PRESSURE MEASUREMENT AND FLOWFIELD CHARACTERIZATION
OF A TWO-DIMENSIONAL, IDEALLY EXPANDED, CONSTANT AREA,
AIR/AIR EJECTOR
By
MICHAEL ANTHONY BENJAMIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1994
Copyright C 1994
by
Michael Anthony Benjamin
To my parents, to Felicity, and to Tristan ...
ACKNOWLEDGEMENTS
I would like to express my appreciation to Professor Vernon Roan for supervising
this interesting and difficult project. Our many discussions have kept the research
focused on the primary objectives, while he has allowed me free choice of the path to
arrive at those goals. He has been my mentor for more than seven years (four years for
this degree), and I consider him a friend.
I have had many fruitful conversations with Professor Calvin Oliver, and I am
much obliged for his help in solving an elusive electrical noise problem. Professor Wei
Shyy always managed to echo the nagging questions that one would like to avoid, but as
such has served my conscience well and inspired me to new heights, for which I thank
him.
The suggestions of Professors Jill Petersen and Jim Klausner on various aspects
of the experimental program are greatly appreciated. I am thankful for the expert
machining skills of Charles Simmons and Tommy Skipper and for the management of
Carl Cox of the Engineering Machine Shop, who went to great lengths to satisfy the
accurate construction of the test rig. I would like to thank the department's
Instrumentation and Innovation Laboratory manager, Robert Harker, for help in the design
and construction of the custom electronic circuits and for his good patience in that
endeavor. Additional thanks go to the department's technicians, Steve Sowa, Howard
Purdy, and Jeff Studstill, for their assistance with fabricating parts and help in the
laboratory.
Among my fellow graduate students, I am especially grateful for the help of Max
Dufflocq in the design of the test rig and the aligning of the optical system and to
Douglas Hyland for his loyal service in assembling the rig and rebuilding a large portion
of the air-supply facility. The contoured centerbodies and walls could not have been
fabricated without the expert CNC machine programming and operation by Britt Cobb and
the access to the Machine Tool Research Center granted by Professor Jiri Tlusty. Much
appreciated assistance with design drawings and facility construction was provided by
undergraduates Brian Holloway, Greg Bass, Chris Covington, Mitch Stokes, Stephanie
Webb, Reed Strickland, Brad Chung, and Mike Cowles.
The experimental facility was made available through research grant NAG3-1187
from the Propulsion Systems Branch of the NASA Lewis Research Center, which is
gratefully acknowledged. Additional financial support, for which I thank Professor Bill
Tiederman, has been provided by the Mechanical Engineering Department.
Lastly, and notably, I would like to thank my wife, Felicity, for her understanding
and patience during my often recondite deeds.
TABLE OF CONTENTS
paNe
ACKNOWLEDGEMENTS ........................................ iv
LIST OF TABLES ...................................... ........ ix
LIST OF FIGURES .................................... ........ x
NOMENCLATURE ............................................ xvi
ABSTRACT ................................................. xx
CHAPTERS
1 INTRODUCTION .................................... 1
1.1 B background .................................... 1
1.2 Ejectors ....................................... 4
1.3 O objective ...................................... 6
2 LITERATURE SURVEY ............................... 9
2.1 Turbulence Theory and Consequences ................. 10
2.1.1 Kolmogorov Theory ....................... 10
2.1.2 Inertial Spectrum Functions .................. 12
2.2 Turbulent-Pressure Investigations in Boundary-Layer Flows .. 16
2.2.1 Incompressible, Zero Pressure-Gradient Flows ..... 16
2.2.2 Compressible, Zero Pressure-Gradient Flow ...... 21
2.2.3 Incompressible, Adverse and Favorable Pressure-
Gradient Flows ......................... 22
2.3 Finite Transducer Size Effects ...................... 24
2.4 Shear-Layer Characteristics ........................ 24
2.5 Jet Acoustics .................................. 29
page
3 EXPERIMENTAL INVESTIGATION-HARDWARE, PROCEDURES
AND MIXING-SECTION INLET CONDITIONS ............ 36
3.1 General Description .............................. 36
3.2 Design M ethodology ............................. 37
3.3 Experimental Apparatus ........................... 38
3.3.1 Gas Storage and Delivery System ............. 39
3.3.2 Primary and Secondary Lines ................ 40
3.3.3 Plenums ............................... 41
3.3.3.1 Primary plenum .................. 41
3.3.3.2 Secondary plenums and flow passages ... 43
3.3.4 Primary Nozzle .......................... 44
3.3.5 M ixing Section .......................... 46
3.3.6 Traversing Impact-Pressure Probe ............. 48
3.3.7 Transition Diffuser and Exhaust ............... 49
3.4 Instrum entation ................................. 50
3.4.1 Impact Pressure .......................... 52
3.4.2 Sidewall Mean Static Pressure ................ 52
3.4.3 Sidewall Time-Accurate Pressure Measurements .. 53
3.4.4 Visualization .......................... .. 55
3.5 Data Collection ................................. 56
3.6 Signal Analysis ................................. 60
3.6.1 Autospectrum ........................... 60
3.6.2 Cross-spectrum .......................... 62
3.6.3 Correlation Functions ...................... 62
3.6.4 Transfer and Coherence Functions ............. 63
3.6.5 W windows .............................. 64
3.7 Boundary-Layer Conditions at the Mixing-Section Inlet ..... 65
4 EXPERIMENTAL RESULTS-mean MEASUREMENTS ........ 83
4.1 Experimental Objectives .......................... 83
4.2 Test Conditions ................................ 83
4.3 Optical Visualizations ............................ 84
4.3.1 W aves in the Flow ........................ 85
4.3.2 Growth Rates ........................... 85
4.3.3 Visible Structures ......................... 86
4.3.4 Laminar-to-Turbulent Transition .............. 88
4.4 Static W all-Pressure ............................. 88
4.5 Impact-Pressure Measurements ...................... 90
4.5.1 Normalized Profiles ....................... 90
4.5.2 Similarity Profiles ........................ 92
4.5.3 Growth-Rates ........................... 94
4.5.4 Entrainment .............................. 97
4.6 Contour Plots and Visualizations ..................... 98
4.6.1 Contour Plots ........................... 99
4.6.2 Color Visualizations ....................... 101
5 EXPERIMENTAL RESULTS-TIME-ACCURATE
MEASUREMENTS ................................. 122
5.1 Experimental Objective ........................... 122
5.2 Second, Third, and Fourth Moments of Pressure ......... 124
5.2.1 Mixing-Section Inlet Measurements ........... 124
5.2.2 Centerline Measurements .................. 127
5.2.3 Measurements at y/2b=-1.64 ...... ........ 129
5.2.4 Whole-Field Measurements ................. 129
5.3 Autospectra .................................. 131
5.3.1 Mixing-Section Inlet Measurements ........... 132
5.3.2 Centerline Measurements .................. 133
5.3.3 Measurements at y/2b=-1.64 ................ 135
5.4 Two-Point Spectral Measurements ................ 136
5.5 Narrowband Measurements ....................... 138
6 CONCLUSIONS AND RECOMMENDATIONS ............ 172
6.1 Conclusions from Significant Results ................ 172
6.2 Suggestions for Further Work ..................... 175
APPENDICES ........... ..... ................... ............ 177
A BOUNDARY-LAYER MODELS ........................ 177
B FLOW PARAMETERS FOR TABLES 4.1 AND 4.2 AND
SUPERSONIC PITOT PROBE DATA REDUCTION ........ 181
C PRESSURE TRANSDUCER PARAMETERS .............. 186
REFEREN CES ........................................... .. 188
BIOGRAPHICAL SKETCH ..................................... 195
LIST OF TABLES
Table page
3.1 Boundary-layer thicknesses of primary and secondary supplies ......... 67
4.1 Experimental flow conditions ............................... 102
4.2 Calculated nondimensional and flow parameters .................. 103
5.1 Calculation of the probability W that the wall pressure p exceeds
the threshold p, at x/2b=0 ................................. 140
5.2 Calculation of the probability W that the wall pressure p exceeds
the threshold pa at (x/2b, y/2b)=(6.89, 0) ...................... 141
LIST OF FIGURES
Figure page
1.1 Definitions of ejector regions and lengths ......................... 7
1.2 Comparison of similar geometry air-primary/air-secondary ejector
impact-pressure contours (kPa) from data of Roan et al. (1992, 1993).
P,=P2=34.5 kPa, Mi=2.20 in both cases.
(a) axisymmetric configuration, M2=0.64;
(b) two-dimensional planar configuration, M2=0.41 ................ 8
2.1 The three-dimensional velocity energy spectrum (after Hinze, 1975) ..... 34
2.2 Vortex-ring model of coherent structure and burst (after Kobashi and
Ichijo, 1986) .......................................... 34
2.3 Flow regimes in the ejector .................................. 34
3.1 Drawing of flow passages, flow-conditioning devices, and typical mixing-
section configuration ...................................... 68
3.2 Gas storage and delivery system .............................. 69
3.3 Flow conditioning sections, settling chambers and nozzles. Flow is
from right to left ........................................ 70
3.4 Nozzle-block dimensions ................................... 71
3.5 Mixing section and impact-pressure probe/pressure transducer/stepper-motor
assembly. Mixing-section extension is not shown ................. 72
3.6 Wall static-pressure plate used for manometer and strain-gauge transducer
measurements ......... ........... .. .................... 73
3.7 Brass transducer-sleeves and plugs used for time-accurate wall-pressure
m easurem ents .......................................... 74
Figure page
3.8 Plates used for time-accurate wall-pressure measurements. Only one
plate was actually installed for a test run ........................ 75
3.9 Top rail of mixing section showing fitted plugs ....... ........... 76
3.10 Plugs for top and bottom rails of the mixing section ................ 77
3.11 Exhaust system ............................. ............. 78
3.12 Schematic of instrumentation and computer interface ............... 79
3.13 Custom-built power supply for Kulite transducer .................. 80
3.14 Custom-built buffer amplifier, gain=l ................... ...... 80
3.15 Schlieren setup ..................... ................... 81
3.16 Optical window installation ................. ................ 82
4.1 Schlieren visualization. Knife edge at 00 (horizontal).
(a) low contrast;
(b) high contrast ................ ...................... 104
4.2 Schlieren visualization. Knife edge at 300 (from horizontal).
(a) low contrast;
(b) high contrast ....................................... 104
4.3 Schlieren visualization. Knife edge at 600 (from horizontal).
(a) low contrast;
(b) high contrast ....................................... 105
4.4 Schlieren visualization. Knife edge at 90 (from horizontal).
(a) low contrast;
(b) high contrast ....................................... 105
4.5 Shadowgraph visualization ................................. 106
4.6 Typical manometer reading. Three reference pressures (one for each
mercury reservoir) are indicated by tubes 1, 11, and 21 from the left.
From left to right, the rest of the tubes indicate increasing pressure in
the downstream direction. .................... .............. 107
Figure page
4.7 Mean static-pressure measurements on the wall. 95% confidence
intervals are shown.
(a) Comparison of the three types of static-pressure measurements
made on the left wall of the mixing section;
(b) Difference between the strain-guage and manometer static-pressure
readings on the wall ............................. .... 108
4.8 Comparison of manometer static-pressure measurements on the left and
right walls of the mixing section ............................. 109
4.9 Impact-pressure traverses.
(a) 0xx/2bs7.91;
(b) 12.6sx/2bs20.2 ............ .................... .... 110
4.10 Unrepeatable impact-pressure traverses.
(a) x/2b=15.8;
(b) x/2b=20.6 ........................................ 111
4.11 Normalized centerline values of velocity, density, and dynamic pressure .112
4.12 Normalized impact-pressure plotted against normalized shear-layer ..... 112
4.13 Shear-layer edges determined from impact-pressure measurements ..... 113
4.14 Measured shear-layer thicknesses, growth rates, and virtual origins ..... 113
4.15 Comparison of normalized growth-rates for ejector shear-layers (current
experiment, and Benjamin et al. (1993)) and free-shear layers ........ 114
4.16 Contour plots generated from impact-pressure measurements.
(a) P,/Po2;
(b) PPo2 ;
(c) P/P2;
(d) U/U2;
(e) M;
(f) Q/Qi2;
(g) P/P .............. .......................... .. ........... 115
4.17 Detailed dynamic-pressure contour plot generated from impact-pressure
m easurem ents ......................................... 117
Figure page
4.18 Color plots of mean flow parameters. From top to bottom are impact
pressure Pi,, stagnation pressure Po, density p, velocity U, Mach
number M, and dynamic pressure Q. The maximum contour levels are
white and the lowest are black ............................. 121
5.1 Normalized turbulent wall-pressure at the mixing-section inlet, x/2b=-0 142
5.2 Skewness at the mixing-section inlet, x/2b=0 .................... 142
5.3 Kurtosis at the mixing-section inlet, x/2b=0 .................... 143
5.4 Probability density function of wall-pressure at the 144
mixing-section inlet, x/2b=0 ........................ ..... 143
5.5 Turbulent wall-pressure along the wall centerline, y/2b=0 ........... 144
5.6 Skewness along the wall centerline, y/2b=0 ..................... 144
5.7 Kurtosis along the wall centerline, y/2b=0 ...................... 145
5.8 Turbulent wall-pressure along the wall at y/2b=-1.64 .............. 145
5.9 Skewness along the wall at y/2b=-1.64 ........................ 146
5.10 Kurtosis along the wall at y/2b=-1.64 ......................... 146
5.11 Comparison of turbulent wall-pressure along the wall at
y/2b=0 and y/2b=-1.64 ..................... ............. 147
5.12 Comparison of skewness along the wall at y/2b=0 and y/2b=-1.64 ..... 147
5.13 Comparison of kurtosis along the wall at y/2b=0 and y/2b=-1.64 ...... 148
5.14 Color plots of turbulent wall-pressure.
(a) ps;
(b) p. P, ;
(c) dB p (re: 20 tPa) ...................... ............ 150
5.15 Color plots of higher-higher order wall pressure statistics.
(a) Skewness;
(b) Kurtosis ....................... ............. ..... 152
Figure page
5.16 Probability density function of wall-pressure at the location of maximum
skewness and kurtosis, (x/2b, y/2b)=(0, 6.89) ................... 153
5.17 Wall-pressure spectra and repeatability at the mixing-section inlet, x/2b-0.
(a) y/2b=-1.08;
(b) y/2b=0;
(c) y/2b=1.08 ....................................... 154
5.18 Wall-pressure spectral decay slopes at the mixing-section inlet, x/2b=0.
(a) In the secondary flow at y/2b=-0.81;
(b) In the shear-layer at y/2b=-0.54;
(c) In the primary flow at y/2b=0.27;
(d) In the secondary flow at y/2b=0.54 ..................... 155
5.19 Development of the wall-pressure spectra along the centerline, y/2b=0 156
5.20 Development of the wall-pressure spectra along the line y/2b=-1.64 .... 159
5.21 Two-point, transverse pressure-spectra at x/2b=0. Measurement locations are
in the upper and lower secondary streams at y/2b=-1.36 and y/2b=1.09 161
5.22 Two-point, transverse pressure-spectra at x/2b=0. Measurement locations are
in the primary flow at y/2b=0 and y/2b--0.27 ................... 162
5.23 Two-point, longitudinal pressure-spectra at y/2b=0. Measurement locations
are just before the end of the primary core at x/2b=8.83 and x/2b=9.15.
Measurement bandwidth is 20 kHz ........................... 163
5.24 Two-point, transverse pressure-spectra just downstream of the end of the
primary core at x/2b=10.34. Measurement locations are either side of the
centerline at y/2b=-0.27 and y/2b=0.27 ....................... 164
5.25 Two-point, transverse pressure-spectra just downstream of the end of the
primary core at x/2b=10.34. Measurement locations are on the lower
side of the mixing section at y/2b=-1.08 and x/2b=-0.27 ........... 165
5.26 Two-point, transverse pressure-spectra at the streamwise location where
the shear layers interact with the wall, x/2b=17.23. Measurement
locations are on the lower side of the mixing section at y/2b=-0.54 and
x/2b=-0.27 .................................. .......... 166
5.27 Two-point, transverse pressure-spectra at the streamwise location where
the shear layers interact with the wall, x/2b=17.23. Measurement
locations are on the lower side of the mixing section at y/2b=-1.87 and
x/2b=-0.81 ................................ ........... 167
5.28 Two-point, transverse pressure-spectra at the streamwise location where
the shear layers interact with the wall, x/2b=17.23. Measurement
locations are on the lower side of the mixing section at y/2b=-1.87 and
x/2b=-0.27 .............................. .............. 168
5.29 Two-point, transverse pressure-spectra at the streamwise location where
the shear layers interact with the wall, x/2b=17.23. Measurement
locations are either side of the centerline at y/2b=-1.87 and x/2b=1.08 169
5.30 Color plots of narrowband measurements.
(a) 312.5 Hz;
(b) 3.625 kHz;
(c) 49.625 kHz where sharp or broad peak is present;
(d) 49.625 kHz where only sharp peak is present ............... 171
Figure
Ra.e
NOMENCLATURE
Symbol Description
Ai Mixing-section inlet area of interest
b Primary nozzle exit half-height
c Sound speed
d Pressure transducer sensor diameter
E(k, t) Velocity spectrum function, see Fig. 2.1
Em Mass entrainment ratio, r i2/?1
Ey Volumetric entrainment ratio, V12/l2
f Frequency
f(M) Convective Mach number function, see Eq. (2.35)
h Primary nozzle throat half-height
H. transfer function of pressure, see Eq. (5.6)
k Wavenumber (customarily based on convective velocity of wave with
frequency f, k=w/uc(f) ).
K Kurtosis of pressure, see Eq. (5.3)
Li Length of region i
max Maximum value at streamwise location
min Minimum value at streamwise location
i Mass flowrate
M Mach number
M Convective Mach number, see Eqs. (2.32) and (2.33)
M Relative Mach number, see Eq. (B.21)
MW Molecular weight
p Static pressure
P Mean static pressure
P Impact pressure
Po Mean stagnation pressure
Q Mean streamwise dynamic pressure, 'pU2
r Secondary to primary velocity ratio, U2/U1
R Gas constant
RP Pressure correlation, see Eq. (5.4)
Re Reynolds number
Re Reynolds number per unit length, see Eqs. (B.26) and (B.27)
Re8 Reynolds number based on visual shear-layer thickness at streamwise
distance x Re =(p, AUS i)/l
Re Reynolds number based on streamwise distance x Rex =(PiU1x )/J1
s Secondary to primary density ratio, p2/ P
S Skewness of pressure, see Eq. (5.2)
S Pressure autocorrelation, see Eq. (5.7)
S Pressure spectral density, see Eq. (5.5)
t Elapsed time from start of measurement
T Static temperature, or time-accurate measurement duration
To Stagnation temperature
u Velocity
u Fluctuating streamwise velocity component
uT Friction velocity, uX=J/j
U Mean velocity component
Uc Convective velocity, see Eqs. (2.30) and (2.31)
v Fluctuating transverse velocity component
V Volumetric flow rate
xvii
W(p
x Streamwise direction
x Position vector
xo Shear-layer streamwise virtual origin
xg Streamwise distance to visible shear-layer growth
y Transverse direction
Greek Symbols
y Constant-pressure specific heat ratio
y Coherence of pressure, see Eq. (5.8)
8 Boundary-layer thickness or characteristic shear-layer length scale defined
in section 5.3
s8 Visual shear-layer thickness
8' Shear-layer growth rate
8/ Vorticity thickness growth-rate, see Eq. (4.7)
8* Boundary-layer displacement thickness
e Time-average energy dissipation per unit mass and time
1
7 Kolmogorov length scale 1=(v3/e)4, or normalized shear-layer thickness,
see Eq. (4.2)
0 Initial boundary-layer momentum thickness
/ Molecular dynamic viscosity
v Kinematic viscosity
rr Normalized impact-pressure, see Eq. (4.3)
I Coles' pressure-gradient parameter
p Mean static density
Po Mean stagnation density
S- Shear stress
I Spectral density of pressure, see Eq. (5.10)
xviii
- Angular frequency, o =2irf
Subscripts
0 Incompressible value at same r and s.
1 High-speed (primary) stream condition
2 Low-speed (secondary) stream condition
ave Based on average of streams 1 and 2
cl Centerline value, y=0
i Stream i condition, or mixing-section inlet condition
,i Mixing-section inlet condition
i,j Stream j condition at inlet plane
pit Impact-probe value
rms Root-mean-square value
th Threshold value
w Value at the wall
oo Boundary-layer freestream condition
Superscripts
* Complex conjugate
+ Non-dimensionalized value using inner variables
' Fluctuating quantity
Other Symbols
< > Ensemble average
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
PRESSURE MEASUREMENT AND FLOWFIELD CHARACTERIZATION
OF A TWO-DIMENSIONAL, IDEALLY EXPANDED, CONSTANT AREA,
AIR/AIR EJECTOR
By
Michael Anthony Benjamin
December 1994
Chairman: Vernon P. Roan
Major Department: Mechanical Engineering
A detailed experimental investigation of a two-dimensional, Mach 1.8 air-primary,
Mach 0.3 air-secondary ejector at high Reynolds number has been performed, from which
a nonintrusive method for whole-field visualization using turbulent wall-pressure has been
developed.
The experiments were conducted using mean and time-accurate wall pressure
measurements, impact-pressure measurements using a traversing probe, and Schlieren and
shadowgraph visualization techniques. The time-accurate pressure measurements were
recorded using a sealed Kulite miniature pressure transducer with a 0.7 mm diameter
sensing diaphragm. For all except the optical methods, measurements were taken from
the initial flow interface to about 13 hydraulic tube-diameters downstream in the constant-
area mixing section.
From the mean measurements, values of stagnation pressure, density, velocity,
static pressure, Mach number, and dynamic pressure were developed and are presented.
Using the time-accurate pressure measurements, a color contour plot of the rms
pressure was developed that definitively shows the regions of the flow in agreement with
the other measurements. Additionally, probability density functions, skewness, and
kurtosis were calculated. Peak values of skewness (S) and kurtosis (K) on the centerline
at about 2.5 hydraulic diameters are S=1.85 and K=1 1.5. The inlet rms pressure values,
normalized by freestream dynamic pressure for the primary (-0.001), were found to be
in fair agreement with previous experimental values; however, those in the secondary
were much higher (-0.2), apparently due to the acoustic radiation from the primary.
Fourier analysis of the time-accurate pressure measurements show that the
autospectra contain k', k'7, and k"' pressure spectrum functions as predicted by
prevailing theory for the overlap layer, turbulence-turbulence interaction, and turbulence-
mean-shear interaction, respectively. It is believed that this is the first experiment in
which the k"' spectral slope has been observed, perhaps as a result of the high Reynolds
number.
Two-point longitudinal and transverse measurements are presented that show the
development of multiple large eddies in the flow.
CHAPTER 1
INTRODUCTION
1.1 Background
To meet the challenge of increasing the performance of modem gas turbines,
designers require knowledge of the smallest details and behavior of the flow. These
effects cannot be modelled using computational methods without some knowledge of what
they are and where they originate. Where they originate is dependent upon several factors
including turbulence behavior, turbulence/shock interaction, the physical shape/design of
the enclosure, and the mean dynamics of the flowfield. The range of details from mean
to Kolmogorov flow scales, as well as wall roughness and slip at the wall, cannot be
closely modelled even with some of our best computational techniques, such as Direct
Numerical Simulation (DNS). Turbulence alone can be modelled using DNS, but DNS
is limited by machine speed and memory so that current DNS flows have, at best, a
Reynolds number of the order of 1,000 and a spatial resolution that can resolve details
up to about 1000 wall units, far short of what is required for complete modelling of a
complex flow.
Experimental noise analysis techniques using piezo-electric pressure transducers
(e.g., Willmarth 1958b, Kistler and Chen 1963) or microphones (Schewe 1983) have been
used for about 40 years to measure turbulent wall-pressure, sparked by the need to
investigate and control the effect of pressure on aircraft fuselages. Throughout the 1950s
and 1960s, most fundamental experimental research focused on turbulent wall-pressure
measurements under a thick flat-plate boundary-layer (e.g., Willmarth 1956). This led to
the familiar result of the turbulent energy cascade from large eddies (integral scale) to
2
small eddies (dissipation scale) via the inertial subrange, which was in accord with
Kolmogorov's first and second hypotheses. Following the end of the Apollo and
supersonic transport (SST) programs of the 1960s, experimental research in this area has
focused on fundamental understanding of the small scale disturbances in the flow near
the wall.
An area of current interest that can be used to further understand some of the
fundamental details associated with internal mixing flows is to study the ejector shear-
layer, which comprises a supersonic primary jet and a subsonic secondary flow.
Presently, the model of a developing ejector-flowfield is typically divided into three
regions preceded by uniform primary and secondary flows, and followed by a region of
fully mixed flow (see Fig. 1.1). Modelling the ejector using a computational process,
which will not be addressed in this work, involves a marching scheme that starts at the
point of initial contact (start of region 1).
Region 1 begins with this initial contact and ends where the potential core of the
primary ends. Region 2 also begins with the initial contact and ends where the secondary
shear layer reaches the wall. Region 3 starts at the end of region 1 or 2 (whichever is
longer) and continues until the streams are fully mixed.
Recent experimental work at the University of Florida (Petersen et al. 1992, Roan
et al. 1992, 1993, Dufflocq et al. 1992, 1993, Benjamin et al. 1993) has focused on
traversing an impact probe across both axisymmetric and two-dimensional planar flows
at several axial positions, which has enabled some understanding of the flow inasmuch
as pumping performance was measured and compared to theory, growth rates were
estimated, and the lengths of regions 1, 2 and 3 were measured or inferred when possible.
This experimental work has led to the development of a semi-empirical model for
predicting the lengths of regions 1 and 2 (Benjamin and Roan 1993). In the axisymmetric
case, the experiments identified large coherent structures in the primary core that probably
3
result from vortex/shock-wave interaction, whereas such structures are not apparent in the
two-dimensional planar case. Figure 1.2 shows a comparison of an axisymmetric and a
two-dimensional planar case at similar flow conditions. The result of the vortex/shock
interactions is seen in region 1 of the axisymmetric case.
Most experimental research of high speed internal flows has relied mainly on
techniques such as laser doppler velocimetry or insertion of some type of pressure probe
into the flow. The current research was undertaken using a different approach to provide
additional detailed measurements necessary to better understand and enhance the ability
to measure and predict the behavior of ejector flowfields. The research was also designed
to provide more insight into the structure of this type of internal flow and, hence, enable
designers to better optimize their designs.
This research investigates compressible shear-flow regions that are sandwiched
between two essentially inviscid streams of supersonic and subsonic fluid, all of which
are surrounded by boundary layers. Far downstream where the shear layers have reached
the walls, the flowfield comprises only shearing fluid in the freestream, which is again
surrounded by boundary layers. There is no reason to expect that, in terms of the
turbulent-pressure field, the inviscid flow regions behave any differently from previously
reported subsonic and supersonic turbulent boundary-layer flows, although jet-noise
contamination may be present. The behavior of the shearing regions is undetermined and
results from this research will characterize them. On the small scales in discrete ranges
of the universal equilibrium range, local isotropy should generally still apply because
compressibility simply acts as a source of dissipation in addition to the viscous stresses
(Hinze 1975, p. 312). Also, if there is significant acoustic radiation emitted or absorbed,
a change in the energy transfer rate will occur that will cause a change in the spectral
slope. Hence, a knowledge of the physics or behavior of the following is required to lay
the foundations for ejector flow:
(1) the fundamental premises of turbulent flow,
(2) the turbulent boundary-layer in subsonic and supersonic flows under the
influence of zero-, adverse-, and favorable-pressure gradients,
(3) mean freestream-pressure and velocity fields,
(4) ejector shear-layer growth rates, and
(5) acoustic phenomena of jets.
The effect of pressure transducer size on resolution of all the pressure scales in
the flow is a concern that has been studied to a fair degree and is also outlined.
1.2 Ejectors
Ejector systems have been investigated for many years due to the interest in their
fluid-pumping ability. These devices are of particular interest to the aircraft industry,
where they have been studied and employed for their enhancement of thrust augmentation,
lift augmentation and engine-noise reduction (Braden et al. 1982). Other possible
applications of ejectors include jet pump compression, extraction of secondary fluids,
ventilation and air conditioning, innovative thermal cycles, and high energy chemical
lasers (Power 1994).
The pumping ability of ejectors is primarily the result of momentum transfer from
the high-velocity primary fluid to the slower secondary fluid. Ultimately, the overall
performance of these devices is influenced by a number of physical processes that
individually and/or collectively govern the development of the associated flowfields.
Among the processes of interest are (a) details of mixing of primary and secondary
streams, (b) viscous dissipation due to mixing, compression waves, and solid boundaries,
and (c) the effects of geometry. Maximizing ejector efficiency translates into enhancing
the mechanisms that result in efficient mass and momentum transfer and inhibiting those
that have a detrimental effect.
5
An ideal ejector is one yielding ideal primary-to-mixed out momentum transfer as
determined from a control volume analysis. In a real ejector, the closest to ideal that can
be obtained is a configuration where the shroud of the ejector is long enough to allow for
near completion of the mixing process and yet short enough to minimize wall friction
losses. (Additionally, there are area ratio trade-offs.) These are conflicting ideals that
have been well substantiated by ejector studies. Therefore, ejector design compels trade-
offs where the goal is to find the optimum balance between these opposing factors for
given requirements and input parameters.
Traditionally, theoretical ejectors studies have been aimed at comparing predicted
overall performance-usually in terms of thrust augmentation, overall pressure ratio, or
entrainment ratio-with results obtained experimentally (Braden et al. 1982, Petersen et
al. 1992). In many cases, performance is predicted using one-dimensional conservation
equations to find flow properties at the "fully mixed" exit plane of the ejector. Such
models show the general trends that are expected under ideal flow conditions (uniform
flow and no losses). However, they generally predict performance superior to those of
real ejectors, and, hence, their value as design tools is limited.
Similarly, semi-empirical models have been used to predict real ejector flowfields
in which undesired dissipation effects are unavoidable. Here, factors such as wall
frictional losses, nonuniform inlet and exit flow, imperfect mixing of the two streams, and
primary nozzle efficiencies, for example, are taken into account using best estimates. The
approach in this case has usually been to adjust parameters within the model ("calibrate")
so that predicted flowfields match the experimental results. These models are only valid
for the range of calibrated ejector geometries and flow parameters, and, therefore, their
potential as design tools is also limited.
Progress in ejector technology largely depends on being able to predict ejector
flows accurately. This ability can only be achieved through a better understanding of the
6
physical processes that govern these flowfields, as well as the ability to measure the
details of these flowfields effectively. The experimental investigation outlined below is
aimed at providing additional detailed data necessary to further understand these
mechanisms and thus improve current ejector models.
1.3 Objective
The flowfield of a two-dimensional, single-nozzle (method of characteristics
profile) ejector was investigated to improve the design optimization process for ejectors.
A series of detailed time-accurate wall-pressure measurements were undertaken using
miniature pressure transducers having 0.7 mm diameter diaphragms and installed in
interchangeable wall-plates. The arrangement allowed plenty of flexibility for a pair of
transducers to be positioned at almost any wall location and with a wide range of spacing
either longitudinally or laterally. One- and two-point measurements were recorded,
enabling calculation of the rms pressure, skewness, kurtosis, autospectra and cross-spectra.
From these measurements, a nonintrusive technique was sought to identify the different
regions of the flow.
Schlieren and shadowgraph visualization were employed for mean fluid-structure
information and to look for evidence of large structures. Also, the mean flowfield was
determined using an impact-pressure probe.
The wall locations of disturbances and their statistics were investigated to see if
their effects, when compared to a similar uncontained flow, indicate the nature of the
fluid mechanisms present in the ejector.
Also, the ends of regions 1 and 2 were located by measuring turbulent wall
pressures and are compared to those obtained from the mean measurements. Finally, two-
dimensional graphics were developed from the mean and turbulent experimental data.
x=x x=L,+x x=L2 +xgr
XIdY- I I I
x=OI I I
Sx=01 I I
Secondary > 2b -I- --
2b I-t
Primary ------- -!- 2h
I ---
Sawdy L%-n
Secondary I "I -
Region I
-- Region 2
Region 3
Figure 1.1 Definition of ejector regions and lengths.
x. (n-)
x. On)
Comparison of similar geometry air-primary/air-secondary ejector impact-
pressure contours (kPa) from data of Roan et al. (1992, 1993).
P,=P2=34.5 kPa, Mi=2.20 in both cases.
(a) axisymmetric configuration, M2=0.64;
(b) two-dimensional planar configuration, M2=0.41.
Figure 1.2
CHAPTER 2
LITERATURE SURVEY
In considering the turbulent flow in ejectors, it is helpful to have a general
background of the behavior of individual, or related, aspects of the flow. Obviously, it
is desirable to first understand the regions of the mean flowfield. The second area of
understanding is of the acoustic behavior of jets. There have been many acoustic studies
of free jets and some on ejectors, but these have been studies that measure the pressure
radiated from the flow. Radiated pressure is difficult to interpret physically because the
intensity of acoustic radiation not only originates from turbulent eddy vorticity but, at
least, is influenced by observation vector, local Mach number, and turbulent velocity.
The third area of understanding is that of turbulent wall-pressure measurements under
boundary layers in zero, adverse, and favorable pressure gradient flows. There are several
excellent incompressible studies in this area that have been made over the last 35 years
that have characterized these flows fairly comprehensively. There have been very few
studies in compressible (especially supersonic) boundary layer flow, and those studies are
limited in scope due to the inherent difficulty of the measurements. Excellent
comprehensive reviews of turbulent wall-pressures in incompressible flows have been
compiled by Willmarth (1975) and Eckelmann (1990).
The background on turbulence, and especially the first and second Kolmogorov
hypotheses (Kolmogorov 1941a, b, hereinafter referred to as K41), provide the motivation
for the experimental methods and results presented in the present research.
2.1 Turbulence Theory and Consequences
The foundation for a statistical description of turbulent fluid mechanics is first
presented, followed by a discussion of its application to spectrum functions.
2.1.1 Kolmogorov Theory
The two K41 hypotheses are derived using the definitions of local homogeneity
and local isotropy. Turbulence is called homogeneous when the spatial derivatives of
mean turbulence quantities are zero, e.g.,
aui(x) i1 U;2(X)
ui(x -0
&. 2 d. (2.1)
The turbulence is isotropic when it is homogeneous and invariant with respect to
rotations and reflections of the original spatial coordinate axes. At sufficiently high
Reynolds numbers, the turbulence can be considered locally isotropic with good
approximation in sufficiently small domains not lying near the boundary of the flow or
its other singularities. There is no rigorous proof for this definition, but the experimental
evidence of approximate isotropy and homogeneity is well established experimentally for
flow downstream from a regular array of rods in a wind tunnel (Taylor 1935).
The first hypothesis can be summarized as follows: For the locally isotropic
turbulence there exists a range of high wavenumbers where the turbulence is statistically
in equilibrium and uniquely determined by the average dissipation by turbulence, s, and
the cinematic viscosity, v. The state of equilibrium is universal. This equilibrium range
is termed universal because the turbulence in this range is independent of external
conditions. Also, any change in the effective length scale and/or time scale of this
turbulence can only be a result of the effect of the parameters e and v.
To put this hypothesis into mathematics (Sreenivasan 1991), the hypothesis can
be restated such that the multivariate probability distributions of the velocity difference
in a direction i, Aui=ui(x) -ui(+f)r, are functions only of r=rIF, e, and v. Specifically,
the nth-order structure functions have the form
Iu, n=(Er)O'f,(ri/ ) (2.2)
Here, 7 =(v3/e)114 is the Kolmogorov length scale and the functions f, are universal; Tr is
on the order of magnitude of the energy dissipation scales.
The second K41 hypothesis states that, at very high Reynolds numbers, there is
a range of scales 7
become independent of viscosity; that is, f,(oo) =K, where K. are universal constants
independent of the flow
Aui n =Kn (r),O (2.3)
The K41 hypotheses arise fundamentally from dimensional arguments, and their
validity rests on experimental support (Sreenivasan 1991).
The idea of intermittency was first pointed out by Landau in 1944 (Landau and
Lifshitz 1987) and arises due to spatial variations ofe over physical volumes larger than
the inertial subrange, i.e., the energy cascade process in the inertial subrange undergoes
fluctuations caused by larger than inertial-subrange scales. Therefore, intermittency
causes nonuniversality of the constants, K,, because the large scales are flow-specific.
Recent experimental measurements of high-order velocity structure functions (Anselmet
et al. 1984) indicate that the scaling function exponent, n, shows strong deviation (i.e.,
non-Gaussian statistics) from the K41 theory for structure functions higher than second-
order. There is currently much discussion and controversy about the fundamental cause
for intermittency, and the reader is referred to Sreenivasan (1991) and Anselmet et al.
(1984) for more detailed discussion.
2.1.2 Inertial Spectrum Functions
Using Kolmogorov's second hypothesis, it can be shown from dimensional analysis
(e.g., George et al. 1984) that the velocity spectrum function in the inertial subrange is
given by
2 5 (2.4)
E(k)=ae3k 3
where E(k) is defined by
E(k)= f fF () do4Q)
(2.5)
where the integral is over spherical shells a of radius k. The cross-spectral densities of
the velocity moments are
F1j(-) 1 fff-el-.rB (f)a 3F_
F (2)3 (2.6)
with the velocity moments defined as
(2.7)
B j(r)= ui(F ) ui(f-+r)
13
Tennekes and Lumley (1987) recommend that experimental data indicate a=1.5,
approximately. Figure 2.1 shows the form of the velocity spectrum function over all
wavenumbers.
George et al. (1984) derived models for turbulent pressure fluctuations by directly
Fourier transforming the integral solution to the incompressible Poisson equation for a
constant-mean-shear-flow (the integral solution is presented in the next section). They
found that the spectrum function
E,(k)= ffF, (i) dao) (2.8)
is composed of three parts as follows:
E (k) =rk) +r,(k) +t(k) (2.9)
where sr, and ir. represent the second- and third-moment turbulence-mean-shear
interaction spectrum functions respectively, and rT, represents the turbulence-turbulence
interaction spectrum function. The inertial subrange for each of these spectrum functions
must be determined by only the parameters k, e, and the mean-shear velocity-gradient K
By dimensional analysis, they arrive at
1 2 11
-2s2( 2)=aK2 3 k 3
P (2.10)
1-1',(k) =a, Kk -3
P 2(2.11)
1
-n-(k)=a ke 3
P2 (2.12)
where ac., a, and a, are constants. These constants are found by assuming that the
turbulence is isotropic, although the turbulence-mean-shear interaction is not. The values
of constants calculated by George et al. (1984) for the one-dimensional spectrum (which
can be measured) are
Pf=0.307a (2.13)
=3-0 (2.14)
P =0.283a2 (2.15)
From their calculations George et al. (1984) suggest that the shear contribution is
impossible to isolate from spectral data, even in flows with moderate-to-high shear rates
relative to the turbulence shear-rate u/l, and conclude that this is the reason that such a
range has not been previously observed. They also note that there is no direct dissipation
of pressure fluctuations.
Bradshaw (1967b) argues by dimensional reasoning, or by direct substitution into
the Poisson equation, that the one-dimensional spectral density is proportional to k' in the
overlap region of the turbulent boundary layer. Panton and Linebarger (1974) calculated
wall pressure spectra using the Law of the Wall and Coles' wake function, which also
revealed an overlap region ofk'. Following Panton and Linebarger (1974), P denotes the
spectrum in outer variables and T the spectrum in inner variables. 11 is Coles' pressure
gradient parameter. Then
P(k1i; n) = +(ki)/yT2 6, P(klv/u) = (k1)u, /T 2 v, Re = u, /v (2.16)
There is an intermediate wavenumber where these expressions are equal: P=P/Re.
Additionally, if there is a range of wavenumbers where the expressions are equal, then
the equality may be differentiated for this range such that
(k,6)2 P'(k,6; n) = (kv/u~2 P'(kiv/u,) = const. (2.17)
For this expression to be true, P must be independent of n in the overlap region, and the
spectra have the form
P(k,6)2 = A(kb6)', P'(klv/u) = A(kv/u)-' (2.18)
This gives a straight line slope of -1 on a log-log scale. Panton and Linebarger (1974)
conclude, along with Bradshaw (1967b), that the overlap region is beyond the reach (too
high frequency) of most measurements.
Recent measurements by Farabee and Casarella (1991) show that in the very low
wavenumber region, the spectrum follows a k' behavior, as suggested by Bradshaw
(1967a).
2.2 Turbulent-Pressure Investigations in Boundary-Layer Flows
The flows in ejectors have regions of zero, adverse and favorable pressure-
gradients that will affect the wall-pressure measurements. Hence, presentation of
pertinent literature for such pressure-gradient flows, as well as the effects of rough and
smooth walls, is covered in the following three subsections.
2.2.1 Incompressible. Zero Pressure-Gradient Flows
The incompressible Navier-Stokes equation may be given by
aui &,uUj 1-p -2
+ +v-U.
at aJ pri afr, (2.19)
and taking the divergence of Eq. (2.19) gives the relationship between the pressure and
velocity fields (Lilley and Hodgson, 1960):
pq
a&'i (2.20)
where
q-
&iAj (2.21)
The right-hand side of Eq. (2.21) can be manipulated (Thomas and Bull, 1983) to give
aui ,j' 2 / / ,/\
q=2 +-- uiu; -uiu
8x x. x,8x .
j J
(2.22)
17
These equations show that the pressure field is related to the velocity field through the
turbulence-mean-shear and turbulence-turbulence shear stress interactions.
The solution for the fluctuating pressure on the wall pw using the integral solution
for the Poisson equation in the half-plane, and neglecting surface integrals (Lilley and
Hodgson, 1960) can be written
_p,) q dV-)
20/x x/ s (2.23)
This indicates that sources over a region of flow (theoretically, a semi-infinite region) will
contribute to the wall pressure fluctuations and that the contributions from various source
regions will fall off rapidly with their distance from the point under consideration
(Thomas and Bull, 1983).
Kraichnan (1956a, b) was the first to report theoretical estimates for the mean-
square wall-pressure and spectra. He overcame the difficulty of having to treat the
turbulent boundary layer as anisotropic by assuming that the turbulent flow is
homogeneous in planes parallel to the wall. The source terms for the turbulence-mean-
shear interactions were estimated based on a rough approximation (-50% accuracy) to the
mean shear and Kraichnan's (1956a) results for the Fourier-transformed velocity and
pressure fields for homogeneous anisotropic turbulence. The prediction of the ratio of
root-mean-square (rms) wall-pressure fluctuations to mean wall shear stress was of order
six, much more accurate than would be expected. He also found that for the
homogeneous and anisotropic cases, the rms turbulent pressure for the turbulence-mean-
shear interactions was 8.25 times larger than that of the turbulence-turbulence interactions.
Incompressible measurements by Willmarth and Wooldridge (1962) under a thick
boundary layer showed that the rms wall pressure was 2.19 times the wall shear stress.
18
Static pressure fluctuations over a flat boundary in the atmospheric boundary layer by
Elliott (1972) were found to produce turbulent pressure values of about 2.6 times the
mean stress at heights up to 6 m. Hinze (1975) recommends that a reasonable average
from all incompressible experimental data available is pW.3r,. Willmarth (1975)
concluded that the ratio of turbulent wall-pressure to freestream dynamic pressure has a
value of 0.0035 for incompressible flow.
Corcos (1964) made some calculations based on pressure and velocity experiments
and found that the inner part of the Law of the Wall region (y+100) seemed to be
substantially free of pressure sources (except perhaps at very high frequencies), and within
that region (a) the pressure can be given in terms of its boundary value, and (b) the local
velocity field is dependent upon but unable to affect appreciably the turbulent pressures.
Using the method of space-time correlation, Willmarth (1958a) discovered that the
random pressure fluctuations under a flat-plate boundary layer were convected with a
speed of approximately 0.8U,. More detailed experiments (Willmarth and Wooldridge
1962, Bull 1963) showed that the convection velocity varies with streamwise spatial
separation of the measuring stations and that for small spatial separation the convection
velocity is low, 0.56U, but increases to 0.83U, for very large spatial separation. The
increase in convection velocity with streamwise separation of measuring points was
attributed to the more rapid decay of the smaller pressure producing eddies. Wills (1970)
measured the wave-number/phase velocity spectrum of the wall-pressure, which showed
the above relation for different wavenumbers (eddy sizes). Willmarth and Wooldridge
(1962) also concluded that a pressure-producing eddy of large or small wavelength X
decays and vanishes after travelling a distance of approximately 6k.
The effect of wall roughness was investigated by Blake (1970), who found that
there was no effect on the ratio of rms wall pressure to wall shear stress. However, the
19
magnitude of the rough-wall rms pressure increased by at least a factor of two over the
smooth wall value.
From an analysis of several experimental investigations, Thomas and Bull (1983)
proposed that the dominant contribution to the rms pressure fluctuations comes from the
source term associated with the turbulence-mean-shear in agreement with Kraichnan and
hence, for two-dimensional flow, Eq. (2.23) becomes
S p r P U av dV' )
2> ay ax -S (2.24)
Schewe (1983) found that the skewness of the turbulent pressure has a value of
-0.2 and a kurtosis of 4.9, compared to values of 0 and 3, respectively, for a Gaussian
distribution. Analysis of the time records showed that the difference from the Gaussian
was caused by wavetrains or pulses that convect at 0.53U., and can therefore be attributed
to pressure structures in the buffer layer at 12sy':21 based on the measured velocity
profile. The measurements also confirmed that there were more negative pulses than
positive ones as indicated by the negative skewness. The characteristic wavelength of the
wavetrains based on their frequency of occurrence and their convective velocity was
.=hXujv=145. Hence, a transducer would have to be smaller than X'=145 to begin to
resolve the wavetrains. Additionally, Schewe calculated from the probability density
function of events ph>3|p~. that they contribute 40% to the rms pressure, although the
events only occur for 1% of the time. Karangelen et al. (1993) found that the same
threshold was exceeded for 5% of the time and contributed 49% to the rms pressure.
Hence it can be seen that the pressure pulses can vary significantly between different
flows. The measured time between burst events for channel flow (Tiederman 1990) was
found to be 90 scaled on inner variables and was independent of Reynolds number. For
20
boundary layer flow the time between events is closer to 300 (Blackwelder and
Haritonidis 1983). Using these values as lower and upper bounds, Karangelen et al.
(1993) conclude that the average time between bursts corresponds to the pressure events
range of 2sp,/jpI\<3, which suggests a direct correspondence between bursts and large
amplitude wall pressure events.
Dinkelacker and Langeheineken (1983) studied the relation between intermittent
high positive wall-pressure fluctuations and velocity fluctuations in a 50 mm diameter
pipe with ReD=9600. The results were conditioned over several thousand events and
clearly show that the high wall-pressure events are related to a steep increase in
measured between y*=5 and y=52. Preceding this event is a comparatively long period
in which is below average and followed by a comparatively long period in which
is above average. From the duration of the periods the streamwise length of the
pattern was calculated to be twice the pipe diameter (Ax'O1000). Similarly, the
measurements showed that the flow moves away from the wall and then towards the wall
either side of the turbulent-pressure peak, but occurs over 1/6 the period of the event
and to a height of at least y=100. The disturbance in the azimuthal direction was found
to have a span of order Aq'=50. It was concluded that these events were "bursts," which
were defined by other investigators to be instantaneous high levels of turbulence at about
y= 15.
Kobashi and Ichijo (1986) measured wall pressure fluctuations in relation to the
coherent motions of the turbulent boundary layer. The low frequencies were found to be
related to the large scale motions of the outer layer and prevail through and outside the
boundary layer, while the high frequencies were found to be related to the bursts in the
wall region. They concluded that the large scale motions are composed of periodic and
nonperiodic components that are initiated by the instability of the mean flow and rotate
in the direction of the mean flow shear (cf. Fig. 2.2). The bursts are vortical in nature
21
and rotate against the mean flow shear. The origin of the bursts is from the interaction
of the periodic large scale motions with the wall. The burst is bound by two pairs of
counter-rotating vortices that are inclined forward, and the ejection and sweep that
characterize these burst phenomena can be explained as the flows induced by the paired
vortices. Kobashi and Ichijo (1986) also hypothesized that the paired vortices are formed
from streamwise vortex pairs that appear in the large-scale periodic motions.
The coupling between high-amplitude, positive wall-pressure peaks and flow
structure in the near wall region was studied by Johansson et al. (1987). They
characterized wall-pressure fluctuations in a two-dimensional boundary layer by making
simultaneous measurements of high-amplitude, positive turbulent wall-pressure peaks with
u' and v'. The results indicated that the amplitude of the pressure peak is linearly related
to the amplitude of the turbulent velocity peak, which in turn indicates that the generation
of high-amplitude pressure peaks is predominantly governed by the turbulence-mean-shear
interaction. Also, these pressure peaks were found to be caused, or the cause of shear-
layer structures ("bursts") in the buffer region (5
pressure peaks were found to be primarily associated with sweep-type motions-flow
parallel to the wall.
2.2.2 Compressible. Zero Pressure-Gradient Flow
Measurements on the wall under a turbulent Mach 0.6 boundary layer by Serafini
(1963) show that the convection velocity varies for different streamwise transducer
spacing in exactly the same way as for the incompressible boundary layer. Lilley (1963)
showed that the wall pressure fluctuations are the result of fluctuations in both the
vorticity and sound modes. Kistler and Chen (1963) found that the rms pressure on a
solid surface for flows in the Mach number range 1.33-5.00 was proportional to the local
skin friction. The value of p"/r, increases fairly linearly from the incompressible value
22
of-3 to -5 for M:2. They also found that the convection speed falls from 0.8 at M=1.33
to 0.6 at M=5. The effect of Reynolds number was found to be pnlQ-Res.4. The
integral scale of the wall-pressure fluctuations changed from 0.166 at M=1.33 to 0.00066
at M-4.54. Additionally, they found that the peak value of the correlation coefficient
drops to one half for a spatial separation of the measuring points of about two tenths of
the boundary layer thickness. Laufer (1964) measured the radiated pressure intensity from
a supersonic boundary layer and found it to be two orders of magnitude less than that
measured on the wall.
2.2.3 Incompressible. Adverse and Favorable Pressure-Gradient Flows
Adverse pressure-gradient flows differ from zero and favorable pressure-gradient
flows in that they are not self-preserving, and, hence, the value of every parameter is a
function of local position. Burton (1973) compared strong adverse and favorable
pressure-gradient flows with smooth and rough walls to a zero pressure-gradient flow.
For a favorable gradient, it was found that rms wall pressure intensity varied in
proportion to mean wall shear-stress with the same value as the zero pressure-gradient
case. The favorable gradient decreased longitudinal spatial decay rates and increased
convection velocities, whereas roughness had the opposite effect.
The adverse gradient slowed convection velocities and increased spatial decay
rates. Pressure statistics were found to depend on local mean flow parameters and
upstream flow conditions, but not on wall roughness. The velocity profiles were
logarithmic over a small region close to the wall, but were generally wake-like, so that
outer variable scaling was appropriate.
Insight into the suitable scaling of boundary layers can be seen from treatment of
the integrated momentum equation for two-dimensional turbulent flow (Burton, 1973):
8* P- _2u)dy- __ dy'dy+
r dx dx- U I dy 0 X 2 (2.25)
For rough walls the terms on the right-hand side of Eq. (2.25) must be retained,
whereas for smooth walls the terms on the right-hand side of Eq. (2.25) are at least three
orders of magnitude less than those on the left-hand side and can therefore be written as
dx dx(2.26)
For a favorable pressure gradient flow, the two terms on the left-hand side of Eq.
(2.26) are of equal magnitude, and their ratio is a valid measure of the magnitude of the
pressure gradient. However, for adverse gradient flow, the wall shear stress is small
compared to the pressure gradient term and the right-hand side of Eq. (2.26). Hence,
scaling with inner wall variables is not appropriate for the adverse pressure gradient case.
Burton also concludes that broadband spatial coherence of the wall-pressure
fluctuations is only moderately affected by imposition of a favorable pressure gradient,
when scaled on outer variables. Coherence is markedly improved in the adverse gradient
flows, which are dominated by disturbances that are large compared to the displacement
thickness and therefore decay more slowly. In only adverse gradient flows, broadband
convection velocities were found to be higher between oblique separations than between
longitudinally separated points. This was also the case for narrowband convection
velocities. Phase velocities were only moderately affected by favorable gradients, but the
effect on phase velocities for adverse gradient flow was strong and complex, but always
lowered the phase velocities considerably, the lowest values being about 0.25U,.
2.3 Finite Transducer Size Effects
The problem of inadequate spatial resolution of a pressure transducer was
recognized by Willmarth (1956). The finite size of a transducer-sensing element limits
its space resolution of a convecting pressure field associated with a turbulent flow.
Consequently, a lack of resolution in space causes an apparent inability to resolve in time.
Corrections were first attempted by Corcos et al. (1959), followed by an improved method
by Corcos (1963). This method was based on new experimental measurements of the
longitudinal and lateral space correlations and predicted that the attenuation at high
frequencies was much greater than predicted from the previous method.
Willmarth and Roos (1965) used several different-sized pressure transducers and
extrapolated the results to obtain the vanishingly small value of rms pressure. The result
was p"z.,=2.66, which would account for Willmarth and Wooldridge's (1962) results to
be increased by approximately 13%. They also concluded that Corcos's (1963) correction
can be used at low frequencies but not at high frequencies.
Schewe (1983) experimentally showed that the dimensionless transducer diameter
d+=du/v=19 is sufficient to resolve the essential structures of the turbulent pressure
fluctuations. If this criterion is assumed to hold for adverse gradient flow, then the
condition d'=19 is easier to obtain in adverse gradient flow due to the reduction in -,.
2.4 Shear-Layer Characteristics
The equations of motion for the two-dimensional, incompressible, turbulent
mixing layer have been solved by Gortler (in White, 1974). The boundary layer equations
are solved with the following antisymmetric boundary conditions.
U(-o)=U U(+o)=U2 U(O)= +U)
(2.27)
The solution is
U= -(U +U2) 1 + erfr4y)
2 U2+U xl (2.28)
where a-13.5.
No analytical solution has yet been found for the compressible mixing layer.
However, functional forms of the mixing layer growth-rate have been suggested by using
order of magnitude estimates. Using this method, Brown and Roshko (1974) suggested
M U (2.29)
Bogdanoff (1984) suggested that the convective velocity of the largest structures
in a mixing layer could be estimated by equating the stagnation pressures from both sides
of the mixing layer with respect to the velocity of the large structures.
{ i-A1 U( -Uc }1 Y- Y2 ( U2)2 -1
2 cI f 2 c2 (2.30)
For the case when Yi=Y2, the above equation can be solved explicitly for U,
2U C1 +1U2
C1 +C2 (2.31)
and the convective Mach numbers, MK, and Mc2, defined as
= u, =U-,
S 1 cC C2 (2.32)
are equal and can be written as
U -U2
M YI=Y2
cl +c2 (2.33)
It can be seen that the relative Mach number (defined in Eq. B.21) M,=2M, when the
average speed of sound is used to calculate Mr.
Papamoschou (1986) proposed that the compressible shear-layer growth rate is
functionally related to the incompressible growth-rate at the same density and velocity
ratios such that
8' fn.(r, s, M
S'0 fn.(r, s, Mc=0) (2.34)
This isolates the effect of compressibility so that the above ratio is a universal function
for free shear-layers. The ratio is given by Benjamin (1990) using a curve fit to the
experimental data as
S/ -3M2:
= f(Mc) = 0.2+0.8e -3
0 (2.35)
Dimotakis (1986) proposed that there are three main phases to shear-layer
entrainment, which are referred to as (1) induction, (2) diastrophy, and (3) infusion.
27
Induction is when fluid in the vicinity of the vorticity-bearing fluid is set in motion by
the Biot-Savart-induced velocity field, which is a kinematic, not diffusive, process.
Irrotational fluid sufficiently close to the vortical fluid will in fact participate in the large-
scale structure motions long before it has acquired vorticity of its own. These motions
appear at the low wavenumber part of the turbulent spectrum, and although irrotational,
can be considered part of the turbulent flow.
The second stage is diastrophy, where the irrotational fluid is strained until its
spatial scale is small enough to put it within reach of the (viscous) diffusive processes.
Viscosity then takes over and causes cascading to the Kolmogorov scale.
The third stage can be associated with other possible diffusive processes, such as
molecular mixing or heat conduction, and may or may not precede diastrophy, depending
on the relative magnitude of the corresponding molecular diffusivity to that of the
cinematic viscosity. In the case of gas-phase entrainment, it would be difficult to
distinguish between infusion and diastrophy because the corresponding diffusion
coefficients are of the same order.
Dimotakis (1986) also derived the incompressible shear-layer growth rate and
accounted for unequal entrainment from either side of the shear layer
S/ 1-r 1/ 1-p ]
0 +ri J 1+2.9(1 +r)/(1-r) (2.36)
where E is a constant.
Nixon et al. (1991) calculated the transverse mass flux around a vortex core using
transonic flow theory and proposed that the normalized growth rate for relative Mach
numbers less than 1.4 is
8'(r, s,M) M ( 1- M'
r,s, M=0) (1 M) (2.37)
where M~.AU/c. Nixon et al. also found that a considerable amount of the incoming flow
energy, which would normally be converted into rotational energy of the eddies within
the mixing layer, is expended compressing the gas.
Papamoschou (1989) attempted to disrupt the structures that he observed in
Schlieren photographs by using various splitter plate trailing-edge devices (vortex
generators, trip wires, and saw-tooth extensions). He found that these devices had little
effect on turbulent structures or shear-layer growth rate.
Goebel and Dutton (1990) made detailed velocity measurements of fully
developed, compressible shear-layers (0.40r M, K1.97) using laser doppler velocimetry
(LDV). They observed that the length for full development generally increased in the
order of mean streamwise velocity, streamwise turbulence intensity, transverse turbulence
intensity, and kinematic Reynolds stress. Fully developed, streamwise mean-velocity
profiles were well approximated by an error function profile, even for the more
compressible cases. Transverse turbulence and normalized kinematic Reynolds stresses
both decreased like the normalized growth rates with increasing Mr. Mixing lengths and
other turbulence quantities with a transverse fluctuation component also decreased with
increasing MK. This suggests that the primary effect of compressibility on the turbulence
field in a shear layer is the suppression of transverse velocity fluctuations. Streamwise
turbulence intensities remained fairly constant with M, which means that the anisotropy
of the turbulence increased significantly with M. They also found that the flowfields
were reasonably two-dimensional based on velocity profiles measured at two parallel
planes in the spanwise direction.
2.5 Jet Acoustics
There has been a great deal of research in the area of acoustics due to the major
increase in commercial air travel over the last few decades. The three areas relevant to
the current research are
(a) acoustics generated by turbulent flow,
(b) acoustics generated at nozzle lips, and
(c) acoustics generated from imperfectly expanded nozzle flow.
A subset of all these categories is the interaction of any one source with a solid wall.
The clearest way to understand acoustics generated by turbulent flow is to consider
the basic equation first derived by Lighthill (1952). This equation is the basis for
virtually all work on noise and can be derived from the mass and momentum conservation
equations:
ap apui Q
at xi (2.38)
and
apui apuu +p.
at 9x x. 5 (2.39)
where Qm is the rate of mass introduction per unit volume and Fi is the external force per
unit volume. The nonhomogeneous wave equation for propagation of sound in a uniform
medium at rest due to sources of matter, external forces or applied fluctuating stresses
a'p 2a2p am aF, a2Ti
S-co +
t2 ax,2 at axi axF, (2.40)
where
30
7=Up,,-c2 (2.41)
is the Lighthill equation, co is the sound speed in the medium, and 6b is the Kronecker
delta. The stress tensor for a Stokesian gas is given in terms of the velocity field by
au, au 2+2a "uk
P 'P" i ax-i 3 xk 1 (2.42)
Effects such as convection of sound by the turbulent flow, or the variations in the speed
of sound within it, are taken into account by incorporation as equivalent stresses (Lighthill
1953).
Each term on the right-hand side of Eq. (2.40) gives the effect of a different
acoustic source mechanism. The first, OQJ/t, gives the effect of mass introduction.
Examples include pulse jets, tip jet rotors, and the random mass fluctuations that can
occur across the exit plane of a jet exhaust (Pao and Lowson 1969). The second term,
aF/axi, gives the effect of external fluctuating forces that can act on the fluid. Examples
include compressors, propellers, helicopter rotors, and the fluctuating forces that exist on
a nozzle lip or on any body in a turbulent flow-stream. The third term, aTi/Oxidxj,
incorporates several different effects. Ti is called the "acoustic stress tensor." At low
Mach number, the most significant fluctuation of the acoustic stress tensor in a free jet
will be caused by the turbulent velocity fluctuations, which affect the puuj product. The
effect of viscous stresses and thermodynamic fluctuations are contained in the remaining
terms of T,.
Curie (1955) investigated the influence of solid boundaries using the above theory
and showed that the fundamental frequency of dipoles near the wall is one half of that
generated by quadrupoles. This is because the quadrupole strength per unit volume, puiuj,
being essentially proportional to (velocity)2, will have double the frequency of the
fluctuating velocity. On the other hand, the fluctuating force exerted on the fluid at the
solid boundaries will have the same frequency as the velocity fluctuations. The frequency
relation can be shown if one considers a flow in which the velocity in the x-direction is
given to a first approximation as
ui()=Ai()cos(nt+(i (2.43)
(where i is also a function of x), then the quadrupole strength per unit volume is
T.= pu ,ui pAiA cos(nt + C\)cos(nt + C)
=pA A, cos(2nt + i+ < )+cos(Ci- Cy)} (2.44)
Also, at sufficiently low Mach numbers the contribution to the sound field from the
dipoles should be greater than from the quadrupoles. Exactly how small the Mach
number must be before this occurs will depend upon the flow in question.
As the velocity in a free jet is increased, the maximum angle of farfield noise
emission starts to move from normal to the jet boundary to a downstream-pointing angle
(Lighthill 1954). For jet velocities considerably greater than the surrounding speed of
sound, the directional maximum becomes the Mach angle of the local convective Mach
number. Lighthill also noted from far-field measurements, that low frequencies are
emitted in a more downstream direction than high frequencies, and the high frequency
sound is associated with the mixing region just downstream of the nozzle where the
quadrupole strength is due to lateral transport of mean momentum across the shear layer.
The low frequencies originate from the fully turbulent region downstream (cf. Fig. 2.3).
32
In supersonic jets with a shock-cell structure, Powell (1953a, b) observed that
regular stream disturbances were assumed to give rise to stationary sources of sound on
traversing the shock-waves and several of these sources would interact with each other
and produce discrete frequencies. A powerful frequency was observed to travel upstream
and interact with the flow at the orifice, giving rise to an amplification process. The
frequency of this amplified screech for the two-dimensional case is roughly
1 c (2.4
where R is the jet pressure ratio and R, is the critical value. The value of d is the smaller
of the nozzle dimensions. Glass (1968) found that under certain circumstances, this
acoustic feedback can result in wide variations in the growth rate and decay of the mixing
layer. Ribner (1969) found that the ratio of acoustic energy flux to turbulence energy flux
for turbulence passing through a shock of finite strength varies almost linearly with shock
density ratio. The values range from 0.0036% at M=1.01 to 6.2% at infinite Mach
number. The value at M=2 is 1.39%. Tam and Tanna (1982) studied the characteristics
of converging-diverging supersonic nozzles at off design points and found the shock
associated noise to be proportional to (Mf -M)2 where Mj is the jet Mach number and
Md is the design Mach number.
Tam (1972) found that the noise of a nearly ideally expanded supersonic jet
emanates from two rather localized regions of the jet which are located at distances quite
far downstream of the nozzle exit. He suggests that large-scale instabilities of the jet flow
are responsible for transferring the kinetic energy of the jet into noise radiation. Tam
proposes that rapid growth of these waves causes the oscillations of the jet to penetrate
33
the mixing layer at two locations and to interact strongly with the ambient fluid there,
hence giving rise to intense noise radiation.
Using shadowgraph visualization, Lowson and Ollerhead (1968) found the sound
sources in small supersonic jets in the following order of dominance: (i) spherical
radiation from the nozzle, (ii) radiation from shock-turbulence interaction in the flow, and
(iii) Mach wave radiation. Chan and Westley (1973) also observed spherical waves from
the nozzle of a supersonic helium jet. The jet transitioned to turbulence within 1 diameter
and radiated weak shocks from this location with a frequency of 85 kHz, and directed
primarily along a cone 600 from the jet axis.
Independent of condition
of formation
Wavenumber. k
Inertial
subrange
Re' >>> 1
The three-dimensional velocity energy spectrum (after Hinze, 1975).
-burst paired vorticies
burstweep ejection
sweep = 8=== 7- ejection
QZt O
Vortex-ring model of coherent structure and burst (after Kobashi and
Ichijo, 1986)
Dependent on
condition of
formation
E(k,t)
Figure 2.1
Figure 2.2
r- Supersonic mixing
I layer. M>1
Secondary > r b Subsonic, turbulent
No- region. M<1
Primary -- Supersonic or
Secondary -----
Flow regimes in the ejector.
Figure 2.3
CHAPTER 3
EXPERIMENTAL INVESTIGATION-HARDWARE, PROCEDURES
AND MIXING-SECTION INLET CONDITIONS
3.1 General Description
An ejector facility was constructed in which two-dimensional, compressible shear
layers can be established at various Mach numbers, Reynolds numbers, density ratios and
velocity ratios. The flow apparatus is a three-stream, blowdown, supersonic primary,
subsonic secondary wind-tunnel with the capability of supplying different gases to the
primary and secondary. In this experimental investigation high pressure air, supplied by
the Gas Dynamics Laboratory compressed-air facility, was used for both primary and
secondary supply. After being brought to the desired (or design) Mach numbers, the three
streams are brought into contact downstream of two identical splitter plates, and mixed
in a rectangular, constant-area mixing (test) section. The downstream end of the
apparatus is connected to a transition diffuser, which transitions and diffuses the flow
from a 50.8 mm by 25.4 mm rectangular cross-section to a 4 in. circular cross-section
which houses a butterfly control valve. The flow is then dumped into a large muffler and
exhausted to the atmosphere. Time-averaged Schlieren and shadowgraph photography
were used to visualize the structure of the supersonic primary jet, the extent of the shear-
layers, and the extent of the laminar-to-turbulent transition regions. Impact pressure
profiles obtained using a traversing mechanism, were taken at twenty-five streamwise
positions with nominal separations of 20 mm. Additional sidewall static pressures were
obtained using a 30-tube manometer bank. Photographs were taken of the bank and
analyzed. Sidewall, one- and two-point, time-accurate pressure records were obtained
using miniature pressure transducers. All of the measurements are accompanied by a
record of upstream stagnation pressures, stagnation temperatures and, except for the
Schlieren and shadowgraph visualizations, sidewall static pressures. The different
measurements were all controlled using a PC-AT computer, which allowed precise
execution of the required functions during the short duration of the runs. The operation
of the blow-down facility was intermittent, with each run lasting from 3 to 6 seconds.
The ejector hardware, pressure measurement and visualization techniques, as well as
boundary-layer calculations are described in detail below.
3.2 Design Methodology
The two-dimensional ejector rig used in this study was a pre-existing facility built
by the author and his coworkers (Benjamin et al. 1993) in 1992 for dissimilar gas, ejector
experiments. The rig allows for variation in mixing section height from 38.1 mm to
51.4 mm with a fixed-height primary nozzle of 12.7 mm. In this study, the maximum
height of 51.4 mm was chosen so that the maximum number of wall measurements could
be made in that direction (see Fig. 3.1).
The minimum allowable dimension of the nozzle exit height was selected based
on: (a) the resolution provided by the traversing pitot-pressure probe, and (b) the number
of traverses necessary to map the various regions of the shear layer. Firstly, it was
estimated that the resolution of the pitot-pressure probe was no less than 0.2 mm, and that
at least ten measurements should fall across the layer (in fact, many more readings were
obtained across the shear layer). The above meant that the layer must grow to a thickness
of about 2 mm before the thickness can be measured with the probe.
Secondly, it was decided that a minimum of three equally-spaced streamwise
traverses were necessary to identify the primary core region of the shear layer (region 1).
For initial estimates it was assumed that along the length of this core (1) the shear layer
38
grows linearly and by equal amounts into the inviscid regions of the primary and
secondary, and (2) the layer grows with the same rate in both directions (relative to the
mixing-section centerline). This meant that the smallest mixing section that could be used
would result in a layer that would be about 6 mm thick at the end of the core (3 mm each
side of the centerline). That in turn meant that the nozzle exit height had to be at least
6 mm. A factor of two was introduced to account for any margin of uncertainty; hence
a primary exit height of 12.7 mm was chosen.
The exit height of the secondary streams was selected based on similar arguments.
Additionally, it was desired that, at least in some test cases, unmixed secondary flow exist
beyond the point where the primary core ends. Using these criteria, it was decided that
secondary flows having exit heights at least equal to the primary jet height would be
adequate. Hence a variable exit height of 12.70 mm to 19.05 mm was chosen for each
half of the secondary stream.
In selecting the depth of the mixing section (see Fig. 3.1) it was desired to
minimize flowrates as well as provide a flowfield that was approximately two-dimensional
(i.e., sidewall boundary-layers are thin compared to the width), and be adequately sized
for Schlieren visualization. The latter required that the mixing section was deep to
resolve small density-gradients. Since these are opposing factors, an acceptable mixing-
section width that reasonably met the criteria was determined to be 25.4 mm. Complete
details of the rig conception and design can be found in the report by Roan et al. (1993).
3.3 Experimental Apparatus
In the mixing section, the supersonic primary center jet is sandwiched between two
symmetrical, subsonic secondary streams and all three discharge into a constant-area duct
(see Fig. 3.1). Primary and secondary streams mix along the shear layers which begin
to develop at the nozzle exit planes where the streams first come into contact. The mean
development of the flowfield is mapped by traversing a pitot-pressure impact probe across
the mixing duct at a number of axial locations.
The primary and secondary flows were supplied by a high-pressure gas storage and
delivery system. Control of the rig, as well as data-acquisition of the mean conditions,
was accomplished using the laboratory PC-AT computer. This control process enabled
short run times and identically-repeatable instrument reading capability. The test
apparatus as well as the interface hardware and instrumentation are described in more
detail in the following sections.
Fine adjustments of primary and secondary supply pressures, as well as the mixing
section back-pressure had to be made for each test case to achieve the desired flow
conditions. To achieve repeatable pressure conditions between test runs, the run event
sequence and timing was carefully planned and implemented to ensure stability and
validity of the measurements while minimizing run time. The details of the set-up, run-
control, and data-acquisition procedures are also discussed later in this chapter.
3.3.1 Gas Storage and Delivery System
Pressurized air was used for supply of the primary and secondary flows. The
filtered and dried air was delivered from a bank of storage tanks, and supplied to the
primary and secondary plenums through pressure-regulating valves as desired. A
schematic of the gas storage and delivery system is shown in Fig. 3.2.
Compressed air was supplied by a Worthington two-stage positive-displacement
340 cfm (150 HP) compressor which delivers a maximum working pressure of 300 psia.
The compressed air is passed through several oil and water traps and a desiccant air-drier
to remove any moisture. The air is then stored in a bank of cylindrical tanks manifolded
together, with a total volume of approximately 420 ft'. In order to maintain an almost
constant supply pressure level, the compressor was operated throughout test runs. In
addition, the pressure-regulating valves were set when the supply pressure reached a
predetermined value; this value is determined after the compressor's second stage is
disengaged at its maximum delivery pressure. Monitoring of the supply pressure is
performed using a transducer connected to the supply line.
Air was delivered to the apparatus through a 2 in. schedule-80 steel pipe sized to
keep the pipe Mach number below 0.4 for the flow rates of interest. A gate valve placed
between the tanks and the test rig was used to isolate the supply and the test rig.
Automatic on/off control of the air flow was accomplished using a solenoid valve
(Omega, model SV207) placed downstream of the gate valve. The solenoid valve could
be controlled either manually with a switch, or electronically using a digital output
channel from the computer. The air could then be delivered to the primary and/or
secondary supply lines. On the secondary branch, control of the secondary flowrate was
effected utilizing 3/4 in. and 1/4 in. needle valves connected in parallel. On the primary
branch, a 3/4 in. needle valve and a 3/4 in. globe valve connected in parallel controlled
the flow rate of primary air. On each branch, the large and the small valves were used
for gross and fine flow-rate adjustment as required.
3.3.2 Primary and Secondary Lines
The gas flows required for the primary and secondary were delivered to the
respective plenums from the primary and secondary branches described in the previous
section. The primary line was constructed from 3/4 in. schedule-80 steel pipe and
attached to the primary plenum by a flexible metal hose with an inside diameter of
19.05 mm. This attachment method isolated the ejector from any mechanical vibration
due to the flow through the supply piping.
The secondary flow was supplied to the secondary plenums through four steel
reinforced PVC hoses. These hoses were connected to a manifold consisting of four
sections of 1 in. pipe welded at 900 from each other around the capped end of the 2 in.
secondary supply line. A schematic of the air supply hardware described above is shown
in Fig. 3.2.
3.3.3 Plenums
The primary and secondary plenums consist of flow-conditioning sections followed
by settling chambers (Figs. 3.1 and 3.3). The walls of these sections were machined from
aluminum stock and formed continuous flow passages of rectangular cross sections. The
primary plenum was in the center, sandwiched between two identical secondary plenums.
Two large rectangular plates 12.7 mm thick served as side walls for the primary and
secondary plenums. The remaining walls were machined from 25.4 mm thick aluminum
plate which were (a) the internal walls separating the primary and secondary plenum
cavities, (b) the rear wall for all plenums, and (c) the outer walls of the secondary
plenums. Two stainless steel dowel pins inserted through each wall in (a) and (c) and
into location holes drilled in the two side walls, accurately position each part of the
plenum assembly. All plenum walls are bolted together.
3.3.3.1 Primary plenum
The primary plenum consists of a flow conditioning section and a settling
chamber. It has a rectangular cross-section 59.7 mm high by 25.4 mm wide. These
dimensions approximately constrain the plenum flow velocity between 3 m/s and 27 m/s
for a wide range of operating conditions, as suggested by Pope and Goin (1978).
The flexible metal hose which delivers the primary gas to the plenum fastens onto
a thick-walled aluminum nipple with a 25.4 mm inside diameter. This inlet nipple is
welded at right angles to one of the plenum side walls. The gas then passes through two
different-sized perforated plates machined from aluminum, a section of aluminum
42
honeycomb, and two different-sized stainless steel wire meshes (cf. Figs. 3.1 and 3.3).
These flow-management devices help ensure uniform flow and low turbulence intensity.
The perforated plates are 3.9 mm thick and spaced 12.7 mm apart (16.6 mm from
center to center). The plates fit into slots machined in all four walls of the plenum
passage. The first plate is 27.4 mm from the inlet nipple centerline, and has a 12x5 array
of 4.42 mm diameter holes; the center-to-center spacing of the holes is 4.98 mm and
5.08 mm along the long and short dimensions of the plenum's cross section, respectively.
The second plate has a 14x6 array of 3.68 mm diameter holes; the spacing in this array
is 4.27 mm and 4.23 mm along the long and short dimensions, respectively.
The honeycomb section is located 34.9 mm downstream from the second plate.
This section is 82.6 mm long, and has a 3.18 mm hexagonal cell size, and fits in a recess
machined in all four walls of the plenum (cf. Figs. 3.1 and 3.3).
The first of the two mesh screens is approximately 25.4 mm downstream of the
honeycomb, and has a mesh of 30x30 wires/in, and a wire diameter of 0.165 mm
resulting in a 35.2% obstruction and a wire-spacing to wire-diameter ratio of 5.13. The
second mesh is 12.7 mm downstream from the first one, and has a mesh of 40x40
wires/in, and a 0.165 mm wire diameter, resulting in a 45.2% obstruction and a wire-
spacing to wire-diameter ratio of 3.85. The screens are soldered to brass frames which
fit in slots machined in the plenum walls. Both the screens and frames are shown in Figs.
3.1 and 3.3.
The modular design of the flow managements devices enabled easy inspection,
cleaning or replacement of any component.
The primary gas enters the settling chamber, which extends 44.8 mm downstream
of the second screen. A 1.59 mm diameter T-type thermocouple is located between the
first and second screens to measure total primary temperature, and a 0.79 mm diameter
pressure tap located on one side wall 25.4 mm downstream from the second screen is
43
used to measure primary-plenum stagnation pressure.' The outside of the wall is tapped
for mounting a pressure transducer.
3.3.3.2 Secondary plenums and flow passages
Each secondary plenum consists of a flow conditioning section and a settling
chamber similar to those of the primary plenum. Their rectangular flow areas are
71.9 mm high and 25.4 mm wide, which results in a local velocity of 26 m/s for the flow
conditions used in this investigation (see Chapter 4 for flow conditions).
The secondary gas is delivered to each plenum through two flexible hoses clamped
to inlet nipples that are welded at 90 on opposite sides of the plenums. The nipples are
made of aluminum and have inside diameters of 25.4 mm. In each plenum, the gas
passes through flow management devices nearly identical to those used in the primary (cf.
Figs. 3.1 and 3.3). Two different perforated plates machined from aluminum, a section
of aluminum honeycomb, and two different stainless steel wire meshes ensure uniform
flow and low turbulence intensity of the secondary stream.
The perforated plates are 3.94 mm thick and are 12.7 mm apart (16.7 mm from
center to center). The plates fit in slots machined in all four walls of the plenum passage.
The first plate is 27.38 mm from the inlet nipples centerline, and has a 14x5 array of
4.45 mm diameter holes; the center-to-center spacing of the holes is 5.18 mm and
5.13 mm along the long and short dimensions of the plenum's cross-section, respectively.
The second plate has a 17x6 array of 3.66 mm diameter holes; the spacing in this array
is 4.24 mm and 4.23 mm along the long and short dimensions, respectively.
SAlthough this method actually measures the static pressure and static temperature, the
plenum velocity is approximately 29 m/s resulting in deviations of 0.5% from the true
stagnation pressure, 0.15% from the true stagnation temperature, and 0.2% below the
true Mach number.
44
The honeycomb section and screens are the same as those described above for the
primary plenum except for the overall height, and are located at the same streamwise
positions.
The secondary gas enters the settling chamber which is 97.5 mm long. A pressure
tap located 25.4 mm downstream of the second screen allowed for measurement of total
pressure in each secondary settling chamber. This consisted of a 0.79 mm diameter hole
drilled through the side wall of each secondary plenum, and tapped on the outside for
mounting pressure transducers.
Following the settling chamber is a converging channel of rectangular cross-
section which accelerates the flow into the mixing section (cf. Figs. 3.1 and 3.3). It has
a constant depth of 25.4 mm and the height varies monotonically from 71.9 mm to
18.9 mm. The passage is formed by the plenums side walls, an "inner" contoured wall,
and an "outer" contoured wall. The "inner" contoured wall corresponds to the outer
surface of the nozzle block. The machining of the "outer" wall provided an accuracy of
approximately 0.05 mm. Both contoured walls have a shape defined by fifth-order
polynomials with zero-valued first and second derivatives at the beginning and end of the
contraction. The length-to-height ratio of the contraction is 2. Each secondary stream
discharged into the mixing section at the end of the contraction.
3.3.4 Primary Nozzle
The primary gas exits the settling chamber and discharges into the mixing section
through a two-dimensional supersonic nozzle. The nozzle was machined from aluminum
to provide the required flow Mach number of 1.8 for gases with a constant-pressure
specific heat ratio of 1.4.
The nozzle consists of two identical blocks bolted to the internal walls of the
plenum and clamped between the side walls (Figs. 3.4). The blocks are mounted on
opposite sides of the plenum centerline, and provide the desired change in flow area.
Two stainless steel dowel pins inserted through each block and into location holes drilled
in the plenum side walls position the blocks accurately. Each block has two contoured
surfaces; the "inner" surface corresponds to the contoured walls of the primary nozzle,
while the "outer" surface serves as the wall for the secondary passages. One of the
plenum's side plates is split into two sections to facilitate easy exchange of different
nozzle blocks.
The initial portion of the nozzle blocks is straight and serves as an extension to
the walls of the constant-area settling chamber. After this section the blocks provide a
contraction region where the flow begins to accelerate. The shape of this region is
defined by a fifth-order polynomial with zero-valued first and second derivatives at the
beginning and end of the contraction. The contraction region has a length-to-height ratio
of 2.5 and ends at the nozzle throat. This profile design provides uniform flow and
ensures that flow separation does not occur (Papamoschou 1986).
Following the contraction is the diverging region of the nozzle, the shape of which
was obtained using the two-dimensional method of characteristics technique, and
neglecting boundary layer corrections. The intersection between the converging and
diverging regions provide a sharp edge at the nozzle throat.
The nozzle blocks have a trailing-edge thickness of approximately 0.5 mm. Once
installed, the nozzle extends 177.8 mm beyond the second screen at the entrance to the
settling chamber. The contoured walls of the nozzle blocks were machined to within
0.05 mm. The installed throat height, 2h,, is 8.790.03 mm compared to the design
height of 8.84 mm, and the installed nozzle exit height, 2b, is 12.650.03 mm compared
to the design height of 12.70 mm. The installed dimensions result in a calculated value
of M,=1.80 based on 1-D isentropic gas-dynamics area relations. The length of the
diverging section of the nozzle is 16.2 mm.
3.3.5 Mixing Section
The primary and secondary streams come into contact at the exit plane of the
plenums (cf. Figs. 3.1 and 3.3). At this point the streams begin to mix in a constant-area
section of rectangular cross-section (Figs. 3.1 and 3.5). The section is 25.4 mm wide and
51.4 mm high. It is formed by two side-walls, a top wall and a bottom wall which bolt
together providing the desired flow passage. The section is bolted at one end to the
plenum, and it extends 500 mm beyond the exit plane of the plenums. For pitot traverse
and wall static-pressure measurements, both side-walls were machined from aluminum.
For turbulent wall-pressure measurements, the measurement side-wall was machined from
brass. The top and bottom walls were machined from brass.
The side wall used for mean, static wall-pressure measurements has 29 holes
drilled through, all of which are 1.59 mm diameter. Twenty-seven of these holes are
spaced in the streamwise direction every 10 mm along the centerline of the wall, with the
first one positioned so that it lies at the exit plane of the plenums on the primary
centerline. This hole is used to measure the primary-exit static pressure. The other two
holes also lie at the exit plane of the plenums and are located 13.2 mm on either side of
the primary centerline, in the secondary flow. These holes are used to measure the
secondary streams' exit static pressure. A short section of stainless steel surgical tubing,
inserted and glued in a counterbore provided with each hole, is used to connect pressure
transducers via 1.59 mm internal diameter flexible plastic tubing. The plate with wall
static pressure taps is shown in Fig. 3.6.
For the time-accurate wall-pressure measurements, an aluminum frame holding
interchangeable brass plates was designed with the same dimensions as the side wall used
for mean, static wall-pressure measurements. Two types of interchangeable brass plates
were constructed: solid "blank" plates, and plates with hole patterns for static-pressure
transducers. The plates can be placed at various locations along the frame. The holes
in the plates with transducer hole-patterns are reamed to 2.64 mm diameter. Brass plugs,
or sleeves that hold pressure transducers of 2.615 mm diameter, fit in the holes and are
shown in Fig 3.7. An O-ring seals the plugs 0.89 mm from the flow surface, while the
sleeve does not have a seal due to constraints of size. This was expected not to be
significant because there is a metal to metal seal where the sleeve head is fastened to the
back of the plate. Acoustic disturbances caused by the gap between the sleeve and the
hole, such as a Helmholtz resonator, have a calculated frequency on the order of 150 kHz
that is well above what is measured here. A typical installation is shown cutaway in Fig.
3.1, while Fig. 3.8 shows the dimensions of the assembled frame and plates used in this
research. The transducer-hole spacing is either 4.09 mm or 8.17 mm in the streamwise
direction, and 3.44 mm in transverse direction.
The top and bottom walls each have 14 ports machined along their centers to
provide access for the stem of the traversing impact probe. The streamwise spacing
between ports is 20 mm and the first port is 25.4 mm downstream from the mixing
section inlet. Custom-made threaded brass plugs, each fitted with an O-ring, provide a
leak-tight seal for the ports. The nonalignment of the plug ends was minimized by
material removal using in turn, 220, 320, 400 and 600 grain size abrasive paper with the
plugs fitted in their respective ports. Final buffing of the surfaces was accomplished
using a mixture of jeweler's rouge and tallow that produced better than a 0.2 lm surface-
roughness finish. The resulting nonalignment of the plug ends can thus be confidently
stated as being less than 0.015 mm from the surface of the wall. Two additional plugs
were used in connection with the traversing probe. These plugs have a 2.29 mm diameter
hole drilled through their center in which the stem of the probe slides. Figure 3.9 shows
the top wall and Fig. 3.10 shows the brass plugs.
3.3.6 Traversing Impact-Pressure Probe
The traversing total-pressure impact probe constructed for this experimental study
is shown in Fig. 3.5. As a result of the variable probe-interference effect noticed during
previous tests (Roan et. al, 1992), a traversing probe arrangement that results in a nearly
constant obstruction of the flow was constructed. This was accomplished by attaching
an extension to the probe stem (i.e., a dummy stem beyond the sting). The stem and the
extension slide through the holes drilled in the plugs that are used with the probe.
To simplify the procedure necessary to change the probe position, a probe was
designed that allows for removal of the sting. This was accomplished with a special
coupling machined from stainless steel. The sting of the probe was soldered to the male
part of the coupling. The male part was screwed into the female part to which the stem
and the dummy stem were soldered. A Teflon bushing sealed the clearance between the
parts.
A probe sting was constructed from hard tempered stainless steel, hypodermic
tubing having outside and inside diameters of 0.90 mm and 0.58 mm respectively. The
sting is 25.4 mm long, and the nose of the sting has a screwdriver wedge-like shape with
a rectangular opening of approximately 0.89 mm by 0.13 mm. To obtain this geometry,
a piece of 0.127 mm thick shim stock was inserted approximately 4 mm into the end of
the tube, and then clamped together. Care was taken to insure that the tube collapsed
evenly about a plane passing through the axis of the tube. The flattened end was then
shaped to a wedge using a sharpening stone, and then polished to a mirror surface with
600 grain size abrasive paper and 348 rouge.
The stem of the probe and its extension are each roughly 100 mm long, and were
made from hard tempered stainless steel tubing having outside and inside diameters of
3.05 mm and 2.39 mm, respectively. The open end of the stem was connected directly
to the pressure transducer with a compression fitting.
The probe is traversed across the mixing section by means of a computer-
controlled, stepper-motor-driven, leadscrew. The leadscrew assembly is mounted on top
of the upper brass wall of the mixing section. The assembly slides in a rail built into the
brass wall, and can be locked at any desired streamwise position. The probe/leadscrew
assembly is shown in Fig. 3.5. The leadscrew has a pitch of 10.16 mm. The stepper
motor provides 200 steps per revolution, while the motor driver allows for half-step
motion. This arrangement translates into a linear resolution of 0.0254 mm for the traverse
of the probe. The time required for the 46.99 mm traverse was 2.00 seconds. The
pressure transducer used in conjunction with the probe (see section 3.4 for
instrumentation) was mounted on a bracket bolted to the traversing nut of the leadscrew
assembly.
3.3.7 Transition Diffuser and Exhaust
Immediately downstream of the mixing section is a 760 mm long transition-
diffuser with a rectangular cross-section inlet of 50.8 mm by 25.4 mm and a circular
cross-section outlet diameter of 102.3 mm. The design has a 4.820 average included
divergence angle, and an area ratio of 6.4 (see Fig. 3.11).
The diffuser is constructed from two identical sections welded together, each made
from 4.76 mm thick 304 stainless steel sheets bent to the desired shape using about 20
longitudinal bends. Flanges were then welded to each end suitable for mating upstream
to the mixing section and downstream to the 4 in. butterfly valve (Keystone 992).
The flow is exhausted to the atmosphere by way of 4 in. diameter PVC schedule
40 pipe, 10 in. diameter schedule 40 steel pipe, and through a Maxim BR31 silencer to
the atmosphere. These components are shown schematically in Fig. 3.11.
3.4 Instrumentation
A schematic of the instrumentation and computer interface is shown in Fig. 3.12.
Pressure and temperature data were measured using various pressure transducers and T-
type thermocouples. These devices were connected to two 16-channel multiplexers
(Acces AIM-16(P)) using shielded twisted-pair wire for the pressure transducers and steel
braid shielding for the thermocouple wires. The multiplexers were connected to a
moderate-speed analog and digital (A/D) computer board (Acces AD12-8), with 12-bit
data resolution, mounted in a 12 MHz AT-type personal computer (PC). Using Quick
Basic version 4.5 software, the maximum A/D throughput was approximately 9,000 single
channel samples/second.
Digital output from the A/D was used to control two electronic relays (Crydom
S440) supplying 110VAC to the flow solenoid shutoff valve (OMEGA SV207), and a
solenoid valve used for venting the air tank to the atmosphere (OMEGA SV207).
Three models of pressure transducers were used at the following locations
(definitions of designations in parentheses):
P01 (Primary stagnation pressure): OMEGA ENGINEERING PX621,
101.3-2170 kPa, 15VDC input 20mA maximum, 1-5VDC output. The
input power was supplied from a Power-One HCBB 75W set at 15.0VDC
with a 1.5A capacity.
PS1, PS2, P1-P9 (Mixing section, inlet stream, static pressures, and wall
static-pressures at variable locations from 10 mm to 436 mm in 10 mm
steps). PS1 (primary static pressure) and PS2 (secondary static pressure):
Micro Switch 140PC, 0-207 kPa differential, 10VDC input at 20mA
maximum, 1-8VDC output. The input power was supplied by an Acopian
V59D6A with a capacity of 1A, set at 10.0VDC.
51
POPR (Traverse impact-pressure): Validyne DB15, 0-414 kPa differential,
connected to a Validyne CD12 carrier demodulator with 0-10VDC output.
P02 (Secondary plenums' average stagnation pressure): Validyne DB15, 0-
103 kPa differential, connected to a Validyne CD15 carrier demodulator
with 0-10VDC output.
DP2 (Average differential pressure between the secondary plenums'
stagnation and static ports): Validyne DB15, 0-12.4 kPa differential,
connected to a Validyne CD15 carrier demodulator with 0-10VDC output.
PSPR (Static pressure on wall centerline at position of impact-pressure
probe tip): Validyne DB15, 0-103 kPa differential, connected to a Validyne
CD 15 carrier demodulator with 0-10VDC output.
PS2bk (Redundant reading of mixing section, secondary inlet, static
pressure): Validyne DB15, 0-69.0 kPa differential, connected to a Validyne
CD 15 carrier demodulator with 0-10VDC output.
PAIR (Supply air pressure): OMEGA ENGINEERING PX241, 101.3-2170
kPa, 15VDC input 20mA maximum, 1-5VDC output. The input power
was supplied from a Power-One HCBB 75W set at 15.0VDC with a 1.5A
capacity.
The specifications of all transducers used, including their calibration curves, accuracies,
and locations are given in Appendix C, Tables C.1 and C.2.
52
Two T-type (copper-constantan) thermocouples, with designations T/C 1 and T/C
2, were installed in the plenums to measure the stagnation temperatures of the primary
and secondary flows, respectively.
3.4.1 Impact Pressure
A 5VDC, 1.0 amp stepper motor (New England Affiliated Technologies 23D-
6102) drove the impact-pressure probe traverse mechanism and developed a maximum
dynamic torque of 0.13 N-m at speeds of less than 380 pulses-per-second (pps). It was
powered by a 0-40VDC, 0-5A variable power supply (Hewlett-Packard 6266B) set at
12.9VDC and 1.OA. During tests the stepper motor was run as fast as practicable
(953 pps = 24.2 mm/s) to minimize unsteady and nonstationary effects in the mixing
section and also to maintain constant inlet-flow conditions during the blowdown.
The probe was traversed across the mixing section using a 2.00 second
trapezoidal-motion velocity profile. This was accomplished using a single-axis stepper-
motor driver card (Industrial Computer Source 6402) and controlled by a single-axis
stepper-motor control board (Industrial Computer Source 5000) mounted in the PC. The
stepper-motor driver and controller cards were isolated from the PC by using an external
5VDC, 6.0A power supply (Power-One HCBB 75W); separate grounding for power and
data was also used.
3.4.2 Sidewall Mean Static Pressure
The sidewall static pressures were measured using a Meriam Instrument Co.
manometer bank (Model 33M1335) which has 30 mercury-filled tubes. Connections to
the sidewall were made using 1/32 in. inside diameter plastic tubing having lengths of
approximately 3 m. The manometer bank has fluorescent backlights that allowed
reasonably fast photographic exposure times (-1/250th second). Before each test run, the
53
lights were switched on for at least 30 minutes prior to running a test, so that thermal
equilibrium was achieved.
3.4.3 Sidewall Time-Accurate Pressure Measurements
Two Kulite Semiconductor (Model XCW-062-25SG) sealed gage miniature
pressure transducers were used for time-accurate pressure measurements. The sealed-gage
transducer was chosen to eliminate the possibility of external pressure influences. The
body of the transducer has a maximum diameter of 1.62 mm and is 31.75 mm in length.
The two transducers were individually mounted in transducer sleeves (described in
Section 3.3.5) using Dow Coming 3145 RTV adhesive/sealant as recommended by Kulite
engineers. When cured, the RTV has a tensile strength of 7 MPa. Using the RTV to fill
the 0.075 mm gap between the transducer outside diameter and the plug wall, it was
possible to mount the transducers within 0.025 mm of the plug ends.
The 0.7 mm pressure sensitive area is composed of a fully active, four-arm
wheatstone bridge diffused into a silicon diaphragm. The diaphragm is located
approximately 0.25 mm behind a metal screen which has 10 holes of 0.152 mm diameter
drilled through it on a 0.89 mm diameter circle. The effective diameter of the transducer
is thus 0.152 mm. The transducers have a range of 170 kPa from the atmospheric sealed-
pressure, and manufacturer's specifications of maximum combined linearity and hysteresis
of 0.5% full scale best fit straight line (FS BFSL), maximum thermal sensitivity shift
of 2% FS/100F, and a flat frequency response to at least 50 kHz. The natural
frequency of the transducers is greater than 250 kHz. They have a compensated
temperature range of 300 K to 355 K, input impedances of 2542 Q and 3431 0, output
impedances of 1322 Q and 1953 0, and at 15.00 VDC excitation have sensitivities of
1.201 iV/Pa and 1.215 gV/Pa, respectively. The transducers were calibrated by the
manufacturer to standards traceable to the National Bureau of Standards.
54
Matched, custom-built power supplies (see Fig. 3.13 for circuit) were used to
excite the transducers at 18.16 VDC nominal, which is within the 20 VDC maximum
excitation level specified by the manufacturer. Using the linearity of the transducers, this
level of excitation raised the transducer sensitivities to 1.450 piV/Pa and 1.467 pV/Pa.
Output from each pressure transducer was split into two, one being indirectly fed
to an Access multiplexer (described above) for static readings, and the other used for
dynamic readings. A custom-built buffer was designed for the split as noise from the
multiplexer was found to seriously contaminate the dynamic side. The buffer is shown
schematically in Fig. 3.14, in which the operational amplifier (Harris HA-5177) has an
ultra-low offset voltage of 20 uV and a low noise-level of 9.0 nV/VHz.
The dynamic split from the pressure transducer was fed to a pre-amplifier
(Stanford Research Systems Model SR 520) and was AC coupled using the pre-amplifier's
0.03 Hz capacitor, resulting in a high-pass filter with -10 dB/decade attenuation. The
pre-amplifier gain was calibrated by the manufacturer to standards traceable to the
National Bureau of Standards (1%, DC to 10 kHz, and 3%, 10 kHz to 1 MHz), and the
quoted input-noise associated with the preamplifier is 4 nV/vHz for gains of 100 or
greater, 10 nV/VHz for a gain of 50, and 11 nV//Hz for a gain of 20. The gain of the
pre-amplifier was varied between twenty and five hundred depending on the flow
conditions. In the AC-coupled mode, the pre-amplifier has a low-pass filter capability of
-20 or -40 dB/decade attenuation, and high- and band-pass capability of -20 dB/decade
attenuation. The filter cutoff frequencies range from 0.03 Hz to 1 MHz in steps of 3 and
10.
The output from the pre-amplifier was AC-coupled (0.8 Hz half-power cutoff
frequency) to a two-channel, 12-bit resolution, Tektronix Fourier Analyzer (Model 2642)
for time and frequency domain analysis. The analyzer has the capability of measuring
up to 1600 spectral lines at a bandwidth up to 200 kHz. The bandwidth was usually set
55
to 100 kHz as it was felt that the transducers would not be reliable at greater frequencies.
The noise floor for the analyzer is at most 4 pVJ..Hz when set at 14 mV full scale
input, and has a full scale input range from 14 mV to 10 V. The analyzer samples at
a rate of 2.56 times the bandwidth to satisfy the Nyquist sampling criterion, and to
compensate for the anti-aliasing filter attenuation.
The analyzer was connected to, and controlled from, the PC-AT computer via a
115.2k bits/second RS-232 connection. Using the local 2 Mbytes memory, the analyzer
was used as a simultaneous sample and hold, remote, high-speed data collection device.
The data in memory could then be subsequently stored and analyzed at the user's
convenience. This method was used for all data collected in this research and assures
continuous data records.
3.4.4 Visualization
Flow visualization was employed to detect the mean structure of the flow and to
estimate the location of laminar-to-turbulent transition. This was accomplished with
continuous light source Schlieren and shadowgraph optical systems. A schematic of the
system is shown in Fig. 3.15. All the components were mounted independently on heavy
steel stands at the same height as the mixing section. All of the components could be
adjusted for correct alignment and focussing except for the mixing section.
The continuous light source is 150-watt short arc xenon lamp (Osram XBO150)
mounted in a fan-cooled lamp housing (Oriel Model C-60-30) with a built-in 32 mm
diameter, condenser. The light source is powered by a Universal Lamp Power Supply and
Ignitor (Oriel Model C-20). The lamp is focused onto a horizontal knife edge at the
focus of the first 400 mm diameter, F8.25 parabolic mirror (J. Unertl Optical Co.). The
parallel light then passes through the mixing section to the other parabolic mirror, and is
focused onto another knife edge that produces a Schlieren image on a finely-ground,
56
glass plate. Photographs of the flow were obtained with the knife edge angled at 0, 30,
600, and 90 counter-clockwise from the horizontal streamwise direction. Also, the knife
edge was removed to obtain a shadowgraph photograph.
The mixing section glass windows are schematically shown mounted in the mixing
section in Fig. 3.16. They are made of precision, optical quality, BK7-P crown glass, and
were manufactured by Schott Glass Technologies to conform to MIL-0-13830A. The
overall dimensions measure 305 mm long by 50.8 mm wide by 12.7 mm thick. The
optical specifications are surface flatness within 6 waves over the total length, parallelism
within 16 arc-seconds, and scratch/dig of 80/50. Mounting in the aluminum frames was
accomplished using a pourable, two-part urethane rubber (Devcon, Flexane 80 Liquid
15800) which cures at room temperature. Curing shrinkage is very small at 0.0018 m/m,
and when cured, it has a tensile strength of 14.5 MPa.
3.5 Data Collection
Data collection was made using the laboratory PC-AT computer and electronic
instrumentation described in section 3.4, and was controlled by three computer programs
written in Quick Basic.
A. The first module (set of programs), called ATFRONT, is used to set up a test run
for the first time and functions as follows:
Al. The conditions for the test are obtained from a predefined test matrix or
typed in manually.
A2. The probe's axial position is recorded.
A3. The wall port corresponding to the impact probe's axial location is
recorded.
57
A3. A set of warnings and checks regarding the test rig were displayed to the
screen for the operator to verify.
A4. The above data was stored in a file called ATFRONT.MAB for use by
other modules.
B. The second module is called ATZERO and measures and records the transducer
zero offsets for subsequent use:
B 1. The test rig is opened to atmosphere and 75 readings from each transducer
are recorded.
B2. The means and standard deviations are calculated and displayed for the
operator to verify.
B3. The means are written to a file called ATZERO.CAL. If the file already
exists, it is renamed ATZERO.OLD before being written.
C. The third module is called ATMAIN and controls the actual test run. The
program has the option of controlling a test run for an impact-pressure traverse or
a wall-pressure. The major details for an impact-pressure traverse are as follows:
C 1. The pitot pressure probe is set up for a trapezoidal velocity-profile traverse
as follows:
1. Initiate traverse using near-maximum motor torque at an
instantaneous (t=0.00 seconds) speed of 400 pulses/second (pps).
2. Accelerate from the top of the mixing section at t=0.00 seconds to
953 pps at t=0.10 seconds.
3. Slew at constant speed of 953 pps until t=1.90 seconds.
4. Decelerate from 953 pps at t=1.90 seconds to 400 pps at
t=2.00 seconds.
5. Instantaneous stop at t=2.00 seconds.
C2. Solenoid valves are opened using the keyboard. Monitoring of the supply
stagnation pressure allows precise repeatability of the start flow-conditions.
User-defined, inlet stagnation and static pressure ranges are automatically
monitored for correct flow conditions. The test will not run if the flow
conditions are outside these ranges. Adjustment of the primary or
secondary valves, or downstream butterfly valve allowed precise control
of the flow conditions.
C3. After 1.00 seconds, ten samples of each pressure and temperature are read
at a sampling rate of 1864 samples/second, and a check for overpressure
is made.
C4. The probe traverse is made with the motor-controller card counter being
discretely sampled as fast as possible. The probe pressure is read
immediately following the a step change in counter reading. In testing this
procedure, it was found that three to eight pressure readings were obtained
for each counter reading.
C5. Ten samples of each system pressure and temperature are read at a
sampling rate of 1864 samples/second.
C6. The gas-supply solenoids are closed.
C7. The probe is returned to its initial position using maximum motor-torque.
C8. The means and standard deviations of the mixing-section conditions are
calculated before and after the run, and are written to the output file along
with the probe data.
C9. The operator then has the option of displaying the profile on the screen or
terminating the test.
For a wall-pressure measurement, the control procedure is as follows:
Cla. The Fourier analyzer is set up to remotely capture the desired number of
data points using the desired bandwidth.
C2a. Solenoid valves are opened using the keyboard. Monitoring of the supply
stagnation pressure allows precise repeatability of the start flow-conditions.
User-defined, inlet stagnation and static pressure ranges are automatically
monitored for correct flow conditions. The test will not run if the flow
conditions are outside these ranges. Adjustment of the primary or
secondary valves, or downstream butterfly valve allowed precise control
of the flow conditions.
C3a. After 1.00 seconds, ten samples of each pressure and temperature are read
at a sampling rate of 1864 samples/second, and a check for overpressure
is made.
C4a. The Fourier analyzer is started. For a 100 kHz bandwidth, this step took
less than 2 seconds.
C5a. Ten samples of each system pressure and temperature are recorded at a
sampling rate of 1864 samples/second.
C6a. The gas-supply solenoids are closed.
C7a. The means and standard deviations of the mixing-section conditions are
calculated before and after the run, and are written to the output file along
with the probe data.
C8a. The raw data is downloaded from the Fourier analyzer to the PC-AT and
stored in a file for post-processing.
3.6 Signal Analysis
This section generically describes the complex, discrete, Fourier transform routines
used by the Tektronix Fourier Analyzer (Model 2642). The computations for the fast
Fourier transform (FFT) algorithm are performed in 16-bit, fixed-point arithmetic, and
uses a block floating basis to scale the data. The data obtained from the discrete Fourier
transform (DFT) of a frame (usually 3200 points in this research) is then converted from
fixed to floating point. Before summation of multiple frames for averaging, the DFT is
converted to 32-bit IEEE floating-point format (8-bit exponent, 23-bit plus sign mantissa
plus the implied hidden bit).
The FFT routine computes the DFT pair:
N-1
Inverse: z(i)= Z(k)Wk ,i=0,...,N-1
k=0 (3.1)
N-1
Forward: Z(k)= z(i)W' ,k=0,...,N-1
i=0 (3.2)
where W=exp(-j27/N),
z(i) is a complex time history, and
Z(k) is a complex frequency function of the same number (N) of data
values.
3.6.1 Autospectrum
The single-sided autospectrum is computed from the convolution of the complex
DFT, X(k):
T ) (3.3)
where Pr=NT=the length of a data block, in seconds,
k=-0,1,2,...,(N/2)-1 and
T= the sampling interval.
The factor T2/PT arises because it is omitted from the DFT computation for simplicity in
scaling. The spectrum is single-sided because the textbook FFT is symmetric about zero
and therefore has "negative" power frequencies. To avoid the negative frequencies, only
(N/2)-1 points of X(k) and X'(k) are used. This doubles the power for the positive
frequencies and results in zero power for negative ones.
In all of the data collected in this research, multiple frames were obtained to
reduce the statistical uncertainty of the measurements. Additive averaging was employed
in the following manner:
S (k)= ( (k) M (3.4)
where S,,(k) is the most recent autospectrum, and
M--total number of frames.
The final result after M averages is
M-1
S '/(k)=_1 ES (k)
Mw -0 (3.5)
The normalized standard deviation (or standard error) of the averaged rms
measurement is given by
2- (36)
2(/M (3.6)
Most of the results that will be presented are averages of 10 frames which have a
standard error of 16%. However, some of the data was compared to 40-frames averages
which have a standard error of 8%, and it was found that 10 frames gave repeatable
results in terms of both frequency amplitude and rms. These results will be discussed in
Chapter 5.
3.6.2 Cross-spectrum
The cross-spectrum is computed in a similar manner to the autospectrum except
that the cross-spectrum is complex. The cross-spectrum of a size N frame is
T2
SX(k) =-X *(k)Y(k)
P (3.7)
and the average cross-spectrum of M frames is calculated by
S (k) =S / ( -)(k) '
-w y(w- M (3.8)
where S,(k) is the most recent cross-spectrum. The final result of the M averages is
thus:
M-1
S '(k) = S (k)
M <=o (3.9)
3.6.3 Correlation Functions
The auto- and cross-correlation are covariance functions and are computed using
inverse Fourier transforms (IFTs). For accurate covariance results, the acquired data is
time-averaged, zero-padded to avoid wraparound error, and has no window applied to the
63
spectrum calculation. The spectrum is modified by changing the sign of the odd-indexed
data values, k=-,3,5,...,(N/2)-1, followed by an IFT to obtain the covariance:
Cross -correlation: sy(i) =IFT[S(k)] (3.10)
Auto -correlation: s (i) =IFT[S (k)] (3.11)
where the subscript "c" distinguishes these functions from the final corrected functions.
For both functions, the lag index, i, runs from
i= -N,..., -1,0,1,...N-1 (3.12)
The auto-correlation function is symmetric, and hence there are only N distinct values for
this function.
Both correlations are then adjusted to account for the linear taper that is induced
by the fact that the spectra are divided by N instead of N minus the lag number. The
statistical result is to provide an unbiased estimate of the correlation function:
s Y(0 = -' i=--'N"-,...'0' N_"-
Si=-,1 (3.13)
3.6.4 Transfer and Coherence Functions
The averaged transfer function is computed from
H' (k)=S k) k=0,1,...,N-1
x S/(k) 2(3.14)
and the averaged coherence function is computed from
S '* (k) S (k) ,
y (k)- yk ,,..., -1
S '(k) S '(k) 2 (3.15)
64
The transfer function compares the gain of the "response" channel to the
"reference" channel, that here would be calculated from two pressure transducers' outputs.
It also contains phase information. Two different results can be obtained by switching
the response and reference channels, and the results can be compared to identify which
direction certain frequencies are travelling.
The coherence function is similar to the transfer function except that it is not
affected by switching the reference and response channels, and contains no phase
information. It is useful for identifying the relative power of frequencies present at two
locations.
3.6.5 Windows
For all measurements except for the correlation measurements, a window is
applied to the data in the frequency domain. The benefit of using a window is to reduce
the effects of discontinuities in the analyzed signal introduced by the FFT process. The
function of the window is to reduce the values of the ends ofa N point frame to zero, and
avoid wraparound error. No windowing is used for the correlation measurements because
the window is applied in the frequency domain, and would therefore distort the time
record, producing an incorrect analysis.
The Hanning window is applied to the data when a window is used, and is
implemented for each FFT operation in a calculation:
Z'(k)= Z(k) -Z(k-1) (3.16)
2 4
where Z'(k) is the windowed FFT.
It is obvious that, compared to using no window, windowing affects the rms power
and amplitudes of the spectral lines. This can be overcome by applying an amplitude
scaling factor of 2 (reciprocal of the first weighting term), or an rms scaling factor of
2.667 to the windowed data. In this research, the rms scaling factor is always used so
that the total power values are correct. As amplitude correction is not employed, the
effect on spectral amplitude on a pure sine wave for example, would be an amplitude
reduction of 1.22 compared to the true value.
A further effect of the Hanning window is to reduce the effective duration of the
record being transformed to half of that represented by the samples themselves. This
reduction can be overcome by overlapping the frames by 50%, and is used here.
3.7 Boundary-Layer Conditions at the Mixing-Section Inlet
The boundary layer thicknesses at the mixing-section inlet were calculated to
quantify the mean inlet-conditions and to validate the impact-pressure measurement. The
thicknesses were calculated using the compressible Thwaites method (Thwaites, 1949) for
momentum thickness of Rott and Crabtree (1952), that is valid for the compressible
boundary under arbitrary pressure gradient with no imminent separation. The necessary
assumptions are : (a) adiabatic wall, (b) the Prandtl number of the gas is unity, and (c)
the viscosity jI is a linear function of temperature.
These calculations were performed for the trailing edge thicknesses of the primary
and secondary on either side of the nozzle block. The momentum thicknesses were
assumed to be zero at the start of the convergence sections leading into the mixing
section, and the calculations are shown in Table 3.1. The details of the method are given
in Appendix B.
66
The initial thicknesses on the side walls were not calculated, but are approximately
the same as the ones calculated presented in Table 3.1 due to the similarity in the
freestream, axial Mach number profiles.
Using the calculated value for 6' to get the effective primary flow height 2(b-6),
the predicted primary Mach number can be estimated using this value in the one-
dimensional, isentropic, gas-dynamics relation for area ratio. Also, it can be calculated
that the measured value should be increased by about 0.2% to account for the static
temperature and pressure measurements being made on the side wall of the primary
plenum. This results in a corrected, measured primary Mach number of 1.79. This
agrees with the primary Mach number calculated from the measured nozzle-throat and
exit heights, and accounting for the displacement thicknesses in Table 3.1, to within the
tolerances of the height measurements.
Table 3.1 Boundary-layer thicknesses of primary and secondary supplies.
(* 0 8*
Location 0 (mm) 8* (mm) 8 (mm) H=
8 8 0
Primary Throat 0.0085 0.027 0.051 0.54 0.17 3.2
Primary Trailing Edge 0.0085 0.045 0.073 0.62 0.12 5.3
Secondary Trailing
Edge-Inner Wall 0.037 0.099 0.20 0.50 0.19 2.7
Secondary Outer Wall 0.039 0.103 0.21 0.50 0.19 2.7
68
I-
0
U
C)
o
'5i
C.
o
0
U
0
o
0
4-
0
o1
I-
CE
G=i
0n
F1
)
:
---- ---- -
Sa,
C
4)0 I
CE
o
|tt e1
It
4)
N v
--c>
S L 0
C
a
c
0
3
]^
0
C,
u
N
Flow conditioning sections, settling chambers and nozzles. Flow is
from right to left.
Figure 3.3
t
2b-12.65 mm
I
--16.15 mm
Nozzle-block dimensions.
Figure 3.4
Mixing section and impact-pressure probe/pressure transducer/stepper-
motor assembly. Mixing-section extension is not shown.
Figure 3.5
Wall static-pressure plate used for manometer and strain-gauge
transducer measurements.
Figure 3.6
TRANSDUCER SLEEVE
01.78
4.57
0-RING -
PLUG
1.40 -
Figure 3.7
0.89
1.12
Dimensions in millimeters.
Brass transducer-sleeves and plugs used for time-accurate wall-pressure
measurements.
3.05
3.05 J
1.47
E
E cs
43
S] I---
1oE
EE c
1 E 0 E
E I E
SE
10
Jo I
EoE
10 0 E)
00
0 7> j|
IE
E
o
E ----a I
-e;
C4
F 4)CUI-
oI~ E C.
La j
Top rail of mixing section showing fitted plugs.
Figure 3.9
Figure 3.10
Plugs for top and bottom rails of the mixing section.
.* ..o *.
:_.
4'
Butterfly
Valvs
From
Test
Section .- 11 .
2.0" X 1.0" rectangular to 4" DIA Transition Diffuser
Discharge to
Atmosphere
4" PVC
4" PVC
10" Pipe
40 DIA x 110" long Maxim BR31
HI-Veloclty Discharge Silencer
Glass Wool Filled
Figure 3.11 Exhaust system.
Figure 3.12 Schematic of instrumentation and computer interface.