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Dynamic modeling and compensation schemes for pressure sensitive paints in unsteady flows

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Dynamic modeling and compensation schemes for pressure sensitive paints in unsteady flows
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Winslow, Neal Andrew
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English
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xii, 110 leaves : ill. ; 29 cm.

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Calibration ( jstor )
Modeling ( jstor )
Oxygen ( jstor )
Parametric models ( jstor )
Pixels ( jstor )
Plugs ( jstor )
Pressure ( jstor )
Supersonics ( jstor )
Time constants ( jstor )
Transducers ( jstor )
Aerospace Engineering, Mechanics and Engineering Science thesis, Ph. D ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 2001.
Bibliography:
Includes bibliographical references (leaves 106-109).
General Note:
Printout.
General Note:
Vita.
Statement of Responsibility:
by Neal Andrew Winslow.

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DYNAMIC MODELING AND COMPENSATION SCHEMES FOR PRESSURE SENSITIVE PAINTS IN UNSTEADY FLOWS














By

NEAL ANDREW WINSLOW


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


2001














ACKNOWLEDGMENTS

I would first like to acknowledge Dr. Joanne Bedlek-Anslow for her time and effort in applying the PSPs used in most of the experiments presented herein. Also, Dr. Paul Hubner is owed many thanks for the numerous times he acted as a sounding board for my various theories and off-the-wall ideas. I would also like to thank Troy Livingston for the use of his digital camera and for his help in acquiring the images of the experimental apparatus shown in this dissertation.

Special thanks go Dr. Bruce Carroll, my advisor, for his extreme patience

throughout my studies. Further appreciation goes to the other members of my committee, Dr. Kirk Schanze, Dr. Andrew Kurdila and Dr. Martin Morris. They have each helped me in many ways.

Lastly, I wish to thank my parents. Without their encouragement, love, and support I would have never finished.


ii















TABLE OF CONTENTS

pMe

A CKN OW LED GM EN TS .................................................................................................. ii

LIST OF TA BLES........................................................................................................ v

LIST OF FIGURES ........................................................................................................... vi

N OM EN CLA TU RE ........................................................................................................ ix

A BSTRA CT....................................................................................................................... xi

CHAPTERS

1 INTRO DU CTION .........................................................................................................1

Basic PSP Physics......................................................................................................... 1
PSP in Steady Pressure Fields.................................................................................... 5
PSP in Unsteady Pressure Fields .............................................................................. 7
Transverse Jets ........................................................................................................ 13

2 PSP M OD EL D EV ELOPM EN T ............................................................................ 17

Em pirical M odels.................................................................................................... 17
Diffusion Based M odels .......................................................................................... 20
Diffusion Based M odel w ith Linear Calibration .............................................. 25
Diffusion Based Model with Stern-Volmer Calibration................................... 30

3 DYN A M IC CO M PEN SA TION ...............................................................................36

M odel Reference Control........................................................................................ 36
Application of M RC to PSP System ........................................................................ 40
MRC Simulations and Compensation of Experimental Data .................................. 44

4 EXPERIMENTAL SETUP AND PROCEDURES................................................ 52

PSP Dynam ics Test Cell.......................................................................................... 52
Supersonic W ind Tunnel.......................................................................................... 56
Im age Processing ................................................................................................... 61
Uncertainty Analysis............................................................................................... 68


iii










5 RESU LTS ....................................................................................................................81

6 CONCLUSIONS AND RECOMMENDATIONS ..................................................94

Conclusions................................................................................................................. 94
Recom m endations.................................................................................................... 94

APPENDICES

A M A TLA B CO D E: PSPM RCPA RA M S .................................................................. 96

B SIM U LIN K CO D E: PSPM RC ..................................................................................101

C PSP U N CERTA IN TY A N A LY SIS...........................................................................103

REFEREN CES ................................................................................................................106

BIO G RA PH ICA L SKETCH ...........................................................................................110



































iv















LIST OF TABLES


Table Page

1. Sample PSP Indicated Pressures............................................................................ 27

2. Comparison of Curvefit Values. ............................................................................. 34

3. Calibration Functions, Constraints, and Typical Values. ........................................66

4. Nominal values and 95% confidence limits of key variables................................. 69


v















LIST OF FIGURES


Figure Page

1.1 Schem atic of a typical PSP layer. ......................................................................... 2

1.2 Transverse injection flowfield schematic ..............................................................14

2.1 Frequency response of a 16-pm-thick PSP sample ............................................. 19

2.2 Comparison of PSP data and empirical model data.............................................. 19

2.3 Simulated PSP unit step response using diffusion-based model with linear
calibration. Notes: A) t5 = 1.1; B) t9, = 12/7[2 ; C) t , = 1.8 .......................28

2.4 Simulated PSP frequency response using diffusion-based model with a linear
calib ration ..................................................................................................................2 8

2.5 Comparison of PSP data and diffusion-based model output with a linear
calib ration . .................................................................................................................2 9

2.6 Com parison of calibrations................................................................................... 31

2.7 Simulated positive step in pressure (model comparison) ......................................34

2.8 Simulated negative step in pressure (model comparison).....................................35

2.9 Positive step in pressure from experimental data (model comparison).................35

3.1 Structure of the Model Reference Control scheme................................................37

3.2 Structure of the Model Reference Control scheme for use as a compensator for
P S P d ata. ....................................................................................................................4 1

3.3 Linear compensation simulation of a positive step in pressure ............................45

3.4 Compensation simulations using nonlinear form of the PSP plant. (a) Positive
step in pressure; (b) Negative step in pressure.....................................................46


vi








3.5 Compensation simulations using nonlinear form of the PSP plant with a
sinusoidal driving pressure. (a) co =1; (b) co = 3; (c) o' =10; (d) &o = 30 ........47

3.6 Compensation of experimental data. (a) Positive step in pressure; (b) Sawtooth
driving pressure...................................................................................................... 49

4.1 PSP Dynamic Test Cell: A) Digital valve to vacuum, B) Pressure transducer,
C) Digital valve to high pressure, D) Thermistor, E) Window to PSP sample,
F) C ooling fins ..................................................................................................... . 70

4.2 Schematic view of the PSP Dynamic Testing Cell................................................71

4.3 Heat Exchanger Unit: A) Inlet, B) Outlet .............................................................. 72

4.4 Oriel Quartz-Tungsten-Halogen (QTH) Lamp .....................................................73

4.5 ISSI LED Lam p ..................................................................................................... 74

4.6 Schematic of the PSP Dynamic Testing Cell Optics ........................................... 75

4.7 Fisher Control Valves: A) 1 1/2" Valve, B) 1" Valve ...........................................76

4.8 Stagnation chamber of the supersonic test stand facility ......................................76

4.9 Mach 1.56 Test Section: A) 3 of 8 Static pressure taps, B) 2 of 4 Thermocouples,
C) Exterior reference marker, D) Bottom edge of circular plug insert..................77

4.10 Detailed front view of plug insert showing the central jet orifice and the 4 static
pressure tap locations............................................................................................ 77

4.11 Detailed rear view of plug insert: A) 2 of 4 Pressure taps, B) Thermocouple,
C) D igital valve controlling the jet ....................................................................... 78

4.12 A) Xenon Strobe, B) Photometrics CCD Camera ................................................78

4.13 Im age Processing Flow chart................................................................................. 79

4.14 Sam ple PSP Static Calibration............................................................................... 80

4.15 Relative uncertainty in the indicated PSP pressure.............................................. 80

5.1 A sequence of 10 uncompensated PSP images taken at 5 ms intervals during a full
cycle of 50ms. Grayscale is such that white = 0.7atm and black = 0.latm.
Dimensions are from -4.5Dj to +5.5Dj in X and from -3.5Dj to +3.5Dj in Y. ......86

5.2 A sequence of 10 compensated PSP images taken at 5 ms intervals during a full
cycle of 50ms. Grayscale is such that white = 0.7atm and black = 0.1 atm.
Dimensions are from -4.5D to +5.5Dj in X and from -3.5Dj to +3.5Dj in Y. .......87


vii








5.3 A three-cycle loop of PSP data along the streamwise centerline through the jet.
(a) Uncompensated; (b) Compensated. Grayscale is such that white = 0.7atm and
black = 0.1 atm. The X-coordinate is oriented along the horizontal direction
(positive to the right) and time is oriented vertically (positive up).......................88

5.4 Comparison of PSP data to pressure tap data. (a) Upstream of the jet; (b)
Downstream of the jet; (c) Spanwise from the jet ................................................89

5.5 Variation in the paint thickness near abrupt edges ................................................90

5.6 Effective back-pressure ratio at the jet................................................................. 93


viii














NOMENCLATURE


a = pressure sensitive paint thickness; Stern-Volmer intercept
A = amplitude
b = Stern-Volmer slope
B = bias uncertainty
c = calibration constants
C = feedback gains of model reference controller
D = diffusivity; denominator polynomial; diameter
E = activation energy
f = calibration function from pressure to intensity
g = calibration function from intensity to pressure
G = transfer function
H = transfer function
I = integrated intensity
j = the complex number, i
J = momentum ratio; intensity per unit thickness
k = rate constant; high frequency gain coefficient
K = Stern-Volmer constant; intensity difference per unit thickness
m = oxygen concentration difference relative to surface condition
M = Mach number
n = oxygen concentration
N = model order; numerator polynomial
P = pressure
Q = calibration parameter
s = Laplace variable
S = sample standard deviation
t = time
T = temperature; 1 st order system time constant
u = uncertainty
U = input to model reference controller
V = velocity; internal state of model reference controller
W = internal state of model reference controller
x = distance from substrate
X = downstream distance from jet center
Y = output of model reference controller; processed inverse intensity; spanwise
distance from jet center
Z = internal state of model reference controller


ix








a = modal states
P = electrons per bit of the CCD A/D process
y = specific heat ratio
A = relative degree of a transfer function or plant
X = eigenvalues
p = density
o = solubility
T = time constant
* = quantum efficiency; phase
x = mole fraction
' = spatial eigenfunctions
Co = frequency

Superscripts and Subscripts


H = Fourier Transform of H
N = time derivative of H
Hl = average value of H a = augmented
C = compensated
drk = dark level er = external reference ext = external marker ff = flat field j =jet
in = input m = model
p = plant
PSP = pressure sensitive paint q = quencher
nr = non-radiative r = radiative
ref = at reference conditions run = at run conditions out = output model = model output
* = dimensionless
00 = freestream conditions
0 = initial
1 = final
O.xx = at xx% response (e.g.: 0.99)


x














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

DYNAMIC MODELING AND COMPENSATION SCHEMES
FOR PRESSURE SENSITIVE PAINTS IN UNSTEADY FLOWS By

Neal Andrew Winslow

August 2001

Chairman: Dr. Bruce F. Carroll
Major Department: Aerospace Engineering, Mechanics and Engineering Science

Pressure sensitive paints (PSP) have been used extensively in the past decade to measure surface pressures in steady flow conditions. There are many unsteady phenomena to which the aerospace community would like to apply PSP as a measurement technique. However, to date there has not been a viable model to describe the dynamic behavior of the PSP system. A rigorous development from a systems dynamics perspective is needed to advance PSP from a static measurement technique to a dynamic one.

With this aim, two models for the dynamic behavior of pressure sensitive paints will be presented. The first of the two models is a purely empirical approach to designing a model and compensator. The second model encompasses the physics of the process by which an unsteady pressure field over the paint layer affects the layer and causes an intensity of fluorescence that is fluctuating in time. Within this second model, two different forms for the static calibration are chosen. The first results in purely linear


xi








system dynamics whereas the second yields a nonlinear system. These models are then compared to experimental data. Next, a dynamic compensation scheme is developed to correct for the time-lag and amplitude-damping behavior of the paint. Finally, the compensation scheme is applied to PSP data taken surrounding an unsteady (periodic) jet injected transversely into a Mach 1.56 freestream.


xii














CHAPTER 1
INTRODUCTION

Pressure Sensitive Paint (PSP) has been an emergent technology over the past five to ten years. Many different groups have made contributions as is summarized in the reviews by Bell et al.,' Liu et al.,2 and Lu and Winnik.3 Much success has been achieved in the application of PSP to steady pressure fields. In comparison, there are but a few papers discussing the application of PSP to relatively unsteady pressure fields. It is the goal of this dissertation to present a physically realistic model for the dynamic response of PSPs and provide a dynamic compensation scheme for PSP data analysis. With this done, a dynamic PSP experiment will be presented and analyzed using the previously derived dynamic model and compensation scheme.

Basic PSP Physics

PSP is a measurement technique that has its roots in photochemistry. It is based on the luminescence quenching effect of oxygen on luminescent molecules in a polymer binder. Shown in Figure 1.1 is a schematic of a typical PSP layer. The schematic shown here is for a PSP applied directly to the substrate without a primer layer. The reason for this is that primer layers will slow down the dynamic response of the PSP due to luminophore molecules diffusing into the less permeable primer. Because the primer generally has a lower oxygen diffusivity than the polymer binder, the luminophores trapped in the primer will exhibit very slow dynamical responses. The photophysical process begins with the excitation. For most PSPs, the excitation is a light source (strobe,


I






2


Excitation Luminescence Oxygen



O0 / O Polymer Binder a,/-Luminophore Molecule

0 0
00
x 00 0 00
0 / Substrate



Figure 1.1--Schematic of a typical PSP layer.

halogen, LED or laser) with high intensity in the blue to ultraviolet region of the spectrum. A given luminophore in its ground state can absorb a photon and transition to an excited state.4 Once in the excited state, the luminophore has primarily three competing routes to return to the ground state.5 The first is referred to as "radiative decay" or "luminescence" which encompasses two phenomena - fluorescence and phosphorescence. For ruthenium-based paints, fluorescence is the primary radiative decay route whereas for the platinum derivative paints it is phosphorescence. In both fluorescence and phosphorescence, the luminophore molecule emits a photon and drops back down to the ground state. This photon is generally in the orange to red region of the spectrum. The intensity of this emitted light is the measurement variable of PSP. The second decay route is called "non-radiative decay" in which the luminophore releases its energy to the surrounding polymer matrix in the form of thermal energy. The third route is the critical one that differentiates an ordinary fluorescent paint from a pressure sensitive paint. In this route, molecular oxygen that has permeated the layer may collide






3


with an excited luminophore and absorb its energy. This third route is termed "oxygen quenching."

Each of the three methods of energy release has an accompanying rate constant kr, knr, kqno2 respectively. The first two rate constants, k, and knr, have units of 1/s . The third rate constant is composed of two parts, kq, which is a constant with units of m3/(mol - s), and n02 which is the concentration of molecular oxygen in mol/m3 within the PSP layer. The quantum efficiency of PSP luminescence is given generally by kr (1.1)
k, + k,, + kqno2

The quantum efficiency is proportional to the observed PSP intensity, = c - I, with c being a proportionality constant. If a ratio is made of the quantum efficiency in the absence of oxygen (n02 = 0) to that in the presence of oxygen (n02 # 0), the following relation is obtained:


00 - - l+Kqno2 (1.2a)
ScI I

where

k
Kq= q (1.2b)
Sk, +k,

Equation (1.2a,b) is familiarly known as the linear Stern-Volmer equation. The equilibrium oxygen concentration within the layer is related to the air pressure above the layer via Henry's law6


n02 = f(P02)= cP02 = cX02P =P. (1.


(1.3)






4
In the above equation, the linear form for Henry's law has been presented. The solubility of gases in polymers is often modeled in this fashion.7 Substituting equation (1.3) into equation (1.2a) yields a relation between the luminescence intensity and the air pressure over the PSP layer,


" =1+ rKqP. (1.4)
I

In aerodynamic testing it is often not possible to measure the luminescent intensity in the absence of oxygen, 10. However, the intensity at some reference pressure, usually taken to be atmospheric, can be measured. Applying equation (1.4) at a reference condition, Iref, and at a general condition, I, and ratioing gives the "aerodynamic testing" form of the linear Stem-Volmer equation:

I. 1+UKqP _P-Q
ref =_ ._ (1.5)
I 1+ KqPref P., - Q

In this equation, Q = -1/OKq is the virtual pressure (negative absolute pressure) at which the luminescent intensity would become infinite, i.e., the singular point of the intensity function. The parameter Q is a purely mathematical construct, but has an important purpose in relation to the use of equation (1.5). Often equation (1.5) is quoted in the form13

I P
a + b . (1.6)
Pre~f

However, in this form, both the slope, b, and the intercept, a, will be functions of Pref. In the form of equation (1.5), the parameter Q is independent of the reference pressure and is hence a more usable form. Typical values for Q are in the range of - 0.1 atm to

- 0.2 atm. A sensitive PSP with a steeply sloped Stem-Volmer response will have a Q






5


value closer to zero, whereas a less sensitive PSP will have a more negative value of Q. Solving for P in equation (1.5) yields the calibration for a given PSP as P = Q +(Pef - Q) I f (1.7)

So, by taking two measurements, one at a known reference pressure and another at an unknown pressure of interest, the intensities of the two measurements can be transformed into a single measurement of pressure.

Today's modem CCD cameras have on the order of a million pixels. One can then achieve on the order of a million simultaneous pressure measurements distributed over a given model instrumented with PSP in a single image. The physical spacing between these measurements is essentially limited only by the magnification of the camera lens. Put another way, to achieve a denser set of measurements, one has only to zoom in tighter on a given region of interest. Using microscopes, sub-micron resolution can even be achieved.

PSP in Steady Pressure Fields

Many experiments have been performed using PSP to measure pressures under

steady conditions. As such, there are several papers that review the existing literature." To summarize these reports, PSP has been used to measure standard aerodynamic loads on aircraft in both subsonic and supersonic regimes, turbomachinery such as fans and compressors, automobile wind tunnel testing, as well as numerous scientific experiments to investigate particular flowfields of interest. One representative study is that of Everett et al.8

In the paper of Everett et al., a sonic jet of air was injected transversely into a supersonic freestream at a Mach number of 1.6. The PSP was applied to the wall






6


surrounding the jet orifice. The full-field wall pressures around the jet were measured at three different injectant pressures. The jet-to-crossflow momentum flux ratio, (pV2), (ypM2)(1
J (pv) ypM2 (1.8)

took on the values of J = 1.2, 1.7, and 2.2 for the three injectant pressures. In addition to the PSP measurements, shadowgraph and surface oil-flow visualization techiques were used to investigate the flow. The shadowgraph images were useful in identifying the shock structure generated as the jet altered the flow. As the momentum flux of the jet was increased relative to the fixed freestream, the size and strength of the shock structure were seen to increase as one would intuitively imagine. With the oil-flow, the streamlines along the wall were characterized. From these streamlines, the locations of the various shocks near the wall could be determined. Placing static pressure taps in strategic locations allowed simultaneous PSP and conventional pressure transducer measurements. Using these simultaneous measurements, an in-situ calibration of the PSP was performed. The calibration function chosen related the indicated PSP pressure to a quadratic function in terms of Ire/I. Spectacular images of the surface pressure were achieved with a spatial resolution of 85.1 p.m/pixel. Integrating the pressure around the periphery of the jet gave a measure of the effective back pressure Pe - 1 .(o)do (1.9)
P2 r P2

With increasing J, the effective back pressure was also found to rise. Because of this, it was concluded that the effective back pressure was not a constant ratio of either the static or stagnation pressure behind a normal shock in the freestream as was reported by other researchers.






7


PSP in Unsteady Pressure Fields

The earliest reference of PSP in unsteady situations is a paper by Cox and Dunn and contains the beginnings of a good model for the response of PSP to unsteady pressure fields.9 In their paper they investigated oxygen transport within poly(dimethyl siloxane) (PDMS) via its effect on the fluorescence of the luminophore 9,1 0-diphenyl anthracene (9,1O-D). The purpose of their work was to determine such physical quantities as the solubility, diffusivity and activation energies of the polymer PDMS. This was accomplished with a cuvette of PDMS (thickness, a = Icm) into which was mixed the luminophore 9,1 O-D. The schematic shown in Figure 1.1 is similar to the geometry of the cuvette used in their experiments. The cuvette was allowed to come to equilibrium with pure nitrogen at 1 atm. Then, the nitrogen environment was replaced with pure oxygen at

1 atm. Over the next 24 hours the fluorescence intensity at the center of the cuvette (x =

4.5mm to 5.5mm) was monitored. For PSPs, the thickness is on the order of 10pm as compared to the Icm thickness of the cuvette. The response time of a PSP will be shown to be proportional to the thickness squared. Thus, PSPs work on much shorter time scales than in ref. 9. The fluorescence intensity within the cuvette was seen to fall off quasi-exponentially. With the use of a static calibration they were able to extract the indicated oxygen concentration as a function of time within the PDMS. They then developed an analytical model for the indicated oxygen concentration using the one dimensional diffusion equation. For a step change in oxygen concentration at the upper surface (x = a) the analytical solution for the oxygen concentration within the cuvette is

n(x,t)-no =14 ' 1 p 2 sn[2 Dt (.x
= L--- exp sm (2i+1) . (1.10)
n, -no 71i= 2i+ 1) 1 4 a2 _ 2 a






8


This model was found to accurately represent the experimental data. The regression analysis yielded values of diffusivity between D = 2.0 x 10-cm2 /s at T = 5'C and D = 6.1 x I15 cm2 /s at T = 45'C for oxygen in PDMS. However, in the general use of PSPs the coating is viewed from above rather than from the side as was done in this paper. That is, the intensity measured from a PSP is the integrated intensity of the entire layer. The integrated intensity as a function of time for an arbitrary unsteady pressure field will be shown later to have a troublesome nonlinearity.

The work of Mills and Changl0 was the first to look at the dynamic response of optical sensor films (to be later called PSPs). The response and recovery times of the optical signal (luminescence output) from the coating were measured. The response time was defined as the time required for the optical signal to equilibrate to a positive step in oxygen concentration, and the recovery time was the time required for equilibrium due to a negative step in concentration. It was shown that the difference in the response and recovery times was related to the hyperbolic function between the fluorescence intensity and oxygen concentration. This paper was the first attempt at characterizing the dynamics of PSP and used a time-domain-based technique.

A paper similar to that of Mills and Changlo was that of Baron et al." Here, the inverse of the PSP intensity was compared to the pressure as the pressure was quickly stepped from near vacuum up to 1 atm of standard air. In this work the assumption was made that the PSP responded as a first order dynamical system with up to two time constants. The given dynamical system was numerically convolved with the pressure signal from a fast conventional transducer to yield a pressure that was compared to the pressure indicated by the PSP. Using nonlinear least-squares curve fitting the time






9
constants and term weights were determined yielding an empirical model for the dynamic response of the PSPs. With these empirical models, they were able to get extremely good comparisons between curvefits and experimental data.

The work of Borovoy et al.12 was the first actual wind tunnel application of PSP in quasi-unsteady conditions found in the literature. In this experiment the pressure on a cylinder in Mach 6 flow was investigated. To achieve the Mach 6 flow, the cylinder was located perpendicular to the exit of a shock tube. It is stated that the 99% relaxation time of a PSP's fluorescence is given by the equation 12 a2
to99 =- 2 . (.11) n2 D

Once the shock had passed the cylinder, the timing system in the experiment would wait an amount of time given in equation (1.11) and then trigger a single flash from a xenon flash lamp. The PSP data were compared to an analytical solution of Euler's equations for nonviscous flow and found to agree to within 10%. It should be noted that this application was not truly an example of unsteady PSP, as the data were taken after the shock had already passed the cylinder and the PSP had been given sufficient time to respond to the step change in pressure. That is, the flow field was unsteady in time but not during the period over which the data were acquired.

In separate, but concurrent developments, the response of PSP to periodic

pressure fields was investigated by Engler13 and Carroll et al.14 In Engler's work, the PSP was subjected to periodic forcing at frequencies from 0.lHz to 50Hz. The main goal of this investigation was the characterization of the dynamic range and the pressure resolution of the PSP. No effort was made here to perform dynamic compensation or to try to mathematically or physically define the dynamics of the PSP system. Carroll et al.14






10
presented experiments similar in form to those of Engler, namely, the response of PSP to sinusoidal pressure fields. Much of the analysis performed here was done in the frequency-domain. It was shown that the PSP responds as approximately a first order dynamic system based on the amplitude response. The phase response of the PSP system did not match well to the standard first order system model.

Carroll et al.15 investigated the response of PSP to a step change in pressure. In this paper are the beginnings of a physically realistic model for the dynamic response of PSP. In summary, the paper reports on an analytical model based on the one-dimensional mass diffusion equation to describe the transport of oxygen within the PSP layer. This model was nonlinear and was implemented numerically. In addition, empirical models based on a single-term first order response and on a two-term first order response were developed. Using nonlinear least-squares curvefitting the two empirical models were fit to the data. The single-term first order model was shown to inadequately fit the data. The two-term and diffusion-based models were both shown to have excellent agreement with the experimental data. An equivalent form of the diffusion-based model will be derived in its entirety in a later section.

Winslow et al.16 developed a linearized dynamics model for PSP based on the assumption that the luminescence intensity from the paint is proportional to the average oxygen concentration within the layer. The frequency response of this model is that of a "1/2-order" dynamic system with an amplitude response of -10dB/decade above the cutoff frequency and a high frequency phase shift of -450. Curvefitting this model to PSP data in the frequency domain gave a value for the time constant of the layer. A dynamic compensator was then developed by inverting the frequency domain response






I 1
and performing an inverse Fourier transform. The compensator took on the form of a six term finite impulse response (FIR) filter. Applying this compensator to experimental data yielded an impressive increase in the dynamic response of the PSP. This was the first instance of dynamic compensation of PSP data.

Nearly concurrent to the paper of Carroll et al.5 was that of Masoumi et al.'7 Effectively, the same dynamical model was used in both works. In the latter work, the luminescence response of PSPs to positive and negative steps in pressure was examined. The coating thicknesses were on the order of 300pim. Upon performing a curvefit of their dynamics model to the data, values for the diffusivity of 1.0 x 10-' cm2 / s for platinum octaethylporphine (Pt(OEP)) in poly(n-butylaminothionylphosphazene) (PATP) and

1.4 x 10-' cm2 / s for ruthenium biphenyl phenanthroline chloride (Ru(ph2phen)3C2) in PATP were obtained.

Carroll et al.8 reapplied the dynamics model of Winslow et al.16 to investigate the diffusivity of PSP coatings as a function of temperature. A PSP coating was subjected to step increases in pressure over a range of temperatures. From the time constants measured at the different temperatures, values for the mass diffusivity were calculated. Then, modeling the diffusivity using an Arrhenius relation D =Do exp(_ IS (1.12)
RT

a value of 2.48kcal/mole was found for the activation energy, E, of the coating. This value compared well with values from other sources.

Hubner et al.19 performed PSP experiments on a flat plate and a wedge in a shock tube. A first series of tests used a flat plate painted with PSP, a continuous illumination






12
source, and a PMT to measure the integrated light output from the whole of the plate. As the initial shock passed the plate, the PSP was subjected to a step in the pressure. When the reflected shock came back past the plate, the pressure stepped up even more. From this data, the model of Winslow et al.16 was used to calculate a time constant of 3.52ms for the coating. A second series of tests involved a wedge obstructing the flow. A photographic flash lamp was used to excite the paint and a CCD camera acquired a single image for each run of the shock tube. The results showed a pressure gradient on the wedge due to the effective area reduction within the duct. Overall, it was demonstrated that thin films of PSP could be used to measure pressures in short-duration facilities.

The final paper to be discussed here was from Hubner et al.2 Pressure and

temperature measurements were made on an elliptic cone model in Mach 7.5 flow. The tests were performed at the 48-inch hypersonic shock tunnel at Calspan-University of Buffalo Research Center. Due to the short duration of the flow, a method similar to that of Borovoy et al.2 was used wherein the PSP was allowed to come to equilibrium with the flow and then a photographic strobe was used to illuminate the paint. The data showed that the pressure on the surface of the model ranged from 3 to 12 kPa. A second set of tests was performed with a high frame rate, 8-bit camera. In these tests the camera acquired several images of a temperature sensitive paint (TSP) coated model during a single flash from the strobe. The temperature data from these tests were used to compute heat transfer rates on the surface of the model. The results compared well to measurements made with thin-film heat transfer transducers.

As can be seen in the literature to-date there have been many papers investigating sinusoidal or step-like pressure fields, but none that have presented a PSP dynamics






13


model in a fashion for use with arbitrary pressure fields. It is the goal then of this dissertation to present a sound mathematical and physical model for PSP dynamics. Moreover, a compensation scheme will be designed based on the dynamics models to correct for the amplitude-damping and phase-shifting behavior of the paints. Finally, a full-field dynamic experiment of PSP will be presented along with the results of the compensation scheme having been applied to the data.

Transverse Jets

The study of a jet of air injected transversely to a supersonic freestream is a

classical problem that has been the subject of many studies. A schematic of the general flowfield is shown in Figure 1.2. The main purposes for which this flowfield have been studied is in relation to thrust vector control and to fuel injection ports in scramjet engines. For thrust vector control, the integrated wall pressure is of primary importance. For scramJet engines it is important for the fuel (injectant) to become fully mixed with the freestream air in as short a time/distance as possible. Additionally, because the injectant is to be burned downstream, it is necessary to push the injectant as far from the wall area as possible to reduce heating of the wall. Hence, the goal of many studies has been on determining the wall pressure distribution or on finding optimum parameters to increase mixing or to maximize penetration of the jet. What follows is a brief introduction to the existing literature on the evolution of transverse jets in supersonic crossflow.

The first work to be published on the transverse jet in supersonic crossflow came from Cubbison et al. out of the NASA Lewis Research Center in 1960.2 In their paper, extensive wall pressure measurements were made surrounding a 0.062"jet. In this study






14


F


Bow SHOCK





'REESTREAM MAC







RECIRCULATION REGioN RECIRCULATION
REGION
INJECTANT


Figure 1.2--Transverse injection flowfield schematic (Gruber et al.21). there were basically three independent parameters which could be varied -jet pressure, freestream Mach number, and pressure altitude. The range of jet pressures was from 50 to 440 psig. Three freestream Mach numbers of 2.92, 4.84 and 6.4 were investigated. The pressure altitude was further varied from 55,000 to 115,000 feet. This range of pressure altitudes combined with the range of Mach numbers resulted in a range of Reynolds numbers between 0.84x106 and 7.78x106 per foot. The wall surrounding the jet was instrumented with on the order of 100 pressure taps arrayed upstream, downstream and spanwise relative to the jet orifice. The general flowfield resulted in a high pressure region immediately in front of the jet orifice and a low pressure region behind the jet. The results of this study indicated that the jet-to-freestream pressure ratio had a large effect on the pressure distribution on the surface surrounding the jet. The pressure distribution was also affected by changing the freestream Mach number at a constant


DISK






15
pressure ratio. The effect of the Reynolds number was small compared to the preceding two effects.

In a later examination by Schetz et al.2 the location of the center of the first Mach disk was correlated to a pressure ratio between the jet pressure and an "effective back pressure." This effective back pressure was an analogy to the back pressure seen by a jet issuing into a quiescent medium. The value chosen for this effective back pressure was defined as 80% of the static pressure behind a normal shock in the freestream.

A paper by Orth et al.2 looked at the effect of changing the shape of the injector port as well as the angle of injection into the freestream. A control volume analysis of the fluid in the jet was also presented. In this study, an effective back pressure equal to 2/3 of the freestream stagnation pressure was used. It was argued that the effective back pressure should be the average pressure field around a cylindrical obstruction on a flat plate. So, the definition used here differed from that of Schetz et al.2 Several experiments were performed to better understand the factors affecting penetration of the jet into the freestream. A first series of tests used a circular-ended rectangular slot oriented along the flow direction and perpendicular to the flow as well as an equal area circular port as the injector shape. These tests were run at a Mach number of 2.1 and a jet to back pressure ratio of 5.23. It was found from these tests that the shape of the injector port had no real effect on the penetration of the jet as was originally hoped. A second series of tests was a parametric study of the injectant pressure and Mach number. It was found that the distance from the wall of the center of the first Mach disk was related to the jet-to-freestream pressure ratio.






16
25 26
Billig et al. 2 and Heister and Karagozian performed detailed vortex modeling of the jet/crossflow interaction. These models were used to predict the effects of varying the freestream Mach number, the velocity of the jet as it exits the wall, and the Mach number of the jet in relation to the amount of penetration. It was determined that the primary variable of interest for enhancing penetration was increasing the jet-to-crossflow momentum ratio (see equation 1.8).

As new measurement techniques were developed, new visual studies of the jet in crossflow problem were examined. Among the measurement techniques employed are planar laser-induced fluorescence (PLIF),27,28 classical Schlieren images,29 planar Mie scattering,2 and laser Doppler velocimetry (LDV).3 Each of these methods was used to primarily to investigate penetration and spreading of the jet. The latest measurement technique used to attack the jet in crossflow problem has been PSP as was described in the earlier section "Static PSP."

It has been shown by the works of Johari and Paduano, 1 Hermanson et al., and Johari et al.3 that both penetration and mixing can be enhanced through the use of a pulsed or modulated jet in crossflow. It should be noted that these studies were all performed in an incompressible medium as opposed to the compressible flows we have previously discussed. However, the analogy may still stand. If this is the case, it should be of interest to investigate a pulsed jet in supersonic crossflow. This then will be the form of the dynamic flowfield to which unsteady PSP measurements will be tested. The results of these tests will be presented in a later chapter of this dissertation.














CHAPTER 2
PSP MODEL DEVELOPMENT

The most important step in performing a dynamic analysis of PSP data is model selection. What follows are two different approaches yielding models increasing in design complexity and accuracy. The first approach is purely empirical in nature. The second is based on solving the one-dimensional mass diffusion equation in conjunction with the relation between intensity and pressure under static conditions.

Empirical Models

Perhaps the easiest model that can be developed is one which does not even

attempt to understand the physics of the system in question, but rather, simply measures the input and output of the system and uses the data as the characterization. This approach is often referred to as the "black box" approach. All one must do is define the input/output mapping for the system as a function of frequency in order to create a crude compensator to correct for the amplitude damping and phase shifting behavior. The only underlying assumptions to this type of model are that the system being modeled must be linear and time-invariant. That is, for an input given by Pin (t)= PO + P, sin(ot). (2.1)

the output from the system must be of the form P, (t)= P0 + P, A(o)sin(ot + 4(o)). (2.2)


17






18
In the above equation A and are the amplitude ratio and phase shift, respectively, of the output relative to the input and are functions of the driving frequency co. Equations (2.1) and (2.2) can be more easily related to one another in the frequency domain via S(jw) = (jo)Pn (jo))= A(o)e- )P (jco). (2.3)

Shown in Figure 2.1 is the frequency response of a 16ptm thick PSP sample based on input/output data. This data was acquired using the experimental setup used by Winslow et al.16 and Carroll et al.'8 The general uncertainties of the measurements are +0.1 kPa for the conventional pressure transducer and 2% of the reading for the PSP measurements. Based on the experimental data, a transfer function of the form


(o )= 'T (2.4)
1 + jorr2

was chosen to model the system. This particular form of transfer function is that of a lag system. The transfer function was applied to the input data, Pin(t), from a high frequency response pressure transducer using the Matlab computer software to achieve a model output pressure, Pmdei(t). The parameters T1 and T2 were varied until the root mean square error (RMSE) between the PSP-indicated pressure, Pou'(t), and the model output was minimized in the time domain. The resulting values OfTI and T2 were 0.012 and 0.15 seconds respectively. The model transfer function is also shown in Figure 2.1. Figure

2.2 contains the time domain data for this experiment. Shown in the figure are the PSPindicated pressure, the pressure measured by the conventional transducer, the model output data and the residual between the PSP indicated pressure and the model output. The absolute error between the PSP data and the model output data is less than 2 kPa (RMSE = 0.97 kPa). As can be seen, the transfer function of equation (2.4) does a










5 0

m -5

-10
4)
3 -15 E -20


-25


-30
10


1


- -:
OD
- - ,

00
e0 0


A PSP Amplitude
- 0 PSP Phase ,0 01
- Model Amplitude * ,
- ---- Model Phase ,' ' ' ' ' '


Figure 2.1--Frequency response of a 16-jm-thick PSP sample.


2
Time (sec)


3


Figure 2.2--Comparison of PSP data and empirical model data.


19


10

0

-10


-20
(n
-30 a -40 a.

-50 2 -60


-2


10^1


100


10


80 60


(L
40 In 20



0


0%
0


PSP
------TransduCOr
'a A Model
0 Resldual
0onnO ( O Onnn o e n)O nO e Q 0 G

I I L


0


1


4


I






20


reasonably good job of approximating the PSP system dynamics. However, the parameters of this curvefit are dependent upon the thickness and chemistry of the PSP layer in an unknown manner. That is, if the thickness or formulation of the PSP were changed, a different transfer function would result. Hence, the empirical approach is useful for a single sample, but does not transfer well to the type of experiments one might actually want to perform. Therefore, a more thorough understanding of the physical processes involved in the dynamics of PSP is required.

Diffusion Based Models

As was mentioned earlier, PSP is based on the luminescence quenching effect of oxygen on luminescent molecules in a polymer binder. This model tracks the process by which an unsteady pressure above the coating affects the oxygen distribution within the PSP and in turn causes the intensity of luminescence to change in time.

Under static conditions, the oxygen concentration within the layer will be constant and related to the pressure over the surface via a linear sorption law (Henry's Law) n = aP. (2.5)

Also under static conditions, the intensity of luminescence is related to the pressure over the surface via a static calibration

I =f(P) (2.6a)

or

P =g(I) (2.6b)

due to the luminescence quenching effect of oxygen within the layer. Note here that the functions f and g are inverse functions. For the moment a generic relation between pressure and intensity is assumed. Later, specific forms for the static calibration function






21


will be used to complete the system dynamics. Letting I* = I/If and P' = P/Pf the calibration equation can be brought into a dimensionless form I* = f(P*). (2.7)

In the above equation, the non-dimensional integrated intensity I* can be replaced by the dimensionless intensity per unit depth, J*, by noting that . I I/a J .
I - a - J (2.8)
I ef 1, /a J e

Also, pressure can be replaced with oxygen concentration by noting that P - P- n -n. (2.9)
Pr, o-P. n.
r,~ Fef nref

A static calibration equation relating the intensity per unit depth to the oxygen concentration can then be given as

J* = f(n*). (2.10)

Neglecting any internal light attenuation, this equation also holds on a local level, that is, at any point within the layer (Cox and Dunn9). Therefore, equation (2.10) can be used within the layer by considering both J* and n* to be functions of depth, x. In addition, this equation holds as a function of time assuming that the kinetics of the luminescence and quenching processes are much faster than the kinetics of the diffusion process. The kinetics of the luminescence and quenching processes are on the order of microseconds for the ruthenium-based paints (Bell et al.,' Liu et al.,2 Schanze et al.,' Bums and Sullivan34) and on the order of 10's to 100's of microseconds for the platinum-based paints (Bell et al.'). Then, since the kinetics of the diffusion process are on the order of milliseconds to seconds for typical coating thicknesses (Baron et al.", Winslow et al.'6,






22
Carroll et al.'8, Hubner et al.19), the assumption is indeed valid. Restating equation (2.10) with the new dependencies one observes that the static calibration provides a dynamic relation between intensity at a given depth within the PSP layer and the oxygen concentration at that depth,

J'(x, t)= f(n*(x, t)). (2.11)

The next step is to solve for the oxygen concentration as a function of depth and time for a given surface pressure input, Pi, (t). This will be done using the one-dimensional mass diffusion equation with boundary conditions and initial conditions of a2 n I an
--- - -n (2.12a)
ax2 D at

n(a, t)= crPi (t) (2.12b)

an(O, t)0 (2.12c)
ax

n(x,0) = aP, (0). (2.12d)

The governing equation, (2.12a), assumes that the PSP layer is infinite in the transverse directions and that the diffusivity is constant. The first boundary condition, equation (2.12b), makes the assumption that the kinetics of the sorption process are much faster than the kinetics of the diffusion process so that the surface of the PSP is in equilibrium with the air above. The second boundary condition, equation (2.12c), represents a nonpenetration condition at the substrate. The initial condition, equation (2.12d), states that the oxygen concentration across the layer is initially constant (i.e. in equilibrium initially). The set of equations (2.12a-d) or their underlying assumptions have been used in many other analyses of the solubility and diffusion of gases in polymers.9,'0,5'-1'35






23


These equations can be non-dimensionalized by letting x = x / a, t* = t / T where T= a2 /D, and making use of equation (2.9) to arrive at a2n* _ an* (2.13a)
(ax* a t*

n*(1,t*)= Pi*(t*) (2.13b)

an*(0,t*)= 0 (2.13c)
ax*

(x *,0)= Pi* (0). (2.13d)

In PSP measurements, the luminescence intensity is the quantity actually measured. Thus it is instructive to recast equations (2.13a-d) in terms of intensity. Inverting equation (2.11) to get n*(x, t*)= g(J'(x*, t*)) and substituting into equations (2.13a-d) yields

aJ* a2j* a 2 g / - 2 =-a + (2.14a)
at* (ax*) ag/aJ (x*

J' (1, t* )= f(Pi*(t*) (2.14b)

a =* 0 (2.14c)
ax =

J'*(x0,0) = f(Pi'n (0)).- (2.14d)
Equations (2.14a-d) represent a nonlinear partial differential equation with nonhomogeneous boundary conditions describing the intensity distribution as a function of depth and time. Notice that recasting the formulation in terms of intensity, J, instead of oxygen concentration, n*, introduces a nonlinear term (last term on right-hand side of

2.14a). The problem can be restated via a change of variables K(x*, t*) = J* (x*, t*)-- f (P* (t' (2.15)






24


to achieve a nonlinear partial differential equation with homogeneous boundary conditions and initial conditions


tK _ (2 g2 2 2 P*(t)) (2.16a)


K(1,t*)= 0 (2.16b)


K, t*= 0 (2.16c)
Ox

K(x*,0)= 0. (2.16d)

The output of a PSP measurement is the integrated intensity (neglecting internal absorption)


I:,(t*)= J'(x,t')x* =JK(x*,t*lx*+f(P1*(t')). (2.17)
0 0

The integrated intensity is then related to the indicated pressure through the static calibration, P*-Jt*)=g(lI*,(*Equations (2.16a-d) are an extremely important set of equations representing the difference between the intensity output by the uppermost layer of the PSP and the intensity output from a given depth within the PSP layer. The term on the left-hand side (LHS) and the first term on the right-hand side (RHS) of equation (2.16a) are both linear terms. The last term on the RHS of that same equation is a forcing term. The middle term on the RHS is in general highly nonlinear and contains a forcing part as well. Because of the nonlinearity of this term it is not possible in general to apply classical controls methods as will be shown later. Using two different assumed calibration functions and modal analysis, a system model now will be developed and simulated.






25


Diffusion Based Model with Linear Calibration

The first calibration function to be investigated will be

I* =f(P*)= c0 + c.P* (2.18a)

or


P- g(I) c 0 (2.18b)
ci

This form of calibration is not typically used in PSP analysis; however, it has one important feature. Using a calibration which is linear between intensity and pressure causes the (normally) nonlinear term from equation (2.16a) to drop out yielding aK(x*,t*) a2K(x*,t*) af(Pi*,(t)) (2.19)
at* (ax* Y t*

which is now a linear partial differential equation with forcing. The solution to this equation will be assumed via separation of variables as K(x*,t*)= D j(t* j(x') (2.20)
i=1

where Tj (x) are sine or cosine functions. This form of solution is based on a method called modal analysis and is often used in linear systems dynamics modeling. The spatial eigenfunctions T (x*) must satisfy the boundary conditions (2.16b,c) giving Ti(x*)= cos(X x*) where Xj = (2i -1) . (2.21)
2

Substituting into equation (2.19), multiplying by T (x*) and integrating across the thickness of the layer gives


ai (t)= -Xai ()- 2 sin(1 (P. (t)) (2.22)






26


Substituting equations (2.20) and (2.21) into equation (2.17) yields the output I:',' (t* = Oc i(t*si + f(Pin (t' . (2.23)
ii

In implementation the infinite sum of equation (2.20) will be truncated after some number of terms, N. After truncation, equations (2.22) and (2.23) can be written in the standard linear systems form



t][]) 0 - 0 a (t -2sin(X )/, (2.24a)


fP1i(t*))

I:, (t*) 9..sin(.),-- a ) +(1,0 .*(t (2.24b)


This system is a multi-input, single-output system with inputs of f(P (t)) and f(P1* (t*)), state vector ai (t*) and output I*, (t*). A useful property of this model is that the system matrix is diagonal with distinct, nonzero eigenvalues. The usefulness of this property is in calculating the observability and controllability matrices. It can be shown that these matrices are both full rank, thus implying that the system is both observable and reachable. These two properties should allow a simple compensator to be constructed. There is one additional point of interest concerning this system. Since it is a linear system, and the relation between intensity and pressure is linear, equations (2.24a,b) can be rewritten via a linear transformation as a system between the input pressure and the indicated output pressure

. 0 0 P:(
6ti~~~t*) s nk .n .~ t)i * (2.25a)






27


P, (t= - ) a + +(1,0 .in (2.25b)


It can be shown that the above output is identical to the solution for n (x t) integrated across the layer, that is,


P:..(t*)= n*(*** (2.26)
0

where n (x* , t*) is obtained by solution of equations (2.13a-d). This system is then identical to that used by Winslow et al.16 and Carroll et al.'8 For a number of inputs P* (t') the output from the system is analytic. Table 1 below lists several of these solutions (Carslaw and Jaeger36).

Table 1--Sample PSP Indicated Pressures P,:(t P. t



PO*+P,'step(t*) P*+P. I- 2exp[ -(2i +) t*



P' + P1* sin((O*t' P * + P:.tanh(p)sin(O*t*), = , =CD




Note that the function step(t), sometimes referred to as the unit step function, is zero for t < 0 and unity for t > 0. Also note that the solution shown for the sinusoidal input is only that due to the steady periodic conditions, not the initial transient period.

Shown in Figure 2.3 is the simulated step response using the first 1, 2, 4, and 20 terms from the analytical solution of Table 1. Labeled on the graph are several points





























0


28


0

C
CL CL


1


0.8 0.6


0.4 0.2


0


0.5


1


1.5


2


t*
Figure 2.3--Simulated PSP unit step response using diffusion-based model with linear calibration. Notes: A) t95 = 1.1, B) t*96 = 12/nC) t99 = 1.8.


5 0 -5
0
IM -10

-15

E -20


-25


10.2


10~1


100


101


1


10

0

-10
0> V
-20

-30 a

-40

-50


'-60
02


0) T


Figure 2.4--Simulated PSP frequency response using diffusion-based model with a linear calibration.


-- C


N=1 " N = 2 N=4 N = 20


- ' ---- -






- -



- Amplitude
------ Phase
' I' " ' ' ' ' ' ' ' '


L


v


I


I


I


-30






29


100


80 -

60 -U IS



C. 20 -- - - - - - Transducer
A Model
0 Residual
1 0 0 0 n p - C)0000


-20 I I
0 1 2 3 4 5 6
Time (sec)
Figure 2.5--Comparison of PSP data and diffusion-based model output with a linear calibration.

throughout the response. It can be seen that the time to reach 99% of the total response is approximately 1.80T as compared to the value of (12/7c2),r put forth in the paper by
-20 2





Borovoy et al.0. It can also be seen that a model order of N = 5 should be sufficient to represent most of the dynamic behavior.

Figure 2.4 shows the simulated frequency response of the PSP. There are two interesting items of note here. First, the amplitude slopes off at --10 dB per decade. Second, the phase shift levels off at -45'. Following the trend of I" and 2"nd order systems it would then seem that this system exhibits the behavior of a "" order system, which has some meaning in the context of fractional calculus in which derivatives and integrals of rational order are defined. In the text by Oldham and Spanier 7 on this subject, the relation between many diffusion-based phenomena and the 2 order






30
derivative, or semiderivative, is shown. An attempt has been made to express this current model using fractional calculus, but no clear results have yet been achieved.

Figure 2.5 shows a set of experimental input and output pressure data. Also included in the figure is the model output pressure based on the linearized diffusion model of equations (2.25a,b) with N = 5. The value of the time constant that minimizes the RMSE between the PSP indicated pressure and the model output pressure is ,= 0.437. The RMSE of the curvefit is 0.91 kPa which is a very good fit. The thickness of this particular sample was 17 tm. Using T -a 2/D a value of D = 6.6 x 10- cm2/s is acquired for the mass diffusivity, which compares well to values from other sources.5'9'38 Diffusion Based Model with Stern-Volmer Calibration

The most widely used calibration for PSP is based on the Stern-Volmer relation. Taking equations 1.5 and 1.7 and letting Q* = Q/Prf we get


I- = f 1(P-) Q. (2.28a)

or

P= g(*)= Q* +(-Q*1/I*). (2.28b)

In the above equations the parameter Q* uniquely determines the static calibration. Figure 2.6 illustrates the basic difference between the linear calibration of equation (2.18) and the Stern-Volmer calibration of equation (2.28). Note that this figure shows the intensity versus pressure, as opposed to the inverse of intensity versus pressure which is normally presented. Applying equation (2.8) to equations (2.28a,b) and substituting into equation (2.16a) gives






31


a)


Pressure
Figure 2.6--Comparison of calibrations.


aK _ a2K 2 K2 .)((t*))2.29)
at* (ax* K+f(P()) x* at

This is the governing equation for the PSP dynamics based on the Stern-Volmer calibration. It can be seen that the middle term on the RHS is highly nonlinear and contains a forcing component as well. Because of this term, a numerical solution to this equation would require a complex numerical algorithm to be developed. However, rather than perform a direct solution to this equation, a different method will be employed. It will begin by solving for the oxygen concentration from equations (2.13a-d). These equations can be transformed to a linear partial differential equation with forcing with homogeneous boundary conditions and initial condition via m(x *, t*) = n*(x *, t *)- P*(t). (2.30)


- Near Cal. Stern-Volmer Cal.






32


Then, modal analysis can be applied (see equations (2.20) and (2.21)). The resulting system equation is identical to equation (2.25a). The system equation together with an output equation designed to give the oxygen concentration distribution is then

- 0 0pi"t
H t') = 0 -? 0 ai (t*) + -2sinX ])/ K" ,t. (2.3 1a)




n*(x*,t*)=(--,COS4x*),--. at*) +(1,0 ." ((2.3 1b)


This system is linear and can be easily solved at a discrete number of points throughout the layer to achieve an oxygen concentration distribution. This oxygen concentration distribution will then be converted to an intensity distribution by using the calibration in a dynamic sense,


J1(x*,t') ,-Q , (2.32)
n*(x,t* )- Q

It may be helpful for the reader to refer once again to the discussion leading from equation (2.7) up to equation (2.11) for the validity of equation (2.32). Then, the intensity distribution can be numerically integrated to get the integrated intensity output. Making use of the static calibration of equation (2.28b) achieves the indicated output pressure.

Figures 2.7 and 2.8 show the simulated response of PSP to a positive and a negative step in pressure respectively. For the positive step, the initial pressure was PO = 0.1, the final pressure was P, =1.0 and the value of Q* was Q* = -0.25. For the negative step the values of PO* and P,' were reversed. Shown in the two figures are







33


comparisons of the two models based on the linear calibration and the Stern-Volmer calibration. For the datasets based on the linear calibration, a model order of N = 5 was used. For the data based on the Stem-Volmer calibration, a model order of N = 5 was used to calculate the oxygen concentration distribution. This was evaluated at 21 evenly spaced grid points throughout the depth of the layer. Next, it was converted to intensity per unit depth via the calibration equation and numerically integrated across the layer using the trapezoidal rule integrator. The integrated intensity was then converted to an indicated pressure by means of the calibration. From Figure 2.7 it can be seen that for a positive step in pressure, the linear model solution leads the Stern-Volmer model solution. In Figure 2.8 this behavior is reversed. There are several items of note with respect to these comparisons. First, the difference between the models is a function of each of the values P., P* and Q*. If the pressure data were scaled via the ratio (P* - PoJ)/(P' - P*) the response of the Stern-Volmer based model would continue to show different behavior for different values of PO, P:*, and Q* whereas the linear calibration based model would always show the same response. Second, as AP of the step decreases, the two models will collapse to the same behavior. Similarly, the models will also collapse to the same behavior for Q* -> -o.

Figure 2.9 shows experimental input and output data for a positive step in

pressure. Also shown is a comparison of curvefits of the two preceding diffusion-based models. The results of the curvefits are shown in Table 2. For the model based on the Stem-Volmer calibration, a value of Q* = -0.163 was used. It is quite clear that the model using the Stern-Volmer calibration does a much better job of approximating the PSP response than the model based on the linear calibration.






34


Table 2--Comparison of Curvefit Values


Calibration T (sec) RMSE (kPa)


Linear 1.25 4.10

Stern-Volmer 1.11 1.39


1.2


1


0.8 a. 0.6


0.4 0.2


0


0


0.5


I


1.5


2


Figure 2.7--Simulated positive step in pressure (model comparison).


- - -A - - Unar Cal.


V . C


i


i


i


I






35


1.2 1
-6- - - Unear Cal e Stem-Vomer Cal
1


0.8


. 0.6

0.4


0.2

0
0 0.5 1 1.5 2
t*
Figure 2.8--Simulated negative step in pressure (model comparison).



140

120 -

100

S 80

2 60 4
4)
40
------ Transducer
20 PSP
A Stem-Volmer Cal.
0 Unear Cal.
0
0 0.5 1 1.5 2
Time (sec)
Figure 2.9--Positive step in pressure from experimental data (model comparison).














CHAPTER 3
DYNAMIC COMPENSATION

The task of designing a compensation system for PSP is comprised of two steps. First, a model must be developed for the response of the PSP to arbitrary dynamic pressures. Such models for the PSP dynamics have already been presented in the previous chapter. The second step makes use of the PSP model in combination with feedback control to generate a compensated output. This chapter will first show a complete presentation of the general compensation scheme known as Model Reference Control (MRC). Then, it will progress into the specifics of the application of MRC to PSP. Finally, the chapter will wrap up with theoretical simulations and compensation of some of the experimental data presented in the previous chapter.

Model Reference Control

Model Reference Control is a generalized scheme in which one can take a

dynamical system (plant, Gp) with known or modeled parameters and with the use of feedback control, give the resulting system (model reference, Gm) a desired type of dynamics. The system shown in Figure 3.1 is similar to that presented in standard Adaptive Controls texts with regard to MRC with two exceptions.39'40 First, there is not a constant gain feedback term from the output Y. It is stated in an exercise later in the presentation of Ioannou and Sun that such a term is unnecessary. Second is the addition of the gain 1/k, in the feedback from Y. This addition was found to simplify some of the later analysis.


36





37


U CO Y







Figure 3.1 --Structure of the Model Reference Control scheme.

The MRC design begins with a set of assumptions (either known or given):39 Al. G, (s) = kN, (s)/D, (s) is a known transfer function. A2. Np(s) is a monic Hurwitz polynomial of degree np. A3. Dp(s) is a monic polynomial of degree dp. A4. The relative degree of Gp(s) is Ap = d, - np. A5. G m(s)= k mNm(s)/D m(s) is a user-defined transfer function. A6. Nm(s) and Dm(s) are monic Hurwitz polynomials of degree nm and dm
respectively, where dm d.

A7. The relative degree of Gm(s) is Am dm -nm, where Am = p . In the above assumptions, the term "monic" means that the coefficient of the highest order polynomial term is unity. The term "Hurwitz" implies that the roots of the specified polynomial all lie in the left complex half-plane. As a result of the assumptions (A 1-7), we will also define H(s)= N(s)/D(s), where N(s)= (sdP-~, s,-2 ... 1)T and D(s) is an as-yet undetermined monic polynomial of degree d = dp. Note that H(s) is a singleinput, multi-output system. Analyzing the system of Figure 3.1 yields





38


y (3.1)
U DD,-DC,-N-NC2 -N


where the Laplace variable, s, has been dropped from the notation for simplicity. It is desired by the user for the Model Reference System (Closed-Loop System) to have the dynamics


Y N_(s)
--= k . ' .(3.2) U "'D.(s)


Equating the systems gives CkN _ kmN. (3.3)
DD --DC,-N-NC2-N Dm


An inspection of the polynomials on the left-hand and right-hand sides reveals that the numerators and denominators are all monic polynomials excepting the values Co, kp and km. Therefore, equating these constant gains gives C = " (3.4)


If we now let

D=DNm, (3.5)

then equation (3.3) can be re-arranged into the form DC1 N+ NpC2 *N = D(NmDp - NpD). (3.6)

This equation will be termed the "Matching Equation" and will be used to calculate the elements of the unknown vectors C and C2. Before proceeding to this solution we must first define the polynomial Do. In general it is best to choose the roots of Do to be on the





39


order of the roots of Dm to keep the feedback gains at reasonable levels. With all of the variables in equation (3.6) having now been defined except for C, and C2, we can proceed to a solution. For the analysis presented here, we shall assume that we are dealing with a plant, GP, with the property np = d, -1, or, A, =1. Assumption (A7) of the MRC scheme requires then that Am 1. The simplest dynamical system that has this property is a Is' order system

1 1
G.(s)= - -(3.7) T s+)4

where T is the time constant of the desired system.

Let us for a moment consider a plant with d, = 3. Let us also assume the following notations

C1. N =a2s 2+CIs+aO C2 -N= P2s2+p3s+po N, =Y2S2 +yjs+Y0 (3.8a-e)
D, = 3S5 +82s2 +81s+80 Do(NmDp - NPDm)=E5s5 +S4s4 +63s +62s2 +ss+60

If we now define x = (X2 I , p2 , 1 )T and b = (65,64,31,2, , )T, the Matching Equation can be transformed into a set of standard linear equations ( Ax = b) 83 0 0 0 0 0 a2 6
62 83 0 Y2 0 0 a1 64 61 82 83 Y1 72 0 aO _ 3 (3.9)
60 81 62 YO Yi Y2 P2 62
0 60 81 0 yo Y7 Ip P 0 0 0 0 0 Yo A R 6Some notes regarding equation (3.9):





40


Bl. Column 1 of matrix A is simply the vector of coefficients of the polynomial D,(s) with trailing zeros added to bring the length of the vector up to 2dp.

B2. Column 2 of A is the same as column 1 but with the last element rotated around to the top of the vector. Similarly for column 3 with respect to column 2.

B3. Column 4 of A is the vector of coefficients of the polynomial N,(s) with one leading zero and with trailing zeros added to bring the length of the vector up to
2dp.

B4. Columns 5 and 6 of A are 1-element rotations of columns 4 and 5 respectively. From this example, the general pattern for the matrix A and vectors x and b for a plant of degree dp can be extrapolated. Solving this system of linear equations gives the values of the ai's and Pi's which are in turn the elements of the vectors C, and C2.

Application of MRC to PSP System

The purpose for which the Model Reference Control scheme will be used, with regard to PSP, is not one for which it was intended. That is, the MRC scheme will be used as a "State Observer". Considering a PSP plant with input P(t) and output Ps (t), we desire to have a measurement of the unknown input as opposed to what we can measure which is the output. Therefore, we have designed the MRC so that the desired measurement is an internal state. We will further specify parameters of the design so that the estimated input to the plant will have very fast dynamics so that it should quickly approach the true pressure P(t).

For the particular application of PSP dynamics, the MRC scheme shown in Figure

3.1 will be changed to the diagram shown in Figure 3.2. We will consider the measured PSP pressure, PPSP (t), to be the input U. The internal state V will be the estimation of the true pressure, P(t) = Pc (t), and will in turn be our value of the compensated pressure. The output Y of the MRC scheme is simply an estimation, PPSP (t), of the input. The





41


PpsGm CoG Pcs









Figure 3.2--Structure of the Model Reference Control scheme for use as a
compensator for PSP data.

other internal states Wi, W2, and Z are of no relevance. The plant, Gp, is one of the dynamics models for the PSP given in Chapter 2. Our aim is to design the MRC parameters so that Gm is a very fast system. Thus, the output Ppsp (t) will be closely mimicking the input Ppsp (). Furthermore, since Psp ( is the output of the plant Gp, the input to the plant should be closely tracking the true pressure. That is, V = P(t)= Pc (t). So, by picking off the input to the plant Gp and making it an output, we will be able to get an approximate measure of the true pressure versus time.

To begin, we will need to use a linearized model for the PSP dynamics. It is

necessary to use a linearized model in order to calculate the values Co, Ci, and C2. After calculating the parameters of the MRC using the linearized plant, the nonlinear plant will be resubstituted into the overall scheme keeping the same values for Co, Ci, and C2. For the linearized model we will use the diffusion model from Chapter 2 with the linear calibration (equations 2.25a,b). Further, for the purposes of presentation, we will assume a 2 order model. The linearized plant specifications are





42


[ -2.47 0.00] 1 [0.00 -1.271 FP(t)1
6c 0.00-22.21_ a 0.00 0.42J [P(t)J (3.10a,b)

P,, ,(0)[ .64 -0.21] + [1.00 0.00] ab


Converting this from State-Space form into Transfer Function form we get s 2 + 24.67s +54.79
- 0.90s -18.22 (
s2 +24.67s +54.79

The numerator of this equation is an array with two rows - one for each of the inputs P(t) and P(t). Notice that the transfer function for P(t) is that of an all-pass system. Making use of the fact that P(t) = s - P(t), we can reduce the system of equation (3.11) from a MISO system (multi-input, single-output) to a SISO system (single-input, singleoutput)

G,(s) =0.0994(s + 10.05Xs + 54.88) 0.0994( 10.05,-54.88) (3.12) (s +2.47Xs +22.21) = (-2.47,-22.21)

The plant expressed in this form has an input of P(t) and output of PpsP (t). This equation will be used to define the plant during the design stage of the MRC development. Later, the nonlinear form of the plant (diffusion model with Stern-Volmer calibration) will be used in its place. The latter portion of equation (3.12) includes a definition of the plant in "Zero-Pole" format where the zeros and poles of the transfer function are listed in vector form. The value of 0.0994 in front is the high-frequency gain of the transfer function.

In the earlier section of this chapter, the MRC scheme was developed for systems with the property A, = 1. Notice however, that the plant definition of equation (3.12) has





43


a relative order of A, = 0. Therefore, we must either re-develop the MRC scheme for systems with A, = 0, or find a way to modify the plant to give it a relative order of A, = 1. There is a simple "trick" that can be employed to give the desired effect of changing the relative order of the plant. By augmenting the plant with a very fast I" order system, the resulting transfer function will have the desired properties. It should be noted that this augmented 0S order system will also have to operate on the actual PSP data prior to becoming the input of the MRC scheme. The question now is how to choose the time constant of the augmented system. Inspection of the poles of the plant shows

2
poles = -[(2i - I)j , i = 1, 2,... (3.13)
21

Following this pattern, the next value in the sequence will be used as the pole of the augmented system yielding:


G (s)= k N,(s) k Na (s) =k spa 6.13 (s +10.05Xs + 54.88) (3 14)
pa " D,(s) a Da (s) p Dpa() (s +2.47Xs+22.21Xs+61.69) (

In using this choice of augmentation there will be no loss of dynamics from the physical system. That is, because the augmented system is in essence a lowpass filter, but with a very high cutoff frequency, the dynamics of the physical system will be relatively unaffected. Considering this augmented plant as the design plant (Gp = Gpa) for the purposes of calculating the Model Reference Controller parameters, we can choose the model system as


G (s)= k1 "' =100 (3.15)
D.n(s) s+100





44


which is a 1" order system with a time constant of 0.01. For the Plant Gpa(s) of equation (3.14) and Model Gm(s) of equation (3.15) and for the choice of Do(s) D.(s)= (s +I ooXs +lI)oXs +120), (3.16)

the solution to the Matching Equation gives Co = 16.32
C, = (-7.86el, 1.33e4, 1.13e6) (3.17a-c)
C2 = (-3.79e4, - 4.39e6, -1.3 1e8) Or, if we redefine the feedback systems in the following manner: HI(s)= C1 * N(s)= k N,(s) D(s) D1(s) (3.18a,b)

H2 (S)- C2-(S)_ N2(s) Ds D2(s)

we get (in zero-pole format) H,(s)= -78.57 (-62.07,231.81) (-100,-110,-120) (3.19ab)

H2 (S)=-37947 (- 57.91 9.73i) (-100,-110,--120)

Following this procedure, the MRC parameters can be calculated for a PSP plant of arbitrary order.

MRC Simulations and Compensation of Experimental Data

It now remains to be seen how well the MRC scheme can be used to compensate the PSP data. Using Matlab and Simulink, programs were developed to implement the MRC scheme (see Appendices A, B). With this code several simulations were performed. In these simulations, a 5th order plant design and its accompanying MRC was used as opposed to the 2"' order design already presented. There were three basic sets of tests performed. The first used the linear form of the Plant to create the input Ppsp(t), and





45


1.2




0.8


: 0.6
a.

0.4


0.2


0
0 0.5 1 1.5 2
t*
Figure 3.3--Linear compensation simulation of a positive step in pressure.


the linear form of the Plant as Gp(s). This test was performed to ensure that the compensator works as it was designed, that is, the compensated output should have the dynamics of the ideal design Model Reference System. The performance of this linear version of the MRC compensator was tested on a positive step from 0.1 atm up to 1.Oatm. The results of this test can be seen in Figure 3.3. Notice that the PSP indicated pressure takes approximately two time units to come to equilibrium, whereas the compensated output levels out in less than 0.05 time units. Therefore, this form of compensation system does indeed work in the linear case and affects a more than forty-fold increase in speed. Thus, the MRC compensation scheme does work in the linear limit. The second set of tests used the full nonlinear form (diffusion model with Stern-Volmer calibration) to create the input and as Gp(s). The compensator parameters, Co, CI, and C2, used in this test were from the linear design. The response of the compensator system to a positive





46


(a)


1.6

1.4 1.2

1

0.8 0.6

0.4 0.2

0





1.2


1


0.8


0


0.5


I


1.5


2


t*
Figure 3.4--Compensation simulations using nonlinear from of the PSP plant. (a) Positive step in pressure; (b) Negative step in pressure.


a
IL


0 0.5 1 1.5 2
t*

(b)


P
E3____ PPS
PC



-e-







- -I


-~ -' :2: 3 3 ~


P
PC


I


1 a.
a.


0.6


0.4 0.2


0


I


I


I


i


-


0 8 8 8 A 9


i V-- - G





47


(a)


1.2


1


0.8 0.6 0.4


0.2


0


0.2


0.4


0.6


0.8


1


t*
Figure 3.5--Compensation simulations using nonlinear form of the PSP plant with a sinusoidal driving pressure. (a) co =1; (b) o' = 3; (c) co =10; (d) o* = 30.


0.
a.


0 0.5 1 1.5 2 2.5 3
t*

(b)


P
PPS
PC---


1 2


a.
IL


.



0.8


0.6 0.4


0.2


P
- ------P
PPSP P Pc


0


0





48


(c)


I 0.
a.


1.2


1


0.8 0.6


0.4


0.2


a.


U


1.2


1


0.8 0.6


0.4 0.2


0


0


0.05


0


Figure 3.5--continued.


0.1


- -p
E3 PP
-s--- --


0.15
t*


0.2


0.25


(d)


0.3


F






- p P c
I I I


0.02


0.04


0.06


t*


0.08


0.1





49


(a)


1.6

1.4 1.2

1

0.8 0.6

0.4 0.2

0





1


0.8


0.6


0.4


0.2


0


2


3


4


5


I


6


Time (sec)
Figure 3.6--Compensation of experimental data. (a) Positive step in pressure; (b) Sawtooth driving pressure.


a.
CL.


i


0 0.5 1 1.5 2
Time (sec)

(b)















P
-e--
I ~ ~i I


-_ _ P
- - PC


0












= =


1


i


I


I





50


and negative step was simulated. As in Chapter 2, the step simulations were between the pressures 0.1 atm and 1.0atm and Q = -0.25 determined the Stern-Volmer calibration. Shown in Figure 3.4 are the results of the simulations for the positive step and negative step. The compensated response to a positive step in pressure contains sizable oscillations, whereas the negative step response is practically that of the ideal linear compensator. Thus, the nonlinearity of the PSP system, as it pertains to the MRC compensator, is more in evidence for positive changes in pressure than for negative changes. The third set of simulations were similar to the second in that the nonlinear form was used for the plant. However, rather than applying impulsive steps in pressure, sinusoidal fluctuations in pressure were simulated. The sinusoidal tests were between the limits 0.1 atm and 1.0atm with the same value of Q. The frequency of the driving pressure was set at four different values, o* = 1, 3, 10, and 30. The results of these tests are shown in Figure 3.5. There are several items to note concerning these simulations. It can be seen that the compensator does a remarkable job even out to driving frequencies ten times the critical frequency of the paint. One thing of note with respect to the sinusoidal simulations is that the phase shift of the compensated data relative to the applied pressure is very nearly 0.01 in each of the four cases (frequencies) shown. Thus, the compensator is nearly a "linear phase" system. Also note that this value of 0.01 is exactly the time constant of the designed Model Reference System.

What remains is to determine how well the compensator works on actual PSP data. Figure 3.6 shows the compensator response from two different experiments. The first is a positive step in pressure from 20 kPa up to 120 kPa (note: Prf= 100 kPa). The time constant of the PSP was 1.11 sec. The second was a sawtooth driving pressure and





51

the time constant was 0.437 sec. In the step test a slight overshoot can be seen, but the compensator is still a great improvement over the raw PSP data. The compensation of the sawtooth data has a peak error of 4 kPa and an RMS error of about 1 kPa. The maximum error occurs on the rising edge of the sawtooth wave where the highest frequencies are contained. From these two sets of experimental data, we can conclude that the MRC compensation scheme does indeed work. The next step will be to apply this to a sequence of full-field images.














CHAPTER 4
EXPERIMENTAL SETUP AND PROCEDURES This chapter will begin with a presentation of the two facilities employed to

acquire the experimental data presented in this dissertation. Both of these facilities are located at the University of Florida, Department of Aerospace Engineering, Mechanics and Engineering Sciences. The first of these is a test cell made for testing the dynamic response of small-sized PSP samples. The second is a test stand for investigation of a pulsed transverse jet in a Mach 1.56 crossflow. The end of this chapter will be a presentation of the image processing techniques used to get a single calibrated PSP image. The method of compensating these calibrated images will be presented in the following chapter.

PSP Dynamics Test Cell

The data presented in Chapters 2 and 3 were performed in the PSP Dynamic Test Cell. This test cell is designed to control the pressure of the air over the PSP, the temperature of the PSP sample, and the temperature of the air entering the test cell. The experimental apparatus consists of the test cell itself (with various accessories), an external heat exchanger, a lamp, a photomultiplier tube (PMT), some optical components, a computer with data acquisition card and various intermediate circuits. Additionally, vacuum and high pressure lines were available to connect to the test section. The entire apparatus rested on an optical bench with an enclosing hood to shut out external light sources.


52





53


A front view of the test cell is shown in Figure 4.1. A schematic view of the test cell is also shown in Figure 4.2. There are six ports (1/8" NPT) around the periphery of the cell to allow access to the small volume (1/8" thick x 1/2" diameter) immediately above the sample. These ports can be instrumented in a variety of ways. As shown in the Figure 4.1, it is instrumented with two digital valves (Asco/Angar AL4112L), a high frequency response pressure transducer (Entran EPV-501X-25AZ), and a thermistor (Omega 4420 1). The two remaining ports are capped. The two digital valves are hardwired to be 1800 out of phase so the test cell is either open to vacuum or open to high pressure. With this feature, a pressure step from vacuum to ambient pressure occurs in just under 20ms. Additionally, an analog valve (Pneutronics VSO, 1.25mm orifice, not shown in figure) can be mounted to one of the ports. With the use of the analog valve, the pressure in the cell can be driven at any arbitrary function the user may desire such as a sine wave, square wave or sawtooth wave. It should be noted that because of the small orifice of the analog valve, the pressure inside the cell was band-limited to less than 80Hz. However, with the dual digital valves controlling the pressure in the cell, frequency content upwards of 250Hz could be achieved, but only in the form of a step or square wave. It was later discovered that using a latex burst diaphragm on one of the ports could create frequency content of the pressure signal in excess of 1kHz, however, no such experiments are presented in this dissertation. The window on the front of the cell is plexiglass and allows optical access to the PSP sample. The sample is a small piece of aluminum (12.7mm x 15.9mm x 3.2mm) with the PSP applied to one of the faces. The size of the samples was chosen so they could be easily and cheaply made from 12.7mm x 3.2mm (1/2" x 1/8") aluminum bar stock. The sample is mounted on top





54


of a raised circular platform. An O-ring around the base of the platform seals the interior of the cell. Behind the platform is a double stack of thermoelectric chillers (TEC's) (Melcor CPl.0-31-05L) to heat or cool the sample. Mounted further behind the TEC's are a set of cooling fins and a small DC fan. Originally the test cell was made with a water-bath behind the circular platform to control the temperature of the sample, but this was later changed out for the TEC's.

The inflow air heat exchanger unit, shown in Figure 4.3, is made of two

aluminum plates (17.8cm x 15.2cm) clamped together. Machined into one of the plates is a zig-zagging channel thru which air can flow. Mounted on top of the the unit are four TEC's (Melcor CPl.4-127-lOL) with cooling fins and fans to control the temperature of the top plate. The entire unit is encased in foam insulation to aid in temperature control. Mounted at the exit is a thermistor (Omega 44201) to measure the temperature of the air leaving the unit.

Two different styles of lamp can be used to illuminate the PSP sample. The first, shown in Figure 4.4, is a quartz-tungsten-halogen lamp with stabilized power supply and photofeedback system (Oriel QTH lamp housing with 68830, 68850, 68855 accessories). The second style of lamp is shown in Figure 4.5 and incorporates an array of super-bright LED's (ISSI LM2).

The PMT used to collect and measure the light from the PSP sample has a

spectral response from 185nm to 900nm and a bandwidth of 20kHz (Hamamatsu HCl2005), although analog RC lowpass filters used to aid in reduction of shot noise lessened the bandwidth.





55


There are several components of optics in the setup - two filters, a mirror and a lens. The first of the two filters is a bandpass filter centered at 450nm (Melles Griot 03FIB004) and is mounted to the lamp. The second filter is also a bandpass filter, but is centered at 650nm (Melles Griot 03FIB014) and is mounted to the front of the PMT. The mirror reflects the excitation from the lamp into the test cell and onto the PSP sample. The light emitted from the sample is then focused thru a lens into the PMT. A schematic for the path of the light is shown in Figure 4.6.

The computer used in the experiment was one of several IBM-compatible Pentium or Pentium II machines. A data acquisition card (DAQ card) (National Instruments AT-MIO- 1 6E-2 with SCB-68 accessory) was used to sample the signals from the various transducers and to control the operation of the digital and analog valves. Data acquisition and control programs were developed using National Instruments' LabView programming language. This program was designed in a very "hands off' fashion and allowed the user to control the digital and analog valves and to perform the data acquisition of the transducers all from a seat in front of the computer. Data from the transducers was acquired at rates on the order of 5kHz and then downsampled with averaging to a rate of 1 kHz.

Several circuits were used to interface the transducers and valves to the DAQ

card. First order analog lowpass filters with a cutoff frequency of 480Hz were applied to the signals from the pressure transducer and from the PMT to aid in noise rejection. Because the DAQ card could only source 5mA of current, power transistors were used to provide current amplification to the voltage signals coming from the analog output and





56


digital output lines of the DAQ card leading to the valves. Finally, resistor circuits were used to create a linear voltage versus temperature output from the thermistors.

Supersonic Wind Tunnel

The second facility used for experimentation was a blow-down wind tunnel. Two large tanks external to the building provide an initial source of approximately 30m3 of air at a maximum pressure of 1375kPa. Supplying air to the tanks is a 300hp rotary screw compressor (Quincy QSI-1000) capable of delivering 34kg/min at 1375kPa (950SCFM at 200psig). Attached to the compressor are dual dessicant dryers (Hydronix) that cycle in turn to remove moisture from the air after compression. A 3" line runs from the tanks, into laboratory, and thru one of two control valves (Fisher ET- 1Y", Fisher ET-1" with Type 667 Actuators, Series 3582 Valve Positioners and Type 67AF Filter Regulators) depending on the desired flowrates or stagnation pressures required at the test stand. These valves are shown in Figure 4.7. The two valves are mounted in parallel on the 3" line. The valves are ultimately user-controlled in one of two fashions. First, they may be controlled manually via a pneumatic controller system (Fisher "Wizard" Controller). Alternatively, the valves may be controlled via a computer connected to an external transducer (Fisher Type 846 Electro-Pneumatic Transducer). The second of these was the method of choice. The final element of the general test stand was a large cast-iron pressure plenum, or, stagnation chamber shown in Figure 4.8. The air flow is from left to right in the image. To this chamber can be mounted whatever test section the user may design such as the Mach 1.56 nozzle used in the experiments to be presented in the next chapter.





57


The test section chosen for experiments reported in this dissertation has a design Mach number of 1.60 and its geometry was calculated using the inviscid nozzle design program of Carroll et al.4' Later pressure measurements made in the tunnel yield a Mach number of 1.56. This difference is due to the growth of the boundary layer downstream of the sonic throat. A slight divergence of the test section downstream of the nozzle exit could have compensated for the boundary layer growth, however, it was decided to use a straight section to ensure a perpendicular viewing of the jet to be described later. The main section of the tunnel downstream of the nozzle is a 5.08cm x 5.08cm (2" x 2") cross-section with optical access via a 15.24cm (6") long plexiglass wall. There are also two opposing sidewalls with interchangeable plates with provisions for additional optical access, or transducer measurements, or for placing obstructions into the flow. An external view of the test section is shown in Figure 4.9. The direction of airflow is from left to right. The main plexiglass window is toward the front. Along the upper removable sidewall are an array of static pressure ports and thermocouples. For a lower viewing angle, a 4.44cm (1.75") diameter circular plug opposite the main plexiglass wall would also be seen. Figure 4.10 shows a closeup view of the front of the plug. For the purposes of the experiments to be presented in the next chapter, this plug has been modified to create a transverse, or normal, jet 3mm (0.118") inches in diameter surrounded by four 0.5mm diameter (0.020" 0) pressure taps at 900 increments on a

4.5mm (0.177") radius about the jet. Additionally, a single thermocouple is imbedded

2.5mm (0.1") from the interior surface of the test section. Figure 4.11 shows a rear view of the plug. The radial location of the thermocouple can be seen. Also shown in the figure is the digital valve used to control the jet. The diameter of the jet was chosen to be





58


approximately equal to the free-stream boundary layer thickness as calculated by Carroll. Other studies have used this same method to choose the diameter of a jet injected normally into a supersonic air stream.8,2' The internal geometry of the jet is simply a straight 3mm diameter cylinder. Other studies have used a converging geometry at the jet orifice to ensure a sonic throat. However, because the valve is being cycled, and because it is desired to have as fast responding a jet as possible, the minimal volume straight cylinder geometry was chosen.

The optical system used to measure the intensity of the PSP consisted of a

camera, a strobe and appropriate filters. The camera was a Photometrics Model CH250 CCD camera as shown in Figure 4.12. The CCD array itself was a 512 x 512 array of pixels. The full-well capacity of the pixels is approximately 320,000 electrons and the pixels were read out using a 14-bit A/D converter. The strobe, also shown in Figure 4.12, was a Xenon DT-301 Digital Stroboscope Tachometer. The strobe could operate at frequencies of 2Hz - 200Hz. The duration of a single flash of the strobe is on the order of 30ps. A composite of blue Schott glass filters (BG12 - 3mm, BG39 - 3mm) was placed on the output of the strobe to dampen its output at red wavelengths and yet allow the blue excitation through. A Melles Griot 03FIB014 bandpass filter centered at 650nm was placed on the input to the camera to filter out the source illumination from the strobe.

Various conventional transducers were used in these experiments. The stagnation pressure was measured with an Omega PX302-200AV pressure transducer. This measurement was used as the process variable of a PID algorithm to control the Fisher

1 2" valve and for calculating the Mach number of the flow. All of the temperature measurements were made using Omega T-type thermocouples. The cold-junction





59


reference temperature will be described later. Static pressure measurements were made with a combination of two different ranges of Kulite piezoresistive pressure transducers. Two Kulite Model XT-190-25A (25psia range) and two Kulite Model XT-190-50A (50psia range) transducers were available for measuring the pressures at the various static pressure taps. One of the the 50psia transducers was positioned at the sidewall pressure port closest to the nozzle exit and measured the static pressure for purposes of calculating the Mach number. It was located far enough upstream of the injected jet to be outside of its influence. A second 50psia transducer was placed at the pressure tap spanwise from the jet. The two remaining transducers with ranges of 25psia were placed at the taps upstream and downstream of the jet orifice. Thus, the two 25's were placed at the locations of highest interest and the two 50's were placed at the less critical locations. The Kulite transducers mounted to the pressure taps on the plug insert were offset from the interior of the test section by a 5.08cm x 1.6mm0 (2.0" x 0.063"0) volume. The frequency response of the pressure transducer/tap volume combination was not calculated, but was assumed to have a bandwidth greater than 1kHz which was the maximum sampling rate used in the experiments presented. The pressure transducers were calibrated using a NIST-traceable pressure standard (Druck DPI-145 Multifunction Pressure Indicator). Additionally, prior to each run, the transducers were "zerocalibrated" to match the ambient pressure indicated by the pressure standard.

The jet could be turned on or off or modulated at a given frequency via a SnapTite digital valve (2W13W-INB-A8E6, 24V, 9W, 1/8 orifice). The valve was capable of operating at frequencies upwards of 100Hz although at the upper range the valve had a tendency to stick in the open position. At frequencies below 80Hz a duty cycle of 40%





60


on and 60% off seemed to work the best in terms of the pressures seen while running the tunnel and on the basis of "non-stick". The pressure upstream of this valve was set using a regulator coming off the 100psi house air supply within the building. This pressure was further measured using the Druck pressure standard.

The experiment was controlled via a data acquisition and control program developed using LabView and running on a Pentium III computer. This program controlled two separate DAQ cards within the computer (AT-MIO-16E-2; PCI-6024E). The first board measured the 5 pressure transducers, the wall thermocouple, and the voltage signals being applied to the jet valve and the strobe. Prior to performing the experiment, a switch was set on the external connector block (SCB-68) that connected channel 0 of the board to a cold-junction temperature sensor on the connector block. A measure of the cold-junction temperature was then recorded and used later to properly convert the thermocouple voltage readings into temperature. After measuring the coldjunction, the switch was reset so that channel 0 could be used to measure one of the external transducers. One analog output on the first board was used to control the main valve responsible for running the tunnel. Using PID control, the stagnation pressure was monitored and set to a desired pressure of 158kPa (23psia) which was just above the pressure necessary to fully start the tunnel. The second DAQ card was used solely for its two analog output lines. These were used to drive the jet valve and the strobe. The phase of the strobe relative to the jet could be controlled down to 1150h of a cycle at a frequency of 100Hz. This allowed the strobe to be flashed consistently at a desired phase relative to the cycling of the jet. Hence, the acquired PSP images were actually ensemble averages of hundreds to thousands of individual strobe flashes.





61


The final element of the tests performed in this test section are the PSP and TSP coatings. The PSP consisted of the luminophore PtTFPP [Pt(II) meso-tetrakis(pentafluorophenyl)porphine] interspersed in the polymer matrix VPDMS [vinyl polydimethylsiloxane]. The TSP was a ruthenium derivative in clear-coat acrylic. These formulations was made and applied to the plug insert by a member of Dr. Schanze's group at the University of Florida Department of Chemistry. The coating thickness varied from painting to painting but was nominally on the range from 1 Opm - 20ptm.

Image Processing

Achieving a calibrated PSP image is a very complex process. Numerous papers have been written on just this subject. The general process is shown in the flowchart of Figure 4.13. To begin, one needs two images - one at the unknown run condition and a second at known reference conditions. Consider the two image intensities to be I,. and Iref respectively. The next step is to subtract the dark image, Idrk, from each of the run and reference images. The dark image is an image of the same duration as the run and reference images but with all excitation sources turned off. Next is a flat field correction which compensates for the pixel-to-pixel variation in sensitivity of the CCD array. This correction takes the form of a matrix, Cif, the size of the CCD array with values all near

1.0. When this matrix is multiplied by the given image, the differences in the pixel gains is cancelled out. The task of generating the matrix Cif is an involved process in and of itself. In the process of running the tunnel, the model of interest may move slightly relative to its position at the reference image. In a later step, the data contained in the two images will need to be ratioed. Therefore, it is of extreme importance that the two images be perfectly aligned prior to the ratioing step. Often, markers are placed on a





62


model to be used as registration points. Alternatively, certain parts of a model may be used as registration points. For example, oftentimes a model will have pressure taps in addition to the PSP to be able to get very acurate measurements at a distrete number of points. In a PSP image these taps will show up as tiny dots of low intensity. If a number of these are scattered over the model, they can be used to calculate the movement of the model relative to its reference position. For certain models, a simple pixel-shifting is sufficient. However, if the model rotates relative to its reference or warps due to the loads it experiences, the registration algorithm must become much more complex. For the model used in this dissertation, warp and rotation are not present. So, a simple linear algorithm will be used to make the run and reference images line up. The run image will be shifted so that it aligns with the reference image: Ir(i', ') = Iu(i+ Ai, j+ Aj) (4.1)

where Ai and Aj are the amount of pixel shifting necessary. Note that these quantities are generally non-integer values in which case a weighted average of the four closest pixels will be used to calculate the required value. The next step is one that is often not possible to perform, but, for the experiment to be presented in this dissertation, it is. A tiny fluorescent marker was applied to the external surface of the test section in a location where it would be within the field of view of the PSP images. Because this marker is on the outside, its intensity will only vary if the intensity of illumination varies. Taking a measure of the average intensity over a region of the marker in the run image (minus the average dark intensity over the same region) and ratioing to the same in the reference image, the run image can be scaled to match the intensity of illumination that was present





63


during the acquisition of the reference image. That is, define the "external reference" scaling constant:

Ir, - I
C er = - rex -r (4.2)
1lfl ,ext -Idrk

When multiplied by the run image, this constant will perform the necessary scaling. With constant light sources such as the tungsten-halogen lamp or LED lamp used in the aforementioned Dynamic Test Cell, such corrections are not generally necessary. However, for the particular application to be presented, a strobe lamp is being used. If the shutter on the camera is not ultra stable in its opening process, closing process and total duration of operation, or if there is variability in the strobe, one image may see an additional strobe flash or two relative to another image depending on the frequency of the strobe flashes. So, it becomes necessary to be able to scale out the intensity of illumination using the external reference technique. It is being assumed here that the spatial variations due to a single flash of the strobe will even out with the hundreds of strobe flashes used to acquire a single image. The next step is to filter the images spatially to remove some of the shot noise inherent in measuring photonic sources. Generally, one is able to take multiple run images at the same conditions and average them to reduce the shot noise. However, for the particular experiment presented, the runtimes were limited to where this luxury was not available. Moreover, the integrated light levels of many strobe flashes was still not sufficient for making use of the full well capacity of the CCD. That is, the measured intensities were only on the order of hundreds of counts (bits). This means that the signal-to-noise ratio of the pixel intensities is relatively low. If the noisy run and reference images were ratioed, the noise would only be amplified on those pixels that just happened to be statistically greater(lesser) and





64


lesser(greater) than average between the run and reference images respectively. So, at the expense of some spatial resolution, the images are filtered prior to being ratioed. The smoothing kernel used was of the form:



K(i, j Co)= 2(7c i02+ j02 n-,Fj where i0, j0 = -n,...,n; n = 1,2,3,... (4.3)
I cos 2(t i2 + j2 2nF)
i=n j=-n

In the above equation, io and jo are the coordinates of the pixel within the kernel being calculated. The kernel has dimensions (2n+l) x (2n+1), thus it is of odd order (3x3, 5x5, ...) and is centered on the pixel ((n + 1)/2, (n + 1)/2). It may be noticed that the corner pixels of this kernel will turn out to be identically zero. This kernel is then 2Dconvoluted with an image to achieve a smoothed image. This form of kernel is superior to a 3x3 or 5x5 "flat-top" kernel in that it gives more weight to pixels near the center and less weight to those farther away. For a flat-top, equal weight is given to all pixels within the area of support of the kernel. This kernel was developed based on ideas found in IDDSP wherein spectral leakage is reduced by using a tapered windowing function on a dataset. Depending on the amount of noise observed (based on the light levels measured), either a 3x3 or 5x5 kernel was used to smooth the data. An estimate for the spatial resolution based on a 95% sum of the kernel values gives a value roughly equal to "n". As an example, a 7x7 kernel for which n=3, roughly 95% of the kernel's weight is contained within a radius of 3 pixels about the center of the kernel. It should be noted that for 3x3 and 5x5 kernels, 100% of the kernel's weight is contained within the estimated area. The next step in the flowchart is to ratio the smoothed images. The steps up to this point may be summarized via the following equations:





65


Y. = K 9[e I(4.4a)
'"" Kf[.C (I r.-Ild, A


Yref = (4.4b)
"'K([Cff *(I. -Idr A

where o is the "array multiplication" operator (C = A e B -+ ci = a i by ), and 0 represents the convolution operation. Ratioing these two equations we finally end up with the equation:


Y* Y._ _ K 9[Cff (Iref - I(4k A = f(PrP., 4.5)
Yf K 9[CerCff (In - Idrk

which represents the static calibration function. Analytically inverting this function gives Pr = g(Y, P,) (4.6)

where f and g are nearly inverse functions.

One fact that will aid in the analysis of the images for this particular application is that we have the capability to perform a static calibration of the PSP sample in the exact spot and with the same lighting as during a run. That is, the exit duct may be unscrewed from the test section and replaced with a cap. The test section is now an enclosed volume which can be drawn down to vacuum or can be pressurized. Thus, we can get a very accurate measure of the static calibration. Following the process outlined in leading up to equation (4.6), a sequence of images taken at different pressures with the tunnel capped may be reduced to a sequence of data points of dimensionless-intensity ratio versus dimensionless-pressure ratio. An analytic function may now be curvefit to this data to give a useful expression for the calibration function. Table 3 lists mathematically equivalent forms of three common calibration functions. The first function in the table is the standard linear Stern-Volmer equation. The second corresponds to a linear





66


Table 3--Calibration Functions, Constraints, and Typical Values


Function Constraintsa Typical Values"b


Y (P-Q1) Q, <0.01 Q, = -0.2
(Pr f -Q.)


Y- _ Y* (P - Q, )(Pref - Q2) Q, < 0.01 Q, = -0.I
(Pef -Q1 XP -Q2) Q2
(P* -1Nr9 Q, <0.01 Q 00
YO - (PQ1XPef -Q2 XP -Q3) Q<.0 Q1-0.05
(Pef -QI1XP -Q2 XPor -Q3) Q2 Q3 a Numerical values of constraints and typical values are given in units of atm. b yo* = 1.00 is the typical value for all three functions. Stern-Volmer response with a pressure-independent bias in intensity. The third entry is the dual-mode or dual-sorption calibration. All of these calibration forms have the benefit that the Qj's will be independent of the reference pressure, Pre. The calibration data of Y* versus P for the PSPs used in the transverse jet experiments had a nonnegligible amount of downward curvature. Therefore, the standard Stem-Volmer calibration was insufficient for fitting the data. It was found that the third function from Table 3:


y* _ y* (P - Q, xpr1 -Q2 XP - Q3) (4.7)
(Poef - Q)P - Q2(Pr - Q3)

most accurately fit the data. The parameter YO is only used during the curvefitting process. It will be a value very near 1.00 and allows for uncertainty in the reference image during the curvefitting process. When implementing this function to calibrate





67


actual PSP data, the parameter Yo is set to 1. The three values Q1, Q2 and Q3 must all satisfy certain constraints to satisfy physics, otherwise the intensity would be allowed to be infinite at some small positive pressure. The three constraints listed in Table 3 are: " Q, < -0.01
" Q2 " Q3
The value of -0.01 on the first constraint is not truly a physical constraint but rather an empirical one. Physically speaking, the true constraint on Q, is that it must be strictly less than zero. However, it has been found that during curvefitting, this parameter will often approach the upper limit and perhaps even fit to the upper limit value. If a limit of 0.00 were chosen, the calibration would be indicating that when the pressure goes to zero the intensity goes to infinity. This cannot possibly be, so, a slightly more negative value was chosen for the upper limit on Q1. One additional note with regard to these parameters is that they are mathematically the poles (Q2) and zeros (Q1, Q3) of the calibration function. The inverse of equation (4.7) with Yo = 1 gives


- B+ B2 -4AC
2A
where
A= (Prf - Q2) (4.9)
B = -(P.f - Q2 XQ1 + Q3)- Y*(Pef - Q1XPf - Q3) C = QQ3,P -Q2)+ Y*(P., -Q)P.f - Q)Q2

A sample calibration curve is shown in Figure 4.14. The data spans pressures from

0.18atm up to 1. 67atm (1 9kPa - 170kPa). The curvefit parameters for this dataset are (Q1, Q2, Q3, Y*)= (-0.0100, -0. 2254, -1. 3211, 1.0066) where Q1, Q2, and Q3 are given in units of atm. Notice that Q, has gone to its upper constraint and that Yo is very close





68


to 1. The RMSE of the curvefit is 0.84% of Pref, or, 0.85kPa. If the constraints are lifted and the data is re-curvefit, the parameters become (0.0484, -0.0759, -1.0761, 1.0054) and the RMSE is 0.77% of Pref, or 0.78kPa. So, with the constraints lifted, the curvefit would indicate that the intensity is infinity when the pressure is P = 0.0484atm, that is, when P = 4.89kPa. Notice that without the constraints, the curvefit only decreases the error of fit by 0.07kPa which is a very negligible amount for most applications.

Uncertainty Analysis

For a function f(x,y) where x and y are independent variables, the general uncertainty in f is given by


uf = -u + --uj . (4.10)


For the independent variables x and y, the 95% confidence limits can be calculated by the formula


U, = B +(2Sx)2 (4.11)

where B represents all sources of systematic or bias error and S is the standard deviation of the random component of the error. Using the formulas of equations (4.10) and (4.11), the uncertainties of the variables for the pulsed jet experiment can be tabulated. Table 4 lists many of these variables, their nominal values and their uncertainties.

A detailed investigation of the uncertainty in the indicated PSP pressure shows

that the relative uncertainty in the measured PSP pressure can be reduced to the following equation:

u, Y* BP 1+Y*
u P - - 1 ap(4.12) P P aY. pIf





69


where P is the function P(Y*) from the static calibration, Y* is the measured intensity ratio, P is the A/D gain of the CCD pixels in electrons per bit, and Irf is the intensity in the reference image. A full derivation of this equation is given in Appendix C. We can


Table 4--Nominal values and 95% confidence limits of key variables.


Variable Nominal Value 95% Confidence Limits


1.555atm 0.390atm


9.780C 1.557


2.000atm


1.134


13.5pm


5.35kJ/mole


0.20s 260cts 160cts


0.390atm


0.390atm


0.02 1atm 0.004atm 0.21*C 0.006


0.014atm


0.057


2.Opm


0.27kJ/mole


0.06s 3.7cts 2.9cts


0.016atm 0.001atm


a Assumes no smoothing (see Figure 4.15); b Upstream and downstream pressure taps.


Pstag Pstat Twa ii


M Pi

J


a


E


I'n - Idrk Irf - Idrk P a


Ptap b





70


further take into account the effect of the smoothing kernel (equation 4.3). Smoothing has the effect of increasing the effective number of counts at a given pixel by a factor related to the size of the smoothing kernel. Specifically, the factor is given by the denominator of equation (4.3). That is, for a 3 x 3 kernel, the effective intensity increases by a factor of 2.032 and by a factor of 5.808 for a 5 x 5 kernel. The result of this in relation to the relative uncertainty in the indicated pressure is shown in Figure 4.15. The effect of smoothing in relation to equation (4.12) is also given in Appendix C.


Figure 4.1--PSP Dynamic Test Cell: A) Digital valve to vacuum, B) Pressure transducer, C) Digital valve to high pressure, D) Thermistor, E) Window to PSP sample, F) Cooling fins.




























Window Sample

1/8" NPT Port
O-Ring


Figure 4.2--Schematic view of the PSP Dynamic Testing Cell.


71


30


jo





72


Figure 4.3--Heat Exchanger Unit: A) Inlet, B) Outlet.





73


Figure 4.4--Oriel Quartz-Tungsten-Halogen (QTH) Lamp.





















'4M my v i. 41l 46 p.









41.





'-



* t


E









0
as CO
















=


Lamp


Lamp Filter


PMT PMT Lens


Filter


Reflector PSP Test Cell


Figure 4.6--Schematic of the PSP Dynamic Testing Cell Optics.


75





76


Figure 4.7--Fisher Control Valves: A) 1 2" Valve, B) 1" Valve.


Figure 4.8--Stagnation chamber of the supersonic test stand facility.






77


Figure 4.9--Mach 1.56 Test Section: A) 3 of 8 Static pressure taps, B) 2 of 4 Thermocouples, C) Exterior reference marker, D) Bottom edge of circular plug insert.


Figure 4.10--Detailed front view of plug insert showing the central jet orifice and the 4 static pressure tat locations.





78


Figure 4.11--Detailed rear view of plug insert: A) 2 of 4 Pressure taps, B) Thermocouple, C) Digital valve controlling the jet.


Figure 4.12--A) Xenon Strobe, B) Photometrics CCD Camera.
















Run Image


Reference Image


Dark Subtraction Dark Subtraction


Flat Field Correction Flat Field Correction



Image Registration 4---- Image Registration


External Reference External Reference


Smoothing Filter Smoothing Filter




Ratio: Iref/Irun


Apply Calibration


Figure 4.13--Image Processing Flowchart.


79











1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
O.A


1.00
Pressure (atm)


1.50


Figure 4.14--Sample PSP Static Calibration.


-e--- No Smoothing
0 3 x 3 Smoothing
- 5 x 5 Smoothing
' ' ' ' ' ' ' ' 'I


0.50


1.00


1.50


Pressure (atm)
Figure 4.15--Relative uncertainty in the indicated PSP pressure.


80


0.50


0 Raw Data
- Uryofit


2.00


0.05


0.04
a.


- 0.03 3 0.02

4'
W 0.01


0.00


0.00


2.00


0














CHAPTER 5
RESULTS

Numerous studies have been done on the effect of a jet of air injected into a

supersonic freestream.8,21-30 All of these tests were performed using a steady blowing jet. As-yet, there have been no experiments involving pulsed jets in supersonic crossflow. This may be because the complexity of the flowfield under steady blowing is already difficult enough to describe and quantify. Or, it may be because there hasn't been a viable measurement technology. However, with the mounting interest in synthetic jets, which on small time scales are cyclic, there may soon be great interest in an investigation of a pulsed jet. This then is the basic idea for an experiment to test the ability of the compensator design to correct wall PSP measurements around the orifice of a pulsed jet.

The equipment used is as described in the supersonic wind tunnel section of the preceding chapter. The tunnel was run at a stagnation pressure of 158kPa (23psi) which was the lowest value possible that would leave the tunnel fully started. That is, if the stagnation pressure dropped much below that value, the normal shock at the exit of the test section would detach from the exit and move upstream into the test section proper. The reason for this low stagnation pressure was to allow as long a runtime as possible. Once the tunnel had been started, there was an approximately 5 minute equilibration period during which the wall temperature of the test section would drop over time and then level off. Depending on the outside temperature (i.e. the temperature of the exterior tanks) the static wall temperature would range from as low as 00C (in the wintertime) up


81






82


to more normal value of around 10*C to 15C. It should be noted that all of these experiments were performed at night. This was because ambient light from outdoors would come through the exhaust duct and into the test section creating a bias intensity in the PSP images.

For the particular tests to be shown here, the digital valve controlling the jet was cycled at a frequency of 20Hz with a 40 percent "open" duty cycle. That is, the jet valve was opened for 20ms and then closed for a duration of 30ms, giving a total period of 50ms. The strobe could be set to any desired phase relative to the 20Hz jet valve control voltage. A single PSP image had a duration of 20 seconds of integration. During this 20 seconds the strobe would have flashed a total of 20Hz x 20s = 400 flashes at exactly one relative phase. Thus, the image acquired was an ensemble average of 400 flashes. The strobe would then be reset to a different phase and another image would be taken.

In other transverse injectant studies, the jet pressure was one of the primary variables. In the experiments presented herein, the jet pressure was set to a value of around 2 atmospheres absolute. Based on this value, the jet-to-crossflow momentum ratio (equation 1.8) was J= 1.134. After a full sequence of images had been taken, the tunnel was quickly shut down. As soon as the flow in the tunnel had stopped, a reference image at ambient pressure was acquired. Because this reference image was acquired soon after shutdown, the temperature of the tunnel wall had not yet had time to rise very far. Typically, this temperature rise was on the order of 0. I*C to 0.2C, which is within the limits of uncertainty quoted in Chapter 4. So, a reference image was available that was taken at the same temperature as during the run images. Performing the calibration procedure outlined in the previous chapter, one now had a sequence of calibrated PSP






83
images. Because of the very low light levels, a 5 x 5 smoothing kernel was used to even out the pixel-to-pixel statistical variations. Based on the uncertainty analysis presented in Chapter 4 and in Appendix C, the uncertainty of the PSP indicated pressures will be reduced by a factor of 5.808 from the value presented in Table 4. From that point, the experiment now proceeds into the compensation phase.

The time constant of the PSP was calculated from a compilation of various

experiments. Recalling the definition of the time constant from Chapter 2, t = a2/D, and assuming an Arrhenius relation for the diffusivity, equation (1.12), the necessary measurements are evident. The thickness was measured at 20 points over the surface of the plug insert using an eddy-current gauge (Positector 6000) and then averaged. At the same time the plug insert was painted, two small samples were also painted for use in the Dynamic Test Cell presented in Chapter 4. The PSP on these samples were roughly 0.4x and 1.2x the thickness of the coating on the plug insert. The thickness of these coatings was also measured using the eddy-current gauge. Using the Dynamic Test Cell, a series of tests were performed on the thicker of the two small samples to measure the time constant of the PSP as a function of temperature. The data was then curvefit using the formula

T1 =tgexp(E/RT), (5.1)

where E is the activation energy of diffusion. Because the thickness of this coating was not equal to the coating thickness on the plug insert, a further step was needed to calculate the value of the time constant to be used in the compensation routine. The formula






84


T = a (5.2)
a,

corrects the time constant for differences in coating thickness between the small sample, a, , and the plug insert, a. Using the measured wall temperature in combination with equations (5.1) and (5.2), a value of T = 0.20s was calculated for the PSP on the plug insert and used for the purposes of dynamic compensation.

Shown in Figure 5.1 are a sequence of 10 calibrated, but uncompensated, PSP images taken at equal time steps throughout the cycle. In terms of a temporal measure, the images were separated in time by 5ms per image. The image in the upper left-hand corner of the figure corresponds to the opening of the jet valve. Progressing from the top down and then to the top of the second column and down spells out a full cycle. The phase delays relative to the opening of the jet valve are shown on each image as a percentage of a full cycle. The gray-scale of the images has been set so that white corresponds to a pressure of 0.7atm and black is 0.1 atm. It can be seen that the area upstream of the jet (left of the jet) is generally a region of high pressure and that the area downstream of the jet (to the right) is a region of low pressure. From the sequence of 10 images, the upstream pressure indicates a peak pressure of between 0.55 and 0.60atm for phase delays of 10% to 50%. The minimum pressure downstream of the jet is between 0.25 and 0.28atm and occurs around a phase delay of 50%. The mean pressure far from the jet is on the order of 0.39atm. So, based on the uncompensated PSP images, the peak wall pressure is 0.1 9atm above the freestream and the minimum wall pressure is 0.13atm below the freestream. Note that the uncertainty in these values is approximately

0.007atm. So, the uncertainty is on the order of 5% of the absolute fluctuation.






85


Progressing now to the compensation data, the sequence of 10 compensated images is shown in Figure 5.2. These images were generated using the Matlab and Simulink codes presented in Appendices A and B. Running on a Pentium III 800MHz machine with 256MB of RAM, compensating a sequence of 10 images (I77pixels x 253pixels) took approximately 2hrs 1 5min to compile. In these compensated images we see the shock strength much clearer. Also note that the pressure very quickly dies down after the jet is shut off as compared to the uncompensated where it appears the downstream pressure never equilibrates. The pressure midway between the front edge of the jet orifice and the upstream pressure tap peaks at around 0.70atm at a phase delay of 0%. The pressure downstream of the jet reaches a minimum around 0.1atm almost immediately downstream of the jet at a phase delay of 20%. This gives a peak wall pressure 0.31 atm above the freestream and a minimum of 0.29atm below freestream. Another interesting feature is that at the initiation of the jet (0% phase delay) the whole of the image, both upstream and downstream, is seen to raise in pressure. This seems to indicate that in the initial instant of the jet, it is filling up the boundary layer in all directions, pumping it up to a higher pressure. Then, the gross flow is able to overcome this initial transient and set up a more clearly defined shock structure. This shock structure persists nearly unchanged until the jet closes. When the jet is then turned off, the flow nearly immediately adjusts, wiping out all evidence of pressure gradients.

Shown in Figure 5.3 is a comparison of the pressures along the streamwise

centerline through the jet for both the uncompensated and the compensated images. The centerline pressures from a set of 10 images were stacked into a 2D array. This 2D array was then duplicated and stacked twice yielding a three-cycle loop, thus giving a clearer






86


Figure 5.1--A sequence of 10 uncompensated PSP images taken at 5 ms intervals during a full cycle of 50ms. Grayscale is such that white = 0.7atm and black = 0.latm. Dimensions are from -4.5Dj to +5.5Dj in X and from -3.5Dj to +3.5Dj in Y.







87


Figure 5.2--A sequence of 10 compensated PSP images taken at 5 ms intervals during a full cycle of 50ms. Grayscale is such that white = 0.7atm and black = 0.latm. Dimensions are from -4.5Dj to +5.5Dj in X and from -3.5Dj to +3.5Dj in Y.







88








































Jet Orifice

Figure 5.3--A three-cycle loop of PSP data along the streamwise centerline through the jet. (a) Uncompensated; (b) Compensated. Grayscale is such that white = 0.7atm and black = 0.1 atm. The X-coordinate is oriented along the horizontal direction (positive to the right) and time is oriented vertically (positive up).




Full Text

PAGE 1

DYNAMIC MODELING AND COMPENSATION SCHEMES FOR PRESSURE SENSITIVE PAINTS IN UNSTEADY FLOWS By NEAL ANDREW WINSLOW 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 2001

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ACKNOWLEDGMENTS I would first like to acknowledge Dr. Joanne Bedlek-Anslow for her time and effort in applying the PSPs used in most of the experiments presented herein. Also, Dr. Paul Hubner is owed many thanks for the numerous times he acted as a soimding board for my various theories and off-the-wall ideas. I would also like to thank Troy Livingston for the use of his digital camera and for his help in acquiring the images of the experimental apparatus shown in this dissertation. Special thanks go Dr. Bruce Carroll, my advisor, for his extreme patience throughout my studies. Further appreciation goes to the other members of my committee. Dr. Kirk Schanze, Dr. Andrew Kurdila and Dr. Martin Morris. They have each helped me in many ways. Lastly, I wish to thank my parents. Without their encouragement, love, and support I would have never finished.

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TABLE OF CONTENTS page ACKNOWLEDGMENTS ii LIST OF TABLES v LIST OF FIGURES vi NOMENCLATURE ix ABSTRACT xi CHAPTERS 1 INTRODUCTION 1 Basic PSP Physics 1 PSP in Steady Pressure Fields 5 PSP in Unsteady Pressure Fields 7 Transverse Jets 13 2 PSP MODEL DEVELOPMENT 17 Empirical Models 17 Diffusion Based Models 20 Diffusion Based Model with Linear Calibration 25 Diffusion Based Model with Stem-Volmer Calibration 30 3 DYNAMIC COMPENSATION 36 Model Reference Control 36 Application of MRC to PSP System 40 MRC Simulations and Compensation of Experimental Data 44 4 EXPERIMENTAL SETUP AND PROCEDURES 52 PSP Dynamics Test Cell 52 Supersonic Wind Tunnel 56 Image Processing 61 Uncertainty Analysis 68 i iii

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5 RESULTS 81 6 CONCLUSIONS AND RECOMMENDATIONS 94 Conclusions 94 Recommendations 94 APPENDICES A MATLAB CODE: PSPMRCPARAMS 96 B SIMULINK CODE: PSPMRC 101 C PSP UNCERTAINTY ANALYSIS 1 03 REFERENCES 106 BIOGRAPHICAL SKETCH 110 iv

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LIST OF TABLES Table Page 1 . Sample PSP Indicated Pressures 27 2. Comparison of Curvefit Values 34 3. Calibration Functions, Constraints, and Typical Values 66 4. Nominal values and 95% confidence limits of key variables 69 V

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LIST OF FIGURES Figure Page 1.1 Schematic of a typical PSP layer 2 1 .2 Transverse injection flowfield schematic 14 2.1 Frequency response of a 16-|im-thick PSP sample 19 2.2 Comparison of PSP data and empirical model data 19 2.3 Simulated PSP unit step response using diffusion-based model with linear calibration. Notes: A) t*,; = 1.1 ; B) t* ,^ = Xlln^ ; C) t*„ = 1.8 28 2.4 Simulated PSP frequency response using diffusion-based model with a linear calibration 28 2.5 Comparison of PSP data and diffusion-based model output with a linear calibration 29 2.6 Comparison of calibrations 3 1 2.7 Simulated positive step in pressure (model comparison) 34 2.8 Simulated negative step in pressure (model comparison) 35 2.9 Positive step in pressure from experimental data (model comparison) 35 3.1 Structure of the Model Reference Control scheme 37 3.2 Structure of the Model Reference Control scheme for use as a compensator for PSP data 41 3.3 Linear compensation simulation of a positive step in pressure 45 3.4 Compensation simulations using nonlinear form of the PSP plant, (a) Positive step in pressure; (b) Negative step in pressure 46 vi

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3.5 Compensation simulations using nonlinear form of the PSP plant with a sinusoidal driving pressure, (a) co* = 1 ; (b) w* = 3 ; (c) co* = 10 ; (d) co* = 30 47 3.6 Compensation of experimental data, (a) Positive step in pressure; (b) Sawtooth driving pressure 49 4.1 PSP Dynamic Test Cell: A) Digital valve to vacuum, B) Pressure transducer, C) Digital valve to high pressure, D) Thermistor, E) Window to PSP sample, F) Cooling fins 70 4.2 Schematic view of the PSP Dynamic Testing Cell 71 4.3 Heat Exchanger Unit: A) Inlet, B) Outlet 72 4.4 Oriel Quartz-Tungsten-Halogen (QTH) Lamp 73 4.5 ISSI LED Lamp 74 4.6 Schematic of the PSP Dynamic Testing Cell Optics 75 4.7 Fisher Control Valves: A) 1 1/2" Valve, B) 1 " Valve 76 4.8 Stagnation chamber of the supersonic test stand facility 76 4.9 Mach 1 .56 Test Section: A) 3 of 8 Static pressure taps, B) 2 of 4 Thermocouples, C) Exterior reference marker, D) Bottom edge of circular plug insert 77 4. 1 0 Detailed front view of plug insert showing the central jet orifice and the 4 static pressure tap locations 77 4. 1 1 Detailed rear view of plug insert: A) 2 of 4 Pressure taps, B) Thermocouple, C) Digital valve controlling the jet 78 4.12 A) Xenon Strobe, B) Photometries CCD Camera 78 4. 13 Image Processing Flowchart 79 4. 14 Sample PSP Static Calibration 80 4. 1 5 Relative uncertainty in the indicated PSP pressure 80 5.1 A sequence of 10 uncompensated PSP images taken at 5 ms intervals during a fiill cycle of 50ms. Grayscale is such that white = 0.7atm and black = 0. latm. Dimensions are from -4.5Dj to +5.5Dj in X and fi-om -3.5Dj to +3.5Dj in Y 86 5.2 A sequence of 10 compensated PSP images taken at 5 ms intervals during a full cycle of 50ms. Grayscale is such that white = 0.7atm and black = 0. latm. Dimensions are from -4.5Dj to +5.5Dj in X and fi-om -3.5Dj to +3.5Dj in Y 87

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5.3 A three-cycle loop of PSP data along the streamwise centerline through the jet. (a) Uncompensated; (b) Compensated. Grayscale is such that white = 0.7atm and black = 0.1 atm. The X-coordinate is oriented along the horizontal direction (positive to the right) and time is oriented vertically (positive up) 88 5.4 Comparison of PSP data to pressure tap data, (a) Upstream of the jet; (b) Downstream of the jet; (c) Spanwise from the jet 89 5.5 Variation in the paint thickness near abrupt edges 90 5.6 Effective back-pressure ratio at the jet 93 viii

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NOMENCLATURE a = pressure sensitive paint thickness; Stem-Volmer intercept A = amplitude b = Stem-Volmer slope B = bias uncertainty c = calibration constants C = feedback gains of model reference controller D = diffusivity; denominator polynomial; diameter E = activation energy f = calibration function from pressure to intensity g = calibration fiinction from intensity to pressure G = transfer fiinction H = transfer fiinction I = integrated intensity j = the complex number, V-T J = momentum ratio; intensity per unit thickness k = rate constant; high frequency gain coefficient K = Stem-Volmer constant; intensity difference per unit thickness m = oxygen concenfration difference relative to surface condition M = Mach number n = oxygen concentration N = model order; numerator polynomial P = pressure Q = calibration parameter s = Laplace variable S = sample standard deviation t = time T = temperature; 1^' order system time constant u = imcertainty U = input to model reference controller V = velocity; intemal state of model reference confroller W = intemal state of model reference controller X = distance from subsfrate X = downstream distance from jet center Y = output of model reference controller; processed inverse intensity; spanwise distance from jet center Z = intemal state of model reference controller ix

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a = modal states = electrons per bit of the CCD A/D process Y = specific heat ratio A = relative degree of a transfer function or plant X = eigenvalues P = density o = solubility T = time constant = quantum efficiency; phase X = mole fraction = spatial eigenfiinctions (0 frequency Superscripts and Subscripts IT n — r ouner i ransiorm oi ii H = time derivative of H H = average value of H a = augmented C = compensated dik = dark level er = external reference ext = external marker ff = flat field j = jet in = input m = model P plant PSP = pressure sensitive paint q = quencher nr = non-radiative r = radiative ref = at reference conditions run = at run conditions out = output model = model output * = dimensionless 00 = freestream conditions 0 = initial 1 = final O.xx = at xx% response (e.g.: 0, X

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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 DYNAMIC MODELING AND COMPENSATION SCHEMES FOR PRESSURE SENSITIVE PAINTS IN UNSTEADY FLOWS By Neal Andrew Winslow August 2001 Chairman: Dr. Bruce F. Carroll Major Department: Aerospace Engineering, Mechanics and Engineering Science Pressure sensitive paints (PSP) have been used extensively in the past decade to measure surface pressures in steady flow conditions. There are many unsteady phenomena to which the aerospace community would like to apply PSP as a measurement technique. However, to date there has not been a viable model to describe the dynamic behavior of the PSP system. A rigorous development fi-om a systems dynamics perspective is needed to advance PSP fi-om a static measurement technique to a dynamic one. With this aim, two models for the dynamic behavior of pressure sensitive paints will be presented. The first of the two models is a purely empirical approach to designing a model and compensator. The second model encompasses the physics of the process by which an unsteady pressure field over the paint layer affects the layer and causes an intensity of fluorescence that is fluctuating in time. Within this second model, two different forms for the static calibration are chosen. The first results in purely linear xi

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system dynamics whereas the second yields a nonlinear system. These models are then compared to experimental data. Next, a dynamic compensation scheme is developed to correct for the time-lag and amplitude-damping behavior of the paint. Finally, the compensation scheme is applied to PSP data taken surrounding an unsteady (periodic) jet injected transversely into a Mach 1.56 freestream. xii

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CHAPTER 1 INTRODUCTION Pressure Sensitive Paint (PSP) has been an emergent technology over the past five to ten years. Many different groups have made contributions as is summarized in the reviews by Bell et al.,' Liu et al.,^ and Lu and Winnik.' Much success has been achieved in the application of PSP to steady pressure fields. In comparison, there are but a few papers discussing the application of PSP to relatively unsteady pressure fields. It is the goal of this dissertation to present a physically realistic model for the dynamic response of PSPs and provide a dynamic compensation scheme for PSP data analysis. With this done, a dynamic PSP experiment will be presented and analyzed using the previously derived dynamic model and compensation scheme. Basic PSP Phvsics ' PSP is a measurement technique that has its roots in photochemistry. It is based on the luminescence quenching effect of oxygen on luminescent molecules in a polymer binder. Shown in Figure 1.1 is a schematic of a typical PSP layer. The schematic shown here is for a PSP applied directly to the substrate without a primer layer. The reason for this is that primer layers will slow down the dynamic response of the PSP due to luminophore molecules diffusing into the less permeable primer. Because the primer generally has a lower oxygen diffiisivity than the polymer binder, the luminophores trapped in the primer will exhibit very slow dynamical responses. The photophysical process begins with the excitation. For most PSPs, the excitation is a light source (strobe. 1

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2 Excitation Luminescence Oxygen Q ^ — Polymer Binder ^ Luminophore Molecule O O Figure 1.1 --Schematic of a typical PSP layer. halogen, LED or laser) with high intensity in the blue to ultraviolet region of the spectrum. A given luminophore in its ground state can absorb a photon and transition to an excited state.'* Once in the excited state, the luminophore has primarily three competing routes to return to the ground state. ^ The first is referred to as "radiative decay" or "luminescence" which encompasses two phenomena fluorescence and phosphorescence. For ruthenium-based paints, fluorescence is the primary radiative decay route whereas for the platinum derivative paints it is phosphorescence. In both fluorescence and phosphorescence, the luminophore molecule emits a photon and drops back down to the ground state. This photon is generally in the orange to red region of the spectrum. The intensity of this emitted light is the measurement variable of PSP. The second decay route is called "non-radiative decay" in which the luminophore releases its energy to the surrounding polymer matrix in the form of thermal energy. The third route is the critical one that differentiates an ordinary fluorescent paint from a pressure sensitive paint. In this route, molecular oxygen that has permeated the layer may collide

PAGE 15

3 with an excited luminophore and absorb its energy. This third route is termed "oxygen quenching." Each of the three methods of energy release has an accompanying rate constant kr, knr, kqno2 respectively. The first two rate constants, kr and k„r, have units of 1/s . The third rate constant is composed of two parts, kq, which is a constant with units of mY(mol • s), and no2 which is the concentration of molecular oxygen in mol/m^ within the PSP layer. The quantum efficiency of PSP luminescence is given generally by The quantum efficiency is proportional to the observed PSP intensity, (|) = c • I , with c being a proportionality constant. If a ratio is made of the quantum efficiency in the absence of oxygen (n^^ = O) to that in the presence of oxygen (no2 O), the following relation is obtained: (b cl I To _ *^^o _ (|) ~ cl ~ I --^ = l + K^no2 (1.2a) where Equation (1.2a,b) is familiarly known as the linear Stem-Volmer equation. The equilibrium oxygen concentration within the layer is related to the air pressure above the layer via Henry's law^ "02 =f(Po2) = cPo2 =cXo2P = tJP(1.3)

PAGE 16

4 In the above equation, the Hnear form for Henry's law has been presented. The solubihty of gases in polymers is often modeled in this fashion.'' Substituting equation (1.3) into equation (1.2a) yields a relation between the luminescence intensity and the air pressure over the PSP layer, ^ = 1 + ctK,P. (1.4) In aerodynamic testing it is often not possible to measure the luminescent intensity in the absence of oxygen, lo. However, the intensity at some reference pressure, usually taken to be atmospheric, can be measured. Applying equation (1.4) at a reference condition, Iref, and at a general condition, I, and ratioing gives the "aerodynamic testing" form of the linear Stem-Volmer equation: I , 1 + ctK P p_o ^ = ^ = J_^. (1.5) I l + aK,P„, P„,-Q In this equation, Q = l/crK^ is the virtual pressure (negative absolute pressure) at which the luminescent intensity would become infinite, i.e., the singular point of the intensity fiinction. The parameter Q is a purely mathematical construct, but has an important purpose in relation to the use of equation (1.5). Often equation (1.5) is quoted in the form''^ ^ = a + b^. (1.6) However, in this form, both the slope, b, and the intercept, a, will be fianctions of Pref. In the form of equation (1.5), the parameter Q is independent of the reference pressure and is hence a more usable form. Typical values for Q are in the range of 0.1 atm to 0.2 atm . A sensitive PSP with a steeply sloped Stem-Volmer response will have a Q

PAGE 17

value closer to zero, whereas a less sensitive PSP will have a more negative value of Q. Solving for P in equation (1.5) yields the calibration for a given PSP as P = Q + (Pref-Q)YSo, by taking two measurements, one at a known reference pressure and another at an unknown pressure of interest, the intensities of the two measurements can be transformed into a single measurement of pressure. Today's modem CCD cameras have on the order of a million pixels. One can then achieve on the order of a million simultaneous pressure measurements distributed over a given model instrumented with PSP in a single image. The physical spacing between these measurements is essentially limited only by the magnification of the camera lens. Put another way, to achieve a denser set of measurements, one has only to zoom in tighter on a given region of interest. Using microscopes, sub-micron resolution can even be achieved. PSP in Steadv Pressure Fields ; • . ; Many experiments have been performed using PSP to measure pressures imder steady conditions. As such, there are several papers that review the existing literature.''^'' To summarize these reports, PSP has been used to measure standard aerodynamic loads on aircraft in both subsonic and supersonic regimes, turbomachinery such as fans and compressors, automobile wind tunnel testing, as well as numerous scientific experiments to investigate particular flowfields of interest. One representative study is that of Everett et al.^ In the paper of Everett et al., a sonic jet of air was injected transversely into a supersonic freestream at a Mach number of 1.6. The PSP was applied to the wall

PAGE 18

surrounding the jet orifice. The full-field wall pressures around the jet were measured at three different injectant pressures. The jet-to-crossflow momentum flux ratio, took on the values of J = 1.2, 1.7, and 2.2 for the three injectant pressures. In addition to the PSP measurements, shadowgraph and surface oil-flow visualization techiques were used to investigate the flow. The shadowgraph images were useful in identifying the shock structure generated as the jet altered the flow. As the momentum flux of the jet was increased relative to the fixed freestream, the size and strength of the shock structure were seen to increase as one would intuitively imagine. With the oil-flow, the streamlines along the wall were characterized. From these streamlines, the locations of the various shocks near the wall could be determined. Placing static pressure taps in strategic locations allowed simultaneous PSP and conventional pressure transducer measurements. Using these simultaneous measurements, an in-situ calibration of the PSP was performed. The calibration fiinction chosen related the indicated PSP pressure to a quadratic function in terms of Iref/I. Spectacular images of the surface pressure were achieved with a spatial resolution of 85. l|am/pixel. Integrating the pressure around the periphery of the jet gave a measure of the effective back pressure With increasing J, the effective back pressure was also found to rise. Because of this, it was concluded that the effective back pressure was not a constant ratio of either the static or stagnation pressure behind a normal shock in the freestream as was reported by other researchers. J = (1.8) (1.9)

PAGE 19

PSP in Unsteady Pressure Fields The earliest reference of PSP in unsteady situations is a paper by Cox and Dunn and contains the beginnings of a good model for the response of PSP to unsteady pressure fields.' In their paper they investigated oxygen transport within poly(dimethyl siloxane) (PDMS) via its effect on the fluorescence of the luminophore 9,10-diphenyl anthracene (9,10-D). The purpose of their work was to determine such physical quantities as the solubility, diffusivity and activation energies of the polymer PDMS. This was accomplished with a cuvette of PDMS (thickness, a = 1cm) into which was mixed the luminophore 9,10-D. The schematic shown in Figure 1.1 is similar to the geometry of the cuvette used in their experiments. The cuvette was allowed to come to equilibrium with pure nitrogen at 1 atm. Then, the nitrogen environment was replaced with pure oxygen at latm. Over the next 24 hours the fluorescence intensity at the center of the cuvette (x = 4.5mm to 5.5mm) was monitored. For PSPs, the thickness is on the order of I0[xm as compared to the 1cm thickness of the cuvette. The response time of a PSP will be shown to be proportional to the thickness squared. Thus, PSPs work on much shorter time scales than in ref 9. The fluorescence intensity within the cuvette was seen to fall off quasi-exponentially. With the use of a static calibration they were able to extract the indicated oxygen concentration as a fimction of time within the PDMS. They then developed an analytical model for the indicated oxygen concentration using the one dimensional diffusion equation. For a step change in oxygen concentration at the upper surface (x = a) the analytical solution for the oxygen concentration within the cuvette is n (x,t)-no ^^ 4y^( 1 ' " i=0 ^ exp -{2i + \y 4 a' sm (2i + l)f2 a (1.10)

PAGE 20

This model was found to accurately represent the experimental data. The regression analysis yielded values of diffusivity between D = 2.0x 10"'cm^ /s at T = 5°C and D = 6.1x lO'^cm^ /s at T = 45°C for oxygen in PDMS. However, in the general use of PSPs the coating is viewed from above rather than from the side as was done in this paper. That is, the intensity measured from a PS? is the integrated intensity of the entire layer. The integrated intensity as a fiinction of time for an arbitrary unsteady pressure field will be shown later to have a troublesome nonlinearity. The work of Mills and Chang'" was the first to look at the dynamic response of optical sensor films (to be later called PSPs). The response and recovery times of the optical signal (luminescence output) from the coating were measured. The response time was defined as the time required for the optical signal to equilibrate to a positive step in oxygen concentration, and the recovery time was the time required for equilibrium due to a negative step in concentration. It was shown that the difference in the response and recovery times was related to the hyperbolic fiinction between the fluorescence intensity and oxygen concentration. This paper was the first attempt at characterizing the dynamics of PSP and used a time-domain-based technique. A paper similar to that of Mills and Chang'" was that of Baron et al." Here, the inverse of the PSP intensity was compared to the pressure as the pressure was quickly stepped from near vacuum up to latm of standard air. In this work the assumption was made that the PSP responded as a first order dynamical system with up to two time constants. The given dynamical system was numerically convolved with the pressure signal from a fast conventional fransducer to yield a pressure that was compared to the pressure indicated by the PSP. Using nonlinear least-squares curve fitting the time

PAGE 21

constants and term weights were determined yielding an empirical model for the dynamic response of the PSPs. With these empirical models, they were able to get extremely good comparisons between curvefits and experimental data. The work of Borovoy et al.'^ was the first actual wind tunnel application of PSP in quasi-unsteady conditions found in the literature. In this experiment the pressure on a cylinder in Mach 6 flow was investigated. To achieve the Mach 6 flow, the cylinder was located perpendicular to the exit of a shock tube. It is stated that the 99% relaxation time of a PSP's fluorescence is given by the equation 12 a^ to.99=— — • (1.11) 71 D Once the shock had passed the cylinder, the timing system in the experiment would wait an amount of time given in equation (1 . 11) and then trigger a single flash from a xenon flash lamp. The PSP data were compared to an analytical solution of Euler's equations for nonviscous flow and found to agree to within 10%. It should be noted that this application was not truly an example of unsteady PSP, as the data were taken after the shock had already passed the cylinder and the PSP had been given sufficient time to respond to the step change in pressure. That is, the flow field was unsteady in time but not during the period over which the data were acquired. In separate, but concurrent developments, the response of PSP to periodic pressure fields was investigated by Engler'^ and Carroll et al.''' In Engler's work, the PSP was subjected to periodic forcing at frequencies from 0.1 Hz to 50Hz. The main goal of this investigation was the characterization of the dynamic range and the pressure resolution of the PSP. No effort was made here to perform dynamic compensation or to try to mathematically or physically define the dynamics of the PSP system. Carroll et al."*

PAGE 22

presented experiments similar in form to those of Engler, namely, the response of PSP to sinusoidal pressure fields. Much of the analysis performed here was done in the frequency-domain. It was shown that the PSP responds as approximately a first order dynamic system based on the amplitude response. The phase response of the PSP system did not match well to the standard first order system model. Carroll et al.'^ investigated the response of PSP to a step change in pressure. In this paper are the beginnings of a physically realistic model for the dynamic response of PSP. In summary, the paper reports on an analytical model based on the one-dimensional mass diffusion equation to describe the fransport of oxygen within the PSP layer. This model was nonlinear and was implemented numerically. In addition, empirical models based on a single-term first order response and on a two-term first order response were developed. Using nonlinear least-squares curvefitting the two empirical models were fit to the data. The single-term first order model was shown to inadequately fit the data. The two-term and diffusion-based models were both shown to have excellent agreement with the experimental data. An equivalent form of the diffusion-based model will be derived in its entirety in a later section. * ! Winslow et al.'^ developed a linearized dynamics model for PSP based on the assumption that the luminescence intensity from the paint is proportional to the average oxygen concentration within the layer. The frequency response of this model is that of a "1/2-order" dynamic system with an amplitude response of-lOdB/decade above the cutoff frequency and a high frequency phase shift of -45°. Curvefitting this model to PSP data in the frequency domain gave a value for the time constant of the layer. A dynamic compensator was then developed by inverting the frequency domain response

PAGE 23

11 and performing an inverse Fourier transform. The compensator took on the form of a six term finite impulse response (FIR) filter. Applying this compensator to experimental data yielded an impressive increase in the dynamic response of the PSP. This was the first instance of dynamic compensation of PSP data. Nearly concurrent to the paper of Carroll et al.'^ was that of Masoumi et al." Effectively, the same dynamical model was used in both works. In the latter work, the luminescence response of PSPs to positive and negative steps in pressure was examined. The coating thicknesses were on the order of 300|im. Upon performing a curvefit of their dynamics model to the data, values for the diffusivity of 1.0 x 10"^ cm Vs for platinum octaethylporphine (Pt(OEP)) in poly(n-butylaminothionylphosphazene) (PATP) and 1.4x10"' cm / s for ruthenium biphenyl phenanthroline chloride (Ru(ph2phen)3Cl2) in PATP were obtained. Carroll et al.'^ reapplied the dynamics model of Winslow et al.'^ to investigate the diffusivity of PSP coatings as a function of temperature. A PSP coating was subjected to step increases in pressure over a range of temperatures. From the time constants measured at the different temperatures, values for the mass diffusivity were calculated. Then, modeling the diffusivity using an Arrhenius relation D = Dp exp (1.12) rt; a value of 2.48kcal/mole was found for the activation energy, E, of the coating. This value compared well with values fi-om other sources. Hubner et al.'^ performed PSP experiments on a flat plate and a wedge in a shock tube. A first series of tests used a flat plate painted with PSP, a continuous illumination

PAGE 24

source, and a PMT to measure the integrated light output from the whole of the plate. As the initial shock passed the plate, the PSP was subjected to a step in the pressure. When the reflected shock came back past the plate, the pressure stepped up even more. From this data, the model of Winslow et al.'^ was used to calculate a time constant of 3.52ms for the coating. A second series of tests involved a wedge obstructing the flow. A photographic flash lamp was used to excite the paint and a CCD camera acquired a single image for each run of the shock tube. The results showed a pressure gradient on the wedge due to the effective area reduction within the duct. Overall, it was demonsfrated that thin films of PSP could be used to measure pressures in short-duration facilities. The final paper to be discussed here was from Hubner et al.^° Pressure and temperature measurements were made on an elliptic cone model in Mach 7.5 flow. The tests were performed at the 48-inch hypersonic shock tunnel at Calspan-University of Buffalo Research Center. Due to the short duration of the flow, a method similar to that of Borovoy et al.'^ was used wherein the PSP was allowed to come to equilibrium with the flow and then a photographic strobe was used to illuminate the paint. The data showed that the pressure on the surface of the model ranged from 3 to 12 kPa. A second set of tests was performed with a high frame rate, 8-bit camera. In these tests the camera acquired several images of a temperature sensitive paint (TSP) coated model during a single flash from the strobe. The temperature data from these tests were used to compute heat transfer rates on the surface of the model. The results compared well to measurements made with thin-film heat transfer transducers. As can be seen in the literature to-date there have been many papers investigating sinusoidal or step-like pressure fields, but none that have presented a PSP dynamics

PAGE 25

model in a fashion for use with arbitrary pressure fields. It is the goal then of this dissertation to present a sound mathematical and physical model for PSP dynamics. Moreover, a compensation scheme will be designed based on the dynamics models to correct for the amplitude-damping and phase-shifting behavior of the paints. Finally, a ftiU-field dynamic experiment of PSP will be presented along with the results of the compensation scheme having been applied to the data. Transverse Jets The study of a jet of air injected transversely to a supersonic freestream is a classical problem that has been the subject of many studies. A schematic of the general flowfield is shown in Figure 1.2. The main purposes for which this flowfield have been studied is in relation to thrust vector control and to ftiel injection ports in scramjet engines. For thrust vector control, the integrated wall pressure is of primary importance. For scramjet engines it is important for the fiiel (injectant) to become fiilly mixed with the freestream air in as short a time/distance as possible. Additionally, because the injectant is to be burned downstream, it is necessary to push the injectant as far from the wall area as possible to reduce heating of the wall. Hence, the goal of many studies has been on determining the wall pressure distribution or on finding optimum parameters to increase mixing or to maximize penetration of the jet. What follows is a brief introduction to the existing literature on the evolution of transverse jets in supersonic crossflow. The first work to be published on the transverse jet in supersonic crossflow came from Cubbison et al. out of the NASA Lewis Research Center in 1960.^^ In their paper, extensive wall pressure measurements were made surrounding a 0.062" jet. In this study

PAGE 26

INJECTAOT Figure 1 .2~Transverse injection flowfield schematic (Gruber et al.^'). there were basically three independent parameters which could be varied jet pressure, freestream Mach number, and pressure altitude. The range of jet pressures was from 50 to 440 psig. Three freestream Mach numbers of 2.92, 4.84 and 6.4 were investigated. The pressure altitude was fiirther varied from 55,000 to 1 15,000 feet. This range of pressure altitudes combined with the range of Mach numbers resulted in a range of Reynolds numbers between 0.84x10^ and 7.78x10^ per foot. The wall surrounding the jet was instrumented with on the order of 100 pressure taps arrayed upstream, downstream and spanwise relative to the jet orifice. The general flowfield resulted in a high pressure region immediately in front of the jet orifice and a low pressure region behind the jet. The results of this study indicated that the jet-to-freesfream pressure ratio had a large effect on the pressure distribution on the surface surrounding the jet. The pressure distribution was also affected by changing the freestream Mach number at a constant

PAGE 27

15 pressure ratio. The effect of the Reynolds number was small compared to the preceding two effects. In a later examination by Schetz et al. the location of the center of the first Mach disk was correlated to a pressure ratio between the jet pressure and an "effective back pressure." This effective back pressure was an analogy to the back pressure seen by a jet issuing into a quiescent medium. The value chosen for this effective back pressure was defined as 80% of the static pressure behind a normal shock in the fi-eestream. A paper by Orth et al.^'' looked at the effect of changing the shape of the injector port as well as the angle of injection into the freestream. A control volume analysis of the fluid in the jet was also presented. In this study, an effective back pressure equal to 2/3 of the fi-eestream stagnation pressure was used. It was argued that the effective back pressure should be the average pressure field around a cylindrical obstruction on a flat plate. So, the definition used here differed from that of Schetz et al.^'' Several experiments were performed to better understand the factors affecting penetration of the jet into the freestream. A first series of tests used a circular-ended rectangular slot oriented along the flow direction and perpendicular to the flow as well as an equal area circular port as the injector shape. These tests were run at a Mach number of 2. 1 and a jet to back pressure ratio of 5.23. It was found fi-om these tests that the shape of the injector port had no real effect on the penetration of the jet as was originally hoped. A second series of tests was a parametric study of the injectant pressure and Mach number. It was found that the distance from the wall of the center of the first Mach disk was related to the jet-to-freestream pressure ratio.

PAGE 28

Billig et al. and Heister and Karagozian performed detailed vortex modeling of the jet/crossflow interaction. These models were used to predict the effects of varying the freestream Mach number, the velocity of the jet as it exits the wall, and the Mach number of the jet in relation to the amount of penetration. It was determined that the primary variable of interest for enhancing penetration was increasing the jet-to-crossflow momentum ratio (see equation 1.8). As new measurement techniques were developed, new visual studies of the jet in crossflow problem were examined. Among the measurement techniques employed are planar laser-induced fluorescence (PLIF), ' classical Schlieren images, planar Mie scattering,^' and laser Doppler velocimetry (LDV).^° Each of these methods was used to primarily to investigate penetration and spreading of the jet. The latest measurement technique used to attack the jet in crossflow problem has been PSP as was described in the earlier section "Static PSP." It has been shown by the works of Johari and Paduano,^' Hermanson et al.,^^ and Johari et al. that both penetration and mixing can be enhanced through the use of a pulsed or modulated jet in crossflow. It should be noted that these studies were all performed in an incompressible medium as opposed to the compressible flows we have previously discussed. However, the analogy may still stand. If this is the case, it should be of interest to investigate a pulsed jet in supersonic crossflow. This then will be the form of the dynamic flowfield to which unsteady PSP measurements will be tested. The results of these tests will be presented in a later chapter of this dissertation.

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CHAPTER 2 PSP MODEL DEVELOPMENT The most important step in performing a dynamic analysis of PSP data is model selection. What follows are two different approaches yielding models increasing in design complexity and accuracy. The first approach is purely empirical in nature. The second is based on solving the one-dimensional mass diffusion equation in conjunction with the relation between intensity and pressure under static conditions. Empirical Models Perhaps the easiest model that can be developed is one which does not even attempt to understand the physics of the system in question, but rather, simply measures the input and output of the system and uses the data as the characterization. This approach is often referred to as the "black box" approach. All one must do is define the input/output mapping for the system as a function of frequency in order to create a crude compensator to correct for the amplitude damping and phase shifting behavior. The only underlying assumptions to this type of model are that the system being modeled must be linear and time-invariant. That is, for an input given by Pin(t) = Po+P,sin(cot). (2.1) the output from the system must be of the form Pou. W = Po + P, A(co)sin(tot + (t)(co)). (2.2) 17

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18 In the above equation A and ^ are the ampHtude ratio and phase shift, respectively, of the output relative to the input and are functions of the driving frequency co. Equations (2.1) and (2.2) can be more easily related to one another in the frequency domain via Pou. M = HO«)Pi„ M = A(co)e-J*('»)p,„ (jco) . (2.3) Shown in Figure 2.1 is the frequency response of a 16|im thick PSP sample based on input/output data. This data was acquired using the experimental setup used by Winslow 16 18 et al. and Carroll et al. The general uncertainties of the measurements are ±0.1 kPa for the conventional pressure transducer and ±2% of the reading for the PSP measurements. Based on the experimental data, a transfer fiinction of the form (2.4) 1 + J(0X2 was chosen to model the system. This particular form of transfer fiinction is that of a lag system. The transfer fiinction was applied to the input data, Pin(t), from a high frequency response pressure fransducer using the Matlab computer software to achieve a model output pressure, Pmodei(t). The parameters xi and T2 were varied until the root mean square error (RMSE) between the PSP-indicated pressure, Pout(t), and the model output was minimized in the time domain. The resulting values of ti and X2 were 0.012 and 0.15 seconds respectively. The model fransfer fiinction is also shown in Figure 2.1. Figure 2.2 contains the time domain data for this experiment. Shown in the figure are the PSPindicated pressure, the pressure measured by the conventional transducer, the model output data and the residual between the PSP indicated pressure and the model output. The absolute error between the PSP data and the model output data is less than 2 kPa (RMSE = 0.97 kPa). As can be seen, the transfer fimction of equation (2.4) does a

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20 reasonably good job of approximating the PSP system dynamics. However, the parameters of this curvefit are dependent upon the thickness and chemistry of the PSP layer in an unknown manner. That is, if the thickness or formulation of the PSP were changed, a different transfer function would result. Hence, the empirical approach is useful for a single sample, but does not transfer well to the type of experiments one might actually want to perform. Therefore, a more thorough imderstanding of the physical processes involved in the dynamics of PSP is required. Diffusion Based Models As was mentioned earlier, PSP is based on the luminescence quenching effect of oxygen on luminescent molecules in a polymer binder. This model tracks the process by which an unsteady pressure above the coating affects the oxygen distribution within the PSP and in turn causes the intensity of luminescence to change in time. Under static conditions, the oxygen concentration within the layer will be constant and related to the pressure over the surface via a linear sorption law (Henry's Law) n = ctP . (2.5) Also under static conditions, the intensity of luminescence is related to the pressure over the surface via a static calibration I = f (P) (2.6a) or P = g(l) (2.6b) due to the luminescence quenching effect of oxygen within the layer. Note here that the functions f and g are inverse functions. For the moment a generic relation between pressure and intensity is assumed. Later, specific forms for the static calibration function

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21 will be used to complete the system dynamics. Letting l' =I/Iref and P* = P/P^f the calibration equation can be brought into a dimensionless form r=f(p*). (2.7) In the above equation, the non-dimensional integrated intensity I* can be replaced by the dimensionless intensity per unit depth, J* , by noting that T' I I/a J I =r= rV = -i= J • (2.8) Also, pressure can be replaced with oxygen concentration by noting that P aP n . P = = = = n . (2.9) A static calibration equation relating the intensity per unit depth to the oxygen concentration can then be given as J*=f(n'). (2.10) Neglecting any internal light attenuation, this equation also holds on a local level, that is, at any point within the layer (Cox and Dunn^). Therefore, equation (2.10) can be used within the layer by considering both J ' and n* to be functions of depth, x. In addition, this equation holds as a function of time assuming that the kinetics of the luminescence and quenching processes are much faster than the kinetics of the diffusion process. The kinetics of the luminescence and quenching processes are on the order of microseconds for the ruthenium-based paints (Bell et al.,' Liu et al.,^ Schanze et al.,'^ Bums and Sullivan^") and on the order of lO's to lOO's of microseconds for the platinum-based paints (Bell et al.'). Then, since the kinetics of the diffusion process are on the order of milliseconds to seconds for typical coating thicknesses (Baron et al.", Winslow et al.'^

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22 Carroll et al.'^ Hubner et al.'^), the assumption is indeed valid. Restating equation (2.10) with the new dependencies one observes that the static calibration provides a dynamic relation between intensity at a given depth within the PSP layer and the oxygen concentration at that depth, J*(x,t)=f(n*(x,t)). (2.11) The next step is to solve for the oxygen concentration as a function of depth and time for a given surface pressure input, (t) . This will be done using the one-dimensional mass diffusion equation with boundary conditions and initial conditions of a'n 1 an n(a,t) = aPjt) (2.12b) an(o,t) ^ -^ = 0 (2.12c) n(x,0) = aP.„(0). (2.12d) The governing equation, (2.12a), assumes that the PSP layer is infinite in the transverse directions and that the diffusivity is constant. The first boundary condition, equation (2.12b), makes the assumption that the kinetics of the sorption process are much faster than the kinetics of the diffusion process so that the surface of the PSP is in equilibrium with the air above. The second boundary condition, equation (2.12c), represents a nonpenetration condition at the substrate. The initial condition, equation (2.1 2d), states that the oxygen concentration across the layer is initially constant (i.e. in equilibrium initially). The set of equations (2. 12a-d) or their underlying assumptions have been used in many other analyses of the solubility and difftision of gases in polymers.''-'"''^"'^'^^

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23 These equations can be non-dimensionalized by letting x' = x/ a, t* =Xlx where T = a V D , and making use of equation (2.9) to arrive at n*(l,f)=P:(f) (2.13b) 5n*(0,t') ^ J^: ^ = 0 (2.13c) n*(x*,0)=P*(0). (2.13d) In PSP measurements, the luminescence intensity is the quantity actually measured. Thus it is instructive to recast equations (2.13a-d) in terms of intensity. Inverting equation (2.1 1) to get n*(x*,t')= g(j*(x*,t*)) and substituting into equations (2.13a-d) yields ^* (5x7"" 5g/ar (2.14a) ax j r(i,t-)=f(p:(t-)) (2.14b) aj*(o,t*) , -^ = 0 (2.14c) J-(x',0)=f(p:(0)). (2.14d) Equations (2.14a-d) represent a nonlinear partial differential equation with nonhomogeneous boundary conditions describing the intensity distribution as a function of depth and time. Notice that recasting the formulation in terms of intensity, J* , instead of oxygen concentration, n* , introduces a nonlinear term (last term on right-hand side of 2.14a). The problem can be restated via a change of variables K(x-,t-)=r(x-,t-)-f(p:(t-)) (2.15)

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24 to achieve a nonlinear partial differential equation with homogeneous boundary conditions and initial conditions ydx J at* ^* (ax*)" 5g/aj* K(l,t*)=0 ax* k(x',o)=o. The output of a PSP measurement is the integrated intensity (neglecting internal absorption) i:,(t*)=jj*(x*,t*)dx*=|K(x*,t*}lx*+f(p*(t*)). (2.16a) (2.16b) (2.16c) (2.16d) (2.17) The integrated intensity is then related to the indicated pressure through the static calibration, P:.(t*)=g(i:„,(t*)). Equations (2.16a-d) are an extremely important set of equations representing the difference between the intensity output by the uppermost layer of the PSP and the intensity output from a given depth within the PSP layer. The term on the left-hand side (LHS) and the first term on the right-hand side (RHS) of equation (2. 16a) are both linear terms. The last term on the RHS of that same equation is a forcing term. The middle term on the RHS is in general highly nonlinear and contains a forcing part as well. Because of the nonlinearity of this term it is not possible in general to apply classical controls methods as will be shown later. Using two different assumed calibration functions and modal analysis, a system model now will be developed and simulated.

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25 Diffusion Based Model with Linear Calibration The first calibration function to be investigated will be l'=f(p')=Co+c,P' (2.18a) or P*=g(I*) = ^-^. (2.18b) c, This form of calibration is not typically used in PS? analysis; however, it has one important feature. Using a calibration which is linear between intensity and pressure causes the (normally) nonlinear term from equation (2.16a) to drop out yielding aK(x-,t-) a'K(x-.t-) af(p:(t-)) St~ (ax7 awhich is now a linear partial differential equation with forcing. The solution to this equation will be assumed via separation of variables as K(x-,t-)=|;a,(t>,(x-) (2.20) i=l where (x* ) are sine or cosine functions. This form of solution is based on a method called modal analysis and is often used in linear systems dynamics modeling. The spatial eigenfiinctions ^i{x') must satisfy the boundary conditions (2.16b,c) giving 4'i(x')=cos(XiX*) where =(2i-l)^. (2.21) Substituting into equation (2.19), multiplying by 'Pk(x') and integrating across the thickness of the layer gives d,(.-)=-X;a,(t-)-2^)f(p;(,-)) (2.22)

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26 Substituting equations (2.20) and (2.21) into equation (2.17) yields the ou^ut C(.-)=Za,(,-)^+f(p-(.-)). (2.23) i=l '^i In implementation the infinite sum of equation (2.20) will be truncated after some number of terms, N. After truncation, equations (2.22) and (2.23) can be written in the standard linear systems form / : 0 0' f : 0 -^^ 0 + V • J 0 0 V • 0 -2sm{h,)/X; f(p:.(f))' (2.24a) C(f)= sm V(p:(.-))J (2.24b) This system is a multi-input, single-output system with inputs of f (p* (t* )) and f (p* (t' )), state vector (t* ) and output C (t* ). A useftil property of this model is that the system matrix is diagonal with distinct, nonzero eigenvalues. The usefiilness of this property is m calculating the observability and controllability matrices. It can be shown that these matrices are both fiill rank, thus implying that the system is both observable and reachable. These two properties should allow a simple compensator to be constructed. There is one additional point of interest concerning this system. Since it is a linear system, and the relation between intensity and pressure is linear, equations (2.24a,b) can be rewritten via a linear transformation as a system between the input pressure and the indicated output pressure / : 0 0" f : •) 0 0 a.(l + V • J 0 0 V • ) 0 -2sin(Xi)/Xi p-(.-)(2.25a)

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p:.(t*)= sin(A,j ) f : \ a,(f) 27 (2.25b) It can be shown that the above output is identical to the solution for n'(x*,t') integrated across the layer, that is, I Po*u.(t*)= Jn*(x',t')dx* (2.26) 0 where n*(x',t*) is obtained by solution of equations (2.13a-d). This system is then identical to that used by Winslow et al.'^ and Carroll et al.'^ For a number of inputs P* (t*) the output from the system is analytic. Table 1 below lists several of these solutions (Carslaw and Jaeger^^). Table 1 -Sample PSP Indicated Pressures Pq +P,*step(t') Po* +P,*sin(a)'t*) exp \2 71 (2i + iy-t ' 4 . . tanh(p) . / . .\ „ j—r . Po +Pi — ^sm(co t j, P = Vjco , CO = Note that the function step(t), sometimes referred to as the unit step function, is zero for t < Oand unity for t > 0. Also note that the solution shown for the sinusoidal input is only that due to the steady periodic conditions, not the initial transient period. Shown in Figure 2.3 is the simulated step response using the fu-st 1, 2, 4, and 20 terms from the analytical solution of Table 1. Labeled on the graph are several points

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28 0.8 0.6 D. 0.4 0.2 Figure 2.3--Simulated PSP unit step response using diffusion-based model with linear calibration. Notes: A) t;,^ = 1.1, B) t;,, = I2/71' , C) t;^ =1.8. 5 0 o -10 1 -15 Q. I -20 -25 -30 1 1 1 II 1 I I 1 1 1 1 11 11 1 III Amplttude Phase 10 -2 ' 10^ 10 -10 -20 10" (OT 10^ 10^ o CO -30 0) (A (0 .C -40 °-50 -60 Figure 2.4--Simulated PSP frequency response using diffusion-based model with a linear calibration.

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29 re Q. 3 (A Q. 100 80 60 40 20 0 -20 C) o PSP Transducer A Model 0 Residual ^OOOOOQOqqqqqOOOOOqqOOO q 2 3 Time (sec) 6 Figure 2.5--Comparison of PSP data and diffusion-based model output with a linear calibration. throughout the response. It can be seen that the time to reach 99% of the total response is approximately 1.80t as compared to the value of (1 2/71^)1 put forth in the paper by Borovoy et al.'l It can also be seen that a model order of N = 5 should be sufficient to represent most of the dynamic behavior. Figure 2.4 shows the simulated frequency response of the PSP. There are two interesting items of note here. First, the amplitude slopes off at -10 dB per decade. Second, the phase shift levels off at ^5°. Following the trend of l" and 2"** order systems it would then seem that this system exhibits the behavior of a '"V" order system, which has some meaning in the context of fractional calculus in which derivatives and integrals of rational order are defined. In the text by Oldham and Spanier" on this subject, the relation between many diffiision-based phenomena and the 14 order

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30 derivative, or semiderivative, is shown. An attempt has been made to express this current model using fractional calculus, but no clear results have yet been achieved. Figure 2.5 shows a set of experimental input and output pressure data. Also included in the figure is the model output pressure based on the linearized diffusion model of equations (2.25a,b) with N = 5 . The value of the time constant that minimizes the RMSE between the PSP indicated pressure and the model output pressure is T = 0.437 . The RMSE of the curvefit is 0.91 kPa which is a very good fit. The thickness of this particular sample was 17 ^m. Using x = a^/D a value of D = 6.6 x 10"^ cm^/s is acquired for the mass difftisivity, which compares well to values fi-om other sources.^'^'^^ Diffusion Based Model with Stem-Volmer Calibration The most widely used calibration for PSP is based on the Stem-Volmer relation. Taking equations 1.5 and 1.7 and letting Q* = Q/P„f we get •=f(''')=fr^ , . <2.28a) or P*=g(r)=Q-+(l-Q')(l/r). (2.28b) In the above equations the parameter Q* uniquely determines the static calibration. Figure 2.6 illustrates the basic difference between the linear calibration of equafion (2.18) and the Stem-Volmer calibration of equation (2.28). Note that this figure shows the intensity versus pressure, as opposed to the inverse of intensity versus pressure which is normally presented. Applying equation (2.8) to equations (2.28a,b) and substittiting into equation (2.16a) gives

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31 Pressure Figure 2.6~Comparison of calibrations. dK^ d'K 2 [dK]' af(p'(t')) 5t*"(ax7"K+f(p:(t-))U*J " a• ^^-^^^ This is the governing equation for the PSP dynamics based on the Stem-Volmer calibration. It can be seen that the middle term on the RHS is highly nonlinear and contains a forcing component as well. Because of this term, a numerical solution to this equation would require a complex numerical algorithm to be developed. However, rather than perform a direct solution to this equation, a different method will be employed. It will begin by solving for the oxygen concentration from equations (2.13a-d). These equations can be transformed to a linear partial differential equation with forcing with homogeneous boundary conditions and initial condition via m(x',t')=n*(x',t*)-P.:(t*). (2.30)

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32 Then, modal analysis can be applied (see equations (2.20) and (2.21)). The resulting system equation is identical to equation (2.25a). The system equation together with an output equation designed to give the oxygen concentration distribution is then r : ^ 0 0" f : 0 0 •) + \ '• ) 0 0 \ '• J 0 -2sm{X.,)/X; •>•(.) m (2.31a) n'(x',t")=(--,cos(>.|X'), • a,(f) (2.31b) This system is linear and can be easily solved at a discrete number of points throughout the layer to achieve an oxygen concentration distribution. This oxygen concentration distribution will then be converted to an intensity distribution by using the calibration in a dynamic sense, (2.32) n (x ,t j-Q It may be helpful for the reader to refer once again to the discussion leading from equation (2.7) up to equation (2.1 1) for the validity of equation (2.32). Then, the intensity distribution can be numerically integrated to get the integrated intensity output. Making use of the static calibration of equation (2.28b) achieves the indicated output pressure. Figures 2.7 and 2.8 show the simulated response of PSP to a positive and a negative step in pressure respectively. For the positive step, the initial pressure was Pq* = 0.1 , the final pressure was P* = 1.0 and the value of Q* was Q* = -0.25 . For the negative step the values of Pj and P,* were reversed. Shown in the two figures are

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comparisons of the two models based on the Hnear caHbration and the Stem-Volmer caHbration. For the datasets based on the hnear caHbration, a model order of N = 5 was used. For the data based on the Stem-Volmer calibration, a model order of N = 5 was used to calculate the oxygen concentration distribution. This was evaluated at 21 evenly spaced grid points throughout the depth of the layer. Next, it was converted to intensity per unit depth via the calibration equation and numerically integrated across the layer using the trapezoidal rule integrator. The integrated intensity was then converted to an indicated pressure by means of the calibration. From Figure 2.7 it can be seen that for a positive step in pressure, the linear model solution leads the Stem-Volmer model solution. In Figure 2.8 this behavior is reversed. There are several items of note with respect to these comparisons. First, the difference between the models is a function of each of the values Pj, P' and Q* . If the pressure data were scaled via the ratio (P* Pq )/(Pi* Pq* ) the response of the Stem-Volmer based model would continue to show different behavior for different values of Pj , P,*, and Q* whereas the linear calibration based model would always show the same response. Second, as AP of the step decreases, the two models will collapse to the same behavior. Similarly, the models will also collapse to the same behavior for Q' -oo . Figure 2.9 shows experimental input and output data for a positive step in pressure. Also shown is a comparison of curvefits of the two preceding diffusion-based models. The results of the curvefits are shown in Table 2. For the model based on the Stem-Volmer calibration, a value of Q* = -0.163 was used. It is quite clear that the model using the Stem-Volmer calibration does a much better job of approximating the PSP response than the model based on the linear calibration.

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Table 2~Comparison of Curvefit Values Calibration T (sec) RMSE (kPa) Linear 1.25 4.10 Stem-Volmer 1.11 1.39 Figure 2.7--Simulated positive step in pressure (model comparison).

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35 0 0.5 1 1.5 f Figure 2.8--Simulated negative step in pressure (model comparison). (0 a. • 3 (/> CO Q. Transducer PSP A Stsm-Volmer Cal. o Linear Cal. 0 0.5 1 1.5 Time (sec) Figure 2.9--Positive step in pressure from experimental data (model comparison).

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CHAPTER 3 DYNAMIC COMPENSATION The task of designing a compensation system for PSP is comprised of two steps. First, a model must be developed for the response of the PSP to arbitrary dynamic pressures. Such models for the PSP dynamics have already been presented in the previous chapter. The second step makes use of the PSP model in combination with feedback control to generate a compensated output. This chapter will first show a complete presentation of the general compensation scheme known as Model Reference Control (MRC). Then, it will progress into the specifics of the application of MRC to PSP. Finally, the chapter will wrap up with theoretical simulations and compensation of some of the experimental data presented in the previous chapter. Model Reference Control Model Reference Control is a generalized scheme in which one can take a dynamical system (plant, Gp) with known or modeled parameters and with the use of feedback control, give the resulting system (model reference, Gm) a desired type of dynamics. The system shown in Figure 3.1 is similar to that presented in standard Adaptive Controls texts with regard to MRC with two exceptions. First, there is not a constant gain feedback term from the output Y. It is stated in an exercise later in the presentation of loannou and Sun that such a term is unnecessary. Second is the addition of the gain 1/kp in the feedback from Y. This addition was found to simplify some of the later analysis. 36

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37 U Y Figure 3.1 --Structure of the Model Reference Control scheme. The MRC design begins with a set of assumptions (either known or given):''^ Al . Gp (s) = kpNp (s)/Dp (s) is a known transfer function. A2. Np(s) is a monic Hurwitz polynomial of degree np. A3. Dp(s) is a monic polynomial of degree dp. A4. The relative degree of Gp(s) is = dp np _ A5. Gm (s) = k„,N^ (s)/D„ (s) a user-defined transfer function. A6. Nm(s) and Dm(s) are monic Hurwitz polynomials of degree Um and dm respectively, where dp . A7. The relative degree of Gni(s) is = d„ n^ ^ ^here ^m=\. In the above assumptions, the term "monic" means that the coefficient of the highest order polynomial term is unity. The term "Hurwitz" implies that the roots of the specified polynomial all lie in the lef^ complex half-plane. As a result of the assumptions (Al-7), we will also define H(s) = N(s)/D(s) , where N(s) = (s"'"', s'''-\ ... , l)^ and D(s) is an as-yet undetermined monic polynomial of degree d = dp . Notg that H(s) is a singleinput, multi-output system. Analyzing the system of Figure 3.1 yields

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38 ° " " (3.1) U DDp-DpC,N-NpC,-N where the Laplace variable, s, has been dropped from the notation for simplicity. It is desired by the user for the Model Reference System (Closed-Loop System) to have the dynamics Y , N^(s) 77 = ^^«.TrTl(3-2) U "Djs) Equating the systems gives CokpNpD k„N, m m DDp-D^C,.N-N^C,N D„ (3.3) An inspection of the polynomials on the left-hand and right-hand sides reveals that the numerators and denominators are all monic polynomials excepting the values Co, kp and km. Therefore, equating these constant gains gives Cm 0 = — • (3.4) If we now let kp D = DoN„, (3.5) then equation (3.3) can be re-arranged into the form DpC, • N + NpC, • N = D„ (n^D^ N^D„ ). (3.6) This equation will be termed the "Matching Equation" and will be used to calculate the elements of the unknown vectors Ci and C2. Before proceeding to this solution we must first define the polynomial Dq. In general it is best to choose the roots of Do to be on the

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39 order of the roots of Dm to keep the feedback gains at reasonable levels. With all of the variables in equation (3.6) having now been defined except for Ci and C2, we can proceed to a solution. For the analysis presented here, we shall assume that we are dealing with a plant, Gp, with the property np = dp 1 , or, Ap = 1 . Assumption (A7) of the MRC scheme requires then that = 1 . The simplest dynamical system that has this property is a 1^' order system T s + X where T is the time constant of the desired system. Let us for a moment consider a plant with dp = 3 . Let us also assume the following notations C, -N = a,s^ +a,s + a. (3.7) N =Y2s' +Y,s + Yo Dp =63S +52S +5,s + 5o (3.8a-e) Do(N„Dp -NpD„)=S5s' +ey+s,s' +S2S' +8,s + eo Ifwe now define x = (a2,a,,ao,P2,p,,pJ^ and b = (85,e4,e3,82,e,,eo)'", the Matching Equation can be transformed into a set of standard linear equations ( Ax = b ) "S3 0 0 0 0 0" 'a,' £5' S2 S3 0 Y2 0 0 a, £4 s, S2 S3 Yi Y2 0 «0 £3 So s, S2 Yo Yi Y2 P2 82 0 So s, 0 Yo Yi P, El 0 0 So 0 0 Yo. (3.9) Some notes regarding equation (3.9):

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40 B 1 . Column 1 of matrix A is simply the vector of coefficients of the polynomial Dp(s) with trailing zeros added to bring the length of the vector up to 2dp. B2. Column 2 of A is the same as column 1 but with the last element rotated around to the top of the vector. Similariy for column 3 with respect to column 2. 33. Column 4 of A is the vector of coefficients of the polynomial Np(s) with one leading zero and with trailing zeros added to bring the length of the vector up to 2dp. B4. Columns 5 and 6 of A are 1 -element rotations of columns 4 and 5 respectively. From this example, the general pattern for the matrix A and vectors x and b for a plant of degree dp can be extrapolated. Solving this system of linear equations gives the values of the tti's and Pi's which are in turn the elements of the vectors Ci and C2. Application of MRC to PSP Svstem The purpose for which the Model Reference Control scheme will be used, with regard to PSP, is not one for which it was intended. That is, the MRC scheme will be used as a "State Observer". Considering a PSP plant with input P(t) and output Ppgp (t), we desire to have a measurement of the unknown input as opposed to what we can measure which is the output. Therefore, we have designed the MRC so that the desired measurement is an internal state. We will further specify parameters of the design so that the estimated input to the plant will have very fast dynamics so that it should quickly approach the true pressure P(t). For the particular application of PSP dynamics, the MRC scheme shown in Figure 3. 1 will be changed to the diagram shown in Figure 3.2. We will consider the measured PSP pressure, Pp^p (t), to be the input U. The internal state V will be the estimation of the true pressure, P(t) = P^ (t) , and will in turn be our value of the compensated pressure. The output Y of the MRC scheme is simply an estimation, Pp^p (t), of the input. The

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41 Ppsp Figure 3.2--Structure of the Model Reference Control scheme for use as a compensator for PSP data. other internal states Wi, W2, and Z are of no relevance. The plant, Gp, is one of the dynamics models for the PSP given in Chapter 2. Our aim is to design the MRC parameters so that Gm is a very fast system. Thus, the output Ppsp (0 will be closely mimicking the input Ppsp W. Furthermore, since Ppsp(t) is the output of the plant Gp, the input to the plant should be closely tracking the true pressure. That is, ^ = H^) = Pc 0). So, by picking off the input to the plant Gp and making it an output, we will be able to get an approximate measure of the true pressure versus time. To begin, we will need to use a linearized model for the PSP dynamics. It is necessary to use a linearized model in order to calculate the values Co, Ci, and C2. After calculating the parameters of the MRC using the linearized plant, the nonlinear plant will be resubstituted into the overall scheme keeping the same values for Co, Ci, and C2. For the linearized model we will use the diffusion model from Chapter 2 with the linear calibration (equations 2.25a,b). Further, for the purposes of presentation, we will assume a 2"** order model. The linearized plant specifications are

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42 P(t) P(.) P(t) P(t)J Converting this from State-Space form into Transfer Function form we get 's' + 24.67s + 54.79' "ct,' -2.47 0.00" "a," "0.00-1.27" + 0.00-22.21 0.00 0.42 *PSP 0 ) = [ 0.64 -0.21] "a," + [1.00 0.00] «2. (3.10a,b) G(s) = -0.90s -18.22 (3.11) + 24.67s + 54.79 The numerator of this equation is an array with two rows one for each of the inputs P(t) and P(t). Notice that the transfer function for P(t) is that of an all-pass system. Making use of the fact that P(t) = s • P(t), we can reduce the system of equation (3.1 1) from a MISO system (multi-input, single-output) to a SISO system (single-input, singleoutput) G,(s)=0.0994 (;^\Q-%Xs^54.88L ( 10.05,-54.88) (s + 2.47Xs + 22.21) (-2.47,-22.21) ^ ^ The plant expressed in this form has an input of P(t) and output of Ppgp (t). This equation will be used to define the plant during the design stage of the MRC development. Later, the nonlinear form of the plant (diffiision model with Stem-Volmer calibration) will be used in its place. The latter portion of equation (3.12) includes a definition of the plant in "Zero-Pole" format where the zeros and poles of the transfer function are listed in vector form. The value of 0.0994 in front is the high-frequency gain of the fransfer function. In the earlier section of this chapter, the MRC scheme was developed for systems with the property Ap = 1 . Notice however, that the plant definition of equation (3.12) has

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43 a relative order of Ap = 0 . Therefore, we must either re-develop the MRC scheme for systems with Ap = 0 , or find a way to modify the plant to give it a relative order of Ap = 1 . There is a simple "trick" that can be employed to give the desired effect of changing the relative order of the plant. By augmenting the plant with a very fast 1^' order system, the resulting transfer function will have the desired properties. It should be noted that this augmented 1^' order system will also have to operate on the actual PSP data prior to becoming the input of the MRC scheme. The question now is how to choose the time constant of the augmented system. Inspection of the poles of the plant shows poles = (2i-l)'l 2 ,i = l,2,... (3.13) Following this pattern, the next value in the sequence will be used as the pole of the augmented system yielding: G (s)=k ^^i^ k (s + 10.05X5 + 54.88) In using this choice of augmentation there will be no loss of dynamics fi-om the physical system. That is, because the augmented system is in essence a lowpass filter, but with a very high cutoff fi-equency, the dynamics of the physical system will be relatively unaffected. Considering this augmented plant as the design plant (Gp = Gpa) for the purposes of calculating the Model Reference Controller parameters, we can choose the model system as Gjs)=k,^ = 100.^ (3.15) D^(s) s + 100 '

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44 which is a l" order system with a time constant of 0.01. For the Plant Gpa(s) of equation (3.14) and Model Gm(s) of equation (3.15) and for the choice of Do(s) Do(s) = (s + lOOXs + 1 lOXs + 120), (3.16) the solution to the Matching Equation gives Co =16.32 C, =(-7.86el, 1.33e4, 1.13e6) (3.17a-c) =(-3.79e4, -4.39e6, -1.3 leg) Or, if we redefine the feedback systems in the following manner: H(3)C.-N(s) N,(s) H,(s)=^ = k.^ D(s) 'D,(s) (3.18a,b) we get (in zero-pole format) H,(s)= -78.57 ,t^?:2MMlL (-100,-110,-120) „,(s).-37,47Jr^Z™L (-100,-110,-120) Following this procedure, the MRC parameters can be calculated for a PSP plant of arbitrary order. MRC Simulations and Compensation of Experimental Data It now remains to be seen how well the MRC scheme can be used to compensate the PSP data. Using Matlab and Simulink, programs were developed to implement the MRC scheme (see Appendices A, B). With this code several simulations were performed. In these simulations, a 5'^ order plant design and its accompanying MRC was used as opposed to the 2"'' order design already presented. There were three basic sets of tests performed. The first used the linear form of the Plant to create the input Ppsp(t), and

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0 0.5 1 1.5 2 t* Figure 3.3~Linear compensation simulation of a positive step in pressure. the linear form of the Plant as Gp(s). This test was performed to ensure that the compensator works as it was designed, that is, the compensated output should have the dynamics of the ideal design Model Reference System. The performance of this linear version of the MRC compensator was tested on a positive step from 0.1 atm up to l.Oatm. The results of this test can be seen in Figure 3.3. Notice that the PSP indicated pressure takes approximately two time units to come to equilibrium, whereas the compensated output levels out in less than 0.05 time units. Therefore, this form of compensation system does indeed work in the linear case and affects a more than forty-fold increase in speed. Thus, the MRC compensation scheme does work in the linear limit. The second set of tests used the full nonlinear form (diffusion model with Stem-Volmer calibration) to create the input and as Gp(s). The compensator parameters, Co, €,, and C2, used in this test were from the linear design. The response of the compensator system to a positive

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46 (a) Q. 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 h 0 3 DO aO O 0 0 O O O O 0 0 0 (J9,fi> | rii | nnnnn P e — Pc 0.5 1.5 1.2 1 0.8 ^ 0.6 CL 0.4 0.2 0 CD Oo Chi (b) P P. P ' OOOOOOOOOOOO OOU^U g) d)! LTuHu i q (J _L _L -L 0.5 1.5 t* Figure 3.4--Compensation simulations using nonlinear from of the PSP plant, (a) Positive step in pressure; (b) Negative step in pressure.

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(a) T 1 1 1 r (b) 0 ' 1 • 1 1 I 0 0.2 0.4 0.6 0.8 t* Figure 3.5--Compensation simulations using nonlinear form of the PSP plant with a sinusoidal driving pressure, (a) to* = 1 ; (b) co* = 3 ; (c) ©* = 10 ; (d) to* = 30 .

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48

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(a) Time (sec) (b) a. a: 5 6 Time (sec) Figure 3.6--Compensation of experimental data, (a) Positive step in pressure; (b) Sawtooth driving pressure.

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50 and negative step was simulated. As in Chapter 2, the step simulations were between the pressures 0.1 atm and l.Oatm and Q = -0.25 determined the Stem-Volmer calibration. Shown in Figure 3.4 are the results of the simulations for the positive step and negative step. The compensated response to a positive step in pressure contains sizable oscillations, whereas the negative step response is practically that of the ideal linear compensator. Thus, the nonlinearity of the PSP system, as it pertains to the MRC compensator, is more in evidence for positive changes in pressure than for negative changes. The third set of simulations were similar to the second in that the nonlinear form was used for the plant. However, rather than applying impulsive steps in pressure, sinusoidal fluctuations in pressure were simulated. The sinusoidal tests were between the limits 0.1 atm and l.Oatm with the same value of Q. The frequency of the driving pressure was set at four different values, to* = 1, 3, 10, and 30 . The results of these tests are shown in Figure 3.5. There are several items to note concerning these simulations. It can be seen that the compensator does a remarkable job even out to driving frequencies ten times the critical frequency of the paint. One thing of note with respect to the sinusoidal simulations is that the phase shift of the compensated data relative to the applied pressure is very nearly 0.01 in each of the four cases (frequencies) shown. Thus, the compensator is nearly a "linear phase" system. Also note that this value of 0.01 is exactly the time constant of the designed Model Reference System. What remains is to determine how well the compensator works on actual PSP data. Figure 3.6 shows the compensator response from two different experiments. The first is a positive step in pressure from 20 kPa up to 120 kPa (note: Pref = 100 kPa). The time constant of the PSP was 1.1 1 sec. The second was a sawtooth driving pressure and

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the time constant was 0.437 sec. In the step test a shght overshoot can be seen, but the compensator is still a great improvement over the raw PSP data. The compensation of the sawtooth data has a peak error of 4 kPa and an RMS error of about 1 kPa. The maximum error occurs on the rising edge of the sawtooth wave where the highest frequencies are contained. From these two sets of experimental data, we can conclude that the MRC compensation scheme does indeed work. The next step will be to apply this to a sequence of full-field images.

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CHAPTER 4 EXPERIMENTAL SETUP AND PROCEDURES This chapter will begin with a presentation of the two facilities employed to acquire the experimental data presented in this dissertation. Both of these facilities are located at the University of Florida, Department of Aerospace Engineering, Mechanics and Engineering Sciences. The first of these is a test cell made for testing the dynamic response of small-sized PSP samples. The second is a test stand for investigation of a pulsed transverse jet in a Mach 1.56 crossflow. The end of this chapter will be a presentation of the image processing techniques used to get a single calibrated PSP image. The method of compensating these calibrated images will be presented in the following chapter. PSP Dynamics Test Cell The data presented in Chapters 2 and 3 were performed in the PSP Dynamic Test Cell. This test cell is designed to control the pressure of the air over the PSP, the temperature of the PSP sample, and the temperature of the air entering the test cell. The experimental apparatus consists of the test cell itself (with various accessories), an external heat exchanger, a lamp, a photomultiplier tube (PMT), some optical components, a computer with data acquisition card and various intermediate circuits. Additionally, vacuum and high pressure lines were available to connect to the test section. The entire apparatus rested on an optical bench with an enclosing hood to shut out external light sources. 52

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A front view of the test cell is shown in Figure 4.1. A schematic view of the test cell is also shown in Figure 4.2. There are six ports (1/8" NPT) around the periphery of the cell to allow access to the small volume (1/8" thick x 1/2" diameter) immediately above the sample. These ports can be instrumented in a variety of ways. As shown in the Figure 4. 1, it is instrumented with two digital valves (Asco/Angar AL41 12L), a high frequency response pressure transducer (Entran EPV-501X-25AZ), and a thermistor (Omega 44201). The two remaining ports are capped. The two digital valves are hardwired to be 180° out of phase so the test cell is either open to vacuum or open to high pressure. With this feature, a pressure step from vacuum to ambient pressure occurs in just under 20ms. Additionally, an analog valve (Pneufronics VSO, 1.25mm orifice, not shown in figure) can be mounted to one of the ports. With the use of the analog valve, the pressure in the cell can be driven at any arbitrary function the user may desire such as a sine wave, square wave or sawtooth wave. It should be noted that because of the small orifice of the analog valve, the pressure inside the cell was band-limited to less than 80Hz. However, with the dual digital valves confrolling the pressure in the cell, frequency content upwards of 250Hz could be achieved, but only in the form of a step or square wave. It was later discovered that using a latex burst diaphragm on one of the ports could create frequency content of the pressure signal in excess of IkHz, however, no such experiments are presented in this dissertation. The window on the front of the cell is plexiglass and allows optical access to the PS? sample. The sample is a small piece of aluminum (12.7mm x 15.9mm x 3.2mm) with the PSP applied to one of the faces. The size of the samples was chosen so they could be easily and cheaply made from 12.7mm x 3.2mm (1/2" x 1/8") aluminum bar stock. The sample is mounted on top

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54 of a raised circular platform. An 0-ring around the base of the platform seals the interior of the cell. Behind the platform is a double stack of thermoelectric chillers (TEC's) (Melcor CP1.0-31-05L) to heat or cool the sample. Mounted further behind the TEC's are a set of cooling fins and a small DC fan. Originally the test cell was made with a water-bath behind the circular platform to control the temperature of the sample, but this was later changed out for the TEC's. The inflow air heat exchanger unit, shown in Figure 4.3, is made of two aluminum plates (17.8cm x 15.2cm) clamped together. Machined into one of the plates is a zig-zagging channel thru which air can flow. Mounted on top of the the unit are four TEC's (Melcor CP 1.4-1 27lOL) with cooling fins and fans to control the temperature of the top plate. The entire unit is encased in foam insulation to aid in temperature control. Mounted at the exit is a thermistor (Omega 44201) to measure the temperature of the air leaving the unit. Two different styles of lamp can be used to illuminate the PSP sample. The first, shown in Figure 4.4, is a quartz-tungsten-halogen lamp with stabilized power supply and photofeedback system (Oriel QTH lamp housing with 68830, 68850, 68855 accessories). The second style of lamp is shown in Figure 4.5 and incorporates an array of super-bright LED's (ISSI LM2). The PMT used to collect and measure the light fi-om the PSP sample has a spectral response from 185nm to 900nm and a bandwidth of 20kHz (Hamamatsu HC12005), although analog RC lowpass filters used to aid in reduction of shot noise lessened the bandwidth.

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There are several components of optics in the setup two filters, a mirror and a lens. The first of the two filters is a bandpass filter centered at 450nm (Melles Griot 03FIB004) and is mounted to the lamp. The second filter is also a bandpass filter, but is centered at 650nm (Melles Griot 03FIB014) and is mounted to the fi-ont of the PMT. The mirror reflects the excitation fi-om the lamp into the test cell and onto the PSP sample. The light emitted from the sample is then focused thru a lens into the PMT. A schematic for the path of the light is shown in Figure 4.6. The computer used in the experiment was one of several IBM-compatible Pentium or Pentium II machines. A data acquisition card (DAQ card) (National Instruments AT-MIO-16E-2 with SCB-68 accessory) was used to sample the signals fi-om the various transducers and to control the operation of the digital and analog valves. Data acquisition and control programs were developed using National Instruments' LabView programming language. This program was designed in a very "hands off fashion and allowed the user to control the digital and analog valves and to perform the data acquisition of the transducers all from a seat in front of the computer. Data from the fransducers was acquired at rates on the order of 5kHz and then downsampled with averaging to a rate of IkHz. Several circuits were used to interface the fransducers and valves to the DAQ card. First order analog lowpass filters with a cutoff frequency of 480Hz were applied to the signals from the pressure fransducer and from the PMT to aid in noise rejection. Because the DAQ card could only source 5mA of current, power transistors were used to provide current amplification to the voltage signals coming from the analog output and

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56 digital output lines of the DAQ card leading to the valves. Finally, resistor circuits were used to create a linear voltage versus temperature output from the thermistors. Supersonic Wind Tunnel The second facility used for experimentation was a blow-down wind tunnel. Two large tanks external to the building provide an initial source of approximately 30m^ of air at a maximum pressure of 1375kPa. Supplying air to the tanks is a 300hp rotary screw compressor (Quincy QSI-1000) capable of delivering 34kg/min at 1375kPa (950SCFM at 200psig). Attached to the compressor are dual dessicant dryers (Hydronix) that cycle in turn to remove moisture from the air after compression. A 3" line runs from the tanks, into laboratory, and thru one of two control valves (Fisher ET-l'/z" , Fisher ET-1" with Type 667 Actuators, Series 3582 Valve Positioners and Type 67AF Filter Regulators) depending on the desired flowrates or stagnation pressures required at the test stand. These valves are shown in Figure 4.7. The two valves are mounted in parallel on the 3" line. The valves are ultimately user-controlled in one of two fashions. First, they may be controlled manually via a pneumatic controller system (Fisher "Wizard" Controller). Alternatively, the valves may be confrolled via a computer connected to an external transducer (Fisher Type 846 Electro-Pneumatic Transducer). The second of these was the method of choice. The final element of the general test stand was a large cast-iron pressure plenum, or, stagnation chamber shown in Figure 4.8. The air flow is from left to right in the image. To this chamber can be mounted whatever test section the user may design such as the Mach 1.56 nozzle used in the experiments to be presented in the next chapter.

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57 The test section chosen for experiments reported in this dissertation has a design Mach number of 1 .60 and its geometry was calculated using the inviscid nozzle design program of Carroll et al.'" Later pressure measurements made in the tunnel yield a Mach number of 1 .56. This difference is due to the growth of the boundary layer downstream of the sonic throat. A slight divergence of the test section downstream of the nozzle exit could have compensated for the boundary layer growth, however, it was decided to use a straight section to ensure a perpendicular viewing of the jet to be described later. The main section of the tunnel downstream of the nozzle is a 5.08cm x 5.08cm (2" x 2") cross-section with optical access via a 15.24cm (6") long plexiglass wall. There are also two opposing sidewalls with interchangeable plates with provisions for additional optical access, or transducer measurements, or for placing obstructions into the flow. An external view of the test section is shown in Figure 4.9. The direction of airflow is from left to right. The main plexiglass window is toward the front. Along the upper removable sidewall are an array of static pressure ports and thermocouples. For a lower viewing angle, a 4.44cm (1.75") diameter circular plug opposite the main plexiglass wall would also be seen. Figure 4. 10 shows a closeup view of the front of the plug. For the purposes of the experiments to be presented in the next chapter, this plug has been modified to create a fransverse, or normal, jet 3mm (0. 1 1 8") inches in diameter surrounded by four 0.5mm diameter (0.020" 0) pressure taps at 90° increments on a 4.5mm (0. 1 77") radius about the jet. Additionally, a single thermocouple is imbedded ' 2.5mm (0. 1 ") from the interior surface of the test section. Figure 4. 1 1 shows a rear view of the plug. The radial location of the thermocouple can be seen. Also shown in the figure is the digital valve used to control the jet. The diameter of the jet was chosen to be

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58 approximately equal to the free-stream boundary layer thickness as calculated by Carroll.''^ Other studies have used this same method to choose the diameter of a jet injected normally into a supersonic air stream.^'^' The internal geometry of the jet is simply a straight 3mm diameter cylinder. Other studies have used a converging geometry at the jet orifice to ensure a sonic throat. However, because the valve is being cycled, and because it is desired to have as fast responding a jet as possible, the minimal volume straight cylinder geometry was chosen. The optical system used to measure the intensity of the PSP consisted of a camera, a strobe and appropriate filters. The camera was a Photometries Model CH250 CCD camera as shown in Figure 4.12. The CCD array itself was a 512 x 512 array of pixels. The full-well capacity of the pixels is approximately 320,000 electrons and the pixels were read out using a 14-bit A/D converter. The strobe, also shown in Figure 4.12, was a Xenon DT-301 Digital Stroboscope Tachometer. The strobe could operate at frequencies of 2Hz 200Hz. The duration of a single flash of the strobe is on the order of 30^s. A composite of blue Schott glass filters (BG12 3mm, BG39 3mm) was placed on the output of the strobe to dampen its output at red wavelengths and yet allow the blue excitation through. A Melles Griot 03FIB014 bandpass filter centered at 650nm was placed on the input to the camera to filter out the source illumination from the strobe. Various conventional transducers were used in these experiments. The stagnation pressure was measured with an Omega PX302-200AV pressure transducer. This measurement was used as the process variable of a PID algorithm to control the Fisher 1 V" valve and for calculating the Mach number of the flow. All of the temperature measurements were made using Omega T-type thermocouples. The cold-junction

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reference temperature will be described later. Static pressure measurements were made with a combination of two different ranges of Kulite piezoresistive pressure transducers. Two Kulite Model XT190-25 A (25psia range) and two Kulite Model XT-190-50A (50psia range) transducers were available for measuring the pressures at the various static pressure taps. One of the the 50psia transducers was positioned at the sidewall pressure port closest to the nozzle exit and measured the static pressure for purposes of calculating the Mach number. It was located far enough upstream of the injected jet to be outside of its influence. A second SOpsia transducer was placed at the pressure tap spanwise from the jet. The two remaining transducers with ranges of 25psia were placed at the taps upstream and downstream of the jet orifice. Thus, the two 25 's were placed at the locations of highest interest and the two 50's were placed at the less critical locations. The Kulite transducers mounted to the pressure taps on the plug insert were offset from the interior of the test section by a 5.08cm x 1 .6mm0 (2.0" x 0.063 "0) volume. The frequency response of the pressure transducer/tap volume combination was not calculated, but was assumed to have a bandwidth greater than IkHz which was the maximum sampling rate used in the experiments presented. The pressure transducers were calibrated using a NIST-traceable pressure standard (Druck DPI145 Multiftinction Pressure Indicator). Additionally, prior to each run, the transducers were "zerocalibrated" to match the ambient pressure indicated by the pressure standard. The jet could be turned on or off or modulated at a given frequency via a SnapTite digital valve (2W13W-1NB-A8E6, 24V, 9W, 1/8 orifice). The valve was capable of operating at frequencies upwards of lOOHz although at the upper range the valve had a tendency to stick in the open position. At frequencies below 80Hz a duty cycle of 40%

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on and 60% off seemed to work the best in terms of the pressures seen while running the tunnel and on the basis of "non-stick". The pressure upstream of this valve was set using a regulator coming off the lOOpsi house air supply within the building. This pressure was further measured using the Druck pressure standard. The experiment was controlled via a data acquisition and control program developed using LabView and running on a Pentium III computer. This program controlled two separate DAQ cards within the computer (AT-MIO-16E-2; PCI-6024E). The first board measured the 5 pressure transducers, the wall thermocouple, and the voltage signals being applied to the jet valve and the strobe. Prior to performing the experiment, a switch was set on the external connector block (SCB-68) that connected channel 0 of the board to a cold-junction temperature sensor on the connector block. A measure of the cold-junction temperature was then recorded and used later to properly convert the thermocouple voltage readings into temperature. After measuring the coldjunction, the switch was reset so that channel 0 could be used to measure one of the external transducers. One analog output on the first board was used to control the main valve responsible for running the tunnel. Using PID control, the stagnation pressure was monitored and set to a desired pressure of 158kPa (23psia) which was just above the pressure necessary to fiilly start the tunnel. The second DAQ card was used solely for its two analog output lines. These were used to drive the jet valve and the strobe. The phase of the strobe relative to the jet could be controlled down to 1/50"" of a cycle at a frequency of lOOHz. This allowed the strobe to be flashed consistently at a desired phase relative to the cycling of the jet. Hence, the acquired PSP images were actually ensemble averages of hundreds to thousands of individual strobe flashes.

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61 The final element of the tests performed in this test section are the PSP and TSP coatings. The PSP consisted of the luminophore PtTFPP [Pt(II) meso-tetrakis(pentafluorophenyl)porphine] interspersed in the polymer matrix VPDMS [vinyl polydimethylsiloxane]. The TSP was a ruthenium derivative in clear-coat acrylic. These formulations was made and applied to the plug insert by a member of Dr. Schanze's group at the University of Florida Department of Chemistry. The coating thickness varied from painting to painting but was nominally on the range from 10|im 20(im. Image Processing Achieving a calibrated PSP image is a very complex process. Numerous papers have been written on just this subject. The general process is shown in the flowchart of Figure 4. 13. To begin, one needs two images one at the unknown run condition and a second at known reference conditions. Consider the two image intensities to be !„,„ and Iref respectively. The next step is to subfract the dark image, Idrk, from each of the run and reference images. The dark image is an image of the same duration as the run and reference images but with all excitation sources turned off Next is a flat field correction which compensates for the pixel-to-pixel variation in sensitivity of the CCD array. This correction takes the form of a matrix, Cfr, the size of the CCD array with values all near 1.0. When this matrix is multiplied by the given image, the differences in the pixel gains is cancelled out. The task of generating the matrix Cfr is an involved process in and of itself In the process of running the tunnel, the model of interest may move slightly relative to its position at the reference image. In a later step, the data contained in the two images will need to be ratioed. Therefore, it is of extreme importance that the two images be perfectly aligned prior to the ratioing step. Often, markers are placed on a

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62 model to be used as registration points. Alternatively, certain parts of a model may be used as registration points. For example, oftentimes a model will have pressure taps in addition to the PSP to be able to get very acurate measurements at a distrete number of points. In a PSP image these taps will show up as tiny dots of low intensity. If a number of these are scattered over the model, they can be used to calculate the movement of the model relative to its reference position. For certain models, a simple pixel-shifting is sufficient. However, if the model rotates relative to its reference or warps due to the loads it experiences, the registration algorithm must become much more complex. For the model used in this dissertation, warp and rotation are not present. So, a simple linear algorithm will be used to make the run and reference images line up. The run image will be shifted so that it aligns with the reference image: I™„0'J')=I™„(i + Ai,j + Aj) (4.1) where Ai and Aj are the amount of pixel shifting necessary. Note that these quantities are generally non-integer values in which case a weighted average of the four closest pixels will be used to calculate the required value. The next step is one that is often not possible to perform, but, for the experiment to be presented in this dissertation, it is. A tiny fluorescent marker was applied to the external surface of the test section in a location where it would be within the field of view of the PSP images. Because this marker is on the outside, its intensity will only vary if the intensity of illumination varies. Taking a measure of the average intensity over a region of the marker in the run image (minus the average dark intensity over the same region) and ratioing to the same in the reference image, the run image can be scaled to match the intensity of illumination that was present

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63 during the acquisition of the reference image. That is, define the "external reference" scaling constant: C. = |""'"|'" (4.2) run.ext drk When multiplied by the run image, this constant will perform the necessary scaling. With constant light sources such as the tungsten-halogen lamp or LED lamp used in the aforementioned Dynamic Test Cell, such corrections are not generally necessary. However, for the particular application to be presented, a strobe lamp is being used. If the shutter on the camera is not ultra stable in its opening process, closing process and total duration of operation, or if there is variability in the strobe, one image may see an additional strobe flash or two relative to another image depending on the fi-equency of the strobe flashes. So, it becomes necessary to be able to scale out the intensity of illumination using the external reference technique. It is being assumed here that the spatial variations due to a single flash of the strobe will even out with the hundreds of strobe flashes used to acquire a single image. The next step is to filter the images spatially to remove some of the shot noise inherent in measuring photonic sources. Generally, one is able to take multiple run images at the same conditions and average them to reduce the shot noise. However, for the particular experiment presented, the runtimes were limited to where this luxury was not available. Moreover, the integrated light levels of many strobe flashes was still not sufficient for making use of the fiill well capacity of the CCD. That is, the measured intensities were only on the order of hundreds of counts (bits). This means that the signal-to-noise ratio of the pixel intensities is relatively low. If the noisy run and reference images were ratioed, the noise would only be amplified on those pixels that just happened to be statistically greater(lesser) and

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64 lesser(greater) than average between the run and reference images respectively. So, at the expense of some spatial resolution, the images are filtered prior to being ratioed. The smoothing kernel used was of the form: cos'f7r^^7 + j7/2nV2] K(io Jo)= ^; — — where i^, = -n,...,n; n = 1,2,3,... (4.3) ^ ^cos'(nVi' +jV2nV2) i=-n j=-n In the above equation, io and jo are the coordinates of the pixel within the kernel being calculated. The kernel has dimensions (2n+l) x (2n+l), thus it is of odd order (3x3, 5x5, . . .) and is centered on the pixel ((n + 1)/2 , (n + 1)/2) . It may be noticed that the comer pixels of this kernel will turn out to be identically zero. This kernel is then 2Dconvoluted with an image to achieve a smoothed image. This form of kernel is superior to a 3x3 or 5x5 "flat-top" kernel in that it gives more weight to pixels near the center and less weight to those farther away. For a flat-top, equal weight is given to all pixels within the area of support of the kernel. This kernel was developed based on ideas found in IDDSP wherein spectral leakage is reduced by using a tapered windowing function on a dataset. Depending on the amount of noise observed (based on the light levels measured), either a 3x3 or 5x5 kernel was used to smooth the data. An estimate for the spatial resolution based on a 95% sum of the kernel values gives a value roughly equal to "n". As an example, a 7x7 kernel for which n=3, roughly 95% of the kernel's weight is contained within a radius of 3 pixels about the center of the kernel. It should be noted that for 3x3 and 5x5 kernels, 100% of the kernel's weight is contained within the estimated area. The next step in the flowchart is to ratio the smoothed images. The steps up to this point may be summarized via the following equations:

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65 where • is the "array multipHcation" operator ( C = A • B ^ = a^bij ), and (8i represents the convolution operation. Ratioing these two equations we finally end up with the equation: _Y^_ K®[C..(I.,-1„)] Y„ K«[c„c„ .(I.. -i„r which represents the static calibration function. Analytically inverting this function gives Pru„=g(Y*,P„r) (4.6) where f and g are nearly inverse functions. One fact that will aid in the analysis of the images for this particular application is that we have the capability to perform a static calibration of the PSP sample in the exact spot and with the same lighting as during a run. That is, the exit duct may be unscrewed fi-om the test section and replaced with a cap. The test section is now an enclosed volume which can be drawn down to vacuum or can be pressurized. Thus, we can get a very accurate measure of the static calibration. Following the process outlined in leading up to equation (4.6), a sequence of images taken at different pressures with the tunnel capped may be reduced to a sequence of data points of dimensionless-intensity ratio versus dimensionless-pressure ratio. An analytic function may now be curvefit to this data to give a useful expression for the calibration function. Table 3 lists mathematically equivalent forms of three common calibration functions. The first fiinction in the table is the standard linear Stem-Volmer equation. The second corresponds to a linear

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66 Table 3~Calibration Functions, Constraints, and Typical Values Function Constraints^ Typical Values^''' . . (P-Q.) Q, <0.01 Q, =-0.2 °(P..-Q,) Y'-Y' (P-Q.XP.r-Q2) °(P.r-Q,XP-Qj Y* = y: Q, <0.01 Q, =-0.1 Q2
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67 actual PSP data, the parameter Yj is set to 1 . The three values Qi, Q2 and Q3 must all satisfy certain constraints to satisfy physics, otherwise the intensity would be allowed to be infinite at some small positive pressure. The three constraints listed in Table 3 are: • Q, <-0.01 • Q2
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68 to 1. The RMSE of the curvefit is 0.84% of P^r, or, 0.85kPa. If the constraints are lifted and the data is re-curvefit, the parameters become (0.0484, -0.0759, -1.0761, 1.0054) and the RMSE is 0.77% of Pref, or 0.78kPa. So, with the constraints lifted, the curvefit would indicate that the intensity is infinity when the pressure is P = 0.0484atm , that is, when P = 4.89kPa . Notice that without the constraints, the curvefit only decreases the error of fit by 0.07kPa which is a very negligible amount for most applications. For a function f(x,y) where x and y are independent variables, the general uncertainty in f is given by For the independent variables x and y, the 95% confidence limits can be calculated by the formula where B represents all sources of systematic or bias error and S is the standard deviation of the random component of the error. Using the formulas of equations (4.10) and (4.1 1), the uncertainties of the variables for the pulsed jet experiment can be tabulated. Table 4 lists many of these variables, their nominal values and their uncertainties. A detailed investigation of the uncertainty in the indicated PSP pressure shows that the relative uncertainty in the measured PSP pressure can be reduced to the following equation: Uncertainty Analysis (4.10) (4.11) (4.12)

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where P is the fiinction p(y* ) from the static caHbration, Y* is the measured intensity ratio, P is the A/D gain of the CCD pixels in electrons per bit, and Iref is the intensity in the reference image. A full derivation of this equation is given in Appendix C. We can Table 4~Nominal values and 95% confidence limits of key variables. Variable Nominal Value 95% Confidence Limits P stag 1.555atm ±0.021atm Pstat 0.390atm +0.004atm Twall 9.78°C ±o.2rc M 1.557 ± 0.006 Pj 2.000atm +0.014atm J 1.134 ±0.057 a 13.5|jm ±2.0^m E 5.35kJ/mole ±0.27kJ/mole T 0.20s ±0.06s Irun " Idfk 260cts ± 3.7cts Iref Idrk 160cts ± 2.9cts Ppsp " 0.390atm ±0.016atm P 0.390atm ± 0.00 latm Assumes no smoothing (see Figure 4. 15); Upstream and downstream pressure taps.

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70 further take into account the effect of the smoothing kernel (equation 4.3). Smoothing has the effect of increasing the effective number of counts at a given pixel by a factor related to the size of the smoothing kernel. Specifically, the factor is given by the denominator of equation (4.3). That is, for a 3 x 3 kernel, the effective intensity increases by a factor of 2.032 and by a factor of 5.808 for a 5 x 5 kernel. The result of this in relation to the relative uncertainty in the indicated pressure is shown in Figure 4.15. The effect of smoothing in relation to equation (4.12) is also given in Appendix C. Figure 4.1--PSP Dynamic Test Cell: A) Digital valve to vacuum, B) Pressure transducer, C) Digital valve to high pressure, D) Thermistor, E) Window to PSP sample, F) Cooling fins.

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71 Window Sample 1/8" NPT Port 0-Ring Figure 4.2~Schematic view of the PSP Dynamic Testing Cell.

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Figure 4.3--Heat Exchanger Unit: A) Inlet, B) Outlet.

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73 Figure 4.4--Oriel Quartz-Tungsten-Halogen (QTH) Lamp.

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Figure 4.5--ISSI LED Lamp.

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PSP Test Cell Figure 4.6--Schematic of the PSP Dynamic Testing Cell Optics.

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Figure 4.7--Fisher Control Valves: A) VA" Valve, B) 1" Valve. Figure 4.8--Stagnation chamber of the supersonic test stand facility.

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Figure 4.9-Mach 1.56 Test Section: A) 3 of 8 Static pressure taps, B) 2 of 4 Thermocouples, C) Exterior reference marker, D) Bottom edge of circular plug insert. Figure 4.10~Detailed front view of plug insert showing the central jet orifice and the 4 static pressure tap locations.

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Figure 4.1 1 -Detailed rear view of plug insert: A) 2 of 4 Pressure taps, B) Thermocouple, C) Digital valve controlling the jet. Figure 4.12--A) Xenon Strobe, B) Photometries CCD Camera.

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Run Image Dark Subtraction I Flat Field Correction Reference Image I Dark Subtraction Flat Field Correction Image Registration < Image Registration i External Reference External Reference I Smoothing Filter I Smoothing Filter Ratio: Iref/Irun Apply Calibration Figure 4.13--Image Processing Flowchart.

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80 0.00 0.50 Figure 4.14--Sample PSP Static Calibration 0.05 1.00 Pressure (atm) 0.00 T 1 \ 1 1 1 T o Raw Data Curveftt — 1 1 1 1 1.50 2.00 I I r— I 1 r 0.00 — e No Smoothing — 0 3x3 Smoothing — B 5x5 Smoothing J 1 -I 1 1 1 1 1 I '''''' 0.50 1.00 1.50 Pressure (atm) 2.00 Figure 4.15--Relative uncertainty in the indicated PSP pressure.

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CHAPTER 5 RESULTS Numerous studies have been done on the effect of a jet of air injected into a supersonic freestream/'^'"^" All of these tests were performed using a steady blowing jet. As-yet, there have been no experiments involving pulsed jets in supersonic crossflow. This may be because the complexity of the flowfield under steady blowing is already difficult enough to describe and quantify. Or, it may be because there hasn't been a viable measurement technology. However, with the mounting interest in synthetic jets, which on small time scales are cyclic, there may soon be great interest in an investigation of a pulsed jet. This then is the basic idea for an experiment to test the ability of the compensator design to correct wall PSP measurements around the orifice of a pulsed jet. The equipment used is as described in the supersonic wind tunnel section of the preceding chapter. The tunnel was run at a stagnation pressure of 158kPa (23psi) which was the lowest value possible that would leave the tunnel fully started. That is, if the stagnation pressure dropped much below that value, the normal shock at the exit of the test section would detach from the exit and move upstream into the test section proper. The reason for this low stagnation pressure was to allow as long a runtime as possible. Once the tunnel had been started, there was an approximately 5 minute equilibration period during which the wall temperature of the test section would drop over time and then level off Depending on the outside temperature (i.e. the temperature of the exterior tanks) the static wall temperature would range from as low as 0°C (in the wintertime) up 81

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to more normal value of around 10°C to 15°C. It should be noted that all of these experiments were performed at night. This was because ambient light from outdoors would come through the exhaust duct and into the test section creating a bias intensity in the PSP images. For the particular tests to be shown here, the digital valve controlling the jet was cycled at a frequency of 20Hz with a 40 percent "open" duty cycle. That is, the jet valve was opened for 20ms and then closed for a duration of 30ms, giving a total period of 50ms. The strobe could be set to any desired phase relative to the 20Hz jet valve control voltage. A single PSP image had a duration of 20 seconds of integration. During this 20 seconds the strobe would have flashed a total of 20Hz x 20s = 400 flashes at exactly one relative phase. Thus, the image acquired was an ensemble average of 400 flashes. The strobe would then be reset to a different phase and another image would be taken. In other transverse injectant studies, the jet pressiure was one of the primary variables. In the experiments presented herein, the jet pressure was set to a value of around 2 atmospheres absolute. Based on this value, the jet-to-crossflow momentum ratio (equation 1 .8) was J=l . 134. After a full sequence of images had been taken, the tunnel was quickly shut down. As soon as the flow in the tunnel had stopped, a reference image at ambient pressure was acquired. Because this reference image was acquired soon after shutdown, the temperature of the tunnel wall had not yet had time to rise very far. Typically, this temperature rise was on the order of 0. 1°C to 0.2°C, which is within the limits of uncertainty quoted in Chapter 4. So, a reference image was available that was taken at the same temperature as during the run images. Performing the calibration procedure outlined in the previous chapter, one now had a sequence of calibrated PSP

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83 images. Because of the very low light levels, a 5 x 5 smoothing kernel was used to even out the pixel-to-pixel statistical variations. Based on the uncertainty analysis presented in Chapter 4 and in Appendix C, the uncertainty of the PSP indicated pressures will be reduced by a factor of V5.808 from the value presented in Table 4. From that point, the experiment now proceeds into the compensation phase. The time constant of the PSP was calculated from a compilation of various experiments. Recalling the definition of the time constant from Chapter 2, x s a , and assuming an Arrhenius relation for the diffiisivity, equation (1.12), the necessary measurements are evident. The thickness was measured at 20 points over the surface of the plug insert using an eddy-current gauge (Positector 6000) and then averaged. At the same time the plug insert was painted, two small samples were also painted for use in the Dynamic Test Cell presented in Chapter 4. The PSP on these samples were roughly 0.4x and 1 .2x the thickness of the coating on the plug insert. The thickness of these coatings was also measured using the eddy-current gauge. Using the Dynamic Test Cell, a series of tests were performed on the thicker of the two small samples to measure the time constant of the PSP as a fimction of temperature. The data was then curvefit using the formula T, =Toexp(E/RT), (5.1) where E is the activation energy of diffiision. Because the thickness of this coating was not equal to the coating thickness on the plug insert, a fiirther step was needed to calculate the value of the time constant to be used in the compensation routine. The formula

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X = a 84 X, (5.2) corrects the time constant for differences in coating thickness between the small sample, a, , and the plug insert, a. Using the measured wall temperature in combination with equations (5.1) and (5.2), a value of x = 0.20s was calculated for the PSP on the plug insert and used for the purposes of dynamic compensation. Shown in Figure 5.1 are a sequence of 10 calibrated, but uncompensated, PSP images taken at equal time steps throughout the cycle. In terms of a temporal measure, the images were separated in time by 5ms per image. The image in the upper left-hand comer of the figure corresponds to the opening of the jet valve. Progressing from the top down and then to the top of the second column and down spells out a fiill cycle. The phase delays relative to the opening of the jet valve are shown on each image as a percentage of a full cycle. The gray-scale of the images has been set so that white corresponds to a pressure of 0.7atm and black is 0.1 atm. It can be seen that the area upstream of the jet (left of the jet) is generally a region of high pressure and that the area downstream of the jet (to the right) is a region of low pressure. From the sequence of 10 images, the upstream pressure indicates a peak pressure of between 0.55 and 0.60atm for phase delays of 10% to 50%. The minimum pressure downstream of the jet is between 0.25 and 0.28atm and occurs around a phase delay of 50%. The mean pressure far from the jet is on the order of 0.39atm. So, based on the uncompensated PSP images, the peak wall pressure is 0.19atm above the freestream and the minimum wall pressure is 0.13atm below the freestream. Note that the uncertainty in these values is approximately 0.007atm. So, the uncertainty is on the order of 5% of the absolute fluctuation.

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85 Progressing now to the compensation data, the sequence of 10 compensated images is shown in Figure 5.2. These images were generated using the Matlab and Simuhnk codes presented in Appendices A and B. Running on a Pentium III 800MHz machine with 256MB of RAM, compensating a sequence of 10 images (177pixels x 253pixels) took approximately 2hrsl5min to compile. In these compensated images we see the shock strength much clearer. Also note that the pressure very quickly dies down after the jet is shut off as compared to the uncompensated where it appears the downstream pressure never equilibrates. The pressure midway between the front edge of the jet orifice and the upstream pressure tap peaks at around 0.70atm at a phase delay of 0%. The pressure downstream of the jet reaches a minimum around 0.1 atm almost immediately downstream of the jet at a phase delay of 20%. This gives a peak wall pressure 0.31 atm above the freestream and a minimum of 0.29atm below freestream. Another interesting feature is that at the initiation of the jet (0% phase delay) the whole of the image, both upstream and downstream, is seen to raise in pressure. This seems to indicate that in the initial instant of the jet, it is filling up the boundary layer in all directions, pumping it up to a higher pressure. Then, the gross flow is able to overcome this initial transient and set up a more clearly defined shock structure. This shock structure persists nearly unchanged until the jet closes. When the jet is then turned off, the flow nearly immediately adjusts, wiping out all evidence of pressure gradients. Shown in Figure 5.3 is a comparison of the pressures along the streamwise centerline through the jet for both the uncompensated and the compensated images. The centerline pressures from a set of 10 images were stacked into a 2D array. This 2D array was then duplicated and stacked twice yielding a three-cycle loop, thus giving a clearer

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86 1 (j) = 0.0 (j) = 0.5 Figure 5.1--A sequence of 10 uncompensated PSP images taken at 5 ms intervals during a full cycle of 50ms. Grayscale is such that white = 0.7atm and black = 0. latm. Dimensions are from ^.5Dj to +5.5Dj in X and from -3.5Dj to +3.5Dj in Y.

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Figure 5.2--A sequence of 10 compensated PSP images taken at 5 ms intervals during a full cycle of 50ms. Grayscale is such that white = 0.7atm and black = 0. latm. Dimensions are from -4.5Dj to +5.5Dj in X and from -3.5Dj to +3.5Dj in Y.

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88 Figure 5.3--A three-cycle loop of PSP data along the streamwise centerline through the jet. (a) Uncompensated; (b) Compensated. Grayscale is such that white = 0.7atm and black = 0. latm. The X-coordinate is oriented along the horizontal direction (positive to the right) and time is oriented vertically (positive up).

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89 (a) 0.65 0.60 0.55 I 0.50 qT 0.45 0.40 0.35 0.30 I I I I I I I I 0.00 0.50 0.45 _ 0.40 I 0.35 0.30 0.25 0.00 -S Uncompensated -• Compensated Tap 0.05 0.10 Time (sec) (b) 0.15 T 1 1 r -O Uncompensated -• Compensated Tap 0.05 0.10 Time (sec) 0.15 Figure 5.4--Comparison of PSP data to pressure tap data, (a) Upstream of the jet; (b) Downstream of the jet; (c) Spanwise from the jet.

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90 0.60 0.55 0.30 0.00 (c) I I I X ' ' ' ' ± I I I I — 9 Uncompensated — • Compensated Tap 0.05 0.10 Time (sec) ' ' ' 0.15 Figure 5.4--continued. Figure 5. 5-Variation in the paint thickness near abrupt edges.

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91 visual presentation of the data. The enhanced features due to the compensator are quite evident. Shown in Figure 5.4 are comparisons of pressures measured by the pressure taps to the measurements of the PSP, both compensated and uncompensated. Once again, a single cycle has been looped 3 times to give a clearer presentation. The pressure tap data is an ensemble average of 10 cycles. The PSP data is an average of 32 values taken around the pressure tap. The 32 values are at equal angular spacing at a radius of 14 pixels (1.6mm; 0.63Dj) from the center of the tap. This large radius was necessary to get away from the corrupted data local to the tap as will be explained shortly. In the figure, a large amount of overshoot is seen at the taps just upstream of the jet and spanwise from the jet. This is likely due to an overestimation of the time constant of the paint. That is, increasing the time constant used in the compensator relative to the "true" value of the time constant tends to cause instability in the compensator and yields data with both overshoot and ringing. Underestimating the time constant does not have this effect and is thus a preferable situation, however, it too will yield an incorrect amount of compensation. Another possible explanation for the overshoot is that we only have 10 samples per cycle. During compensation, the Simulink routine of Appendix B runs on a finer temporal grid spacing by linearly interpolating between the data it is given. If there is large uncertainty in the data given to the Simulink routine, which is true in this case, there can be large discontinuities in the time derivative of the data. The compensator will then amplify these resulting in data that appears to have overshoot or ringing. The data immediately surrounding the jet is corrupted for one primary reason paint thickness variations. Figure 5.5 shows an image of the PSP surface taken at a

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glancing angle. It can be seen that the paint thickness around the taps and the jet orifice varies to a noticeable degree. Starting from the edge of a tap, the paint increases in thickness as the distance from the tap increases. Then, at a point approximately 7 pixels (0.8mm; 0.27Dj) fi-om the center of the tap, the paint reaches a maximum thickness. The thickness then decreases slightly down to the nominal thickness. These thickness variations are due to surface tension effects at the time of paint application. Because the time constant of the paint is a strong function of the thickness, the compensated data in the extreme near-field region of the taps cannot be trusted. The last comparison to be done is that of the effective back-pressure ratio shown in Figure 5.6. This data was calculated in a similar manner to the data surrounding the pressure taps. At a radius of 19pixels (2.2mm; 0.74Dj) fi"om the jet center, 32 values around the circumference of the jet orifice were averaged. This value was then divided by the pressure behind a normal shock in the Mach 1.56 fi^eestream. The calculated ratio shown in the figure is nominally on the order of 0.39 except for a 10ms window immediately upon the opening of the jet at which time the ratio peaks at around 0.50. It should be noted that the value of J= 1.1 34 quoted earlier is only valid during the period when the jet is open. When the jet is shut off, the momentum ratio should go to zero and the effective back-pressure ratio should approach a value of 0.379. This back-pressure ratio would be a characteristic value for the M=1.56 flow. Evaluating the uncertainty in the measured back-pressure ratio reveals an uncertainty of 0.017 relative to the value of 0.39. Thus, the characteristic value of 0.379 is within the level of uncertainty present in the experiment. A detailed presentation of the effective back-pressure ratio as a function of the momentum ratio was performed by Everett et al.* Fitting a straight line to their data

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and extrapolating to a value of J= 1.1 34 yields a value for the effective back-pressure o 0.339. This value does not compare particularly well to the values measured here, however, it does compare well in relative to values from other studies 23,43,44 0.55 0.50 0.35 I I I 0.00 I I I -Q Uncompensated -• Compensated ' ' ' ' 0.05 0.10 Time (sec) Figure 5.6~Effective back-pressure ratio at the jet. 0.1!

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CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS Conclusions A thorough investigation of the response of PSP to unsteady pressures was presented. A physically realistic model for the dynamic behavior of the coatings was developed and then used to design a compensation scheme. This compensation scheme was based on the Model Reference Controller and was in effect designed to be state observer. Compensation of data acquired under controlled laboratory conditions yielded excellent results. The ambitious task of applying the compensation scheme to a sequence of full-field images was undertaken. Owing to the large experimental uncertainties, particularly in the overall intensity from the paints and from the spatially varying thickness of the PSP coating, the compensated images could only be used in a semiquantitative sense. However, the designed compensator has the capability to be a great advancement in the use of PSPs. Recommendations From the uncertainty analysis performed for the full-field dynamic PSP experiment presented, the following recommendations are suggested to increase the accuracy of the PSP measurements. 1 . Re-design the existing test section to reduce it in size. This will have the effect of lowering the mass flowrate through the tunnel. If the correct size is chosen, it will allow the existing compressor system to keep up with the demands of the tunnel 94

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95 and result in the ability to run the tunnel continuously. With unlimited runtimes, many more cycles of the pulsed jet can be summed and averaged, thus giving the PSP images a much improved signal-to-noise ratio. 2. Develop a method to measure the coating thickness on a pixel-by-pixel basis. This will result in a better estimate for the time constant of the paint. From the current experiments, a single value of the time constant was used for the entire image. Allowing the time constant to vary spatially should result in much better compensations. 3. Install a pressure tap inside the jet between the valve and the jet orifice. Then, install another Kulite transducer at this location to get a better measurement for the jet pressure. This should then allow a calculation to be made as to whether the jet is sonic at the exit at every point in the phase of the jet. 4. Look for a brighter strobe or perhaps one with a slightly longer flash duration. Currently, the flash duration is about an order of magnitude shorter than it needs to be to freeze the flow for the range of fi-equencies capable by the jet valve. 5. Acquire more than just 10 images per cycle. Finer temporal resolution will result in less noise in the compensation process. A minimum of 20 images per cycle is recommended. 6. A final recommendation is to rotate the plug by a small angle of perhaps 10 or 15 degrees. Assuming the flowfield is symmetric about the streamwise centerline, the paint data from one side of the centerline can be used to approximate the pressure indicated by the tap on the opposite side of the centerline. Thus eliminating the need for averaging the paint data around a tap.

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APPENDIX A MATLAB CODE: PSPMRCPARAMS % PSP Linear and Nonlinear System Models % and Model Reference Controller (MRC) % for Importing into Simulink format short e % basic parameters % ord=5; % SS Model Order nx=20; % # X Grids minus 1 (must be even) dx=l/nx; % X Grid Spacing tm=0.01; % Model System Time Constant Q. "S % calc lin and nlin systems SS form % a=zeros (ord) ; b=zeros (ord, 2 ) ; c=zeros (nx+1, ord) ; d=zeros (nx+1 , 2 ) ; alin=zeros (ord) ; blin=zeros (ord, 2 ) ; clin=zeros ( 1, ord) ; dlin=zeros (1,2) ; for i=l:ord, lam(i)=(2*i-l) *pi/2; a (i, i) =-lam(i) ^2; alin (i, i) =a (i, i) ; b(i,2)=-2*sin(lam(i) ) /lam(i) ; blin(i,2)=b(i,2) ; clin(l,i)=-b(i,2) /2; for j=l:nx+l, x=(nx+0.999999-j) *dx; % use 0.999 instead of 1 so that xmax
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end end dlin=[l,0] ; % convert from SS form to TF form o o [numl, denl ]=ss2tf(a,b,c,d,l) ; [num2,den2]=ss2tf (a,b,c,d,2) ; [numlinl, denlinl] =ss2tf (alin, blin, clin, dlin, 1 ) [numlin2, denlin2 ] =ss2tf (alin, blin, clin, dlin, 2 ) % num2(:,l)==0 so remove them len=size (num2 , 2 ) ; num2=num2 ( : , 2 : len) ; % numlin2 (1) ==0 so remove it len=length (numlin2) ; numlin2=numlin2 (2 : len) ; % convert from MIxO w/ U = [P,Pdot] % to SIxO w/ U = [P], i.e. get rid % of Pdot(t) as an input % for i=l:nx+l, num ( i , : ) =den2+conv ( [ 1 0 ] , num2 ( i , : ) ) ; end den=den2 ; numlin=denlin2+conv ( [1 0] , numlin2) ; denlin=d :i2; "5 % define linear part of nlin plant [ z , p , k ] =t f 2 zp ( num, den ) ; % ; % define lin plant % [zp, pp, kp] =tf 2zp (numl in, denl in) ; % define augmented system using next % highest pole in sequence from denlin(s)

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% note: 0(1) system = ka/(s+ka) Q, O ka=( (2* (ord+l) -1) *pi/2) ^^2; numa= [ ka] ; dena= [1 ka] ; [ za, pa, ka] =tf 2zp (numa, dena) ; o o % augment lin plant w/ fast 0(1) system o o numlin=conv (numa, numlin) ; denlin=conv (dena, denlin) ; % make numlin (s) a monic polynomial % klin=numlin (1) ; numlin=numlin/klin; dp=length (denlin) -1 ; % (klin, numlin, denlin) define augmented % plant for MRC calculation purposes 0, G define model system a km=l/tm; '. numm= [ 1 ] ; denm= [ 1 km] ; , . Q, "O — — % dO is of degree dp and is % arbitrarily chosen as fast or faster % than the model system % dO=l; for i=l:dp dO=conv([l 10* (i+9) ] ,dO) ; end o o % calc matrix aa

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% % column 1 of aa acl= [denlin, zeros (1, dp-1) ] ; % column dp+1 of aa ac2= [0, numlin, zeros (1, dp-1) ] ; for 1=1: dp, aa (i, : ) =acl ; acl= [0, acl] ; acl=acl (1: (2*dp) ) ; aa (i+dp, : ) =ac2; ac2=[0,ac2] ; ac2=ac2 (1: (2*dp) ) ; end aa=aa ' ; % % calc vector bb % bb=conv (numm, denlin) -conv (numlin, denm) bb=conv (dO, bb) ; % note that bb(l)==0 so remove it len=length (bb) ; bb=bb(2:len) ; bb=bb ' ; O — — — — — — — — — — — — — % calc soln vector xx % xx=aa\bb; relerr= (aa*xx-bb) . /bb % % calc feedback vectors cl and c2 % cO=km/klin; cl=xx ( 1 : dp) ; c2=xx(dp+l:2*dp) ; [zl,pl,kl]=tf2zp(cl' ,dO) ; [z2,p2,k2]=tf2zp(c2',d0) ; % Simpson's rule vector coeffs % integrated intensity =

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% (simp) dot (int of x) % for i=l:nx+l, if mod(i,2)==0 simp (i) =4 ; else simp (i) =2; end end simp (1) =1; simp (nx+1) =l; simp=simp/sum (simp) ; o o % calc butterworth filter params a [zb,pb,)cb]=butter(5,l/tm, 's' ) ;

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APPENDIX B SIMULINK CODE: PSPMRC Main Routine: pspmrc.mdl oCDf(u) 1 exp ( I " 2 PToducl4 te-za(s) pa(8) PIX«) ^ "J l«'za(i) * P»(t) ugL.iyilA'nl k2'z2|s) p2(l) -f5PiDductl Subroutine: nlin system.mdl CD-H p 02 iy«em (1-d diffuaon) p2int coefft Dot Product pv Subroutine: p2int.mdl CD— +Q — -KID 101

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102 • Subroutine: int2p.mdl • Notes: 1 . The following Simulink blocks are continuous time systems in pole-zero form: aug system, HI, H2, butterworth, 02_system. 2. The following parameters are loaded from the Matlab workspace into Simulink: ka, za, pa, kp, zp, pp, kl, zl, pi, k2, z2, p2, km, k, z, p, q, simp. 3. Follow these steps from within the Main Routine, pspmrc.mdl, in order to load the parameters from Note 2: Simulation > Parameters > Real-Time Workshop > Inline Parameters > Tunable Parameters. Enter each of the variables from Note 2 as a tunable parameter.

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APPENDIX C PSP UNCERTAINTY ANALYSIS What follows is a derivation of equation (4. 12). It should be noted that the result derived here is only valid for the case of low light output from the PSP. That is, when the light levels are low the uncertainty due to photon counting statistics will be the dominant source of error in the measurement. All other sources of uncertainty will be ignored as they are assumed to be negligible in comparison to the photonic noise. To begin, consider the uncertainty in the indicated PSP pressure Up = fdp ^ 2 (d? ] + — u, Ui 'J 1 51. 'J (C-1) where I is the intensity of the run image and Ir is the intensity of the reference image. If we define Y = I^/I, the chain rule can be employed to change the partial derivatives in equation (C-1) to arrive at Up = ap I 5YI 7^. + dP 1 -u, aYi (C-2) Simplifying (C-2) yields u =Y p dY] hi 2 + uJ J (C-3) The full-well capacity of the pixels of the CCD camera used in the experiments presented herein is 320,000 electrons. The CCD further has a 14-bit digitizer on the A/D process. 103

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104 Dividing the total electrons by the number of discrete levels of the AID gives a pixel gain of the CCD of p = 19.53 e'/bit. (C-4) The intensities measured by the CCD camera, i.e. the counts or bits, is actually then a number of electrons counted e" = pi . (C-5) Photonic sources may be described by the Poisson distribution.'*^ For Poisson statistics, the variance is equal to the square root of the measurement. Therefore, the uncertainty in the number of electrons is given by u,=V7 = VpI (C-6) which gives the uncertainty in the intensity as 1^= 1 (C-7) Beginning from equation (C-3) we can then say the following Un = Y , " BY ay dY I + 1 1. J J ( rrV ( + 1 V fx PIr + ) = Y 1 + Y dYi pi, (C-8)

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105 Or, we have the relative uncertainty as Up _ Y ap [ l+Y (C-9) P ? dY\ pi. Equation (C-9) is identically the equation (4.12) which ends the proof. One additional point that can be added is the effect of smoothing the images. As mentioned previously in Chapter 4, the smoothing kernel has the effect of boosting the effective number of counts at a give pixel, thus enhancing the statistics. We will define k as the denominator of equation (4.3), The variable Y will be unaffected as it will multiply into both the numerator and the denominator. However, it will scale the denominator of the square root of equation (C-9) yielding Thus, the smoothing kernel will improve the uncertainty in the measured pressure by a (C-10) (C-U) factor of Vic .

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REFERENCES 1. Bell, J. H., Schairer, E. T., Hand, L. A., and Mehta, R. D., "Surface Pressure Measurements Using Luminescent Coatings," Annual Review of Fluid Mechanics, Vol. 33, 2001, pp. 155-206. 2. Liu, T., Campbell, B. T., Bums, S. P., and Sullivan, J. P., "Temperatureand Pressure-Sensitive Luminescent Paints in Aerodynamics," Applied Mechanics Review, Vol. 50, No. 4, April 1997, pp. 227-246. 3. Lu, X. and Winnik, M. A., "Luminescent Quenching by Oxygen in Polymer Films," Organic, Physical and Materials Photochemistry, edited by Ramamurthy, V. and Schanze, K. S., Marcel-Dekker, New York, Vol. 6 of Molecular and Supramolecular Photochemistry, 2000. 4. Ingle, J.D. and Crouch, S.R., Spectrochemical Analysis, Prentice Hall, Englewood Cliffs, NJ, 1988. 5. Schanze, K. S., Carroll, B. F., Korotkevitch, S., and Morris, M. J., "Temperature Dependence of Pressure Sensitive Paints," AIAA Journal, Vol. 35, No. 2, February 1997, pp. 306-310. 6. Crank, J. and Park, G. S., "Diffusion in Polymers," Academic Press, New York, 1968. 7. Roger, C. E., "Permeation of Gases and Vapours in Polymers," Polymer Permeability, Comyn, J. (ed.), Elsevier Applied Science Publishers, London, New York, 1985. 8. Everett, D. E., Woodmansee, M. A., Dutton, J. C, and Morris, M. J., "Wall Pressure Measurements for a Sonic Jet Injected Transversely into a Supersonic Crossflow," yowrwa/ of Propulsion and Power, Vol. 14, No. 6, Dec. 1998, pp. 861-868. 9. Cox, M. E. and Dunn, B., "Oxygen Diffusion in Poly(dimethyl Siloxane) using Fluorescence Quenching. I. Measurement Technique and Analysis," Journal of Polymer Science, Part A: Polymer Chemistry, Vol. 24, 1986, pp. 621-636. 10. Mills, A. and Chang, Q., "Modelled Diffusion-controlled Response and Recovery Behaviour of a Naked Optical Film Sensor With a Hyperbolic-type Response to Analyte Conentration," ^/la/y^/. Vol. 117, Sept. 1992, pp. 1461-1466. 106

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107 11. Baron, A. E., Danielson, J. D. S., Gouterman, M., Wan, J. R., Callis, J. B., and McLachlan, B., "Submillisecond Response Times of Oxygen-Quenched Luminescent Coatings," ^ev. Sci. Instrum., Vol. 64, No. 12, 1993, pp. 3394-3402. 12. Borovoy, V., Bykov, A., Mosharov, V., Orlov, A., Radchenko, V., and Phonov, S., "Pressure Sensitive Paint Application in Shock Wind Tunnel," 16"" ICIASF Record, IEEE-95CH34827, 1995, pp. 34.1-34.4. 13. Engler, R. H., "Further Developments Of Pressure Sensitive Paint (OPMS) For Non Flat Models In Steady Transonic Flow And Unsteady Conditions," 76'^ ICIASF Record, 1EEE-95CH34827, 1995, pp. 33.1-33.8. 14. Carroll, B. F., Winslow, N., Abbitt, J., Schanze, K., and Morris, M., "Pressure Sensitive Paint: Application to a Sinusoidal Pressure Fluctuation," 16'^ ICIASF Record, 1EEE-95CH34827, 1995, pp. 35.1-35.6. 15. Carroll, B. F., Abbitt, J. D., Lukas, E. W., and Morris, M. J., "Step Response of Pressure-Sensitive V^inXs," AIAA Journal, Vol. 34, No. 3, March 1996, pp. 521526. 16. Winslow, N. A., Carroll, B. F., and Setzer, F. M., "Frequency Response of Pressure Sensitive Paints," AIAA Paper No. 961 967, June 1 996. 17. Masoumi, Z., Stoeva, V., Yekta, A., Pang, Z., Maimers, I., and Winnik, M. A., "Luminescence Quenching Method for Probing the Diffusivity of Molecular Oxygen in Highly Permeable Media," Chemical Physics Letters, Vol. 261, Oct. 1996, pp. 551-557. 18. Carroll, B., Winslow, N., and Setzer, F., "Mass Diffusivity of Pressure Sensitive Paints via System Identification," AIAA Paper No. 97-0771, Jan. 1997. 19. Hubner, J. P., Carroll, B. F., Schanze, K. S., and Ji, H. F., "Pressure-sensitive paint measurements in a shock tube," Experiments in Fluids, Vol. 28, Sept. 2000, pp. 21-28. 20. Hubner, J., Carroll, B., Schanze, K., Ji, H., and Holden, M., "Temperatureand Pressure-Sensitive Paint Measurements in Short-Duration Hypersonic Flow," AIAA Journal, Vol. 39, No. 4, April 2001, pp. 654-659. 21. Gruber, M. R., Nejad, A. S., Chen, T. H., and Dutton, J. C, "Mixing and Penetration Studies of Sonic Jets in a Mach 2 Freestream," Journal of Propulsion and Power, Vol. 11, No. 2, March-April 1995, pp. 315-323. 22. Cubbison, R. W., Anderson, B. H., and Ward, J. J., "Surface Pressure Distributions with a Sonic Jet Normal to Adjacent Flat Surfaces at Mach 2.92 and 6.4," NASA TND-580, 1961.

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108 23. Schetz, J. A., Hawkins, P. F., and Lehman, H., "Structure of Highly Underexpanded Transverse Jets in a Supersonic Stream," AIAA Journal, Vol. 5, No. 5, May 1967, pp. 882-884. 24. Orth, R. C, Schetz, J. A., and Billig, F. S., "The Interaction and Penetration of Gaseous Jets in Supersonic Flow," NASA CR-1386, 1969. 25. Billig, F. S., Orth, R. C, and Lasky, M., "A Unified Analysis of Gaseous Jet Penetration," ^7^4 Journal, Vol. 9, No. 6, 1971, pp. 1048-1058. 26. Heister, S. D. and Karagozian, A. R., "Gaseous Jet in Supersonic Crossflow," AIAA Journal, Vol. 28, No. 5, 1990, pp. 819-827. 27. Lee, M. P., McMillan, B. K., Palmer, J. L., and Hanson, R. K., "Planar Fluorescence Imaging of a Transverse Jet in a Supersonic Crossflow," Journal of Propulsion and Power, Vol. 8, No. 4, 1992, pp. 729-735. 28. Wang, K. C, Smith, O. I., and Karagozian, A. R., "In-Flight Imaging of Transverse Gas Jets Injected into Compressible Crossflows," AIAA Journal, Vol. 33, No. 12, December 1995, pp. 2259-2263. 29. Papamoschou, D. and Hubbard, D. G., "Visual Observations of Supersonic Transverse Jets," Experiments in Fluids, Vol. 14, 1993, pp. 468-476. 30. Santiago, J. G. and Dutton, J. C, "Velocity Measurements of a Jet Injected into a Supersonic Crossflow," JoMrwa/ of Propulsion and Power, Vol. 13, No. 2, March-April 1997, pp. 264-273. 3 1 . Johari, H. and Paduano, R., "Dilution and Mixing in an Unsteady Jet," Experiments in Fluids, Vol. 23, No. 4, 1997, pp. 272-280. 32. Hermanson, J. C, Wahba, A., and Johari, H., "Duty-Cycle Effects on Penetration of Fully Modulated, Turbulent Jets in Crossflow," Journal, Vol. 36, No. 10, October 1998, pp. 1935-1937. 33. Johari, H., Pacheco-Tougas, M., and Hermanson, J. C, "Penetration and Mixing of Fully Modulated Turbulent Jets in Crossflow," AIAA Journal, Vol. 37, No. 7, July 1999, pp. 842-850. 34. Bums, S. P. and Sullivan, J. P., "The Use of Pressure Sensitive Paints on Rotating Machinery," I ICMSF Record, IEEE-95CH34827, 1995, pp. 32.1-32.14.

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109 35. Evnochides, S. K. and Henley, E. J., "Simultaneous Measurement of Vapor Difftision and Solubility Coefficients in Polymers by Frequency Response Techniques" Journal of Polymer Science: PartA-2, Vol. 8, 1970, pp. 1987-1997. 36. Carslaw, H. S. and Jaeger, J. C, Conduction of Heat in Solids, Oxford Univ. Press, London, 1959. 37. Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, Inc., New York, 1974. 38. Pauly, S., "Permeability and Diffusion Data," Polymer Handbook, 4th Edition, edited by Brandrup, J., Immergut, E. H. and Grulke, E. A., Wiley-Interscience, New York, 1999, pp. VI/543 VI/568. 39. loannou, P. A. and Sun, J., Robust Adaptive Control, Prentice-Hall, Upper Saddle River, NJ, 1996. 40. Tao, G. and Kokotovic, P. V., Adaptive Control of Systems with Actuator and Sensor Nonlinearities, Wiley-Interscience, New York, 1996. 41 . Carroll, B. F., Dutton, J. C, and Addy, A. L., "N0ZCS2: A Computer Program for the Design of Continuous Slope Supersonic Nozzles," Univ. of Illinois at Urbana-Champaign, Report No. UILU ENG 86-4007, Urbana, IL, August 1986. 42. Carroll, B. F., "A Numerical and Experimental Investigation of Muhiple Shock Wave/Turbulent Boundary Layer Interactions in a Rectangular Duct," Ph.D. Dissertation, Dept. of Mechanical and Industrial Engineering, Univ. of Illinois at Urbana-Champaign, Urbana, IL, 1988. 43. Schetz, J. A. and Billig, F. S., "Penetration of Gaseous Jets Injected into a Supersonic Stream," ^7^4 Journal, Vol. 3, No. 11, 1966, pp. 1658-1665. 44. Orth, R. C. and Funk, J. A., "An Experimental and Comparative Study of Jet Penetration in Supersonic Flow," Journal of Spacecraft and Rockets, Vol. 4, No. 9, 1967, pp. 1236-1242. 45. Meyer, S. L., Data Analysis for Scientists and Engineers, John Wiley & Sons, Inc., New York, 1975.

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BIOGRAPHICAL SKETCH Neal Andrew 'Andy' Winslow was bom and raised in his hometown of Columbia, Missouri. He graduated from David H. Hickman High School in 1987 where he had excelled in math and the physical sciences. After high school, he attended the University of Missouri in Columbia. As an undergraduate, Andy dual-majored in Math and Physics. In December of 1991 he graduated with a B.S. Degree in Physics and a B.A. Degree in Mathematics. He stayed on at the University of Missouri and in August of 1994 he received his M.S. in Physics under the advisorship of Dr. Fred Ross. His masters thesis was entitled "Gamma-Ray Induced XRF Measurement of Lead with Low Resolution Detectors." He then found himself at the University of Florida in Gainesville pursuing a Ph.D. in Engineering Mechanics. After finishing up his doctoral degree, Andy has no plans other than to travel and enjoy some of what life has to offer. 110

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jruce F. Carroll, Chairman Associate Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. tOU SKirk S. Schanze Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Andrew J. fCurdila Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Martin J. Morris V Associate Professor of Mechanical Engineering, Bradley University

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This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. fPramod P. Khargonekar Dean, College of Engineering August 2001 Winfred M. Phillips Dean, Graduate School


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