A Computational Study of Mixing Downstream
of a Lobed Mixer with a Velocity Difference
Between the Co-Flowing Streams
by
David Early Tew
B.S. Aerospace Engineering, University of Michigan (1990)
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics and Astronautics
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at the
Massachusetts Institute of Technology
August 1992
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@ Massachusetts Institute of Technology, 1992. All Rights Reserved.
Signature of Author
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Department of Aeronautics and Astronautics
August 25, 1992
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Certified by
Dr. C. S. Tan
Thesis Supervisor
Accepted by
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Professor Harold Y. Wachman
Chairman, Departmental Graduate Committee
A Computational Study of Mixing Downstream
of a Lobed Mixer with a Velocity Difference
Between the Co-Flowing Streams
by
David Early Tew
Submitted to the Department of Aeronautics and Astronautics
in August 1992 in partial fulfillment of the
requirements for the degree of Master of Science
in Aeronautics and Astronautics
Abstract
A computational study of lobed mixer flow fields with both streamwise and
spanwise vorticity has been conducted using a three-dimensional unsteady Euler solver.
The results of this study indicate that the effects of the two vorticity components can be
(approximately) superposed in the region downstream of the lobe in which most of the
mixing occurs. Within this region, the streamwise vorticity stretches the mean cross-flow
interface, and the rate at which it stretches is shown to be relatively unaffected by
spanwise vorticity associated with a velocity difference between the two streams. This
spanwise vorticity initiates a Kelvin-Helmholtz type instability on the stretching crossflow interface, and the amplitude of the undulations resulting from this instability grow
with distance from the trailing edge.
Using the idea of the superposition of the roles of each vorticity component, a
simple expression to predict the mixing rate downstream of these devices is derived. This
expression describes the parametric dependence of the mixing rate on the streamwise and
spanwise circulation, length of the trailing edge, and distance from the trailing edge.
Further, it allows one to interpret the effect of spanwise vorticity as an enhancement of
the cross-stream transport properties. The mixing rates predicted agree with the results of
the numerical simulations conducted in this study, and both predicted and computed
mixing rates initially increase with distance from the trailing edge. This increase in the
mixing rate is largely a consequence of the stretching of the mean cross-flow interface by
streamwise vorticity.
However, as the amplitude of the undulations resulting from the KelvinHelmholtz instability approaches the length scale of the cross-flow interface, the
stretching of the cross-flow interface slows and eventually ceases; this marks the end of
the region in which the effects due to the individual vorticity components can be
superposed, and hence the end of applicability of the above simple prediction of the
mixing rate. Furthermore, the cessation of the growth of the mean cross-flow interface
signifies the end of the mixing augmentation due to the presence of streamwise vorticity.
Henceforth, the mixing layer behaves much like its planar counterpart, growing largely as
a result of the undulations, and the resulting mixing rate slows.
In summary, these results indicate that the range of the mixing augmentation due
to the streamwise vorticity generated by a lobed mixer is limited to the region where the
mean cross-flow interface is being stretched and the roles of the vorticity components
may be superposed. In the current simulations, this range is limited to the region 5-10
wavelengths downstream of the trailing edge of the mixer.
Thesis Advisor:
Choon S. Tan
Principal Research Engineer
Acknowledgments
The completion of this thesis would have been impossible without the guidance
and friendship of a number of people. In particular, I would like to thank Dr. Choon S.
Tan for his assistance, friendship, and persistence in editing this thesis; Professor Edward
M. Greitzer for his helpful suggestions and constructive criticism; and Professor Ian A.
Waitz for his aid and encouragement.
I would also like to thank my colleagues in lobed mixer research for their help and
friendship: Yuan Qiu, Ted Manning, Mike O'Sullivan, and in particular Josh Elliott for
his aid in getting this research effort started.
Furthermore, I would like to thank my friends for making my stay at MIT
enjoyable. Thanks goes to the GTL nerds for the many lunches, field trips, and softball
games:
Aaron, Taras, Pete, Dan, S.J., Martin, Jon, Scott, Bill, Mike, Fred, Eric,
Constantine, Joel, Earl, and Knox. Many thanks to my house mates in the 'Michigan
House' for their friendship: Norm, Mak, Mark, Dave 1, Derek, Henry, Steve, Jeff, and
honorary members Kathy and Charissa. A special thanks to Stacy for her company and
help with this thesis.
Finally, I would like to thank my family for their love and support: Mom, Dad,
Jon, and Rob. This thesis is dedicated to them.
Support for this project was provided by Naval Air Systems Command, under
contract N0001988-C0029 and supervision by Mr. George Derderian and Dr. Lewis
Sloter II. Support for the author was provided by the Air Force Research in Aero
Propulsion Technology (AFRAPT) program, AFOSR-85-0288 E.
Additional
computational resources were provided by NASA Lewis Research Center. This support
is greatly appreciated.
Contents
Nomenclature
Introduction
14
1.1
Motivation ............................................
14
1.2
1.3
Background ............................................
Technical Objective ...................................
1.3.1 Hypothetical Model ..................................
15
18
18
1.3.2
18
1.4
1.5
1.6
2
12
Research Goals ...................................
Technical Approach ............................
.......
....
Contributions..........................................
19
19
1.5.1
Role of Spanwise Vorticity .........................
19
1.5.2
Initial Superposition of Effects of Two Vorticity Components
20
1.5.3 Quantitative Prediction of Initial Mixing Rate ............
Overview of Thesis .....................................
20
20
Parametric Dependence of the Spatial Mixing Rate on Streamwise and
Spanwise Circulation
24
2.1
Introduction ..........................................
24
2.2
Definitions ............................................
24
2.2.1
Geometric Characterization of a Lobed Mixer ...........
24
2.2.2
2.2.3
Non-Dimensionalization of Flow Quantities in this Study ..
Measure of Total Vorticity Shed from Lobe Trailing Edge .
25
26
26
2.3
2.4
2.5
Mixing Layers with only Spanwise Vorticity ................
2.3.1 Growth of Mixing Layers Due to Spanwise Vorticity .....
2.3.2 Mixing with only Spanwise Vorticity ..................
Mixing Layers with only Streamwise Vorticity ................
Hypothetical Model to Predict the Growth of Mixing Layers with
Streamwise and Spanwise Vorticity ........................
27
27
28
29
2.5.1 Quantitative Prediction of the Spatial Mixing Rate
2.5.2 Implications of Hypothesis ..........................
2.5.3 Limitations of Hypothesis ..........................
......
30
32
34
3
Computational Method
3.1
Introduction.......................................
3.2
Test Matrix .....................................................
3.3
Numerical Algorithm ....................................
3.4
Boundary Conditions ..................
..................
3.5
Grids and Grid Generator .....................................
3.6
Interface Tracking .......................................
Definition of Mixed Fluid .................
...............
3.7
39
39
40
42
43
43
44
44
4
Numerical Experiment: Results
4.1
Introduction .................
.........................
4.2
Region 1 ........................................
4.2.1 Roll-Up of Cross-Flow Interface Associated with
Streamwise Vorticity ..............................
4.2.2 Growth of Undulations Associated with
Spanwise Vorticity.................................
4.2.3 Discussion of Results in terms of Model ...............
4.2.4 End of Region 1 ..................................
4.3
Region 2 ..............................................
4.4
Region 3 ..............................................
50
50
51
53
59
62
63
64
Summary and Conclusions
5.1
Summary and Conclustions .............................
5.2
Design Suggestion .................................................
5.3
Recommendations for Future Work .........................
97
97
99
99
5
51
List of Figures
1.1
Generation of streamwise vorticity by a lobed mixer[25]
22
1.2
Growth of undulations associated with spanwise vorticity
23
1.3
Roll-up of cross-flow interface by streamwise vorticity
23
2.1
Geometric characterization of a lobed mixer
34
2.2
Path of integration on x=0 plane to determine
2.3
Experimental flow visualization of planar mixing layers[6]
35
2.4
Roll-up of interface in cross-flow planes downstream of a lobed
mixer with equal velocity streams (F,= 0) for LPM on
cross-flow planes x = 0, 4, 8, 12, 16[9]
36
Cross-flow interface length vs. distance from trailing edge for
LPM with F,= 0.0[9]
37
Spatial growth rate of cross-flow interface length vs.
streamwise circulation for LPM and HPM' with /, = 0.0[9]
37
Constant streamwise circulation for LPM and HPM'
with F,= 0.0[9]
38
2.8
Spanwise undulation about mean cross-flow interface
38
3.1
Three mixer configurations used in the numerical experiments:
LPM,HPM,ADM
46
2.5
2.6
2.7
3.2
Fx
34
Computational domain for all mixer configurations with the LPM trailing
edge cross-flow plane shown to illustrate the location of the lobes in
the domain
47
3.3
Grids on a symmetry plane, z=0, for the LPM, HPM, and ADM
3.4
Grids on a cross-flow plane in the mixing region, 0 < x < 16, for the
LPM,HPM,ADM
4.1
49
Typical computational results illustrating the growth of spanwise
undulations on the z=O plane for the LPM with I",=0.7 and the
roll-up of the cross-flow interface on the x=O, 2, 4, 6 planes for the
ADM with r, =0.7
66
4.2
Roll-up of the cross-flow interface on the x=O plane for the LPM with
F, =0.7: Velocity Vectors(a), Vorticity Magnitude(b),
Passive Scalar(c)
67
4.3
Roll-up of the cross-flow interface on the x=2 plane for the LPM with
F,=0.7: Velocity Vectors(a), Vorticity Magnitude(b),
Passive Scalar(c)
68
4.4
Roll-up of the cross-flow interface on the x=4 plane for the LPM with
F, =0.7: Velocity Vectors(a), Vorticity Magnitude(b),
Passive Scalar(c)
69
4.5
Roll-up of the cross-flow interface on the x=6 plane for the LPM with
F,=0.7: Velocity Vectors(a), Vorticity Magnitude(b),
Passive Scalar(c)
70
4.6
Disturbances due to spanwise undulations on the roll-up of the
cross-flow interface for the LPM in passive scalar fields on the x=4
plane: (a)Steady case with F,= 0.0, (b)Unsteady case with F,=0.7
at time 1, (c)Unsteady case with F,=0.7 at time 2, (d)Range
of movement for unsteady case with F,=0.7
71
4.7
Stretching of the mean cross-flow interface with distance from the
trailing edge and growth of spanwise undulations on this
interface in the mean passive scalar field on the x=O, 2, 4, 6
planes for the LPM with
4.8
4.9
F,=0.7
Mean cross-flow interface length vs. distance from trailing edge in
Region 1 for the LPM with F,=0.0, 0.7
72
73
Instantaneous velocity vectors on the x=O, 2, 4, 6 planes illustrating the
cross-flow circulation associated with streamwise vorticity for the
HPM with 1 ,=0.7
74
4.10
4.11
Instantaneous passive scalar field on the x=O, 2, 4, 6 planes
illustrating the roll-up of the cross-flow interface associated with
streamwise vorticity for the HPM with F, =0.7
Instantaneous velocity vectors on the x=O, 2, 4, 6 planes illustrating the
cross-flow circulation associated with streamwise vorticity for the
ADM with
4.12
4.13
4.14
4.15
75
F, =0.7
76
Instantaneous passive scalar field on the x=O, 2, 4, 6 planes
illustrating the roll-up of the cross-flow interface associated with
streamwise vorticity for the ADM with F, =0.7
77
Mean cross-flow interface length vs. distance from the trailing edge
in Region 1 for all tested cases
78
Spatial growth rate of cross-flow interface length in Region 1 vs.
streamwise circulation for all tested cases
78
Velocity vectors at four time intervals on the x=4 plane illustrating
the unsteadiness of the flow field associated with spanwise vorticity
in Region 1 for the LPM with F, =0.7
79
4.16
Vorticity magnitude at four time intervals on the x=4 plane illustrating
the unsteadiness of the flow field associated with spanwise vorticity
in Region 1 for the LPM with F, =0.7
80
4.17
Passive scalar field at four time intervals on the x=4 plane illustrating
the undulation of the interface associated with spanwise vorticity
in Region 1 for the LPM with F, =0.7
4.18
81
Velocity vectors at four time intervals on the z=0 plane illustrating
the unsteadiness of the flow field associated with spanwise vorticity
for the LPM with F, =0.7
82
4.19
Vorticity magnitude at four time intervals on the z=0 plane illustrating
the development and convection of vortical structures associated
with spanwise vorticity for the LPM with F, =0.7
83
4.20
Passive scalar field at four time intervals on the z=0 plane illustrating
the growth and convection of undulations in the interface associated with
the spanwise vorticity for the LPM with F, =0.7
84
4.21
Mean passive scalar field on the z=0 plane illustrating the growth
of the spanwise undulations and the region of mixed fluid with distance
from the trailing edge for the LPM with F, =0.3, 0.5, 0.7
85
4.22
Mean passive scalar field on the x=4 plane illustrating the increasing
spatial growth rate of the spanwise undulations and region of mixed fluid
with increasing
F, for the LPM with F, =0.3, 0.5, 0.7
86
4.23
Mean passive scalar field on the x=4 plane illustrating the increasing
spatial growth rate of the spanwise undulations and region of mixed fluid
with increasing
4.24
F, for the HPM with F,=0.8, 1.2, 1.8
Mean amplitude of spanwise undulations vs. distance from trailing edge
in Region 1 for the LPM with
4.25
87
, = 0.3, 0.5, 0.7
88
Mean amplitude of spanwise undulations vs. distance from trailing edge
in Region 1 for the HPM with F,=0.7, 1.2, 1.8
88
4.26
Predicted and computed mass flow rates of mixed fluid vs. distance from
the trailing edge in Region 1 for the LPM with F,=0.3,0.5, 0.7
89
4.27
Predicted and computed mass flow rates of mixed fluid vs. distance from
the trailing edge in Region 1 for the HPM with FI,=0.7,1.2, 1.8
89
4.28
Streamwise interface structure scale vs. distance from
trailing edge in Region 1 for the LPM, HPM, and ADM
90
Streamwise interface structure scale and amplitude of spanwise
undulations vs. distance from trailing edge in Region 1 for the LPM
91
Streamwise interface structure scale and amplitude of spanwise
undulations vs. distance from trailing edge in Region 1 for the HPM
91
Mean cross-flow interface length vs. distance from trailing edge
in all regions for the LPM with F, = 0.0, 0.3, 0.5, 0.7
92
Mass flow rate of mixed fluid vs. distance from trailing edge
in all regions for the LPM with F, = 0.0, 0.3, 0.5, 0.7
92
Mean cross-flow interface length vs. distance from trailing edge
in all regions for the HPM with F, = 0.7, 1.2, 1.8
93
Mass flow rate of mixed fluid vs. distance from trailing edge
in all regions for the HPM with F, = 0.7, 1.2, 1.8
93
4.29
4.30
4.31
4.32
4.33
4.34
4.35
Velocity vectors at four time intervals on the x=14 plane illustrating the
unsteadiness of the flow field associated with the large scale
spanwise undulations in Region 3 for the LPM with F, =0.7
94
4.36
Passive scalar field at four time intervals on the x=14 plane illustrating
the large scale movement of the interface associated with the spanwise
undulations for the LPM with F, =0.7
95
4.37
Mean passive scalar field on the x=O1, 12, 14, 16 planes illustrating the
planar like growth of the region of mixed fluid in Region 3 for the LPM
with F, =0.7
96
5.1
Two mixer configuration to further enhance mixing by reintroducing
streamwise vorticity after growth of mean cross-flow
interface has ceased
102
List of Tables
3.1
Test Matrix
3.2
LPM Test Cases
4.1
Constants in Exponential Fits for
3mean in Region 1
Nomenclature
A
Area
A
Area Flow Rate
ADM
Advanced Mixer
h
Lobe Height
HPM
High Penetration Mixer
k
Arbitrary Constant
Interface Structure Scale
L
Cross-Flow Interface Length
7
Vector Arc Length
LPM
Low Penetration Mixer
Mass Flow Rate
Mc
Convective Mach Number
r,
Ratio of Amplitude of Spanwise Undulations to Streamwise Interface
Structure Scale
TE
Trailing Edge
Velocity
x
Distance from Lobe Trailing Edge
y
Height Above or Below Mid-Line Lobe
z
Coordinate Parallel to Array of Lobes
a
Penetration Angle
F
Circulation
8
Amplitude of Spanwise Undulations
AU
Velocity Difference Between Co-Flowing Streams
Lobe Wavelength
h1'
e'
Passive Scalar
Generic Linear Transport Coefficient
Vorticity
Vorticity Magnitude
Subscripts
x
Axial/Streamwise Direction
t
Transverse/Spanwise Direction
mix
Mixed Fluid
TE
Lobe Trailing Edge
Chapter 1
Introduction
1.1
Motivation
Lobed mixers augment mixing between two co-flowing streams of fluid in a wide
range of applications(Figure 1.1). The devices mix the core and bypass streams in
turbofan engines[2,4], fuel and air in combustion chambers[34], and reactants in a
chemical laser[8].
The lobed mixer provides this mixing augmentation via two mechanisms: the
increase in initial interface area due to the presence of the lobes[23] and the introduction
of streamwise vorticity to the mixing process[10,28,30]. This streamwise vorticity is
generated along the lobe trailing edge due to a variation in aerodynamic loading along its
span[10] and thus is present in addition to any spanwise vorticity component associated
with a velocity difference between the two streams.
In previous work, the mixing rate has been shown to increase with increasing
streamwise vorticity; the mechanism for this augmentation has been shown to be the
stretching of the mean cross-flow interface with distance from the trailing edge by
streamwise vorticity[9,10,23,29]. On the other hand, planar mixing layer studies have
demonstrated that the mixing rate, at a fixed mean velocity, increases with an increasing
velocity difference between the two streams, i.e. spanwise vorticity, and the mixing
mechanism associated with spanwise vorticity is the growth of an instability in the
interface with distance from the trailing edge[6,8].
However, a detailed understanding of mixing with both of these vorticity
components present is still lacking. Specifically, downstream of a lobed mixer and in the
presence of strong streamwise vorticity, the parametric dependence of mixing on
spanwise vorticity has yet to be clarified. Thus, this study will focus on the role of the
spanwise vorticity in the mixing process downstream of a lobed mixer.
1.2
Background
Despite the lobed mixer's range of uses, its history is firmly rooted in its use in
aircraft engines. The concept of lobed mixers can be traced to the early 1940's and the
dawn of jet engine development[12].
However, it was not until the 1960's that these
devices saw widespread use in jet engines to promote the mixing of exhaust jets with
ambient air, to decrease jet velocity and thereby decrease jet noise[30]. More recently, as
a mean of noise reduction, the lobed mixer has been proposed for use in an ejector nozzle
system in engines for the High Speed Civil Transport[34].
A brief review of previous work in the development of the lobed mixer will now
be presented, but before embarking on this history lesson, a quote from Sir William
Hawthorne in the first Minta Martin Lecture will help to set up the framework for the
discussion.
In the creation of a new machine, man's inventiveness often outstrips his
understanding. It is surprising how often technological advances have
been made from a state of knowledge which in retrospect we would be
tempted to regard as cripplingly inadequate. Someone has remarked that
thermodynamics owes more to the steam engine than the steam engine to
thermodynamics. .. We are told in fact that it was the existence of the
steam engine which inspired Carnot to puzzle over the problem of
determining the highest efficiency one could get from an engine, and led
to the formulation of the second law of thermodynamics[13].
In the development of the lobed mixer, a similar trend may be observed. The engineering
gains have certainly outpaced a detailed understanding of the mixing flow fields
downstream of these devices. This disparity between the capabilities of this device and
our understanding of the mechanisms of its operation becomes evident with a brief
review of past mixer research.
Most of the early work on lobed mixers focused on engineering performance. In
the experimental study by Kozlowski and Kraft as part of NASA's Energy Efficient
Engine program, gross thrust coefficients were measured for a series of mixer
configurations[16]. In another experimental study, Presz et al tested a series of lobed
mixers in ejector configurations, and a large performance benefit was found over
conventional 'flat-plate' designs[29].
As this early work demonstrated the potential mixing performance benefits of
lobed mixers, attention turned toward an understanding of the mechanisms through which
these devices augment mixing, with the intent to use this knowledge to optimize their
design. Based on experimental measurements and a series of calculations, Povinelli and
Anderson suggested that the streamwise vorticity generated by lobed mixers plays a role
in the mixing process[28]. In experiments conducted at UTRC, Paterson demonstrated
that the mixing process is dominated by large secondary flows with a scale of roughly
that of the lobe[27]. Further experimental work by Werle and Paterson provided some
insight into these secondary flows; in particular, their results indicate that the convoluted
lobe trailing edge generates an array of axial vortices of approximately the same strength
but of alternating sign[35]. These studies demonstrated that lobed mixers augment
mixing through the presence of large scale (-lobe) stirring motions associated with
streamwise vorticity.
More recently, research activities have been focused on the basic fluid mechanic
issues associated with lobed mixers that may have a direct bearing on the development of
guidelines for the rational design of these devices[9,10,23,24,30].
For instance, the
relative importance of streamwise vorticity compared to the increase of initial interface
area associated with the convoluted trailing edge was determined in water tunnel
experiments by Manning; his results showed quantitatively that streamwise vorticity does
indeed augment mixing[23]. Furthermore, in these experiments, Manning also noted the
presence of a Kelvin-Helmholtz type instability associated with spanwise vorticity[23].
This instability was also observed in experiments conducted by Qiu[30].
Additional computational work by Qiu focused on the mechanism through which
streamwise vorticity augments mixing:
the stretching of the mean cross-flow
interface[30]. This cross-flow interface stretching was also observed in computational
work by Elliott[9]. Furthermore, Elliott's results show that this process is only marginally
affected by compressibility. The stretching of the cross-flow interface is also seen in the
most recent experiments conducted at UTRC by McCormick. In these experiments, he
has provided flow visualization on the detailed structure of mixing layers downstream of
lobed mixers with spanwise vorticity[24], and in these pictures, undulations due to a
Kelvin-Helmholtz type instability were seen to shed from the lobe trailing edge.
These studies have provided a foundation for the current work. The lobed mixer
has been demonstrated to provide a mixing augmentation[16,29], and the mechanisms
through which this augmentation arises have been shown to be the stretching of the mean
cross-flow interface by streamwise vorticity and the increase in initial interface area due
to the shape of the lobe trailing edge[9,10,23,30].
Furthermore, spanwise vorticity,
associated with a velocity difference between the co-flowing streams, has been shown to
result in the development of undulations in the interface separating the streams; the
development of these undulations is a consequence of the Kelvin-Helmholtz
instability[23,24,30].
1.3
Technical Objective
This study, building on the work reviewed in Section 1.2, will attempt to assess
the role of spanwise vorticity in the evolution of the mixing layer downstream of a lobed
mixer, to provide a foundation for the design and optimization of these devices. In order
to develop a framework for this work, a hypothetical model has been developed to
describe the influence of spanwise vorticity in the mixing process, and the objectives of
this study are cast in terms of a test of this model.
1.3.1
Hypothetical Flow Model
In the model, the roles of streamwise and spanwise vorticity in the mixing process
are superposed. A Kelvin Helmholtz instability associated with spanwise vorticity is
assumed to grow on a mean cross-flow interface which is being stretched by streamwise
vorticity. The assumed superposition of the roles of the vorticity components allows a
simple quantitative prediction of the rate of growth of the mixing layer with distance
from the trailing edgel to be derived.
1.3.2
Research Goals
In light of the above concerns, the goals of this thesis are as follows-* To understand the role of spanwise vorticity in a mixing layer
downstream of a lobed mixer,
* To determine the conditions under which the roles of each vorticity component
in the mixing process can be superposed,
* To assess the usefulness of the model as a predictive tool.
1 The rate of growth of the mixing layer with distance from the trailing edge will be referred to as the
spatial mixing rate.
1.4
Technical Approach
In pursuit of the above research goals, a series of numerical experiments have
been conducted to assess this model and to isolate the role of spanwise vorticity. The
experiments are performed with an unsteady, time-accurate, three-dimensional Euler
code. In essence, a series of numerical simulations have been conducted, in which the
circulations associated with streamwise and spanwise vorticity are varied. These
numerical simulations are used to track the unsteady movement of the interface
separating the two streams; this unsteadiness is primarily the result of the KelvinHelmholtz instability associated with spanwise vorticity.
The unsteady mixing process may be described at the largest scales as the roll-up
of a vortex sheet, an inviscid phenomena which leads to an increase in interface area
between the co-flowing streams. The simulations model this unsteady roll-up, and
knowing that mixing occurs across the stream interface, an estimate of the mixedness is
made from the range of interface movement[6,7].
1.5
Contributions
The following contributions have been made to the understanding of the mixing
process downstream of a lobed mixer.
1.5.1 Role of Spanwise Vorticity
One role of spanwise vorticity in the mixing process downstream of a lobed mixer
has been identified. A Kelvin-Helmholtz type instability distributed along the mean
cross-flow interface is initiated at the trailing edge of the lobe. The resulting undulations
grow on the stretching mean cross-flow interface. 2 However, as the spanwise undulations
2 The growth of the spanwise undulations will be referred to as the spanwise process. (Figure 1.2)
grow, they will begin to interact with the streamwise winding process. 3 These
undulations eventually grow large enough to dominate the mixing layer and give it a
planar-like appearance.
1.5.2
Initial Superposition of Effects of Two Vorticity Components
The results of the numerical simulations indicate that in the region within 5-10 X
of the lobe trailing edge, the hypothesized superposition of the roles of the two vorticity
components is valid. The axial extent of this region is limited by the interaction between
the spanwise undulations and streamwise roll-up.
1.5.3 Quantitative Prediction of Initial Mixing Rate
Furthermore, the simple expression for the spatial mixing rate downstream of a
lobed mixer deduced from the superposition of the roles of the vorticity components does
provide a rough estimate of the computed results in the region where the roles of the
vorticity components may be superposed.
1.6
Overview of Thesis
This thesis is organized as follows.
In Chapter 2, the key relevant technical results from studies on planar mixing
layers with only spanwise vorticity and lobed mixers with only streamwise vorticity will
be reviewed. The results of these studies will used to develop a model that describes the
parametric dependence of the spatial mixing rate on streamwise and spanwise vorticity.
Numerical experiments designed to assess the model are described in Chapter 3,
as well as details of the numerical algorithm, grid generator and boundary conditions
3 The winding and stretching of the mean cross-flow interface will be referred to as the streamwise process.
(Figure 1.3)
employed in the experiments. The technique used to track the interface separating the
two streams is also described.
The results of the numerical experiments are presented in Chapter 4. It is found
that the mixing layer is divided into three regions based on the pertinent length scales of
the spanwise and streamwise processes. With the aid of the model presented in Chapter
2, the role of spanwise vorticity is elucidated.
The conclusions of this study are presented in Chapter 5. Based on these
conclusions several design guidelines are suggested. Recommendations for future work
are also made in light of the results presented in this thesis.
CDI ITTED
DI ATE
TRAILING
EDGE
V %n I MAc
Figure 1.1: Generation of streamwise vorticity by a lobed mixer[25]
U1
U2
I
I
11
Meanflow direction is from left to right.
Figure 1.2: Growth of undulations associated with spanwise vorticity
x=O
x>O
x >> 0
Meanflow direction is into page.
U (8
Figure 1.3: Roll-up of cross-flow interface by streamwise vorticity
Chapter 2
Parametric Dependence of the Spatial
Mixing Rate on Streamwise and Spanwise
Circulation
2.1
Introduction
Mixing layers with streamwise or spanwise vorticity are discussed, and the effects
of the vorticity components in these flows are superposed to develop a hypothetical
model which describes the parametric dependence of the spatial mixing rate on the
distance downstream of the lobe trailing edge, the length of the trailing edge, the
streamwise vorticity generated by the mixer lobe, and the spanwise vorticity associated
with any velocity or total pressure difference between the co-flowing streams.
2.2
Definitions
Before proceeding further, the terms and quantities used in the remainder of this
work will be defined.
2.2.1
Geometric Characterization of a Lobed Mixer
A lobed mixer may be described as a plate with a periodic array of convolutions
which shed a streamwise vorticity distribution when placed in a moving stream of fluid,
as indicated in Figure 1.1. The geometry of a typical lobed mixer is shown in Figure 2.1
and can be characterized in terms of the wavelength(k) of these convolutions, the
amplitude or height(h), the penetration angle(ac), and the lobe shape (sinusoidal, square,
etc.).
2.2.2
Non-Dimensionalization of Flow Quantities in this Study
All flow quantities will be made appropriately dimensionless as described below.
Lengths will be made dimensionless with respect to the lobe wavelengthl , velocities with
respect to the mean inflow velocity, and densities with respect to the mean inflow
density. The mean inflow quantities are defined as
P1+ P2
UU + U2
2
2
(2.1)
where an overbar has been used to denote a mean inflow quantity. The non-dimensional
length L', non-dimensional velocity V', and the non dimensional density p' are
respectively
L
L'=
V'= =•
V
U
P p'-P
=p.'
5,
(2.2)
where a prime has been used to denote a non-dimensional flow quantity. However, for
convenience, the prime will be dropped, and henceforth, all flow variables should be
interpreted as non-dimensional.
1 For convenience, lengths in the planar mixing layer discussion will be non-dimensionalized with respect
to the wavelength of the lobed mixers used in this study.
2.2.3
Measure of Total Vorticity Shed from Lobe Trailing Edge
The vorticity generated by the lobe can be described in terms of the strength of the
streamwise and spanwise vorticity components. In the mixers examined in this study 2,
the total streamwise vorticity shed from a half wavelength span of a mixer may be
approximated by the circulation associated with the axial vorticity[23]. As illustrated in
Figure 2.2, this circulation may be found by integrating the velocity field about a contour
on the axial plane enclosing the lobe trailing edge:
-x=
.d.
(2.3)
The spanwise circulation is here defined as the integral of the spanwise vorticity along the
lobe trailing edge. For the mixers examined in this study, the local velocity difference
across the lobe is relatively constant along the span. Hence the total spanwise vorticity
shed from the lobe may be approximated by multiplying the velocity difference between
the two streams by the length of the trailing edge, .i.e.
F =AULTE.
2.3
(2.4)
Mixing Layers with only Spanwise Vorticity
Theoretical and experimental studies of planar mixing layers will be reviewed to
give the reader an appreciation of the parametric dependence of mixing on the
characteristic flow quantities of planar mixing layers.
2
-5
- 100
2.3.1
Growth of Mixing Layers with only Spanwise Vorticity
From experiments[6,7,24] as well as theoretical studies[3], it has been observed
that the interface between two co-flowing streams with different velocities is unstable,
and this instability bears the names of its discovers:
Kelvin and Helmholtz. Small
perturbations in this interface grow in amplitude with distance from the splitter trailing
edge and develop into large scale undulations in the interface[3,6,7] (Figure 2.3). This
process may be described as the roll-up of a vortex sheet.
In inviscid hydrodynamic stability calculations, the growth of these undulations is
shown to be[3,9]
2D
-
k ek2AUx
45inviscid
(2.5)
(2.5)
Note that the exponent grows linearly with AU.
On the other hand, in experimental mixing studies, these undulations are found to
grow linearly with distance from the trailing edge[6,7,24], i.e.
2D
AU
--= k3 U
x.
(2.6)
It is noted that the spatial growth rate of these structures is proportional to the velocity
difference between the co-flowing streams--i.e., the initial spanwise vorticity.
2.3.2
Mixing with only Spanwise Vorticity
In flow visualization pictures, these undulations are seen to represent the upper
end of a spectrum of interface structure scales in the mixing layer[6,7] (Figure 2.3). The
scales range from the size of these large undulations down to the Kolmogorov[6,7,17].
The smaller structures in this spectrum are observed to develop on the larger
undulations[6,7], and their development has been shown to enhance mixing, as they
dramatically increase the interface area across which molecular mixing occurs[7].
The large scale undulations span the region of fluid in which the smaller interface
structures reside. As molecular mixing occurs as a diffusion process across the stream
interface, it may be argued that these large scale structures span the region of fluid that
may potentially be mixed. In this study the extent of the region of fluid populated by
these large structures is used to estimate the mixedness. Specifically, a constant
fractional portion of this region is assumed mixed[7].
From this assumption, the mass flow rate of mixed fluid in planar shear layers can
be approximated by[6]
* 2D
2D
Sri = k 4
,2D
(2.7)
where the constant k4 is between zero and one. The spatial mixing rate may then be
found by differentiating Equation 2.7 with respect to distance from the trailing edge:
3
* 2D
dx
= k4
2
D
- ksAU.
(2.8)
This expression indicates that the spatial mixing rate depends solely on the velocity
difference between the two streams, i.e. spanwise vorticity, and is constant for a given
flow field.
2.4
Mixing Layers with only Streamwise Vorticity
To ascertain the parametric dependence of mixing on streamwise vorticity, the
flow field downstream of a lobed mixer with equal velocity streams will be reviewed.
The contoured lobe surface of the mixer generates a streamwise vorticity distribution
along the trailing edge, and as the streams have equal velocities, no spanwise vorticity
component is present.
Streamwise vorticity winds and stretches the interface between the co-flowing
streams in axial, or cross-flow, planes[9,10,23,30] (Figure 2.4). In low Reynolds Number
Navier-Stokes calculations by Qiu[30] and Euler calculations by Elliott[9], the mean
cross-flow interface grew linearly with distance from the trailing edge as shown in Figure
2.5 for a relatively low penetration mixer(LPM) examined by Elliott[9]. Furthermore, the
growth rate was found to be proportional to the streamwise circulation as shown in Figure
2.6. Thus, from these results, it may be deduced that
dL
-
= k6
x.
(2.9)
Upon integrating Equation 2.9, the length of the mean cross-flow interface may be shown
to be
L =k6 Fxx+ LTE.
(2.10) 3
Furthermore, as a consequence of Kelvin's Theorem, the streamwise circulation is
constant for a given mixer, as shown in Figure 2.7, and hence, the growth rate of the
cross-flow interface length is constant for a given mixer.
2.5
Hypothetical Model to Predict the Growth of Mixing Layers with
Streamwise and Spanwise Vorticity
A model describing the parametric dependence of the spatial mixing rate on the
streamwise and spanwise circulation, lobe trailing edge length, and distance from the
3 The length of the cross-flow interface at the lobe trailing edge is the length of the lobe trailing edge.
trailing edge will now be postulated. This postulate is based on the results of the mixing
studies discussed in Sections 2.3 and 2.4 with spanwise or streamwise vorticity.
The basic assumption is that the effects associated with spanwise and streamwise
vorticity, can be superposed. In words,
the streamwise vorticity stretches the cross-flow interface while the
spanwise vorticity initiates a Kelvin-Helmholtz type instability on this
stretching interface. Hence, the cross-flow interface will undergo
undulations of the same nature as the two dimensional Kelvin-Helmholtz
type seen in planar mixing layers, and the region of mixed fluid, which is
spanned by these undulations, spreads from the stretching cross-flow
interface. (Figure 2.8)
From this hypothetical model, a quantitative prediction of the spatial mixing rate may be
deduced.
2.5.1
Quantitative Prediction of the Spatial Mixing Rate
Using the quantitative results of the mixing discussions in Sections 2.3 and 2.4,
the growing undulations will be superposed on the stretching cross-flow interface to
deduce a predictive model for the spatial mixing rate.
In Section 2.4, the cross-flow interface is shown to grow linearly with distance
from the lobe trailing edge[9,30]. The growth rate is proportional to the streamwise
circulation shed from the mixer(Equations 2.9-10). On the other hand, in Section 2.3, the
width of the mixing layer at a given location is seen to increase linearly with the spanwise
vorticity[6,7] (Equation 2.6).
The expression for the stretching cross-flow interface (Equation 2.10) may be
combined with the expression for growth of the planar mixing layer (Equation 2.6) to
obtain an expression for spatial mixing rate downstream of lobed mixers similar to the
expression for planar shear layers in Equation 2.7:
lý,ix = k4 5 2D Lmean
(2.11)
After substituting the expressions in Equations 2.6 and 2.10 into Equation 2.11, and using
the definition of
F, in Equation
2.4, an expression for the mass flow rate of mixed fluid
in terms of four characteristic parameters is achieved:
,x=k4k3XE [Ft[k6Fxx+LTE]
(2.12)
Upon taking the derivative of 14mix with respect to x, the spatial mixing rate may be
shown to be
Sm"x =k4k [F,][2k6Fxx +LTE].
(2.13)
Thus, the above model provides a quantitative estimate of the mixing rate downstream of
the lobed mixer.
2.5.2
Implications of Hypothesis
From Equation 2.13, several key points on the roles of each vorticity component
may be deduced.
(1) The spatial mixing rate, as given in Equation 2.13, increases with distance
from the trailing edge; this increasing growth rate is in contrast to the constant mixing
rate in planar shear layers . (See Equation 2.8)
(2) The spatial mixing rate increases linearly with Fx and
F,.
Furthermore, the
effects of each component are augmented by the presence of the other.
This
augmentation becomes more evident upon differentiation of Equation 2.13 with respect to
Fx and F,:
C
o dMx 2xk3k4k6
F,
dF x
(2.14)
(214)
d,=
(2.15)
2xk3k4 k6,
x+ k3 k4
In Equation 2.14, the growth of the spatial mixing rate with increasing
by
F,; likewise, the growth
by the presence of
of the spatial mixing rate with increasing
Fx is augmented
F,
is augmented
'x (Equation 2.15).
(3) The role of spanwise vorticity in the mixing process may be identified. As
spanwise vorticity increases, the spatial growth rate of the spanwise undulations
increases. These undulations span the region of mixed fluid, and hence, as their spatial
growth rate increases, the region of mixed fluid spreads more rapidly from the mean
cross-flow interface. Drawing an analogy to the comparison between the linear and
turbulent transport coefficients, the spanwise vorticity may be considered to increase the
effective cross-stream transport properties, i.e.
(linear
Oeffecuve = (k3Fr)
(2.16)
where 19 denotes the transport coefficient.
(4) On the other hand, the role of streamwise vorticity may also be identified. The
streamwise vorticity stretches the mean cross-flow interface on which the region of mixed
fluid grows, i.e.
L,,,ea= k6 Fx
+L
(2.17)
To summarize, the above model provides a guide for examining the dependence
of the mixing downstream of a lobed mixer on four characteristic parameters:
'x,
Ft'
LTE, and x.
2.5.3
Limitations of Hypothesis
The superposition of the roles of the two vorticity components cannot be expected
to hold throughout the entire downstream flow field. This limitation becomes apparent
upon consideration of the interface structure scales of the two processes. The amplitude
of the spanwise undulations grow with distance from the trailing edge, while the
streamwise vorticity winds and stretches the cross-flow interface, leading to the formation
of ever smaller interface structures(Figures 1.3 and 2.4).
As the amplitude of the
spanwise undulations approaches the scale of the windings in the cross-flow interface,
some interaction between the two processes must occur. As will be shown in Chapter 4,
despite this limitation, the hypothesis does provide a useful estimate of the mixing rate in
the initial region of the flow field. The axial extent of this region is limited by the
structure scale limitations discussed above, and in the numerical experiments, the model
adequately describes the mixing process within 5-10 X of the lobe trailing edge.
Symmetry Plane (Side View)
Trailing Edge
...........
..........
.................
................
...................
....................
......................
........................
.........................
............................
...........................
.............................
...............................
...
..........................
.................................
.....
I............................
....................................
....................
............
.......................
4
...
~L:::::::::::::·:·:·::::::::::::::::::;:
:·:::
h
:·:·:·:·:·:·:·:·:·:
::·:
-- Lobe Height
~i··::::::::::::::::::::::::::::::::::::
L::·:·:·:·:·:·:·:·:·:·:·:·:·:·:···:·····
:·:·:·::·
:·:
;;I~::::::::::::::::::::::::::::::::::::
s::::::::::::::::::::::::::::::r
~,:·:·:·:·:·:·:·:·:·:·:·
::·:·······-·
;·:·:·:
Cross-Flow Planes (Trailing Edge View)
X-
Wavelength
I
I
I
X - Wavelength
I
h--Lobe Height --h
Square Lobe Mixer
Sinusoidal Lobe Mixer
Figure 2.1: Geometric characterization of a lobed mixer
Path
diling Edge
Meanflow direction is into page.
White indicates upperstream, and gray indicates lower stream.
Figure 2.2: Path of integration on x = 0 plane to determine Fx
Figure 2.3: Experimental flow visualization of planar mixing layers[6]
35
x=O
x=4
x=8
x=12
x=16
Mean flow direction is into page.
White indicates upper stream, and gray indicates lower stream.
Figure 2.4: Roll-up of interface in cross-flow planes downstream of a lobed mixer
with equal velocity streams (F,= 0) for
LPM on cross-flow planes x = 0, 4, 8, 12, 16[9]
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
12
14
16
x
Figure 2.5: Cross-flow interface length vs. distance from trailing edge for
LPM with Fr = 0.0
0.4
0.6
Figure 2.6: Spatial growth rate of cross-flow interface length vs.
streamwise circulation for LPM and HPM' with r,= 0.0[9]
L.PM
HPM'
Mixer
Figure 2.7: Constant streamwise circulation for
LPM and HPM' with F,= O.O[9]
Figure 2.8: Spanwise undulation about mean cross-flow interface
Chapter 3
Computational Method
3.1
Introduction
To assess the usefulness of the model proposed in Chapter 2, a series of numerical
simulations of the flow field downstream of the lobed mixer are conducted with a threedimensional unsteady Euler code[9]. In each of these simulations,
Fx and F, are
varied, and the results are reduced to a form that can be cast in terms of the trends
predicted by the model.
One may question the use of an Euler code to measure mixing as it would occur in
a real flow. However, if the inviscid flow processes are considered to act in a manner as
to provide opportunity for mixing to occur, then the use of an Euler code for these
simulations is justified. For instance, the roll-up of the interface downstream of the lobe
trailing edge leads to an increase in the interface area between the co-flowing streams.
While this roll-up is an inviscid phenomena, it acts to provide additional opportunity for
mixing to occur.
In these simulations, the Euler code computes a flow field that closely
approximates the actual unsteady flow field and the movement of the interface is tracked
via a passive scalar transport equation. However, grid resolution and the use of artificial
viscosity (although it is minimal) place a lower limit on the scale of the interface
structures that can be resolved. Nevertheless, as the spatial mixing rate will be estimated
from the range of interface movement, only the largest scale structures need be accurately
modeled, and in these simulations the grid resolution and artificial viscosity are such that
these large structures are adequately represented.
3.2
Test Matrix
To assess the model, three mixers, each generating a streamwise circulation of
different magnitude, are examined while maintaining the same spanwise circulation, and
one mixer is examined for three different spanwise circulations. The three mixers are
shown in Figure 3.1.
Two of the mixers, LPM and HPM, have been used in numerical experiments by
Elliott. They are selected for this study such that the current results may be compared to
those of Elliott; the third mixer is the advanced design mixer, as its square lobes are
thought to be more in line with current design trends. Furthermore, a primary application
of these mixers is the mixing of the core and bypass streams within the exhaust nozzle of
turbofan engines[2,4], the flow regimes modeled in these simulations are chosen to be
representative of those typically found in modem turbofan engines.
The simulations performed are encapsulated below.
Table 3.1: Test Matrix I
F,=0.7
FX
= 0.1 (LPM)
AU= 0.50
Fx= 0.4 (HPM)
AU= 0.32
Fx= 0.9(ADM)
AU= 0.27
F
F,= 1.2
F,= 1.8
AU= 0.50
AU= 0.80
A key element in the proposed hypothesis is the assumption that the roles of the
individual vorticity components (streamwise and spanwise) can be superposed.
Agreement between the results of these simulations and the hypothetical flow model
would strongly imply that the effects of spanwise and streamwise vorticity components
can indeed be superposed.
In all the test cases, the co-flowing streams were of equal density, temperature,
and pressure. The mean velocity and Mach number were kept constant at 0.5. Only the
individual stream velocities/Mach numbers/total pressures were varied but always such
that the mean velocity remained at 0.5. Hence, density and compressibility effects should
not be an issue in the results of the numerical experiments as only FT or I, is varied in
each case.
1 In the table, F,
grows from left to right, and Fx grows from top to bottom. The velocity difference
between each stream required to achieve the desired F, marks the numerical simulations conducted in this
study.
In addition to the five test cases discussed above, several other test cases
involving the low penetration mixer (tabulated in Table 3.2) were used for the
preliminary assessment of the flow model. Indeed, it is in these initial assessments that
the superposition of the individual effects of the streamwise and spanwise vorticity
components was recognized. The LPM mixer has the smallest streamwise circulation of
the three mixer configurations, and thus, the mean cross-flow interface rolls up more
slowly than in the other two configurations.
As a result, the streamwise interface
structure scale decreases at a slower rate, and therefore, any interaction with the spanwise
undulations occurs farther downstream. Hence, the region where the proposed model is
valid is expected to be larger.
Table 3.2: LPM Test Cases
F,
x= 0.04 (LPM)
3.3
F,= 0.70
,--0.23
_',= 0.14
AU= 0.50
AU-0.33
AU=0.20
Numerical Algorithm
The code used in the numerical experiments is a three dimensional, unsteady Euler
code; it is based on a finite-volume flux-corrected transport algorithm and was developed
by Elliott[9]. The algorithm limits the extrema generated by a high order scheme with a
monotonicity preserving low order scheme. The high order scheme uses the LeapfrogTrapezoidal method with minimal fourth order dissipation upstream of the lobe trailing
edge. No dissipation is used downstream of the trailing edge so as not to contaminate the
details of the roll-up process. The low order method used is the Euler method with a
zeroth order dissipation term. This dissipation term is used throughout the flow field to
stabilize the numerical algorithm, and the amount of dissipation used is just sufficient to
stabilize the scheme. Additional details of the numerical scheme may be found in
Elliott[9].
3.4
Boundary Conditions
The boundary conditions used are as follows. Symmetry boundary conditions are
used at the two side walls, and characteristic based boundary conditions are used at the
inflow and outflow boundaries. Furthermore, the outflow pressure is set equal to the
inflow pressure. Solid wall boundary conditions are used on the upper and lower walls as
well as on the lobe surface. The solid wall conditions used permitted no velocity
component normal to the wall. From this stipulation and the normal momentum equation
at the wall, the wall pressure may be determined.
3.5
Grids and Grid Generator
The structured grids used in these calculations are generated with an elliptic solver
written by Liu[21]. Poisson's equation is solved on a series of axial planes, and these
planes are then stacked to form the three dimensional grid.
Three mixers, and hence three grids, were utilized in the study.
The
computational domain, shown in Figure 3.2, covers an extent of 20 X from inflow to
outflow (x-direction). The mixing region between the lobe trailing edge and the outflow
plane spans 16 X. In an axial plane, the domain covers an extent of one-half X from side
wall to side wall (z-direction) and 4 X from the upper to lower walls (y-direction).
The size of the computational domain is limited by grid resolution and the
availability of computer resources. However, as will be shown in the discussion of the
computational results, mixing augmentation associated with streamwise vorticity appears
complete within 5-10 wavelengths downstream of the trailing edge. Thus, the chosen
computational domain is more than adequate for the present investigation.
In some cases, the interface moves throughout a large portion of this domain.
Thus, an effort was made to keep the grid resolution constant. Consequently, grid lines
are not clustered about the lobe and in the region near the lobe trailing edge. To illustrate
the resolution, a symmetry plane from each grid is shown in Figure 3.3, and an axial
plane in the mixing region from each of the three grids is pictured in Figure 3.4.
3.6
Interface Tracking
In these simulations, the interface is tracked via a passive scalar transport
equation:
D__ = 0.
Dt
(3.1)
The upper stream is seeded with the value 0, while the lower stream is seeded with the
value 1. The interface is identified with the V = 0.5 contour.
3.7
Definition of Mixed Fluid
The mixing layer in the presence of strong spanwise vorticity is characterized by
an unsteady undulating interface. The range of the interface movement is used as a
measure of mixing, and a time-averaging process is used to determine the full range of
this movement.
In these numerical simulations, a period of these undulations is found to be a
fraction (-1/4 ) of a flow through cycle; here, a flow through cycle is defined as the time
taken by the fluid particles traveling at the mean velocity to traverse the entire axial
extent (20k) of the domain. To determine the full range of these undulations, and hence
estimate the mixing, the location of the interface is averaged over at least 4 periods of
spanwise undulation, i.e. at least one flow through cycle. Averaging the interface
location over one period of undulation would be adequate to determine the full range of
interface movement, as in this one period the interface will have spanned its entire range.
However, the precise period of undulation in each simulation has not been monitored, and
hence, to ensure that the full range of interface movement is determined, the averaging
process is conducted over at least several periods of undulation.
In practice, to determine the range of movement, the passive scalar field is
averaged over these periods. As the two streams are seeded with the P values of 0 and
1, any point in the mean scalar field with a value of P between 0 and 1 has been spanned
by the interface at some time during the averaging process. However, in practice, the
computed scalar field appears slightly noisy (~0.01) about the value Y = 0, and hence, to
alleviate any noise contamination, any value of '
between 0.1 and 0.9 is defined as
mixed for this study. As alluded to in Chapter 2, the only stipulation is that a constant
percentage of the region spanned by the interface be considered mixed, and this
requirement has been met by the above definition of mixed fluid. For instance, if any
value of Y between 0.3 and 0.7 is defined mixed, then the calculated mixedness would
differ; however, the spatial growth rate of the mixedness would not.
To calculate the normalized mass flow rate of mixed fluid in a given cross-flow
plane, the value of Y in each grid cell in that plane was checked to see if it fell between
0.1 and 0.9. If so, the mass flow through that cell was added to the total mass flow of
mixed fluid for that cross-flow plane, i.e.
Mmix -1/
M
inflow
pUdA.
0.1<W0.9
(3.2)
LPI
I
Figure 3.1: Three lobed mixers used in the numerical experiments:
LPM, HPM, and ADM
(0,0, 0)
T
0.5X
z
I
x
Figure 3.2: Computational domain for all mixer configurations with the LPM trailing
edge cross-flow plane shown to illustrate the location of the lobes in the domain
LPM
HPM
..........
............
ADM
............ . ..................................................... -------I.
a.
I.
II
..
I.
..
NI
......
..
..
..
..
..
....I ----IIIIII--- -III.
I.
II
II
II
..
-1
.........N..
..
.I
.I
.I
..
.I
..
..
..
..
..
...
.I
.I
..
.I
..
.
III.........IIIIII....... ..III
The lobe location is indicatedin bold.
Figure 3.3: Grids on a symmetry plane, z = 0, for the LPM, HPM, and ADM
LPM
HPM
ADM
The lobe location is indicatedin bold.
Figure 3.4: Grids on a cross-flow plane in the mixing region, 0 < x < 16, for the
LPM, HPM, and ADM
Chapter 4
Numerical Experiment: Results
4.1
Introduction
A vortex sheet with streamwise, and possibly spanwise, components is
continually shed from the trailing edge of a lobed mixer. The inviscid mixing processi
downstream of the lobe may be described as the roll-up of this sheet. This process is
illustrated in spanwise and cross-flow planes in Figure 4.1. In this figure, the cross-flow
interface stretches and winds, while spanwise undulations grow and convect downstream.
The growth and convection of these undulations results in the unsteady movement of the
stream interface, and in these simulations, the range of this movement is used to estimate
the extent of the mixing layer.
The computational results presented in this chapter will show that this mixing
layer may be divided into three regions characterized by the ratio of the length scale
associated with the spanwise undulations to that associated with the cross-flow interface
structure. This ratio will be denoted by r,. In the region stretching 5-10 X downstream
of the lobe, hereafter referred to as Region 1, the spanwise undulations are much smaller
than the streamwise interface structures, and the model provides a reasonable description
of the mixing process. However, the amplitude of the spanwise undulations grows with
distance downstream. As the ratio r, approaches unity, a change in the nature of the
1 Increase in interface area
mixing layer occurs, and the model no longer adequately describes the mixing process.
The region in which the ratio of the streamwise structures and spanwise undulations is
roughly unity is designated Region 2. However, the spanwise undulations continue to
grow with distance from the trailing edge and eventually become large enough to
dominate the mixing layer. The region characterized by these large scale undulations will
be designated Region 3.
4.2
Region 1
In this section, the computational results will be presented to demonstrate that the
region immediately downstream of the lobe is characterized by the linear growth of the
cross-flow interface due to streamwise vorticity and the growth of a spanwise KelvinHelmholtz type instability on this interface.
4.2.1
Roll-Up of Cross-Flow Interface Associated with Streamwise Vorticity
The role of streamwise vorticity in the mixing process downstream of a lobed
mixer in the absence of spanwise vorticity has been discussed in some detail in Section
2.4. In summary, the cross-flow interface grows linearly with distance from the trailing
edge, and its growth rate is proportional to the streamwise circulation.
In the presence of spanwise vorticity, a similar roll-up occurs. This process is
illustrated for the LPM with
',=0.7 in four axial planes in Figures 4.2-4.5. In these
views the streamwise vorticity, delineated by the vector plot on the left, winds and
stretches the vortex sheet/interface separating the co-flowing streams. The winding of the
interface is illustrated by the vorticity magnitude plot in the middle of the figures and by
the passive scalar contour on the right.
However, as shown in Figure 4.6, flow disturbances due to the Kelvin-Helmholtz
instability associated with spanwise vorticity result in the unsteady movement of the
cross-flow interface. These unsteady disturbances are illustrated for the LPM with
',--0.7 in Figures 4.2-4.5. Passive scalar contours in four cross-flow planes at x=4 are
shown: one without spanwise vorticity(AU=0) shown on the left, two instantaneous
views with spanwise vorticity illustrating the unsteadiness of the interface in the middle,
and a time mean view illustrating the range of the movement on the right.
This movement can conceal the streamwise stretching process as the
instantaneous cross-flow interface length changes with time. However, a time mean
interface may be defined as the W= 0.5 contour in the mean scalar field. This contour is
shown in Figure 4.7 for the LPM with
F,=0.7
Furthermore, in this figure the mean
interface is seen to correspond approximately to the mid-line of the cross-flow region
bounded by the spanwise undulations and is observed to stretch with distance from the
trailing edge. In Figure 4.8, the interface is shown to grow linearly with distance from
the trailing edge; furthermore, it is shown to grow at the same rate as the cross-flow
interface for which F,=O.
A similar roll-up of the cross-flow interface occurs downstream of the HPM and
the ADM.
In Figure 4.9 for the HPM, the circulation associated with streamwise
vorticity that winds and stretches the cross-flow interface is illustrated via the velocity
vector field; the actual roll-up process is shown in Figure 4.10. Likewise for the ADM,
the streamwise circulation and associated interface roll-up are pictured in Figures 4.11
and 4.12 respectively.
A comparison between the results presented for the LPM in
Figures 4.2-4.5 and those shown for the HPM in Figures 4.9-4.10 demonstrates that a
smaller streamwise circulation is associated with the LPM than the HPM, and as a result,
the roll-up process occurs at a slower rate downstream of the LPM. An additional
comparison between the results of the HPM and the ADM presented in Figures 4.9-4.10
and Figures 4.11-4.12 respectively, illustrates the stronger streamwise circulation
associated with the ADM and the resulting more rapid evolution of the cross-flow
interface downstream of that mixer. The more rapid stretching of the interface associated
with the ADM implies that, in comparison to the LPM and HPM, a greater opportunity
for mixing to occur exists downstream of this mixer.
In Figure 4.13, the growth of the mean cross-flow interface with distance from the
trailing edge in Region 1 is illustrated for all the numerical simulations conducted in this
study: three LPM, three HPM, and one ADM. The interface grows linearly with x.
However, the range of this linear growth is limited, and in these results, the region of
linear growth ranges from about 4 X in the ADM to about 10 Xin the LPM. Furthermore,
the growth rate is independent of the spanwise vorticity. In Figure 4.14, the growth rate
is shown for each of the simulations conducted, and
proportional to
ea
dx
is seen to be roughly
FX-
The above computational results are in good agreement with the model presented
in Chapter 2. Hence, the growth of the mean cross-flow interface can be represented by
Lmean
(2.16)
= k6 FxX + LFE,
where the constant, k6 , has been determined to be 1.3 from the computational results.
Based on the results presented in this section, the role of streamwise vorticity in
the mixing process may be stated as follows: if the spanwise undulations are considered
to ride on the mean cross-flow interface, the stretching of this interface by streamwise
vorticity creates more area for the growth of the spanwise Kelvin-Helmholtz undulations.
4.2.2
Growth of Undulations Associated with Spanwise Vorticity
In the discussion that follows, the spanwise vorticity is shown to initiate a Kelvin-
Helmholtz type instability on the mean cross-flow interface.
The amplitude of the
undulations grow with distance from the trailing edge, and the spatial growth rate of the
undulations increases with the velocity difference between the two streams.
Qualitative Description of Spanwise Process
The unsteadiness of the flow field downstream of a mixer with a velocity
difference between the co-flowing streams, and thus shedding spanwise vorticity, is
illustrated in Figure 4.15, where the velocity vector field is shown on the x=4 plane at
four equal time intervals. This unsteadiness is due to an instability associated with the
spanwise vorticity, as the large scale inviscid flow field downstream of a mixer without
spanwise vorticity is steady[9]. In Figure 4.16, where views of the vorticity field in the
x=4 plane are shown at the same four equal time intervals, this instability is seen result in
the movement of the cross-flow vortex sheet between the streams with time. In Figure
4.17, the resulting undulation of the cross-flow interface is illustrated in the passive scalar
field at the same four time intervals used in Figures 4.15 and 4.16.
In Figure 4.18, the velocity vector field in the z=O plane is shown at four equal
time intervals to illustrate the growth of region of the fluid exhibiting this unsteady
behavior with distance from the trailing edge. This unsteadiness results in the growth and
convection of large scale vortical structures, as shown in the z=O plane in Figure 4.19.
These vortical structures mark the stream interface, and in the passive scalar field shown
in Figure 4.20, the instability associated with spanwise vorticity is clearly seen to result in
the growth and convection of large scale structures/undulations in the interface. Indeed,
the development of these undulations appears to be Kelvin-Helmholtz like. The growth
of these structures with x is more clearly illustrated in the mean passive scalar fields
shown in the z=O plane in Figure 4.21. Furthermore, in this figure, their spatial growth
rate is seen to increase with AU, as does the spatial growth rate of undulations resulting
from the Kelvin-Helmholtz instability.
This instability is shown to develop with distance from the trailing edge along the
entire mean cross-flow interface downstream of the LPM in four axial planes in Figure
4.7, and as shown in Figures 4.22 and 4.23 for the LPM and HPM respectively, the
spatial growth rate of the undulations increases with AU along the entire cross-flow
interface as well.
The computational results presented in Figures 4.15-4.23 allow one to deduce the
qualitative variation in the spatial growth of the spanwise undulations with distance from
the trailing edge and the velocity difference between the two streams: the undulations
grow with x, and their rate of growth increases with AU. Indeed, the evolution these
undulations is seen to be a Kelvin-Helmholtz type process. In the next section, using
these computational results and taking the analysis a step further, a quantitative measure
of the growth of the undulations resulting from this Kelvin-Helmholtz type instability is
derived in terms of these two parameters: x and AU.
Quantitative Analysis of Growth of Spanwise Undulations
In the scalar field shown in Figure 4.7, it may be argued that the cross-flow
interface undulates about a mean. If this undulation is imagined to be a series of local
Kelvin-Helmholtz instabilities, one may then say that the amplitude of these local
undulations varies along the mean interface. From knowledge of this variation, an
average amplitude of undulation may be determined. In these simulations, this amplitude
in a given cross-flow plane is calculated by dividing the area bounding these undulations
by the length of the mean cross-flow interface, i.e.
6mean
Amix
(4.5)
3mean may be considered to be an equivalent planar shear layer width. 2
Shown in Figure
4.24 is the downstream variation of this width for the LPM flows examined in this study;
Smean is seen to increase with distance from the trailing edge. In Figure 4.25, 5mean is
also see to grow with x for the three HPM simulations.
The result of the inviscid hydrodynamic stability analysis of a planar mixing layer
mentioned in Section 2.3 predicts that the spanwise undulations will grow exponentially
with distance from the trailing edge and that the exponent will increase linearly with
AU[3,9]. The model, introduced in Section 2.5, suggests that the spanwise undulations in
mixing layer downstream of a lobed mixer should grow in a manner similar to their twodimensional counterparts. Hence, if the model adequately describes the mixing layer
downstream of a lobed mixer, the spanwise undulations should be expected to grow
exponentially with distance from the trailing edge, and the exponent should be expected
to increase linearly with AU.
The results in Figures 4.24-25 show an exponential
variation, which may be adequately described by
mea
= k7ek x .
(4.1)
The constants, k7 andk 8 , are tabulated below in Table 4.1 for all the LPM and HPM
flows examined in this study.
2 Recall that the expression for the growth of the planar shear layer is given in Equation 2.6 as
AU
2D
=
k3 -U,P
Table 4.1: Constants in Exponential Fits for
3 mean in Region
LPM
k9
1
HPM
AU=0.2
AU=0.33
AU=0.5
AU=0.32
AU=0.5
AU-0.8
0.014
0.014
0.011
0.033
0.028
0.024
0.16
0.29
0.38
0.15
0.25
0.35
0.8
0.9
0.8
0.5
0.5
0.4.
If the spanwise undulations do indeed grow like their two dimensional counterparts, the
exponent, and hence k, in the above table, should increase linearly with AU. To
determine whether the computational results do show this variation, the ratio of k8 to AU
is calculated from the results in Table 4.1, and this ratio is denoted as k9:
k, =
"
AU
(4.2)
The ratio, k 9 , is shown in Table 4.1 as well. For each mixer, k9 is nearly invariant with a
change in AU, implying that k s , and hence the exponential growth rate, increases with
AU. Thus, the predicted trend from the hydrodynamic stability analysis is seen in the
computational results.
One can see that for a given mixer, and hence initial streamwise circulation,
spanwise undulations grow exponentially with distance from the trailing edge and that the
exponent increases linearly with AU. The disparity between the exponential growth rates,
i.e. k9 , for the LPM and HPM indicates that the streamwise vorticity does indeed
influence the spanwise process. In this case, one may argue that the presence of strong
streamwise vorticity appears to have diminished the growth of the spanwise undulations
somewhat.
Nevertheless, these results imply that one may consider the spanwise process as a
Kelvin-Helmholtz type instability distributed along the mean cross-flow interface.
Furthermore, they imply that the growth of the spanwise undulations for a given mixer
may be described by
5mean = kIek2AUx,
(2.5)
where k1 and k are characteristic constants for that mixer.
Based on observations from planar shear layer experiments[6,7], the hypothetical
model predicted that the spanwise undulations downstream of a lobed mixer would grow
linearly like their planar counterparts. This linear growth appears to be in contrast to that
seen in these inviscid computational results and Equation 2.5. However, the Taylor
expansion of Equation 2.5 yields an expression in which the spanwise undulations grow
linearly for small x. This linear growth is indeed seen in the data within the region a
few(1-3) wavelengths downstream of the trailing edge. However, in the computational
results, these undulations continue to grow beyond this initial linear region, and their
growth is best approximated by Equation 2.5. Hence, using the knowledge gained in this
analysis, the model will be modified to include this exponential growth of the spanwise
undulations in order to extend its range of usefulness. This modification will be made in
Section 4.2.3.
In summary, the computed results indicate that the spanwise undulations on the
mean cross-flow interface grow in a manner similar to their two dimensional
counterparts, thereby supporting the model presented in Section 2.5 and the superposition
of the effects of spanwise and streamwise vorticity in the mixing layer.
Role of Spanwise Vorticity in the Mixing Process
Despite the limited influence of the streamwise vorticity, the spanwise
undulations qualitatively grow like their two dimensional counterparts along the mean
cross-flow interface. The spanwise undulations grow and act to spread the region of
mixed fluid from this stretching cross-flow interface. As mentioned previously in the
discussion of the hypothetical model in Section 2.5, the spanwise vorticity may be
considered to increase the effective cross-stream transport properties. For small x, as the
spatial growth rate of the spanwise undulations grows linearly with AU, the local
effective transport properties may be considered to increase linearly with the spanwise
vorticity, i.e.
Oeffective = (k odU)
1 Olinear
(4.3)
For larger x, the spatial growth rate of the spanwise undulations grows exponentially with
AU, and such a simple relation may not be deduced. However, the spanwise vorticity
may still be considered to increase the effective transport properties. In the absence of
any growing spanwise undulations, mixing would proceed, albeit at a much slower rate,
as a diffusion process across the stretching cross-flow interface. The growing spanwise
undulations can enhance this mixing rate.
4.2.3
Discussion of Results in Terms of Model
In Section 4.2.1, the streamwise vorticity has been shown to stretch the mean
cross-flow interface with distance from the trailing edge. This spatial stretching rate is
proportional to
Fr
and is relatively unaffected by spanwise vorticity. In Section 4.2.2,
the spanwise vorticity has been shown to initiate a Kelvin-Helmholtz type instability on
this mean cross-flow interface. The undulations resulting from this instability grow
exponentially with distance from the trailing edge. In these simulations, the presence of
strong streamwise vorticity is seen to slow the growth of these undulations. However, for
a given mixer and streamwise circulation, the exponent alluded to above increases
linearly with AU.
From these computational results, and with the aid of the hypothetical model
presented in Chapter 2, quantitative expressions for the growth of the mean cross-flow
interface and the growth of the spanwise undulations can be deduced. These expressions
are indeed quite similar to those proposed with the model in Chapter 2, with a
modification to the spanwise expression to extend its range of usefulness. Using these
expressions, and defining the various constants based on the computational results, a
predicted mass flow rate of mixed fluid may be deduced.
In Chapter 2 the proposed superposition of the effects of the vorticity components
leads to an expression for Mmix in terms of Lmean and
5mean
mean Lmean,,
=
Smix
The expressions for
3mean
(4.4)
and L,,,mea deduced in Sections 4.2.1 and 4.2.2 may then be
substituted into Equation 4.4 to arrive at an expression for a predicted Mmix:
A•mix
-= kekAx(k 3 fxx + LrE).
(4.5)
Upon defining k1 , k2 , and k3 based on the results presented in Sections 4.2.1 and
4.2.2, the predicted mass flow rates of mixed fluid are plotted with the results of the
numerical experiments for the LPM in Figure 4.26. In this figure, the expression in
Equation 4.5 does quite well in predicting the computational results, indicating that the
roles of the two vorticity components may be superposed. In Figure 4.27, the predicted
and computed mass flow rates of mixed fluid are shown for the HPM. In this figure, the
mass flow rates predicted by Equation 4.5 do not agree as well with the computational
results. As the HPM generates a larger streamwise circulation, these results indicate that
the spanwise and streamwise processes are coupled to some extent in the presence of
strong streamwise vorticity.
Based an these results, an interesting observation regarding the spatial mixing rate
may be made: the spatial mixing rate downstream of a lobed mixer with both streamwise
and spanwise vorticity increases with distance from the trailing edge! In the inviscid
limit, as the spanwise undulations grow exponentially in Region 1, the spatial mixing rate
increases exponentially with distance from the trailing edge--
dimix = kIek2AUx(k
dx
2
AUL + k6
x).
(4.6)
However, due to the stretching of the mean cross-flow interface by streamwise vorticity,
this increasing spatial mixing rate is observed even in the region of linear growth(1-3 X's)
of the spanwise undulations. This increasing spatial mixing rate is in contrast to the
constant rate observed in experiments for planar mixing layers[6,7]. The streamwise
vorticity stretches the mean cross-flow interface, and hence, the interface length available
for the spanwise undulations to grow increases with distance from the trailing edge.
However, as will be discussed in the next section, the region of increasing mixing
rate is limited to the first 5-10 X downstream of the mixer.
4.2.4
End of Region 1
The discussion in this sub-section will show that the superposition of the effects
of each vorticity component is limited to the first 5-10 wavelengths downstream of the
trailing edge of the mixer. This limitation has been noted in Section 2.5.
Streamwise Scale
The stretching and winding of the cross-flow interface creates ever smaller
interface structures. (Figures 1.3, 4.2-4.5, 4.10,4.12) Thus, the average structure scale in
the streamwise interface decreases with distance from the trailing edge. As the crossflow interface stretches and winds more rapidly with increasing streamwise circulation,
the average structure scale can be expected to decrease more rapidly with x with
increasing
"x. Quantitative values of this scale are difficult to measure. However,
estimates from the cross-flow pictures in Figures 4.2-4.5, 4.10, and 4.12 have been made
by averaging the largest and smallest cross-flow interface structures in selected planes,
i.e. planes on which a large and small scale structure could be easily identified and
measured. The results of the measurements are presented in Figure 4.28, and from this
figure, an expression approximating the variation in the streamwise structure scale with
Fx and x may be deduced:
(4.7)
tx ~ Y2- klo l-xx.
Thus, tx is seen decrease with distance from the trailing edge as the cross-flow interface
rolls up and is seen to decrease more rapidly as the rate of the roll-up process (~Fx)
increases.
Comparison Between Streamwise and Spanwise Length Scales
The variation of the spanwise and streamwise scales with downstream distance is
illustrated in Figure 4.29 and 4.30 for the LPM and HPM respectively. Relevant portions
of the streamwise structure scale plot in Figure 4.28 are shown with the
6
mean
data from
Figures 4.24 and 4.25 to highlight a comparison between the two processes. As shown in
this figure, the spanwise to streamwise scale ratio approaches one with distance from the
trailing edge, i.e.
rA-
(4.8)
meanl
In Figure 4.29 for the LPM, the rate at which this ratio approaches unity grows with
increasing AU, as the spatial growth rate of
3mean increases with the velocity difference
between the two streams. This same observation may be made from the data presented in
Figure 4.30 for the HPM. Furthermore, a comparison between Figures 4.29 and 4.30
illustrates that this ratio approaches unity more rapidly in the HPM than in the LPM, as
the cross-flow interface winds more rapidly due to the higher streamwise circulation
generated by the HPM. In the LPM and HPM numerical experiments conducted in this
study, the scale ratio approaches one at a distance of 7 to 10 wavelengths downstream of
the trailing edge in the LPM cases and at a distance of 5 to 8 wavelengths downstream in
the HPM cases.
When r, approaches a value of unity, the interaction between the vorticity
components becomes significant and a change in the nature of the mixing layer occurs.
This change in the mixing layer will be the subject of discussion in the next section.
4.3
Region 2
This region of the mixing layer is characterized by a flow field having
the order of
£x,
3 mean of
i.e.
r- ~ 1,
(4.9)
and the discussion in this section will show that the roles of the two vorticity components
may not be superposed.
As the scale ratio approaches unity with x, the growth of the cross-flow interface
slows and eventually ceases. This slowing growth is illustrated in Figure 4.31 for the
LPM and in Figure 4.33 for the HPM. In a comparison between these figures and the
results of Figures 4.29 and 4.30, the point marking the end of linear growth of the crossflow interface is seen to roughly coincide with a unity scale ratio; this indicates that as r,
approaches one, a change in the nature of the mixing layer does indeed occur.
For the LPM in Figure 4.32 and for the HPM in Figure 4.34, the decrease in the
growth of the mean cross-flow interface is marked by a decrease in the spatial mixing
rate. This diminished spatial mixing rate should be anticipated if one recalls that the
mixing augmentation associated with the streamwise vorticity is a result of the stretching
of the mean cross-flow interface. Hence, as this stretching rate slows and ceases, the
spatial mixing rate should be expected to decrease as well. Thus, as the ratio of the
spanwise and streamwise scales approaches unity, the effects of the two vorticity
components begin to interact, and their roles may no longer be superposed.
When growth of the mean cross-flow interface ceases, streamwise vorticity will
no longer be effective in promoting mixing enhancement.
Nevertheless, mixing
continues due to the growth of the spanwise undulations. These undulations continue to
grow and eventually dominate the nature of the mixing layer. The mixing layer in the
presence of these large scale undulations will be further examined in the following
section.
4.4
Region 3
The discussion in this section will demonstrate that the mixing layer in Region 3
is characterized by large scale spanwise undulations, .i.e.
r, >> 1.
(4.10)
r, is so large in this region that the mixing layer assumes a planar character. Some
movement of the interface associated with streamwise vorticity is still seen, but on the
whole, this region may be characterized by the dominance of the spanwise process.
These large scale spanwise undulations are illustrated in spanwise planes in
Figure 4.20 and axial planes in Figures 4.35-36.
In Figure 4.35, the large scale
unsteadiness in the flow field is illustrated via the velocity vector field at x=14 at four
equal time intervals. In Figure 4.36, the unsteady movement of the interface is illustrated
in scalar fields at the same axial location and at the same four time intervals. In this
figure, some small scale movement of the interface associated with streamwise vorticity
is seen, as the interface does not undulate in unison. These computed results show that
the large scale spanwise undulations dominate the character of Region 3. The spanwise
undulations continue to grow with distance from the trailing edge; however, as is evident
in the mean scalar fields in Figure 4.21, their growth rate slows.
As the amplitude of the spanwise undulations has grown so large that they
completely overwhelm the streamwise interface structures. From the definition of mixed
fluid used in this study, the streamwise structures are completely mixed, and the region of
mixed fluid simply increases in size due to the growing spanwise undulations. This
growth is illustrated in time mean views of the scalar field at four axial locations in
Figure 4.37 for the LPM with F,=0.7. As the region of mixed fluid expands only to the
growth of the undulations, the associated spatial mixing rate slows, as shown in Figure
4.32 for the LPM and in Figure 4.34 for the HPM.
LPM
z=O
i~ijij~i~ iiiijii~iiii..........~i
..............
.........
iii
ijij~
iiiiiiii~~i
. . .ii~i~
....
..............
T
TE
416A
NOTE: Mean flow direction is from left to right.
ADM
x=O
x=2
x=4
2A
NOTE: Meanflow direction is into page.
................
'::':::::::'
................
'
...........
'::::::::::::::::
i~jiiiiiiiiiiiii::
:. ..........
................
Stream 1
Stream 2
Figure 4.1: Typical computational results illustrating the growth of spanwise undulations
on the z=O plane for the LPM with ",=0.7 and the roll-up of the cross-flow
interface on the x=O, 2, 4 planes for the ADM with F, =0.7
I
LPM
Fx =0.1
A4I
A
-;.
2A
\
WIV
Ilk \
I
-
-
p
1
I
(a)
(b)
(c)
NOTE: Meanflow direction is into page.
LEGEND
Vector Vectors(a)
V=1=Mean Inflow Velocity
Vorticity Magnitude(b)
10
8
6
4
2
Passive Scalar(c)
I::::::ijiijiijiiij:::::i ::::::ii1
~~~iiititi:::~
i
Stream 1
Stream 2
x=O
Figure 4.2: Roll-up of the cross-flow interface on the x=O plane for LPM with F,=0.7:
Velocity Vectors(a), Vorticity Magnitude(b), Passive Scalar(c)
LPM
F.,=O.l
V
A
*
I
W/
I
I
'
t
2).
i
\
1,
/
I
p
,
, 0.5) .40
(a)
(b)
(c)
NOTE: Meanflow direction is into page.
LEGEND
Vector Vectors(a)
V=1=Mean Inflow Velocity
Vorticity Magnitude(b)
I
10
8
6
4
2
Passive Scalar(c)
Stream 1
............
Stream 2
x=2
Figure 4.3: Roll-up of the cross-flow interface on the x=2 plane for LPM with ",=0.7:
Velocity Vectors(a), Vorticity Magnitude(b), Passive Scalar(c)
LPM
F,,=O.l
a
a
a
i
t
2l
t
t
I
I'
F
-
(a)
(b)
(c)
NOTE: Mean flow direction is into page.
LEGEND
Vector Vectors(a)
V=1=Mean Inflow Velocity
Vorticity Magnitude(b)
10
8
6
4
PassiveScalar(c)
:·:·:1:·::i:i:Si:i:3~i:3i48
s::~:
iii~s:i:Iii
rima1~11iiiiiiij8iiiji:::iiiiii'''''
'i::·:ii
:~::I:·;:
:.·:.:·:·:·:~·
Stream 1
Stream 2
x=4
Figure 4.4: Roll-up of the cross-flow interface on the x=4 plane for LPM with F,=0.7:
Velocity Vectors(a), Vorticity Magnitude(b), Passive Scalar(c)
LPM
F.,=O.J
I&I
(a)
(b)
NOTE: Meanflow direction is into page.
LEGEND
Vector Vectors(a)
V=1=Mean Inflow Velocity
Vorticity Magnitude(b)
10
8
6
4
2
II
PassiveScalar(c)
..
Stream 1
Stream 2
x=6
Figure 4.5: Roll-up of the cross-flow interface on the x=6 plane for LPM with F,=0.7:
Velocity Vectors(a), Vorticity Magnitude(b), Passive Scalar(c)
LPM
Fx and F,
F- and F,
Time=1
Time=2
Mean
(b)
(c)
(d)
F, Only
2A
~ 0.5 )A4
All Time
(Steady)
(a)
NOTE: Mean flow directionis into page.
LEGEND
InstantaneousPassive Scalar(a),(b), (c)
.·.·.·...·.·.·...:.:.·;·;·.·.·.·.·.·.·.·
'''' ' ' ' ' ' ' ';''
r:::·:::::::::::::::2:::i:::·:i:::::
.... ....·...·...·.·.·.·.·.·.·.·.·.·.·.·.·.·.
Zi~iiSiZiiiRil::::::::::::::'"'I:·:·:·:
Stream 1
Stream 2
Mean Passive Scalar(d)
....
...................
:'g
...
.................
::::::*::
:1X:V,::
:"
Stream 1
Mixed Fluid
Stream 2
x=4
I
Figure 4.6: Disturbances due to spanwise undulations on the roll-up of the cross-flow
interface for the LPM in passive scalar fields on the x=4 plane:
(a)Steady case with F,=0.0,(b) Unsteady case with F,=0.7 at time 1,
(c) Unsteady case with F,=0.7 at time 2,
(d) Range of movement for unsteady case with F,=0.7
LPM
r, =0.1
x=O
0
x=2
x=4
x=6
(b)
(c)
(d)
S0.5 A)4
(a)
NOTE: Meanflow direction is into page.
LEGEND
Mean PassiveScalar(a),(b), (c), (d)
Figure 4.7: Stretching of the mean cross-flow interface with distance from the trailing
edge and growth of spanwise undulations on this interface in the mean passive scalar field
on the x=O, 2, 4, 6 planes for the LPM with F,=0.7
2
''''
I
`
`
'
'
I
'
'
'
'
I
'
I
....
'
'
'
I
'
'
'
'
1.6
1.2
0.8
0.4
0
I
Figure 4.8: Mean cross-flow interface length vs. distance from trailing edge in Region 1
for the LPM with F-,=0.0, 0.7
HPM
F• =0.4
Figure 4.9: Instantaneous velocity vectors on the x=O, 2, 4, 6 planes illustrating the
cross-flow circulation associated with streamwise vorticity
for the HPM with
,--0.7
HPM
Frx =0.4
x=O
x=2
(a)
(b)
x=4
x=6
22A
(d)
NOTE: Mean flow direction is into page.
LEGEND
Passive Scalar(a),(b), (c), (d)
....
iii~iiiii
ii:::··::·~·:
~ii
........
...
. .:~
''''rii
.................
j~iiiiijili
Stream 1
Stream 2
Figure 4.10: Instantaneous passive scalar field on the x=O, 2, 4, 6 planes illustrating the
roll-up of cross-flow interface associated with streamwise vorticity for the HPM with
F, =0.7
ADM
Fx =0.9
x=O
x=2
(a0.5
)
(a)
(b)
x=4
x=6
(d)
NOTE: Mean flow direction is into page.
LEGEND
Velocity Vectors(a), (b), (c), (d)
V=1=Mean Inflow Velocity
Figure 4.11: Instantaneous velocity vectors on the x=O, 2, 4, 6 planes illustrating the
cross-flow circulation associated with streamwise vorticity
for the ADM with F,=0.7
ADM
-x=0.9
x=O
x=2
4 0.5 A
(a)
(b)
x=4
x=6
(d)
NOTE: Mean flow direction is into page.
LEGEND
Passive Scalar(a), (b), (c), (d)
..............
.....
................
ii:::::··::::::i:·::
-:
::··
~jiiiiiiiii::ii
..
Stream 1
:·.:ii~iii
.. .........
Stream 2
Figure 4.12: Instantaneous passive scalar field on the x=O, 2, 4, 6 planes illustrating the
roll-up of cross-flow interface associated with streamwise vorticity
for the ADM with F,=0.7
14
12
10
8
6
4
2
0
0
2
4
6
8
10
Figure 4.13: Mean cross-flow interface length vs. distance from trailing edge
in Region 1 for all tested cases
-
--
O
LPM, rt =0.3
O
LPM, Ft =0.5
X
LPM, Ft =0.7
O
1HPM, Ft =0.7
+
HPM, r =1.2
A
HPM, Ft=1.8
8
ADM, Ft =0.7
- dL mean /dx = -0.067806 +12719Fx
Linear Fit
Figure 4.14: Spatial growth rate of cross-flow interface length in Region 1 vs. streamwise
circulation for all tested cases
LPM
lx =0.1
Time= t0
I
to +12(~)
t0 + 8(,•/•)
"
I
I
\(
I
Ij
,
\\\
t
r
t
• t
k~
c
r
J
/
/
2A.
I
t tj
/
t
!\!
I
Ji4
-
(40.5
a)
(a)
-,
d
\ •
\
IN
V
4.__._
(d)
(b)
NOTE: Mean flow direction is into page.
LEGEND
Vector Vectors(a), (b), (c), (d)
V=1=Mean Inflow Velocity
th
x=4
Figure 4.15: Velocity vectors at four time intervals on the x=4 plane illustrating the
unsteadiness of the flow field associated with spanwise vorticity
in Region 1 for the LPM with F,=0.7
LPM
F• =0.1
Time= to
to +4( )
A
I
p_
21I,
-40.5aL
(a)
(b)
(c)
(d)
NOTE: Mean flow direction is into page.
LEGEND
Vorticity Magnitude(a), (b); (c), (d)
10
8
6
4
2
Iz
x=4
Figure 4.16: Vorticity magnitude at four time intervals on the x=4 plane illustrating the
unsteadiness of the flow field associated with spanwise vorticity
in Region 1 for the LPM with ,=0.7
LPM
l, =0.1
Time= to
to +4(U)
to + 8
U)
A
2A
4 0.a5 A"
(a)
(c)
(b)
(d)
NOTE: Meanflow direction is into page.
LEGEND
Passive Scalar(a),(b), (c), (d)
I
I
Stream 1
I
Stream 2
x=4
Figure 4.17: Passive scalar field at four time intervals on the x=4 plane illustrating the
undulation of the interface associated with spanwise vorticity
in Region 1 for the LPM with
F,=0.7
Time= t0
I
T-
m.
.
a
.
.
..
o -----
0
ON
.
0,
...
-0--
16
TE
- -
0"
-
-
-
P".
-
0 - -
---
--
-
-
- --
6A
Time= to + 4(YU)
-
- ---0- •
-0-0-e-0------0----0-
---0--
--- 0
Y
---
-- - --0--0-0-0---0 --0
-0---0---"-0--0---0-
•------•---•.-
.--
-
--
-----0-0-o
0--0-
-•
-.
-
j--o-b-----------
--
40
--
-0 -D-
----0---00 0-.--_
-_.. -........
im
--
- 0
--- -
--
.------0--0-
po-
-
100
Time= to +8(u
327Z0
-go
-
ON---
-0---
-M-
-0-
-- 0-------om
0BI---0--
-
-- 0
-0
-0
0----0-----m-.--
0
m-
.-
-0-
-0-
-
-
-r
-
Ps
-
0
-0
0-
-0--L
I BO.
-
---
- ---------
0
0g---0-- -0-D---_
-0-
-C
-
ON
-C
-
-C
-U-
-0
-e
-r
-C
s~
eI
-.
Tme= to t-12
7
.-----.--.--
-0---
-0-•Do-0-
-----
-0
-
-- -
----
-- --
--
g-o-
04---0--o
-.
-
-'V-r14
-r
-r
-w
A
-r
-
0-)
-0--
-D- -D
-b-.W.--o.
-
0
~~-------
----
~--------------------------4
----
--- 0
-- -
---------
---------_--b -• -
.
....•
---
-t
-------------
~---.
- ------
·.
-
----•
-
-
0-
•
-
.-
-------
~-kL
-4h.e
--
ON.
-~0
A
-.-
-
/
-0-
- _
-
U-
LEGEND
Velocity Vectors
-----
--
--
- -
- -
NOIvE: Mean flow direction isfrom left to right.
z=0
Figure 4.18: Velocity vectors at four time intervals on the z=0 plane illustrating the
unsteadiness of the flow field associated with spanwise vorticity
for the LPM with 1,=0.7
82
Time= to
9 F-I
TE
Time= t0 +4()
Time= to+
Time= to + 12(Y4j)
Ip-Zv
IFdFAfl________________________
_______________________
vur
· u
II
5
4
3
2
1
NOTE: Mean flow direction is from left to right.
z=O
Figure 4.19: Vorticity magnitude at four time intervals on the z=0 plane illustrating the
development and convection of vortical structures associated with
spanwise vorticity for the LPM with 1,=0.7
Time= to
t
~i~iiiiii..........e
··.·;· 1.:-:..-;·;. ·............
·:
.......
2A
.....................
~·
..
..................
...
.....
......
*
..........
....
....
........
...............
··.. ......
...·.
..
.
.
.....
..
............
iiji
i~
......
. ......
2
TE
r;';~"':' '''~""`
`"`'''..............'''
16 A
4
Time= to + 4('~
)
-~~iiiiiiji
~..........i~
. . . ... . .. .. ..
.......
...
...
.........
..
...
..
.....
... .... . . ........
.
......
........
.....
......
..
....
.............
....................
::::::::::::::::::::::i
.......
......
......
.....
.....
.....
.....
...............
.....
:·
:.............
·:·:·:··:·:·
·
............
::::ij:::~i:i,,,
........
:tiiiii
· :·:::::::::::::
·
..........
........
':
............
..........................................
........
.......
........
· :·:·:·:··:·:..
·. ..
.................
...
.................
...................... ..
.....
..
.....
..........
...........................................
...........................
..
......................
..
...................
..................:
.............
... ......................................................
.............. ......
.....
....................
...............
................
....
............
............
.......
.....
..........
..............
............
..................
..............
.............
..
.......
........
Time= to +8( (
................
...................
)
.::
.......
- - .........
..............
............
......
I•GEND.'':
PassiveScalar
................
:.............
·::_·::::··:·::··
..............
....
::·:
...................
.................
:::::::
Stream 1
Stream 2
NOTE: Mean flow direction is from left to right.
z=O
Figure 4.20: Passive scalar field at four time intervals on the z=O plane illustrating the
growth and convection of undulations in the interface associated with the spanwise
vorticity the LPM with F,=0.7
LPM
Fx =0.1
F,=0.3,AU=0.20
TE 4-.
16A
T',=0-5, AU=0.3
............
.......
--..
...
....
:·:............
·.
··............
.`;
:~i::j~i::
....
................
:i::...
. . .:.~ ....
"f#:j~ij:::~:-:~:I:·:.......
:~:~:~·~·~~·:·:·~·;=....
...................
.............
.....
..........
,,,
:·:··~·~'
:
·::·:.·:·......
·
·~;~;.
·
..
...............
"'`' :··:·..............................................
·:·:·:·....................................
: ~:-·:·:~ ·~··:·::·:·'.~::·: ·.·~·~;~:·· ····..........
Fr=0.7, AU=0-50
··:;;.r·;;;......·~-~-.·.·-·....................·····
;.;·.;·;;.....::~·.;;
......
..
..
..
.
..
..
...
...
..
..
...
..
..
..
...
..
...
..
..
---...
..
..
..
..
...
..
...
..
...
..
...
...
:-~·:-·:-:·~:·:·:-·
:·:
·..
·I
...................
......
.............
I............
...........
.........
I
LEGEND
Mean Passive Scalar
iii
!i
X. .~·:,···:·~::
.......................
. .. .. ............
. .·f·i:..'Y·';'.~~
. . .....
. ·.: :;.... ....
..............................
. .iiiiiiiii~ii
. . .:. :. :. :. :~i. .~.~~i. .iiiiiitiiiiii
:::::~::
...........
Stream 1
Mixed Fluid
Stream 2
NOTE: Meanflow direction is from left to right.
Figure 4.21: Mean passive scalar field on the z=0 plane illustrating the growth
of the spanwise undulations and the region of mixed fluid with distance
from the trailing edge for the LPM with I,=0.3, 0.5, 0.7
z=0
LPM
J, =0.1
F,=0.3
1, =o.s
F,=0.7
It
42,
IF
4 0.5 ALEGEND
Mean Passive Scalar
...
"::''
~i
:::·i::iiiiiiil~ ~::''::;"
........................
~
''
::..:::...
...........
Stream 2
Mixed Fluid
Stream 1
NOTE: Mean flow direction is into page.
x=4
Figure 4.22: Mean passive scalar field on the x=4 plane illustrating the increasing spatial
growth rate of the spanwise undulations and region of mixed fluid
with increasing F, for the LPM with F,=0.3,0.5, 0.7
HPM
Fx =0.4
,=1.8
,= 1.2
LEGEND
Mean Passive Scalar
Stream··::·
2~~~':~5~::::·0~::iii
Mij.i~iixe
j~i:
Fluid·
~ ~ ~~~8i ~~I
Stream 2
Mixed Fluid
Stream 1
x=4
NOTE: Mean flow direction is into page.
Stream 1
:i::.:~ :iiiii~i
· · · · ·
I...
Figure 4.23: Mean passive scalar field on the x=4 plane illustrating the increasing spatial
growth rate of the spanwise undulations and region of mixed fluid
with increasing I, for the HPM with
-,=0.7, 1.2, 1.8
0.25
0.2
0.15
0.1
0.05
N
0
2
4
6
8
10
Figure 4.24: Mean amplitude of spanwise undulations vs. distance from trailing edge in
Region 1 for the LPM with F,=0.3,0.5, 0.7
0.25
_
---0.2
t --0.7,
AU=0.32
t =1.2,
AU--0.5
rt=1.8, AU=--0.8
0.15
0.1
0.05
0
-
a
-
x
Figure 4.25: Mean amplitude of spanwise undulations vs. distance from trailing edge in
Region 1 for the HPM with I,=0.7,1.2, 1.8
2
4
6
8
Figure 4.26: Predicted and computed mass flow rates of mixed fluid vs. distance from
trailing edge in Region 1 for the LPM with F,=0.3,05, 0.7
2
4
6
8
x
Figure 4.27: Predicted and computed mass flow rates of mixed fluid vs. distance from
trailing edge in Region 1 for the HPM with
,= 0.7, 1.2, 1.8
0.4
0.3
0.2
0.1
A
0
2
4
6
8
Figure 4.28: Streamwise interface structure scale vs. distance from trailing edge in
Reg ion 1 for the LPM, HPM, and ADM
0.5
0.4
0.3
0.2
0.1
0
2
4
6
8
Figure 4.29: Streamwise interface structure scale and mean amplitude of spanwise
undulations vs. distance from trailing edge in Region 1 for the LPM
0.5
0.4
S0.3
0.2
0.1
0
2
4
6
8
Figure 4.30: Streamwise interface structure scale and mean amplitude of spanwise
undulations vs. distance from trailing edge in Region 1 for the HPM
2.5
2
1.5
1
0.5
A
0
2
4
6
8
10
12
14
16
Figure 4.31: Mean cross-flow interface length vs. distance from trailing edge in
all regions for the LPM with
F,=0.0, 0.3, 0.5, 0.7
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
x
10
12
14
16
Figure 4.32: Mass flow rate of mixed fluid vs. distance from trailing edge in
all regions for the LPM with I,=0.3,0.5, 0.7
5
4
3
2
1
0
0
2
4
6
8
10
12
14
16
x
Figure 4.33: Mean cross-Flow interface length vs. distance from trailing edge in
all regions for the HPM with 1,=0.7,1.2, 1.8
0
2
4
6
8
10
12
14
16
Figure 4.34: Mass flow rate of mixed fluid vs. distance from trailing edge in
all regions for the HPM with I",=0.7,1.2, 1.8
LPM
Fx =0.1
Time= to
to0 + 4 YU
to +8( U)
to + 12U)
LEGEND
V=1=Mean Inflow Velocity
Figure 4.35: Velocity vectors at four time intervals on the x=14 plane illustrating the
unsteadiness of the flow field associated with the large scale spanwise
undulations in Region 3 for the LPM with
,--0.7
LPM
Frx =0.1
Time= to
t0+4( U)
to +8 U)
to + 12
U
4A
LEGEND
Stream 1
Stream 2
NOTE: Mean flow direction is into page.
x=14
Figure 4.36: Passive scalar field at four time intervals on the x=14 plane illustrating the
large scale movement of the interface associated with the spanwise undulations
in Region 3 for the LPM with F,=0.7
LPM
F• =0.1
x=10
x=14
x=12
x=16
A
41
I
.0.5
A..
LEGEND
Mean Passive Scalar
Stream 2
Stream 1 Mixed Fluid
NOTE: Mean flow direction is into page.
Figure 4.37: Mean passive scalar field on the x=10, 12, 14, 16 planes illustrating the
planar-like growth of the region of mixed fluid in Region 3 for the LPM with F,=0.7
96
..
Chapter 5
Summary and Conclusions
5.1
Summary and Conclusions
A hypothetical model has been presented, in which the roles of streamwise and
spanwise vorticity in the mixing process downstream of a lobed mixer are superposed. In
this model, the streamwise vorticity stretches the mean cross-flow interface, and the
spanwise vorticity initiates a Kelvin-Helmholtz type instability on this stretching
interface.
The superposition of these two roles permits the derivation of a simple
expression predicting the variation of the mixing rate downstream of a lobed mixer, and
furthermore, it allows one to interpret the effect of spanwise vorticity as an enhancement
of the cross-stream transport properties.
A series of numerical experiments to test this model have been performed. From
the model and the results of these experiments, the following conclusions can be drawn:
(1)The roles of streamwise and spanwise vorticity in the mixing process may indeed
be superposed. The streamwise vorticity stretches the mean cross-flow interface
with distance from the trailing edge.
The rate at which it is stretched is
proportional to the streamwise circulation shed from the mixer, and this rate is
relatively unaffected by the presence of spanwise vorticity.
The spanwise
vorticity initiates a Kelvin-Helmholtz type instability on the stretching mean
cross-flow interface. In these inviscid simulations, the undulations resulting from
this instability have been shown to grow at an exponential rate. For a given
mixer, the exponent increases linearly with the velocity difference between the
two streams. The presence of streamwise vorticity has been shown to slow the
growth of these undulations.
(2)From the superposition of the roles of the two vorticity components, a simple
expression predicting the spatial mixing rate downstream of a lobed mixer is
derived. This expression describes the parametric dependence of the mixing rate
on streamwise and spanwise circulation, length of the trailing edge and distance
from the trailing edge. In this expression, the spatial mixing rate is initially seen
to grow linearly with streamwise and spanwise circulation.
The effect of
spanwise vorticity may be interpreted as an enhancement of the cross-stream
transport properties.
(3)The validity of the first two conclusions is limited to the region 5-10 X
downstream of the trailing edge of the mixer. The limitation becomes apparent
upon consideration of the scales of the cross-flow interface structures and the
amplitude of the spanwise undulations. Spanwise undulations continue to grow
with distance from the trailing edge, and as the amplitude of these undulations
approaches the scale of the cross-flow interface structures, the two processes
begin to interact. This interaction results in the slowing, and eventual halt, of the
growth of the cross-flow interface in the current simulations.
(4)Downstream of the point where the growth of the cross-flow interface has
ceased, the region of mixed fluid behaves like a planar shear layer of roughly the
lobe height and grows largely as a result of the spanwise undulations.
resulting mixing rate is also seen to decrease.
The
(5)The presence of streamwise vorticity provides a mixing augmentation in the
region where the cross-flow interface is being stretched and the roles of the two
vorticity components may be superposed. In this region, largely as a consequence
of the stretching of the cross-flow interface, the spatial mixing rate is seen to
increase with distance from the trailing edge. Downstream of this region, which
extends 5-10 X from the trailing edge, the streamwise vorticity is no longer
effective in promoting mixing enhancement.
5.2
Design Suggestion
When the stretching of the mean cross-flow interface ceases, the mixing
augmentation due to the presence of streamwise vorticity is complete, and a decrease is
the spatial mixing rate is observed. The addition of a second set of lobes, downstream of
the first set, near the point where the growth of the mixing layer slows, would reintroduce
streamwise vorticity to the layer. If the second set of lobes were placed near the outer
edge of the region of mixed fluid, the streamwise vorticity should stretch the cross-flow
interface between the region of mixed fluid and one of the initial streams, and thereby
further enhance the mixing rate. This two mixer configuration and conjectured mixing
layer are illustrated in Figure 5.1.
5.3
Recommendations for Future Work
(1)A two-pronged investigation of the effects of compressibility on the mixing
process. In the current study, the mean velocity/Mach number remained constant
at 0.5, as only the variation in the growth of the mixing layer with streamwise and
spanwise circulation was studied. A next logical step in the investigation of these
flow fields would be to study the influence of mean velocity(Mach number) on of
growth of the mixing layer. In planar shear layer experiments the spatial growth
rate of the shear layer diminishes with increasing mean velocity[6,7]. Secondly,
the velocity difference between the co-flowing streams in this study was limited
to situations where the convective Mach number is subsonic: the convective
Mach number(Mc) is defined as the velocity difference between the two streams
divided by the mean inflow sonic speed. In planar mixing layers, shocks are seen
to form in cases where the convective Mach number is supersonic[7], and hence,
the associated flow fields are more complex. As these mixers are likely to be used
in situations where Mc>1, an investigation of these flow regimes is necessary.
(2)An investigation of the interaction between the roles of streamwise and spanwise
vorticity, as in this study some interaction between the effects of the two vorticity
components was observed. In particular, strong streamwise vorticity was seen to
slow the growth the undulations resulting from the Kelvin-Helmholtz type
instability associated with spanwise vorticity, and the reason for this diminished
growth rate has not been identified. As the rate of stretching of the cross-flow
interface increases with increasing streamwise circulation but the rate of growth
of the spanwise undulations slows with this growing streamwise circulation, an
understanding of the interaction between these processes is necessary to optimize
the design of these devices.
100
U2
Figure 5.1: Two mixer configuration to further enhance mixing by reintroducing
streamwise vorticity after growth of mean cross-flow interface has ceased
101
Bibliography
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[3]
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Elliott, J., A Computational Investigation of the Fluid Dynamics of a Three
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102
[10] Elliott, J., Manning, T., Qiu, Y., Greitzer, E., Tan, C., Tillman, G., Computational
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