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AN EXPERIMENTAL INVESTIGATION OF
HIGH COMPRESSIBILITY MIXING LAYERS
TECHNICAL REPORT TSD-138
THERMOSCIENCES DIVISION
DEPARTMENT OF MECHANICAL ENGINEERING
STANFORD UNIVERSITY
STANFORD, CALIFORNIA 94305-3032
By
Tobias Rossmann
December 2001
ii
© Copyright by Tobias Rossmann, 2001
All Rights Reserved
iii
ABSTRACT An investigation of the effects of high compressibility conditions on the two-
stream, planar turbulent mixing layer is performed in a unique shock tunnel driven,
supersonic mixing layer facility. Compressibility levels, previously unattainable in
traditional blowdown wind tunnels, are reached to examine their effect on the growth and
development of large-scale structures which dominate the entrainment behavior of
mixing layers. The free stream density ratio, s = ρ2/ρ1, and the average convective Mach
number, Mc = (U1-U2)/(a1+a2) are the principal parameters varied in this study.
Visualizations of the shear layer are achieved by Schlieren imaging and planar laser
induced fluorescence (PLIF) of two, seeded tracer species, acetone and nitric oxide. Side,
plan, and end view visualizations of three compressibility conditions (Mc = 0.85, 1.71,
and 2.64) provide information on large-scale structure character and dimensionality.
The first focus of this work is an accurate measurement of the shear layer growth
rate at compressibility conditions greater than Mc = 1. Ensemble averaged Schlieren
images provide a measure of the visual growth rate and are captured for a wide
compressibility range (Mc = 0.85 to 2.84). Second, spatially resolved, non-intrusive,
laser-based imaging techniques are employed to probe the underlying three-dimensional
structure of highly compressible shear layers that is masked by line-of-sight integrated
imaging. A final emphasis is on an extension of a previous PLIF technique, cold
chemistry imaging of mixed fluid, to low pressure mixing layer environments to examine
whether the mixedness of shear layers continues to increase with compressibility and
Reynolds number, as was seen in previous low Mc experiments.
Consistent with the trends seen in previous mixing layer experiments and
computations, all the imaging techniques reveal that the high compressibility shear layer
is dominated by three-dimensional streamwise oriented structures with limited transverse
dimension. The growth rate of these structures, and thus the mixing layer, tends to
asymptote to 18% of its incompressible value above Mc = 1.5. PLIF imaging results
iv
uncover three-dimensional shock structures which are caused by slow scalar structures
convecting in a supersonic flow, but do not confirm the existence of shocklets in highly
compressible turbulence. Visualizations of the mean and instantaneous scalar field at Mc
= 2.64 suggest the applicability of gradient transport mixing models over structure based
techniques at high compressibility. Also, the mixedness of the shear layer at this higher
compressibility condition is seen to slowly increase with compressibility and Reynolds
number when compared to prior cold chemistry results.
v
TABLE OF CONTENTS
Abstract …………………………………………………………………………… iii
Table of Contents …………………………………………………………………. v
List of Tables ……………………………………………………………………… ix
List of Figures ……………………………………………………………………... x
Nomenclature ……………………………………………………………………… xv
CHAPTER 1. INTRODUCTION …………………..……………………. 1 1.1 Background and Motivation ………………………………………….. 1
1.2 Literature Review of Past Work ………………………………………. 3
1.2.1 Incompressible Mixing layers …………………………….. 3
1.2.1.1 Growth Rate ………………………………………. 3
1.2.1.2 Coherent Structures ……………………………….. 5
1.2.1.3 Scalar Mixing ……………….……………………... 6
1.2.2 Compressible Mixing Layers ……………………………… 10
1.2.2.1 Growth Rate ……………………………………….. 10
1.2.2.2 Coherent Structures ……………………………….. 12
1.2.2.3 Velocity and Scalar Measurements ……………….. 14
1.2.2.4 Acoustic Radiation in Mixing Layers……………… 16
1.2.3 Simulations and Modeling of Mixing Layers ……………... 18
1.2.3.1 Linear Stability……………………………………...18
1.2.3.2 Direct Numerical Simulations ……………………...20
1.2.3.3 Shocklets and Turbulent Kinetic Energy Transport ..21
1.3 Present Research Work and Objectives ……………………………….. 22
CHAPTER 2. EXPERIMENTAL APPARATUS………...…………………. 27 2.1 Shock Tube ……………………………………………………………. 27
2.2 Mixing Layer Facility …………………………………………………. 29
2.3 Optical Diagnostics ……………………………………………………. 32
2.3.1 CO2 Emission Setup ……………………………………….. 32
vi
2.3.2 Schlieren Imaging Setup …………………………………... 33
2.3.3 Acetone PLIF Imaging Setup ………………………………34
2.3.4 NO PLIF Imaging Setup …………………………………... 35
CHAPTER 3. MIXING LAYER TEST CONDITIONS .……………………. 46 3.1 X-T Diagrams for Shock Tunnels ……………………………………... 46
3.1.1 Shock Tube Performance and Shock Attenuation ………… 50
3.1.2 Contact Surface Acceleration ……………………………... 52
3.1.3 Correction for Nozzle Mass Flow Rate …………………… 54
3.1.4 Setup of Steady Nozzle Flow ……………………………… 57
3.2 Prediction of Mixing Layer Test Conditions ………………………….. 57
3.2.1 Back Pressure ……………………………….……………... 58
3.2.2 Side-Two Mass Flow Rate …………..……………………. 60
3.2.3 Wall Deflection ……………………………………………. 61
CHAPTER 4. SCHLIEREN IMAGING AND MIXING LAYER GROWTH RATE …………………...71
4.1 Schlieren Image Corrections and Specifics ………...…………………. 71
4.2 Schlieren Imaging Results (Mc = 0.85 to 2.84) ………………………...73
4.2.1 Growth Rate ……………………………………………….. 73
4.2.2 Instantaneous Images …………………………………….... 74
4.2.3 Large-Scale Structures …………………………………….. 75
4.3 Schlieren Image interpretation ……………………………………….... 75
4.3.1 Acoustic Field Observations ………………………………. 75
4.3.2 Shear Layer Stability ……………………………………….78
4.3.3 Convection Velocity ………………………………………. 79
4.3.4 Shocklets …………………………………………………... 82
4.3.5 Co-Layers ………………………………………………….. 83
CHAPTER 5. ACETONE PLIF IMAGING OF MIXING LAYERS ………. 94 5.1 Acetone PLIF Test Conditions …………………………………………94
vii
5.1.1 Acetone Seeding …………………………………………... 94
5.1.2 Acetone PLIF in Shock Tunnel Flows …………………….. 96
5.1.3 High Temperature Acetone Chemistry ……………………. 96
5.1.4 Applicability of Acetone PLIF in Shock Tunnels ………….98
5.1.5 Seeded Stream Selection for Mixing Layer Imaging ……… 100
5.2 PLIF Modeling for Flowfield …………………………………………..101
5.3 Imaging of the Mc = 0.85 Test Condition ……………………………... 105
5.3.1 Schlieren …………………………………………………... 105
5.3.2 Acetone PLIF, Side View …………………………………. 106
5.3.3 Plan View ………………………………………………….. 108
5.3.4 End View ………………………………………………….. 109
5.4 Imaging of the Mc= 1.71 Test Condition …………………………..…. 110
5.4.1 Schlieren …………………………………………………... 110
5.4.2 Acetone PLIF, Side View…………………………….……. 111
5.4.3 Plan View ………………………………………………….. 113
5.4.4 End View ………………………………………………….. 115
5.4.5 Shocklet Production ……………………………………….. 116
CHAPTER 6. NO PLIF IMAGING OF MIXING LAYERS …..……………. 131 6.1 Tracer Selection and LIF Modeling …………………………………… 131
6.1.1 NO Seeding Strategies …………………………………….. 132
6.1.2 Transition Selection ……………………………………….. 135
6.1.3 Iso-Quenching Environments ……………………………... 137
6.1.4 Pressure and Saturation Behavior …………………………. 138
6.2 NO PLIF Imaging of the Mc = 2.64 test condition ……………………. 140
6.2.1 Side View ………………………………………………….. 141
6.2.2 Plan View ………………………………………………….. 143
6.2.3 End View ………………………………………………….. 144
6.3 Mixed Fluid Fraction Imaging ………………………………………... 146
6.3.1 Experimental Approach …………………………………… 146
6.3.2 Interpretation of Cold Chemistry Results at Low Pressures . 149
viii
6.3.3 Mixed Fluid Fraction Results ……………………………... 149
6.3.4 Error Analysis of Mixed Fluid Fraction Results …..………. 152
6.4 Additional Discussion of Large-Scale Structure and Entrainment ……. 154
CHAPTER 7. CONCLUSIONS ………………………..……………………. 171 7.1 Summary of Results …………………………………………………… 171
7.2 Recommendations for Future Work …………………………………....175
APPENDICES …………………………………...………………………………. 178 A Side Two Flow Initial Conditions ……………………………………. 178
A.1 Injection Pressure vs. Actual Flow Rate ……………..………. 178
A.2 Inlet Velocity Profile ………………………………………….. 182
B Results From Non-optimized Flow Conditions ………………..……. 187
B.1 Non-Pressure Matched ………………………………………... 187
B.2 Non-Mass Flowrate Matched …………………………………. 190
C Four Level LIF Modeling ……………………………………………. 196
D Growth Rate Data ……………………………………………………. 202
REFERENCES …………………………………..………………………………. 203
ix
LIST OF TABLES
Table Page
4.1 Convection velocity bounds based on Mach wave radiation 80
5.1 Test conditions for calculation of acetone pyrolysis 96
5.2 Rate coefficients for acetone decomposition 98
5.3 Mixing layer test conditions for acetone PLIF imaging 103
6.1 Temperature ranges for mixture fraction measurements 136
6.2 Mixing layer test conditions for NO PLIF imaging results 141
6.3 Comparison of similarity conditions for the cold chemistry technique 152
A.1 Comparison of front tracking, Pitot probe and tank blowdown
measurements of velocity
183
x
LIST OF FIGURES Figure Page CHAPTER 1
1.1 Schematic of mixing layer flowfield 25
1.2 Historical normalized growth rate vs. convective Mach number 25
1.3 Schematic of shock creation in supersonic shear layers 26
CHAPTER 2
2.1 Shock tunnel schematic 38
2.2 Photograph and schematic of mixing layer test section 39
2.3 Infrared emission setup and sample data trace 40
2.4 Schlieren imaging setup 41
2.5 Acetone PLIF imaging setup 42
2.6 Laser timing diagram for synchronization of laser firing 43
2.7 NO PLIF imaging setup 44
2.8 Excitation spectra of rotational lines in the NO A2Σ
+←X2
Π½ transition 45
2.9 LIF spectra for broadband collection of NO fluorescence 45
CHAPTER 3 3.1 Schematic of shock/contact surface interaction 62
3.2 Theoretical X-T diagram for a Ms =5.0 shock 62
3.3 Shock Mach number versus pressure ratio for shock tunnel 63
3.4 Shock Mach number vs. reduced pressure ratio 63
3.5 Incident shock velocity profile for a Ms = 4.06 shock 64
3.6 Shock velocity attenuation results 64
3.7 Variation in shock speed for 80 independent shots 65
3.8 Incident shock induced boundary layer schematic 65
3.9 Comparison of experimental hot test gas length to Mirels’ prediction 66
3.10 X-T diagram for Ms = 3.41 shock 66
xi
3.11 Predicted and actual pressure trace for Ms = 3.41 shock 67
3.12 Comparison of shock tube test times with predicted values 67
3.13 Schematic of control volume used for shock tunnel flows 68
3.14 Solution of control volume formulation for stagnation conditions 68
3.15 Schematic of wedge rotation method for back-pressure calculation 69
3.16 Comparison of the rotation method of predicting back-pressure 69
3.17 Static pressure traces from runs with and without wall deflection 70
CHAPTER 4
4.1 Region of images from which growth rate data is taken 84
4.2 Normalized mixing layer growth rate vs. convective Mach number 85
4.3 Normalized growth rate showing density ratio variation 86
4.4 Instantaneous Schlieren images from Mc = 0.85 to 1.91 87
4.5 Instantaneous Schlieren images from Mc = 1.93 to 2.84 88
4.6 Schlieren images showing acoustic radiation from mixing layer 89
4.7 Schematics for the creation of shocks by mixing layers 89
4.8 Acoustic radiation regime plot for upper stream versus Mc
and density ratio
90
4.9 Acoustic radiation regime plot for lower stream versus Mc
and density ratio
90
4.10 Magnified images showing character of acoustic radiation 91
4.11 Normalized convection velocity versus convective Mach number 92
4.12 Empirical convective Mach numbers versus Mc, symm 92
4.13 Time sequence image of “shocklet” in flow 93
4.14 Mixing layer with Mach waves visible in both streams 93
CHAPTER 5
5.1 Schematic of pressurized acetone seeder 118
5.2 Minimum seeding level possible in shock tunnel using seeder 119
5.3 Acetone pyrolysis time constant vs. P, χ, and T 119
5.4 Maximum test nozzle Mach number using acetone PLIF 120
xii
5.5 Required stagnation pressure for high SNR acetone PLIF 120
5.6 Variation in absorption cross section of acetone with temperature 121
5.7 Variation in acetone PLIF signal with temperature 121
5.8 Instantaneous side view Schlieren image for Mc = 0.85 122
5.9 Instantaneous side view acetone PLIF images for Mc = 0.85 123
5.10 Instantaneous plan view acetone PLIF images for Mc = 0.85 124
5.11 Instantaneous end view acetone PLIF images for Mc = 0.85 124
5.12 Instantaneous side view Schlieren image for Mc = 1.71 125
5.13 Instantaneous side view acetone PLIF images for Mc = 1.71 126
5.14 Schematic of side view images with shocks and large structures 127
5.15 Instantaneous plan view acetone PLIF images for Mc = 1.71 128
5.15 Elevated plan view acetone PLIF images for Mc = 1.71 129
5.16 Instantaneous end view acetone PLIF images for Mc = 1.71 129
5.17 Computational shocklet production at Mc = 1.80 130
CHAPTER 6
6.1 Simulation of shock heating 2% NO in Argon to 3600K and 30 atm 156
6.2 Simulation of shock heating 5% Air in Argon to 3600 K and 30 atm 156
6.3 Variation of LIF signal for Nitric Oxide with temperature 157
6.4 Simulation excitation spectra of the A2Σ
+←X2
Π (0,0) band of NO 157
6.5 Variation of the normalized LIF signal with temperature 158
6.6 Relative error in NO PLIF signal for iso- and non iso-quenching
environments
158
6.7 Variation of the NO LIF signal with pressure 159
6.8 Computed and experimental LIF signal versus pressure 159
6.9 Computed LIF intensity versus laser spectral power 160
6.10 Computed LIF intensity compared to weak excitation model 160
6.11 Instantaneous NO PLIF side view images at Mc = 2.64 161
6.12 NO PLIF side view images with transverse and streamwise cuts 162
6.13 Instantaneous NO PLIF plan view images at Mc = 2.64 163
6.14 NO PLIF plan view images with cross-stream and streamwise cuts 164
xiii
6.15 Instantaneous NO PLIF end view images at Mc = 2.64 165
6.16 NO PLIF plan view images with cross-stream and transverse cuts 166
6.17 Variation in fluorescence quantum yield versus low-speed fluid
fraction
167
6.18 Variation in overall LIF signal intensity versus low-speed fluid fraction 167
6.19 Scaled mixture fraction profiles for three quenching partners 168
6.20 Profiles of low-speed fluid fraction (χL), mixed high-speed fluid (δH),
and mixed low-speed fluid (δL)
168
6.21 Profiles of the pure and mixed fluid states in a Mc = 2.64 mixing layer 169
6.22 Mixing efficiency as a function of Mc 169
6.23 Mixing efficiency as a function of Re 170
6.24 Mixed mean composition profile for the Mc= 2.64 170
APPENDIX A
A.1 Lower speed side velocity derived by front tracking 183
A.2 Comparison of tank blowdown times with varying stagnation pressure 184
A.3 Comparison of the efficiency (τtheor/τexp) of the tank blowdown process 184
A.4 Pitot probe traces for several tank stagnation pressures 185
A.5 Comparison of Pitot velocity data with tank blowdown velocity data 185
A.7 Two-dimensional velocity map for low-speed side 186
APPENDIX B
B.1 Schlieren images of Mc = 0.85 showing the effect of poorly matched
nozzle back pressure
192
B.2 Schlieren image showing the effects of planar oblique shock
impingement on a Mc = 0.85 mixing layer
193
B.3 Schlieren image showing the effects of planar rarefaction fan
impingement on a Mc = 0.85 mixing layer
193
B.4 Flow instability of Mc = 0.85 mixing layer 194
B.5 Mixing layer facility wall pressure histories 194
B.6 Affect of low-speed mass flowrate on mixing layer development 195
xiv
APPENDIX C
C.1 Excitation and de-excitation pathways for the four level LIF model 201
C.2 Simulation of LIF signal using 4-level LIF model 201
xv
NOMENCLATURE
Greek symbols
α Mach wave angle χi mole fraction of species i δ mixing layer growth rate δinc incompressible mixing layer growth rate
δvis visual mixing layer growth rate δH unmixed fraction of high speed fluid δL unmixed fraction of low speed fluid δM mixed fluid thickness of mixing layer εp Pitot probe characteristic time εp side-two injection tank characteristic time φ lineshape function [cm], fluorescence quantum yield γ ratio of specific heats, collisional line broadening parameter η normalized mixing layer cross-stream coordinate
ηopt optical collection efficiency λ wavelength λB Batchelor scale ν vibrational quantum level, frequency, Prandtl-Meyer angle ∆νc collisional line width [cm-1] θ mixing layer global angle ρ density
σ collision or absorption cross-section
τ characteristic time ξ high-speed fluid mixture fraction ξm mixed mean mixture fraction
Roman Symbols a speed of sound A* choked mass flow rate area Ai area of zone I A21 spontaneous emission B12,21 Einstein coefficients for stimulated absorption and emission c speed of light Cδ growth rate constant CQ dimensionless quenching rate C characteristic time ratio Da Damkohler number
xvi
e internal energy Ev volumetric entrainment ratio, vibrational energy f/# lens or mirror f-number fB Boltzman fraction g overlap integral h Planck’s constant, enthalpy, height J rotational quantum level k Boltzman’s constant ki rotational energy transfer K adjustable constant L characteristic length L1 driven length L4 driver length mD mass flow rate Mc convective Mach number M Mach number Ms incident shock Mach number n number density P pressure P0 stagnation pressure Pback nozzle back pressure, test section pressure Pij pressure ratio from zone i to zone j Q21 collisional queching r velocity ratio R gas constant (J/kg K) Re Reynolds number s density ratio Sf fluorescence signal Sc Schmidt number t time T temperature T0 stagnation temperature U streamwise velocity Uc convection velocity ∆U velocity difference {v} relative collisional velocity V tank volume W12 stimulated absorption W21 stimulated emission x streamwise direction y cross-stream direction z spanwise direction
xvii
Abbreviations BBO Beta Barium Borate CCD Charge Coupled Device DNS Direct Numerical Simulation KDP Potassium Dihydrogen Phosphate LDV Laser Doppler Velocimetry LIF Laser-Induced Fluorescence OH hydroxyl radical NO nitric oxide PDF Probability Density Function PIV Particle Image Velocimetry PLIF Planar Laser-Induced Fluorescence PMT Photomultiplier Tube RET Rotational Energy Transfer SLPM Standard Liters Per Minute SNR Signal to Noise Ratio TiO2 Titanium Dioxide VET Vibrational Energy Transfer Superscripts and Subscripts 0 stagnation 1 fast speed, upper stream, driven section, initial 2 low speed, lower stream, behind incident shock 3 behind contact surface 4 driver, initial 5 reflected / (prime) upper energy state // (double prime) lower energy state abs absorption, absolute avg average B rotational bath c convection, collisional H high speed L low speed las laser noz nozzle symm symmetric T total vis visual
1
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND AND MOTIVATION
The study of fluid mixing by turbulent motions at high Mach numbers is of
fundamental importance to the creation of new propulsion systems for high-speed
vehicles and to the elucidation of the physical processes in compressible turbulence. As
an increase in compressibility tends to stabilize turbulent flows, the efficiency of the
mixing process is reduced and mixing lengths tend to increase. Although mixing
strategies in real systems are typically complex and three-dimensional, the two-
dimensional shear layer (the simplest compressible shear flow able to be created in a
laboratory environment) offers an opportunity to attain insight into the complex effects of
compressibility on mixing.
Many applications rely on the efficient mixing of two streams. Scramjet
combustion engines seek to maximize combustion efficiency and thrust while minimizing
weight and length of the engine (Gutmark et al., 1995). Though, a large increase in
mixing efficiency is usually accompanied by some thrust penalty. Mixing is also
important in the performance of supersonic ejectors, which play a crucial role in chemical
lasers (Siegman, 1986), entrainment of metal powders into supersonic jets for metal
deposition (Wei et al., 1992), and in noise and infra-red signature detection for
supersonic military aircraft (Gutmark et al., 1995; Papamoschou, 1999).
This thesis describes an experimental investigation of the effects of very high
compressibility conditions on non-reacting turbulent mixing layers. The work stems from
a long history of mixing layer experimentation at Stanford University. Clemens (1991)
studied the effects of compressibility on scalar mixing and the appearance of large-scale
structures at moderate convective Mach numbers. Miller (1994) investigated the effects
of compressibility on combustion in a shear layer in which finite-rate chemistry effects
2
were significant. Island (1997) explored the effects of compressibility on scalar mixing
using a fully resolved scalar imaging technique for the same flow conditions as those
studied by Clemens. Finally, Urban (1999) examined the effects of compressibility on the
instantaneous velocity field using particle image velocimetry. A similarly prodigious
effort on the simulation of this flowfield using computational methods has also occurred
at Stanford and will be highlighted in the literature review.
All these previous results point to trends in the mixing behavior of the shear layer,
which were changing as the compressibility was increased. However, the limitations of
the facilities did not allow for many of these trends with increasing compressibility to be
fully investigated. Thus, the direct goal of this thesis was to extend experimental
conditions into higher compressibility levels, which are not attainable in traditional
blowdown wind tunnel facilities, to explore potentially new mixing physics suggested by
the numerical effort and to confirm some of the trends observed in lower compressibility
experimental studies. Accordingly, these results could then be used to formulate more
complete models of compressible turbulence for use in engineering applications.
In the present experiments, the behavior of the two-stream planar shear layer is
studied at numerous high compressibility conditions. The supersonic, high-speed side of
the layer is generated by a stagnation reservoir created by a shock tunnel, while the low-
speed stream is injected from a subsonic manifold. Flowfield data are obtained using
Schlieren imaging and planar laser-induced fluorescence of either acetone or nitric oxide,
which act as a conserved marker of high-speed fluid. The images provide mixing layer
growth rate information along with the instantaneous character of the large-scale
structures present in the flow. Comparison of large-scale structures at different
compressibilities allows for an assessment of the mechanisms through which
compressibility affects the growth, evolution, and entrainment behavior of vortical
structures. Also, the development and implementation of useful laser-based diagnostics
for low-pressure compressible mixing environments is also explored.
3
1.2 LITERATURE REVIEW OF PAST WORK Several excellent reviews have been performed recently on many aspects of
turbulent mixing layer experiments and simulations. Clemens (1991), Clemens and
Mungal (1995), and Dimotakis (1991) have focused on the structure of incompressible
and compressible shear layer structures. Scalar mixing was examined by Karasso and
Mungal (1996), Koochesefahani and Dimotakis (1986), and Island (1997). Mungal and
Dimotakis (1984) and Miller (1994) reviewed thoroughly the effects of reaction on
compressible and incompressible mixing layers. Urban and Mungal (2000) and Goebel
and Dutton (1991) both studied the relevant literature for compressible effects on fluid
and structure velocity through the examination of the instantaneous velocity field. Day
(1999) and Freund (1997) extensively reviewed prior direct numerical simulation (DNS),
linear stability, and other simulations performed on mixing layers. Here, since this thesis
is concerned with the effects of very high compressibility conditions applied to the planar
mixing layer, the review concentrates on the previous studies and experiments which are
germane to the foundations of these new results. For convenience in summarizing such a
large amount of work, the review is sectioned into three parts: results stemming from
incompressible mixing layer experiments, compressible mixing layer experiments, and
computational simulations of compressible mixing layers. Appropriate reviews for
different diagnostic techniques will occur in the relevant thesis chapters for clarity.
1.2.1 INCOMPRESSIBLE MIXING LAYERS
1.2.1.1 Growth Rate
The plane mixing layer is one of the simplest turbulent shear flows in which to
examine the effect of various flow parameters and boundary conditions on the mixing
efficiency of all the shear flows. Figure 1.1 is a schematic of the two-dimensional
turbulent mixing layer developing downstream of a splitter tip. High- and low-speed
streams meet after the splitter plate, forming a turbulent mixing region with thickness
δ(x), which increases linearly with downstream distance. For this thesis, δ will be defined
4
as the visual thickness of the shear layer, as it can be discerned from Schlieren images.
The growth of the mixing layer is linear in the far field, after the effects of the near field
and boundary layers on the splitter plate decay, and flow field variables become self-
similar. For equal density shear layers, the growth of the mixing layer with downstream
distance has been shown to be
rrC
dxd
+
−
=
11
δ
δ (1.1)
where r = U2/U1 is the velocity ratio and Cδ is a constant which depends on which flow
field variable is used to judge the shear layer thickness (Abramowich, 1963 and Sabin,
1965). Solution of the stability equations for an incompressible vortex sheet yields the
same result. However, Birch and Eggers (1972) showed significant scatter when fitting
all the existing experimental data on a single curve, because the destabilizing effect of
increasing density ratio was not taken into account. In their landmark study, Brown and
Roshko (1974) investigated the effect of density ratio (s = ρ2/ρ1) on the spreading rate of
shear layers and arrived at a more complete incompressible growth rate scaling:
( )( )sr
srCdxd
+
+−
=
111
δ
δ (1.2)
A similar scaling was proposed by Papamoschou and Roshko (1988) and can also be
derived by examining the growth of instability waves in a compressible vortex sheet.
Dimotakis (1991) derived a correction to Equation 1.2 due to the fact that experimental
shear layers grow in space and not with time; however, this improvement was found to
have a small effect for the growth rate predictions of the mixing layers examined in this
study, such that Equation 1.2 was deemed sufficient. Papamoschou and Roshko (1988)
state that Cδ = 0.17 provides the best fit to their visual thickness data (δvis), but a recent
analysis of the large set of experimental growth rate data by Slessor et al. (2001) shows
that Cδ ranges from 0.14 to 0.18. The value set by Papamoschou and Roshko will be used
throughout this study to facilitate the comparison of growth rate data obtained herein with
the historical literature.
5
1.2.1.2 Coherent Structures
With the discovery of large-scale spanwise structures in shear layer research by
Brown and Roshko (1974) at high Reynolds numbers, the role of large-scale structures in
entrainment, growth, and mixing of shear layers has been intensely studied. These rollers
are consistent with the dominant Kelvin-Helmholtz instability, which plays a role in
every shear flow. Conclusive proof of the dominance of this instability was shown by
various investigators (Michalke, 1965; Monkewitz and Heurre, 1982; Corcos and
Sherman, 1976) using linear stability analysis. Rogers and Moser (1992) used DNS to
elucidate the development of the planar shear layer in terms of pairings of Kelvin-
Helmholtz rollers. The experimental discovery of these Kelvin-Helmholtz induced large-
scale structures by Brown and Roshko gave rise to the term “Brown-Roshko rollers” or
“structures”. This term is often used interchangeably with Kelvin-Helmholtz rollers or
spanwise rollers.
Many other studies have confirmed the existence and examined closely the
dynamics and behavior of large-scale structures. Winant and Browand (1974) reported
that the growth mechanism in shear layers is primarily due to the pairing of dominant
vortices (large-scale structures). Dimotakis and Brown (1976) showed the existence of
large-scale structures up to very high Reynolds numbers (Re = 4 x 105) by noting that
pure unmixed fluid could be transported across the layer due to the motion of these
entraining structures and that entrainment and mixing seemed to be independent
processes. Mungal et al. (1985) demonstrated the existence of coherent structures up to
Re = 1.9 x 105, for both laminar and turbulent initial boundary layers. Browand and Trout
(1985) also captured the existence of two-dimensional structures using two-point
correlations at similarly high Re conditions. This study was also one of the first to
observe a transitional region where initial boundary layer effects, which propagate due to
the engulfing of the boundary layer by the shear layer immediately after the splitter tip,
decay. Beyond this transition, large spanwise structures develop and propagate
downstream.
Visualizations of the scalar field in mixing layers also yield information on large-
scale structures due to the important role they play in scalar transport. Flow visualizations
6
of scalar fields (Konrad, 1976; Breidenthal, 1981; Koochesfahani and Dimotakis, 1986;
Karasso and Mungal, 1998) show that coherent structures dominate the flow topology,
even when spatially resolved imaging techniques are used. These periodic structures span
the thickness of the shear layer, with tight braid regions connecting the large vortices.
Further links between large-scale structure and entrainment have been established by
Dimotakis (1989) and Konrad (1976). Since molecular mixing is effected by the
development of fine scales and the cascade of energy to these scales, these studies
describe the important relationship between large-scale structures and mixing.
Another type of instability, secondary to the Kelvin-Helmholtz, manifests itself in
planar shear layers as streamwise vortical structures. Streamwise streaks have been
observed in the flow visualizations of Bradshaw (1966), Brown and Roshko (1974), and
Konrad (1976). Breidenthal (1981) also examined streamwise structures in a water shear
layer and found that they grow and develop as they convect downstream and have a
nearly fixed spanwise spacing when normalized by the local Brown-Roshko roller
streamwise spacing. Measurements by Bernal and Roshko (1986), Jimenez et al. (1985),
and Jimenez (1983) all proved the importance of these secondary vortical structures
which produce and maintain fine-scale turbulence through their interaction with the
larger, spanwise vortices, creating counter-rotating vortex pairs. Huang and Ho (1990)
found that the production of small-scale eddies which control mixing is a result of the
interactions between these two different vortical structures. While these secondary
structures do not strongly participate in the entrainment of pure free-stream fluid into the
shear layer, their role in the small-scale mixing of the shear layer is fairly well
understood.
1.2.1.3 Scalar Mixing
While the large-scale dynamics play a large role in the ability of a shear layer to
mix a conserved scalar, other issues arise which make scalar mixing a more complicated
experimental issue. First, mixing, in this study, will always refer to the mixing at a
molecular level, effectively controlled by diffusion. Mixing is clearly different from
stirring (Eckart, 1948), where material lines are stretched, increasing the interfacial area
7
between the two streams, allowing for increased molecular diffusion per volume. In many
previous studies of incompressible mixing layers, mixing is measured by examining the
local concentration of a conserved scalar (usually seeded into one of the two streams) or
by the formation of a chemical product which occurs due to mixing and reaction between
the two streams. However, these methods, collectively known as passive scalar and
chemistry methods respectively, can be severely affected by either measurement
resolution or finite-chemical rate effects (small Damkohler number).
The relative resolution for passive scalar measurements is typically defined by the
ratio of the probe dimensions and the mass diffusion scale (Batchelor scale), which
represents the smallest physical scale at which turbulent mixing is effected by the
velocity field. Breidenthal (1981) showed that under-resolved passive scalar
measurements will tend to over-predict the amount of mixed fluid in the shear layer. The
Batchelor scale (λB) is approximated as λB=δRe-0.75Sc-0.5, which places stringent
requirements on the size of the probe, especially for aqueous flows (for a water shear
layer at Re = 3 x 104, λB = 1 µm). To circumvent this strict resolution requirement,
chemical reaction techniques, which are sensitive to the average amount of product (and
thus mixed fluid), are used (Breidenthal, 1981; Mungal and Dimotakis, 1984;
Koochesfahani and Dimotakis, 1986; Karasso and Mungal, 1998). This class of
techniques relies on a chemical reaction taking place between the molecularly mixed
fluid streams to produce a product which can then be sensed either through direct product
measurement or through a small temperature increase. While chemistry methods do not
yield spatially resolved measurements of mixing, since the probe volume is still
nominally larger than the Batchelor scale, they are sensitive to the mean product scalar
value within the sampled volume. In the limit of fast chemistry (Da » 1), the product
concentration may be related to the amount of mixing.
Many quantitative measurements of scalar mixing have been made in
incompressible shear layers to ascertain the composition field across the layer or the
average amount of mixed fluid within the layer. Most researchers have performed high-
resolution passive scalar measurements (L/λB < 100, where L is the resolution of the
experimental probe), while a handful have looked at chemically reacting experiments.
The aim of these experiments is usually to determine the probability density function
8
(PDF), which is the probability of finding a particular mixture fraction at a particular
cross-stream location in the shear layer. The mixture fraction denotes the volume fraction
of fluid from one stream in the shear layer. Computed moments of the PDF can be related
to many quantities, such as the mean fluid profile, the mixed-mean fluid profile, and the
mixed fluid profile.
Fiedler (1974) made measurements in a single-stream mixing layer, showing large
structures with streamwise ramps in the concentration profile and more uniform in the
cross-stream direction, consistent with the results of Mungal and Dimotakis (1984), who
performed chemically reacting experiments in gaseous layers (H2 – F2). Konrad (1976)
and Koochesfahani and Dimotakis (1986) found structures, which were of an overall
more uniform concentration in both directions. These structure observations are mirrored
in the determined PDF’s for these experiments.
Konrad measured a “non-marching” PDF at a Reynolds number of 32,000 in a
gaseous shear layer using an accurate concentration probe (L/λB = 20, which is
considered satisfactory for predicting mixing quantities (Dowling and Dimotakis, 1991).
A “non-marching” PDF is one in which the most probable value of the mixture fraction
of mixed fluid is invariant in the cross-stream direction. This type of PDF along with the
observations of large-scale structures suggests that mixing models must incorporate the
fact that these large vortices control the entrainment process, improving on more
traditional gradient transport models, which are consistent with mixing by small-scale
turbulence. Konrad observed a “mixing transition” at Re ~ 10,000 featuring a sharp
increase in the amount of mixed fluid in the shear layer and also found that the shear
layer tends to entrain fluid asymmetrically from each of the two free streams after this
transition. “Non-marching” PDFs were also measured by Koochesfahani and Dimotakis
(1986) and by Matsutani and Bowman (1986).
Another possible mixture fraction PDF is the “marching” type, where the most
probable mixture fraction of the mixed fluid is close to the local value of the mean
mixture fraction. Batt (1977) measured a “marching” PDF in a single stream mixing layer
at high Reynolds number (70,000) using temperature as a marker of mixed fluid, but with
poorer probe resolution (L/λB = 60) than Konrad. Frieler (1992) measured chemical
product in gaseous shear layers at Re = 15,000, 36,000, and 62,000 and was able to invert
9
his results to mathematically reconstruct the mixture fraction PDF. He found that the PDF
was non-marching at the low Re condition, but evolved to a transitional behavior between
non-marching and marching at the higher Re conditions. Karasso (1994) found that a
minimum number of vortex pairings were required for this transition from a “non-
marching” to a “marching” distribution.
Mungal and Dimotakis (1984) performed chemical product measurements using
temperature as a conserved scalar in a fast, low-heat release chemical system
(hydrogen/fluorine). By performing “flip” experiments where the same reactants are
exchanged on the high and low speed sides, they discovered that the product composition
field across the layer as well as the total amount of chemical product in the layer were
different for a pair of “flip” experiments. This led to the conclusion that the shear layer
entrains asymmetrically from each stream, as the chemical product concentration varied
depending on which stream held the lean reactant, consistent with the results of Konrad
(1976). Koochesfahani and Dimotakis (1986) performed passive scalar measurements in
a water shear layer and confirmed the previous findings of asymmetric entrainment. Also,
using a passive scalar measurement, they measured a non-marching PDF at Re = 23,000,
though with poor probe resolution (L/λB = 1200).
Dimotakis (1986) proposed a theory for the entrainment ratio in mixing layers
based on the observation of the large-scale structures and their streamwise spacings. His
estimate (Equation 1.3) gives the volume flux entrainment ratio.
( )( )( )
2
1
11168.01,
VV
srsrssrEV =
+
+−+= (1.3)
This formula correlates very well with the previous incompressible entrainment data over
a wide range of density ratios (Frieler and Dimotakis, 1988; Frieler, 1992; Karasso,
1994), even though it is only experimentally correlated with scalar measurement data,
and not with velocity and scalar data as a flux term should be. Unfortunately, this formula
does not predict the entrainment behavior as the shear layer becomes compressible, as it
is based on two-dimensional Brown-Roshko rollers controlling the entrainment process.
10
1.2.2 COMPRESSIBLE MIXING LAYERS 1.2.2.1 Growth Rate
Many recent studies have examined the change in the growth rate of shear layers
as the compressibility or shear Mach number (∆U/aavg) was increased. Early works were
performed in the mixing layer regions of axisymmetric jets (Birch and Eggers, 1972;
Sirieix and Solignac, 1966; Wagner, 1973) and the growth rate data were reduced to a
single curve which displayed a clear decrease with increasing jet Mach number. Brown
and Roshko (1974) further improved the correlation of the data by examining the role of
the density ratio on the growth rate, which could then be accounted for in a normalizing
factor (the incompressible growth rate, Equation 1.2). However, a better collapse of the
data was possible with the invention of a compressibility similarity parameter.
The concept of the convective Mach number, Mc, was first developed by
Bogdanoff (1983) and further refined by Papamoschou and Roshko (1988) to collapse the
experimental growth rate data from many experiments and simulations to a single curve.
This approach assumes that large-scale structures, which convect at velocity Uc, form a
local stagnation point flow in a reference frame traveling at this convection velocity. If
the fluid entrained by the large-scale structures is brought to rest isentropically in the
convecting reference frame, a pressure matching condition exists at the stagnation point:
−
−
−+=
−+
122121 2
2
2
1
1
1 211
211
γ
γ
γ
γ
γγ
cc MM (1.4)
Then using the definitions of the convective Mach number as seen by each of the free
streams:
2
2
1
121
and a
UUMa
UUM cc
cc
−
=
−
= (1.5)
Solving these three relations yields a system of equations for Mc1,2 and Uc. However, if γ1
= γ2, then
ssrU
aaUaUaUc
+
+
=
+
+
=
11
121
2112 , (1.6)
and
11
21
2121 aa
UUMM cc+
−
== . (1.7)
For the experiments in the current study, Mc will connote an average level of
compressibility. Consequently, the convective Mach number quoted is calculated using
the average of the value of Mc1 and Mc2 from Equation 1.5. Uc is calculated by solving
the system of equations defined by Equations 1.4 and 1.5, because for the results
considered in this thesis, there is often a considerable difference between the free stream
values of γ. Even though the error associated with Equation 1.7 is quite small when
compared to Equation 1.7, (~4%), the more accurate method of determining Mc (solving
Equations 1.4 and 1.5) will be employed.
All compressible shear layer experiments since Papamoschou and Roshko (1988)
have quoted a normalized growth rate and an average convective Mach number. The
normalization of the actual growth rate by its incompressible component effectively
isolates the effect of compressibility on the planar shear layer. In Figure 1.2, the growth
rate data from many investigators are plotted versus the average convective Mach
number. The figure shows the general trend of reduced normalized growth rate with
increasing convective Mach number, though there is considerable scatter in the data. This
scatter may be attributed to growth rate measurement being made at various stages of
mixing layer development, variation of boundary layer conditions on the splitter tip
between the studies, or to possible acoustic instabilities, which feed back to the splitter tip
and alter the far field growth rate.
The most important idea behind the simple formulation of the convective Mach
number is the idea that large-scale structures travel at a fixed speed, which is not
necessarily the arithmetic mean of the two free stream velocities. Papamoschou (1991)
was the first to attempt direct measurement of the convection velocity using a two spark
Schlieren imaging system for Mc = 0.1 to 1.7. The results showed that the large-scale
structures had a convection velocity, which was biased toward one of the free streams, as
the convective Mach number was increased beyond 0.7. At incompressible conditions the
convection velocity well matches equation 1.6, but as the layer instabilities becomes
more skewed (a phenomenon which will be described in the numerical review section),
this formulation no longer holds. Many other investigators (Fourgette et al., 1990; Poggie
12
and Smits, 1996; Smith and Dutton, 1999) have also produced results that are consistent
with the convective velocity preference for one of the free streams. Papamoschou and
Bunyajitradulya (1997) attributed this asymmetry to lab frame effects, showing a
correlation for slower convection velocities than predicted by Equation 1.6 for mixing
layers made up of two supersonic streams and faster velocities for supersonic-subsonic
combinations. All the previously mentioned studies have been performed using double-
pulse imaging techniques and two-dimensional cross-correlations to find a single
structure convection velocity per imaged area.
Other convection velocity measurements have relied on one-dimensional cross-
correlation techniques to examine the variation of Uc with cross-stream location. This
type of approach usually produces a convection velocity that closely matches the local
mean velocity. Papamoschou and Bunyajitradulya (1997) argued that the transverse
parameterization of the convection velocity removes the ability of the cross-correlation
technique to recognize a two-dimensional structure and biases the measurement towards
the small-scale structures, which propagate at the local mean velocity. However,
visualization techniques that highlight the interface between the shear layer and one of
the free streams tend to return convection velocities which are close in speed to the
closest free stream velocity. Seitzman et al. (1994) observed this spatial bias in their
reacting flow study, which tracked the high-speed interface using imaging of the OH
radical and the low-speed interface by examining a passive scalar (acetone vapor) seeding
in the lower stream. Clearly, imaging methods that highlight free stream interfaces or
one-dimensional correlation techniques can add significant bias errors to convection
velocity measurements due to the sampling of a limited range of structure sizes. Many
explanations for the convection velocity asymmetry have been advanced, but the
solutions have almost uniformly been sought in the computational domain, so those
results are treated in the following section.
1.2.2.2 Coherent Structures
Since the discovery of large-scale structures in mixing layers by Brown
and Roshko (1974), many investigators have sought to describe the effects of
13
compressibility on the shear layer in terms of the change of the shape and extent of these
large-scale structures. The study of Papamoschou and Roshko (1988) demonstrated the
presence of well-defined large-scale structures at their lower compressibility conditions
(Mc <0.7) in their Schlieren images. Clemens and Mungal (1995) were able to clearly
characterize the effects of compressibility on structures. They showed that well-defined
spanwise-oriented, Brown-Roshko type rollers exist below Mc of 0.3. Above this, the
structures transition into less-organized, diffuse elements. Above Mc ~ 0.5, oblique
instability modes begin to skew the spanwise rollers so that strong three-dimensionality is
present in the layer.
Plan view images of both passive scalars and product formation methods
confirmed this decrease in spanwise uniformity and the increased obliquity. Island et al.
(1996) were able to rapidly acquire plan-view images to reconstruct three-dimensional
visualizations of the shear layer which clearly demonstrated the inclination of oblique
structures at Mc = 0.62. Messersmith and Dutton (1996) illustrated the increased three-
dimensionality of structures with convective Mach number using an oblique imaging
strategy. Two-point correlation techniques (Samimy et al., 1992; Martens et al., 1996)
confirm this trend with their ability to detect the presence and shape of large-scale
structures.
Other aspects of the effect of compressibility on large-scale structures have been
investigated. Oertel (1979) found Mach wave radiation from turbulent structures in
mixing layers similar to that seen from supersonic jets. Papamoschou (1989) saw that few
vortex pairing events occurred in compressible shear layers, whereas this phenomenon
was quite common in incompressible layers. Fourgette et al. (1990) used planar Rayleigh
scattering from the mixing layer region of a supersonic jet to examine the coherent
structures at Mc = 0.7 and found that structures deform and rotate but do not pair as they
propagate downstream. Samimy et al. (1990) studied a fairly high compressibility
condition (Mc = 0.86) using static pressure correlations to determine that structure
correlation lengths decreased at high compressibilities as evidence of increasing three-
dimensionality. However, the Schlieren images of Hall et al. (1993) reveal very little
large-scale structure above Mc = 0.5, most likely due to the spanwise integration of the
imaging technique smearing out the three-dimensional structures. Messersmith et al.
14
(1991), using diagnostic techniques similar to Clemens and Mungal, discovered that the
large-scale structures in the shear layer became more three-dimensional with
compressibility and not with increasing Reynolds number, a result also shown in
Clemens (1991). Naughton et al. (1997) showed that reduced pressure communication
across the shear layers inhibited vortex roll-up and pairing events, thereby causing weak
elongated streamwise structures to be preferred at high compressibilities (above
convective Mach number of one). Morkovin (1992) found that streamwise vortical
structures are not subject to communication problems like spanwise rollers as they
primarily interact in the spanwise direction, rather than across the mixing layer. Thus
these structures continue to enlarge and extract energy from the mean flow and entrain
fluid into the layer.
1.2.2.3 Velocity and Scalar Measurements Studies of turbulent velocity fluctuations and mean scalar fields have been made
to examine the underlying physics behind the reduction of growth rate with increasing
compressibility. Elliot and Samimy (1989) performed LDV measurements in medium
compressibility shear layers (Mc = 0.51, 0.64, and 0.86) and found that peak streamwise
and transverse turbulence intensities decreased with increasing compressibility. Also, the
Reynolds stress followed the same trend and was 50% smaller than the incompressible
results of Oster and Wygnanski (1982). They also showed a decrease in the two-point
velocity correlations in the spanwise direction with increasing convective Mach number.
Goebel and Dutton (1990) also performed sets of LDV measurements in a similar
convective Mach number range (0.2 – 0.99) and found consistent results, except for the
streamwise turbulence intensity. This variable was found to be nearly constant in this
compressibility range, but this result could be due to differing normalizations between the
two studies.
A more extensive set of LDV measurements was made by Gruber et al. (1993)
who obtained all velocity perturbations using a three-component measurement system.
They found that the spanwise and streamwise turbulence intensities did not decrease
appreciably with compressibility, but that the Reynolds stress and the total turbulent
15
kinetic energy did decrease. Using kinetic energy anisotropy correlations, they also
showed that three-dimensional turbulence plays an increasing roll in compressible shear
layers. Independent LDV results of Chang et al. (1993) and Debisschop et al. (1994)
support these conclusions.
A more complete instantaneous picture of the velocity field can be provided by
planar image velocimetry (PIV). Urban and Mungal (2000) examined the velocity fields
of Mc = 0.28 and 0.62 mixing layer conditions using PIV of TiO2. They found that the
vortical shapes tend to flatten, incline in the flow direction and breakdown as
compressibility increases. Also, transverse turbulent intensity and Reynolds stress are
strongly suppressed, while the streamwise turbulent intensity drops only slightly. Finally,
this study also showed the importance of the lab-frame sonic velocity in high gradient
regions leading to an communication barrier between streamwise-oriented vortical
elements, consistent with the sonic eddy concept of Breidenthal (1992).
Numerous scalar measurements for mixing layers at different compressibilities
have also been published. Clemens and Mungal (1995) using PLIF (planar laser-induced
fluorescence) of a tracer species, which was seeded into the lower stream, found that the
scalar transverse fluctuations decreased with increasing compressibility. Also, they noted
that a marching PDF was seen for both cases (Mc = 0.28 and 0.62), with the peaks of
mixed fluid composition about 20% wider for the low compressibility case. Dutton et al.
(1990), using planar Mie scattering, also found marching PDFs for his conditions of Mc =
0.3 and 0.77 and a slight increase in mixing efficiency. Messersmith and Dutton (1992)
also observed marching PDFs at Mc = 0.50 and 0.75 and narrowing of the mixed fluid
peaks at the higher compressibility condition. However, all of these results were based on
under-resolved measurements. As was shown previously, the Batchelor scale is inversely
proportional to the Reynolds number, and as high compressibility mixing layers have
large unit Reynolds numbers, the imaging requirements become more stringent at high
velocities. Thus, resolution insensitive techniques must be employed.
All the resolution insensitive mixing measurements made in compressible mixing
layers have been performed using fast, low-heat release chemical reactions or the
quenching of fluorescence of a tracer particle due to its molecular mixing with a
quenching partner, also known as “cold chemistry”. Hall (1991) made quantitative
16
product formation measurements at Mc = 0.51 and 0.92 using the H2-F2 reaction system.
With the addition of flip experiments, this study showed a decrease of mixing efficiency
with compressibility, rather than Reynolds number which was nearly equal between the
two cases. Dimotakis and Leonard (1994) found no strong dependence of mixing
efficiency on Reynolds number for compressible mixing layers.
As an alternative to chemical product methods requiring fast chemistry, some
recent studies have utilized molecular quenching as the product marker. Clemens and
Paul (1995) using the “cold chemistry” technique showed an increase of mixing
efficiency with Reynolds number in an axisymmetric shear layer. Island (1997)
performed extensive “cold chemistry” measurements on Mc = 0.28 – 0.76 mixing layers
and determined that the mixing efficiency increased slightly with both Reynolds number
and compressibility, but the correlation was stronger with Reynolds number. He also
showed that the mean mixed fluid profile more strongly tracks the mean composition at
higher compressibilities, which is consistent with the marching PDFs seen by Clemens
and Mungal (1995) at the same conditions. The difference between the mixing efficiency
data of Hall and that of the cold chemistry results is possible due to finite chemistry
effects in the hydrogen-fluorine chemical system. As the compressibility is increased, the
eddy roll-over times decrease rapidly causing the local Damkohler number to decrease
unless faster chemistry is used. However, fast chemistry usually relies on larger
equivalence ratios, which can also skew the product formation results. The “cold
chemistry” technique does not suffer from such effects as its characteristic time is nearly
equal to the inverse of the collisional frequency in the mixing layer (i.e. as soon as the
fluorescent tracer and the quencher collide, the signal is quenched with a characteristic
time of ~ 10-100 nsec).
1.2.2.4 Acoustic Radiation and Mixing Layers
As compressibility conditions are increased, a point will be reached where the
convection velocity of large-scale structures will exceed the local speed of sound of the
low-speed stream, or the difference between the convection velocity and that of the fast
stream will be greater than the local speed of sound. When this occurs, large-scale
17
structures tend to act as bluff bodies in supersonic flow, with curved bow shocks
surrounding them, or if the disturbances (structures) are weak enough, then they act as a
wavy interface in supersonic flow, with Mach waves radiating from the interface (Smits
and Dussage, 1994). Illustrative schematics of the two behaviors are shown in Figure 1.3.
Few experimental planar mixing layer results have achieved high enough compressibility
conditions to see these oblique waves, which are common in supersonic jet flows.
Naughton et al. (1997) examined mixing layers in the near field of jets up to convective
Mach number 1.9, but saw no radiated waves into the free stream. However, their results
were all for layers with the density ratio greater than 4, and since the convection velocity
depends strongly on the density ratio (Equation 1.6), they may not have created
conditions present in Figure 1.3.
An important question still left unresolved in the literature is whether the acoustic
radiation from planar layers is due to this bluff body effect or due to a supersonic
instability mode, which does not travel at the convection velocity. Blumen et al. (1976)
showed that the compressible shear layer has supersonic instabilities which are amplified
at Mc > 1. Papamoschou (1993) argued that curved streamlines present in the shear layer
generated waves, which coalesce into oblique shocks. Barre et al. (1994) interpreted his
results to mean that compressibility inhibits entrainment and causes the creation of
oblique shocks, which radiate from the shear layer. Also in an earlier work, Barre (1993)
showed that the speed of the radiating sound sources is different from the convection
velocity of the large-scale structures. Alvi et al. (1997) showed in a Mc = 2 counter-
current shear layer that highly convoluted structures existed with compression waves
about the structures, but no eddy shocklets were seen internal to the layer (the spatial
resolution of his study was 15 µm, which was the same order of magnitude as the
Kolmogorov scale for the layer). Also, their scalar scale structures underwent large
transverse motions when entraining free stream fluid. However, this flowfield is
absolutely unstable, unlike the convective instability of the planar shear layer, which
makes direct comparison tenuous. Krothapalli (1996) showed that shock-inducing
streamwise vortices were the results of curved streamlines, which give rise to Taylor-
Görter type instabilities. However, these results were for underexpanded supersonic
annular jets, which have additional helical instabilities at high compressibilities, which
18
limits the comparison. Erdos et al. (1992) examined a very high compressibility shear
layer (Mc = 2.8), but their Schlieren images did not show shock wave radiation and their
study did not include any quantitative results derived from their images.
1.2.3 SIMULATIONS AND MODELING OF MIXING LAYERS Due to the fundamental nature of the mixing layer flowfield, there is a
considerable body of numerical analysis at both incompressible and compressible
conditions produced in parallel with the large amount of experimental research. The
computational efforts can be grouped in linear stability analyses, which seek to explore
the amplification and growth and instability modes, and direct numerical simulations
(DNS), which seek to resolve the smallest scales of the turbulent flow accurately. The
linear stability analysis can be further sub-divided into two- and three-dimensional
simulations (an important distinction as the layer instability modes become more three-
dimensional with increasing compressibility). Finally, many of these simulations have put
forth conclusions about the existence of local shocklets and the nature of the transport of
turbulent kinetic energy at compressible conditions.
1.2.3.1 Linear Stability
The first linear stability analysis began with a vortex sheet or hyperbolic tangent
velocity profile and simple patterns of linear waves. Hama (1962) and Michalke (1965)
both demonstrated that this construction would produce vortical structures similar to
those seen in experiments. Since there is a tremendous body of work on this subject, this
review of linear stability analysis will be restricted to only highly compressible
simulations.
The stabilizing effect of compressibility on the growth rate of mixing layers was
first observed in the linear stability results of Dunn and Lin (1955), Lessen, et al. (1965),
and Gropengeiser (1970). All of these studies show similar reductions in growth rate with
increasing compressibility when compared to experimental growth rate measurements in
the excellent review paper by Ho and Huerre (1984). The computation growth rate
definition is the linear growth rate (α) of the central mode of the mixing layer, a mode
19
that is centered inside the shear layer and spans the transverse extent. Lu and Lele (1994)
were able to compare their maximum linear amplification rates to experimentally
observed growth rates with very good agreement, which is surprising considering the
presence of non-linear effects such as mode saturation and interaction of instability
modes to produce sub-harmonics. These studies also showed that three-dimensional
disturbances have larger growth rates as compressibility is increased. Sandham and
Reynolds (1990) observed this structural shift at Mc > 0.6, where oblique modes begin to
dominate the two-dimensional Kelvin-Helmholtz instability. The same shift to three-
dimensional structures was seen in most experimental results described in the previous
section.
Linear stability analysis also has the ability to measure the structure convection
speeds of instability modes as the mode phase speed is a solved variable in a linear
decomposition technique. As discussed previously, Equation 1.6 fails to reproduce the
experimentally observed structure convection velocities seen at high convection Mach
numbers. Sandham (1989) theorized that the structure convection speed could be
estimated by examining the phase speeds of neutral stability modes. His estimates, based
on asymmetric velocity profiles show much better agreement with the results of
Papamoschou (1989), as the neutral mode phase speeds tend to be well correlated with
outer mode solutions (Jackson and Grosch, 1989) whose convection speeds are nearer
free stream values.
The concept of outer modes, or instability modes which have both a spatial and
convective velocity preference for one side of the shear layer, was first discovered by
both Lessen et al. (1966) and Gropengeiser (1970), who approached the discussion by
looking for supersonic modes, or linear modes which convect supersonically with respect
to one of the free streams. Two types of modes were observed: fast modes, which are
supersonic with respect to the lower stream, and slow modes, which are supersonic
relative to the fast stream. Jackson and Grosch (1989) provided a complete
characterization of the outer modes, but found that the central mode had a significantly
higher growth rate than the outer modes in their convective Mach number range of
interest.
20
The only linear stability study which fully investigated the compressibility regime
where these outer modes show larger linear amplification rates than the central mode is
that of Day et al. (1998). He showed that outer mode solutions can be dominant above Mc
= 2, with the density ratio proving to be an important parameter in determining the
relative amplification rates of the fast and slow modes. Also, co-layer modes were shown
to produce marching PDFs, but the transition from a broad central mode to a more
compact, off-center outer mode structure would negatively impact mixing ability. Finally,
his simulations showed that outer modes continually compete with the central mode
likely making them difficult to discern in experimental conditions and that mode
communication at high compressibility levels is severely depressed.
1.2.3.2 Direct Numerical Simulations
The use of both two- and three-dimensional DNS simulations of mixing layers has
proven fundamental to determining the physical mechanisms behind growth rate
suppression with increasing compressibility. Several investigators have seen good
quantitative comparison between computed growth rates and experiments (Sandham and
Reynolds, 1991; Planche and Reynolds, 1992; Vreman et al., 1996; Freund et al., 2000a).
These simulations also showed a similar shift from primarily two-dimensional
instabilities to fully three-dimensional oblique modes at higher compressibility
conditions.
Further physical results are available from some of the better resolved DNS
results. Papamoschou and Lele (1993) examined vortex generated pressure disturbances
in compressible shear layers and found that cross-stream pressure gradients are
significantly reduced at high Mc, leading to the conclusion that reduced pressure
communication across large eddies tend to decrease mixing and inhibit shear layer
growth. Vreman et al. (1996) showed that simulated Reynolds stress and growth rate
were suppressed with increasing compressibility, consistent with the previously reviewed
experimental results. Rogers and Moser (1994) examined mixing in incompressible shear
layers and saw that the type of PDF generated by the flow was dependent on the initial
boundary layers in the near field. However, after attaining self-similar behavior, they
21
found that the pairing of organized vortical structures gave rise to a “non-marching” PDF,
while a lack of coherent structures and a dominance of small-scale eddies (esp. those
which mixed fluid at the edges of the shear layer) produced a “marching” PDF.
In the most comprehensive analysis of compressibility effects in a shear layer
using DNS, Freund et al. (2000a) showed that smaller streamwise structures dominate the
shear layer topology at high compressibility conditions. Also, suppressed pressure
fluctuations were seen to be a leading factor in the reduction of the growth rate. For the
passive scalar field, Freund et al. (2000b) demonstrated that the PDF shape shifts from
non-marching to marching with an increase in Mach number, concomitant with a
reduction in eddy sizes found from two-point velocity correlations. Also, as
compressibility was further increased, the mixed mean profile was seen to further
resemble the mean scalar profile, suggesting that gradient diffusion models for scalar
transport and mixing may be more successful at higher convective Mach number
conditions than large-scale structure based models (Broadwell and Breidenthal, 1982;
Broadwell and Mungal, 1991; Dimotakis, 1991).
1.2.3.3 Shocklets and Turbulent Kinetic Energy Transport
Direct numerical simulations also have the ability to closely examine the
underlying physical mechanisms of the reduction in growth rate due to the complete
resolution of the turbulent eddies. Initially, two-dimensional simulations (Lele, 1989;
Sandham and Yee, 1989) observed that dilatational effects including dilatational
dissipation were increasing with compressibility and that eddy shocklets were formed at
moderate compressibility levels. Eddy shocklets are small shocks associated with
turbulent eddies and were first observed in two-dimensional decaying turbulence
simulations of Passot and Pouquet (1997). The results of Lee et al. (1991), who saw
shocklet production in three-dimensional isotropic turbulence, prompted the development
of models where enhanced dilatational dissipation was cited as the cause for decreased
turbulent kinetic energy production and thus lower growth rates (Zeman, 1990).
However, as the complexity of DNS simulations increased, the appearance of shocklets
shifts to high convective Mach numbers. Shocklets were not identified in two-
22
dimensional shear flows until Mc = 0.7 and not in three-dimensional simulations until Mc
= 1.2 (Vreman et al., 1996). Freund et al. (2000a) did not see large pressure disturbances
until Mc = 1.54, a higher threshold than in all other previous simulations. Consequently,
since the main region of growth rate suppression occurs between Mc = 0 and 1.5, shocklet
production and large dilatational dissipation are assumed to not affect the production of
turbulent kinetic energy in this low compressibility range. Also, Colonius et al. (1997)
saw radiation from forced sources inside the shear layer strengthen with increasing
compressibility, so that the resulting waves resembled the sound radiated from supersonic
jets. However, his results are consistent with shock generation from eddy interaction with
the free stream, rather than local turbulent shocklets.
Since dilatational effects in the turbulent energy transport equation have been
shown to be relatively small at large Mach numbers, other kinetic energy reduction
methods have been sought to explain the reduction in growth rate with increasing
compressibility. From the early results of Papamoschou and Lele (1993) which confirmed
that reduced pressure fluctuations in the cross stream direction seemed to be the physical
manifestation of the “sonic eddy” concept, according to which eddies with a Mach
number difference across them which is greater than one will not play a role in the
entrainment of fluid. Therefore, as the compressibility is increased, the eddies must
shrink in size in order to remain intact and continue to entrain fluid. Vreman et al. (1996)
found that pressure-strain rate correlations were suppressed with increasing Mach
number, and Freund et al. (2000a) confirmed their result and showed that the reduction
was primarily due to reduced pressure fluctuations. Modeling of the pressure-strain
correlation term in the turbulent kinetic energy equations with reduced communication
reasoning has led to consistent correlation with experimental growth rate data (Burr and
Dutton, 1990; Freund, 1997).
1.3 RESEARCH OBJECTIVES AND THESIS OVERVIEW The main objective of this research is to further the understanding of turbulent
mixing at compressible conditions and to investigate the state of large-scale structures at
higher compressibility conditions than previously attained experimentally. Along with
this directive, the previous literature suggests further goals for this thesis:
23
• Design and construction of a shock tunnel to create high stagnation enthalpy
conditions for use by supersonic nozzles.
• Design and construction of a supersonic mixing layer facility in which a
supersonic nozzle fed by a shock tunnel and a subsonic manifold are used to
create higher compressibility shear layers than are possible in blowdown
facilities.
• Examine the changes in growth rate of compressible mixing layers with variable
density ratios and compressibility.
• Observe the effect of compressibility on the character and motion of large-scale
structures in shear layers by obtaining PLIF images in three orthogonal imaging
planes.
• Observe the effects of off-design conditions on the development of highly
compressible shear layers.
• Quantify the effectiveness of acetone as a tracer species in shock tunnel mixing
flows.
• Improve existing NO PLIF diagnostics including “cold chemistry” techniques for
use in low-pressure supersonic mixing environments.
• Measure mixing efficiency and mean scalar profiles at mixing layer conditions
above Mc = 2.0.
In order to completely describe the fulfillment of these objectives, this thesis will be
divided into seven chapters. Chapter One has described the motivation for this thesis and
examined the previous relevant literature in order to put into focus the efforts of this
research study. Chapter Two explains the experimental facility, imaging hardware, lasers,
and data acquisition systems used for this work. Chapter Three highlights the important
issues for shock tunnel design, the tailoring of stagnation conditions, and the creating of
useful test time along with the methods for generating suitable mixing layers. Chapter
Four presents experimental results for Schlieren imaging of mixing layer flows, along
with the measurements of growth rate. Chapter Five shows acetone PLIF imaging results
for a low and medium compressibility condition along with comments on the
applicability of acetone as a tracer molecule in shock tunnel flows. Chapter Six presents
24
images of a high compressibility condition using nitric oxide as a tracer species along
with measurements of mixedness and mixed mean composition. Finally, in Chapter
Seven, the thesis is summarized and recommendations for future research are given.
25
αU1, T1, ρ1
U2, T2, ρ2
y xδ
Side 1
Side 2
Figure 1.1 Schematic of mixing layer. Fast side refers to side 1 and is always supersonic in this
study. Slow side refers to side 2 and is always subsonic in this study. δ refers to the visual
thickness of the mixing layer. α is the local Mach wave angle. z is the spanwise
coordinate of the mixing layer (into the page).
1.2
1.0
0.8
0.6
0.4
0.2
0.0
δ/δ
inc
3.02.52.01.51.00.50.0Mc1
Papamoschou and Roshko (1988) Goebel and Dutton (1992) Elliot and Samimy (1990) Hall and Dimotakis (1991) Chinzei et al. (1986) Clemens and Mungal (1995) Wagner (1973) Naughton et al. (1997) Dimotakis Fit (1991) Central Mode, Day (1999) Fast Mode, Day (1999) Slow Mode, Day (1999)
Figure 1.2 Selected historical normalized growth rate data for previous compressible mixing layer
experiments plotted versus the relevant compressibility parameter (convective Mach
number, Mc). Also shown are the recent results of a linear stability analysis that
considered the role of very high compressibility in mixing layer physics (Day, 1999).
26
(a) (b)
Free Stream
Mixing Layer
U1
Uc
Θ = sin-1(a1/(U1-Uc))
Uc
U1
Free Stream
Mixing Layer
CurvedBowShock
Figure 1.3 Schematics of the creation of shocks in compressible mixing layers due to slow moving
structures, which protrude into the free stream. Case (a) shows a weak disturbance
interface where regular Mach waves are radiated. Case (b) shows a strong disturbance
where a locally curved shock is launched as the structure acts more like a bluff body in
supersonic flow.
27
CHAPTER 2
EXPERIMENTAL APPARATUS
This section describes the design of the shock tube, built for supersonic fluid
dynamic research, and of the mixing layer facility, built specifically for this study. Also
covered are the many optical diagnostics used to non-intrusively probe the flow in both
facilities.
2.1 SHOCK TUBE
A schematic diagram of the shock tube facility is shown in Figure 2.1. The
dimensions of the tube were selected both to allow substantial test times over a wide
range of stagnation temperatures and to fit inside the room in which it was to be housed.
The shock tube driver section consists of a 2.6 m driver section with an internal diameter
of 18 cm, and can handle pressures of up to 120 atm. The driver is not honed for a
smooth wall finish, but has been electro-polished for gas passivation. The driver contains
a variable length insert, which can reduce the overall internal length of the driver by up to
two feet to save on gas costs if the attendant drop in test time is allowable. At the
diaphragm edge of the driver, a contoured area change from 18 cm to 14 cm in diameter
is used to mate the round driver with the double diaphragm section.
The double diaphragm section consists of a flange that has a diaphragm mounted
on each side and is of similar dimensions as the driver flanges. An 11 x 11 cm passage
through the 7 cm thick flange acts as the buffer section between the driver and driven
sections of the shock tube. The thickness of the flange is minimized so that the petals
from the upstream diaphragm do not interfere with those from the downstream.
Diaphragms which petal into a square cross-section send less debris into the driven
section after firing than do those that shatter into a round cross-section. This section is
typically filled with helium to ½ P4 and quickly vented to atmospheric pressure using a
28
pneumatically actuated valve. The delay from vent initiation to diaphragm rupture is
typically 200 – 300 msec, which is necessary to allow sufficient time for the startup of
secondary flow devices in the test section and laser firing circuits.
The driven section is 9.1 m long with an 11.4 cm diameter (an L/D ratio of 80).
To minimize wall friction and gas adsorbtivity, the inside diameter was honed (to 32 µin)
and electro-polished. The head of the driven section contains a square to round transition
insert that allows the petals of the downstream diaphragm to open into a square cross
section, and then the overall shock flow proceeds down a cylindrical tube. The driven
section is split into three 2.4 m sections, which are designed to withstand 50 atm pressure
loads. The final two sections, a 1.5 m incident section and a 0.4 m reflected section, are
designed to accommodate instantaneous pressures of up to 150 atm. The incident section
has four pressure ports, spaced 30.5 cm apart for measuring the incident shock velocity
and attenuation. This section also contains two horizontally opposed ports, 1.2 meters
from the endwall, allowing optical access for absorption and emissions studies.
The shorter reflected section has 4 opposed ports, which can either be 5 mm
(shock tube configuration) or 19 cm (shock tunnel configuration) from the endwall,
depending on whether the false endwall insert is used. Two vertically opposed ports are
mounted into the last flange, allowing pressure measurements 5 cm from the endwall
when the tube is in the shock tunnel configuration. This section also contains horizontally
opposed ports for optical access to the reflected region of the shock tube, used for
measuring the duration of hot test gas time in this study. A two-ton block of steel and
sand is attached to this portion of the driven tube to serve as an inertial mass to reduce
vibrations and recoil from diaphragm rupture and shock reflection.
Located in the pressure port plugs are six fast-response (< 1 µsec) piezoelectric
pressure transducers (PCB 113A21) that are mounted flush with the shock tube inner
wall, and which trigger five Phillips PM 6666 timer counters, which have an accuracy of
± 0.1 µsec. The time interval measurements made from these counters are used to
generate an incident-shock velocity profile, which is used both to characterize the
attenuation of the incident shock wave and to evaluate the initial velocity of the reflected
shock wave. The reflected-shock test pressure is monitored by a dynamic pressure
transducer (Kistler 603B1) mounted at the bottom port nearest the endwall. The data
29
acquisition system for these pressure transducers consists of 4 Gagescope 1012M (Gage
Applied Sciences) boards mounted in an external housing. A total of eight channels are
available with 12-bit resolution at 10 MHz per channel.
High-pressure helium gas is used in the driver and is supplied from a 170 atm
manifold, regulated using Autoclave high-pressure valving, and measured using both a
Setra 206 high-pressure transducer and an Ashcroft Bourdon tube pressure gage (0 –
3000 psi). Test gas is inserted into the driven section through metering valves, and the fill
pressures (P1) are monitored using an MKS Baratron 122B (SN 00088110) static pressure
transducer (0 – 1000 Torr) with a PDR C-1C signal conditioner for low pressure fills (P1
< 1 atm) and a Setra Model 280E pressure transducer for P1 values greater than 1 atm.
The vacuum system is located at the head of the driven section, 0.5 meters after
the downstream diaphragm. Low vacuums (10 mTorr) are achieved in the driven and
driver sections using Welch 1397 roughing pumps. A turbomolecular pump (Varian
V1000) is also available at this station and can achieve ultimate pressures of 10-6 torr in
the driven tube with a leak rate of 10-5 torr/min achievable after overnight pumping.
However, higher vacuums were not used in this study and the shock tube was only
roughed out to low vacuum before each shot, as trace contaminants in test gas mixtures
were not a significant error source.
Test gas mixtures are created in a high-pressure aluminum mixing cylinder (150
psia maximum) using a stand-alone mixing panel. A Welch 1397 roughing pump is used
to evacuate the mixing cylinder to 10 mTorr prior to filling (measured with a Varian DST
531S Thermocouple gage). The relative amount of each constituent gas is controlled
using partial pressures measured by a Setra 280E (0 – 250psia) pressure transducer. The
mixing cylinder is equipped with a brass stirrer rod, affixed with five mixing blades and a
magnetic base, which can be rotated up to 200 rpm using a magnetic stirrer (Fisher-
Scientific Thermix 220T), reducing mixing times to less than 20 minutes.
2.2 MIXING LAYER FACILITY
In order to create high Mach number cross flows for the generation of
compressible mixing layers, a test section similar to those used in blowdown type
30
windtunnels (Clemens and Mungal, 1995; Papamoschou and Roshko, 1988) was
constructed. However, instead of using plenum sections used to house the high-pressure
gas, which feeds the supersonic nozzle, this facility uses the reflected portion of the shock
tunnel as the high enthalpy reservoir. A lower pressure aluminum cylinder is used to
contain the low-speed side gas.
As shown in Figure 2.2, the test section consists primarily of a two-dimensional
supersonic nozzle block, a splitter plate, a subsonic nozzle and a low-speed manifold
used to inject the subsonic portion of the layer. Designed to house nozzles in the range of
Mach 3 to 6, the test section for this shock tunnel is 10 cm wide, 40 cm high and 1.2m in
length. The maximum throat height for any nozzle, 5 mm, is fixed by the 10 cm x 5 mm
slot in the endwall flange of the shock tube. The minimum throat height is 2 mm due to
separation in the subsonic portion of the nozzle. Since the throat height is so controlled,
the exit height of each nozzle dictates the internal height dimension of the test section. A
set of false upper and lower walls allows for the accommodation of a range of nozzles.
The two dimensional nozzles are designed using the inviscid NOZCS2 code
(Carroll et al., 1986) which has been modified slightly to handle higher design Mach
numbers (M > 3) by increasing the number of characteristic waves in the computation.
Corrections for compressible boundary layers that are based on the empirical relations of
Anderson (1990) were used to correct for the boundary layer displacement thickness. If,
for example, the target exit Mach number was 5.0 using argon as the working fluid, the
nozzle form would be cut to a Mach number of 5.35 to allow for the displacement of the
boundary layer to reduce the area ratio. Also, since the code utilizes a fixed specific heat
ratio, γ, to calculate the local speed of sound, an appropriate average γ for vibrationally
relaxing test gases is necessary when performing nozzle design calculations. Thus, a
simple Landau-Teller formulation (described in Section 6.2) can be used to estimate the
frozen vibrational temperature for any nozzle and stagnation condition combination. A
frozen specific heat ratio is simply found for this set of P5 conditions and then is
compared to the range of γ that are possible considering all the potential stagnation
conditions to be used with a fixed Mach number nozzle. This range of specific heat ratios
is very small as the vibrational temperature in these high Mach number nozzles is frozen
very quickly. Consequently, a fixed, average γ is a very good assumption for the design
31
of high Mach number nozzles for vibrationally relaxing gases. The final nozzle profile is
cut into a solid aluminum block using a CNC (computer controlled) milling machine.
For mixing layer studies, a splitter tip plate acts as both the lower plane for the
half nozzle expansion and as the separating body between the fast and slow streams
(Figure 2.2). The splitter tip thins at a 4° angle to a final thickness of 0.5 mm in order to
minimize the wake region of the tip. The removable splitter plate is held in place and
aligned to the test section by ten self-sealing bolts.
The slow-speed stream is injected from the downstream high-pressure tank by the
use of dual solenoid valves (ASCO 8210G1), which are activated simultaneously with the
venting of the double diaphragm section. The low-speed gas then flows through
approximately 1 meter of 12 mm copper tubing into the test section. The gas is injected
into the low speed side by a perforated plate (10 x 11.4 cm) with 2 mm holes (150 total)
and 4% open area to ensure choked flow at each orifice. The low speed flow is then
conditioned by a 55%-open perforated plate and two 500 µm screens before flowing
through a subsonic minimum length contraction designed for fully attached boundary
layer flow (Chmielewski, 1974) of area ratio 1.8. This contraction is mounted 6.4 cm
below the splitter tip to allow for mixing layer spreading into the low-speed side.
Correlations for side two mass flow rates and velocities with injection pressure, as well as
spatial uniformity, can be found in Appendix A.
The angles of the test section upper and lower internal walls can be adjusted ± 2°
to minimize the streamwise pressure gradient dP/dx caused by the mixing layer. For these
studies, the upper wall was fixed at 0° to minimize waves launched from upper
wall/nozzle interfaces, while the lower wall was mounted at an angle of – 0.5° downward
to control the pressure gradients. Four flush wall-mounted PCB 111A23 pressure
transducers (2 on the upper wall, 2 on the lower) measure the instantaneous wall static
pressure variations.
Optical access to the test section is provided by both acrylic and fused silica
windows (ESCO Products S1-UVA). The use of removable and reversible window
mounts allows for a total viewing area of 66 x 9 cm (side view) or 66 cm x 5 cm (plan
view) in three 22 cm, non-overlapping sections. Step windows epoxied into aluminum
frames were required to withstand the large forces due to the low test section pressures
32
involved in this study. The silica windows are flush mounted into these frames and
carefully mounted so as to minimize the mismatch between the window and the frame.
The frame is then flush mounted into the test section sidewall, along with Buna-N O-ring
seals to maintain low pressure.
The mixing test section then expands into a 3 m3 volume dump tank. This tank is
evacuated using a large Hereaus E135 (135 m3/hr) roughing pump, and the overall test
section and dump tank leak rate is 2 Torr/min. The large size of this dump tank is to
ensure shock tube resting pressures are below 35 psia after firing the maximum
stagnation pressure conditions possible to protect the imaging windows and to guarantee
that the test section pressure rise before the test time due to slow side injection is less
than 2% at the lowest pressure conditions. The large size of the dump tank also allows for
the heavy dilution (10:1) of any noxious or hazardous gases seeded or created in the
shock tunnel before venting.
2.3 OPTICAL DIAGNOSTICS 2.3.1 CO2 EMISSION SETUP
The CO2 detection system is shown schematically in Figure 2.3, and is similar to
that used by Flower (1976). This system utilizes an indium-antimonide (InSb) detector
(Judson Series J-10, liquid N2 cooled), along with a large off-axis parabolic mirror (30
cm, f/2.4, used 18° off-axis), gold-coated plane mirror, variable slits, and a wide band-
pass IR filter (4.18 – 4.74 µm). The detector (1 x 3 mm active area) is imaged at the
inner surface of the near window of the shock tube at a magnification of 1.5. Spatial
resolution of the optical system is controlled by a vertical slit of adjustable width placed
47 cm from the detector image in the shock tube. The slits are set to 5 mm to match the
spatial resolution of the detection scheme (15 mm spatial resolution at the far wall along
the shock tube along with ~1000 m/sec incident shock velocity) to the 1.5 µsec temporal
resolution of the detector, measured using a 1 MHz modulated red LED.
Flat sapphire windows (12.5 mm diameter, 3 mm thick) are mounted tangentially
to the inner contour of the removable plugs, which match the curvature of the shock tube,
33
for use in emission/absorption experiments. The windows are epoxied in place (EPO-
TEK 353ND) to hold high vacuum. The system is aligned using a He-Ne laser and then
focused using a LED mounted at the shock tube near inner wall whose image is then
focused on the detector face. The far side plug in the shock tube also contains a sapphire
window to minimize the background thermal emission, which at the operating pressures
of this shock tube is minimal (Peterson, 1998). CO2 is seeded into the test gas of the
shock tube at levels of 0.5 – 1% allowing for high signal to noise ratios with minimal
self-absorption.
2.3.2 SCHLIEREN IMAGING SETUP
Toepler Schlieren imaging is used to visualize the overall flowfield and growth
rate of the mixing layer using the index-of-refraction gradient across the shear layer. A
long-duration Xenon spark lamp is used as the light source with variable illumination
times of 20, 50, and 200 µsec. The light source is collimated through the test section and
then focused to a point by a pair of 190 cm, f/6 parabolic mirrors. A horizontal knife-edge
is inserted at the focus to make the system sensitive to vertical density gradients. In order
to allow for the imaging of both bright and dark regions (i.e. both positive and negative
density gradients), the sensitivity is adjusted so that the un-deviated image is one-half of
the maximum illumination allowed by the camera at the chosen gain setting. Thus, both
compression and expansion waves are imaged by this system.
Downstream of the knife-edge, the light is focused by either a 105 mm, f/2.8 or a
200 mm, f/4 Nikon lens, depending on the desired magnification. Due to the different
focusing needs of a Schlieren system, the camera lens is not mounted at the typical focal
length for F-type lenses. Since the lens is focusing rapidly diverging rays that have been
focused by the second mirror (Figure 2.4), and not nearly parallel rays as in normal
imaging applications, the lens must be placed further forward in the optical path than it
would be in a traditional imaging setup. Light leakage between the lens and the camera is
not a significant noise source in the images.
A Hadland Photonics 468 high framing-rate camera then images the focused light.
This camera has eight independent ICCD arrays (576 x 385), which can be
34
asynchronously fired. All Schlieren images presented in this paper have been gated at an
exposure time of 250 nsec or less, which corresponds to a 1 pixel blur at the highest
magnification. The synchronization of the camera and the light source is accomplished by
a Stanford Research Systems DG-535 pulse delay generator, using the pressure pulse of
the last pressure transducer as the beginning of the delay cycle.
2.3.3 ACETONE PLIF IMAGING SETUP
The frequency quadrupled output of a Nd:YAG laser is used as the excitation
source (Spectra Physics Quanta-Ray GCR-3, 1.2 Joules at 1064 nm). The laser has a
pulse width of 6 nsec and pulse energy of 16-18 mJ at the test section. The pulse energy
was measured using a Molectron PowerMax 5100 large area pyroelectric energy detector,
and the pulse width is measured using a weak beam reflection directed onto a phosphor
mask imaged by a Thorlabs 1 mm2 photodiode. Fourth harmonic conversion efficiency
tended to drift with time due to the internally heating of the KDP crystal, but could be
stabilized for 1 minute by carefully angle-tuning the crystal just prior to running a shock
experiment.
The beam is formed into a sheet (160 µm x 50 mm) and focused into the test
section using a –25 mm cylindrical and 120 mm spherical lens combination (Figure 2.5).
The sheet is directed into the test section either from above (for side view imaging) or
from the side (for plan and end view imaging) using high reflectors optimized for 45°
reflection of 266 nm laser light. Sheet thickness and size do not vary significantly
between these orientations.
In order to synchronize the laser operation to the shock tunnel operation, a special
digital clock switching circuit is created. Prior to the shock tunnel firing, the Nd:Yag
laser is operated at 10 Hz on Clock A. When the double diaphragm is vented, the laser
switches to a hold mode for approximately 200 msec where no pulses are sent to the
flashlamps. Then, as the shock tunnel is fired, the pressure transducer pulse from the
Kistler 603B1, delayed by a Stanford Research Systems DG-535 Timing Delay
generator, activates the flashlamps of the laser 205 µsec before laser light is needed in the
test section. A further delayed pulse triggers the Q-switch of the laser, which removes
35
significant temporal jitter if the Q-switch is triggered by the Quanta-Ray laser itself. After
the firing of the data collection pulse, the laser is switched onto another 10 Hz clock to
idle until the next shock tunnel firing. Operating the laser in this fashion prevents laser
misfires and minimizes shot-to-shot laser energy fluctuations. A laser timing diagram is
shown in Figure 2.6.
The resulting acetone fluorescence signal is imaged onto an unintensified interline
CCD camera (Princeton Instruments MicroMax RTE/CCD- 1300Y with Double Image
Feature). The array is 1300 x 1030 with 6.8 µm pixels. The camera is equipped with a
Nikkor 50 mm f/1.2 lens. The all-glass construction of the imaging lens effectively
blocks the entire potential UV elastic scatter from the test section, but allowed the red-
shifted acetone fluorescence feature (330 to 600 nm, 450 nm peak) to be imaged. The
exposure of the CCD array is gated to 2 µsec by using the electronically shuttered first
image of the frame transfer mode. All images are binned 3 x 3 into larger 20 µm pixels to
improve the signal to noise ratio. No spectral filters are used for these images, as the lens
cuts all elastically scattered light (based on comparison of in situ background signals with
unseeded background signals) and the flowfield has no visible natural luminosity.
2.3.4 NO PLIF IMAGING SET-UP
Quantitative scalar measurements are made with PLIF of seeded nitric oxide. The
excitation source consists of a frequency doubled, Nd:YAG pumped dye laser (Lumonics
YM-1200, HD-500, HT-1000) tuned to a rotational transition in the A←X (0,0)
electronic transition of the γ bands of NO. The frequency-tripled output of the YAG laser
(355 nm, 280 mJ) pumps a blue dye (Coumarin 2). The dye laser output (452 nm, 30 mJ)
is doubled with a BBO crystal and the harmonic is separated by a set of three high
reflecting mirrors ( > 99% reflectivity at 226 nm and < 2% at 452 nm). This yields a UV
pulse of 226 nm, 0.2 mJ, 6.8 nsec duration, and a 0.5 cm-1 spectral width. Laser energy
measurements are made using an Ophir Nova PE-50BB pyroelectric energy detector. The
pulse energy is intentionally kept lower than that of previous NO PLIF measurements
(Island, 1997; Clemens and Mungal, 1995; McMillin et al., 1994) to avoid saturation of
the absorption transition at the lower test section pressures used in this study. Such
36
saturation would alter the linearity of the LIF signal (saturation behavior of NO PLIF is
covered is Chapter 6). The flashlamps and Q-switch of this YAG laser are controlled
using the same 10 Hz clock switching circuit as described in the acetone fluorescence
setup. However, with this laser, the Q-switch pulse is generated by an Ortec 416A pulse
delay generator, instead of by the Stanford Research Systems DG-535.
As shown in Figure 2.7, a fraction of the output beam of the HT-1000 is split with
two uncoated fused silica windows for wavelength tuning. Tuning is accomplished by
passing the beam through a LIF cell filled with 0.2% NO in Argon to a pressure of 0.1
atm, which roughly simulates the absorption linewidth at the test section conditions. The
cell has two opposed fused silica windows and a 50 mm f/2 fused silica lens mounted
perpendicularly to the beam path to capture the resultant fluorescence. The input laser
beam is attenuated using a 1 mm Schott UG-5 filter to keep the fluorescence signal linear
with laser energy. A photomultiplier tube (Hamamatsu R166UH) fitted with a 3 mm UG-
5 filter collects the LIF signal as the laser is scanned across the NO spectrum. The PMT
is biased to -650 V using a high-voltage power supply (Power Designs Inc., 3K10B) A
secondary computer controlled system for the HT-1000 allows for simultaneous tuning of
the dye laser wavelength along with the doubling crystal angle to minimize power
fluctuations due to changing phase-matching conditions inside the BBO crystal. The
PMT signal is boxcar-averaged (Stanford Research Systems BA 101) over a gate of 200
nsec, synchronized with the laser Q-switch.
The excitation spectra of the NO A2Σ
+←X2
Π½ (0,0) Q1 + P21 (18.5) transition at
225.857 nm and its nearest neighbors is shown in Figure 2.8. This signal is a 3-shot
running average of the boxcar-integrated LIF signal (collected at 10 Hz) recorded by a
Tektronics TDS 3032 (300 MHz, 2.5 Giga-sample) digital oscilloscope over a 100 second
time interval. The averaging of the LIF signal is set equal to the data collection interval of
the oscilloscope to avoid biasing errors. The experimentally acquired spectrum is
compared to a numerically calculated spectra (Bessler, Private Communication) to
calibrate both the laser dial wavelength and estimate the laser spectral width. Since the
absorption lines at this low pressure are mostly Doppler broadened (∆ν = 0.16 cm-1) and
fairly narrow, the de-convolution of the excitation spectrum with the absorption lineshape
yields the spectral line width of the laser. During the course of this study, the spectral
37
position of the dye laser system shifted less than 0.001 nm per day with no change in the
average spectral lineshape, but tuning was still performed daily.
The beam is formed into a 140 µm x 45 mm sheet using the same –25 mm
cylindrical and 120 mm spherical lens combination. The NO fluorescence is captured by
an intensified CCD camera (Roper Scientific IMAX 512-T, 512 x 512 array with 20 µm
pixels) coupled to a UV Nikkor lens (105 mm, f/4). The exposure of the CCD array is
gated to 400 nsec to capture the NO fluorescence (characteristic time of ~ 100 nsec at the
shock tunnel conditions) plus the time associated with the combined temporal jitter of the
laser pulse and camera triggering circuits (80 nsec). The gate begins 100 nsec before the
arrival of the LIF signal to allow uniform charge distribution on the micro-channel plate.
A 1 mm Schott UG-5 filter placed before the camera blocks any elastic scatter, but allows
fluorescence from the (0,1) through (0,7) bands of NO into the camera (Figure 2.9). The
camera is interfaced to a Pentium II computer using a ST-133 controller with
programmable timing generator (PTG) built in. Linearity of this camera response with
different intensifier gain settings has been previously measured (Ben-Yakar, 2001).
(b) Driver Section and Control Panels (c) Driven Section and Dead Mass
Figure 2.1 Shock tunnel schematic (a) showing all sections and support structures. Driver section is shown at the left along with the driven section at the
center of the schematic. (b) Photograph of shock tube driver section, driven section and control panels. (c) Photograph of end of shock tube driven section and dead mass, pressure ports can be seen on the top of the tube.
(a)
(a) Mixing Section with wall removed and Dump Tank
Figure 2.2 (a) Photograph of Mixing Test Section with side wall removed to show nozzle, splitter plate, low-speed injection, and internal guidewalls. (b) Schematic of Mixing Section with wall removed to show the location of pressure taps and optical access along with flow dimensions.
Side 2 Injection50 15 20 25cm10
120 cm
End of Shock Tube
(b) Schematic of Mixing Section with wall removed
40
(a)
Vs
Plane Mirror
Parabolic Mirror
InSb Detector
Slit
Band Pass Filter
Sapphire Window
(b)
0.30
0.25
0.20
0.15
0.10
0.05
0.00
CO2 S
igna
l (A
rb.)
2.01.51.00.50.0-0.5-1.0Time (msec)
70
60
50
40
30
20
10
0
P 2 (ps
ia)
CO2 Signal Pressure at same location
Figure 2.3 (a) CO2 Imaging setup for detection of hot test gas in the shock tunnel. Design of system
is modeled after Flower, 1976. (b) Sample infrared emission trace for 1% carbon dioxide in methane from 4.1 to 4.7 µm, shock heated to 930 K and 3.8 atm.
30cm f/6 mirrors
Horizontal Knife Edge
Long Duration Light Source20 µsec/50 µsec/200 µsec
High Framing Rate Camera
Imacon 4688 independent ICCD arrays
Image Gate (10 nsec - 1 msec)
Figure 2.4 Schlieren imaging setup showing the Xenon spark gap light source, large parabolic focusing mirrors, the horizontal knife edge, and the high framing rate camera. Inset photo shows the actual Imacon 468 camera (approx 0.5 m in length).
Shock Tunnel Flow
80 mJ @ 266 nm
Spectra Physics GCR-3Nd:Yag Laser
Interline CCD Camera1300 x 1030 array2 µsec gate of first exposure
-25mm Cyl
120 mm Sph
4th Harmonic
Figure 2.5 Acetone PLIF imaging setup. All reflectors shown are optimized for 266 nm high power laser radiation
incident at 45º. Fused silica sheet forming optics are mounted to a vertical rail (not shown). Only side view imaging camera and laser position is shown. For end and plan view imaging, laser sheet is brought into the test section from the side window.
43
τ = -300 msec
τ ~ -200 msec
DD Vent Pulse
Output
A
B
Wait for firing signal
Output
A
B
10 Hz Clock
10 Hz ClockOutput
B
A
Output
B Delayed Transducer Signal
Flashlamp Pulse
A
τ ~ -206 µsec
τ ~ 100 msec
Figure 2.6 Laser timing diagram for synchronization of laser firing with shock tunnel firing in order
to minimize temporal jitter and shot-to-shot power fluctuations from the laser.
Shock Tunnel Flow
0.2 mJ @ 226 nm
Roper ScientificIntensified CCDCamera (512 x512)
Lumonics YM-1200Nd:Yag Laser
HD-500Dye Laser
HT-1000SHG
-25 mm Cyl
3mm UG5 filterHamamatsuR166UHPMT
PD
BS
HR
HR
HR
HR
HR
PD
LIF Cell
BS
1mm UG5 filter
120 mm Sph
105 mm, f/4UV Nikkor lens
Figure 2.7 NO PLIF imaging setup using for this thesis. HR = High reflector, optimized for input wavelength and 45º reflection; BS = Beam splitter, fused
silica glass slides giving 8% reflection; PD = Photodiode, fitted with a phosphor mask to make it sensitive to UV radiation; PMT = Photomultiplier Tube; Cyl = Cylindrical lens for beam expansion; Sph = Spherical lens for laser sheet collimation and focusing.
45
10
8
6
4
2
0
LIF
Inte
nsity
(Arb
.)
225.87225.86225.85225.84225.83Vacuum wavelength (nm)
R1+
Q21
(12.
5)
Q1+P 21(18.5)
S 21(7
.5)
R2(1
7.5)
LIF Cell Data NO LIF Simulation (Bessler and Sick, 2001)
Figure 2.8 Excitation spectra of several rotational lines near the A2
Σ+←X2
Π½ (0,0) Q1 + P21 (18.5)
transition in the γ band of NO compared with experimentally achieved excitation spectra. A Gaussian laser spectral profile of 0.5 cm-1 is convolved with the molecular lineshape (0.2% NO, 99.8% Argon at 296 K and 0.1 atm) in the simulated spectra.
1.0
0.8
0.6
0.4
0.2
0.0
LIF
sign
al (A
rb.)
300280260240220Vacuum Wavelength (nm)
(0,0)
(0,1) (0,2)
(0,3)
(0,4)
(0,5)(0,6)
Q1 + P21 (18.5) fluorescence 1mm UG5 filter
Figure 2.9 LIF spectra from pumping the A2Σ
+←X2
Π½ (0,0) Q1 + P21 (18.5) transition of nitric
oxide and utilizing a broad-band collection scheme. Fluorescence from the (0,1) to (0,7) bands is collected, but the elastic scatter and signal from the (0,0) band is highly attenuated by the UG5 filter.
46
CHAPTER 3
MIXING LAYER TEST CONDITIONS
When performing compressible mixing experiments with a test facility which is
coupled to a high pressure shock tunnel, careful examination of both the temporal
dynamics of the shock tunnel and the startup and attainment of steady flow conditions in
the test section is fundamental to creating useful experimental flow conditions. In this
chapter, issues pertaining to the creation of large, steady test-times in the shock tunnel are
discussed for their utility in creating high Mach number nozzle flows for mixing
experiments. While details of the shock tunnel facility and the mixing layer test section
have been covered in chapter two, the underlying principles behind their design and
operation are discussed in this chapter. First, the principles of creating accurate X-T
diagrams, which track the pertinent shock and rarefaction waves in shock tunnels, are
discussed. An overview of both shock attenuation and contact surface acceleration is
provided and used to predict the test-time and variation of the stagnation conditions in the
reflected portion of the shock tunnel. This information is then combined with
measurements of free stream conditions in the test section to forecast accurate mixing
layer test conditions, including nozzle back-pressure, required mass flow rates, and wall
deflections.
3.1 X-T DIAGRAMS FOR SHOCK TUNNELS
The equations governing the ideal one-dimensional wave propagation and the
development of steady flow test-time in the reflected region of shock tunnels are well
known (Liepmann and Roshko, 1957). For ideally propagating shocks and rarefactions,
the equations of motion, initially, before any reflections or interactions, can be written
simply: wave front positions are merely the propagation velocities multiplied by time
(Equation 3.1a-d). The complication occurs as these waves interact with the boundaries
47
of the shock tunnel, causing either shock reflections/transmissions or the self-interaction
of expansion fans as they reflect. These flow relations are still straightforward to
calculate using the appropriate boundary conditions or the method of characteristics. The
equations for the X-T paths for non-simple interactions are shown in Equations 3.1e-f.
44
11
_
4_
11
_
1_
211
12
11
2
cM
Max
tax
tM
Max
taMx
sstailrare
headrare
sssurfcont
sshockinc
−
+
−
−=
−=
−
−=
=
γ
γ
γ
( ) ( ) tutaa
ua
uLx
taLtaLx
shockrefl
reflheadrare
22
5.02
2
22
2
22_
4
413
4
4
4
44__
14
14
11
12
11 4
4
+
+
++
+−=
−+
−
+−=
+
−
γγ
γγ
γ γ
γ
For simple regions of the flow (where characteristics are straight lines and do not cross),
the waves are assumed to travel at constant velocities (though at real shock tube
conditions, waves are constantly accelerating or decelerating as they propagate).
When the reflected shock reaches the contact surface, the conditions across the
contact surface (constant pressure and velocity) must be maintained. These boundary
conditions are met by the generation of a re-reflected and transmitted shock pair, which
propagate back into the stagnation conditions and the driver gas, respectively; see
example ideal X-T diagram (Figure 3.1). The solution for these two generated shocks is
created by imposing the contact surface boundary conditions and solving the associated
shock jump conditions. Following the region-numbering conventions in Figure 3.1, the
solutions for the re-reflected shock and transmitted shock strengths are shown in
Equation 3.2. A53 denotes the ratio of the speeds of sound in region 5 (the stagnation
condition) and region 3 (the post-contact surface region of driver gas), P53 is the pressure
ratio across the reflected shock, and P65 is the pressure ratio across the wave which is
reflected towards the endwall. Depending on the sound speed ratio across the contact
(3.1a) (3.1b) (3.1c) (3.1d)
(3.1e) (3.1f)
48
surface, the re-emitted wave, which travels back towards the stagnation region, will be
either a shock or rarefaction wave following the selection criterion of Equation 3.3. The
transmitted wave is always a compression wave. Furthermore, if both sides of equation
3.3 are equal, then no wave is re-reflected towards the endwall. This is termed the
“tailored interface” condition, as it applies to the instance where the induced pressure and
velocity by the reflected shock is the same on either side of the contact surface, i.e. there
is effectively no barrier “seen” by the reflected shock and thus no reflection from that
boundary occurs. This situation is exactly analogous to electromagnetic waves that
interact with a step change in resistance, or vibration waves in solids with a change in
local propagation speed, causing a reflection of energy from that boundary. This occurs
for exactly one shock Mach number for each driven gas choice (e.g. helium driving argon
has a tailored interface Mach number of 3.75). Equation 3.2 is solved for the reflected
wave pressure ratio using a secant root finding method, appropriate for non-linear
algebraic equations.
( )( )
( )( )
( )( )165
65
153
5323
46553
6553
1
4
53 1
11
µ
µµβ
β
+
−
+
−+
+
−
−
=
PP
PPA
PPPP
A Shock Branch
( )( )
( )( )
[ ]
1
1
11
2
21
65
153
53231
46553
65534
53
22
−
−
+
−
−
+
−
=
−
γ
µ
β
µ
β
γ
γ
P
PPA
PPPP
A Rarefaction Branch
( ) 11 and
121 where
+
−
=
+
=
i
ii
iii µ
γ
γ
γγβ
Branchn Rarefactio
BranchShock
1
4
453
15323
1
4
453
15323
+
+<
+
+≥
β
β
µ
µ
β
β
µ
µ
PPA
PPA
(3.3)
Using these relations outlined in Equations 3.1-3.3, an elementary shock tube X-T
diagram can be calculated using the initial conditions of the shock tube and the incident
shock speed. An example of such a diagram is shown in Figure 3.2. The rarefaction head
(3.2)
49
and tail are shown initially, but after the rarefaction head reflects from the driver endwall,
only the head is shown, as it is the first to reach the stagnation region at the end of the
driven tube. The rarefaction displays a curved trajectory in the driver section caused by
the non-simple interaction of the rarefaction head with the remaining rarefaction
characteristics. For this simple scheme, there has been no attempt to model shock wave
attenuation or contact surface acceleration due to boundary layer growth. Also, since the
shock tunnel in this experiment has an area change between the driver and the driven
sections, the effect of this ratio on the shock wave strength and the propagation of the
rarefaction wave must be taken into account. Using the variables calculated for the
creating of the wave diagram, a simple formula for the test-time can be generated,
( )
+
+−
+=
rs
rs
rrer VUVVV
VVLtimetest
2_1 111 (3.4)
where Vs is the velocity of the incident shock, Vr is the velocity of the reflected shock, U2
is the drift velocity imposed after the incident shock, and Vre_r is the velocity of the re-
reflected wave that propagates away from the contact surface towards the endwall.
Equation 3.4 applies for high temperature shocks where the test time is limited by the
interaction of the reflected shock with the contact surface. For shocks where the test time
is limited by the rarefaction wave, Equation 3.5 applies.
( )
++
++
+=
r
srr
VcutcVVL
cVtimetest
22
231
5
211 (3.5)
where ( )12
1
4
34
4
42
4
4
21
1−
−
−−=
γ
γ
γ
au
aLt
The determining factor for whether an experimental test time is limited by the rarefaction
or by a re-reflected shock is whether the rarefaction overtakes the contact surface before
the reflected shock interacts with the it. Equation 3.6 shows the inequality that exists for
( ) ( )
r
sr
r
sr
VcutcVVL
VuVVL
++
++
>
+
+
22
231
2
1 211 (3.6)
rarefaction-limited shocks. Again, these equations may be used to obtain a quick
approximation of the test time, but the estimate is typically much higher than actual
experimental test times due to viscous effects.
50
Despite its limitations, this type of simple wave diagram is fundamental when
designing a new shock tube facility. The utility of being able to simultaneously simulate
both the shock tube thermodynamic conditions (in both the post-incident and post-
reflection shock regions) and the associated ideal test times at each part of the tube is
critical for the correct sizing of the tube (L4/L1) for the temperature and pressure ranges
of interest. The implementation of wave tracking in shock tube flows allows for a very
simple tool to predict, with some accuracy, the fundamental parameters of the tube
design. More accurate predictions of test time (< 10% error) require more complete
modeling of the wave behavior and experimental correlations for two-dimensional and
viscous effects.
3.1.1 SHOCK TUBE PERFORMANCE AND SHOCK WAVE ATTENUATION A common measure of shock tube performance is the shock strength (Ms)
achieved for a given driver pressure. An efficient shock tube will have a driver-to-driven
pressure ratio (P4/P1) that is close to the ideal prediction. Figure 3.3 shows the
experimental shock Mach number versus diaphragm pressure ratio for the shock tunnel
using a variety of test gases. Also shown is the ideal pressure ratio calculation assuming
equal driver and driven diameter (A4/A1 = 1) and the actual driver-to-driven area ratio of
(A4/A1 = 2.42, using the theory of Alpher and White, 1957). The incident shock Mach
numbers correspond to values extrapolated to the endwall of the tube using the shock
speed measurement technique described in Chapter 2. The diaphragm pressure ratios
required to achieve a particular Mach number are almost exactly those predicted by the
equal area theory for shock tubes, but the data are consistently above the actual area ratio
curve. Consequently, for this shock tunnel, the large area ratio counteracts the non-ideal
effects of diaphragm shatter, finite diaphragm opening time, finite shock formation time,
boundary layer growth and other viscous dissipative mechanisms, and heat conduction
from the shock interface. The P4/P1 data is correlated as a function of incident Mach
number in Equation 3.7.
819.3
1
4 573.0 sMPP
= for argon shocks (3.7)
51
Another collapse of the data is possible when the pressure ratio is normalized by its ideal
prediction, which takes into account the variation in speed of sound and compressibility
of different driven mixtures, Figure 3.4. Using this technique, a prediction may be made
for various driven mixtures. Equation 3.8 shows the linear fit for all the shock results in
the study.
7.123.0
14
14
+−=
s
theor
actual M
PP
PP
(3.8)
However, non-idealities of shock wave attenuation and shot-to-shot differences in
diaphragm breakage between different gas mixtures cause this data collapse to be less
effective than the single power law for each test gas mixture.
One of the largest non-idealities when dealing with confined traveling shock
waves is the attenuation of the shock wave due to viscous wall friction effects as a
boundary layer is initiated just behind the shock interface. Shock attenuation also
depends on diaphragm breakage, opening time, and shock formation distance. The
opening time and formation distance of shock waves has been found to scale inversely
with diaphragm pressure difference (P4-P1) (Rothkopf and Low, 1976) and directly with
the driven diameter (Ikui et al., 1969). The opening times are typically on the order of
several hundred microseconds. The shocks tend to accelerate in the first 10 – 30
diameters of the shock tube as the individual compression waves coalesce into a shock
front, and then decelerate as the boundary layer growth becomes increasingly important
(White, 1958). Thus, for the shock speed measurement location used in this study, all the
shocks examined are decelerating due to viscous effects. Though the shock formation
distance and diaphragm opening time are not measured for each shot, the above
description qualitatively agrees with the observed variation in incident shock attenuation.
Figure 3.5 shows the axial Mach number profile in the shock tunnel for an incident shock
Mach number of 4.06 in argon. A linear fit is shown through the measured data, which is
consistent with viscous shock attenuation models (Mirels, 1963; Petersen, 2001).
A more confusing result, however, is generated when the attenuation values are
plotted against the viscous scaling proposed by Mirels (1955) as in Figure 3.6. Here,
almost no correlation is evident as wide bands of attenuation values are present at a single
52
incident Mach number. This range of values is due only to shot-to-shot variations in the
attenuation and not to variation of another flow field variable. In fact, the three vertical
bands that are clear represent over 80 shocks each at a particular shock tube condition
(test gas composition and diaphragm pressure ratio), used for repeated imaging results
presented in Chapters 4-6. The lack of any correlation from Figure 3.6 shows that viscous
effects are not the driving force behind shock wave attenuation (as was noted by Petersen
(2001) at much higher Reynolds numbers). It appears that only ~ 20% of the
experimental attenuation can be attributed to viscous effects, while the other 80% varies
from shot-to-shot due to the variability of diaphragm opening times and resulting shock
formation distances.
This large variation in shock wave attenuation values fortunately does not
translate into large shot-to-shot uncertainties of the shock Mach number at the endwall..
Figure 3.7 shows the deviation of incident shock speed for approximately eighty shots of
a Ms = 4.14 test condition (helium driving argon). The standard deviation of the shock
velocity at the endwall is ±15 m/sec, which translates into a T5 uncertainly of 1% and a P5
uncertainty of 3%, which is reasonable for flow field repeatability in this study. Also, the
variation of the incident shock speed and attenuation does not significantly alter the
estimates of test time, causing a deviation in test time of less than 5% for the range of
attenuations seen experimentally. Consequently, estimations of test time can be made
accurately if the shock speed at the endwall is known, even though there is a wide range
of possible shock attenuation values at that location.
3.1.2 CONTACT SURFACE ACCELERATION
When operating the shock tube at high reflected temperatures (T5), the test time is
limited not by the arrival of the reflected rarefaction at the endwall but by the reflection
of the reflected shock off the contact surface. This is referred to as contact surface limited
test time. The shock wave reflects from the endwall and further raises the pressure in the
stagnation conditions, effectively ending the constant pressure test-time. Thus, it is
fundamental to the precise prediction of test time to have accurate knowledge of the
location of the contact surface with respect to the incident shock wave. This distance is
53
known as the hot test-gas length and can be predicted by coupling the effects of shock
wave attenuation with boundary layer growth behind the shock. The growth of the
boundary layer has an associated mass flow rate taking hot test gas from the hot test-gas
length and bleeding it into the boundary layer. Figure 3.8 shows the boundary layer
velocity profile in shock fixed coordinates. The boundary layer model shown here is
based on the original laminar theory proposed by Mirels (1966). Experimental
verification has been established in several low-pressure shock tube studies (Duff, 1955;
Davies and Wilson, 1969).
Figure 3.9 shows a comparison of the Mirels’ theory with a simple estimation of
test-gas length using no viscous effects with actual hot-test gas lengths found in the shock
tunnel. The Mirels formulation (and its implicit laminar boundary layer assumption)
match the experimental data well, while the simple estimation shows large scatter and
does not appear to correctly scale the experimental data. The test-gas lengths are taken
from CO2 emission data, multiplying the emission signal duration by the local shock
induced velocity (τCO2*u2). The emission signal is recorded at the first set of incident
windows, 1.2 meters from the endwall and 8 meters from the diaphragm location. The
“simple estimation” shown in Figure 3.9 is the inviscid one-dimensional estimate of the
test gas length at the diagnostic location. The experimental data are taken from shocks
with argon and nitrogen used as the test gas, over a Mach number range of 2.9 to 4.1. A
simple linear relation therefore exists between the Mirels’ solution and the actual
experimental data. This correlation is simple to measure (~10 – 15 shocks) and allows for
the very accurate computation of test time for contact surface-limited cases.
An example of an X-T diagram calculated using the above correlations for
viscous effects (shock wave attenuation and contact surface acceleration) is shown in
Figure 3.10. A further detail included in this calculation is the area ratio between the
driver and driven sections of the shock tube. The rarefaction wave is initiated in a smaller
diameter tube (the driven tube) than the one into which it propagates. Thus, two
rarefaction fans are needed to describe the flow in the driver section. Using the theory of
Alpher and White (1957), the diaphragm location is assumed to be the location of a sonic
throat where the Mach number is set to unity if the induced velocity in the driver gas is
supersonic. In this case, a rarefaction wave propagates into the driver, setting the gas in
54
motion subsonically. Then, at the diaphragm location, the flow is accelerated to sonic
conditions by the area change and then to supersonic speed by a rarefaction located just
downstream of the diaphragm. For subsonic values of the induced velocity (u3<c3), there
is only a single rarefaction wave, which propagates into the driver. The existence of the
area change in the driver tends to slow the rarefaction wave as it propagates backward
from the diaphragm, which leads to longer test times for rarefaction limited shocks.
The calculated test time for the condition shown in Figure 3.10 agrees very well
with the experimentally achieved test time of 1.4 msec as shown in a pressure trace taken
5 cm from the endwall, Figure 3.11 Further evaluation of the predictive capabilities of the
viscous X-T diagram calculator is shown in Figure 3.12. Here, the above relations have
been used to predict the test times of several different cases run in the shock tunnel. An
excellent correlation is seen for both rarefaction and contact surface limited shocks. Since
test time is a fundamental parameter for shock tube design and experiment planning, such
a predictive tool is invaluable.
3.1.3 CORRECTION FOR NOZZLE MASS FLOW RATE
While all the above relations are relevant for both shock tube and shock tunnel
calculations, the coupling of a nozzle to a shock tube (thus creating a shock tunnel) adds
the mass flow rate through the nozzle as a perturbation to the stagnation conditions in the
reflected portion of the shock tube. This can be modeled by a simple control volume
analysis using an expanding control volume which contains all the fluid processed by the
reflected shock that remains in the shock tunnel, as in Figure 3.13. For this control
volume, the motion of the reflected shock to the left adds mass to the control volume at
the local stagnation conditions (P5(x) and T5(x)), while flow through the supersonic
nozzle removes mass from the control volume. From the solution of the one-dimensional
equations of mass and energy, an estimate for the reduction in the stagnation pressure and
temperature can be reached for the shock tunnel. Equations 3.9 and 3.10 state the
conservation of mass and energy for the control volume shown in Figure 3.13.
( ) Mass 15 tmtAVtV nozzr ∆−∆= �ρρ (3.9)
( ) Energy 515 thmteAVetV nozzr ∆−∆= �ρρ (3.10)
55
Here the conditions created by the reflected shock wave are assumed to be constant (i.e.
the reflected wave is not attenuating as it propagates upstream). This assumption is used
in order to arrive at a closed form solution, but a more accurate estimate of the conditions
behind in the reflected region would take into account the attenuation of the reflected
shock and potential non-uniformities in the region into which it propagates. Solving
equations 3.9 and 3.10 under the assumption of constant reflected velocity (V(t) =
VrA1∆t) and calorically perfect gas leads to
( )TPP
βγ 55 1+= (3.11)
( )( )T
TTT
β
βγ
+
+
=
11 55 (3.12)
where ( )
−
+
+=
121
55
5
1
5 5
5
12* γ
γ
γ
γβ
RVARA
r
(3.13)
In this set of equations, P5 and T5 are the ideal stagnation conditions due to the constant
reflected shock wave speed (Vr). A* is the area at the nozzle throat, and A1 is the cross-
sectional area of the shock tube. T and P are the stagnation conditions taking into account
the nozzle mass flow rate. The ratios in Equations 3.11 and 3.12 are always greater than
unity, coincident with the removal of mass and energy from the stagnation conditions by
the nozzle flow.
Unfortunately, solution for the fixed reflected shock speed is not representative of
the actual loss of stagnation temperature and pressure in the reflected region of the shock
tunnel. The attenuation of the reflected shock is quite large (>5%/m) which causes the
assumption of a constant velocity shock to break down. Thus, Equations 3.9 and 3.10
must be rewritten as time integrals.
( ) ( ) ( ) ( ) ( )( ) Mass
1
∫∫ ∫ ∂−=
timetest
onozzrr
timetest
o
t
o
ttmAtVMdVt �ρττρ (3.14)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) Energy
1
∫∫ ∫ ∂−=
timetest
onozzrrr
timetest
o
t
o
tthtmAtVMeMdVtet �ρττρ (3.15)
56
and can be solved numerically if the attenuation of the reflected shock is prescribed. An
example of a pressure trace of a M = 3.45 shock is shown in Figure 3.14 along with the
estimates of the pressure in the stagnation region of the tube. This more complete control
volume analysis attains a value for the change in stagnation pressure due to the nozzle
mass flow that is more consistent with the pressure trace than the simpler model.
However, the more complex model does not take into account the deformation of the
reflected shock due to the presence of the nozzle throat, which can cause further non-
uniformities in the stagnation flow. Palmer et al. (1992) showed that a starting vortex
propagates into the stagnation region after the reflected shock due to the impulsive start
to the nozzle, and significant shock curvature just after reflection is caused by a
swallowing of a small part of the incident shock wave by the nozzle throat. The
associated relaxation of the curved shock front into a planar reflected shock can also
generate non-uniform stagnation conditions.
Despite this potential complication, the complex and simple models both do a
reasonable job predicting the loss of stagnation pressure and temperature in the shock
tunnel due to nozzle flow. Even the simpler model contains most of the correct flow field
physics, and this model’s accuracy is increased if an adjustable constant is multiplied to
the factor β in Equation 3.13, as it contains most of the correct physics of the flow field.
This adjustable constant is always greater than one, effectively decreasing the average
speed of the reflected shock, and is related to the typical attenuations seen in the shock
tunnel. For test gases that are primarily argon, this constant should be 2.2, while for
nitrogen diluents, the constant is 4.5. The reason for the difference between the constants
for different carrier gases lies in the fact that bifurcated shock structures which strongly
attenuate reflected shock waves are more prevalent in diatomic gases than monatomic
ones (Mark, 1958). These shock structures initiate at the wall in the boundary layer and
propagate obliquely into the stagnation region. Some of the fluid in the stagnation region
is thus processed not by the planar oblique shock, but by the double shock foot system of
the bifurcated shock (Davies and Wilson, 1969). This can lead to further non-uniformities
in the reflected shock region and entropy mixing. Observed minor deviations of the
experimental signal from the simple control volume formulation are likely due to flow
perturbations caused by bifurcated shocks and other viscous effects.
57
3.1.4 SETUP OF STEADY NOZZLE FLOW TIME
The establishment of steady flow at the nozzle exit is dependent on two separate
events. First, the incident shock from the shock tube is swallowed and propagates to the
end of the nozzle, then an unsteady expansion fan propagates back through the nozzle to
set up a steady supersonic flow. The time associated with these two events is easily
calculated. The shock simply propagates at the incident speed from inside the shock tube
to the nozzle. Then the associated nozzle flow time can be calculated using the
formulation of Dunn (1967) where the starting process is due to a set of backward-
running expansion characteristics.
Nozzle Start time = ∫−
L
o
dxxaxu )()(
1 (3.16)
This estimation of nozzle start time agrees very well with Schlieren images of the startup
process and with pressure traces inside the mixing test section (Appendix B). The growth
of boundary layers inside the nozzle did not seem to greatly affect the estimation of the
nozzle start time because the displacement thickness associated with the boundary layer
was not sufficient to significantly alter the operating Mach number of the nozzle.
3.2 PREDICTION OF MIXING LAYER CONDITIONS
For the steady growth of zero-pressure gradient mixing layers, several external
flow field variables must be correctly set before the firing of the shock tunnel to ensure
well-behaved experimental conditions. For the short test times available in shock tunnels,
there exists insufficient time for pressure and/or mass flow rate conditions to equilibrate
as they are allowed to do in blowdown wind tunnels. Instead, variables such as test
section and nozzle back-pressure, side-two mass flow rate, and wall deflection are
adjusted so as to allow the development of accurate shear layer conditions.
58
3.2.1 BACK-PRESSURE The single most important flow field variable that must be set prior to the firing of
the shock tunnel is the test section pressure, which also is the effective back-pressure of
the supersonic nozzle. As is known from all textbooks on compressible flow, expansion
and compression waves will result from mis-matched nozzle back-pressures, launched
from the edge of the nozzle, which in this test configuration would be the splitter plate.
The issue is further complicated by the growth of a mixing layer, which begins at the
splitter tip. This shear layer effectively acts as a wedge in the flow field, an oblique
obstruction with lower velocities than the nozzle exit. These concerns make the matter of
setting the correct back-pressure important for every convective Mach number
condition/density ratio in these experiments.
For blowdown wind tunnels, the nozzle back-pressure is initially atmospheric and
then allowed to decrease as an equilibrium is reached between the two mass flow rates of
the high and low speed stream and the pressure in the test section. Urban (1999)
documents the tradeoff in blowdown tunnels between the static test section pressure and
the high-speed mass flow rate. The nozzle exit pressure increases linearly with high-
speed mass flow rate, but the test section static pressure is inversely proportional to the
high-speed mass flow rate due to the supersonic ejector effect (Dutton et al., 1992). Also,
the low-speed side flow tends to increase the test section pressure, albeit slowly in this
experimental work due to the short test times and the low side-two flow rates. So there
exists an equilibrium for each test condition between the two mass flow rates and the
pressure of the test section. Again, blowdown wind tunnels have the luxury of long test
times where mass flow rates are set and the test section pressure is allowed to relax to a
sub-atmospheric level over the course of 1-2 seconds due to the interplay of the mass
flow rate and ejector effects on the test section pressure.
For shock tunnels, the solution to the test section pressure problem requires a
number of experiments at constant convective Mach number but variable test section
pressure in order to eliminate the existence of strong planar waves at the nozzle exit. If
the nozzle back-pressure is too high, shocks will be launched at the splitter tip, while an
expansion fan will occur for pressures too low. The same supersonic ejector effect is at
59
work in the high compressibility mixing layer, but additionally the finite thickness of the
mixing layer, acting as a wedge in supersonic flow, can cause large pressure disturbances.
Schlieren images showing the effects of mis-matched back-pressure on the mixing layer
flow field are shown in Appendix B. Shocks (bright line emanating from the splitter tip at
the left of the images) or rarefaction waves (dark regions within and angular range, again
coming from the splitter tip) are seen when exact pressure matching conditions are not
met. The goal of trial and error is to minimize the strength of this launched wave, or
eliminate it altogether.
Some modeling of the appropriate test section pressure is possible, if the mixing
layer is considered a solid wedge in the flow. In order to eliminate an attached oblique
shock on such a solid wedge, the layer should be rotated so that the upper face is aligned
with the flow direction. Using this as a model, the appropriate test section pressure for
highly compressible shear layers is hypothesized to be that pressure which is consistent
with expanding the Prandtl-Meyer flow angle by the actual size of the mixing layer,
Equation 3.17, where ν is the Prandtl-Meyer angle of the flow.
xP nozzletiontesttiontest
∂
∂+=∝
δνν
21
sec sec (3.17)
Schematically, this is shown in Figure 3.15. This allows for any waves which might have
been launched by the existence of a wedge in the flow to be eliminated due to the turning
of the flow angle by Prandtl-Meyer expansion. In the last images of Figure B.1, no
expansion fan is seen emanating from the splitter tip, consistent with a turning of the flow
angle. Consequently, there is no unsteady expansion anchored at the nozzle exit, but
rather the existence of a zero-pressure gradient mixing layer with no attached waves at
the splitter tip. The correlation of this hypothesis with the actual test section pressures
used is shown in Figure 3.16. The wedge angles are taken directly from experimental
visual growth rate data. There is assumed to be no effect of the low-speed mass flow rate,
though the entrainment appetite of the shear layer is must be satisfied (another operating
parameter, described in the following section). For this correlation, the rotation angle (ν)
is assumed to be the full growth angle of the shear layer (dδ/dx), as it more nearly
recovers the experimentally set back-pressure. The correlation has a slope near, but not
equal, to unity suggesting that the rotational angle must be a larger fraction of the
60
experimental growth rate. However, because of the reasonable agreement and good
correlation with experimental observations, this model does capture most of the processes
in establishing proper mixing layers without requiring long settling times.
3.2.2 SIDE-TWO MASS FLOW RATE As described in the previous section, the flow rate of the lower speed stream is
also a crucial variable that must be correctly set for each test condition in order for a
well-behaved mixing layer to develop within the test time. If the mass flow rate is
insufficient, the entrainment appetite of the layer will not be satisfied. This will cause the
layer to deflect into the low speed stream seeking more fluid to entrain and will also
cause the evacuation of the test section due to the ejector effect. If the mass flow rate is
too high, the test section may be over-pressurized prior to the arrival of the test gas in the
test section, causing oblique shocks to be launched from the splitter plate. Many studies
have examined the effects of adverse pressure gradient and planar shock impingement on
the mixing layer. These non-ideal effects are typically generated by pressure mismatching
the shear layer, or increasing the low-speed flow rate above the correct value. Thus, along
with the overall test section pressure, the low-speed side mass flow rate is an important
control parameter of the compressible mixing layer.
Unfortunately, a simple theory for predicting the required mass for mixing layer
entrainment does not exist for high compressibility conditions. Dimotakis (1986)
proposed a volumetric entrainment mechanism based on the motion of large-scale
structures seen at low compressibilities. This model correlates well with the
experimentally required mass flow rates below Mc = 0.5. However, once the three-
dimensionality of large-scale structures becomes significant, the assumptions in the
model break down and its predictive capabilities are lost. Thus, for this study, the
required mass flow rate for each test condition is determined by trial and error. Test
conditions are first optimized for the correct back-pressure. If the shear layer does not
cause the launching of a wave from the splitter plate, but does not grow parallel to the
side walls, this is likely the cause of insufficient side-two flow rate, as in Figure B.6. In
all cases of low flow rate, the mixing layer dives into the lower speed stream. However,
61
no adverse affects are discovered when the mass flow rate was increased above the level
necessary to satisfy the entrainment appetite of the shear layer. Consequently, the time
that the side-two flow rate is running before the test time (~ 300 msec) is not sufficient to
raise the test section pressure enough to cause an effect on the flow, for the range of mass
flow rates allowed by the injection tank scheme described in Chapter 2. Thus, the
experimental method used in this study involves setting the back-pressure to minimize
the wave structures launched from the splitter tip, and then the increasing of the side-two
mass flow rate until the mid-line of the shear layer is just parallel to the upper wall. This
methodology usually requires two to three shots to achieve the optimum back-pressure
and then one to two more to achieve the correct mass flow rate, if it was initially too low.
3.2.3 WALL DEFLECTION The upper and lower guidewalls inside the test section can be adjusted to various
small angles with respect to the flow. This is normally done to minimize streamwise
pressure gradients, which occur due to the growth of the mixing layer in a finite size test
section. For all the runs presented in this thesis, the upper wall remains horizontal, while
the lower wall is diverged by approximately 0.5 degree. Comparisons of upper and lower
wall streamwise pressure traces for flows with and without wall deflection are shown in
Figure 3.17. The upper plots show the growth of a small pressure gradient during the test
time of the run. The pressure transducers are 20 cm apart on both the upper and lower
walls. The divergence of the lower wall effectively removes the pressure gradient and
allows for optimum mixing layer growth conditions. The pressure gradient does not vary
significantly with increasing convective Mach number; thus, the positions of the upper
and lower walls are maintained for all the runs.
62
Figure 3.1 Enlarged schematic of a shock/contact surface interaction zone on an X-T diagram,
labeling regions of the flow with constant properties consistent with the shock tube
literature.
Figure 3.2 Theoretical X-T diagram for a Ms =5.0, P1 = 0.24 atm, T1 = 297 K shock tube condition
with helium driving argon. Calculated using simple model with no viscous effects.
1
2
3 5
67
Incident Shock
Contact Surface Reflected Shock
x
t
63
P41 = 0.573 Ms3.819
R2 = 0.9756
0
50
100
150
200
250
300
350
400
450
2.0 3.0 4.0 5.0
Ms
Dia
phra
gm P
ress
ure
Ratio
(P41
)
Argon Shock Data
Correlation (Equation 3.7)
T heoretical,
T heoretical,A4/A1 = 1A4/A1 = 2.42
Figure 3.3 Shock Mach number versus pressure ratio for argon as the test gas.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
2.0 2.5 3.0 3.5 4.0 4.5 5.0Ms
P41_
act
/ P41
_the
o
All Shock Data
Linear Fit P41_norm = -0.23Ms+1.7
Figure 3.4 Shock Mach number vs. reduced pressure ratio taking into account variation in the test
gas composition between shock tube runs.
64
1300
1305
1310
1315
1320
1325
1330
1335
1340
7.8 8 8.2 8.4 8.6 8.8 9 9.2 9.4
Distance from Diaphragm (m)
Shoc
k Ve
loci
ty (m
/sec
)
DataLinear Fit
Figure 3.5 Incident shock velocity profile for a Ms = 4.06, T5 = 3650 K, P5 = 30.7 atm shock with
argon as the test gas.
0
1
2
3
4
1.5 2 2.5 3 3.5 4(M s
0.5)(P1-0.14)
Att
enua
tion
(%/m
)
All Shock Data
Figure 3.6 Linear attenuation results for many shocks versus the purely viscous scaling proposed by
Mirels, 1955.
65
0
5
10
15
20
25
1305 1310 1315 1320 1325 1330 1335 1340 1345 1350
Shock Velocity (Vs ) m/sec
Freq
uenc
yσ = 12 m/sec
1.7% variation for 95% confidence
Figure 3.7 Variation in shock speed of 80 different shots for the same P4/P1 ratio and the same test
gas composition. Gaussian fit with a 12 m/sec standard deviation is fit to the histogram
data.
u =u -ue s 2
us
ue
-vboundary
layery
x u =uw e
incidentshock
shock fixed coordinates
Figure 3.8 Incident shock induced boundary layer schematic showing the relative location of the
incident shock and the growing boundary layer along with the velocity surplus near the
wall in shock fixed coordinates.
66
y = 0.7053x + 0.2682
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.5 0.6 0.7 0.8 0.9Actual test gas length (m)
Estim
ated
test
gas
leng
th (m
)
Mirels' TheorySimple EstimateLinear Correlation
Figure 3.9 Comparison of experimental data for hot test gas length as recorded for several shock
strengths using CO2 emission with a simple inviscid model and the laminar viscous
model proposed by Mirels, 1966.
Figure 3.10 Typical X-T diagram for Ms = 3.41, T1 = 297 K, P1 = 0.25 atm, helium driving argon.
Calculated using the viscous formulation with correlations for shock attenuation and
contact surface acceleration.
67
250
200
150
100
50
0
Pres
sure
(psi
a)
4x10-3 3210-1Time (sec)
Arrival of Expansion WaveTest Time
Experimental Theoretical
Figure 3.11 Pressure trace of the test condition shown schematically in Figure 3.10. Ms = 3.41, T1 =
297 K, P1 = 0.25 atm. Pressure transducer is located 5.7 cm from the endwall. Rounding
of pressure trace is due to startup of the supersonic nozzle and reflected shock/boundary
layer effects. Experimental test time is accurately predicted within 5%.
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Act
ual T
est T
ime
(mse
c)
3.02.52.01.51.00.50.0Computed Test Time (msec)
Shock Tube Data y = x
Figure 3.12 Comparison of some test times with predicted values using calculations of test time from
the viscous model.
68
Vr
A*mdot
P ,T5 5
reflected shock
controlvolume
x
Figure 3.13 Schematic of deforming control volume used to calculate the variation in the stagnation
properties behind the reflection due to the nozzle mass flow rate.
500
400
300
200
100
0
Pres
sure
(psi
a)
4x10-3 3210-1time (sec)
Experimental Theoretical w/ attenuating shocks Theoretical w/ no nozzle
Figure 3.14 Solution of control volume system of equations with reflected shock attenuation as
compared to a real pressure trace. Mi = 3.45, helium driving argon. Control volume
analysis captures all of the reduction of P5 due to the nozzle mass flow rate and some of
the pressure rise effects when incident shock attenuation is considered. Further pressure
rise in the stagnation conditions is due to two-dimensional shock/boundary layer
interaction effects.
69
U1
U2
hypothesized location
desired location
rotation
Figure 3.15 Schematic of the method of predicting required operating back-pressure by rotating the
mixing layer (“wedge”) down to propagate parallel to the splitter plate by reducing the
pressure in the test section below the ideally expanded back-pressure.
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
Calcuated Nozzle Back Pressure (psia)
Act
ual N
ozzl
e B
ack
Pres
sure
(psi
a)
Experimental Data
y = 0.89x
Figure 3.16 Comparison of the rotation method of predicting back-pressure conditions for supersonic
mixing layer experiments to actual data. For these calculations the rotation angle was
assumed to be ν = dδ/dx. Since the slope of the linear fit is not unity, the assumed
rotation angle needs to be increased slightly.
70
6
4
2
0
Pres
sure
(psi
a)
2.5x10-3 2.01.51.00.50.0-0.5time (sec)
Image Location
0.4 psia
0.05 psia
Upstream, 0 degrees Downstream, 0 degrees
Upstream, -0.5 degrees Downstream, -0.5 degrees
Figure 3.17 Static pressure traces from runs with and without wall deflection, so as to see the ability
to control the possible streamwise pressure gradient. Upper set of curves from Mc = 2.8
condition with no wall deflection. Lower set from same condition with -0.5 degree
deflection on the lower wall. Pressure readings are from the upper wall, 2 cm (upstream)
and 22 cm (downstream) after the imaging region. Zero degree traces have been
vertically displaced 5 psia for clarity.
71
CHAPTER 4
SCHLIEREN IMAGING AND MIXING LAYER GROWTH RATE
This chapter contains Schlieren imaging results using the high-framing rate
camera for a range of convective Mach numbers (0.85 to 2.84) with a variety of density
ratios. These images are required both operationally, for the setting of a particular shock
tunnel condition to optimize the shear layer, and scientifically to examine the visual
growth rate and shock behavior of the mixing layer. The technique is also used to judge
the character of local large-scale structures, their interaction with the freestreams, and the
convection speeds of structures.
4.1 SCHLIEREN IMAGE CORRECTIONS AND SPECIFICS
Images are obtained in the first part of the test section, where x = 0 – 23 cm. The
knife-edge position is horizontal to sensitize the system to vertical density gradients, and
pushed into the focus such that the mixing layer appears dark and shock waves emanating
from the layer will appear light. Images are corrected for non-uniform illumination by
subtracting the acquired image from a flat-field image, taken just before the firing of the
shock tunnel. Since the Schlieren effect acts to intensify or weaken the illumination
locally due to density gradients, flat-field subtraction and not division is the more
appropriate data correction technique. It is assumed that differences in the illumination
(caused by small perturbations in the spark gap of the light source) are somewhat
repeatable. However, the corrected images display instances of variation in the
illumination from flat-field to signal images, which manifest themselves as areas of the
freestream that appear lighter than others when there is no density gradient present.
The images have also been inverted (negative) to show the mixing layer bright on
a dark background. Images taken with the knife-edge position reversed (coming into the
72
focus from the bottom) displayed less fidelity due to the limited dynamic range of the
camera’s intensifier arrays. The mixing layer, at the usual imaging sensitivity, was often
brighter than the maximum digitization value for the camera, causing the brightness
recorded by the camera to be clipped. This resulted in poor image quality. However,
images taken with the knife-edge coming into the top of the focus showed much better
fidelity as well as the ability to clearly view local shocks in the flow (which appear as
bright streaks). Of the images of the layer shown in this chapter, most have been inverted
to highlight the shear layer. For shock wave visualization, though, it is more effective to
view the un-inverted images, as the bright streaks are easily distinguished.
Some isolated dark regions inside the mixing layer are noticeable in the inverted
and corrected side view Schlieren images presented in this chapter. These are caused by
the pitting and scratching of the fused silica windows due to the collision of fast moving
diaphragm particles with the window surface. The window imperfections deflect the light
opposite to the density gradients in the shear layer. Thus, in the corrected and inverted
images, the window flaws will appear as small dark areas and are most noticeable when
they occur in the bright region of the mixing layer. Unfortunately, there was no available
way to correct for these regions of signal dropout without replacing the very expensive
fused silica viewing windows. Since the rate of generation of window imperfections was
much greater when metal diaphragms were used with the shock tunnel, later runs were
executed with plastic diaphragms. The plastic would tend to streak the windows with a
film as it collided with them instead of scratching or pitting them. Therefore, the problem
of window scratches and signal dropout did not worsen with time.
The resolution of the Schlieren image is set primarily by the focal length of the
imaging lens but also depends weakly on the distance of the imaging plane from the
focus (Kodak Schlieren Photography, 1977). The image resolution for the 105 mm lens is
500 µm x 500 µm x 10 cm (since Schlieren is a line-of-sight integrated technique) and
230 µm x 230 µm x 10 cm for the 200 mm lens. Since the Schlieren images are only used
for qualitative analysis of the scalar structures, this resolution is sufficient. The images
were gated to 250 nsec and spaced 4 or 8 µsec apart depending on the apparent structure
convection velocity. Consequently, the motion of the fastest velocity scales is
73
approximately one pixel during the gating time, and 16 to 32 pixels during the inter-
frame time.
4.2 SCHLIEREN IMAGING RESULTS 4.2.1 GROWTH RATE The mixing layer growth rates were determined from an ensemble average of
eight Schlieren images taken with either 4 or 8 µsec inter-frame time separations. The
visual growth rates were obtained by fitting lines to the edges of these time-averaged
Schlieren images. Growth rates were determined in the region of x = 6 to 23 cm for the
full test section images (taken with the 105 mm lens) to allow the effects of initial
development of the shear layer and splitter tip to diminish (Figure 4.1). Growth rates
from the magnified images (taken with the 200 mm lens) were obtained in the region of x
= 12 to 23 cm. Although the procedure is somewhat subjective, repeated measurements
using different nozzles and low-speed side gas composition give low scatter. Growth rate
data from Mc = 0.85 to 2.25 were taken using a Mach 3.3 nozzle with nitrogen on the fast
side and from Mc = 1.07 – 2.84 using a Mach 5.2 nozzle with argon as the high-speed
gas.
Uncertainty in the convective Mach number is minimized by careful measurement
of the pressures and Mach numbers in each section of the shock tunnel. Shot-to-shot
variations in the incident Mach number in the shock tunnel can vary the nozzle stagnation
pressure by as much as 3% and the stagnation temperature by 2%. However, since the
Mach number of each nozzle is precisely calibrated using Pitot probe and Mach angle
data, the uncertainty of the convective Mach number can be reduced to 1% for most
cases.
The growth rate data normalized by the corresponding incompressible growth rate
is presented in Figure 4.2. The data have been convective Mach number-averaged to
include experimental realizations of similar Mc but different density ratios. Over ninety
experimental realizations were obtained for this study. Up to a convective Mach number
of 1.5, the shock tunnel data are consistent with both previous blowdown wind tunnel
data and the computational results of Day et al. (1998). From Mc = 1.5 to 2.84, the
74
growth rate curve asymptotes to a constant value of δ/δinc = 0.18. Linear stability results
predict a decrease in the central mode growth rate along with a corresponding increase in
the growth rate of outer modes. The experimental results show a general decrease of
growth rate up to Mc = 2.0, but do not show any strong evidence of an increasing trend
for Mc > 2.5.
Using the previous linear stability results as a guide, we would expect to see some
variation in the growth rates with density ratio at high convective Mach number. Co-layer
modes from the linear stability analysis have different growth rates, which depend on the
density ratio. The slow modes should be enhanced for layers with Mc > 2, s > 1, and the
fast modes should dominate for layers with Mc > 2, s < 1. The presence of shocks
emanating from the shear layer is also density ratio dependent, as will be discussed in
Section 4.3.1. Accordingly, the growth rate data is presented again in Figure 4.3.
However, the data have not been convective Mach number-averaged, and points with
differing density ratios have been separated. There is no apparent density ratio bias to the
growth rate trend. This suggests that the growth rate mechanism of the shear layer is
unaffected by local shocks emanating from the layer and that if co-layer modes are in
existence, that the varying density ratio is not altering the mode-dependent growth rates
in this convective Mach number range.
4.2.2 INSTANTANEOUS IMAGES
Shown in Figure 4.4 and Figure 4.5 are instantaneous side-view Schlieren images
of high compressibility mixing layers taken in the convective Mach number range of 0.85
to 2.84. Each has been corrected for non-uniform illumination and inverted to show the
mixing layer as a bright region. Both the convective Mach number and density ratio are
noted at the bottom of each image, and the images are shown in order of increasing
convective Mach number. Since the conditions for these shear layers are created with
different shock tunnel conditions, supersonic nozzles, and low-speed gases, conditions
with the same convective Mach number but different density ratios are possible.
75
4.2.3 LARGE-SCALE STRUCTURE
A wide variety of instantaneous mixing layer states are seen in Figures 4.4 and
4.5. For the lower convective Mach numbers examined in this study (up to Mc = 1.40),
the layers are very uniform and show almost no evidence of two-dimensional large-scale
structures. The lowest convective Mach number images (Mc = 0.85) exhibit some
inclined bands, which are evident near the end of the viewing window. These banded
structures do not appear to be roller-like or coherent as in lower convective Mach number
studies. However, similar indistinct structures and bands have been observed in other
experiments such as those of Goebel and Dutton (1991) and Clemens and Mungal (1995).
As the convective Mach number is increased above Mc = 0.85, the shear layer
remains uniform with very little Mach wave radiation in the upper free stream. However,
above Mc = 1.4 there are two distinct visual forms that the mixing layer can take. When
the density ratio is below ~ 0.8, the layer maintains its streamwise uniform appearance
with little free stream disturbances. However, for all convective Mach number conditions
with s > 0.8, significant Mach wave radiation can be seen in the upper free stream. Since
shocks appear as bright streaks in the raw images, they will appear as dark and light
bands in these corrected and inverted images. No images above Mc = 1 show any
evidence of two-dimensional large-scale structure.
4.3 SCHLIEREN IMAGE INTERPRETATION 4.3.1 ACOUSTIC FIELD OBSERVATIONS
Shown in Figure 4.6 is a comparison of several experimental runs involving
different combinations of convective Mach number and density ratio. For the low
compressibility, low-density ratio case (upper left hand corner, Mc = 1.4, s = 0.5), the
upper free stream contains a pattern of weak Mach waves in the upper stream. The same
is true for the highly compressible Mc = 2.24 case shown to the right. No strong shocks
emanating from the layer are in evidence. Hall et al. (1993) observed a similar pattern of
Mach waves in the low-speed free stream of an Mc = 0.96 shear layer. He reasoned their
76
strength was due to the weak disturbances radiating from the shear layer, manifesting
themselves in the side-wall pressure traces.
However, when the convective Mach number is increased above 1.6 and the
density ratio is also high, strong shocks emanating from the shear layer at a variety of
angles are seen (Figure 4.5) in this study. These shocks are not the result of pressure
disturbances, emanating either from the splitter tip or the nozzle, reflecting between the
upper wall of the test section and the shear layer. Instead, they are caused by the local
structure convection velocity decreasing to the point where the upper stream is supersonic
with respect to the moving large-scale structures. This is shown schematically in Figure
4.7a. If the structures do not protrude too far into the free stream, they will form a wavy
interface of slower convecting fluid. The velocity difference between the free stream and
the convection velocity can become greater than the local speed of sound causing the
flowfield to resemble a wavy wall in supersonic flow, a venerable example of linearized
supersonic flow described in many compressible flow textbooks. However, if the
structures have a large streamwise component and protrude significantly into the free
stream, then the linearized assumption no longer holds. Individual structures now act as
bluff bodies in a supersonic flow, with curved bow shocks attached to their leading edge
(as shown in Figure 4.7b).
The interaction of the wave and large-scale structure is most evident in the Mc =
1.71 case, shown in false color in Figure 4.6 to enhance the contrast. Here the inverted
image (which shows the mixing layer as bright) is overlain with the raw image, which
highlights the shock waves. Clearly, there are curved shocks that can be associated with
particular structures. Also, the large-scale structures appear to have a preferred
orientation with respect to the upper free stream, suggesting that the shock waves interact
with the structures to elongate them in the streamwise direction. The existence of these
local shocks appears to be associated with a certain range of both the convective Mach
number and the density ratio.
The density ratio, recalling Equation 1.6, sets the ideal convection velocity for the
mixing layer. Though it has been widely shown that structures do not convect at exactly
this speed (Fourgette et al., 1991; Papamoschou and Bunyajitradulya, 1997), the basic
scaling is important (Uc ~ U1/(1+s½) for low velocity ratio mixing layers). For low
77
density ratios (s < 1), the symmetric convection velocity is nearer the fast-side velocity.
Thus, we would expect that lower density ratio mixing layers would tend to lack strong
acoustic fields, due to the similarity of the free stream and convection velocities.
Unfortunately, real mixing layer convection velocities do not follow the symmetric
formula and attempts have been made to make empirical correlations for the convective
velocity (Papamoschou and Bunyajitradulya, 1997).
For the symmetric convection velocity, the compressible scaling would dictate
that any mixing layer with a convective Mach number above one would radiate shock
waves into the free stream because, by definition, the velocity difference between the free
stream and the convection velocity is greater than the local speed of sound. However, as
is clear from the image at Mc = 2.24 in Figure 4.5, it is possible to generate highly
compressible shear layers which do not emit shock waves even when the free stream is
traveling more than twice the speed of sound as compared to the symmetric convection
velocity. Clearly, the structures in this case are traveling with a speed nearer to the upper-
stream value.
By examining all the Schlieren images taken to determine at what conditions
strong shocks are present, a basic correlation between density ratio and the existence of
strong waves can be made. For the experiments performed in this study, the experimental
conditions where s > 0.75 exhibited shock waves in the high speed stream, and those
below showed either very weak Mach radiation (as for Mc = 1.40, in Figure 4.6) or no
radiation at all (Mc < 1.17). The conditions are plotted in convective Mach
number/density ratio space, where the experiments have been divided into images with
and without waves (Figure 4.8). A similar plot created for the existence of Mach wave
radiation in the lower stream is shown in Figure 4.9. Here, no correlation in the parameter
space is noted, with the presence of waves apparent in both high and low density ratio
shear layers across the spectrum of convective Mach numbers. The existence or absence
of Mach wave radiation in the Schlieren images places bounds on the convection
velocity. Velocity bounds based on the Mach radiation schematics of Figure 4.7a,b will
be derived from these regime plots to bracket the large structure convection velocity in
Section 4.3.3.
78
4.3.2 SHEAR LAYER STABILITY
The stability of the shear layer is also closely connected with the density ratio. For
example, by comparing the instantaneous images for the Mc = 1.62-1.65 conditions with
the Mc = 1.71-1.72 conditions, we see a fundamental change in the character of the shear
layer with different density ratios but similar compressibilities. For the Mc = 1.62-1.65
images, the layer is relatively uniform, displaying some instances of local large-scale
structure (especially apparent at the edges of the shear layer). However, the Mc = 1.71
shear layer is highly disorganized with large broken structures throughout the layer. The
structures in these images also exhibit a preferred orientation with respect to the free
stream flow angle, with streamers of fluid noticeable in both the upper and lower streams.
What is remarkable about the Mc = 1.71, s = 1.56 condition is that the strong
instability associated with it is surprising. If the exact free stream used for the Mc = 1.71
case is allowed to mix with helium instead of air, the convective Mach number drops to
0.85 along with the density ratio to 0.23. This condition is shown in the upper left hand
corner of Figure 4.10, which exhibits a very smooth and uniform shear layer. Thus, the
instability is not caused by some unsteadiness from the shock tunnel or nozzle, but
simply due to the increase in the convective Mach number and density ratio. The
controlling parameters for this onset of instability appear to be a combination of both Mc
and density ratio.
If we examine all the Schlieren images from Figures 4.4 and 4.5, inspecting for
the overall stability of the shear layer, two relationships become clear. First, as shown
with the acoustic radiation, increasing density ratio appears to have a destabilizing effect
on the high compressibility shear layer. In most cases, the layer becomes less uniform
and exhibits rougher edges (caused by the interaction of the acoustic waves with the
scalar field) with higher density ratio. Second, increasing convective Mach number
appears to have a stabilizing effect on the layer. While the existence of local shocks is
independent of Mc, the strength and effect that the shocks have on the layer overall seem
to diminish as the compressibility is increased. Whether this is due to changes in the
convection velocity, a transformation to co-layer behavior and the appearance of multiple
convection velocities, or the decrease in interaction timescales between the pressure and
79
scalar fields is unclear. However, the behavior is most consistent with the last item, as
shown by Freund et al. (2000a) in an annular mixing layer, but without accurate
convection velocity measurements, the other possibilities cannot be ruled out.
4.3.3 CONVECTION VELOCITY
With multiple framing capabilities, a cross-correlation of two Schlieren images of
known time separation should yield the convection velocity of large-scale structures.
However, the large-scale structures must be discernable in both images in order to
provide an accurate estimate of the displacement. Of the images collected in this study,
very few displayed strong evidence of large-scale structure, due likely to the increased
three-dimensionality of the shear layer at these convective Mach numbers. Since
Schlieren is a line-of-sight integrated technique, high levels of three-dimensionality will
cause the signals of individual structures to be stacked on each other, causing a more
uniform image to occur.
For those images which did have some distinguishable structure, the correlation
peaks were far too broad to support a reasonable estimate of the structure displacement.
This broadening often occurs when the structure changes shape between images or the
signal strength from the structure varies between images. The latter is very likely in
Schlieren images of highly three-dimensional phenomena, where one element in the line-
of-sight may be “brighter” in one image and then fade in another. Thus, due to both its
difficulty in determining structures and poor correlations, the cross-correlation technique
for convection velocity measurements was not used.
Another promising method for measuring the structure speed is to measure the
local angle of the Mach wave radiation in the free stream. From Figure 4.7a, the local
Mach angle is directly related to the convection speed if the fast-stream conditions are
known. Unfortunately, there is a wide variation in the Mach angles in the free stream.
Figure 4.10 shows a magnified view of four different shear layer conditions, three of
which show considerable acoustic radiation. The main difficulty in finding the average
Mach angle is that due to the curved nature of the shocks in the high compressibility
mixing layers, the weak Mach wave limit is not necessarily reached for each wave. Thus,
80
significant bias errors when measuring the local angle lead to large standard deviations in
the measurement. For example the convection velocity as measured using the Mach wave
radiation technique for the Mc = 2.25 condition gives Uc = 450 ± 100 m/sec.
Consequently, accurate measurements of convection velocity are not possible using this
technique, even for a relatively well-behaved set of Mach waves.
Even though convection velocity measurements are difficult to obtain by
measuring local wave angles, upper and lower bounds can be placed on the structure
convection velocity, based on the appearance or absence of Mach waves in the free
stream. For Mach waves to reside in a particular free stream, the difference between the
convection velocity and the freestream has to be greater than the far field speed of sound.
Thus, the presence of Mach waves allows the placing of bounds on the convection
velocity. The boundaries derived from this assumption are shown in Table 4.1.
UPPER STREAM – NO WAVES UPPER STREAM – WAVES PRESENT Mc1< 1 or Uc > U1-a1 Mc1> 1 or Uc < U1-a1
LOWER STREAM – NO WAVES LOWER STREAM – WAVES PRESENT Mc2< 1 or Uc < U2+a2 Mc2> 1 or Uc > U2+a2
Table 4.1 Convection velocity bounds based on the existence of Mach wave radiation in both the
fast and slow streams.
Each Schlieren image was examined thoroughly for evidence of Mach waves
generated by the shear layer in either stream. Then, using the above formula, upper and
lower bounds on the convection velocity for each convective Mach number condition can
be found. In some cases, such as that of waves in the upper stream and none in the lower
stream, there is no lower bound revealed for the convective velocity as the inequalities
are of the same type. The results of this analysis are presented in Figure 4.11. Both the
upper and lower bounds are shown for each Mc value, joined with a vertical bar.
Two distinct groups of data are apparent: those which set both upper and lower
bounds on the convection velocity, and those which set only upper limits. A correlation
for the convection velocity as a function of convective Mach number is possible (thick
line) if the following assumption is made. If a single convective velocity exists for each
81
Mc, then the convective velocity limits dictate a rough correlation. This line lies between
lower bounds generated by the absence of waves in the lower stream for high density
ratio conditions and the upper bounds generated by the absence of waves in the upper
stream for low density ratio runs at the same convective Mach number. The correlation
strongly resembles the normalized growth rate curve shown in Figure 4.2, with the
convection speed decreasing to a constant value as the compressibility is increased. No
such correlation is possible if the same data are plotted versus density ratio, suggesting
that this convective velocity dependence is purely due to compressibility effects and not
the traditional mixing layer scaling (Equation 1.6). Also, the correlation of convection
velocity appears to improve as the compressibility is increased above Mc = 1.5.
Using this correlation for the actual convection speed, computation of the true
convective Mach numbers for both streams is possible. Results from this computation are
shown in Figure 4.12. This plot reflects that the high-speed convective Mach number is
always supersonic for shots above Mc,symm=1 and that Mc,2 typically hovers around unity
for all of the results shown. This pinning of the convection velocity such that the lower
stream is always just sonic with respect to the large-scale structures implies a velocity
preference for the low-speed stream that cannot cross the layer sonic line. Urban and
Mungal (2001) show that high gradient regions and sheared vortical structures in the
velocity field are strongly tied to the low-speed sonic line defined by the stagnation
temperature of the lower stream. The behavior here is consonant with those results as the
inferred convective speed is seemingly confined at the side-two sonic velocity.
However, for highly three-dimensional, high compressibility layers, only spatially
resolved techniques will yield truly accurate structure velocity information. Fluorescence
or scattering techniques from a seeded tracer species have been effective in creating
image pairs which yield cross-correlation peaks and estimates of structure displacement
(Papamoschou and Bunyajitradulya, 1997; Smith and Dutton, 1999). Nonetheless, the
results from the Mach wave bounding technique detailed above do agree fairly well with
these higher accuracy results in that the actual convective Mach number is quite different
from the symmetric formulas and the convection velocity tends to favor one of the
freestream speeds. However, this analysis strictly assumed a single convection velocity,
82
as multiple velocities would decouple the measurements (top and bottom waves) such
that no correlation could be derived.
4.3.4 SHOCKLETS
Small shocks associated with turbulent eddies, which have come to be known as
“eddy shocklets”, have been observed in numerical simulations of turbulent mixing flows
(Lele, 1989; Vreman et al., 1996; Freund et al., 2000a) and in a counter-flowing
supersonic shear layer (Papamoschou, 1995). Significant importance has been placed on
the role shocklets might play in redistributing kinetic energy through dilatational
dissipation (Zeman, 1990) and altering the structure convection velocity (Papamoschou,
1991). The high compressibility conditions in the shock tunnel mixing layer facility allow
for the investigation of potential shocklet behaviors at high convective Mach numbers,
where numerical analyses have shown they exist.
Figure 4.13 shows a time history of a shock structure that looks very much like
the eddy shocklet shown by Papamoschou (1995) in the only other experimental evidence
known to the author. The mixing layer conditions that accompany the image are Mc =
2.0, s = 1.28, which are comparable to those used by Papamoschou. In this time series, a
bright bow shock, seemingly embedded in the shear layer, convects downstream as it
ascends through the layer. The shock becomes less curved as it rises within the layer,
implying that the scalar structure to which it is attached is accelerating or its size is
decreasing. Note also that the structure associated with the shock is drawn into the upper
free stream as a result of the turning of the flow locally by the curved shock surface.
Whether this shock structure is indeed a shocklet contained within the layer, or
simply a bow shock structure attached to an eddy with a significant cross stream velocity,
is unclear. However, as has been discussed previously and will be shown concretely in
Chapters 5 and 6, the structures and shocks associated with the shear layer are highly
three-dimensional. Also, though shocks may appear imbedded within the layer in
Schlieren images, spatially resolved techniques will show that the shocks indeed exist
only on the edge of large structures which entrain large amounts of low-speed fluid.
Thus, the structures will appear in the line-of-sight integrated Schlieren technique to be
83
deep within the layer and rise to the upper stream. However, their actual structure is no
different from attached bow shocks, which radiate from the high-speed interface into the
upper free stream.
4.3.5 CO-LAYERS
As was described more fully in Chapter 1, high convective Mach number mixing
layers have been shown numerically to exhibit multiple convection velocities, with
turbulent eddies having a preference for a fast or slow velocity dependent on their
position within the layer. No concrete experimental evidence for co-layer behavior has to
date been seen in a compressible mixing layer, but detailed linear and parabolized
stability analyses (Sandham and Reynolds, 1991; Day, 1999) have detailed their
existence at very high levels of compressibility or heat release.
Shown in Figure 4.14 is an instantaneous side-view Schlieren image from the Mc
= 2.84, s = 1.38 shear layer condition. Shock waves are clearly visible in both the upper
and lower streams. This will occur, according to the same model used before, if the
structures are supersonic with respect to the lower stream as well as the upper stream.
Using the Mach wave radiation technique outlined above, the lower convection velocity
is estimated to be 500 ± 50 m/sec, while the upper convection velocity is 710 ± 90 m/sec.
Since the difference in the velocities is greater than the associated uncertainties with the
Mach wave radiation technique, it is likely that two separate convection velocities exist in
this mixing layer condition.
Unfortunately, no other mixing layer conditions examined in this study provided
shocks in both streams. Thus, no further measurements of dual convection velocities are
possible. If multiple convection speeds are prevalent in high compressibility mixing
layers, further experimental evidence must be sought in a higher convective Mach
number range (Mc > 3) by examining the dependence of the growth rate on the density
ratio, which is not possible using the shock tunnel driven mixing layer facility in its
current configuration.
84
a)
x = 0 – 23 cm
b)
x = 12-23 cm
Figure 4.1 Schematic showing the regions of mean mixing layer Schlieren images where growth rate
data was taken in order to avoid development effects and waves launched from the splitter tip.
85
1.2
1.0
0.8
0.6
0.4
0.2
0.0
δ/δ
inc
3.02.52.01.51.00.50.0Mc1
Papamoschou and Roshko (1988) Goebel and Dutton (1991) Elliot and Samimy (1990) Hall et al. (1993) Chinzei et al. (1986) Clemens and Mungal (1995) Wagner (1973) Naughton et al. (1997) Nuding (1996) Brummund and Mesnier (1999) Dimotakis Fit (1991) Central Mode, Day (1998) Fast Mode, Day (1998) Slow Mode, Day (1998) Mc averaged data
Figure 4.2 Current normalized growth rate data compared with previous experimental and
computational efforts. The data from this study has been convective Mach number averaged over multiple density ratios.
86
1.0
0.8
0.6
0.4
0.2
δ/δ
inc
3.02.52.01.51.00.50.0Mc1
Papamoschou and Roshko (1988) Goebel and Dutton (1992) Elliot and Samimy (1990) Hall and Dimotakis (1991) Chinzei et al. (1986) Clemens and Mungal (1995) Wagner (1973) Naughton et al. (1997) Central Mode, Day (1999) Current study, ρ
2/ρ
1 > 1
Current study, ρ2/ρ
1 < 1
Figure 4.3 Current normalized growth rate data compared with previous experimental and computational efforts. The data from this study has not been convective Mach number averaged over multiple density ratios to explore any bias effects due to the existence of fast or slow modes above Mc = 2.0.
87
Figure 4.4 Instantaneous Schlieren images taken from many different experiments in the convective
Mach number range of 0.85 to 1.91. The convective Mach number and density ratio for each experimental condition is shown at the bottom of that image.
88
Figure 4.5 Instantaneous Schlieren images taken from many different experiments in the convective
Mach number range of 1.93 to 2.84. The convective Mach number and density ratio for each experimental condition is shown at the bottom of that image.
89
Figure 4.6 Instantaneous Schlieren images that have not been inverted so as to highlight the acoustic radiation field in the upper free stream, which is caused by the interaction of the large scale structures and the supersonic flow. The lower right shows the inverted Schlieren signal from the mixing layer (yellow) and the non-inverted signal from the radiated waves (blue) so as to highlight the wave/structure interaction.
Free Stream
Mixing Layer
U1
Uc
Θ = sin-1(a1/(U1-Uc))
Uc
U1
Free Stream
Mixing Layer
CurvedBowShock
Figure 4.7 Schematics of the creation of shocks in compressible mixing layers due to slow moving structures, which protrude into the freestream. Case (a) shows a weak disturbance interface where regular Mach waves are radiated. Case (b) shows a strong disturbance where a locally curved shock is launched as the structure acts more like a bluff body in supersonic flow.
90
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5
Mc
dens
ity ra
tio (s
)
Waves Present
No Mach Waves
Figure 4.8 Regime plot of density ratio and convective Mach number showing the appearance of acoustic radiation in the fast stream for high compressibility shear layers.
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5
Mc
dens
ity ra
tio (s
) Waves Present
No Mach Waves
Figure 4.9 Regime plot of density ratio and convective Mach number showing the appearance of
acoustic radiation in the slow stream for high compressibility shear layers.
91
Figure 4.10 Magnified Schlieren images displaying the differing character of the acoustic radiation field with increasing convective Mach number.
92
1.0
0.8
0.6
0.4
0.2
0.0
Uc/U
1
3.02.52.01.51.00.5Mc (avg)
Ucmax
Ucmin
Uc (theorertical) Uc (approximate fit)
Figure 4.11 Normalized convection velocity plotted versus symmetric convective Mach number. Bounds (circles and triangles) on convection speed are derived based on appearance or lack of Mach wave radiation in either of the freestreams. Approximate fit is drawn between the maximum and minimum convection velocity bounds.
0
0.5
1
1.5
2
2.53
3.5
4
4.5
5
0 0.5 1 1.5 2 2.5 3 3.5 4
Mcsymm
Mc1,M
c2 McMc
M = 5.0
M = 3.0 Nozzle 1
2
Figure 4.12 Empirical convective Mach numbers plotted versus the symmetric convective Mach
number. Convection speeds are taken from the approximate fit from Figure 4.11.
93
Figure 4.13 Schlieren image time sequence of a “shocklet” type behavior in a Mc = 2.0 mixing layer.
Motion of the shock across the layer suggests that it is attached to a local eddy and not imbedded inside the layer.
Figure 4.14 Instantaneous side-view Schlieren image from the Mc = 2.84, s = 1.38 shear layer
condition showing shock waves visible in both the upper and lower streams. Wave angles from each side give rise to differing convection velocities, suggesting co-layer behavior.
94
CHAPTER 5
ACETONE PLIF IMAGING OF MIXING LAYERS
Acetone was first suggested as a flow tracer for these mixing layer experiments
due to its ease of handling, simple laser excitation set-up, and well known spectroscopy.
Also, little was known about its applicability in high-temperature supersonic flows,
especially due to its tendency to pyrolyze at even moderate shock tunnel stagnation
conditions. Thus, the first main PLIF imaging thrust of this study was to examine
spatially resolved images of large-scale structure in moderate compressibility conditions
and to investigate the use of acetone in shock tunnel flows.
5.1 ACETONE PLIF TEST CONDITIONS 5.1.1 ACETONE SEEDING
Acetone vapor is seeded to a mole fraction of 22% into the shock tunnel test gas
by the use of a pressurized acetone bubbler (Figure 5.1). The carrier gas was injected into
the bottom of an acetone bath in a 15 cm diameter, 1 m tall aluminum seeder through a
12 mm copper manifold tube with 32, 6 mm holes. The rows of holes are separated by 90
degrees on the top, bottom, and sides of the manifold tube, and groups are spaced 2 cm
apart. The manifold tube is then covered with 3 layers of 1 mm2 stainless steel mesh to
break up the large bubbles. This created smaller bubbles, which enhanced the seeding
capability. By keeping the bubble sizes small and allowing a large bubbling path (0.5 m)
through the acetone bath, vapor concentrations equal to the saturated vapor pressure of
acetone at room temperature (190 mTorr, Lozano et al., 1992) are attainable up to carrier
gas flow rates of up to 20 SLPM. These high flow rates are necessary to facilitate the
quick filling (~ 3 min) of the shock tunnel driven section, which would take a much
longer time ( > 20 min) using the simpler seeding strategies used in other acetone PLIF
experiments.
95
The quick seeding ability unfortunately causes the acetone bath to decrease in
temperature rapidly at the highest flow rates. To keep the bath temperature roughly
constant and in equilibrium with the vapor above it, a copper cooling circuit is inserted
into the liquid acetone running laboratory cooling water. The bath temperature was
continuously monitored using a sealed Type K thermocouple probe and was maintained
at 290 K during seeding. For the typical filling procedure, the seeder was run for 1 minute
to the room exhaust system at low flow rates to initiate bubbling and acetone flow in the
piping. Then the seeder was run for 2 minutes to a vacuum to allow thermal equilibrium
to be reached in the seeder and equilibrium to be reached in the flow metering valves and
pressure gages. Finally, the seeded gas is allowed to fill the driven section of the shock
tunnel. Since the seeder is designed for filling to a vacuum instead of to atmospheric
pressure, the choked orifices on the flow meters are positioned on the downstream side.
The pressure in the seeder and these flow control valves must be kept such that the flow
in the metering valves is always choked, even as the back-pressure (the pressure inside
the shock tunnel) is rising. This effectively sets the maximum seeding level possible for
the shock tunnel environment, as shown in Figure 5.2. For the experiments outlined in
this chapter, the seeder pressure was maintained at 17 psia to ensure choked output of the
flowmeters for the entire filling process of the shock tube.
Calibration of the seeder is performed by several simple methods. By examining
the ratio of the inlet and outlet mass flow rates to the seeder, an instantaneous measure of
the seeding fraction is known, as the downstream flowmeter reading will differ from the
upstream due to both the increased mass flow and the change in molecular weight of the
seeded gas. The seeder is calibrated in a time-averaged sense by measuring the total
volume of acetone lost per hour at a fixed flow rate. This is used as a check on the
instantaneous technique. Both yield seeding mole fractions that differ by less than 5%,
which is accurate enough for the purposes of this study. As a final check, the incident
shock pressures computed from the known shock speeds are compared to the actual post-
shock instantaneous pressures in the shock tube to discover the mixture properties (γ, R)
from basic normal shock relations. From these mixture properties, the mole fraction of
acetone is easily computed. Again, all three techniques offer measurement accuracy of 3-
5%, which is sufficient for this experiment. A more careful calibration of the seeder must
96
be performed for more accurate knowledge of the exact seeding fraction of acetone,
necessary for more fundamental photophysics experiments, but was not necessary for the
results attained in this thesis.
5.1.2 ACETONE PLIF IN SHOCK TUNNEL FLOWS
The test condition used for all the acetone PLIF experiments in this thesis is
generated by shock heating an acetone/air mixture (22 % acetone, 78% air) with an
incident shock wave, Ms = 3.7, creating the nozzle reservoir conditions listed in Table
5.1. The use of acetone as a tracer species in shock tunnel flows necessitates further
examination of the possible error sources that arise due to the high temperatures and
pressures present in supersonic flows.
Test Gas Mixture 22 ±1 % Acetone, balance Air
T5, P5 1130 K, 22.2 atm
T2, P2 775 K, 3.7 atm
τPyrolosis 38 msec (3.2% loss)
τtest time 1.4 msec
τnozzle start 800 µsec
Table 5.1 Test gas conditions in the stagnation region of the shock
tunnel for calculation of acetone pyrolysis.
5.1.3 HIGH TEMPERATURE ACETONE CHEMISTRY
For high temperatures (T > 1000 K), acetone can pyrolyze in a unimolecular
decomposition reaction and associated radical attack reactions.
CH3COCH3 → CH3CO + CH3 (5.1)
R + CH3COCH3 → RH + CH3COCH2 (5.2)
The general radical attack reaction is a hydrogen abstraction reaction where R represents
a radical species in the flow (O, OH, CH3, etc.). Since equation 5.1 is a unimolecular
reaction, the reaction rate will be a function of pressure as well as temperature. Ernst et
al. (1976) measured the pyrolysis rate of acetone in the fall-off pressure regime. They
97
reported rates for acetone decomposition between temperatures of 1350 and 1650 K for
number densities below 10-3 mol/cm3. Their measurements show that for number
densities of 3 x 10-4 mol/cm3, the pyrolysis reaction is in the fall-off pressure regime
where the reaction rate is a factor of three lower than the high pressure limit. The high
pressure limit reported by Ernst et al. (1976) agrees well with other reported
measurements in lower temperature ranges (Benson and O’Neal, 1970).
The products of the unimolecular reaction plus oxygen radicals generated in the
high stagnation enthalpy region of the shock tunnel will also interact with the acetone
molecule. A summary of the reaction rate coefficients for the acetone unimolecular
reaction and radical attack reactions is shown in Table 5.2. Since the stagnation
temperatures used in this study with acetone PLIF are below 1400 K, the generation of
free oxygen atoms in shock-heated air is not significant. However, due to the pyrolysis of
acetone, methyl radical generation will be appreciable. These methyl radicals will then be
oxidized by the well-known set of chemical reactions associated with methane
combustion, producing greater amounts of NO, H, OH, and O radicals that can then
attack acetone through the list of radical reactions shown in Table 5.2. The measured
reaction rates for the attack of acetone by radicals have all been made at temperatures
below 1000K, but extrapolation up to 1300 K is reasonable for the purposes of estimating
their effect on the overall acetone pyrolysis time constant.
An overall acetone pyrolysis reaction rate including the unimolecular
decomposition, radical attack, and finite rate chemistry between the decomposition
products and radical pool has been formulated by Prioris et al. (1999). The correlations
are based on 80 separate kinetic simulations of acetone combustion using the large
hydrocarbon kinetic simulation tool developed at Lawrence Livermore Labs (Curran et
al., 1991).
×= −−−
TP 000,37exp1018.4 216.0184.013χτ [msec] (5.3)
In Equation 5.3, χ is the seeding level in %; P is the overall gas pressure in atm; T is the
gas temperature in Kelvin. The correlation is valid for P= [1,30] atm, χ = [1,25] % and T
= [1000,1500] K. This lifetime result represents a refinement of the simple unimolecular
decomposition reaction rate quoted in Lozano et al. (1992) to include full finite rate
98
chemistry between the decomposition products. A complete review of the simulation
tool, mechanism, and pyrolysis time correlation procedure can be found in Prioris et. al,
(1999).
Reaction A Ea
(kcal/mol)
Ref
CH3COCH3 → CH3CO + CH3 [s-1] ↓
a) high-pressure limit 2.7 x 1016 81.6 Ernst. et al. (1976)
b) 20 atm estimate 8.5 x 1015 81.6 ↓
c) LLNL calculation (22 atm, 22 % acetone + air) 4.7 x 1015 73.2 Prioris et al. (1999)
R + CH3COCH3 → RH + CH3COCH2 [cm3/s] ↓
R = O 1.1 x 1013 5.96 Herron (1988)
R = CH3 4.0 x 1011 9.78 Kinsman and Roscoe (1994)
R = H 1.9 x 1013 6.36 Ambridge et al. (1976)
R = OH 1.0 x 1012 1.19 Atkinson et al.(1989)
Table 5.2 Rate coefficient information for fundamental acetone decomposition and radical attack
reactions which influence the rate of acetone pyrolysis at shock tube conditions.
5.1.4 APPLICABILITY OF ACETONE PLIF IN SHOCK TUNNELS
Using the shock tunnel conditions shown in Table 5.1 and Equation 5.3, the
acetone pyrolysis lifetime is approximately 20 msec in the stagnation conditions for the
acetone PLIF experiments. Since the shock tunnel test time is only 1.4 msec,
approximately 7% of the initially seeded acetone is lost to pyrolysis reactions if the
acetone remains in the reservoir state (Table 5.1) for the full test time. Above 1200K,
37% of the seeded acetone will decompose at this pressure, and at 1250K, 80% has
decomposed. Due to the high activation energy of the acetone unimolecular
decomposition reaction, the variation in pyrolysis timescales with temperature is quite
large, which effectively puts a stringent limitation on the high temperature applicability
of acetone in shock tunnel flows.
99
Since there are many products of acetone decomposition whose composition will
vary with shock tunnel residence time, reservoir conditions that are conducive to
significant pyrolysis will cause problems for image analysis. While spectroscopic
constants and quantum yields are well known for acetone, the potential decomposition
products (ketene, formaldehyde, etc.) will have shifted absorption features and different
fluorescence yields. These molecules would exist post the nozzle flow in the test section,
which is being imaged using PLIF; accordingly, their undesirable LIF signals would limit
the ability to quantitatively interpret the acetone PLIF images.
The pyrolysis also effectively limits the maximum Mach number that can be used
with supersonically seeded acetone PLIF. Figure 5.3 shows a summary of the acetone
pyrolysis equation (Eqn. 5.3) in graphical form. Also noted on the figure is the limiting
line of 15 msec, below which, significant pyrolysis ( > 10% removal) will occur during a
1.5 msec test time. Acetone is clearly not useful in shock tunnel flows above 1200 K for
reasonable stagnation pressures. However, compared to the use of acetone PLIF in
atmospheric, subsonic conditions (where 1000 K is the usual cutoff temperature,
representing a pyrolysis lifetime ~ 2 sec), a larger temperature range is permissible for
transient flow experiments. This allows rather surprisingly high Mach number flows to be
imaged in shock tunnels using acetone PLIF.
In order to estimate the maximum allowable Mach number for acetone PLIF in
shock tunnel flows, some boundary conditions in the nozzle and shock tunnel must be
specified. For the recommendations made in this thesis, there must be less than 10%
pyrolysis in the stagnation region and the free stream static temperature in the test section
must not dip below 295 K so as to not lose large amounts of acetone to condensation in
the expanded state. The maximum test section Mach number will then be a function of
the stagnation conditions (T5, P5) and the acetone seeding level. The maximum nozzle
Mach number versus acetone seeding fraction for several different stagnation
temperatures is shown in Figure 5.4. The Mach number is computed using the usual
isentropic relations (Liepmann and Roshko, 1957), taking into account the changes in γ
with variable amounts of acetone. The associated stagnation pressures, which allow for
reasonable PLIF signal to noise ratio (SNR > 5), are shown in Figure 5.5 for the same
conditions using the maximum allowable Mach number from Figure 5.4. Estimates for
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PLIF SNR are scaled from experimentally achieved SNRs in this study. Clearly, as either
the temperature or the seeding fraction is increased, the maximum allowable Mach
number will also increase; however, concomitantly, the pyrolysis time constant will be
decreasing as those flowfield variables are increased. Thus, the appropriate
recommendation from this study is that for acetone PLIF in shock tunnel flows with large
seeding levels, the stagnation temperature should not exceed than 1150 K and the
maximum nozzle Mach number allowable varies from 3.9 to 4.4 for seeding level of 5%
to 25%, respectively.
5.1.5 SEEDED STREAM SELECTION FOR MIXING LAYER IMAGING
Since seeding in the fast stream has been shown to be problematic, especially with
the removal of the tracer species at the high stagnation enthalpies associated, a reasonable
assumption might be that the seeding acetone in the lower stream would be simpler, yield
higher signal levels, and allow for straightforward shock tunnel flow analysis. The main
difficulties to seeding the slower stream are caused by the limitations of the low-speed
side injection system. First, the low-speed injector length is quite short, less than 25 cm,
which does not allow for the use of an atomizing spray nozzle to seed liquid acetone into
the flow or for droplet evaporation prior to the mixing layer (as was used for ethanol by
Clemens and Mungal, 1995). Because of this short injector length, the acetone must be
seeded prior to injection into the test section, i.e. in the side-two injection tank. In order
to satisfy the entrainment appetite of the shear layer, reasonably large mass flow rates are
required from the low-speed stream, forcing large injection pressures from the holding
tank (> 4 atm). At these injection pressures, the maximum acetone seeding level is less
than 5%. Though the lower velocity stream of the mixing layer is at a lower temperature,
and thus the signal from the acetone PLIF in this stream would be stronger than similar
seeding in the fast-side stream where the number density is lower, the maximum seeding
levels achievable in the low-speed stream result in a reduction of the PLIF SNR by a
factor of 2.5. For the low-pressure, variable temperature environment of compressible
mixing layers, this drop in signal is too great to allow for reasonable PLIF imaging.
101
Finally, if acetone were injected into the low-speed stream, the variation of the
convective Mach number would have to occur primarily by altering the shock tunnel
conditions or by changing the carrier gas used in the acetone seeder. These two methods
would have required significantly more calibration shocks per convective Mach number
condition and a more extensive calibration of the acetone bubbler. For the above reasons,
only imaging of acetone seeded in the fast stream is used in this study as it allows for the
examination of a reasonably large convective Mach number regime (0.85 – 1.71) with
large PLIF SNR and relatively simple flowfield calibration procedures.
5.2 PLIF MODELING FOR FLOWFIELD
As for any PLIF measurement, the ultimate goal is often to translate the
fluorescence signal from the flowfield into a representation of a meaningful flowfield
variable. For acetone PLIF, this is a challenge for non-isothermal mixing flowfields.
Thurber (1999) performed sensitive simultaneous imaging of both temperature and
mixture fraction in a heated co-flowing jet with a dual laser, single camera technique.
However, the high SNR of those experiments relied on a table-top experimental setup
which allowed for higher laser fluences along with higher overall pressures. Since the
supersonic mixing flowfields examined in the current experiment have high temperatures
coupled with lower pressures (thus low tracer species number densities) and more
complicated beam paths/optical access, ease of interpretation of the images must yield to
the more important problem of attaining images with high SNR.
The fluorescence signal from the broadband absorption feature of acetone, which
extends from 225-300 nm, in the weak excitation limit is given by Equation 5.4.
( )
= ∑
ii
acetonecoptf PTT
kTP
dVhc
ES χλφλσχ
ηλ
,,,, (5. 4)
Here, E is the laser fluence [J/cm2], (hc/λ) is the energy [J] of the photon at the excitation
wavelength λ, ηopt is the overall collection efficiency of the optical path, and dVc is the
collection volume [cm3]. The final two quantities are the absorption cross section of
acetone (σ) which is a function of wavelength and local temperature, and the
fluorescence quantum yield (φ) which is also a function of excitation wavelength and
102
local conditions (including potential collisional quenching partners). Thurber et al. (1999)
made a thorough study of the effects of wavelength, temperature, pressure, and local
quenching on acetone LIF, and these results will assist in the interpretations of the images
acquired in this study.
In order to consider the potential difficulties in image interpretation for
compressible mixing layers imaged with acetone fluorescence, the conditions of the
flowfield must be established. Table 3 shows the mixing layer conditions used in
conjunction with acetone PLIF. Note that the mixing layer conditions occur at
approximately one-third of atmospheric pressure and range in temperature from 295 to
~600K (when one considers the effects of local shocks). These flowfield conditions will
lead to absolute acetone LIF levels in the upper free stream which are ~ 12% of those at
standard conditions (295 K, 1 atm) with the same laser energy. So while simpler
experimental setups allow for higher laser fluences and number densities yielding larger
signals, supersonic flowfields generated in shock tunnel environments will generate
smaller LIF signals than table-top experiments using acetone PLIF. This demonstrates
that overall signal level is often the controlling parameter in the imaging of supersonic
flowfields, and that ease of image interpretability must be somewhat sacrificed.
In this study, acetone is excited by 266 nm laser radiation, because this is the
highest power laser excitation source available in the broadband absorption feature.
Acetone has a higher absorption cross-section at ~ 280 nm (Figure 5.6), but laser sources
at this wavelength are typically frequency-doubled dye lasers or based on sum frequency
mixing of a dye laser output, which provide significantly lower laser energies (< 20 mJ at
the laser source). Thus, the benefit of a higher absorption cross-section value is offset by
lower laser intensities. Because we expect that this experiment will suffer from low
acetone LIF signal levels, the strongest laser source available is employed, as the
difference between the absorption cross-sections for 266 and 280 nm has been shown to
be smaller than the relative power of the laser sources available at those wavelengths (80
mJ available at 266 nm at the laser source). The laser fluence used in this experiment is
approximately 37 MW/cm2, which is well below the saturation limit proposed in Lozano
(1992).
103
Mc avg 1.71 0.85
Mc1, Mc2 1.79, 1.65 0.93, 0.79
Ptest section [atm] 0.29 0.37
Gas1 22% Acetone, 78% Air 22% Acetone, 78% Air
Gas2 100% Air 100% Helium
T2, T1 [K] 564, 296 564, 295
V1, V2 [m/sec] 1290, 15.5 1290, 93
Mexit 1 3.0 3.0
M1, M2 3.25, 0.045 3.1, 0.09
s 1.568 0.228
r 0.012 0.073
21, mm �� 1.26, 0.034 (kg/sec) 1.26, 0.037 (kg/sec)
(δ)x = 18 cm [cm] 2.0 2.5
(Reδ) x = 18 cm 1,2 5.1 x 105, 4.8 x 105 5.9 x 105,7.4 x 104
(Rex) avg 4.4 x 106 9.5 x 105
(δ/∆V) x = 18 cm [µs] 16 21
2x/(U1-U2) [µs] 283 301
Table 5.3 Mixing layer test conditions for acetone PLIF imaging results
By choosing an excitation wavelength, the temperature dependence of the
absorption cross-section and fluorescence quantum yield is fixed. Using data collected by
Thurber et al. (1998), the solution to Equation 5.4 for a range of temperatures is shown in
Figure 5.7, where the fluorescence signal has been normalized by its value at 295 K.
Clearly a sizeable temperature dependence of the fluorescence signal will occur for this
excitation wavelength. Also shown in Figure 5.7 is the expected signal if the cross-
section and fluorescence quantum yield were constant with temperature, which would
force the LIF signal intensity to scale linearly with the local density. Signals with reduced
temperature dependencies are possible if the laser excitation wavelength is ~ 320 nm;
however, the PLIF signal levels will be prohibitively low as high power laser sources are
not available at that wavelength.
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For the mixing layer flowfields generated in this study, the pressure is nominally
constant across the layer (except in the proximity of local shock waves). Thus the
normalized signal intensity should scale as
( )
( ) ( )( )( )
( )( )( )( ) H
H
f
f
TT
TT
TT
SS
χφ
χηφ
σ
ησ
ηη
η
11
1 ,1=
=
(5.5)
where χH is the local mole fraction of high-speed fluid (which is seeded with acetone)
and η is the normalized cross-stream coordinate. While Equation 5.5 is a complicated
function of the local temperature, it can be reduced to an equation dependent on only the
local mole fraction of high-speed fluid if the temperature also acts as a conserved scalar
in the flow field (e.g. T(η) = χH(η)T1 + (1-χH(η) )T2). The results of Fiedler (1974), Batt
(1977), and Konrad (1977) show that temperature and mixture fraction are well
correlated throughout the mixing layer at low compressibility conditions, with both acting
as conserved scalars. These studies were all carried out at much lower compressibility
levels, Mc < 0.05, than the current work, and the result should not be extended to high
convection Mach numbers. For highly compressible shear layers, temperature does not
act as a local conserved scalar, as high levels of aerodynamic heating are possible as
upper free stream fluid is decelerated to the convection velocity. Consequently, the local
mixed temperature is a function of both the local scalar and velocity fields, which show
decreased levels of correlation as compressibility is increased (Urban and Mungal, 2001).
Therefore, the LIF signal is a non-linear function of the local high-speed mole
fraction, conditioned by the local temperature field. The signal levels from the layer will
tend to increase with decreasing temperature, so regions of mixed fluid nearer the low-
speed stream temperature will provide a larger signal than similar regions at the same
mixture fraction nearer the high-speed temperature. This results in a low-speed side bias
to the images, and structures mixed at low mixture fraction levels (which will be closer to
the low-speed side of the shear layer) should be enhanced. Therefore, no mixture fraction
profiles or statistics will be compiled for the acetone PLIF results as there are significant
bias errors near the low-speed stream. However, the overall character of the large scale
structures and the acoustic radiation field are weakly affected by the bias errors, and thus
the acetone PLIF diagnostic is a good choice for visualizing the behavior of large-scale
105
structure in supersonic mixing flowfields as it allows for simple seeding strategies and
the potential for high SNR.
5.3 IMAGING OF THE MC = 0.85 TEST CONDITION
5.3.1 SCHLIEREN
An instantaneous side view Schlieren image of the lower convective Mach
number case examined using acetone PLIF is shown in Figure 5.8. The imaged area is the
full window length of 23 cm, beginning at the splitter tip. There is little discernable large-
scale structure evident in the mixing layer using this line-of-sight integrated technique.
Brown-Roshko type rollers are not readily identifiable; however, some darker bands,
which are apparent towards the right-hand side of the image, suggest some large-scale
structure is possibly contained inside the layer and is obscured by smaller scale turbulent
eddies to which the Schlieren technique is more sensitive. These banded structures do not
appear to be either roller-like or coherent as seen in incompressible mixing layers.
Similar indistinct structures have been observed in experiments by Clemens and
Mungal (1995), Papamoschou and Roshko (1988), and Hall et al. (1993). Some evidence
of weak streamwise structures is frequently prevalent in the upper free stream, especially
towards the end of the frame, as the local Reynolds number increases (Figure 5.8). The
streamwise vortical structures and the visual bands that are present in the layer all appear
to have a similar angle of inclination with respect to the mean flow of approximately 25
degrees. This shows a preferred streamwise orientation that is consistent with the
transitions from two-dimensional to three-dimensional structures seen previously
(Clemens and Mungal, 1992) and the increased role of streamwise structures (Freund et
al., 2000a). As the layer becomes more compressible, the roller/braid structure breaks
down and the large scale rollers are greatly stretched in the streamwise direction, causing
the appearance of inclined structures.
Comments on the dimensionality of the structures are difficult to extrapolate from
Schlieren images due to the spatial integration of the imaging technique. Only structures
that are strictly two-dimensional will appear so in Schlieren images, as three-
106
dimensionality tends to mask structure information as the signal from individual
structures begin to stack, muddying the signal. Also, the sensitivity and contrast of the
Schlieren technique (i.e. how far the knife edge is pushed into the focus of the light)
affect the quality of the images, especially which density gradients appear the brightest in
the flow. Small-scale turbulence can completely mask the gradients of the large-scale
structures and conceal coherent structures from visualization. Thus, spatially resolved
techniques are fundamental to judge the character and evolution of large-scale structures.
5.3.2 ACETONE PLIF, SIDE VIEW
Shown in Figure 5.9 is a set of single-shot acetone PLIF images taken at 18 cm
downstream of the splitter plate, where x ranges from 16 to 20.5 cm and y from -2.0 to
2.0 cm. All side view images presented in this thesis were made through the centerline of
the test section (z = 0 cm), no systematic differences were seen for images taken at (z =
1.5 cm). The images in Figure 5.9 are from separate firings of the shock tunnel and the
flow is from left to right. The images are corrected for instantaneous laser sheet non-
uniformities by using the cross-stream averaged streamwise PLIF signal in the uniformly
seeded fast stream. Localized bright spots are likely plastic diaphragm particles, which
absorb the laser radiation at 266 nm and then fluoresce in the visible part of the spectrum.
These hot spots do not affect the dynamic range of the measurement or the SNR as they
did not saturate the CCD array.
Some structure is evident within the images, but a mostly jagged interface is seen
between the high-speed side and the mixing layer, which is consistent with previous
studies (Clemens and Mungal, 1995; Papamoschou and Bunyajitradulya, 1997). The
intermittency of the images shows that large-scale structures are still important, but they
appear highly disorganized and three-dimensional. The mixing layer is composed of large
amounts of mixed fluid but there is limited evidence of larger rollers and thin braid
regions, which are common in low compressibility mixing layers. However, a spatially
resolved planar imaging technique tends to highlight two-dimensionality while de-
emphasizing three-dimensional structures; thus, plan and end views are necessary to
comment on the apparent dimensionality of the large scale structures.
107
For passive scalar visualization techniques, the low-speed interface is emphasized
and the false color table is used to attempt to distinguish the high-speed interface. A
simple grayscale color table makes image interpretation on the high-speed side of the
layer more difficult. Product formation methods are typically superior for visualizing
shear layers due to the natural contrast on both the high- and low-speed sides of the layer.
However, the passive scalar tends to emphasize the intrusion of free-stream fluid into the
layer, an aspect which will become more useful as the compressibility is increased. The
images of Figure 5.9 show many instances of “streamers and jets” of mixed fluid which
protrude into each of the free streams, past the average mixing layer boundaries.
The regions internal to the mixing layer appear to be fairly well mixed (i.e. islands
of pure fluid are not often found internal to the mixing layer). Also, few regions of pure
low-speed fluid near the high-speed interface are found in these images, suggesting that
while some small streamwise structures are in evidence, they do not contribute strongly
to the entrainment mechanism of this shear layer. Thus, the entrainment mechanism of
this layer is still predominantly controlled by large roller structures, though they are more
three-dimensional in nature than lower convective Mach number conditions (Clemens
and Mungal, 1995). Entrainment by strictly two-dimensional structures tends to lead to
structures with a linear or “ramped” behavior in the streamwise direction (Fiedler, 1974;
Mungal et al. 1985), while relatively uniform in the transverse direction. The ramps are
due to the large-scale structure concentration that is biased toward the side from which it
entrains fluid. The side view images shown in Figure 5.9 are more consistent with a mix
of both streamwise ramped and streamwise uniform structures. Also, the cross-stream
direction is most certainly ramped in nature and not uniform as the two-dimensional
mixing model would suggest.
The bias errors cited previously would have likely generated a more cross-stream
uniform shear layer as the low speed side of the mixing layer would have enhanced signal
due to the temperature field. So the conclusion that the cross-stream profiles are ramped
is not weakened by the non-idealities of the imaging technique. Rather, the notion that
there are cross-stream ramps is buoyed by the fact that the imaging technique would tend
to smooth out the cross-stream profiles, yet the ramps are clearly apparent in the side
view images.
108
While the imaging region is not as large as other researchers have used in the past
with higher signal scattering techniques, some estimates of the length scales or observed
structures involved can be made. Though biased by the limited extent of the window, the
mean large-scale structure dimension in the streamwise direction appears to be
approximately twice the cross-stream extent of the structures (δ). Also, since no image
captures more than what appears to be a single large scale structure, the mean eddy
spacing must be greater than twice the visual thickness, which is higher than that
measured by Brown and Roshko (1974) but consistent with the results of Clemens and
Mungal (1995) and Goebel and Dutton (1991).
5.3.3 PLAN VIEW
Plan view imaging allows for the observation of a dominant disturbance angle of
the large-scale structure and the assessment of spanwise two-dimensionality. In previous,
less compressible work, this view demonstrated to researchers the transition to oblique
downstream propagation of the rollers at Mc > 0.6 (Sandham and Reynolds, 1990;
Fourguette et al., 1991). Plan views for the Mc = 0.8 case are shown in Figure 5.10, 18
cm downstream of the splitter plate. The imaged region ranges from x = 16 to 20.5 cm, z
= -2.0 to 2.0 cm, and y = 0 cm. The flow direction is left to right. The instantaneous
images are corrected for laser sheet energy non-uniformities using a 20-frame average
sheet intensity profile. These images reveal a highly three-dimensional structure when
compared to lower compressibility cases. No dominant two-dimensional structures are
apparent, as we would expect to see alternating bright and dark spanwise oriented bands
as have been seen at lower convective Mach number conditions (Clemens and Mungal,
1995). In general, these plan view cuts appear highly three-dimensional and lacking in
any spatial regularity or organization. Some evidence of purely streamwise structures is
clear, which show up as either light or dark patches that are elongated in the streamwise
direction (esp. images c and b). These passive scalar images will emphasize turbulent
motions that bring pure fluid into the layer. The structures in these images appear to
entrain either pure high- or low-speed fluid, depending on their location inside the layer,
and bring it into the image plane. Also, the layer appears to be more well mixed than
109
lower compressibility shear layers, owing to the fact that as the roller/braid structure
begins to break down, more streamwise uniform plan views are seen, in agreement with
the observations from the side views.
5.3.4 END VIEW
Figure 5.11 shows instantaneous end view PLIF images of the Mc = 0.85 case
imaged at the same downstream location (18 cm) as the plan and side views. The field of
view is from y = -2.0 to 2.0 cm, z = -5 to 5 cm, and x = 18 cm. Due to shot-to-shot
variations in the laser intensity profile, these images exhibit some sheet correction noise,
which appears as horizontal bands. The instantaneous images were corrected using a 20-
frame average sheet intensity profile, but might have been improved by correction using
instantaneous sheet profiles, as there were significant shot-to-shot spatial energy
fluctuations of the laser in this imaging plane. Also, since the laser is brought in through
the side window, streaks on the window caused by impact of high-speed plastic
diaphragm shrapnel can alter the laser sheet intensity profile if the two regions overlap.
Furthermore, because of the non-linear stretching of the image due to the angle between
the imaging axis and the laser sheet, the right-hand side of the images appears a bit out of
focus. Reducing the off-axis imaging angle to the smallest value allowed by the
experimental setup minimized this problem. A complete discussion of off-axis imaging
and its associated image setup is found in Merklinger (1996) and Scheimpflug (1904).
All the end views shown display highly convoluted, three-dimensional structures.
There are both larger (~2δ) and smaller size spanwise structures in evidence, consistent
with the findings of Clemens and Mungal (1992) at convective Mach number of 0.62. As
were seen in the plan and side views, images again feature small streamwise structures
reaching up into the fast stream (especially in image c). Both the small and large-scale
structures appear to be mixed internally, rather than jets of pure high-speed or low-speed
fluid protruding into the free stream, consistent with the observations made in the side
view plane. There also appears to be a wide variation of the streamwise structure spacing
due to the wide range of structure scales present in these images. Jimenez (1983) and
Bernal and Roshko (1986) found that the mean streamwise structure spacing normalized
110
by δ is approximately 0.25, at incompressible conditions. From this small subset of
images (both plan and end views) a streamwise structure spacing of 0.4 – 0.5 δ is
estimated.
5.4 IMAGING OF THE MC = 1.71 TEST CONDITION
5.4.1 SCHLIEREN
Figure 5.12 shows an instantaneous side-view Schlieren image of the Mc = 1.7
case, from x = 0 to 23 cm downstream. The mixing layer does not resemble any previous
lower compressibility conditions, as there appears to be little large-scale centralized
structure. When compared to the only other Schlieren images in this convective Mach
number range known to the author (Papamoschou and Roshko, 1988), the flow appears
more dominated by streamwise vortices in the present study. Also, the prevalent Mach
wave radiation field shown in Chapter 4 is much stronger in the images of this study than
in those of Papamoschou and Roshko. However, Papamoschou and Roshko only show
images up to Mc = 1.4 and do not display an image for their highest compressibility case
of Mc = 1.8, which may explain the differences in appearances. It is interesting to note
that the increase in compressibility and density ratio have caused the complete
destabilization of this layer and the marked growth of streamwise structures which
protrude into both the high- and low-speed streams. Previously, the same fast side
conditions when coupled with a side two flow of helium gave rise to a more typical
mixing flowfield (Mc = 0.85).
Much of the structure evident in the Mc = 1.7 Schlieren image is in the form of
streamwise tongues of mixed fluid, which protrude into both the high- and low-speed
side. These jets also appear to have a preferred orientation that is approximately equal to
the Mach wave angle of the convective Mach number. This dominant angle appears to
decrease as the structures propagate downstream, suggesting either a Reynolds number
dependence of the structure angle or an amplification/feed-back loop to the structures
from reflections of the Mach waves off the test section walls and back onto the shear
layer. This change in preferred streamwise orientation was not seen in the unbounded
111
simulations of Freund et al. (2000a) or Day et al. (1998), so it is most likely a result of
the shear layer confinement. Strong coupling between the radiation field and the scalar
structure has been shown previously in Chapter 4, consistent with the computations of
Freund, et al. (2000a) and Lessen, et al. (1966). However, further evidence of the
streamwise nature of the structures and their interaction with radiated Mach waves is
available in the PLIF views.
5.4.2 ACETONE PLIF, SIDE VIEW
Figure 5.13 shows several single-shot acetone PLIF images taken at 18 cm
downstream of the splitter plate. The imaged region is from x = 16 – 20.5 cm, y = -2.0 to
2.0 cm, and z = 0 cm. The images were corrected for laser sheet non-uniformity by using
the cross-stream averaged streamwise PLIF signal in the uniformly seeded fast stream.
Unfortunately, due to the strength of the local shocks, the thermodynamic field in the
upper free stream experiences some variation, reducing the effectiveness of the laser
sheet correction, and some vertical ghosting is unavoidable. Again, an instantaneous laser
sheet monitor would be needed if these were truly quantitative images; however, for
judging the character of the large-scale structures and acoustic field, the errors generated
by non-ideal laser sheet correction are tolerable.
Immediately apparent in the image, different from the Mc = 0.85 case, are the
bright streaks, which emanate from the mixing layer and propagate into the free stream.
These bright streaks are Mach wave radiation, which originates from the shear layer. This
radiation occurs because the difference between the convection velocity of the
propagating large-scale structures and the velocity of the fast stream is greater than the
local free stream speed of sound. Thus, the large structures act locally as bluff bodies in
the flow and the entire fast-side interface can be thought of as a “wavy” wall in
supersonic flow, radiating Mach waves whose angle is defined by the local velocity
difference. These are exactly the same waves seen in Chapter 4, but this time in a
spatially resolved sense. Accordingly, the position of the shock is correct with respect to
the large-scale structures that appear in the image. The strength of these waves and their
apparent interaction with the scalar field of the shear layer is much greater than those
112
witnessed in the Mc = 0.91 case of Hall et al. (1993), where distinct Mach wave radiation
was seen in the lower stream, but did not appear to be strongly interacting with the shear
layer dynamics.
The side views also demonstrate that large amounts of pure low-speed fluid can
be found adjacent to the high-speed interface (images a,b,c,e). Streamers of both high-
and low-speed fluid are penetrating well into the lower and upper streams, respectively.
This large variance of scalar values at the high speed interface is consistent with rapid
entrainment motions which sweep large amounts of free stream fluid (low-speed fluid
entrainment is highlighted due to the color table in this set of images) quickly to the other
side of the layer before the structure has a chance to mix internally. The concept of a
“sonic eddy”, Breidenthal (1992), is used to describe reduced communication across a
large-scale eddy due to a velocity difference across the eddy which is larger than the local
speed of sound. Thus, there is a communication timescale, which slows the structures’
ability to mix. DNS simulations (Freund et al., 2000b) found that this effect manifested
itself in reduced production of turbulent kinetic energy through a reduced pressure-strain
correlation as the convective Mach number was increased. This correlation is large for
locally subsonic eddies, but drops rapidly as the eddy size forces this communication
timescale on the flow. The structures in evidence in the images are typically 0.5 to 1.5 δ
in streamwise extent, which is a marked decrease from the lower compressibility
condition. Freund et al. (2000b) and Day et al. (1998) also report this effect as a decrease
in the two-point velocity and scalar correlations with increasing compressibility denoting
smaller structure sizes. This type of scalar structure is consistent with three-dimensional
streamwise structures whose motions are closer to the edges of the shear layer, thus
allowing for the entrainment of large amounts of pure fluid. However, the true three-
dimensionality of these structures is only evident in the other imaging views.
The spatially resolved acoustic field in the upper free stream basically resembles
what was seen in the line-of-sight integrated Schlieren images of the same mixing layer
conditions. However, in this planar view, the shocks are located on the leading edge of
the scalar structures (especially in image e) and can be seen to be three-dimensional bow
shocks in the near field and not planar waves. Also, these are not simple acoustic waves
as large amounts of curvature are present in the near field (closer to the shear layer) as the
113
Mach wave is curving around the structure, akin to a three-dimensional bow shock on a
bluff body. Also, since these waves are highly three-dimensional in nature, depending on
the location of the image plane relative to the centerline of the structure, the shock will
appear differently. Figure 5.14 shows a schematic of the bow shock structure seen in two
spanwise imaging planes with respect to a three-dimensional streamwise structure. Near
the centerline of the structure, the bow shock will appear very strong, nearly normal, and
will have a high degree of curvature. Also the shock will appear to be well correlated
with the shape of the structure (image e, Figure 5.13). However, if the image plane is not
near the center of the structure, the shock will appear weaker and poorly correlated with
the mixing layer structure in the near vicinity (images c and d). The local acoustic field
appears to affect the scalar field by forcing large-scale structures to assume streamwise
structure angles that match the local acoustic wave angles. Also, the streamers in the low
speed side of the shear layer, transporting high-speed fluid to the low-speed interface,
appear to have the same dominant flow angle.
5.4.3 PLAN VIEW
Plan views for the Mc = 1.7 case are shown in Figure 5.15, 18 cm downstream and
at the height of the splitter plate. The imaged region is from x = 16 – 20.5 cm, z = -2.0 to
2.0 cm and y = 0 cm. Flow direction is from left to right. The images presented here show
highly convoluted three-dimensional structures similar to those at the Mc = 0.85
condition. However, here the structures are composed primarily of mixed fluid, stacking
on top of each other so as to appear layered in the plan view; this observation is
consistent with numerical simulations at a similar convective Mach number (Freund, et
al., 2000a).
Little evidence of two-dimensional structure (oblique or spanwise oriented) is
present. Nearly all the images display some streamwise oriented structures, whose length
is greater than the mixing layer visual thickness at this imaging station. Most often,
streamwise structures composed of mostly high- or low-speed fluid are prevalent, snaking
through the images. These structures are most often aligned with the flow direction (and
not taking on some oblique propagating angle with respect to the mean flow like scalar
114
structures for Mc ~ 0.6 – 1.0), and seem to be evenly divided between structures primarily
composed of low-speed fluid and those of high-speed fluid. Also, the mean
concentrations of the individual structures (i.e. the structural mixed mean) is closer to the
mean scalar value of the entire image (i.e. the local mean concentration) than in the Mc =
0.85 case. This suggests that the mixed fluid profile of the shear layer is more like the
average scalar profile of the shear layer, a trend which agrees with resolved scalar field
measurements in a lower convective Mach number regime (Island, 1997) and will be
explored more completely in Chapter 6 at a higher convective Mach number condition.
Notice that no shocks are present in the images taken at the y = 0 cm plane. While
there were many instances of local shocks in the side view images, all of those emanated
externally from the layer in the side view. Thus, since none of those shocks appear in the
plan views, no shocks must be created internally to the mixing layer which then radiate
out into the free stream. A more informative impression of the nature of the radiated
Mach waves can be obtained from plan view cuts taken above the shear layer centerline.
Figure 5.16 shows instantaneous images taken near the high-speed interface of the shear
layer (y = 0.5 cm) and just above it (y = 1.0 cm). The true three-dimensionality of the
streamwise structures is unmistakable in the left image, where several elongated
structures are clearly aligned with the mean flow direction. Also, the structures distinctly
act as bluff bodies in the flow, with clear bow shock type waves surrounding the large-
scale structures and conforming to the structures’ shape. Further into the free stream,
above the level of the mixing layer, the complexity of the radiation field is evidenced by
many Mach wave structures, which emanate from the mixing layer, contained and
reflecting in the test section. Even though at the imaging station the shear layer is only
slightly confined (htest-section/δvis = 6), the reflections of shocks emanated from the mixing
layer further upstream have reflected off the upper wall of the test section and are
beginning to interact with the shear layer at this downstream location. Also, since the
shocks are now three-dimensional in nature, the shocks also reflect off the sidewalls of
the test section and interact with the shear layer and other generated shocks (Figure 5.16).
115
5.4.4 END VIEW
Figure 5.17 shows instantaneous end view PLIF images of the Mc = 1.7 case
imaged at 18 cm downstream from the splitter tip. Due to shot-to-shot variations in the
laser intensity profile, these images exhibit some sheet correction noise, which appears as
horizontal bands. The field of view is y = -2.0 to 2.0 cm, z = -5 to 5 cm, and x = 18 cm.
The instantaneous images are corrected using a 20-frame average sheet intensity profile,
and exhibit the same image correction noise as described in the Mc = 0.85 case. The
horizontal streaks due to variation in the sheet intensity profile here are more pronounced
than the Mc = 0.85 case due to decreased SNR (lower test section operating pressure for
this case).
In this set of images, the streamwise character of the structures is pronounced, and
elevated levels of spanwise asymmetry are prevalent. Consistent with the other imaging
planes, three-dimensional scalar structures consisting of mostly low-speed side fluid are
seen reaching into the fast-side free stream (as low scalar value jets are highlighted by the
passive scalar technique); however, some evidence of high speed fluid streamers into the
low-speed stream is also visible. The spanwise extent of the streamwise structures in this
set of images is typically much smaller than the Mc = 0.85 case (Figure 5.11), in good
agreement with the structure information from DNS simulations (Freund, 2000a; Vreman
et al., 1996). Also, the higher level of spanwise asymmetry with compressibility has also
been seen by Clemens and Mungal (1995), and has now been extended to even higher
convective Mach number in this work. Furthermore, the protrusion distance of the
structures into the fast and slow streams is much larger than in the lower compressibility
case. Clemens and Mungal (1995) also noticed that the scalar streamer and jet structures
they saw at Mc = 0.62 and 0.79 tended to travel farther into the free streams of the mixing
layer than those at lower compressibilities. The streamwise structures in Figure 5.11
show very large penetration depths (some greater than δvis) causing the rapid transport of
free stream fluid across the layer, in what is likely a sweeping motion. Based on this
small set of images, it appears that the structures initially have small spanwise extent as
they sweep up from the lower speed stream (as in images a, c, and e), and then they begin
to mix internally and spread as they propagate downstream (as in images b, d, and f).
116
Three-dimensional bow shock structures are also in evidence in the upper
freestream in the images of Figure 5.17 (esp. images b, c, and d). The appearance of the
shocks in these images highlights the fact that the shock front conforms to the shape of
the scalar structure as seen in the side and plan views, making the shock fronts quite
curved. Most of the shocks imaged in these views are relatively weak, compared to the
shocks evident in the side view images, because the stronger part of the bow shock (i.e.
the part which is normal to the flow direction) usually resides at the front of the structure,
which is not visible in the end views.
5.4.5 SHOCKLET PRODUCTION
As described in Chapter 1, it is difficult to define and experimentally identify a
shocklet. Following the simulations of Passot and Pouquet (1987), Lee et al. (1991) and
Vreman et al. (1996) as well as many other numerical simulations, shocklets have been
observed and studied in two- and three-dimensional simulations of mixing layers.
Experimental evidence remains elusive because the physical manifestation of a shocklet
is a bit ambiguous. Typically shocklets are defined as areas of strong negative dilatation
in mixing layer simulations, as shown in an example of shocklet marking in the DNS
results of Freund et al., 2000a (Figure 5.18). They show shocklets that emanate from
inside the mixing layer and radiate into the potential core of a supersonic jet. It is clear
that shocklets should form internal to the mixing layer as a result of turbulent motions
and large-scale structure interactions, and should not be formed at the edges of the layer
(as this could be easily confused with weak oblique shocks generated by the local shear
velocity). In Chapter 4, Schlieren images of what appeared to be a shocklet in a Mc = 2.0
mixing layer were presented in Figure 4.10, with the caveat that the results of Chapter 5
would shed some light on the existence of these shocklet structures.
Shocklets are noticeably absent from all the instantaneous images at this high
compressibility case. All the waves apparent in the instantaneous images occur only in
the free stream, or are radiated from the local high-speed interface into the free stream.
No localized high gradient regions internal to the mixing layer are found in any of the
images, especially in the plan views where it would be likely that shocks, which exist
117
inside the mixing layer, would be imaged. Furthermore, in the plan view, shocks are only
visible in the free stream fluid surrounding large-scale streamwise structures. The same
statement is also true for the side view images, where the bow shocks are clearly attached
to the face of large-scale structures, but do not radiate from the mixed regions of the
shear layer. Thus, there is likely no shocklet production at this convective Mach number,
because there is no evidence of them in many spatially resolved image realizations of the
flow.
It could be argued that the current imaging technique lacks the dynamic range or
spatial resolution to capture the shocklet structures. However, if the PLIF images are able
to capture weak oblique shock waves which have normal Mach numbers < 1.05, then, the
technique should also be able to image shocklets of that strength. Furthermore, if
shocklets exist but are only very weak (such that Mshocklet < 1.05), then their previously
stated importance in high compressibility turbulence has been exaggerated. Also, the
importance of viewing shocklet behavior in spatially resolved imaging techniques is
highlighted by the comparison of the Schlieren images of Chapter 4 with the PLIF images
contained in this chapter. As was shown here, strong streamwise structures rapidly sweep
large amounts of low-speed fluid up into the layer toward the high-speed interface, before
the structure has a chance to mix internally. This causes the transport of low-momentum
fluid into a region of high-speed flow, which will cause bow shocks to form on the
structure, as was seen in the side, plan, and end views of the Mc = 1.71 condition. Using a
line-of-sight integrated technique, this would appear as a bow shock type structure
moving up through the shear layer with time, as shown in the time sequence images of
Figure 4.10. However, using a PLIF technique, the shock structures clearly exist only on
the leading edge of large-scale structures. Due to the highly three-dimensional nature of
the shocks and the structure of the scalar field, true shock locations with respect to
structures are masked when viewed using Schlieren. By this reasoning, the previous
experimental evidence in a Mc = 2.0 counter-flowing shear layer (Papamoschou, 1995) is
believed to be a three-dimensional bow shock existing in a scalar field similar to those
imaged in the Mc = 1.7 case here, and not a shocklet imbedded in the flow.
118
Digital thermocouple
Filling Vent
Level
Drain
Water flow
Tank pressure gage
Flow meters
Pressure Measurement
Carrier Gas
To the shock tube
Bypass Line
Figure 5.1 Schematic of pressurized acetone bubbler and seeder unit for fast, uniform seeding into shock tunnel test gases.
Bubbling Manifold
Water flow
Flow Meters
Vent
Level
Drain
119
2
3
456
10
2
3
4
Pres
sure
(psi
a)
0.7 0.6 0.5 0.4 0.3 0.2 0.1Maximum Seeded Acetone Mole Fraction
Driven Tube Pressure Seeder Pressure
Figure 5.2 Maximum seeding level possible in shock tunnel using the high pressure seeder as a function of P1.
1
2
4
6
10
2
4
6
100
2
4
6
1000
Pyro
losi
s tim
e sc
ale
(τ,
µse
c)
0.250.200.150.100.05Acetone Mole Fraction ( χ )
> 10% loss for1.5 msec test time
1100 K
1200 K
1150 K
1 atm 5 atm 30 atm
Figure 5.3 Acetone pyrolysis time constant as a function of temperature, pressure, and acetone mole fraction for acetone – air mixtures as set by Equation 5.3. Line denotes regions where loss of tracer due to pyrolysis becomes significant for typical test times.
120
5.0
4.8
4.6
4.4
4.2
4.0
3.8
3.6
Max
imum
Noz
zle M
ach
Num
ber
0.250.200.150.100.05Acetone Mole Fraction ( χ )
T5 = 1100 K T5 = 1150 K T5 = 1200 K
Figure 5.4 Maximum supersonic nozzle Mach number achievable using acetone seeding when considering the effects of pyrolysis and condensation in the expanded state.
60
50
40
30
20
10
Requ
ired
P 5 for
reas
onab
le P
LIF
S/N
(atm
)
0.250.200.150.100.05Acetone Mole Fraction ( χ )
T5 = 1100 K T5 = 1150 K T5 = 1200 K
Figure 5.5 Required supersonic nozzle stagnation pressure for good SNR in acetone PLIF
experiments in the expanded state while maintaining low pyrolysis in the stagnation region.
121
250 260 270 280 290 300 310 3200
1
2
3
4
5
6
7
8
T = 600°C T = 400°C T = 200°C T = 100°C T = 23°C T= 25°C,
Hynes et al. (1992)
σab
s ( ×
10-2
0 cm
2 )
Wavelength (nm)
Figure 5.6 Variation in the absorption cross-section and the red-shift of the absorption feature of acetone with temperature (Thurber et al., 1998).
1.0
0.8
0.6
0.4
0.2
0.0
S f (T)
/ S f (
295
K)
1000800600400200Temperature (K)
Experimental Temperature Range
λ = 266 nm λ = 289 nm 295/T (~ density)
Figure 5.7 Variation in the acetone fluorescence signal with changing local temperature for two different excitation wavelengths (266 nm and 289 nm), using data from Thurber et al., 1998.
122
Figure 5.8 Instantaneous side view Schlieren image for the Mc = 0.85 case. The imaged region begins at the splitter tip, and has ∆x = 23 cm,. ∆y = 7.5 cm.
123
Figure 5.9 Instantaneous passive scalar side view images at Mc = 0.85 with fast-side seeding. Lower right hand corner is a 6-frame average image. Flow is left to right with the high-speed stream on top. The imaged region is ∆x = 4.5 cm and ∆y = 4.0 cm and centered 18 cm downstream of the splitter tip and aligned with the centerline of the test section. The images are uncorrelated in time.
124
Figure 5.10 Instantaneous passive scalar plan view images at Mc = 0.85 with fast-side seeding. Flow
is left to right. The imaged region is ∆x = 4.5 cm and ∆z = 4.0 cm and centered 18 cm downstream of the splitter tip and aligned with the centerline of the mixing layer (y = 0). The images are uncorrelated in time.
Figure 5.11 Instantaneous passive scalar end view images at Mc = 0.85 with fast-side seeding. Flow is into the page with the high-speed stream on top. The imaged region is ∆z = 10 cm and ∆y = 4.0 cm and centered 18 cm downstream of the splitter tip. The images are uncorrelated in time.
125
Figure 5.12 Instantaneous side view Schlieren image for the Mc = 1.71 case. The imaged region
begins at the splitter tip, and has ∆x = 23 cm,. ∆y = 7.5 cm.
126
Figure 5.13 Instantaneous passive scalar side view images at Mc = 1.71 with fast-side seeding. Lower right hand corner is a 6-frame average image. Flow is left to right with the high-speed stream on top. The imaged region is ∆x = 4.5 cm and ∆y = 4.0 cm and centered 18 cm downstream of the splitter tip and aligned with the centerline of the test section. The images are uncorrelated in time.
127
BowShock
Side Views
Plan View
a)
a)
b)
b)
Figure 5.14 Schematic of side view images which contain shocks and structures, highlighting that the three-dimensionality of the bow shock can change the character of the interaction (from weak Mach wave to curved shock) depending on the orientation of the structure with respect to the imaging plane.
128
Figure 5.15 Instantaneous passive scalar plan view images at Mc = 1.71 with fast-side seeding. Flow
is left to right. The imaged region is ∆x = 4.5 cm and ∆z = 4.0 cm and centered 18 cm downstream of the splitter tip and aligned with the centerline of the mixing layer (y = 0). The images are uncorrelated in time.
129
Figure 5.16 Instantaneous passive scalar plan view images at Mc = 1.71 with fast-side seeding. Flow
is left to right. The imaged region is ∆x = 4.5 cm and ∆z = 4.0 cm and centered 18 cm downstream of the splitter tip. Image heights with respect to the shear layer centerline are (a) y = 0.5 cm and (b) = 1.0 cm. The images are uncorrelated in time.
Figure 5.17 Instantaneous passive scalar end view images at Mc = 1.71 with fast-side seeding. Flow is into the page with the high-speed stream on top. The imaged region is ∆z = 10 cm and ∆y = 4.0 cm and centered 18 cm downstream of the splitter tip. The images are uncorrelated in time.
130
Figure 5.18 Shocklet production, or regions of large negative dilatation in annular mixing layer (DNS simulation of a round jet, Mc = 1.80 (Freund et al., 2000a).
131
CHAPTER 6
NO PLIF IN HIGH COMPRESSIBILITY MIXING LAYERS
When imaging large-scale structures in mixing layers, it is desirable to have an
optical diagnostic whose signal is proportional only to the local mole fraction of one of
the free streams, independent of both local pressure and temperature. This becomes even
more important in high compressibility shear layers where temperature no longer acts as
a conserved scalar and thus cannot be corrected for in the post-processing of fluorescence
images using a mixture fraction correlation. In order to create such a diagnostic, the
issues of tracer selection and seeding, transition selection, saturation, quenching
environment, and other potential systematic errors must be examined, and in general
compromises between error sources must be made. This chapter will describe the design
and application of a nitric oxide PLIF measurement technique for examining the local
mixture fraction field in a high compressibility mixing layer.
6.1 TRACER SELECTION AND LIF MODELING
Nitric oxide was chosen as a fluorescent tracer for this high compressibility
condition for several reasons. The spectroscopy of NO has been previously well
characterized allowing for accurate modeling of the laser-induced fluorescence with
respect to variations in flow field variables. Also, since NO has been widely used as a
tracer gas in several previous PLIF experiments, the literature and prior experience at
Stanford for NO and other PLIF tracers provides a useful base of experiences (Kychakoff
et al., 1984; Lee et al., 1994; McMillin et al., 1991). Furthermore, NO has good thermal
stability and low reactivity at the mixing layer conditions examined in this study. While
NO is extremely toxic, as will be discussed later, its use in pulsed high-stagnation
enthalpy facilities is much simpler than in continuous flow situations. Lastly, NO PLIF
has been shown to be an excellent candidate species for mole-fraction imaging in non-
132
isothermal supersonic environments (McDaniel and Graves, 1988; O'Byrne et al., 1999;
Palmer et al., 1992).
6.1.1 NO SEEDING STRATEGIES
In order to image NO fluorescence, first, nitric oxide must be seeded into one of
the free streams of the supersonic shear layer. One choice would be to seed the lower
stream with nitric oxide prior to injection into the test section. This would allow for easy
calibration of the optical system (as NO would be available without running the shock
tunnel). However, there are some strong disadvantages to this option, mostly owing to the
extremely hazardous nature of nitric oxide. Since the side two mass flow is initiated 200
msec before the firing of the shock tunnel and continues for 1-2 seconds after the shock
tunnel firing, a large amount of the NO seeded gas will end up in the dump tank after the
running of the tube. A conservative estimate for the final NO level in the shock tunnel
after firing using a 2500 ppm seeding level in the low speed stream is 500 ppm. This
means that a safety venting system with NO scrubbers and dilution air would have to be
installed to handle the exhaust gas, since levels of NO above 25 ppm are considered
health-threatening. Also, increased safety precautions would have to be in place for the
loading of seeded gas into the downstream injection cylinder. Thus, the complications
stemming from gas handling rules out simple injection of NO from the lower speed side.
Seeding of NO from the shock tunnel and thus into the high speed side of the
mixing layer is far more promising. Since seeding into the test gas represents a fixed
volume of nitric oxide in the system, there will be a large dilution of the NO as it mixes
after the firing of the shock tunnel with both the driver gas and the gas initially in the
dump tank. A conservative estimate for the final NO level in the shock tunnel with a
2500 ppm seeding level in the pre-shocked test gas is 32 ppm, which can be vented in a
much simpler manner. The difficulty of handling mixtures of NO in the laboratory is still
present, but since much smaller amounts are in use the management issues are less
important.
Unfortunately, a distinct problem occurs when shocking mixtures of nitric oxide
and argon to high temperatures and pressures, as is required to achieve very high
133
compressibility mixing layers. Though nitric oxide is very stable at room temperature in
the absence of oxygen (which causes the slow conversion of NO to NO2), at high
temperatures, the NO molecules can dissociate with the resultant oxygen radicals strongly
attacking other NO molecules. A very simplified mechanism for the thermal NO removal
at these high pressures can be stated (Wray and Teare, 1962) as:
NO + M → N + O + M
NO + O → N + O2
NO + N → N2 + O
O + O ↔ O2
N + N → N2
This simple mechanism shows that nitric oxide is dissociated into a free oxygen and
nitrogen atom in an initiation step, whereby these radicals can then attack another NO
molecule in a propagation step. The recombination of radical atoms into molecular
oxygen and nitrogen represents the termination reactions.
This simplified model is supported by a more complete simulation of the
reflection shock conditions. Shown in Figure 6.1 are the mole fraction histories of a 2%
NO in argon mixture shocked to 3600 K and 31 atm as calculated using CHEMKIN and
the GRIMECH 3.0 model. The temporal history shows the gas mixture at the reflected
condition in the shock tunnel for the entire test time (1.5 msec) before the mixture travels
through a Mach 5 nozzle. Initially, there is strong removal of NO along with the equal
production of both molecular and atomic oxygen. After 255 msec, the removal process
has reached steady state, with 93% of the initially seeded NO lost to the production of
oxygen. Further loss of NO in the nozzle is due to the production of NO2, which occurs at
lower pressure and temperature. Thus, the final yield of NO in the test section for this
seeding strategy is 6% of the initial NO mole fraction. Since this seeding technique also
requires some enhanced safety measures for handling nitric oxide mixtures prior to
injection into the shock tunnel, it is also ruled out due to the low yield of NO in the test
section.
134
The most promising seeding technique for nitric oxide in high temperature shock
tunnel flows is to utilize the high pressure/temperature stagnation conditions to produce
NO in the reflection portion of the shock tunnel. At 1 atm, the peak production of NO
from air is at 3100 K, and this peak production shifts to 3500 K at 30 atm (Liepmann and
Roshko, 1957). Thus, the shock tunnel conditions for high compressibility mixing layers
are ideally suited to creating NO from air in the reflected region. Figure 6.2 shows the
mole fraction histories of a 5% air in argon mixture shock heated to 3600 K and 31 atm.
As in the previous seeding strategy, atomic and molecular oxygen quickly come into
equilibrium with each other; however, in this case, some of the nitrogen atoms combine
with the free oxygen to form nitric oxide. The conversion efficiency of air to NO at these
conditions is 5.5%, with an induction time of 160 µsec, which is an order of magnitude
lower than the test time. Again, NO is lost in the nozzle as the rapidly dropping static
pressure and temperature is conducive to NO2 formation. Thus, the final yield of NO in
the test section for this seeding strategy is 4.6%.
The advantages of the technique are many. First, since air is now the seeded gas
in the shock tube, no precautions are necessary for the preparation and loading of the test
gas mixtures. Also, because nitric oxide is now only produced in the reflected portion of
the shock tunnel, the final mole fraction of NO in the shock tunnel after firing is
approximately 13 ppm, a safe level. Finally, the production of NO is less costly than
seeding a large amount of nitric oxide, only to lose most of the seeded amount in the
reflected portion of the shock tunnel.
There are also some disadvantages to creating NO into the shock tunnel versus
seeding it into the low speed side. Because the nitric oxide is formed at high temperature
and then quickly passes through the nozzle, the vibrational temperature remains high due
to the slow vibrational relaxation rates at the nozzle free stream conditions. A simple
simulation of the vibrational relaxation of NO can be performed as it moves through the
nozzle using the Landau – Teller formulation (1936):
( )( )PT
ETEdt
dE vvv,
*
τ
−
= (6.1)
where E*v is the local equilibrium vibrational energy, Ev is the actual local vibrational
energy, and τ is the vibrational relaxation time, which depends on the local conditions.
135
Vibrational relaxation rates are taken from Kamimoto and Matsui (1970) and Millikan
and White (1963). For the M = 5 nozzle used to achieve the Mc = 2.64 mixing layer
condition, the solution states that the vibrational temperature of NO will be frozen at
2000 K. This results in a 33% reduction in the lower electronic state Boltzman fractions
for the PLIF technique and will cause lower overall signal levels. Another disadvantage is
the addition of diatomics to the test gas which increases the reflected shock/boundary
layer interaction and reduces overall test time as described in Chapter 3. Although no
seeding technique is without its complications, the choice is made to create NO in the
shock tunnel due to its demonstrable advantages.
6.1.2 TRANSITION SELECTION
The selection of the pumping transition (and thus the variation of its Boltzman
fraction with temperature) depends on the temperature range of interest for the
experiment. For the Mc = 2.64 condition examined in this chapter, the free stream static
temperatures are 390 K and 289 K for the fast and slow streams respectively. Thus, the
mixed states are expected to range in temperature from 300 K up to a hot zone in the
center of the mixing layer where the fast stream has been decelerated and the slow stream
accelerated. In the initial analysis of this diagnostic, little a priori knowledge of this
aerodynamic heating is documented as the heating depends on the relative entrainment
ratios of the two streams and the local convection velocity of structures.
Assuming an upper boundary of 700 K on the temperature range of the
experiment, the solution to the 4-level model fluorescence equation (Appendix C) as a
function of temperature will lead to the choice of the most temperature-insensitive
pumping transition. In this study, potential transitions are selected from the A2Σ
+←X2
Π½
(0,0) band of NO, due to the advantages of this system over the B←X system as
discussed by Lee et al. (1992). Shown in Figure 6.3 are the relative fluorescence signals
from some Q1 + P21 transitions. This family of transitions has the highest spontaneous
emission rate (A21) of all the possible transitions (e.g. Q21 + R1, Q2 + R12, P2 + Q12, R2,
P1). Immediately obvious from this plot is that the LIF signal will decrease with
increasing rotational quantum number due to decreasing Boltzman populations. Also,
136
there are regions of temperature for each transition shown where the curves are nearly
flat, which permits LIF signals that are insensitive to temperature in that range. As a
check of the simulation produced by this study, a similar examination of the J´´ = 18.5
and surrounding rotational transitions was performed using the NO spectra simulation
tool of Bessler and Sick (Private communication, 2001). Those results are shown in
Figure 6.4 and the overall changed in the absorption feature as a function of temperature
well match the results in this thesis.
Figure 6.5 shows the relative change in the fluorescence signal when normalized
by its peak value in the temperature range of 300 to 700 K. Each rotational transition has
a temperature range where the LIF signal shows small variation (< 5 %). For the J´´ =
18.5 transition, this 5% error window ranges in temperature from 340 to 540 K which is
expected to encompass a large fraction of the mixed temperatures expected in the Mc =
2.64 mixing layer. Table 6.1 shows the temperature ranges for the 5% and 10% errors in
interpreting the LIF signal as a direct measurement of local concentration. As the
rotational quantum number is increased, a larger range of temperatures falls into these
error bands; however, this increased temperature independence comes at the cost of
overall signal levels (Figure 6.3).
Transition 5% Error 10% Error
J´´ = 17.5 310 to 490 K, 280 to 540 K
J´´ = 18.5 340 to 540 K, 310 to 595 K
J´´ = 19.5 380 to 600 K, 350 to 675 K
J´´ = 20.5 415 to 660 K, 380 to 730 K
Table 6.1 Comparison of temperature range for temperature
insensitive measurements of mixture fraction for
several different lower rotational states.
Due to the selection of the A2Σ
+←X2
Π½ (0,0) Q1 + P21 (J´´ = 18.5) transition, the
NO PLIF diagnostic will have reduced temperature sensitivity at the mixed temperatures
of the mixing layer. However, the technique will systematically under-predict the local
high-speed stream mole fraction in areas of the mixing layer where the local temperature
137
is high. Since temperature no longer acts as a conserved scalar in this flowfield, no
straightforward correction of this temperature dependence is possible. Another possible
choice for the excitation transition would be the Q1 + P21 (J´´ = 17.5). However, the
expected mixed temperatures are mostly greater than the supersonic free-stream
temperature of 390 K. Thus, with minimal a priori knowledge of the temperature range
of the mixing layer, the higher rotational state seems the better choice.
6.1.3 ISO-QUENCHING ENVIRONMENTS
Another important consideration for making accurate measurements of local high-
speed fluid mole fraction is the change in the quenching environment as the seeded
stream mixes with the unseeded, lower speed stream. If the packet of mixed fluid has a
different quenching environment than the upper free stream, the fluorescence quantum
yield will not be the same locally. Thus, the LIF signal will not scale linearly with the
local mole fraction of high-speed fluid, as shown in Equation 6.2.
( )( ) ( )∑+
≅∝
speciesalliii
iHfTvT
kTPA
AwhereTPS
21
21 ,,σχ
φχφχ (6.2)
However, if the lower speed stream mixture can be constructed such that the
product of ΣχI σI ⟨vi⟩ is equal to that of the high stream, then the quenching environment
will be the same in all the mixed states. This will be termed an “iso-quenching”
environment. The effect of using an iso-quenching environment is shown in Figure 6.6.
Using the quenching cross sections from McDermid and Laudenslager (1982), the
quenching rate in the mixture 98.8% argon, 1.2% O2 exactly balances the quenching in
the supersonic free stream (95% argon, 3.97% N2, 0.8% O2, 0.23% NO). The relative
error in the mole fraction measurement increases as the lower speed mole fraction (χL)
increases in the mixed states for non-matched quenching environments. The error bars on
the two curves denote the variation of error in the temperature range of 300 to 600 K for a
fixed lower speed mole fraction. The mixture fraction error due to mixing using an iso-
quenching environment is reduced to 0.3% at the minimum detectable level. Thus, it is
clear that the potential errors associated with the mixing of the two streams to different
138
local mole fractions and temperatures are minimized by using an iso-quenching mixing
environment.
6.1.4 PRESSURE AND SATURATION BEHAVIOR
Though mixing layer flowfields are essentially isobaric, local shock waves
emanating from large scale structures have been shown to occur at high compressibilities
in Chapters 4 and 5. These local shock waves cause sizeable local number density
perturbations, and thus the effect of variable pressure on the LIF signal must be
examined. Also, lower pressures have smaller lower rotational energy transfer rates,
which exacerbate saturation effects. Thus, running these experiments in a shock tunnel
with lower test section pressures than other mixing layer experiments, which were
conducted at near-atmospheric pressures, is a distinct disadvantage which necessitates the
careful examination of pressure and saturation effects on the LIF signal.
From the pressure scaling in the standard, weak excitation two-level fluorescence
equation (Equation 6.3),
( ) ( )∑+
∝
speciesalliii
HfTvT
kTPA
AkTPS
21
21
σχ
χ (6.3)
it is clear that the LIF signal will scale linearly with pressure if the quenching term is
much smaller than the spontaneous emission. Also, the fluorescence signal will be
independent of pressure if the quenching term is much larger than the spontaneous
emission. Unfortunately, for the supersonic flowfield of interest in this chapter, the ratio
of the spontaneous emission rate (A21 = 1.0 x 107 sec-1) to the quenching rate (Q21 = 4.9 x
106) is about 2, placing these experiments in the “roll-off” regime where the local LIF
signal scales as ~ P/(1+P).
Using the 4-level LIF model, the response of the fluorescence signal to changing
pressure can be simulated (Figure 6.7). Clearly, for the temperature range of interest,
these experiments exist in the regime where the LIF signal scales non-linearly with
pressure. This would pose a challenge for image interpretation if regions of strong
139
pressure variation were found internal to the mixing layer; these regions would be
interpreted as local mixture fraction variations if pressure effects were not considered.
However, as was seen in the acetone PLIF images of Chapter 5, there are no
instances of strong imbedded shocklets internal to the mixing layer at the Mc = 1.7
condition. Instead, all of the shocks are attached to structures and propagate into the
upper free stream. Therefore, if the same type of flowfield is expected to exist at the
higher compressibility condition (Mc = 2.64), the non-ideal pressure effects will be
limited to the shocks, which propagate from the upper side of the mixing layer into the
supersonic free stream. Consequently, the slight signal variation with pressure is actually
useful. The local shocks will occur in regions where the high-speed side mole fraction is
1, and the LIF signal from those shocks will be greater than the average signal from the
free stream. Thus, the signal variation with pressure allows for imaging of the local
shocks in the mixing layer flowfield.
Validation of the pressure scaling is shown in Figure 6.8, where the model is
compared to static cell measurements, showing excellent agreement. The actual LIF
intensity differs in this 300K simulation from that shown in Figure 6.7 due to a difference
in the equilibrium vibrational temperature. The LIF signals in Figure 6.7 are calculated
using the frozen vibration temperature of 2000 K as discussed in Section 6.1.1, while the
LIF cell data are compared to calculations made with an equilibrium vibrational
temperature of 300 K.
Another difficulty with lower pressure LIF experiments is the potential of
saturation of the electronic transition and the significant perturbation of the ground state
rotational distribution. Saturation will cause a non-linear LIF response to a changing laser
intensity, which poses great problems for both image correction and interpretation
considering the Gaussian spatial energy profile of the laser beam. Variation of the LIF
intensity with increasing spectral power (W/cm2 cm-1) in the temperature range of interest
is shown in Figure 6.9. Temperature effects play a very minor role in saturation of this
transition and can be neglected.
Figure 6.10 shows the comparison between the 4-level model and the weak
excitation model. Clearly, saturation effects become important above a spectral power of
1 MW/cm2 cm-1 for the shock tunnel free stream conditions. Consequently, the laser
140
fluence in this experiment is kept below 0.5 MW/cm2 cm-1 to ensure that saturation
effects do not play a role. The results shown here are in good agreement with a LIF
modeling analysis done by Lee (1991). The onset of saturation versus the signal gain
from reduced quenching represents a potentially difficult tradeoff, because as the test
section pressure is reduced, the overall number of absorbing molecules is also decreased,
lowering the potential image SNR. Adding to that the restriction placed on the maximum
laser intensity, which lowers as the pressure is decreased due to the decreasing RET rates,
the potential SNR of the images is strongly linked to the overall test section pressure of
these mixing layer experiments. At the high Mach number used in this experiment,
raising the test section pressure means both more stress on the shock tunnel and higher
helium gas costs. So, as expected, quality images cost more. Since these experimental
conditions occur at low pressures, a balance between mixture fraction sensitivity and
overall signal levels must be achieved. Maximization of signal quality will also
necessitate the tradeoff between of seeding levels, laser attenuation in the field of view,
fluorescence saturation, shot-to-shot laser fluctuations, and measurement resolution.
6.2 NO PLIF IMAGING OF THE MC = 2.64 TEST CONDITION
For these measurements, NO is produced in the shock tunnel at initial conditions
of 3650 K and 31 atm using the air chemistry outlined in section 6.1.1. A mixture of 95%
argon and 5% dry air is used as the test gas and shocked by a Ms = 4.10 incident wave.
The test time for this shock condition is 1.95 msec and nozzle start time is 450 µsec.
Consequently, there is 1.5 msec for the layer to develop before PLIF images were taken.
The maximum laser fluence for these experiments is 0.45 MW/cm2 cm-1 which is just
below the saturation level for these conditions. The test section walls are set at the same
angles as the lower compressibility cases (upper, 0 degrees; lower, -0.5 degrees) to
minimize the streamwise pressure gradient. The mixing layer variables for this condition
are shown in Table 6.2.
141
Mc avg 2.64
Mc1, Mc2 2.64, 2.64
Ptest section [atm] 0.10
Gas1 95% Argon, 5% Air
Gas2 98.8% Argon, 1.2% O2
T1, T2 [K] 390, 289
V1, V2 [m/sec] 1880, 94
Mexit 1 4.87
M1, M2 5.12, 0.30
s 1.369
r 0.05
21, mm DD 0.69, 0.10 (kg/sec)
(δ)x = 18 cm [cm] 1.2
(Reδ) x = 18 cm 1,2 9.6 x 104, 1.6 x 105
(Rex) avg 1.88 x 106
(δ/∆V) x = 18 cm [µs] 6.7
2x/(U1-U2) [µs] 201
Table 6.2 Mixing layer test conditions for NO PLIF imaging results
6.2.1 SIDE VIEW
Eleven instantaneous, corrected images are shown for the Mc = 2.64 condition in
Figure 6.11. Nitric oxide was seeded in the upper high-speed stream by its creation in the
stagnation conditions of the shock tunnel and flow is from left to right. The field of view
extends from 16 to 20.3 cm downstream of the splitter plate and from –2.7 cm < y < 2.0
cm. These side view images area taken on the centerline of the test section (z = 0). The
peak SNR for these images is 12. The field of view encompasses approximately 3 mixing
layer thicknesses in both height and width. As was observed in lower compressibility
layers (Clemens, 1991), limited fields of view of the scalar field can foster incorrect
assumptions about the character of the large-scale structure field, as the infrequency of
structures at Mc < 0.5 causes their appearance in less than 30% of those images.
However, as structures sizes decrease with increasing compressibility, more large-scale
structures can be contained in this field-of-view.
142
Very few instances of true Brown-Roshko type large-scale rollers are seen in
these images. The layer interface on the high-speed side appears very ragged and the
layer thickness does not vary greatly with downstream distance in the field of view,
suggesting very few streamwise correlation structures, which were so common at low
compressibilities. There also are very few regions of finding pure low-speed fluid near
the high-speed stream interface, in contrast to the lower compressibility results in Chapter
5. Engulfing entrainment motions do not appear to be as prevalent at this compressibility
condition. The layer also appears internally well mixed, with what seems to be a
reasonably smooth variation of mixture fraction in the cross-stream direction for most
structures.
The layer thickness in this set of images varies widely from approximately 0.6 cm
in image (e) to over 1.3 cm in images (h) and (i). This type of behavior was witnessed in
the end views of lower compressibility conditions, where large structures are separated
by thin braid regions in the streamwise direction. Thus, end view images can capture
either large structures (where the mixing layer appears at its full average thickness) or the
braid regions (where the layer appears much thinner than the average thickness). The
same type of effect is seen in the side view images of Figure 6.11. Consequently, large-
scale streamwise structures that have spanwise periodicity must exist in order to provide
side view images where the layer thickness varies greatly.
Free stream shocks can also be found in this suite of images. Prevalent shocks can
be discerned in images (a), (d), (e), and (k). The shocks in these images are less well
defined than those in the Mc = 1.7 case visualized with acetone PLIF. Since the NO PLIF
technique is sensitive to mixture fraction, which is conserved across a shock, only the
variation in local pressure due to the shock can be imaged. And since the sensitivity of
the technique to pressure is rather weak (especially for weak Mach waves), the ability to
discern local shocks is much reduced. Also, in the images in which shocks appear, it is
difficult to correlate a wave with a structure that launched it.
Cross-stream and streamwise cross sections through the side-view images are
shown in Figure 6.12. Each graph of local mixture fraction corresponds to the closest
parallel line through the layer (which denotes the position of the cross section). All graph
axes are in units of cm downstream (in the case of the streamwise cuts) or cm from the
143
origin set at the 50% level of the average mixture fraction profile (in the case of the
cross-stream cuts). The upper streamwise cuts are taken at the 50% level of the average
mixture fraction profile, while the lower streamwise cuts are taken at the 20% level.
In all of the streamwise cuts, very few streamwise ramps are in evidence,
suggesting that there are few significant spanwise-oriented structures, like those seen at
low compressibilities. Some very slight streamwise ramps are present in images (b) and
(c) and are perhaps caused by some spanwise coherent structures of limited extent. The
cross-stream cuts show almost uniformly that the structures in the mixing layer are
strongly ramped in the cross-stream direction. These ramps suggest that the dominant
entrainment motions are streamwise, rather than spanwise.
6.2.2 PLAN VIEW
Five instantaneous plan view images for the Mc = 2.64 condition are shown in
Figure 6.13. The field of view extends from 16 to 19.7 cm downstream from the splitter
tip and from z = -1.9 cm to 1.9 cm, and the flow is from left to right. These images are
taken at the y = 0 level, through the 50% mixture fraction level of the average mixing
layer profile. Images are corrected for laser sheet non-uniformity using the streamwise
intensity distribution of the average image. The peak SNR for these images is 15 owing
to the slightly smaller field of view as compared to the side view images.
These images reveal a set of strong streamwise structures (about one per image),
which are oriented nearly parallel to the flow direction. The streamwise structures appear
to be at a nearly uniform value of mixture fraction along their length, which is usually
lower than the average mixture fraction of the image. These streamwise structures are
more dominant on the low speed side of the mixing layer, entraining low speed fluid and
rapidly transporting it across the layer. However, there is also a probability that the
streamwise structure has entrained mostly high-speed fluid into the layer, as shown in
image (b). Outside of the streamwise structures, the mixture fraction field appear
generally to be ramped in the spanwise direction and relatively constant in the streamwise
direction. No local shocks interacting with the streamwise structures are evident in these
images.
144
Spanwise and streamwise cross sections through the plan-view images are shown
in Figure 6.14. Here, there is substantial evidence to support the impression that the
streamwise structures are mostly streamwise uniform (images (a), (b), and (c)). The few
streamwise cross-sections, which exhibit some ramps, are mostly correlated with the
edges of these streamwise structures existing within the cut (images (e) and (d)).
The layer also shows significant spanwise ramps in mixture fraction profile in all
the images. These ramps are consistent with spanwise-localized streamwise structures
engulfing and mixing low-speed fluid in streamwise oriented stream tubes. Therefore, as
these structures continue to propagate downstream, the stepped spanwise profiles caused
by the gross entrainment behavior of the streamwise structures will be smoothed out into
spanwise ramps. It is interesting to note that for every spanwise cross section that exhibits
these ramps, there is another that does not. This reinforces the idea that these streamwise
structures, while longer than one to two average mixing layer thicknesses, are still
bounded in their downstream extent, and the effect of their mixing is still localized in the
streamwise direction.
6.2.3 END VIEW Five instantaneous end view images for the Mc = 2.64 condition are shown in
Figure 6.15. The field of view extends from z = -2.6 cm to 2.6 cm and from y = -1.8 cm
to 0.9 cm, and the flow is into the page. These images are taken at x = 18 cm
downstream of the splitter tip. Images are corrected for laser sheet non-uniformity using
the cross-stream intensity distribution of the average image normalized to the cross-
stream scalar distribution as found in the side view images. The peak SNR for these
images is 20, owing to the smaller field of view and differences in the imaging setup
from the side view images. In the end view setup, the laser light and resulting
fluorescence both travel through the single pane side window, whereas in the side view
setup, the laser must pass through two windows of fused silica. Similar to the previous
acetone PLIF end view images, these are corrected for off-axis imaging using the same
technique as described in Chapter 5.
145
The images in Figure 6.15 demonstrate the varied states of the streamwise
structures as they propagate downstream. Image (e) shows two streamwise structures that
have likely just entrained large amounts of low-speed fluid and transported it in an
engulfing motion to the high-speed side of the layer. Image (b) shows the same
characteristics, but the streamwise structures imaged here are more mixed. This trend of
structures appearing more mixed as they propagate downstream is reinforced by images
(a), (c), and (d). Though each image is from a separate firing of the shock tunnel and not
correlated with the others, the images do display streamwise structures at varying degrees
of internal mixedness, which is consistent with the engulfing entrainment and subsequent
mixing model.
Little organized shock structure is evident in this view when compared with the
Mc = 1.7 end views. Some individual structures appear to have “halos” of signal levels
above the χ = 1 level (images (a), (c), and (d)), which suggests that bow shocks are
forming around structures that protrude into the high-speed stream. However, no well-
defined bow shocks are in evidence. This is probably due to the reduced strength of these
shocks, which yields minimal local pressure perturbations in the flowfield, and thus
smaller local increases in the LIF signal.
Cross-stream and spanwise cross sections through the end view images are shown
in Figure 6.16. The cross-stream cuts demonstrate streamwise structures in various stages
of internal mixedness. The cuts from images (b) and (c) display sharp drop-offs in cross-
stream scalar profiles consistent, with the engulfing streamwise structures that have yet to
mix internally. Cross-stream sections from the other images show a layer, which has well
defined cross-stream ramps, consistent with the results from the side view images.
The upper spanwise cuts are taken at the 50% level of the average mixture
fraction profile, while the lower spanwise cuts are taken at the 20% level. Both spanwise
ramps and spanwise uniform cross-sections are evident in the spanwise mixture fraction
profiles, which is similar to the results of the plan view images. Furthermore, now these
spanwise-localized structures that cause the spanwise ramps are clearly seen.
146
6.3 MIXED FLUID FRACTION IMAGING In order to determine the reaction rates in a chemically active mixing layer, the
level of mixedness of the two streams must first be known. Dimotakis (1991) stated that
the process of chemical reaction in shear layers occurs in three distinct steps, as discussed
in Chapter 1. In this section, measurements of the mixing efficiency (δm/δ) are made at
the high compressibility condition using a cold chemistry technique with NO PLIF.
6.3.1 EXPERIMENTAL APPROACH
Purely passive scalar imaging techniques measure the extent of a scalar as it
mixes in the flow. Many previous passive scalar experiments (Batt, 1977; Koochesfahani
and Dimotakis, 1986; Clemens, 1991; Fiedler, 1974) suffer from poor resolution
measurements of molecular mixing because the Batchelor scale (or scalar mixing length)
is much smaller than the largest length scale in the probe volume. The Batchelor scale is
given by
2/14/3Re −−
≈ ScCB ωωδδλ (6.9)
where Cδ is an adjustable constant (usually 25 for mixing layers, but assumed here to be
unity as a conservative estimate), δω is the vorticity thickness of the shear layer, Reω is
the Reynolds number based on the vorticity thickness, and Sc is the Schmidt number
(Breidenthal, 1981; Dowling and Dimotakis, 1990). For the Mc = 2.64 condition and
using the effective resolution of the NO PLIF technique, L/λB ~ 100 for this flowfield.
This represents a resolution very similar to the aforementioned passive scalar
measurements. A danger of under-resolved passive scalar measurements is that they tend
to over-predict the amount of mixed fluid, since the probe will average over the mixed
and unmixed fluid. Thus, the technique cannot discern between a stirred volume (where
the two streams have not molecularly mixed) and a mixed volume.
To combat the severe resolution requirements for mixed fluid measurements,
chemical reaction techniques were developed in which a reactant is seeded into each
stream. Then, where molecular mixing takes place, a reaction generates product, and the
147
product can be detected (i.e. temperature rise, new species, etc.). Even though the
resolution has not changed, the correct amount of mixed fluid can be measured,
independent of resolution. However, while chemistry can serve as an accurate marker in
low-speed flows, Damkohler number effects become important at supersonic speeds
where chemical time scales may no longer be fast as compared to fluid mechanical time
scales. The eddy rollover time (δ/∆U) for the Mc = 2.64 condition is 7 µsec, which places
a stringent requirement on the chemical timescale (τchem < 700 nsec) in order to retain the
fast chemistry assumption (Da > 10).
Since the requirements for fast chemistry using chemical reactions are quite
severe, a cold chemistry method (Clemens and Paul, 1995) is used to attain resolution
independent mixing measurements. Cold chemistry refers to the quenching of the NO
fluorescence by another gas, which occurs on a timescale of ~ 10 nsec (Paul and
Clemens, 1993), thus mimicking a fast, fuel lean chemical reaction with no heat release.
The quenching process typically requires 100-1000 collisions to occur, which means for
the high compressibility conditions in this study, the effective Da is greater than 20 for
the cold chemistry technique.
Previous cold chemistry measurements of Clemens and Paul (1995) and Island
(1997) utilized the fact that at near atmospheric pressure, the quenching effects in mixing
layers cause a near binary signal to occur. Mixed seeded fluid is both diluted and
quenched by mixing with the lower stream, which contains a strong quenching partner. In
fact, signal reductions of a factor of 15 are possible for just 10% mixing of the fast stream
with the slow stream using oxygen in the lower air stream as the primary quencher
(Island, 1997). For that technique, fluid volume elements, which contained high-speed
mixture fractions less than 90%, were viewed by the imaging system as providing no
signal, giving a binary signal associated with mixed fluid. Consequently, a large number
of images were required in order to compute the mean of the signal, interpreted as the
probability of finding pure high-speed fluid at that transverse location. Flip experiments,
where the lower stream is seeded with NO, provide the probability for finding low-speed
fluid in the mixing layer. The difference between the sums of these two probabilities
from unity yields the mixed fluid fraction in the mixing layer.
148
In a shock tunnel experiment, where repetition rates, and thus data yields, are low
and shot costs are high, a variation on the cold chemistry technique is necessary to
determine the mixed fluid fraction in the mixing layer. Figure 6.17 shows the variation in
the fluorescence quantum yield with increasing low-speed fluid fraction. The calculation
is performed for P = 0.1 atm at T = 400 K, with the error bars denoting the fluorescence
yield variation in the 300 – 550 K temperature range. The quenching cross sections are
taken from Paul, 1993 and McDermid and Laudenslager, 1982 with σO2 = 25.5 Å2
(independent of temperature), σN2 = 0.02 Å2, and σCO2
= 69.3 - 0.028T [K] Å2.
However, the actual cold chemistry signal depends on both the dilution of the
high-speed stream due to mixing and the quenching of the fluorescence. Figure 6.18
shows the change in the fluorescence signal with increasing low-speed fluid fraction. The
high-speed stream is at exactly the free-stream conditions in the Mc = 2.64 mixing layer.
The iso-quenching environment shows that passive scalar signal decreases linearly with
mixture fraction simply due to dilution. The two other quenching environments produce a
non-linear signal decrease with increasing mixing. For example, a parcel of fluid which is
100% mixed at a mixture fraction of 0.3 will give a normalized fluorescence signal of
70% for the iso-quenching environment, 30% for air mixing, and 4% for CO2 mixing.
Since the SNR for these images is approximately 12, the cold chemistry results will be
ambiguous for fluid parcels mixed above 93% low-speed fluid for the iso-quenching
environment, above 70 % for 100% air, and above 17% for 100% CO2 as these regions
will not provide sufficient LIF signal. However, what is important to note from these
simulations is that the LIF signal from these quenching environments cannot be assumed
to be binary at these low pressures. Appreciable signal will be achieved for some to most
of the transverse positions in the layer, even using a very strong quencher like CO2.
Consequently, the previous use of cold chemistry and flip experiments to generate local
probabilities of high- and low-speed fluid cannot be implemented. Thus, a new
interpretation of the cold chemistry results is needed to yield information regarding the
mixed fluid fraction in the layer.
149
6.3.2 INTERPRETATION OF COLD CHEMISTRY RESULTS AT LOW PRESSURES
Since the instantaneous cold chemistry images can no longer be assumed to be
binary when operating at lower test section pressures, the average mixedness of the shear
layer must be approached by looking at the comparison of the ensemble averaged
transverse LIF profiles, normalized by the high-speed signal. Since the quenching term
from the 4-level model appears to be well fit by 1/(1+CQχL), where CQ is related to the
total quenching rate (Q21) of the lower free stream (Figure 6.17), the fluorescence signal
from a region of fluid can be modeled quite simply by Equation 6.10,
( )( )( )
( )
LLQ
HLHL
Lf
f
iCSS
δχ
δχδχ
χ +
−
+−−=
= 11
110
(6.10)
where χL is the total local mole fraction of low-speed fluid (mixed and unmixed), δH is
the amount of local high-speed fluid which is molecularly mixed, and δL is the amount of
local low-speed fluid which is molecularly mixed. Accordingly, the two terms in
Equation 5.8 can be understood quite simply. The first term arises from parcels of diluted
high-speed seeded fluid that is not molecularly mixed. These regions in the collection
volume sense only the dilution of the tracer molecule and not the quenching of the LIF
signal due to the molecules from the lower stream. The second term in Equation 5.8 is the
signal that occurs from regions of fluid where the high- and low-speed molecules are
mixed, and thus the region feels both the effects of dilution and quenching.
6.3.3 MIXED FLUID FRACTION RESULTS
Since Equation 6.10 contains three unknown variables, its solution requires three
separate realizations of the high compressibility mixing layer condition using different
quenching partners. In a shock tunnel facility, the switch of quenching partners, and thus
low-speed side gas, is very simple and cost effective due to the low volumes required.
The same free stream from the Mc = 2.64 condition is allowed to mix with other free
streams containing 100% Air and 100% CO2. Eight images for each alternative
quenching partner are taken and ensemble averaged. The average images are then column
averaged and normalized by the high-speed value, providing a single, averaged and
150
normalized transverse LIF profile for each quenching partner, representing an average of
1400 image columns or ~ 31 mixing layer thicknesses. The results are shown in Figure
6.19. As expected, the fluorescence signal from the 100% CO2 mixing partner is heavily
quenched, while the signal from the 100% Air mixing is less so. The profiles have been
corrected for the different growth rates of the three mixing layer conditions. Profile
similarity is discussed further in the error analysis in Section 6.4.4.
Using these three profiles, and the quenching constants CQ taken from Figure 6.17
(CQ (CO2) = 52.5, CQ (Air) = 4.54), the values of χL, δH, and δL can be solved for as a
function of transverse position (Figure 6.20). Using combinations of these variables as
suggested by Equation 6.10, the amount of mixed and pure, high- and low-speed fluid
plus the overall amount of mixed fluid can be obtained (Figure 6.21). The mixed fluid
fraction exhibits asymmetric behavior with respect to the center of the mixing layer,
which is consistent with other cold chemistry and passive scalar results (Island 1997;
Clemens and Paul, 1995; Frieler, 1992). Also, the probability of finding purely mixed
fluid at the center of the layer reaches 100% and the probability of finding mixed fluid is
greater than 90% for a broad region of the layer (-0.2 < η < 0.18). The results are in
agreement with trends seen by previous researchers (Island, 1997; Hall, 1991), where the
mixed mean composition near the high-speed edge of the mixing layer increases with
compressibility, thus leaning the mixed fluid profile towards the high speed stream.
The integral of the mixed fluid fraction (Pm) over all space yields the mixing
efficiency of the layer. The value of this efficiency is achieved using Equation 6.11:
( )( )∫∞
∞−
∂−+= ηδχδχδ
δHLLL
m 1 (6.11)
where η is the transverse direction variable (y) normalized by the shear layer thickness,
as discussed in Chapter 1. The mixing efficiency determined for the Mc = 2.64, Re = 1.88
x 106 condition is plotted along with other resolution-independent results from various
researchers versus Reynolds number in Figure 6.22 and versus convective Mach number
is Figure 6.23.
The apparent trend of increased mixing efficiency with compressibility seen in the
experiments shown is likely attributable to both Reynolds number effects and Mc. As
shown in Figure 6.22, the present result along with previous experimental results show a
151
strong correlation of increased mixing efficiency with Re, approximately 25% per
decade. In the study by Island (1997), several of the experiments were conducted at
different compressibilities, while holding the Re const, and this study concluded that the
strong correlation between δm/δ and Re seemed to largely account for the change in
efficiency seen with compressibility. The result from the current work is consistent with
that finding, reinforcing the conclusion that mixing efficiency is only slightly dependent
on compressibility.
The conclusion that mixing efficiency increases with increasing Re is contradicted
by some other studies (Hall, 1991; Bond et al., 1998) Those data in Figure 6.22 which
show a decrease in mixing efficiency with Re are believed to suffer from finite rate
chemistry effects. Island (1997) explored fully the effect of finite rate chemistry on mixed
fluid measurements and concluded that slow chemistry and low φ results in a major
under-prediction of mixed fluid, which can be as high as 40% near the edges of the shear
layer. In higher compressibility mixing layers like the ones examined here, the fluid is
well mixed throughout the layer including the edges of the layer, thus slow chemistry will
have an increasingly detrimental effect on the accuracy of the mixed fluid fraction at
increasing compressibility. The cold chemistry technique shows a much reduced bias due
to the very fast nature of the chemistry (Island, 1997, Clemens and Paul, 1995); thus the
results from this technique are relied on for the conclusion of increased mixing efficiency
with Re.
Finally, a profile for the mixed mean composition can be calculated by Equation
6.12, where P1 is the local probability of pure high-speed fluid and ξ is the total amount
of high-speed fluid at any location.
( )( ) ( )
( )( ) ( )( )( )
( ) HLLL
HLL
mm P
Pδχδχ
δχχ
η
ηηξηξ
−+
−−−−−
=
1111
or 1 (6.12)
This profile is shown in Figure 6.24. Significantly, the shape of this profile nearly
matches the shape of the overall high-speed mole fraction in the shear layer, because the
two stream dependent mixing efficiencies (δH and δL) rapidly increase to unity towards
the center of the layer. At this very high compressibility condition, the mixed mean is
more continuous than at lower compressibilities, when it tends to a preferred value. It is
also worth noting that little of the mixed fluid, on average, is comprised of more than
152
90% free stream fluid. The mean of the mixed mean composition for the Mc = 2.64
mixing layer is 0.45.
6.3.4 ERROR ANALYSIS OF MIXED FLUID FRACTION RESULTS
In addition to the excellent error analysis presented by Island (1997) in his thesis
on the accuracy of cold chemistry techniques, the current measurements of mixed fluid
fraction are subject to slightly different systematic error sources caused by variations in
some of the flow parameters due to the substitution of new quenching partners in the
lower stream. Since the low-pressure quenching technique is not required to generate a
binary signal from pure and mixed fluid parcels, the errors associated with under-
predicting the amount of mixed fluid near the edges of the layer are removed. However,
errors created by the imaging system’s dynamic range and signal-to-noise ratio are still
important.
By substituting the different quenching partners into the lower stream of the
mixing layer, several of the important similarity parameters of the mixing layer are
altered slightly. Table 6.2 shows the comparison of these different similarity parameters.
Side 2 Mc Re s
CO2 2.86 2.45x106 1.51
1.2% O2 +Argon 2.64 1.88x106 1.37
Air 2.55 1.75x106 1.10
Table 6.3 Comparison of similarity conditions for three different quenching
partners for the low pressure, cold chemistry technique.
For this particular cold chemistry technique, error sources due to incomplete similarity
will come from variations of the total high-speed fluid profile (χH(η)) with changing
convective Mach number, Reynolds number and density ratio. Clemens and Paul (1995)
show weak evidence for the steepening of the mean composition profile as Mc is
increased from 0.3 to 1.3, but even over this wide compressibility range, the difference is
very slight and within the error of their measurement. Island (1997) saw mean
153
composition profiles which exhibited similar shapes for Mc = 0.25 to 0.79 and Reynolds
numbers from 100,000 to 1 million with no clear trend with Mc or Re in evidence. Both
sets of results also experience variations in the density ratio of greater than 30%, similar
to the 34% variation in this study. Frieler and Dimotakis (1990) also showed no density
ratio dependence on their mixed fluid fraction results. The mixed fluid profiles displayed
a slightly increasing degree of asymmetry at large density ratios, but the integral stayed
constant. Thus, it is possible to conclude that passive scalar profiles show little variation
over wide variations in Mc, Re, and some slight variation with density ratio.
Since the scalar profiles will exhibit good similarity for the three different low-
speed gas choices, the variation in the recorded LIF profiles will be due solely to the
quenching term. Therefore, the accuracy of the quenching term in Equation 5.8 plus the
accuracy of the imaging system dictate the accuracy of the mixed fluid measurement in
Section 6.4.3. The error sources in the quenched signal are primarily due to resolution
bias and systematic temperature errors. Using the mean temperature profile as expected
from the Crocco-Buseman solution for a highly compressible shear layer, the average
temperature in the shear layer could be as high as 1000 K for this Mc = 2.64 case.
However, since the same average temperature profile occurs for each of the quenching
partners, only the variations in the quenching terms of CO2 and O2 with temperature are
important. This would mean that the error in the fluorescence quantum yield for the CO2
profile would be 15% and 10% for the Air profile at χL = 0.8 when compared to the iso-
quenching environment, owing to variations in the CQ value with temperature. These
local errors scale directly with low-speed mole fraction, and non-linearly with
temperature. Consequently, the amount of mixed fluid will be under-predicted for local
temperatures above 400 K and over-predicted for temperatures below that value. For the
high-speed side of the layer, the error associated with the temperature variation will be
small (< 5%) due to low levels of mixing with low-speed fluid, causing the amount of
mixed fluid to be under-predicted by less than 4%. And for the lower edge of the layer (η
< -0.3), the amount of mixed fluid will be over-predicted by 8-12%.
The imaging system’s dynamic range and SNR also play a role in the accuracy of
the cold chemistry technique. On the low-speed side of the layer, where the signal is
reduced below the imaging threshold (which for these experiments is 8% high-speed
154
fluid), these areas of the layer will be interpreted as being 100% low speed fluid in the
instantaneous cross-stream profiles. Thus for the averaged profiles, the local amount of
high-speed fluid in the layer on the low-speed edge will be slightly under-predicted.
Improvement could be achieved by collecting more images to increase the SNR by more
ensemble-averaged images.
Taking these errors from the temperature dependent quenching and imaging
system SNR and integrating them across the mixed fluid profile shown in Figure 6.21,
this cold chemistry technique using the three quenching partners listed in Table 6.3 will
likely over-predict the mixed fluid fraction in the shear layer by 3-4% and amplify the
mixed fluid profile on the far low-speed side of the layer by as much as 8%. This error
will then skew the mixed fluid profile slightly, but the overall mixedness of the layer is
captured within the SNR of the imaging system.
6.4 DISCUSSION OF LARGE-SCALE STRUCTURE AND ENTRAINMENT
Both the imaging results in Chapter 5 for the Mc = 1.71 condition and the results
shown in this chapter suggest that as the convective Mach number is increased, highly
three dimensional streamwise structures, which preferentially entrain fluid from the
closest free stream, dominate the flow topology. This view is strongly supported by the
simulations of Sandham and Reynolds (1989), Chen (1991) and to some extent the co-
layer simulations of Day (1999). They all report the dominance of highly three-
dimensional streamwise structures at high compressibilities, which have both a spatial
and convective velocity preference in the layer. The large-scale scalar structures seen in
all three views are thin in the spanwise dimension and elongated in the streamwise
direction, probably stretched downstream by the bow shocks that emanate from
interaction between the structure and the fast stream. This type of streamwise structure is
consistent with those seen in the DNS of a highly compressible annular mixing layer by
Freund et al. (2000a).
The larger the structures relative to the shear layer thickness, the less well mixed
they appear to be, in agreement with the “sonic eddy” concept proposed by Breidenthal
(1992). Also, the structure sizes in the transverse (cross-stream) direction appear to
155
decrease with increasing convective Mach number. A decrease in transverse length scale
is consistent with the development of “co-layers” predicted by linear stability analysis
(Day et. al, 1998); however, little experimental evidence of the separation of the mixing
layer into distinct fast and slow modes is observed. Similar to the difficulty in identifying
distinct oblique modes in moderately compressibility shear layers, it is possible that
distinct co-layers may be masked in highly compressible turbulent flow. However, the
presence of co-layers would lead to a decrease in the transverse structure size and an
increase in three-dimensionality, which is experimentally observed.
The entrainment motions perceived from the images recorded in this study show
that streamwise structures tend to entrain pure fluid from the closest free stream and
rapidly traverse the shear layer before the structure has become well mixed. However, as
the convective Mach number is increased, the streamwise structures tend toward a scalar
composition similar to the mean composition of the shear layer, implying that as the
structure size continues to decrease, the scalar variance must also decrease as the mixing
distances have shortened and structures tend to reside nearer the free streams.
156
20x10-3
15
10
5
0
χ (M
ole
Frac
tion)
1.5x10-3 1.00.50.0Time (sec)
3500
3000
2500
2000
1500
1000
500
Tem
pera
ture
(K)
NO O O2 Gas Temperature
Minimum Residence Time = 255 µsec
Figure 6.1 CHEMKIN simulation of shock heating 2% NO in Argon to 3600 K and 30 atm. Simulation follows a Lagrangian particle which resides in the stagnation conditions for the full test time (1.5 msec) before traveling through the changing thermodynamic conditions of the nozzle.
10x10-3
8
6
4
2
0
χ (M
ole
Frac
tion)
1.5x10-3 1.00.50.0Time (sec)
3500
3000
2500
2000
1500
1000
500
Tem
pera
ture
(K)
Beginning of Nozzle
NO O O2 NO2*100 Gas Temperature
Minimum Residence Time = 160 µsec
Figure 6.2 CHEMKIN simulation of shock heating 5% Dry Air in Argon to 3600 K and 30 atm. Simulation follows a Lagrangian particle which resides in the stagnation conditions for the full test time (1.5 msec) before traveling through the changing thermodynamic conditions of the nozzle.
157
1500
1000
500
0
LIF
Inte
nsity
(Arb
.)
700600500400300Temperature (K)
Q1+P21 (12.5) Q1+P21 (14.5) Q1+P21 (16.5) Q1+P21 (18.5) Q1+P21 (19.5) Q1+P21 (20.5)
Q1+P21 (21.5)
Figure 6.3 Variation of LIF signal for NO tracer species pumped by a resonant broad laser pulse at different A←X, Q1+P21 rotational transitions as computed using the 4-level LIF model from Appendix C. Simulations are performed at 0.1 atm with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2.
10
8
6
4
2
0
LIF
Inte
nsity
(Arb
.)
225.90225.88225.86225.84225.82225.80Vacuum wavelength (nm)
350K 450K 550K 650K
Q1+P 21(18.5)
Q1+
P 21(1
9.5)
Figure 6.4 Simulation excitation spectra of the A2
Σ+←X2
Π (0,0) band of NO in the neighborhood of the Q1 + P21 (J´´ = 18.5) transition for different temperatures using the NO spectra simulation code of Bessler and Sick (Private communication, 2001).
158
1.10
1.05
1.00
0.95
0.90
0.85
0.80
Scal
ed L
IF In
tens
ity (A
rb.)
700600500400300Temperature (K)
Q1+P21 (16.5) Q1+P21 (17.5) Q1+P21 (18.5) Q1+P21 (19.5) Q1+P21 (20.5) Q1+P21 (21.5)
Figure 6.5 Variation of LIF signal normalized by the maximum signal for each transition to show the applicability of each line for temperature insensitive imaging of mole fraction. Computed using the 4-level LIF model presented in Appendix C. Simulations are performed at 0.1 atm with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2.
30
20
10
0
% E
rror i
n χ
H
0.80.60.40.20.0χL
Iso-quenching environment (98.8% Argon, 1.2% 02) 100% Argon
x10
Figure 6.6 Computation of the relative error in determining the mixture fraction of high-speed side fluid (χH) in the shear layer as it mixes with low speed fluid (χL) when not using an iso-quenching environment. Quenching cross-sections are taken from McDermid and Laudenslager (1982), and the simulations are performed at 0.1 atm with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2. The error bars denote the range of errors for 300 – 600 K.
159
1400
1200
1000
800
600
400
200
LIF
Inte
nsity
(Arb
.)
0.50.40.30.20.1Pressure (atm)
300 K 350 K 400 K 450 K 500 K 550 K
Figure 6.7 Variation of the LIF signal intensity with changing pressure. The simulations are performed for a variety of temperatures with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2. The vibrational temperature of the NO is fixed at 2000 K, which is supplied by a simple vibrational relaxation formulation and simulation (Equation 6.1)
1600
1400
1200
1000
800
600
400
200
LIF
Inte
nsity
(Arb
.)
0.50.40.30.20.1Pressure (atm)
LIF Cell Data (296 K) 4 - level Model (300 K)
Figure 6.8 Variation of the LIF signal intensity with changing pressure. The simulations are performed for 2000 ppm in Argon at 300 K and compared to experimental static LIF cell data of the same mixture at 296 K. The vibrational temperature of the NO is 300 K.
160
10x103
8
6
4
2
LIF
Inte
nsity
(Arb
.)
4x106 321Spectral Power (W/cm2cm-1)
300 K 350 K 400 K 450 K 500 K
Figure 6.9 Variation of the LIF intensity with increasing spectral power (W/cm2 cm-1) in the temperature range of interest (300 – 600). Simulations are performed at 0.1 atm with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2.
14x103
12
10
8
6
4
2
LIF
Inte
nsity
(Arb
.)
4x106 321Spectral Power (W/cm2cm-1)
4-level Model (400 K) Weak Excitation Model
Figure 6.10 Comparison of the LIF intensity as computed using the 4-Level LIF model and the weak excitation model (Equation 6.3) to show the effects of saturation in low pressure flowfields. Simulations are performed at 0.1 atm with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2.
161
Figure 6.11 Instantaneous passive scalar side view images at Mc = 2.64 with fast-side seeding. Lower
right hand corner is a 12-frame average image. Flow is left to right with the high-speed stream on top. The imaged region is ∆x = 4.3 cm and ∆y = 4.7 cm and centered 18 cm downstream of the splitter tip and aligned with the centerline of the test section. The images are uncorrelated in time.
0
1 Mole Fraction
162
(a) (b)
0 1-3
-2
-1
0
1
0 1-3
-2
-1
0
1
16 18 20
0
1
16 18 200
1
0 1-3
-2
-1
0
1
0 1-3
-2
-1
0
1
16 18 20
0
1
16 18 200
1
(c) (d)
0 1-3
-2
-1
0
1
0 1-3
-2
-1
0
1
16 18 20
0
1
16 18 200
1
0 1-3
-2
-1
0
1
0 1-3
-2
-1
0
1
16 18 20
0
1
16 18 200
1
Figure 6.12 Instantaneous passive scalar side view images at Mc = 2.64 with fast-side seeding. Flow
is left to right with the high-speed stream on top. The imaged region is ∆x = 4.3 cm and ∆y = 4.7 cm and centered 18 cm downstream of the splitter tip and aligned with the centerline of the test section. The images are uncorrelated in time. Plotted alongside each image is the high-speed mixture fraction versus distance (in cm) along the streamwise and cross-stream lines indicated. Color map interpretation of mixture fraction is shown to the right of images b and d.
0
1 Mole Fraction
163
Figure 6.13 Instantaneous passive scalar plan view images at Mc = 2.64 with fast-side seeding. Lower right hand corner is a 6-frame average image. Flow is left to right. The imaged region is ∆x = 4.7 cm and ∆z = 3.8 cm and centered 18 cm downstream of the splitter tip and aligned with the centerline of the test section. The images are uncorrelated in time.
0
1 Mole Fraction
164
(a) (b)
0 1
-1
0
1
0 1
-1
0
1
-1 0 1
0
1
-1 0 10
1
0 1
-1
0
1
0 1
-1
0
1
-1 0 1
0
1
-1 0 10
1
(c) (d)
0 1
-1
0
1
0 1
-1
0
1
-1 0 1
0
1
-1 0 10
1
0 1
-1
0
1
0 1
-1
0
1
-1 0 1
0
1
-1 0 10
1
(e)
0 1
-1
0
1
0 1
-1
0
1
-1 0 1
0
1
-1 0 10
1
Figure 6.14 Instantaneous passive scalar plan view images at Mc = 2.64 with fast-side seeding. Flow is left to right. The imaged region is ∆x = 4.7 cm and ∆z = 3.8 cm and centered 18 cm downstream of the splitter tip and aligned with the centerline of the test section. The images are uncorrelated in time. Plotted alongside each image is the high-speed mixture fraction versus distance (in cm) along the streamwise and spanwise lines indicated. Color map interpretation is shown to the right of images b and d.
0
1 Mole Fraction
165
Figure 6.15 Instantaneous passive scalar end view images at Mc = 2.64 with fast-side seeding. Lower
right hand corner is a 6-frame average image. Flow is into the page with the high-speed stream on top. The imaged region is ∆y = 2.7 cm and ∆z = 5.2 cm and centered 18 cm downstream of the splitter tip. The images are uncorrelated in time.
0
1 Mole Fraction
(a)
(b)
01
-10
01
-10
-2-1
01
2
01
-2-1
01
201
01
-10
01
-10
-2-1
01
2
01
-2-1
01
201
(c)
(d)
01
-10
01
-10
-2-1
01
2
01
-2-1
01
201
01
-10
01
-10
-2-1
01
2
01
-2-1
01
201
Fi
gure
6.1
6 In
stan
tane
ous p
assi
ve sc
alar
pla
n vi
ew im
ages
at M
c = 2
.64
with
fast
-sid
e se
edin
g. F
low
is in
to th
e pa
ge w
ith th
e hi
gh-s
peed
stre
am o
n to
p. T
he
imag
ed re
gion
is ∆
y =
2.7
cm a
nd ∆
z =
5.2
cm a
nd c
ente
red
18 c
m d
owns
tream
of t
he s
plitt
er ti
p. T
he im
ages
are
unc
orre
late
d in
tim
e. P
lotte
d al
ongs
ide
each
im
age
is t
he h
igh-
spee
d m
ixtu
re f
ract
ion
vers
us d
ista
nce
alon
g th
e st
ream
wis
e an
d sp
anw
ise
lines
ind
icat
ed.
Col
or m
ap
inte
rpre
tatio
n is
show
n in
bet
wee
n im
ages
(c) a
nd (d
).
01Mole Fraction
167
2
3
4
56
0.1
2
3
4
56
1
φ (χ
L)/φ
{χ
L= 0
)
0.80.60.40.20.0χL
98.8% Argon, 1.2% O2 100% Air 100% CO2
1/(1+CAirχL) 1/(1+CCO2
χL)
Figure 6.17 Variation in the fluorescence quantum yield with increasing low-speed fluid fraction (χL). The calculation is performed at 0.1 atm and 400 K with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2. The error bars denote the fluorescence yield variation in the 300 – 550 K temperature range.
4
68
0.01
2
4
68
0.1
2
4
68
1
S f(χL)
/Sf(χ
L =
0)
0.80.60.40.20.0χL
100% Air 100% CO2 98.8% Argon, 1.2% O2
Figure 6.18 Variation in the overall LIF signal intensity including the effects of mixing with increasing low-speed fluid fraction (χL). The calculation is performed at 0.1 atm and 400 K with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2 mixing with different quenching partners. The error bars denote the fluorescence yield variation in the 300 – 550 K temperature range.
168
1.0
0.8
0.6
0.4
0.2
0.0
Scal
ed L
IF S
igna
l
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6(y-y0.5)/δ
98.8% Argon, 1.2% O2 100% Air 100% CO2
Figure 6.19 Scaled mixture fraction profiles showing the effect of strong quenching partners on the relative LIF signal for a Mc = 2.64 mixing layer. The high speed stream conditions are identical in each case (390 K, 0.1 atm with 2300 ppm NO in 95% Argon, 3.9% N2, and 0.8% O2)
1.0
0.8
0.6
0.4
0.2
0.0
Mol
e fra
ctio
n or
Mixe
dnes
s
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6(y-y0.5)/δ
χL δH δL
Figure 6.20 Profiles of the three important parameters involved in Equation 6.10: the mean mixture fraction profile (χL), the mean profile of high speed fluid which exists in a mixed state (δH), and the mean profile of low-speed fluid which exists in a mixed state (δL). Profiles are for a Mc = 2.64 mixing layer.
169
1.0
0.8
0.6
0.4
0.2
0.0
Mol
e Fr
actio
n
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6(y-y0.5)/δ
T otal Mixed Fluid Pure Low-Speed Pure High-Speed Mixed Low-Speed Mixed High-Speed
Figure 6.21 Profiles of the pure and mixed fluid states in a Mc = 2.64 mixing layer using the multiple quenching partners at low-pressure, cold chemistry technique.
1.0
0.8
0.6
0.4
0.2
0.0
Mix
ing
Effic
ieny
(δm
/δ)
3 4 5 6 7 8 9
105
2 3 4 5 6 7 8 9
106
2 3 4
Re
Mungal and Dimotakis (1985) Hall (1991) Frieler (1992) Clemens and Paul (1995) Bond et . al. (1996) Island (1997) Rossmann et . al. (2001)
Figure 6.22 Mixing efficiency as a function of Re for the present and other resolution-insensitive measurements.
170
1.0
0.8
0.6
0.4
0.2
0.0
Mix
ing
Effic
ieny
(δm
/δ)
0.12 3 4 5 6 7 8 9
12 3
M c
Hall (1991) Dimotakis and Leonard (1994) Clemens and Paul (1995) Island (1997) Rossmann et . al. (2001) Freund et .al. (2000)
Figure 6.23 Mixing efficiency as a function of Mc for the present and other resolution-insensitive measurements.
1.0
0.8
0.6
0.4
0.2
0.0
ξm(η
) or χ
L(η
)
-0.4 -0.2 0.0 0.2 0.4(y-y0.5)/δ
ξmavg = 0.45
ξm 1−χL
Figure 6.24 Mixed mean composition profile for the Mc= 2.64 condition using the multiple quenching partners at low-pressure, cold chemistry technique. Mean mixture fraction profile shown for comparison. Mean mixed mean composition is also shown.
171
CHAPTER 7
CONCLUSIONS AND FUTURE RECOMMENDATIONS 7.1 SUMMARY OF RESULTS
Turbulent mixing layers were investigated experimentally under a variety of
conditions to examine the effects of very high compressibility on the growth rate of the
shear layer and on the character and dynamics of large-scale structures. The experiments
contained in this thesis were conducted in a unique shock tunnel fed supersonic mixing
layer facility, which was able to reach compressibility levels previously unattainable in
conventional wind tunnels. The mixing layers in this study were formed using the output
of a supersonic nozzle, whose stagnation plenum conditions were created by the transient
flow of the shock tunnel, along with a subsonic manifold injector to create a planar, two-
stream shear layer. Because of the low gas volumes required for transient flow research, a
wide variety of high purity gases and gas mixtures were employed in both the fast and
slow streams (Air, N2, O2, CO2, Argon, He, NO, acetone vapor). The convective Mach
number and density ratio of the mixing layer were varied by using different shock tube
test gases, nozzle Mach numbers, and low-speed gases. Visualizations of the growth rate
and large-scale structures were performed by the use of a high-framing rate Schlieren
system along with PLIF of different tracer species in order to compare their behavior at
different compressibilities.
The main emphasis of this work was to gather information on the behavior of
highly compressible shear layers as compared to lower convective Mach number
conditions (Mc < 1), which have been extensively researched. The main results are:
1) A new shock tunnel was devised and constructed with the purpose
of generating high stagnation enthalpy conditions for long test-times. The
reflected region in the shock tunnel was then used to feed a two-
172
dimensional supersonic nozzle which created the high-speed side of a
mixing layer. Simple wave dynamics were used to create X-T diagrams to
predict test-times in the shock tunnel. The addition of viscous boundary
layer models with some experimental correlations for shock attenuation
and contact surface acceleration led to the ability to predict shock tunnel
test times to within 10% over a wide range of incident shock Mach
numbers and test gas compositions.
2) Measurements of the linear growth rate of the mixing layers in the
compressibility range of Mc = 0.85 to 2.84 were achieved using multiple
exposure, high framing rate Schlieren imaging. The visual thickness was
used as a measure of the growth rate. For the experiments, two different
nozzle Mach numbers were used (Mnoz = 3.5 and 5.1) giving growth rate
information at the same convective Mach number and different density
ratios to investigate the possibility of density ratio dependent compressible
growth rates. The growth rate was seen to decrease steadily to δ/δinc = 0.18
until Mc = 1.5, and then remain constant out to the highest level of
compressibility examined in this study. This behavior is consistent with
both computational and experimental results in the lower convective Mach
number regime, but suggests that at higher compressibilities, new large-
scale structures which are formed have a growth rate which is independent
of compressibility effects caused by the mean local shear.
3) Schlieren imaging of the shear layer showed that strong acoustic radiation
is possible for high compressibilities. The existence of these waves
correlated well both the density ratio and convective Mach number of the
shear layer, suggesting that the effect was solely due to changes in the
convection velocity of structures or instability waves. Using the
appearance or absence of acoustic radiation in either the upper or lower
streams, estimates of the convection velocity bounds were made over a
wide range of convective Mach numbers. Some evidence of the
173
importance of the lab frame sonic velocity on the instantaneous convective
Mach number was seen.
4) Using acetone PLIF, imaging of large-scale structures in a Mc = 0.85 and
1.71 shear layer was performed in three views (side, plan, and end) to
determine the character and dimensionality of the scalar field structures.
The lower compressibility conditions exhibited a ragged high-speed
interface with high levels of three-dimensionality. Elongated structures
that appeared to be a superposition of spanwise rollers and purely
streamwise structures dominated the topology. Little evidence of spanwise
uniformity was present in the plan and end views, consistent with previous
results. At the higher compressibility condition, the character of the shear
layer was quite different. Elongated streamwise structures were much
more prevalent, and the structures appeared to have fast entrainment
motions and slow internal mixing, consistent with the “sonic eddy” model.
Spanwise asymmetry was prevalent and large protrusions into the upper
free stream were observed in both the end and side views. Strong acoustic
radiation, stemming from the large-scale structures acting as bluff bodies
in supersonic flow, emanated from the high-speed/shear layer interface.
The shock structure was highly three-dimensional and was shown to exist
only above the shear layer boundary (plan and end views). No shocklets
were seen internal to the mixing layer, but the resolution of the images
was an order of magnitude greater than the Kolmogorov scale of the flow.
5) Simultaneously with the acetone PLIF results, an investigation was made
into the use of acetone as a tracer species in low-pressure shock tunnel
flows. The effects of condensation, pyrolysis, and low signal level on
overall image signal to noise ratio, were considered both theoretically and
experimentally. Thus, acetone is recommended as a flow tracer for shock
tunnel flows where the stagnation temperature is kept below 1150 K and
the nozzle Mach number is below 4.4. However, the difficulties in
174
interpreting single wavelength excitation acetone PLIF signals in a non-
isothermal environment limited the imaging results of this study to
qualitative examinations of large-scale structure.
6) Variations on established nitric oxide PLIF diagnostics were developed
and employed to quantitatively image mole fraction in a non-isothermal,
low pressure, mixing flowfield. Careful selection and modeling of the
electronic transition, quenching environment, and LIF signal allowed for a
straightforward diagnostic which yielded reasonable SNR and small
mixture fraction uncertainty over a wide temperature range. Extension of
this NO diagnostic to a modified cold chemistry method achieved
measurements of mixedness and mixing efficiency.
7) At the highest compressibility condition imaged by PLIF, Mc = 2.64,
streamwise structures, inclined in the flow direction from the side view,
were seen to dominate the scalar field of the shear layer. Mach wave
radiation was in evidence, though it did not appear as strong as in the
lower compressibility case. The streamwise structures appeared to engage
in engulfing entrainment motions, scooping large amounts of free-stream
fluid and transporting it across the shear layer before the structures were
internally mixed. The structures showed strong instantaneous cross-stream
ramps suggesting that gradient transport models for scalar mixing are
likely appropriate at high compressibilities where the structure size is
smaller and reduced pressure communication inhibits large-scale mixing.
Strong spanwise ramps were apparent in large-scale structures which had
yet to mix internally, further suggesting that the streamwise structures had
some associated spanwise motions.
8) The mixing efficiency of the Mc = 2.64 layer was captured using a
quantitative PLIF technique. The mixing efficiency of 0.64 at a Reynolds
number of 2.7 x 106 shows a slight increase with Re when compared to
175
other cold chemistry results. The result is also consistent with trends from
simulations of scalar mixing at high compressibility. Unfortunately, the
effects of high compressibility and high Reynolds number were not
decoupled, but prior data suggest that mixing efficiency increases
primarily with Re. The mixed mean fluid profile closely followed the
mean scalar profile, strongly suggesting that the scalar PDF for this
flowfield is “marching” with narrow peaks.
7.2 RECOMMENDATIONS FOR FUTURE WORK
The results described in this thesis suggest some new topics for study in
compressible mixing layers. Many instantaneous images of large-scale structure are
presented, but time sequence structure imaging proved elusive. A high power, high
repetition rate laser and framing camera PLIF setup could be utilized to create movies of
the highly three-dimensional structures as they propagate downstream. The framing
Schlieren movies, while providing a higher data yield of instantaneous images, did not
provide much information on the motion and character of large scale-structures due to
Schlieren’s line-of-sight integration. Using rapid imaging in the end view, full three-
dimensional reconstructions of structure could be performed by taking several snapshots
as the flow convects through the imaging plane. The need for movies is clear to be able to
judge the convection velocity of structures (Papamoschou and Bunyajitradulya, 1997)
and for the reconstruction of the flow field (Island et al., 1996; Thurrow et al., 2001).
Along the line of more complicated laser diagnostics, dual pulse PLIF imaging of
conserved scalar fields should be employed. Two-line instantaneous temperature imaging
could be performed using either acetone or nitric oxide as the tracer species. These
species could be seeded to equal number densities in the high and low speed streams,
allowing for high signal to noise ratio imaging of temperature, or only seeded on one side
of the mixing layer to provide an instantaneous measurement of both mixture fraction and
temperature (though at a weaker SNR on the unseeded side of the mixing layer). Low
signal difficulties in this supersonic flowfield can be reduced by operating the test section
at higher overall pressures. This comes at the expense of higher gas and diaphragm costs,
176
as the shock tunnel must be run with higher stagnation pressures; however, this change
has a very small effect on the test time of the shock tunnel. Experiments which attempt to
reconstruct the PDF of mixture fraction will meet with the frustration of low imaging data
yields with shock tunnels. Typically, greater than 2 kHz imaging systems are required to
achieve multiple realizations per run. The use of higher test-time wind tunnel devices,
such as free piston shock tunnels, could increase the amount of useful test time by a
factor of 4, but these devices still require fairly high data rates. While scattering
techniques (Mie or Rayleigh) provide an easy route to faster data collection, their non-
linear dependence on the local temperature makes them less attractive for large-scale
structure imaging.
While the imaging of large-scale structures performed produced some interesting
results, several additional questions have been raised. Large-scale streamwise structures
appear to entrain in engulfing motions with some helical motion, a hypothesis which
could be resolved by three-dimensional imaging. Also, the structure/acoustic field
interaction appears to be a function of density ratio as well as convective Mach number,
but a more complete study of this phenomenon coupled with dual-pulse PLIF imaging to
measure the true structure convection velocity would be useful to determine the character
of the instability waves contained in the shear layer. Also, extending experiments to very
high convective Mach number (Mc > 3) to look for the manifestations of co-layers would
also be worthwhile. Finally, a further examination of shocklet behavior must be
employed to differentiate between imbedded shocklets and radiated oblique shock waves;
this examination could also be undertaken in the Mc = 1.5 to 4 range. Finely resolved
PLIF measurements of the pressure/density field using a two-line technique could search
for local shocked regions in the flow, which would by systemic of shocklets. Also, higher
convective Mach number growth rate measurements could be used to determine the
existence of new instability modes, as the normalized curve for the outer modes begins to
increase above Mc = 3.0.
Additionally, NO PLIF diagnostics for mean mixed scalar quantities in shock
tunnel flows proved their utility at a single convective Mach number condition in this
thesis, and for low-compressibility shear layers in the work of Island (1997). This low-
pressure interpretation of cold-chemistry results could be employed in the compressibility
177
range of Mc = 1 – 3, extending the range of resolved scalar measurements and filling in
the trend of Reδ > 4 x 105. Coupled with higher test section pressures, the current shock
tunnel experimental setup could attain Reδ ~ 7 x 106 to determine whether the mixedness
of the shear layer attains an asymptotic value or continues to increase.
Finally, the effect of very high compressibility on combustion in shear layers
could be examined using this shock tunnel facility. Because of the low gas volumes and
short run times, combustion experiments are ideally suited for pulsed facilities. Building
on the work of Miller (1994) and Day (1998), the effect of compressibility and density
ratio coupled with heat release on large-scale structure dynamics and combustion
efficiency could be investigated. Also, the similarity between the effects of high
compressibility and high heat release levels, as suggested by linear stability analysis,
could be investigated. PLIF imaging of the hydroxyl radical (OH) could be employed to
look at zones of heat release, and CH fluorescence used to examine the flame sheet.
These visualization techniques at high compressibility levels could seek to realize the
structural changes to the mixing layer predicted by Planche (1992) and Day et al. (1998).
178
APPENDIX A
SIDE TWO FLOW INITIAL CONDITIONS
One potential source of systematic error in the shock tunnel driven mixing
experiments is the characterization of the fast and slow side velocities of the mixing layer
and any non-uniformities or fluctuations contained within their spatial profiles. This
section deals with the experimental calibration and analysis of the low-speed side
injection system. Data on the velocities achieved versus the tank stagnation pressure and
the resulting spatial velocity profile are shown and discussed.
A.1 INJECTION PRESSURE VS. ACTUAL FLOW RATE
For the low-speed side of the mixing layer, a fast injection system is required to
initiate the slow side flow rate before the arrival of the reflected shock at the endwall of
the shock tunnel. In the initial designs of the shock tunnel driven mixing layer facility,
the possibility of using a second shock tunnel to inject a variable Mach number flow for
the slow side of the mixing layer was suggested. While allowing further variability in
convective Mach number and density ratio, this design would have added significant
complexity to the already involved issues of flow timing and space considerations. Thus,
the dual-firing shock tunnel idea was abandoned for a fast injection system.
In order to make the injection system as responsive as possible, a large volume
gas cylinder (Praxair, ALM – 1030, 1 ft3 internal volume) was installed with two 12 mm
diameter lines between the tank and the test section. The large diameter and short lengths
(~ 0.75 m) of the injection lines allowed for a quick temporal response. The dual solenoid
valves had a nominal response time of 15 – 20 msec. As was described in Chapter 2, the
fill lines led to a manifold injection plate inside the shock tunnel.
The temporal response of the injection system was examined using the already
installed Schlieren system. Front tracking was employed using a helium-air mixture as
179
the injection gas. Velocities were estimated using the progress of the front versus time
from valve opening. This offered a first order estimate of the velocity on the lower speed
side versus the injection pressure in the tank as well as information on the initiation time
of the flow inside the test section. The results of the front tracking velocity method are
shown in Figure A.1.
Another method of estimating the velocity of the low-speed side gas in the test
section is to measure the decrease in stagnation pressure in the gas storage tank. If a
choked orifice exists in the flow (nominally the minimum diameter of the solenoid
valves, 9mm), the mass flow rate will scale linearly with the stagnation pressure. Thus,
the conservation of mass can be transformed into a differential equation describing the
decrease of stagnation pressure with time. However, due to the difficulty in measuring
the exact choking location and to potential losses in stagnation pressure along the
injection tubing due to viscous effects, the stagnation pressure must be continuously
monitored in the injection tank to relate it to the mass flow rate. The formula governing
the adiabatic discharge of a high pressure tank is shown in Equation A.1.
−
+
+−=
∂
∂ 11
000
12* γ
γ
γγ
VARTKP
tP (A.1)
Here, P0 is the stagnation pressure, T0 is the stagnation temperature, A* is the choked
orifice area, and V is the tank volume. The temporal solution of this equation yields a
simple exponential as long as the stagnation temperature inside the tank does not vary
greatly, which is consistent with a fast, low flowrate blow-down process. The right hand
side of Equation A.1 can be reduced to a characteristic time for the discharge of the tank
(Eqn. A.2),
τ
00 KPtP −
=∂
∂ (A.2)
where K is an efficiency coefficient. Since the exact measurement of the choking
diameter is inexact and the viscous loses are not well characterized, the coefficient will
be less than one, effectively limiting the maximum mass flow rate. A fit of the tank
pressure data (taken with a Setra 280E pressure transducer mounted to the tank with a ¼”
diameter pressure tap) to an exponential decay is shown in Figure A.2. The frequency
180
response of the pressure transducer, 200 Hz, is more than sufficient to resolve the change
in pressure. A single exponential decay rate matches the blowdown data with excellent
agreement. The decay rate should fundamentally be independent of stagnation pressure in
the tank if no viscous mechanisms are taken into account. However, as the stagnation
pressure, and thus the flow rate is increased, viscous effects will change for larger tank
stagnation pressures. As a result, the efficiency coefficient increases slightly with an
increase in stagnation pressure for all gases tested, Figure A.3.
The utility of the direct stagnation pressure measurements is the ability to convert
loss of stagnation pressure to mass flow rate. Then, assuming standard conditions for the
low speed stream in the test section, the velocity of the slow stream can be computed
from the conservation of mass. As a check of this velocity estimation technique, two
different methods of direct velocity measurements were used to verify the conservation of
mass results. The first method utilized was front tracking of a helium-air gas mixture
using the Schlieren system. Images of the test section were taken at different time delays
after the activation of the injection solenoid valves, and the average position of the
helium/air interface was estimated. At least six images taken at different time delays per
stagnation pressure condition were taken to estimate the velocity of the front. The front
tracking results for helium are compared with the velocity estimations from the
stagnation pressure monitoring technique in Table A.1. The agreement between the
predicted and measured velocity is very good for the helium case. However, this
technique is not available for argon or air injection as there is not a large index of
refraction disparity between these gases and the displaced air in the test section.
Another technique for measuring the flow velocity in the test section just
upstream of the splitter plate on the low speed side is a Pitot probe (Dwyer Instruments,
1mm tube) coupled to a low range differential pressure transducer (Setra Model P7D, 0 –
0.1” water). This technique offers a high accuracy and low noise measurement. However,
the rise time associated with the Pitot probe is fairly large (measured to be 0.83 ± 0.06
sec), so the signal derived from the probe will have a double exponential type behavior.
Since the test section velocity increases nearly instantaneously after the valves open, and
then decreases as the tank pressure decreases, the signal from the pressure transducer
should resemble equation A.3.
181
( ) ( ) ( )( )
pt
p tC
CC
V
ε
τ
ε
ε
τττ
==
−−−
−
=
and where
expexp1
1
(A.3)
In this equation, εp and εt represent the characteristic times (either rise or fall) of the Pitot
probe and the tank pressure respectively. The data from the Pitot probe are shown in
Figure A.4 for several different tank stagnation pressures. The time history of the probe
pressure is also fit very well by the double exponential formula of Equation A.3, yielding
simultaneous measurement of the probe response time and the tank blowdown time.
Results for the tank blowdown times agree very well with those measured inside the tank.
The voltage from the transducer is linearly related to the stagnation pressure in the probe,
which is related by to the flow velocity by P0 - P∞ = ½U2. For the helium mixture case,
the Pitot measured flow velocity is compared to that derived from the flow tracking
results in Table A.1, and the agreement is very good and within the uncertainties of each
measurement technique. Also, the velocities computed from the Pitot probe
measurements agree very well with the velocities computed from the tank blowdown data
(overall mass flow rate), Figure A.5. Consequently, the assumption of standard
conditions in the test section, and no large temperature changes, is validated. Thus for all
the shock tunnel runs used in this thesis, knowledge of the tank stagnation pressure and
gas composition was sufficient to determine the low speed velocity through these
correlations to within 5%.
One important issue germane to this appendix is the response of the efficiency
coefficient (K) to different mixtures of gases, especially those consisting of helium and
air. Using the same techniques described above of monitoring the stagnation pressure
time constant for several different stagnation pressures the following mixture rule was
∑=
igasiitot KK
χ (A.4)
derived using the data from mixtures of 75% helium, 25% air and 50% helium, 50% air
(Equation A.4). Since mixtures of other gases were rarely used, this correlation for
helium-air mixtures is sufficient for the results presented in this thesis. The comparison of
this mixture rule to the time constant data is shown in Figure A.3. There is excellent
agreement between the fit to the data and the fit derived from the mixture rule.
182
A.2 INLET VELOCITY PROFILE
After using the Pitot probe to measure the average velocity at the center of the
low-speed flow, the uniformity of the slow flow is also of great interest, so as not to
introduce some bias into all the mixing layer results due to a poorly-behaved side two
flow. Thus a set of stagnation pressure data was taken using a Pitot traverse for entire
low-speed flow area, after the subsonic contraction but before the splitter plate. The goal
was to try to quantify any large scale unsteadiness and any separation regions caused by
poor flow conditioning. The Pitot tube traces (as shown previously in Figure A.4) were fit
with the double exponential formula (Equation A.3) with the known tank discharge time
constant for the gas type and stagnation pressure and the known time response of the
Pitot tube. This allowed for an estimation of the flow velocity as a function of time.
Figure A.6 shows a two-dimensional map of the local velocity in the low-speed side of
the layer at 200 msec after the opening of the solenoid valves for 5 atm stagnation
pressure using air as the injection gas. This delay time of 200 msec is an average double
diaphragm opening time, thus the supersonic nozzle flow from the shock tunnel usually
meets the low speed flow near this instant in time. The spatial resolution of this velocity
measurement is 5 mm x 5mm.
Due to the limitations of the probe size, shape, and insertion strategy, not all
regions of the low-speed flow velocity were measured. However, the flow is fairly
uniform, with over 85% of the points within 5% of the maximum velocity value
measured. The standard deviation for all velocities is 6% of the maximum value. This
velocity map was measured for two different stagnation pressures with the injection of
either air or helium gas and was not seen to vary significantly for any of those
combinations of conditions. No significant velocity variations were seen during any of
the Pitot probe traces, suggesting that no large-scale unsteady behavior was present. The
hot spot (a region of high velocity) seen near the center of the channel was minimized
with the addition of the screens and perforated plate described in Chapter 2, but could not
be completely eliminated in the rather short flow development section of the low-speed
injection inside the mixing facility.
183
Collection Technique Stagnation Pressure (psia)
Front Tracking (m/sec)
BlowdownTime Constant (m/sec)
Pitot Probe (m/sec)
40 2.4 � 0.1 2.55� 0.06 2.5� 0.3 70 4.6� 0.1 4.67� 0.04 4.8� 0.2 90 6.1 � 0.2 6.02� 0.03 6.2� 0.2
Table A.1 Comparison of Front tracking, Pitot probe and tank blowdown measurement techniques
for determining the flow velocity on the lower speed side of the mixing layer.
0
0.04
0.08
0.12
0.16
0.2
0.24
40 50 60 70 80 90 100 110 120
Time (msec)
Dis
tanc
e (m
eter
s)
Po = 90 psia
Po = 70 psia
Po = 40 psia
Figure A.1 Lower speed side velocity derived by front tracking. A 10% helium, 90% air mixture is
injected from various stagnation pressures in the tank (P0) and the Schlieren system is
used to visually discern the progress of the front. The static pressure in the test section is
maintained at 1 atm.
184
5
4
3
2
1
0
Tank
Pre
ssur
e Tr
ansd
ucer
(Vol
ts)
43210Time (sec)
Tank Stagnation Pressure = 40 psia 45 psia 50 psia 55 psia 60 psia 65 psia 70 psia 75 psia 80 psia 85 psia 90 psia 95 psia
Figure A.2 Comparison of side two tank blowdown time constants with varying initial tank pressure.
Test section pressure maintained at 1 atm. Injection mass flow rate remains choked down
to the lowest injection pressure studied.
0.26
0.27
0.28
0.29
0.3
0.31
0.32
0.33
0.34
30 40 50 60 70 80 90 100
Stagnation Pressure (psia)
Effic
iency
(tth
eor/t
exp)
100% Air100% He100% CO250% He, 50% AirEquation A.4
Figure A.3 Comparison of the efficiency (τtheor/τexp) of the tank blowdown process for several
injection gases and stagnation pressures. The test section static pressure was maintained
at 1 atm during all tests.
185
5
4
3
2
1
0
Pito
t Vol
tage
(Vol
ts)
43210Time (sec)
Tank Stagnation Pressure = 40 psia 45 psia 50 psia 55 psia 60 psia 65 psia 70 psia 75 psia 80 psia 85 psia 90 psia 95 psia
Figure A.4 Pitot probe traces from the slow speed side of the test section. Tank stagnation pressure is
varied from 40 to 90 psia, and the test section static pressure is 1 atm. A clear double
exponential functional form as described in Equation A.3 is evident.
6
5
4
3
Velo
city
from
Pito
t Tub
e D
ata
(m/s
ec)
6543Velocity fron Tank Blowdown Data (m/sec)
Side Two Velocity Data y=x
Figure A.5 Comparison of Pitot velocity data with velocity data derived from tank blowdown
measurements.
186
0 1 2 3 4
Width (inches)
-2.5
-2
-1.5
-1
-0.5
0
Height(inches)
Splitter Plate
1.04
1.01
0.98
0.96
0.93
0.90
0.88
0.85
Bottom Plate
V/Vavg
Figure A.6 Two-dimensional velocity profile map taken with point Pitot measurements of velocity
for many independent firings of the low-speed side injection system. y = 0 is the location
of the splitter tip and –6.3 cm is the bottom of the test section, while the spanwise extent
is from z = -5 cm to 5 cm.
187
APPENDIX B
RESULTS FROM NON-OPTIMIZED FLOW CONDITIONS
In the course of searching for optimized run conditions of the mixing layer test
section, many mixing layers where the free-stream pressures are mis-matched or the low-
speed side flowrate is too low were recorded using the Schlieren diagnostic. Since the
PLIF experiments were conducted for fixed stream conditions well after the setup of a
steady flow environment, no non-optimized results were available from the PLIF images.
However, as described in Chapter 3, two to three shots of the shock tunnel were required
before exact conditions for reasonable flow occur inside the test section. These topics
were covered in Chapter 3 in their relation to creating a zero-pressure gradient optimal
mixing layer conditions, but in this appendix, the results were interpreted in order to
elucidate some of the behaviors of off-design conditions as there were many such
instances achieved experimentally.
B.1 NON-PRESSURE MATCHED The most fundamental parameter to set for the proper evolution of the mixing
layer is the test section pressure. If the pressure is too low, an expansion fan will exist at
the exit of the nozzle. Conversely, for high back pressures, a shock wave will emanate
from the splitter tip into the fast stream. Shown in Figure B.1 are several shocks taken
with a back pressure that was too high. The experimental back pressure is shown
normalized by the actual, optimal back pressure for all cases. The same trends for splitter-
launched waves are noticed. However, the strength of the shock or expansion fan depends
on the mismatch between the desired nozzle back pressure and that which is set inside the
test section prior to firing of the shock tunnel.
The strength of the wave launched at the splitter tip also determines the angle at
which the mixing layer will propagate. For oblique shocks, the mean flow is turned
188
tangentially towards the shock, causing the layer to grow toward the upper wall of the test
section. This usually results in strong confinement effects as the strong oblique shocks
impinge on the shear layer further downstream and alter the linearity of the turbulent
growth rate. Expansion waves launched from the splitter tip cause the layer to dip down
into the lower speed portion of the test section. Since these waves are of a weak type,
they do not strongly interact with the shear layer to cause instability. However, the gross
protrusion into the slow speed side will cause a local depressurization of the test section,
similar to the ejector effect, which will give a non-zero adverse pressure gradient. The
existence of this gradient in pressure produces mixing layers with non-linear growth rates
and much different entrainment behavior.
If the launched wave from the splitter plate is strong enough, it can cause an
extreme breakdown of the mixing layer. This phenomenon is shown in Figures B.2 and
B. 3. In the first case, the mismatched pressure fields cause a shock wave launched from
the splitter plate to reflect off the top wall and impinge on the shear layer downstream.
This shock/layer interaction pushes the shear layer downward rapidly, forming a local
step in the visual thickness on the high speed side. The shock then reflects from the
constant pressure interface of the mixing layer as a rarefaction wave. Though the shock
impingement clearly displaces the shear layer, it is not clear whether this interaction also
changes the growth rate. Hall et al. (1991) has shown previously that the mixing layer
recovers the original growth rate quickly after interaction with a plane wave. Also, the
effects of the wave in the shear layer are felt upstream of where it interacts with the layer,
a consequence of the subsonic communication pathway available in the shear layer. Since
the shear layer is partially subsonic, information from the free streams, which propagates
into the layer, can be communicated upstream. This is likely the case for this particular
shot where the downward motion on the low-speed side precedes the interaction of the
high-speed side with the wave.
Another type of wave launched at the splitter tip was caused by the nozzle back
pressure being too low. A strong rarefaction fan is launched from the splitter plate and
then reflected at the upper wall/boundary layer. The rarefaction then coalesces into a
shock after reflection from the constant pressure surface of the mixing layer. This mixing
layer/rarefaction interaction is shown in Figure B.3. Here, instead of a downward shift in
189
the shear layer, a bubble type instability is evident. The layer has been displaced upward,
causing a separated flow to exist on the back (slow) side of the layer. The initial
downward displacement of the layer is consistent with the flow turning of a rarefaction
wave, but then as the reflected rarefaction fan arrives to turn the flow back, a bubble-type
instability occurs. Again, due to the possible subsonic communication channels in the
shear layer, the effect of the wave is felt upstream of the disturbance interaction on the
high-speed side. These two described instability patterns for shocks and rarefactions are
stationary in time (for at least 40 µsec) and space, leading to the belief that both are
caused by the local behavior of the shock and not by an amplified instability mode which
is initiated by the structure interaction and then propagates upstream to the splitter plate.
For the shock/layer interaction, the downward displacement is not the only
instability case possible. Another type of instability pattern caused by a planar shock
interacting with the shear layer is a typical forced Kelvin-Helmholtz instability. Figure
B.4a shows the undulation of the shear layer (Mc = 1.17), which is possible if the
information from the shock interaction is allowed to propagate back to the splitter tip.
The deflection of the shear layer will now be amplified at the splitter tip into a strong
instability wave, which will cause the fluttering of the layer. This behavior is not
stationary in time, with the layer flapping several periods over the course of the Schlieren
illumination time. Because this condition is susceptible to this fluttering instability, the
shear layer was always run with a slight expansion fan attached to the splitter tip at this
condition, as this did not cause the amplification of any instabilities and allowed the shear
layer to grow in a linear manner (Figure B.4b).
Static pressure histories for these cases do not illustrate the complexity of the
flowfields illuminated by Schlieren imaging. The pressure traces (Figure B.5) are not
appreciably different at any of the monitoring positions for the upper and lowermost
images in Figure B.1. This is likely due to the fact that the depressurization of the test
section is a small effect, below the resolution of the gage (~0.1 psia), or to the difficulty
in reproducing instantaneous negative gage pressure for the piezo-electric pressure
sensing crystal. The pressure sensors are not in the field of view of the Schlieren, and are
in fact, placed 5 cm downstream of the window edge. Consequently, the pressure traces
show little information about the pressure signals inside the test section. However, the
190
pressure traces from the 1.5 Ptank, both the upper and lower walls, are consistently larger
and take a longer time to settle to zero than those from the Ptank case. This is primarily
due to the fact that strong oblique shock waves are passing the pressure transducer and
altering the flowfield and causing pressure disturbances. Consequently, the pressure
signals for the matched pressure field are much more uniform and show a 1 msec
duration of steady flow time after the arrival of the shock at the endwall.
B.2 NON-MASS FLOWRATE MATCHED
The second most important parameter to be set correctly for optimal mixing layer
growth is the low speed mass flow rate. The entrainment appetite of the shear layer must
be satisfied by mass supplied by the slow-speed stream. If the flow is insufficient, the
shear layer will dip into the low speed stream to achieve the required balance of flow
rates. This depressurizes the test section due to the ejector effect and causes poor
development of the mixing layer. Figure B.6 shows Schlieren results from several shots
where the fast side conditions are held constant, but the low speed injection pressure (and
thus the mass flow rate) is continuously increased. In the upper image, the layer clearly
dips into the low-speed stream seeking further entrainment mass. As the flow rate is
increased, the layer deviates less into the slow stream. Finally, in the bottom image, the
flow rate achieved just satisfies the entrainment appetite of the shear layer, as it
propagates parallel to the guidewalls and grows linearly downstream. No specific
instabilities or interactions were noted for cases where the pressures were sufficiently
matched but the mass flow rate was too low. There is some evidence of an expansion fan
launched from the splitter tip, especially in the top image of Figure B.6, caused by the
rapid depressurization of the test section, but this rarefaction fan does not interact with
the layer in the viewing window. Also this fan does not seem to amplify any instability in
the layer that propagates back to the splitter tip. The expansion fan disappears as the
entrainment appetite of the shear layer is met. As the mass flow rate is further increased,
no changes are seen. One would expect that if the mass flow rate were greatly increased,
the test section would over-pressurize and an oblique shock would be launched from the
splitter tip. However, such mass flow rates were not possible with the side-two tank
191
injection setup employed in this thesis, so that such behavior was not observed. Static
pressure traces from the mass flow rate mismatched cases again do not exhibit many
strong features, except that the pressure in the test section is slightly lower for the low
mass flow rates. No significant streamwise pressure gradients were caused by this
mismatch.
192
Figure B.1 Instantaneous Schlieren images where ∆x = 23 cm showing the effect of poorly matched
nozzle back pressure for development of a Mc = 0.85 mixing layer. Pback here is the ideal
back pressure for this case. All images are independent.
193
Figure B.2 Instantaneous Schlieren image showing the effects of planar oblique shock impingement
on a Mc = 0.85 mixing layer.
Figure B.3 Instantaneous Schlieren image showing the effects of planar rarefaction fan impingement
on a Mc = 0.85 mixing layer.
194
Figure B.4 Instantaneous Schlieren image of a Mc = 0.85 mixing layer showing the potential
undulation behavior of an oblique shock impinging on the shear layer. Top image (a)
shows the instability behavior and lower image (b) does not.
2.0
1.5
1.0
0.5
0.0
∆P
(psi
)
3.02.52.01.51.00.50.0time from reflected shock arrival (msec)
1.5 pback pback
Lower Wall
Upper Wall
Figure B.5 Pressure histories from static pressure taps in the mixing layer test section 30 cm
downstream from the splitter tip, mounted on both the upper and lower walls. Upper wall
traces have been displaced up 1.5 psi.
195
Figure B.6 Instantaneous Schlieren images showing the change of mixing layer behavior with low
speed mass flowrate. The lower side mass flow rate increases in each image from top to
bottom for this Mc = 1.4 shear layer. The mass flow rate increases from 0.01 kg/sec in the
upper images to 0.04 kg/sec in the lower images in increments of 0.01 kg/sec.
196
APPENDIX C
FOUR LEVEL LIF MODEL
The basic physical model of laser-induced fluorescence (LIF) has been described
previously both by quantum mechanical analysis (Demtroder, 1982) and by semi-
classical rate-equation analysis (Berg and Shackelford, 1979; Lucht, 1987, and Eckbreth,
1988). Typically, a two level rate equation approach is used to describe this process, and
is generally valid for atmospheric-pressure combustion and gas dynamic measurements
(Daily, 1977) where the stimulated absorption rate is slow compared to the characteristic
rotational relaxation/quenching rates.
For the case of highly compressible mixing layers driven by a shock tunnel, the
flowfield has both low static temperatures (200 – 500 K) and low static pressures (0.05 –
0.2 atm). Thus, the assumptions used in the simpler two-level models are not valid.
Saturation phenomenon due to smaller rotational energy transfer rates is prevalent at
lower laser fluences and must be modeled carefully. In the following treatment, a quasi-
four level analysis of the NO A2Σ
+←X2
Π system is used (following Lee at. al., 1992) to
accurately capture the effects of the low-pressure environment on the LIF signal. All
spectroscopic constants are computed using relations from the literature (DiRosa, 1996;
Engleman et al., 1970; Freedman and Nicholls, 1980), and the reader is directed to the
excellent explanation of rotational term energies and line strengths for both the upper and
lower electronic states found in DiRosa (1996).
The important excitation and de-excitation mechanisms involved in a typical low-
pressure LIF experiment are shown by the vertical arrows in Figure C.1. The energy
transfer mechanisms shown are stimulated absorption (W12), stimulated emission (W21),
spontaneous emission (A21), collisional quenching (Q21), and rotational energy transfer
(ki). Since the rotational energy transfer rates are comparable to de-excitation pathways,
energy may be distributed among the upper level rotational states throughout the
manifold, and thus the spontaneous emission and collisional quenching originate from the
197
entire upper state rotational manifold rather than simply the laser-coupled upper state.
Spontaneous emission to other vibrational states in the electronic ground state is also
possible, making the overall spontaneous emission rate the summation of the spontaneous
emission rates from the upper electronic state to all the lower vibronic states.
No vibrational energy transfer mechanisms (VET) are shown on the energy level
diagram for the quasi-four level fluorescence model. This is due to the fact that the
vibrational relaxation times are all greater than 10 µsec for NO-NO, NO-O2, NO-Ar, and
NO-N2 interactions at the shock tunnel conditions used in this study (Kamimoto and
Matsui, 1970). Since the ratio of the VET rates to the fluorescent lifetimes is at least 50,
the VET process will be discarded in this LIF model. If this particular NO diagnostic
were to be used in a high-pressure combustor environment, then these energy transfer
rates would become more important.
In the quasi-four level model, the levels considered are the individual rovibronic
levels in the ground and excited state and the rotational baths associated with both the
upper and lower electronic states. The governing equations for the model, referencing the
states in Figure C.1, can be written as equations C.1 – C.5 with initial conditions in
equation C.6.
( )( ) ( ) 2212212211112221
1 nQnAnknkntWntWdt
tdn+++−−= (C.1)
( )( ) ( ) ( ) 22122121213322221112
2 nQnAAAnknkntWntWdt
tdn BT−++−+−−= (C. 2)
( ) ( ) 3213212133443 nQnAAnknkdt
tdn TB−+−−= (C. 3)
( )( ) 32132214411
4 nQnnAnknkdt
tdn B+++−= (C. 4)
( ) ( )∑∑≠′′′′
′′=′′′′==′′=′′′=′=
0 2121
2121 ,1,, 0,1,,
vall
T
Jcoupledallu
B vvJJAAandJJJAAwhere νν (C. 5)
( ) ( )
( ) ( ) totJ
B vtotJ
B v
totBtotB
ntnntn
nftnnftnwith
ff ∑∑=′′
=′′=′′
=′′====
====
87
1 1287
1 02
31
0 0
0 0 31 (C. 6)
198
The summation of all the spontaneous emission rates from the upper laser-
coupled state to all possible radiation-coupled states in the lower bath is represented as
AB21. It is assumed that the spontaneous emission rates for the entire set of bath states are
given by this fixed number (A(ν′=0→ν′′=0) = 9.526 x 105 sec-1, Laux and Kruger, 1992),
termed the “band Einstein A coefficient”. This is the case for Honl-London factors which
sum to one for a particular lower rotational level (Zare, 1988).
The summation of all the spontaneous emission rates for the upper laser-coupled
state to all other lower vibrational states in the lower electronic state is represented by
AT21. It is also assumed in this model that since the VET processes are slow, that
molecules, which decay into lower energy states where v´´≠ 0, will have no pathway to
reach the v´´= 0 state before the end of the fluorescence lifetime. Thus the overall number
of molecules is not conserved in this simulation.
It is assumed that the line broadening of these NO transitions stems entirely from
rotational energy transfer (RET). The rate of transfer out of a specific rotational state (ki)
may be inferred from line broadening using the uncertainty principle where
ici ck νπ ∆= and where ∑
=∆
ispecies
n
iicT i
i P 295
2 χγν (C.7)
In Equation C.7, c is the speed of light [cm/sec], ∆νci [cm-1] is the FWHM collisional
broadening of the pumped line with the given temperature (T, [K]), pressure (P, [atm]),
and bath gas parameters (2γI [cm-1 atm], χi, and ni). Also necessary in RET is the notion
of detailed balance (Yardley, 1980), where RET rates from a particular state to a
rotational bath must be scaled on the Boltzman fraction of the particular state, i.e. (1-fB1)
k1 = fB(J”,v’’) k2, to ensure that as one molecule transitions from the bath to the lower
laser-coupled state, exactly as one travels in the other direction. All broadening
parameters are taken from Chang et al. (1992) and DiRosa and Hanson (1994). The
results of DiRosa also show that there is no J′′ dependence of the 2γ broadening
coefficient for the colliding species of interest in this study, so that one rotational energy
transfer rate is used for all the RET in this model. The RET rates in the upper electronic
state are assumed to be equal to those in the lower electronic state.
199
In writing equations C.1 and C.2, the stimulated emission and absorption rate
coefficient have been spectrally integrated to include the effects of the finite linewidth of
the laser and absorption lineshapes (overlap integral). The stimulated rate coefficients are
given by
( ) ( ) ( )laslasabsabsgtIBtW νννν ∆∆= ,,,1212 (C.8a)
( ) ( ) ( )laslasabsabsgtIBtW νννν ∆∆= ,,,2121 (C.8b)
where B21 and B12 are the Einstein coefficients for stimulated absorption and emission
[cm2 J-1 cm-1], I is the laser intensity [W/cm2] and g is the overlap integral of the laser and
absorption lineshape [1/cm-1]. The overlap integral is a function of the laser and the
absorption lineshape and is given by (Gross et al. 1987)
( ) ( ) ( ) ννννφνννφνννν dg laslaslasabsabsabslaslasabsabs ∆∆=∆∆ ∫∞
,,,,,,,0
(C.9)
The laser lineshape is assumed to be a Gaussian profile characterized by ∆νlas, which is
the FWHM of the profile. The absorption lineshape is the convolution of the
inhomogeneous, Doppler broadened linewidth (∆νd = 7.16235x10-7ν0 (T [K]/M [amu])0.5)
and the homogenous collision broadened linewidth (∆νc), where this linewidth is defined
by equation C.7. Collisional shift effects have also been taken into effect with parameters
again taken from both Chang et al. (1992) and DiRosa and Hanson (1994). The laser
pulse energy is modeled temporally as a Gaussian, in fairly good agreement with the
average experimentally measured temporal mode shape of the YAG pumped dye laser.
The overall quenching parameter is dependent on the local mole fraction of the
colliding species and the quenching cross-section. The temperature-dependent cross-
section data are taken from the measurements of Paul et al. (1996) and McDermid and
Laudenslager (1982). The overall quenching rate is determined by the sum of the
independent quenching rates as shown in Equation C.10,
∑=
ispeciesiiitot vnQ
21 σχ (C.10)
where ntot [cm-3] is the total number density, σi [Å2] is the quenching cross section, χi is
the local collider mole fraction, and <vi> [cm/sec] is the relative velocity of the collision
pair (Vincenti and Kruger, 1986).
200
The total fluorescence signal is determined by integrating the spontaneous
emission rate over the time interval of the laser pulse duration and fluorescence decay.
The total, temporally-integrated fluorescence signal, Sf, can be evaluated using
( )( )VdtAAnnS BTf ∫
∞
++=
0212143 (C.11)
where V is the measurement volume.
This model is integrated using a fourth order Runge-Kutta scheme using variable
time steps. The code is implemented in the MATLAB programming environment for ease
of programming and data display. A set of sample results is shown in Figure C.2. Here a
mixture of 2300 ppm NO, 95% Argon, 3.9% N2 and 0.8% O2 at 300 K and 0.1 atm is
irradiated with a spectra energy of 0.5 MW/cm2cm-1 on line center of the A2Σ
+←X2Π
(0,0) Q1+P21 (J′′ = 18.5) transition. The lower state population is slightly perturbed (~
5%) during the laser pulse but not sufficiently to cause any saturation effects. The
calculated fluorescence temporal decay of ~ 80 nsec compares well with the decay seen
experimentally for the same mixture.
201
W21
W12
Q21A21
k1
k2
k3
k4
AB21 AB
21Q21
1
2 3
4
AT21 AT
21
v´´≠ 0v´´≠ 0
Figure C.1 Excitation and de-excitation pathways for the four level LIF model employed in this
thesis. Symbols are defined in the text.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (micro-sec)
N1/N1(t=0)N2/N1(t=0)Normalized Laser PowerNormalized Photons/cm3/sec
Figure C.2 Simulation of LIF signal and state populations for 2300 ppm NO, 95% Argon, 3.9% N2
and 0.8% O2 at 300 K, 0.1 atm J” = 18.5, Iν = 0.5 MW/cm2cm-1. State populations and
LIF signal are normalized to either initial or peak values.
202
APPENDIX D
GROWTH RATE DATA
Mc* Normalized
Growth Rate (δ/δδ/δδ/δδ/δinc)† 0 0.97
0.8 0.482 0.864 0.399 0.92 0.4 0.99 0.347 1.07 0.257 1.18 0.288 1.3 0.24 1.40 0.183 1.64 0.195 1.7 0.212 1.73 0.207 1.80 0.188 1.93 0.193 1.99 0.186 2.03 0.161 2.13 0.146 2.16 0.151 2.2 0.173 2.25 0.172 2.39 0.183 2.49 0.172 2.59 0.178 2.63 0.228 2.68 0.166 2.73 0.175 2.77 0.177 2.8 0.162 2.84 0.178 2.89 0.218
* Uncertainty for convective Mach number is ± 0.05 for Mc < 1.5 and ± 0.07
for Mc > 1.5
† Uncertainty for normalized growth rate is ± 0.03 for Mc < 1.5 and ± 0.04 for Mc > 1.5
203
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