7.5 and (b) x/D = 12.5.

Table 7. Sample percent error for mean axial velocity, h
Axial Location Radial Location Case 1 Error Case 2 Error Case 3 Error

 

(x/D) (r/ro)
2.5 0.3 0.42 0.42 0.45
5 1.2 12.31 9.72 11.64
12.5 0.3 9.75 4.16 1.52
12.5 2.3 5.92 9.15 1.56

 

54

The spatial development of the velocity field of the carrier gas and dispersed phase for
each of the two-phase cases listed in Table 5 is shown in Figure 28. Once again, note the
striking similarities between Case 3 and the experimental results. It is easily seen that the

LES captures the two-way coupling effects, as there is good agreement for both carrier

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

gas and particle phases.
x Particle x Particles
— Carrier Gas : — Carrier Gas
1
> >
l
(a) Experimental (b) Case 1
x Particle x Particles
— Carrier Gas t — Carrier Gas
> x
l
I
(c) Case 2 ((1) Case 3

Figure 28. Spatial Development of Axial Velocity Profiles at x/D = 2.5, 5 and 12.5

55

It is significant to note that the experiment was conducted with relatively large particles,
2
pp "’p

with r =
p 18,11

= 57.4. This indicates that the particles have a high inertia to drag

force ratio, and tend to follow their own path rather than the path of the fluid (although
they still decelerate and grossly affect the carrier gas). It is also important to note that
there is a significant variation in the calculation of the radial quantities. The experimental
results represent one cross-sectional slice of the jet, while the LES results represent an
azimuthally averaged radial profile. Considering the precision of the centerline quantities
(where there is no significant difference in sampling), it is not unreasonable to assume
that much of the increased errors seen in the radial profiles can be attributed to the
difference in sampling between the experimental and LES results. This is mentioned only
in an effort to remind the reader that there are fundamental differences in the way that
data can be interpreted, especially in the case where there is a significant amount of data
that is available for use in calculations. It would be very difficult for an experiment to
measure the whole field, whereas the numerical simulation allows for the calculation of

the variables at all points in the field.

Figure 29 shows the profiles of the root-mean-square of axial velocity at x/D = 5 and
12.5. Note the significant increase in agreement as the jet develops, indicating that the
effects of the two-way coupling are captured with increasing accuracy as the jet develops.
It is also an indication that the initial region is highly dependent on the inlet conditions.
Table 8 is a collection of various percent errors in the RMS values. Again, notice the

increased accuracy of the Case 3 results at farther downstream locations.

56

 

  
 
 
   

 

 

 

 

0.15 — 1-way - (a)
,- ’ --- 2-way. 1:,=1.0
i" " \ - -' 2—way. 15:10.0
5 ' // —- 2—way, tp=57 .4
i /
> 0.1 " /
3
‘8
g 0.05 -
0 I g I
0 0.5 1
0.15 I I I I I t I
—- 1—way (b)
I --- 2-way. 1:,=1.0 .
‘\ - -- 2-way, 45:10.0

—- 2-way, 1:,=57.4

 

 

 

0 A l A L A I A I A I A I

0 0.5 1 1.5 2 2.5 3 3.5
Radial Position

Figure 29. Radial profiles of RMS of axial velocity at (a) x/D = 5 and (b) x/D = 12.5

 

Table 8. Sample percent error calculations for arms

 

 

Axial Location Radial Location Case 1 Case 2 Case 3
(Km) (r/ro)
2.5 0.9 20.14 28.27 27.78
5 1.2 21.13 15.93 12.50
12.5 0.3 0.57 22.92 5.81

 

The last statistical quantity that is examined in this section is the turbulence intensity, Tu.

The turbulence intensity can be calculated as

57

12 12 '2
ra=“ +v +W 4mm. am

r2+vz+wz

 

Figure 30 shows the radial profiles of the turbulence intensity for the experimental results
and the LES results of Case 3 at x/D = 5 and 12.5, and Table 9 gives some sample
percent error calculations. Again, it is observed that the LES predictions are in good
agreement with the experimental results. It is important to note the relative magnitude of

the turbulence intensity, the values are very small, so slight Tu variations cause large

 

 

 

 

 

error deviations.
>5 0.16 '- OXID=5,Em -(
*" , .x/D=125.Exp
3 i — - x/D=5. LES
> — 00:12.5. LES
I 01
3 0.12
‘6
.5“
g 0.08
.5
a 0.04
E
t-
0 —T"d-i A l 4 J A I A I A I A I“:-
-2 —l.5 —1 -0.5 0 0.5 1 1.5 2

Radial Position
Figure 30. Radial profiles of turbulence intensity at x/D = 5 and 12.5

Table 9. Sample percent error calculations for Tu

 

 

Axial Location (x/D) Radial Location (r/ro) Case 1 Case 3
5 0.4 32.8 7.38
5 0 93.7 3.12
12.5 0.9 35.9 7.71
12.5 0.1 22.9 2.05

 

To further validate the accuracy of the SGS models, comparisons between experimental

and LES data are made for the case of single—phase flow. This enables the verification of

58

the effects of the particles on the carrier gas. As shown in Figures 31 and 32, there is
good agreement between both the single- and two- phase flow configurations, a clear
indication that the SGS models employed are viable for both. Both the experiment and
the simulation show that adding heavy particles to the flow causes the centerline velocity
to decay at a slower rate. This can be rationalized in that the heavy particles effectively
pull the carrier gas along at their rate, slowing the deceleration due to viscous effects. An
increase in turbulence intensity was also noted when particles are added to the flow,

which also correlates well with the experimental results.

 

Mean Axial Velocity
.o .o
‘1 00

.o
as

 

 

0.4

 

 

Axial Position
Figure 31. Mean centerline axial velocity of single- and two-phase flow

59

0.13.....,.r

 

 

 

 

A l

91.25 —0.75 A -025 A 0.25
Radial Position
Figure 32. Radial profiles of mean axial velocity at x/D = 7.5

0.75 1.25

 

3.2.2 Additional physical observations

In Section 3.2.1, we have validated our LES models and numerical scheme by comparing
the numerical results with experimental data. In this section, we use the LES to delve into
the physics of two-phase turbulent jet flows. For example, further analysis of the particle
number density indicates that the total number of particles integrated over the jet cross-
stream direction increases along the jet direction. This is shown in Figure 33 and is due to
accumulation, clustering and overall lag of the particles by the particle drag force. The
phenomenon can be pictured in other ways as well. Figure 34 shows the velocity of the
carrier gas decreasing much more rapidly than the particles. The ‘faster’ particles near the
inlet move downstream quickly, effectively ‘catching up’ with the particles near the exit

plane.

60

 

m V ' " V V I v V I

— integrated over y and z
__ y = 5111 +1984, r = 0.68

p

“a?

 

Particle Number Density
“8’

g
l
l

 

 

 

O 4* I A A L A A l A A
0 3 6 9 12
Axial Position

Figure 33. Particle number density versus downstream distance

 

 

O

 

Axial Velocity

 

a...
O

 

 

0.15 —1‘-0.5‘ 6 ‘03‘ 1 ‘rs
Radial Position

Figure 34. Carrier gas and particle velocity versus radial position at (a) x/D=2.5, (b)

x/D=5.0 and (c) xlD=12.5

 

Figure 35 shows the variation of the particle velocity with particle size or inertia. At the
inlet of the jet, the particles are given a velocity that is nearly uniform and close to the
local canier gas velocity. The particle size/mass is randomly chosen and there is no initial

correlation between particle mass and location and/or velocity in any way. Farther

61

downstream, the larger particles tend to stay at their initial velocity, while the smaller
particles start to deviate from their initial velocity due to drag effects. Hence, a

relationship between the particle mass and velocity can be shown to develop.

 

 

.’ . . OXID=5.0

 

 

 

02 _ . . .XID = 12.5
. . -- - lin. reg. xID=5. slope=-4.8
. — lin. reg, x/D=12.5. slope=102.0
. o
0 A I A l A
0 0.001 0.002 0.003

Particle Mas
Figure 35. Correlation between particle axial velocity and particle mass at different

downstream locations
It is not only the axial velocity of the particles that is affected by the carrier gas. In the
LES conducted herein, the initial velocity of the particles is exclusively in the axial
direction. If the carrier gas has no effect on particle dispersion, one would expect the
particles to remain at their initial (inlet) radial location throughout the jet. Figure 36
shows otherwise. Several of the smaller particles deviate from their original location as
they move downstream. This indicates that they must have gained a radial velocity
component fiom the carrier gas two-way interaction drag effects. However, the larger
particles tend to remain in the core of the flow. The linear regression shown in the figure

is intended to show a general trend, not an empirical relationship.

62

0.003 - r - r

 

3 OxlD=5
- “8' Cx/D=l2.5 J
‘3 0 . --- lin. reg. xlD=5. slope=0.00007

O . — lin. reg. xlD=12.5. slope=-0.0004
0.002

d

Particle Mass

0.001 ‘

 

 

 

 

3

Radial Position
Figure 36. Particle mass versus radial position at different downstream locations

Another way to see the preferential concentration of particles in the jet is to look at a
comparison between the particle number density profile and the average particle mass
profile, as in Figure 37. The total number of particles entrained in the core of the jet
increases in the axial direction, forcing the average particle mass to decline. This does not
mean that the larger particles are being replaced by smaller ones. It seems more likely
that the average mass is decreasing due only to the addition of smaller particles, not the
loss of large particles. This also may be due to the fact that the larger particles are swept
downstream at a high velocity while some of the smaller particles are caught in low
vorticity regions near the core of the jet. It is interesting to note that there are peaks in the
profiles of both the average particle mass and the particle number density near the shear
layer. In addition, both plots indicate that there is noticeable particle dispersion for the

x/D=12.5 location, as evidenced by the smaller peaks beyond r/D=0.7.

63

 

 

 

 

 

 

 

0.0015 . . . . . . . 25
— xlD = 2.5 — n’, xlD=2.5
—- mrxlD =12.5 —- nyxlD=125 '
' - 20
0.001 ~ ‘ j
i ‘1;
3 .1
0.0005 - . a
AA
4 5
0 up A A A l A "_._". o
-15 -1 . 1 1
Radial Position
Figure 37. Particle mass and number density radial variation at different axial
locations

The clipped Gaussian particle size distribution applied at the inlet of the jet was with
respect to the particle diameter. The probability distribution of the particle size by mass is
shown in Figure 38. Of particular interest is the positive skewness of the pdf, indicating
that there are more excursions from the mean on the larger particle side. This distribution
remains nearly constant at different axial locations. The pdfs are integrated over the entire
plane perpendicular to the jet; so local particle distribution is not distinct, only general

axial trends.

64

 

025 - r . r fi r

 

— inlet
--- x/D=5
0.2 ' “ —- x/D=12.5 ‘

0.15

0.1

PDF of Particle Mass

005

 

 

 

0.004

 

Figure 38. Probability distribution function of particle mass

Velocity correlations can be very insightful as to the physics of two-phase jet flows. One
can see how carrier gas production, transport and dissipation are affected by the particle
phase by looking at particular correlations. In the computation of a two-phase jet,
different correlations become available and relevant. Consider Figure 39, which shows
the axial variation of the particle-carrier gas velocity correlations, which are integrated

over the plane perpendicular to the jet at each axial location. The correlation between the

 

o A D A * n o
axial velocrtres of the carrier gas and the particles, u pu , where the asterisk (*) again

denotes the carrier gas property interpolated to the particle position, is the most
significant, meaning that the ability to predict the carrier gas axial velocity knowing the
particle axial velocity is better than the other velocity correlations. The velocity
correlations decay along the axial direction as expected, indicating that there is increased
randomness in the slip velocity farther downstream than there is in the near-field. Also,

the radial components of the particle and carrier gas velocities are not very well

65

a avA-‘li’

 

'rl". "L.-.

 

 

 

It , , =1: :1:
correlated (v pv z 0 ), while the cross-correlations, u pv and vpu , are both the same.

 

They are opposite in sign of up; and of lower magnitude.

 

 

 

 

 

 

 

 

0.01 . r . fi .
a W
. “5“ .
8 0.005 —v,v"
i .,
8 Ov’u‘
5‘
i 0 _ _._ — .
;>
06400-00
<>-o~e-o-e‘o-0ve~o-@‘°‘°‘e’°e'e’o
—0.005 ‘ ‘ ‘ ' ‘
0 5 10 15
AxiaiLocation

Figure 39. Axial variation of correlation between particle and carrier gas velocities

Figure 40 shows the same velocity correlations as Figure 39, but here the focus is on the
radial variation of the velocity particle-carrier gas correlation, as the correlations are
integrated over the jet length for each radial location. The sharp peaks appearing farther

away from the centerline are caused by a low particle density in those regions. There are

 

interestmg differences between the axral variation and the radial variations. The upu

 

correlation is still the most Significant and the vpv correlation is still very poor.

 

 

 

:1: =1: _ =1: ,
However, the u pv and vpu are no longer equal, and it seems as though the u pu rs

 

 

 

balanced by the u pv* , while the vpv* is balanced by the vpu* . A portion of the

disparity in Figures 39 and 40 can be attributed to the integration; the particle dispersion

66

effects are not accounted for on a local level when the correlations are integrated over the
entire jet length or cross-plane. Additionally, the sample size varies greatly in regions of
low particle concentrations. Figure 40 implies that the particle axial velocity correlates
well with both components of the carrier gas velocity, whereas the radial velocity of the
carrier gas has minimal correlation with either component of the carrier gas. Recall that
the particles considered in this study are quite large, and that they are not given any radial
component of velocity initially. Any radial velocity that they obtain is thus very small
and somewhat random, which helps to explain the observed low correlation. It is
interesting that the radial velocity of the carrier gas correlates with the axial velocity of
the particles better than it does with the radial velocity of the particles when integrated
over the length of the jet, as in Figure 40. Conversely, it is equally as interesting that the
two cross-correlations are nearly identical when integrated over the jet cross—plane, as in

Figure 39.

67

I rm: 53.45.372.21. firmlrflfiijfliaf

 

0.01 . r . r

0.005

Velodty Correlation

 

 

 

 

—0.005 0 ‘ L ‘ '
Radial Location

Figure 40. Radial variation of correlation between particle and carrier gas velocities
One of the more difficult problems facing experimentalists is that of accurately resolving
both temperature and velocity fields of highly turbulent flows. Figure 41 can be used to
gain some insight about the thermal inertia of the particles. It is obvious that the larger
particles will respond to the flow in quite different ways than the smaller ones. Not only
will they act as larger momentum sources and sinks, but they will also act as thermal
sources and sinks. As the smallest scales of turbulence are dissipated into the internal
energy of the carrier gas, a significant part of that energy will be transferred to the
particles (1998a). Figure 41 shows that the temperature distribution of the smaller
particles will, in fact, spread out as the jet develops. In the near field, all particles are at
nearly the same temperature, while further downstream the smaller particles attain higher
temperatures in comparison to the larger ones. This can affect the carrier gas temperature

field and, for sufficiently large temperature variations, the fluctuating velocity field.

68

 

1.008 . r . .

 

_ o arr/D = 5.0
8 Ox/D = 12.5
— lin.reg..x/D=5. slope=—0.43
1'00“ _ o -- - lin. reg. x/D=12.5. slope=—1.44 '
(O
o
o

l .004

Particle Temperature

1.002

 

 

 

 

ParticleMass
Figure 41. Particle temperature as a function of particle mass

The results shown in Figures 33-41 are for particles with relatively large average 1;,
similar to that in the experiment. Experimentally, it is difficult to consider smaller
particles due to the interference of the particle phase with the seeded particles. It is not
however very difficult to perform numerical simulations with smaller average rp. Figure
42 shows the difference in the root—mean-square of axial velocity of the carrier phase due
to the introduction of different average particle sizes. The figure shows that the

attenuation of the turbulence is greater for the cases with larger average particle sizes.

Figure 43 is further evidence that the size of the particles in two-phase jets can be very
important and should never be neglected. It is a plot of the mean axial velocity versus
axial position at the centerline and versus radial position at x/D = 7.5. Note that in (a) the
effects of the particles are minimized by decreasing their size. This is due to the inlet
conditions imposed upon both phases. In all three two-phase cases, the particles are

initialized at a velocity that is less than that of the carrier gas. In the cases with larger

69

particles, the particles have a large inertia, and effectively drag the carrier gas along at

their velocity, while the smaller particles generally interact in a more complicated way.

0.15

 

   
 

 

' I ' I ' I

— l-way ..
-- 2—way, 3:1.0
- -- 2-way, 1:,=10.0

—— 2—way. 3:57.41

 

 

 

 

— 1-way

--- 2-way, 3:10

‘ - -- 2-way, 'c,=10.0
\\ ,. — - 2—way, 15:57.4

 

 

 

O 0.5 1 1.5 2
Radial Position

35

(a)

(b)

Figure 42. Mean axial velocity versus radial position for varying average particle

inertia at (a) x/D = 5 and (b) x/D = 12.5

The smallest particles appear to have no effect on the mean flow, while the larger

particles significantly decrease the decay of the centerline velocity. The plot of radial

variation of axial velocity can be interpreted in a similar way. The smaller particles seem

70

 

to augment the axial velocity at this particular axial location, while the larger particles

seem to attenuate the carrier gas velocity.

 

r ' I

   
 

— l-way

--- 2-way. 3:1.0
--- 2-way, 5:100 _
-- 2-way. 15:57.4

  

.0
00
l

MeanAxlalVelodty
o
k:

 

 

 

 

0.6 - -
l d
0.5 " . ' s "
x
P \x. s
0.4 ‘ l ‘ ‘ ‘
O 5 10 15
Axial Position
' I V 1 r r r I r
0.8 ~ -
— l-way
-- 2-way.t,=1.0 .
/A\ --- 2-way.1:’=10.0
/ —- - =
0.6 _ / \\ 2 way, 1, 57.4‘

Mean Ardal Velocity
,o
a.

.0
to

 

 

 

A

A 1 g l a l

0 A l
—l.25 -0.75 -O.25 0.25 0.75 1.25

 

RadialPositlon

Figure 43. Mean axial velocity (a) versus axial position at the centerline and (b)
versus radial position at x/D = 7.5

71

 

CHAPTER 4
CONCLUSIONS
In Chapter 2, direct numerical simulations (DNS) of two-dimensional, droplet-laden,
harmonically forced jets were conducted in an effort to better understand the underlying
physics involved in such flows. Full two-way mass, momentum and energy coupling
between phases was considered. The effects of particle time constant, carrier gas
temperature and ‘degrees of coupling’ on various turbulent properties were numerically

measured.

The results indicate that the downstream particle dispersion is nearly independent of
particle injection for one-way coupling cases. The previous findings related to particle
size and preferential concentration were confirmed, as smaller particles tend to follow the
flow and therefore do not preferentially concentrate, while larger particle are largely
unaffected by the carrier gas and also do not preferentially concentrate. The phenomenon
of local particle dispersion was noted, as moderately sized particles tend to cluster
together in regions of low-strain, while either large or small particles tend to spread out in

the flow (by their own large- or small-particle mechanisms).

The cases for two-way coupled, non-evaporating droplets indicate that temperature
effects are still very significant. The finite thermal inertia of the particles significantly
alters the density profile of the jet, causing a modification of the instabilities that govern
jet grth rate. Thus, cooler particles injected into a hot jet will tend to stabilize the jet,

decreasing the growth rate. In addition to temperature effects, the effects of mass loading

72

 

were investigated. The results indicate that as the mass-loading ratio increases, the slope
of the jet half-width increases as well. This relationship is hypothesized to be linear for
the range of mass loading ratios considered in this study. The results also indicate that the
production of turbulent kinetic energy is greatly affected by the presence of solid
particles (and also evaporating droplets). The production of turbulent kinetic energy
(TKE) was found to be of opposite sign on the positive transverse side of the jet,
indicating a significant sink of TKE due to the particles. The relative magnitude of the

Reynolds stress was found to decrease with increased mass loading.

Perhaps the most valuable findings are those of the evaporating cases. An effort has been
made to clearly separate the turbulence modification due to temperature effects, particle
drag and heat transfer and evaporative effects (e. g., added mass). Evaporative effects
contributed to the increased stability of the jets in this study, as the evaporation decreased
the temperature and increased the density of the carrier gas. The Reynolds stress was
damped by the addition of the evaporation mechanism. Hence, the production of TKE
was also decreased. By considering the most physically complicated case, the effects of

interpolation scheme on solution accuracy were determined to be insignificant.

The long—range goal of this and other similar works is to accurately predict the behavior
of multiphase turbulent flows in complex geometries using numerical methods. This
study is part of a larger ongoing effort to develop more accurate and robust turbulence
closure models. Specifically, the DNS results (for these two-dimensional and filture

three—dimensional cases) will be used in a priori analyses to develop sub-grid scale

73

 

(SGS) closure models for the large-eddy simulation (LES) methods of computing
multiphase turbulent flows. The full analysis of the three-dimensional results will be the

subject of future work.

In Chapter 3, the abilities of large eddy simulation (LES) methods to predict multiphase
turbulent flows was investigated via an a posteriori study, correlating experimental and
numerical results for an axisymmetric turbulent round jet laden with heavy particles. The
results indicate that the subgrid-scale (SGS) carrier gas stress model and stochastic SGS
model employed for particles in the LES are viable and the latter increase the accuracy of
the numerical prediction by as much as 10%. Three different simulations of the flow were
conducted. Of these, the simulation utilizing both the stochastic particle SGS model and
the non-uniform (Gaussian) particle size distribution (Case 3) most accurately predicted
the various measured turbulent quantities, while the simulation that did not incorporate
either of these two (Case 1) was the least precise. The simulation that included the SGS
particle model but not the Gaussian particle size distribution (Case 2) was reasonably
accurate, but less so than the previously mentioned Case 3. In addition to the results for
two-phase flow, results were considered for the single-phase flow, for which the LES

utilizing the SGS closures was also accurate.

Additional results indicated that the physics of the two-phase round jet are complex
indeed. How the particles affect the turbulent velocity field and the thermal field leads to
interesting connections between the two. The results indicate that as the carrier gas

velocity fluctuations are damped by the particles the dissipation decreases, but the

74

 

particle temperature increases, showing a true two-way coupling effect. The analysis of
the thermal inertia of particles shows that the temperature distribution of the small
particles widens as the jet develops, while the temperature distribution of the large
particles remains nearly constant. The velocity distribution of the particles follows a
similar pattern, as expected. The larger particles tend to keep their initial velocity, while
the smaller particles tend to accelerate and/or decelerate more readily, and thus have a
more disperse correlation between mass and velocity. Analysis of particle number density

and average particle mass profiles supports these findings.

Analysis of the probability distribution firnctions of the particle mass indicates that the
size density remains relatively the same in the axial direction of the flow. Analysis of the
particle-carrier gas velocity correlations indicate that they decay in the axial and the
outward radial directions, leading to increased slip velocities in those locations.
Interestingly, the axial velocity correlations do not follow the same trends as the radial
velocity correlations. Both indicate that the dominant correlation is the correlation
between the axial velocities of the carrier gas and the particles, and that the correlation
between the radial velocities of the carrier gas and the particles is approximately zero.
However, the correlations integrated over the plane perpendicular to the jet indicate that
both the cross-correlations are similar, whereas the correlations integrated over the length
of the jet indicate that the correlation between the axial velocity of the particles and the
radial velocity of the carrier gas is significant, and the correlation between the radial

velocity of the particles and the axial velocity of the carrier gas is minimal.

75

 

Further analysis based on modifying the average particle inertia indicates that the choice
of particle size can significantly affect the turbulence fields of two-phase jets. The growth
of the jet can be amplified with the addition of tiny (micro-) particles, or greatly
attenuated with the addition of very large particles to the flow. It has been shown that the
numerical simulation of two-phase jets using LES methods is an efficient and accurate
method to conduct experiments in order to help to determine real physical parameters that

govern a particle-laden flow.

76

 

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77

 

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