Pressure and Temperature Sensitive Paints 9783540222415, 3540222413, 9783540266440, 3540266445

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Experimental Fluid Mechanics R.J. Adrian · M. Gharib · W. Merzkirch D. Rockwell · J.H. Whitelaw

T. Liu J.P. Sullivan

Pressure and Temperature Sensitive Paints

Tianshu Liu NASA Langley Research Center MS 493, Hampton, VA 23681-0001 USA Prof. John P. Sullivan Purdue University School of Aeronautics and Astronautics 315 N. Grant St. Grissom Hall West Lafayette, IN 47907-2023 USA

ISBN 3-540-22241-3

Springer Berlin Heidelberg New York

Library of Congress Control Number: 2004109333 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com

c Springer-Verlag Berlin Heidelberg 2005
Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by the authors. Final processing by PTP-Berlin Protago-TeX-Production GmbH, Germany Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 61/3020Yu - 5 4 3 2 1 0

Series Editors Prof. R.J. Adrian University of Illinois at Urbana-Champaign Dept. of Theoretical and Applied Mechanics 216 Talbot Laboratory 104 South Wright Street Urbana, IL 61801 USA Prof. M. Gharib California Institute of Technology Graduate Aeronautical Laboratories 1200 E. California Blvd. MC 205-45 Pasadena, CA 91125 USA Prof. Dr. W. Merzkirch Universit¨ at Essen Lehrstuhl f¨ ur Str¨ omungslehre Sch¨ utzenbahn 70 45141 Essen Germany Prof. Dr. D. Rockwell Lehigh University Dept. of Mechanical Engineering and Mechanics Packard Lab. 19 Memorial Drive West Bethlehem, PA 18015-3085 USA Prof. J.H. Whitelaw Imperial College Dept. of Mechanical Engineering Exhibition Road London SW7 2BX UK

Preface

The aim of this book is to provide a systematic description of pressure and temperature sensitive paints (PSP and TSP) developed since the 1980s for aerodynamics/fluid mechanics and heat transfer experiments. PSP is the first global optical technique that is able to give non-contact, quantitative surface pressure visualization for complex aerodynamic flows and provide tremendous information on flow structures that cannot be easily obtained using conventional pressure sensors. TSP is a valuable addition to other global temperature measurement techniques such as thermographic phosphors, thermochromic liquid crystals and infrared thermography. This book mainly covers research made in the United States, Japan, Germany, France, Great Britain and Canada. Excellent work on PSP in Russia has been described in the book “Luminescent Pressure Sensors in Aerodynamic Experiments” by V. E. Mosharov, V. N. Radchenko and S. D. Fonov of the Central Aerohydrodynamic Institute (TsAGI). We are truly grateful to our colleagues in the field of PSP and TSP for kindly providing their paper drafts, offering comments, and allowing us to use their published results. Without their helps, this book cannot be completed. Especially, we would like to thank the following individuals and organizations: T. Amer, K. Asai, J. H. Bell, T. J. Bencic, O. C. Brown, G. Buck, A. W. Burner, S. Burns, B. Campbell, B. F. Carroll, L. N. Cattafesta, J. Crafton, R. C. Crites, G. Dale, R. H. Engler, R. G. Erausquin, W. Goad, L. G. Goss, J. W. Gregory, M. Gounterman, M. Guille, M. Hamner, J. M. Holmes, C. Y. Huang, J. P. Hubner, J. Ingram, H. Ji, R. Johnston, J. D. Jordan, M. Kameda, M. Kammeyer, J. T. Kegelman, N. Lachendro, J. Lepicovsky, Y. Le Sant, X. Lu, Y. Mebarki, R. D. Mehta, K. Nakakita, C. Obara, D. M. Oglesby, T. G. Popernack, W. M. Ruyten, H. Sakaue, E. T. Schairer, K. S. Schanze, M. E. Sellers, Y. Shimbo, K. Teduka, S. D. Torgerson, B. T. Upchurch, A. N. Watkins. NASA, ONR, AFOSR, Boeing, Raytheon, Japanese NAL.

Table of Contents

1.

2.

3.

4.

5.

Introduction .................................................................................................... 1 1.1. Pressure Sensitive Paint ....................................................................... 2 1.2. Temperature Sensitive Paint ................................................................ 8 1.3. Historical Remarks ............................................................................ 11 Basic Photophysics....................................................................................... 15 2.1. Kinetics of Luminescence.................................................................. 15 2.2. Models for Conventional Pressure Sensitive Paint ............................ 18 2.3. Models for Porous Pressure Sensitive Paint ...................................... 24 2.3.1. Collision-Controlled Model .................................................... 26 2.3.2. Adsorption-Controlled Model................................................. 27 2.4. Thermal Quenching ........................................................................... 31 Physical Properties of Paints ........................................................................ 33 3.1. Calibration ......................................................................................... 33 3.2. Typical Pressure Sensitive Paints ...................................................... 34 3.3. Typical Temperature Sensitive Paints................................................ 45 3.4. Cryogenic Paints................................................................................ 50 3.5. Multi-Luminophore Paints................................................................. 54 3.6. ‘Ideal’ Pressure Sensitive Paint ......................................................... 56 3.7. Desirable Properties of Paints............................................................ 58 Radiative Energy Transport and Intensity-Based Methods .......................... 61 4.1. Radiometric Notation......................................................................... 61 4.2. Excitation Light ................................................................................. 62 4.3. Luminescent Emission and Photodetector Response......................... 65 4.4. Intensity-Based Measurement Systems ............................................. 69 4.4.1. CCD Camera System .............................................................. 70 4.4.2. Laser Scanning System ........................................................... 74 4.5. Basic Data Processing........................................................................ 75 Image and Data Analysis Techniques........................................................... 81 5.1. Geometric Calibration of Camera...................................................... 82 5.1.1. Collinearity Equations............................................................. 82 5.1.2. Direct Linear Transformation ................................................. 85 5.1.3. Optimization Method .............................................................. 87 5.2. Radiometric Calibration of Camera ................................................... 92 5.3. Correction for Self-Illumination ........................................................ 95 5.4. Image Registration........................................................................... 102 5.5. Conversion to Pressure .................................................................... 105 5.6. Pressure Correction for Extrapolation to Low-Speed Data.............. 107 5.7. Generation of Deformed Surface Grid............................................. 112

X

6.

7.

8.

Table of Contents

Lifetime-Based Methods ............................................................................ 115 6.1. Response of Luminescence to Time-Varying Excitation Light ....... 116 6.1.1. First-Order Model ................................................................. 116 6.1.2. Higher-Order Model ............................................................. 118 6.2. Lifetime Measurement Techniques.................................................. 118 6.2.1. Pulse Method ........................................................................ 118 6.2.2. Phase Method........................................................................ 119 6.2.3. Amplitude Demodulation Method ........................................ 121 6.2.4. Gated Intensity Ratio Method ............................................... 123 6.3. Fluorescence Lifetime Imaging ....................................................... 128 6.3.1. Intensified CCD Camera ....................................................... 128 6.3.2. Internally Gated CCD Camera .............................................. 130 6.4. Lifetime Experiments ...................................................................... 131 Uncertainty ................................................................................................. 137 7.1. Pressure Uncertainty of Intensity-Based Methods........................... 137 7.1.1. System Modeling .................................................................. 137 7.1.2. Error Propagation, Sensitivity and Total Uncertainty ........... 138 7.1.3. Photodetector Noise and Limiting Pressure Resolution........ 141 7.1.4. Errors Induced by Model Deformation ................................. 144 7.1.5. Temperature Effect ............................................................... 145 7.1.6. Calibration Errors.................................................................. 145 7.1.7. Temporal Variations in Luminescence and Illumination ...... 146 7.1.8. Spectral Variability and Filter Leakage ................................ 146 7.1.9. Pressure Mapping Errors....................................................... 146 7.1.10. Paint Intrusiveness ................................................................ 147 7.1.11. Other Error Sources and Limitations .................................... 148 7.1.12. Allowable Upper Bounds of Elemental Errors...................... 149 7.1.13. Uncertainties of Integrated Forces and Moments.................. 150 7.2. Pressure Uncertainty Analysis for Subsonic Airfoil Flows ............. 151 7.3. In-Situ Calibration Uncertainty ....................................................... 156 7.3.1. Experiments .......................................................................... 156 7.3.2. Simulation ............................................................................. 159 7.4. Pressure Uncertainty of Lifetime-Based Methods ........................... 163 7.4.1. Phase Method........................................................................ 163 7.4.2. Amplitude Demodulation Method ........................................ 164 7.4.3. Gated Intensity Ratio Method ............................................... 166 7.5. Uncertainty of Temperature Sensitive Paint .................................... 169 7.5.1. Error Propagation and Limiting Temperature Resolution..... 169 7.5.2. Elemental Error Sources ....................................................... 170 Time Response ........................................................................................... 175 8.1. Time Response of Conventional Pressure Sensitive Paint............... 175 8.1.1. Solutions of Diffusion Equation............................................ 175 8.1.2. Pressure Response and Optimum Thickness......................... 178 8.2. Time Response of Porous Pressure Sensitive Paint ......................... 182 8.2.1. Deviation from the Square-Law............................................ 182 8.2.2. Effective Diffusivity ............................................................. 183 8.2.3. Diffusion Timescale.............................................................. 186

Table of Contents

XI

8.3. 8.4.

Measurements of Pressure Time Response...................................... 187 Time Response of Temperature Sensitive Paint .............................. 194 8.4.1. Pulse Laser Heating on Thin Metal Film .............................. 195 8.4.2. Step-Like Jet Impingement Cooling ..................................... 198 9. Applications of Pressure Sensitive Paint .................................................... 201 9.1. Low-Speed Flows ............................................................................ 201 9.1.1. Airfoil Flows ......................................................................... 201 9.1.2. Delta Wings, Swept Wings and Car Models......................... 207 9.1.3. Impingement Jet.................................................................... 214 9.2. Subsonic, Transonic and Supersonic Wind Tunnels........................ 215 9.2.1. Aircraft Model in Transonic Flow ........................................ 215 9.2.2. Supercritical Wing at Cruising Speed ................................... 221 9.2.3. Transonic Wing-Body Model ............................................... 223 9.2.4. Laser Scanning Pressure Measurement on Transonic Wing . 225 9.2.5. Boundary Layer Control in Supersonic Inlets....................... 226 9.3. Hypersonic and Shock Wind Tunnels.............................................. 230 9.3.1. Expansion and Compression Corners ................................... 230 9.3.2. Moving Shock Impinging to Cylinder Normal to Wall......... 236 9.4. Cryogenic Wind Tunnels ................................................................. 237 9.5. Rotating Machinery ......................................................................... 242 9.5.1. Laser Scanning Measurements.............................................. 242 9.5.2. CCD Camera Measurements................................................. 247 9.6. Impinging Jets.................................................................................. 249 9.7. Flight Tests ...................................................................................... 256 9.8. Micronozzle ..................................................................................... 260 10. Applications of Temperature Sensitive Paint ............................................. 263 10.1. Hypersonic Flows .......................................................................... 263 10.2. Boundary-Layer Transition Detection ........................................... 270 10.3. Impinging Jet Heat Transfer .......................................................... 275 10.4. Shock/Boundary-Layer Interaction ................................................ 280 10.5. Laser Spot Heating and Heat Transfer Measurements ................... 282 10.6. Hot-Film Surface Temperature in Shear Flow ............................... 289 References .................................................................................................. 293 Appendix A. Calibration Apparatus ........................................................... 313 Appendix B. Recipes of Typical Pressure and Temperature Sensitive Paints................................................. 317 Appendix C. Vendors ................................................................................. 319 Color Plates ................................................................................................ 321 Index........................................................................................................... 327

1. Introduction

Quantitative measurements of surface pressure and temperature in wind tunnel and flight testing are essential to understanding of the aerodynamic performance and heat transfer characteristics of flight vehicles. Pressure data are required to determine the distribution of aerodynamic loads for the design of a flight vehicle, while temperature data are used to estimate heat transfer on the surface of the vehicle. Pressure and temperature measurements provide critical information on important flow phenomena such as shock, flow separation and boundary-layer transition. In addition, accurate pressure and temperature data play a key role in validation and verification of computational fluid dynamics (CFD) codes. Traditionally, surface pressure is measured by utilizing a pressure tap or orifice at a location of interest connected through a small tube to a pressure transducer (Barlow et al. 1999). Hundreds of pressure taps are needed to obtain an acceptable pressure field on a complex aircraft model. Manufacturing, tubing and preparing such a model for wind tunnel testing is very labor-intensive and costly. For thin models such as supersonic transports, military aircraft and small fan blades, installation of a large number of pressure taps is impossible. Furthermore, pressure measurements at discrete taps ultimately limit the spatial resolution of measurements such that some details of a complex flow field cannot be revealed. Similarly, a surface temperature field is traditionally measured using temperature sensors such as thermocouples and resistance thermometers distributed at discrete locations (Moffat 1990). Since the 1980s, new optical sensors for measuring surface pressure and temperature have been developed based on the quenching mechanisms of luminescence. These luminescent molecule sensors are called pressure sensitive paint (PSP) and temperature sensitive paint (TSP). Compared with conventional techniques, they offer a unique capability for non-contact, full-field measurements of surface pressure and temperature on a complex aerodynamic model with a much higher spatial resolution and a lower cost. Therefore, they provide a powerful tool for experimental aerodynamicists to gain a deeper understanding of rich physical phenomena in complex flows around flight vehicles. Both PSP and TSP use luminescent molecules as probes that are incorporated into a suitable polymer coating on an aerodynamic model surface. In general, the luminophore and polymer binder in PSP and TSP can be dissolved in a solvent; the resulting paint can be applied to a surface using a sprayer or brush. After the solvent evaporates, a solid polymer coating in which the luminescent molecules are immobilized remains on the surface. When a light of a proper wavelength illuminates the paint, the luminescent molecules are excited and the luminescent

2

1. Introduction

light of a longer wavelength is emitted from the excited molecules. Figure 1.1 shows a schematic of a generic luminescent paint layer emitting radiation under excitation by an incident light. Detector (CCD, PMT, PD)

Excitation light (UV, laser, LED)

Optical Filter

Calibrated output for pressure & temperature

Emission

Luminescent molecule Binder (polymer, porous solid)

Fig. 1.1. Schematic of a luminescent paint (PSP or TSP) on a surface

The luminescent emission from a paint layer can be affected by certain physical processes. The main photophysical process in PSP is oxygen quenching that causes a decrease of the luminescent intensity as the partial pressure of oxygen or air pressure increases. The polymer binder for PSP is oxygen permeable, which allows oxygen molecules to interact with the luminescent molecules in the binder. For certain fast-responding PSP, a mixture of the luminophore and solvent is directly applied to a porous solid surface. In fact, PSP is an oxygen-sensitive sensor. By contrast, the major mechanism in TSP is thermal quenching that reduces the luminescent intensity as temperature increases. TSP is not sensitive to air pressure since the polymer binder used for TSP is oxygen impermeable, while due to the thermal quenching PSP is intrinsically temperature-sensitive. After PSP and TSP are appropriately calibrated, pressure and temperature can be remotely measured by detecting the luminescent emission. PSP and TSP are companion techniques because they not only utilize luminescent molecules as probes, but also use the same measurement systems and similar data processing methods.

1.1. Pressure Sensitive Paint The basic concepts of pressure sensitive paint (PSP) are simple. After a photon of radiation with a certain frequency is absorbed to excite the luminophore from the ground electronic state to the excited electronic state, the excited electron returns

1.1. Pressure Sensitive Paint

3

to the unexcited ground state through radiative and radiationless processes. The radiative emission is called luminescence (a general term for both fluorescence and phosphorescence). The excited state can be deactivated by interaction of the excited luminophore molecules with oxygen molecules in a radiationless process; that is, oxygen molecules quench the luminescent emission. According to Henry’s law, the concentration of oxygen in a PSP polymer is proportional to the partial pressure of oxygen in gas above the polymer. For air, pressure is proportional to the oxygen partial pressure. So, for higher air pressure, more oxygen molecules exist in the PSP layer and as a result more luminescent molecules are quenched. Hence, the luminescent intensity is a decreasing function of air pressure. The relationship between the luminescent intensity and oxygen concentration can be described by the Stern-Volmer relation. For experimental aerodynamicists, a convenient form of the Stern-Volmer relation between the luminescent intensity I and air pressure p is

I ref I

= A+ B

p p ref

,

(1.1)

where I ref and p ref are the luminescent intensity and air pressure at a reference condition, respectively. The Stern-Volmer coefficients A and B, which are temperature-dependent due to the thermal quenching, are experimentally determined by calibration. Theoretically speaking, the intensity ratio I ref / I can eliminate the effects of non-uniform illumination, uneven coating and nonhomogenous luminophore concentration in PSP. In typical tests in a wind tunnel, I ref is taken when the tunnel is turned off and hence it is often called the wind-off intensity (or image); likewise, I is called the wind-on intensity (or image). Figures 1.2 and 1.3 show, respectively, the luminescent intensity as a function of pressure at the ambient temperature and the corresponding Stern-Volmer plots for three PSPs: Ru(ph2-phen) in GE RTV 118, Pyrene in GE RTV 118 and PtOEP in GP 197. A measurement system for PSP or TSP is generally composed of paint, illumination light, photodetector, and data acquisition/processing unit. Figure 1.4 shows a generic CCD camera system for both PSP and TSP. Many light sources are available for illuminating PSP/TSP, including lasers, ultraviolet (UV) lamps, xenon lamps, and light-emitting-diode (LED) arrays. Scientific-grade chargecoupled device (CCD) cameras are often used as detectors because of their good linear response, high dynamic range and low noise. Other commonly-used photodetectors are photomultiplier tubes (PMT) and photodiodes (PD). A generic laser-scanning system, as shown in Fig. 1.5, typically uses a laser with a computer-controlled scanning mirror as an illumination source and a PMT as a detector along with a lock-in amplifier for both intensity and phase measurements. Optical filters are used in both systems to separate the luminescent emission from the excitation light.

4

1. Introduction

7 Ru(ph2-phen) in GE RTV 118 Pyrene in GE RTV 118 PtOEP in GP 197

6 5

I/Iref

4 3 2 1 0 0.0

0.2

0.4

0.6

0.8

1.0

p/pref Fig. 1.2. The luminescent intensity as a function of pressure for three PSPs at the ambient temperature, where pref is the ambient pressure and Iref is the luminescence intensity at the ambient conditions. 1.0

0.8

Iref/I

0.6

0.4

Ru(ph2-phen) in GE RTV 118

0.2

Pyrene in GE RTV 118 PtOEP in GP 197

0.0 0.0

0.2

0.4

0.6

0.8

1.0

p/pref Fig. 1.3. The Stern-Volmer plots for three PSPs at the ambient temperature, where pref is the ambient pressure and Iref is the luminescence intensity at the ambient conditions.

1.1. Pressure Sensitive Paint

5

Fig. 1.4. Generic CCD camera system for PSP and TSP

Once PSP is calibrated, in principle, pressure can be directly calculated from the luminescent intensity using the Stern-Volmer relation. Nevertheless, practical data processing is more elaborate in order to suppress the error sources and improve the measurement accuracy of PSP. For an intensity-based CCD camera system, the wind-on image often does not align with the wind-off reference image due to aeroelastic deformation of a model in wind tunnel testing. Therefore, the image registration technique must be used to re-align the wind-on image to the wind-off image before taking a ratio between those images. Also, since the SternVolmer coefficients A and B are temperature-dependent, temperature correction is certainly required since the temperature effect of PSP is the most dominant error source in PSP measurements. In wind tunnel testing, the temperature effect of PSP is to a great extent compensated by the in-situ calibration procedure that directly correlates the luminescent intensity to pressure tap data obtained at welldistributed locations on a model during tests. To further reduce the measurement uncertainty, additional data processing procedures are applied, including image

6

1. Introduction

summation, dark-current correction, flat-field correction, illumination compensation, and self-illumination correction. After a pressure image is obtained, to make pressure data more useful to aircraft design engineers, data in the image plane should be mapped onto a model surface grid in the 3D object space. Therefore, geometric camera calibration and image resection are necessary to establish the relationship between the image plane and the 3D object space. Besides the intensity-ratio method for a single-luminophore PSP, lifetime measurement systems and multi-luminophore PSP systems have also been developed. Theoretically speaking, the luminescent lifetime is independent of the luminophore concentration, illumination level and coating thickness. Hence, the lifetime method does not require the reference intensity (or image) and it is ideally immune from the troublesome ratioing process in the intensity-ratio method for a deformed model. Similarly, one of the purposes of developing the multipleluminophore PSP system is to eliminate the need of the wind-off reference image and reduce the error associated with model deformation. Another goal of using the multiple-luminophore PSP system is to compensate the temperature effect of PSP.

Computer

Laser Modulator

Lock-in Amplifier

PMT

2D Scanner

Laser Beam

Luminescence

Painted Model

Fig. 1.5. Generic laser scanning lifetime system for PSP and TSP

1.1. Pressure Sensitive Paint

7

Most PSP measurements have been conducted in high subsonic, transonic and supersonic flows on various aerodynamic models in both large production wind tunnels and small research wind tunnels. PSP is particularly effective in a range of Mach numbers from 0.3 to 3.0. Figure 1.6 shows a typical PSP-derived pressure field on the F-16C model at Mach 0.9 and the angle-of-attack of 4 degrees, which was obtained by Sellers and his colleagues (Sellers 1998a, 1998b, 2000; Sellers and Brill 1994) at the Arnold Engineering Development Center (AEDC). For PSP measurements in large wind tunnels, the accuracy of PSP is typically 0.02-0.03 in the pressure coefficient, while in well-controlled experiments the absolute pressure accuracy of 1 mbar (0.0145 psi) can be achieved. In short-duration hypersonic tunnels (Mach 6-10), measurements require very fast time response of PSP and minimization of the temperature effect of PSP. Binder-free, porous anodized aluminum (AA) PSP has been used in hypersonic flows and rotating machinery since it has a very short response time of 30–100 µs in comparison with a timescale of about 0.5 s for a conventional polymer-based PSP. Furthermore, because AA-PSP is a part of an aluminum model, an increase of the surface temperature in a short duration is relatively small due to the high thermal conductivity of aluminum. Since a porous PSP usually exhibits the pressure sensitivity at cryogenic temperatures, AA-PSP and polymerbased porous PSP have been used for pressure measurements in cryogenic wind tunnels where the oxygen concentration is extremely low and the total temperature is as low as 90 K.

Fig. 1.6. PSP image for the F-16C model at Mach 0.9 and the angle-of-attack of 4 degrees. From Sellers (2000)

8

1. Introduction

PSP measurements in low-speed flows are difficult since a very small change in pressure must be resolved and the major error sources must be minimized to obtain acceptable quantitative pressure results. Some low-speed PSP measurements were conducted on delta wings where upper surface pressure exhibited a relatively large change induced by the leading-edge vortices. In addition, experiments were conducted on airfoils, car models and impinging jets at speeds as low as 20 m/s. The pressure resolution of PSP in low-speed flows is ultimately limited by the photon shot noise of a CCD camera. Instead of pushing PSP instrumentation to the limit in low-speed flows, the pressure-correction method was proposed to recover the incompressible pressure coefficient from PSP results obtained in subsonic flows at suitably higher Mach numbers by removing the compressibility effect. Since PSP is a non-contact technique, it is particularly suitable to pressure measurements on high-speed rotating blades in rotating machinery where conventional techniques are difficult to use. Both CCD camera and laser scanning systems have been used for PSP measurements on rotating blades in turbine engines and helicopters. Impinging jets were used in some studies as a canonical flow for testing the performance of PSP systems. Flight test is a challenging area where PSP has showed its advantages as a non-contact, optical pressure measurement technique. The pressure distributions on wings and parts of aircrafts have been measured using film-based camera systems in early in-flight experiments and a laser scanning system in recent flight tests.

1.2. Temperature Sensitive Paint Temperature sensitive paint (TSP) is a polymer-based paint in which the temperature-sensitive luminescent molecules are immobilized. The quantum efficiency of luminescence decreases with increasing temperature; this effect associated with temperature is thermal quenching that serves as the major working mechanism for TSP. Over a certain temperature range, a relation between the luminescent intensity I and absolute temperature T can be written in the Arrhenius form

ln

E §1 I( T ) 1 = nr ¨ − ¨ I ( Tref ) R © T Tref

· ¸, ¸ ¹

(1.2)

where Enr is the activation energy for the non-radiative process, R is the universal gas constant, and Tref is a reference temperature in Kelvin. Figures 1.7 and 1.8 show, respectively, the temperature dependencies of the luminescent intensity and the Arrhenius plots for three TSPs: Ru(bpy) in Shellac, Rodamine-B in dope and EuTTA in dope. The procedure for applying TSP to a surface is basically the same as that for PSP. Not only does TSP use the same measurement systems shown in Figs. 1.4 and 1.5, but also most data processing methods for TSP are similar to those for PSP. Ideally, TSP can be used in tandem with PSP to correct

1.2. Temperature Sensitive Paint

9

the temperature effect of PSP and simultaneously obtain the temperature and pressure distributions. Compared to conventional temperature sensors, TSP is a global measurement technique that is able to obtain the surface temperature distribution with reasonable accuracy at a much higher spatial resolution. o A family of TSPs has been developed, covering a temperature range of –196 C o o to 200 C. The accuracy of TSP is typically 0.2-0.8 C. TSP has been used in various aerodynamic experiments to measure the temperature and heat transfer distributions. In hypersonic wind tunnel tests, TSP not only visualized flow transition patterns, but also provided quantitative heat transfer data calculated based on quasi-steady and transient heat transfer models. Figure 1.9 shows a windward-side heat transfer image of the lower half of the waverider model at Mach 10 at 0.57 s after the wind tunnel started to run (Liu et al. 1995b), where the 2 gray intensity bar denotes heat flux in kW/m . TSP is an effective technique for visualizing boundary-layer transition from laminar to turbulent flow. Due to a significant difference in convection heat transfer between the laminar and turbulent flow regimes, TSP can visualize a surface temperature change across the transition line. In low-speed wind tunnel tests, a model is typically heated or cooled to increase the temperature difference. However, in high-speed flows, friction heating often produces a sufficient temperature difference for TSP transition visualization. Cryogenic TSPs have been used to detect boundary-layer transition on airfoils in cryogenic wind tunnels over a range of the total temperatures from 90 K to 150 K. Complemented with other techniques, TSP has been used to study the relationship between heat transfer and flow structures in an acoustically excited impinging jet. The mapping capability of TSP allows quantitative visualization of the impingement heat transfer fields controlled (enhanced or suppressed) by acoustical excitation. The heat transfer fields in complex separated flows induced by shock/boundary-layer interactions have also been studied using TSP. A novel heat transfer measurement technique has been developed, which combines a laser scanning TSP and a laser spot heating units into a single non-intrusive system. An infrared laser was used to generate local heat flux and convection heat transfer was determined based on a transient heat transfer model from the surface temperature response measured using TSP. This system was applied to quantitative heat transfer measurements in complex flows on a 75-degree swept delta wing and around an intersection of a strut and a wall. Through an optical magnification system, TSP can achieve a very high spatial resolution over the surface of a small object like MEMS devices. TSP has been used to measure the surface temperature field of a miniature flush-mounted hotfilm sensor in a flat-plate turbulent boundary layer.

1. Introduction

1.5

EuTTA-dope

I(T)/I(Tr)

Ru(bpy)-Shellac

1.0

0.5 Rhodamine-B-dope

0.0 0

20

40

60

80

T (deg. C)

Fig. 1.7. Temperature dependencies of the luminescent intensity for three TSPs

0.5 0.0 ln[I(T)/I(Tr)]

10

Ru(bpy)-Shellac

-0.5 -1.0

Rodamine-B-dope

-1.5 EuTTA-dope

-2.0 -2.5 -0.6

-0.4

-0.2 3

0.0 -1

(1/T - 1/Tr )10 ( K )

Fig. 1.8. The Arrhenius plots for three TSPs

0.2

1.3. Historical Remarks

11

Fig. 1.9. Heat transfer image obtained using TSP on the windward side of the waverider at Mach 10. From Liu et al. (1995b)

1.3. Historical Remarks The working principles of PSP are based on the oxygen quenching of luminescence that was first discovered by H. Kautsky and H. Hirsch (1935). The quenching effect of luminescence by oxygen was used to detect small quantities of oxygen in medical applications (Gewehr and Delpy 1993) and analytical chemistry (Lakowicz 1991, 1999) before experimental aerodynamicists realized its utility as an optical sensor for measuring air pressure on a surface. J. Peterson and V. Fitzgerald (1980) demonstrated a surface flow visualization technique based on the oxygen quenching of dye fluorescence and revealed the possibility of using oxygen sensors for surface pressure measurements. Pioneering studies of applying oxygen sensors to aerodynamic experiments were initiated independently by scientists at the Central Aero-Hydrodynamic Institute (TsAGI) in Russia and the University of Washington in collaboration with the Boeing Company and the NASA Ames Research Center in the United States. The conceptual transformation from oxygen concentration measurement to surface pressure measurement was really a critical step for aerodynamic applications of PSP, signifying a paradigm shift from conventional point-based pressure measurement to global pressure mapping. G. Pervushin and L. Nevsky (1981) of TsAGI, inspired by the work of I. Zakharov et al. (1964, 1974) on oxygen measurement, suggested the use of the oxygen quenching phenomenon for pressure measurements in aerodynamic experiments. The first PSP measurements at TsAGI were conducted at Mach 3 on a sphere, a half-cone and a flat plate with an upright block that were coated with a long-lifetime luminescent paint excited by a flash lamp. Their PSP was acriflavine or beta-aminoanthraquinone in a matrix consisting of silichrome, starch, sugar and polyvinylpyrrolidone. A photographic film camera was used for imaging the luminescent intensity field. The results obtained in these tests were in reasonable agreement with the known theoretical solution and pressure tap data (Ardasheva et al. 1982, 1985). Another TsAGI group consisting of A. Orlov, V.

12

1. Introduction

Mosharov, S. Fonov and V. Radchenko started their research in 1983 to improve the accuracy of PSP by measuring the lifetime (the decay time). In their first tests on a cone-cylinder model at Mach 2.5 and 3.0, they used a photomultiplier tube as a detector and a pulsed Argon laser mechanically scanned over a surface to excite a newly developed PSP (Radchenko 1985). Unfortunately, they found that the lifetime measurements suffered from very strong temperature sensitivity (about o 7%/ C) and a very long lifetime (about several minuets) of their first PSP. As a result, their effort has been exclusively focused on the development of intensitybased techniques since 1985. At TsAGI, a number of proprietary PSP formulations have been developed and applied to various subsonic, transonic, supersonic, shock, dynamic tunnels, and rotating machinery (Bukov et al. 1992, 1993; Troyanovsky et al. 1993; Mosharov et al. 1997). The imaging devices used at TsAGI covered a range of photographic film cameras, TV cameras, scientific grade CCD cameras and photomultiplier tubes with laser scanning systems. In the later 1980s, TsAGI marketed its PSP technology through the Italian firm INTECO and issued a one-page advertisement in the magazine ‘Aviation Week & Space Technology’ in February 12, 1990. Interestingly, scientists in the Western World were not aware of Russia’s work on PSP until reading the advertisement. Then, TsAGI’s PSP system was demonstrated in several wind tunnel tests at the Boeing Company in 1990 and Deutsche Forschungsanstalt fur Luft- und Raumfahrt (DLR) in Germany in 1991, which attracted widespread attention of researchers in the aerospace community (Volan and Alati 1991). PSP was independently developed by a group of chemists led by M. Gouterman and J. Callis at the University of Washington (UW) in the late 1980s (Gouterman et al. 1990; Kavandi et al. 1990). The chemists at UW were initially interested in use of porphyrin compounds as an oxygen sensor for biomedical applications. After stimulating discussions with experimental aerodynamicists J. Crowder of the Boeing Company and B. McLachlan of NASA Ames, Gouterman and Callis understood the important implication of oxygen sensors in aerodynamic testing and started to develop a luminescent coating applied to surface for pressure measurements. Their classical PSP used platinum-octaethylporphorin (PtOEP) as a luminescent probe molecule in a proprietary commercial polymer mixture called GP-197 made by the Genesee Company. In 1989, using PtOEP in GP-197, M. Gouterman and J. Kavandi conducted PSP measurements on a NACA 0012 airfoil model (3-in chord and 9-in span) in the 25×25 cm wind tunnel at NASA Ames Fluid Dynamics Laboratory. The model was spray coated with a commercial white epoxy Krylon base-coat and then sprayed with PtOEP in GP-197. An UV lamp was used for excitation, and an analog camera interfaced to an IBM-AT computer with an 8-bit frame grabber for image acquisition. The model was set at o the angle-of-attack of 5 and the Mach numbers ranged from 0.3 to 0.66. Their data showed very favorable agreement with pressure tap data, clearly indicating the formation of a shock on the upper surface of the model as the Mach number increases (Kavandi et al. 1990; McLachlan et al. 1993a). More importantly, this work established the basic procedures for intensity-based PSP measurements such as image ratioing and in-situ calibration. Following the tests at NASA Ames, Kavandi demonstrated the same PSP system in the Boeing Transonic Wind

1.3. Historical Remarks

13

Tunnel on various commercial airplane models, which was briefly discussed by Crowder (1990). Several proprietary paint formulations have been developed at UW, and successfully applied to wind tunnel testing at the Boeing Company and NASA Ames (McLachlan et al. 1993a, 1993b, 1995; McLachlan and Bell 1995; Bell and McLachlan 1993, 1996; Gouterman 1997). Excellent work on PSP was also made at the former McDonnell Douglas (MD, now the Boeing Company at St. Louis) (Morris et al. 1993a, 1993b; Morris 1995; Morris and Donovan 1994; Donovan et al. 1993; Dowgwillo et al. 1994, 1996; Crites 1993; Crites and Benne 1995). MD PSPs were mainly based on Ruthenium compounds that were successfully used in subsonic, transonic and supersonic flows for a generic wing-body model, a full-span ramp, F-15 model, and a converging-diverging nozzle. Other major PSP research groups in the United States include NASA Langley, NASA Glenn, Arnold Engineering Development Center (AEDC), United States Air Force Wright-Patterson Laboratory, Purdue University, and University of Florida. European researchers in DLR (Germany), British Aerospace (BAe, UK), British Defense Evaluation and Research Agency (DERA, UK), and Office National d’Etudes et de Recherches Aerospatiales (ONERA, France) have been active in the field of PSP (Engler et al. 1991, 1992; Engler and Klein 1997a, 1997b; Engler 1995; Davies et al. 1995; Lyonnet et al. 1997). In Japan, the National Aerospace Laboratory (NAL), in collaboration with Purdue and a number of Japanese universities, developed cryogenic and fastresponding PSPs (Asai 1999; Asai et al. 2001, 2003). More and more research institutions all over the world are becoming interested in developing PSP technology because of its obvious advantages over conventional techniques. Brown (2000) gave a historical review with personal notes and recollections from some pioneers on early PSP development. Before the advent of polymer-based luminescent TSPs, thermographic phosphors and thermochromic liquid crystals have been used for measuring the surface temperature distributions in heat transfer and aerothermodynamic experiments. Thermographic phosphors are usually applied to a surface in the form of insoluble powder or crystal in contrast to polymer-based luminescent TSPs although both techniques utilize the temperature dependence of luminescence. A family of thermographic phosphors can cover a temperature range from room temperature (293 K) to 1600 K, which overlaps with the temperature range of polymer-based TSPs from cryogenic temperature (about 100 K) to 423 K. In this sense, thermographic phosphors and polymer-based luminescent TSPs are complementary to cover a broader range from cryogenic to high temperatures. L. Bradley (1953) explored aerodynamic application of thermographic phosphors mixed with binders and ceramic materials to measure surface temperature. Then, thermographic phosphors were used for temperature measurements in high-speed wind tunnels (Czysz and Dixon 1969; Buck 1988, 1989, 1991; Merski 1998, 1999), gas turbine engines (Noel et al. 1985, 1986, 1987; Tobin et al. 1990; Alaruri et al. 1995), and fiber-optic thermometry systems (Wickersheim and Sun 1985). Allison and Gillies (1997) gave a comprehensive review on thermographic phosphors. Thermochromic liquid crystals applied to a black surface selectively reflect light and hue varies depending on the temperature

14

1. Introduction

of the surface, which allows measurement of the surface temperature in a o relatively narrow range from 25 to 45 C. After E. Klein (1968) used liquid crystals in aerodynamic testing, this technique for global temperature measurement has been used in turbine machinery (Jones and Hippensteele 1988; Hippensteele and Russell 1988; Ireland and Jones 1986), hypersonic tunnels (Babinsky and Edwards 1996), and turbulent flows (Smith et al. 2000). Polymer-based TSPs are relatively new compared to thermographic phosphors and thermochromic liquid crystals. P. Kolodner and A. Tyson (1982, 1983a, 1983b) of the Bell Laboratory used a Europium-based TSP in a polymer binder to measure the surface temperature distribution of an operating integrated circuit. A family of TSPs have been developed at Purdue University and used in low-speed, supersonic and hypersonic aerodynamic experiments (Campbell et al. 1992, 1994; Campbell 1994; Liu et al. 1992b, 1994a, 1994b, 1995a, 1995b, 1996, 1997a, o 1997b). Two typical TSPs are EuTTA in model airplane dope (-20 to 100 C) and o o Ru(bpy) in Shellac (0 to 90 C). Several cryogenic TSPs (-175 to 0 C) were first discovered at Purdue University (Campbell et al. 1994) and used for transition detection in cryogenic flows (Asai et al. 1997c; Popernack et al. 1997). Further development of cryogenic TSPs was made at Purdue (Eransquin 1998a, 1998b), NAL in Japan (Asai et al. 1997c; Asai and Sullivan 1998) and NASA Langley. TSP formulations were also studied at the University of Washington (Gallery 1993) and one of the paints was used for boundary-layer transition detection at NASA Ames (McLachlan et al. 1993b). PSP and TSP have become an active and growing interdisciplinary research area, offering the promise of quantitative pressure and temperature mapping on the one hand and giving new technical challenges on the other hand. Useful reviews were given by Crites (1993), McLachlan and Bell (1995a), Crites and Benne (1995), Liu et al. (1997b), Mosharov et al. (1997), Bell et al. (2001), and Sullivan (2001). This book provides a systematic and detailed description of all the technical aspects of PSP and TSP, including basic photophysics, paint formulations and their physical properties, radiative energy transport, measurement methods and systems, uncertainty, time response, image and data analysis techniques, and various applications in aerodynamics and fluid mechanics.

2. Basic Photophysics

2.1. Kinetics of Luminescence Pressure sensitive paint (PSP) and temperature sensitive paint (TSP) are, respectively, based on the oxygen and thermal quenching processes of luminescence which are reversible processes in molecular photoluminescence. The general principles of luminescence are described in detail by Rebek (1987), Becker (1969) and Parker (1968). The different energy levels and photophysical processes of luminescence for a simple luminophore can be clearly described by the Jablonski energy-level diagram shown in Fig. 2.1. The lowest horizontal line represents the ground-state energy of the molecule, which is normally a singlet state denoted by S0. The upper lines are energy levels for the vibrational states of excited electronic states. The successive excited singlet and triplet states are denoted by S1 and S2, and T1, respectively. As is normally the case, the energy of the first excited triplet state T1 is lower than the energy of the corresponding singlet state S1. A photon of radiation is absorbed to excite the luminophore from the ground electronic state to excited electronic states ( S 0 → S 1 and S 0 → S 2 ). The excitation process is symbolically expressed as S 0 + !ν → S 1 , where ! is the Plank constant and ν is the frequency of the excitation light. Each electronic state has different vibrational states, and each vibrational state has different rotational states. The excited electron returns to the unexcited ground state by a combination of radiative and radiationless processes. Emission occurs through the radiative processes called luminescence. The radiation transition from the lowest excited singlet state to the ground state is called fluorescence, which is expressed as S 1 → S 0 + !ν f . Fluorescence is a spin-allowed radiative transition between two states of the same multiplicity. The radiative transition from the triplet state to the ground state is called phosphorescence ( T1 → S 0 + !ν p ), which is a spinforbidden radiative transition between two states of different multiplicity. The lowest excited triplet state, T1, is formed through a radiationless transition from S1 by intersystem crossing ( S 1 → T1 ). Since phosphorescence is a forbidden transition, the phosphorescent lifetime is typically longer than the fluorescent lifetime. Luminescence is a general term for both fluorescence and phosphorescence.

16

2. Basic Photophysics

Singlet Excited States Internal Conversion

Triplet Excited State

Vibrational Relaxation

S2

Interstystem Crossing S1

Energy

T1

Adsorption

Fluorescence

Internal and External Conversion

Phosphorescence

So Ground State Vibrational Relaxation

Fig. 2.1. Jablonsky energy-level diagram

Radiationless deactivation processes mainly include internal conversion (IC), intersystem crossing (ISC) and external conversion (EC). The internal conversion (IC) is a spin-allowed radiationless transition between two states of the same multiplicity ( S 2 → S 1 , S 1 → S 0 ). Typically, this process is expressed as

S 1 → S 0 + ∆ , where ∆ denotes heat released. IC appears to be particularly efficient when two electronic energy levels are sufficiently close. The intersystem crossing (ISC) is a spin-forbidden radiationless transition between two states of the different multiplicity, which are expressed as S 1 → T1 + ∆ and T1 → S 0 + ∆ . Phosphorescence depends to a large extent on the population of the triplet state ( T1 ) from the excited singlet state ( S 1 ) by the intersystem crossing. In addition, deactivation of an excited electronic state may involve interaction and energy transfer between the excited molecules and the environment like solutes, which are called external conversion (EC). The excited singlet and triplet states can be deactivated by interaction of the excited molecules with the components of a system. These bimolecular processes are quenching processes, including collisional quenching (diffusion or nondiffusion controlled), concentration quenching, oxygen quenching, and energy transfer quenching. The oxygen quenching of luminescence is the major photophysical mechanism for PSP. Due to the oxygen quenching, air pressure on

2.1. Kinetics of Luminescence

17

an aerodynamic model surface is related to the luminescent intensity by the SternVolmer equation that will be further discussed. The quantum efficiency of luminescence in most molecules decreases with increasing temperature because the increased frequency of collisions at elevated temperatures improves the possibility for deactivation by the external conversion. This effect associated with temperature is the thermal quenching, which is the major photophysical mechanism for TSP. The population of the excited singlet states ( S 1 ) and triplet states ( T1 ) at any given time depends on the competition among different photophysical processes listed in Table 2.1. The singlet state population [ S 1 ] and triplet state population

[ T1 ] are described by the following first-order kinetic model d [ S1 ] = I a − ( k f + k ic + k isc( s1 −t1 ) + k q( s ) [ Q ])[ S 1 ] dt d [ T1 ] = k isc ( s1 −t1 ) [ S 1 ] − ( k p + k isc( t1 − s0 ) + k q( t ) [ Q ])[ T1 ] dt

,

(2.1)

where I a is the light absorption rate of generating the excited singlet states, [ Q ] is the population of the quencher Q, k f and k p are, respectively, the rate constants for fluorescence and phosphorescence, k isc( s1 −t1 ) and k isc( t1 − s0 ) are, respectively, the rate constants for the intersystem crossings S 1 → T1 and

T1 → S 0 , k ic is the rate constant for the internal conversion, and k q( s ) and k q( t ) are the rate constants for the quenching in the singlet states and triplet states, respectively. The light absorption rate I a = k s 1 [ S 0 ] is proportional to the population [ S 0 ] in the ground state and the rate constant of excitation k s 1 . After a pulse excitation, the times required for the populations in the excited singlet state and triplet state to decay to 1/e of the initial value are, respectively,

τ f = 1 / ( k f + k ic + k isc( s1 −t1 ) + k q ( s ) [ Q ])

τ p = 1 / ( k p + k isc( t1 − s0 ) + k q( t ) [ Q ]) .

(2.2)

The time constants τ f and τ p are defined as the fluorescent and phosphorescent lifetimes, respectively. Usually, the lifetime of a specific photophysical process is defined as the reciprocal of the corresponding rate constant. Typical values of the lifetimes for different photophysical processes are listed in Table 2.1. When the intersystem crossing from T1 back to S 1 ( T1 → S 1 + ∆ ) is included in the kinetic model, extra terms k isc ( t1 − s1 ) [ T1 ]

and − k isc( t1 − s1 ) [ T1 ]

should be added,

respectively, to the right-hand sides of Eq. (2.1) for [ S 1 ] and [ T1 ] , where

k isc( t1 − s1 ) is the rate constant for the intersystem crossing T1 → S 1 . In this case,

18

2. Basic Photophysics

the kinetic model becomes a coupled system of equations (Mosharov et al. 1997; Bell et al. 2001). Since S 1 is a higher energy state than T1 , this intersystem crossing is thermally activated and therefore the rate constant for the process T1 → S 1 is temperature-dependent. Table 2.1. Photophysical processes involving electronically excited states Step

Process

Rate

Lifetime (s)

Excitation

S 0 + !ν → S 1

k s1 [ S0 ]

10 −15

Fluorescence (F)

S 1 → S 0 + !ν f

k f [ S1 ]

10 −11 − 10 −6

Internal Conversion (IC)

S1 → S 0 + ∆

k ic [ S 1 ]

10 −14 − 10 −11

Intersystem Crossing (ISC)

S 1 → T1 + ∆

k isc( s1 −t1 ) [ S 1 ]

10 −11 − 10 −8

Phosphorescence (P)

T1 → S 0 + !ν p

k p [ T1 ]

10 −3 − 10 2

Intersystem Crossing (ISC)

T1 → S 0 + ∆

k isc( t1 − s0 ) [ T1 ]

2.2. Models for Conventional Pressure Sensitive Paint From a standpoint of engineering application, it is unnecessary to analyze all the intermediate photophysical processes and their interactions. Therefore, a lumped model for luminescence (fluorescence and phosphorescence) is given here by considering the main processes: excitation, luminescent radiation, non-radiative deactivation, and quenching. The luminophore is excited by a photon from a ground state L0 to an excited state L* , i.e., L0 + !ν → L* . The excited state L* returns to the ground state L0 by either a radiative process (emission) or a radiationless process (deactivation). In the radiative process, the luminescent kr emission releases energy of !ν l , that is L* ⎯⎯→ L0 + !ν l , where k r is the rate constant for the radiation process and ν l is the frequency of the luminescent emission. In the deactivation process, L* returns to L0 by releasing heat, which k nr is expressed as L* ⎯⎯ ⎯→ L0 + ∆ , where k nr is the rate constant for the combined effect of all the non-radiative processes. If temperature around a luminophore molecule increases, the deactivation rate increases, reducing the radiative process from L* . Thus, the rate constant k nr for the non-radiative processes is temperature-dependent. The quenching process by a quencher Q is expressed as kq L* + Q ⎯⎯→ L0 + Q* , where k q is the rate constant of the quenching process

2.2. Models for Conventional Pressure Sensitive Paint

19

and Q* denotes the excited quencher. The molecular oxygen O2 in the ground state is an efficient quencher for both the excited singlet and triplet states. The molecular oxygen is excited to O2* once it quenches luminescence, i.e.,

L* + O2 → L0 + O2* . By combining the rates of emission, deactivation and quenching processes, the rate of change of the population of the excited state [ L* ] is given by the first-order equation d [ L* ] = I a − ( k r + k nr + k q [ Q ])[ L* ] . dt

(2.3)

The rate of excitation is I a = k s 1 [ L0 ] , where [ L0 ] is the population in the ground state and k s 1 is the rate constant for excitation.

At a steady state

d [ L* ] / dt = 0 , without quenching ( [ Q ] = 0 ), we have * I a = ( k r + k nr )[ L ] .

(2.4)

The amount of luminophore molecules in a given excited state is described by the quantum yield of luminescence defined by

Φ=

rate of luminescence . rate of excitation

(2.5)

The quantum yield Φ for the luminescent emission from L* with the quencher Q is expressed by [ *] I kr , (2.6) Φ = kr L = = k r + k nr + k q [ Q ] I a Ia where I is the luminescent intensity. The quantum yield without quenching is

[ *] Φ0 = k r L = Ia

I kr = 0 k r + k nr I a

,

(2.7)

where I 0 is the luminescent intensity without quenching. Dividing Φ 0 by Φ , we obtain the well-known Stern-Volmer relation

Φ0 I 0 kq = = 1+ [ Q ] = 1 + k qτ 0 [ Q ] , Φ I k r + k nr where τ 0 = 1 /( k r + k nr ) is the luminescent lifetime without quenching. luminescent lifetime with the quencher is

τ =

1 + k r k nr + k q [ Q ]

.

(2.8) The

(2.9)

20

2. Basic Photophysics

Thus, Eq. (2.8) can be written as

Φ0 / Φ = τ 0 / τ .

(2.10)

When the quencher is oxygen, the Stern-Volmer equation is

I0 τ 0 = = 1 + k qτ 0 [ O2 ] . τ I

(2.11)

In general, the rate constants k nr and k q for the non-radiative and quenching processes are temperature-dependent. The temperature dependency of k nr can be decomposed into a temperature-independent term and a temperature-dependent term modeled by the Arrhenius relation (Bennett and McCartin 1966; Song and Fayer 1991), i.e., E nr (2.12) ), k nr = k nr 0 + k nr 1 exp( − RT where k nr 0 = k nr ( T = 0 ) and k nr 1 are the rate constants for the temperatureindependent and temperature-dependent processes, respectively, Enr is the activation energy for the non-radiative process, R is the universal gas constant, and T is the absolute temperature in Kelvin. The temperature dependency of the rate constant k q for the quenching process is related to oxygen diffusion in a homogenous polymer layer used for a conventional PSP. According to the Smoluchowski relation, the rate constant k q for the oxygen quenching can be described by k q = 4π R AB N 0 D (2.13) where R AB is an interaction distance between the luminophore and oxygen molecules, and N 0 is the Avogadro's number. The diffusivity D has the temperature dependency modeled by the Arrhenius relation D = D0 exp( −

ED ), RT

(2.14)

where E D is the activation energy for the oxygen diffusion process. Therefore, from Eq. (2.9), the reciprocal of the luminescent lifetime is

1

τ

= k r + k nr 0 + k nr 1 exp( −

E nr E ) + 4πR AB N 0 D0 exp( − D ) [ O2 ] polymer RT RT

(2.15)

According to Henry's law, the oxygen population [ O2 ] polymer in a polymer binder is proportional to the partial pressure of oxygen pO2 or air pressure p , i.e.,

[ O2 ] polymer = S pO2 = S φ O2 p

(2.16)

2.2. Models for Conventional Pressure Sensitive Paint

21

where S is the oxygen solubility in a polymer binder layer and φ O2 is the mole fraction of oxygen in the testing gas. The mole fraction of oxygen φ O2 is 21% in the atmosphere, but it varies depending on testing facilities. For example, φ O2 is -4 only a few ppm (1ppm = 10 %) in a cryogenic wind tunnel where the working gas is nitrogen. Defining the permeability P0 = SD0 , from Eq. (2.15), we have

1

τ

= ka + K p ,

(2.17)

where the coefficients k a and K are defined as

k a = k r + k nr 0 + k nr 1 exp( −

E E nr ) and K = 4πR AB N 0 P0 exp( − D )φ O2 . (2.18) RT RT

In aerodynamic applications, it is difficult to obtain the zero-oxygen condition since the working gas in most wind tunnels is air containing 21% oxygen. Thus, instead of using the zero-oxygen condition, we usually utilize the zero-speed (wind-off) condition as a reference. Taking a luminescent intensity ratio between the wind-off and wind-on conditions, we obtain the Stern-Volmer equation suitable to aerodynamic applications

I ref I

=

τ ref p . = A polymer ( T ) + B polymer ( T ) p ref τ

(2.19)

The Stern-Volmer coefficients in Eq. (2.19) are

A polymer = A polymer ,ref

K ka , , and B polymer = B polymer , ref k aref K ref

(2.20)

where the reference coefficients are defined as

A polymer , ref =

1 + K ref

p ref 1 . and B polymer , ref = p ref / k aref k aref / K ref + p ref

(2.21)

The subscript ‘polymer’ specifically denotes a conventional polymer-based PSP; it will be seen that porous PSPs have somewhat different forms of the SternVolmer coefficients. Eq. (2.19) indicates that a ratio between the luminescent intensities in the wind-on and wind-off conditions is required to determine air pressure. This intensity-ratio method is commonly employed in PSP and TSP measurements. Using the expressions for k a and K , we can write A polymer and B polymer as a function of temperature ª 1 + ξ exp( − E nr RT ) º » A polymer = A polymer , ref « «¬ 1 + ξ exp( − E nr R T ref ) »¼

22

2. Basic Photophysics

ª E D B polymer = B polymer , ref exp «− «¬ R T ref

·º § Tref ¨ − 1¸ » , ¨ T ¸» © ¹¼

(2.22)

where the factor ξ is defined as ξ = k nr 1 /( k r + k nr 0 ) . For ( T − Tref ) / Tref represents the effect of reflection and scattering of the luminescent light at the wall, which is defined as

< M > = ȍ −1

³

ȍ

M(µ ) dȍ = 0.5 + ȡ Ȝwp2 ( µ1 + µ 2 ) ,

where µ1 = cos ș1 and µ 2 = cos ș 2 are the cosines of two polar angles in the solid angle ȍ . Imaging system aperture area, A0 = ʌ D2 /4

Source area

Image of source area AI

As

R1

R2

Fig. 4.3. Schematic of an imaging system

68

4. Radiative Energy Transport and Intensity-Based Methods

The response of a photodetector to the luminescent emission can be derived based on a model of an optical system (Holst 1998). Consider an optical system located at a distance R1 from a luminescent source area, as shown in Fig. 4.3. The collecting solid angle with which the lens is seen from the source can be approximated by ȍ ≈ A 0 / R12 , where A0 = ʌ D 2 /4 is the imaging system entrance aperture area, and D is the effective diameter of the aperture. Using Eq. (4.19) and additional relations A s / R12 = A I / R22 and 1 / R1 + 1 / R2 = 1 / fl , we obtain the radiative energy flux onto the detector (QȜ 2 )det =

ʌ AI Top Tatm ȕ Ȝ h Φ ( p, T ) q0 E Ȝ 2 (Ȝ 2 ) K 1 < M > , 4 F 2 (1 + M op )2 1

(4.20)

where F = fl / D is the f-number, M op = R2 / R1 is the optical magnification, fl is the system’s effective focal length, AI is the image area, and Top and Tatm are the system’s optical transmittance and atmospheric transmittance, respectively. The output of the detector is

V =G

³

∞

0

Rq (Ȝ 2 ) (Q Ȝ 2 )det Ft 2 ( Ȝ2 )dȜ2 ,

(4.21)

where Rq (Ȝ 2 ) is the detector’s quantum efficiency, G is the system’s gain, and

Ft 2 ( Ȝ2 ) is a filter function describing the optical filter for the luminescent emission. The dimension of V/G is [V/G] = J/s. Substitution of Eq. (4.20) into Eq. (4.21) yields V =G

AI ʌ ȕ Ȝ h Φ ( p, T ) q0 K 1 K 2 , 4 F 2 (1 + M op ) 2 1

(4.22)

where

K2 =

³

∞

0

Top Tatm E Ȝ 2 (Ȝ 2 )< M > Rq (Ȝ 2 ) Ft 2 ( Ȝ2 )dȜ2 .

The term K 2 represents the combined effect of the optical filter, luminescent light scattering, and system response to the luminescent light. The above analysis is made based on an assumption that the radiation source is on the optical axis. In general, the off-axis effect is taken into account by multiplying a factor cos 4 θ p in the right-hand side of Eq. (4.22), where θ p is the angle between the optical axis and light ray through the optical center (McCluney 1994). Eq. (4.19) gives the directional dependency of the luminescent radiant flux

Q Ȝ+2 ∝ 1 + 2 ρ λwp2 [cos( θ − ∆θ / 2 ) + cos( θ + ∆θ / 2 )] ,

(4.23)

4.4. Intensity-Based Measurement Systems

69

where ∆θ = θ 2 − θ 1 is the difference between two polar angles in the solid angle ȍ . Clearly, the luminescent radiant flux contains a constant irradiance term and a Lambertian term that is proportional to the cosine of the polar angle θ . Le Sant (2001b) measured the directional dependency of the luminescent emission of the OPTROD’s B1 PSP composed of a derived Pyrene dye and a reference component. Figure 4.4 shows the normalized luminescent intensity as a function of the viewing polar angle for the B1 paint and the B1 paint with talc compared with the theoretical distribution Eq. (4.23) with ρ λwp2 = 0.5 and ∆θ = 4 degrees. The experimental directional dependency remains nearly constant for both paints o until the viewing polar angle is larger than 60 . The theoretical distribution for a non-scattering paint fails to predict the flatness of the experimental directional distributions of the luminescent emission. This is because the simplified theoretical analysis does not consider scattering particles (e.g. talc and solid reference component particles) re-directing and re-distributing both the excitation light and luminescent light inside the paints. A more complete analysis of the radiative energy transport in a luminescent paint with scattering particles requires a numerical solution of an integro-differential equation (Modest 1993).

Normalized Luminescent Intensity

1.2

1.0

0.8

0.6

Theory B1 PSP without talc B1 PSP with talc

0.4

0.2 -100

-80

-60

-40

-20

0

20

40

60

80

100

Polar Angle (degree)

Fig. 4.4. Directional dependency of the luminescent emission from the B1 paint and B1 paint with talc, compared with the theoretical directional distribution for a non-scattering paint. Experimental data for the B1 paints are from Le Sant (2001b)

4.4. Intensity-Based Measurement Systems The photodetector output V responding to the luminescent emission, Eq. (4.22), is re-written as (4.24) V = Ȇ c Ȇ f ȕ Ȝ1 h q0 Φ ( p, T ) .

70

4. Radiative Energy Transport and Intensity-Based Methods

The parameters Ȇ c and Ȇ f

are Ȇ c = ( ʌ / 4) G AI [ F 2 (1 + M op ) 2 ] −1 and

Ȇ f = K 1 K 2 , which are related to the imaging system (camera) performance and filter parameters, respectively. The quantum yield Φ ( p, T ) is described by

Φ ( p,T) = k r /( k r + k nr + k q S φO2 p ) , where kr is the radiative rate constant, knr is the radiationless deactivation rate constant, kq is the quenching rate constant, p is air pressure, S is the solubility of oxygen, and φ O2 is the volume fraction of oxygen in air. In PSP applications, the intensity-ratio method is commonly used to eliminate the effects of spatial variations in illumination, paint thickness, and molecule concentration. Without any model deformation, air pressure p is related to a ratio between the wind-off and wind-on outputs by the Stern-Volmer relation

Vref V

= A( T ) + B( T )

p . p ref

(4.25)

The essential elements of a measurement system for PSP and TSP include illumination sources, optical filters, photodetectors and data acquisition/processing units. In terms of the detectors and illumination sources used, measurement systems can be generally categorized into CCD camera system and laser scanning system with a single-sensor detector. Since each system has advantages over the other, researchers can choose one most suitable to meet the requirements for their specific experiments.

4.4.1. CCD Camera System A CCD camera system is most commonly used for PSP and TSP measurements in wind tunnel tests. Figure 1.4 shows a schematic of a CCD camera system. The luminescent paint (PSP or TSP) is applied to a model surface, which is excited to luminesce by an illumination source such as UV lamp, LED array or laser. The luminescent emission is filtered optically to eliminate the illuminating light before projecting onto a CCD sensor. Images (wind-on and wind-off images) are digitized and transferred to a computer for data processing. In order to correct the dark current in a CCD camera, a dark current image is acquired when no light is incident on the camera. A ratio between the wind-on and wind-off images is taken after the dark current image is subtracted from both images, resulting in a luminescent intensity ratio image. Then, using the calibration relation for the paint, the distribution of the surface pressure or temperature is computed from the intensity ratio image. Scientific grade cooled CCD digital cameras are ideal imaging sensors for PSP and TSP, which can provide a high intensity resolution (12 to 16 bits) and high spatial resolution (typically 512×512, 1024×1024, up to 2048×2048 pixels). Because a scientific grade CCD camera exhibits a good linear response and a high signal-to-noise ratio (SNR) up to 60 dB, it is particularly suitable to quantitative measurement of the luminescent emission (LaBelle and Garvey 1995). The major

4.4. Intensity-Based Measurement Systems

71

disadvantages of a scientific grade CCD camera are its high cost and a very slow frame rate. Less expensive consumer grade CCD video cameras were used in early PSP and TSP measurements (Kavandi et al. 1990; Engler et al. 1991; McLachlan et al. 1992); the intensity resolution of a CCD video camera is typically 8 bits with a conventional frame grabber. When there is a large pressure variation over a model surface, a consumer grade video CCD camera can be used as an alternative to give acceptable quantitative results after the camera is carefully calibrated to correct the non-linearity of the radiometric response function of the camera (see Chapter 5). The low SNR of a video camera can be improved by averaging a sequence of images to reduce the random noise. In addition, film-based camera systems were occasionally used in special PSP measurements like flight tests (Abbitt et al. 1996). The performance of a CCD array is characterized by the responsivity, charge well capacity and noise. From these quantities, the minimum signal, maximum signal, signal-to-noise ratio and dynamic range can be estimated (Holst 1998; Janesick 1995). These performance parameters are critical for quantitative radiometric measurements of the luminescent emission, which can be estimated based on the camera model and noise models (Holst 1998). Here, the most relevant concepts are briefly discussed. The responsivity, the efficiency of generating electrons by a photon, is determined by the spectral quantum efficiency The full-well capacity specifies the number of Rq ( λ ) of a detector. photoelectrons that a pixel can hold before charge begins to spill out, thus reducing the response linearity. The maximum signal is proportional to the fullwell capacity. Normally, the well size is approximately proportional to the pixel size. Therefore, in a fixed CCD area, increasing the effective pixel size to enhance the SNR may reduce the spatial resolution. The dynamic range, defined as the maximum signal (or the full-well capacity) divided by the rms readout noise (or noise floor), loosely describes the camera’s ability to measure both low and high light levels. The minimum signal is limited by the camera noises, including the photon shot noise, dark current, reset noise, amplifier noise, quantization noise, and fixed pattern noise. The photon shot noise is associated with the discrete nature of photoelectrons obeying the Poisson statistics in which the variance is equal to the mean. The dark current is due to thermally generated electrons, which can be reduced to a very low level by cooling a CCD device. The reset noise is associated with resetting the sense node capacitor that is temperature-dependent. The amplifier noise contains two components: 1/f noise and white noise; the array manufacturer usually provides this value and calls it the readout noise, noise equivalent electrons, or noise floor. By careful optimization of the camera electronics, the readout noise or noise floor can be reduced to as low as 4-6 electrons. The quantization noise results from the analog-to-digital conversion. The fixed pattern noise (the pixel-to-pixel variation) is due to differences in pixel responsivity, which is called the scene noise, pixel noise, or pixel nonuniformity as well.

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4. Radiative Energy Transport and Intensity-Based Methods

Noise Electrons (rms)

10000

Total Noise

1000

100 Noise Floor 10

Fixed Pattern Noise Photon Shot Noise

1 10

100

1000

10000

100000

Photoelectrons

Fig. 4.5. Noise curves of CCD for the noise floor = 50e and non-uniformity U = 0.25%

Although various noise sources exist, for many applications, it is sufficient to consider the photon shot noise, noise floor, and fixed pattern noise due to pixel nonuniformity. Thus, according to the Poisson statistics, the total system noise < nsys > is given by 2 < nsys > = < nshot > + < n 2floor > + < n 2pattern > = n pe + < n 2floor > + ( Un pe )2

, (4.26)

2 where < nshot > , < n 2floor > and < n 2pattern > are the variances of the photon shot

noise, noise floor and pattern noise, respectively, n pe is the number of collected photoelectrons, and U is the pixel nonuniformity. Accordingly, the signal-to-noise ratio (SNR) is

SNR = n pe / n pe + < n 2floor > + ( Un pe ) 2

.

(4.27)

Figure 4.5 shows the total noise, photon shot noise, noise floor (readout noise), and fixed pattern noise of a CCD as a function of the number of photoelectrons for < n 2floor > 1 / 2 = 50e and U = 0.25%. For a very low photon flux, the noise floor dominates. As the incident light flux increases, the photon shot noise dominates. At a very high level of the incident light flux, the noise may be dominated by the fixed pattern noise. When the photon shot noise dominates, the SNR asymptotically approaches to

SNR = n pe , and the dynamic range is

( n pe )max / < n floor > , where ( n pe )max is the full-well capacity. The dark current only affects those applications where the SNR is low. In most applications of PSP

4.4. Intensity-Based Measurement Systems

73

and TSP, the pressure and temperature resolutions are limited by the photon shot noise. Table 4.1, which is adapted from Crites (1993), lists the performance parameters of some CCD sensors. Table 4.1. Characteristics of CCD Sensors CCD

TH7883PM

TH7895B

TH896A

TK512CB

TK1024F

Pixel array

384×586

512×512

1024×1024

512×512

1024×1024

1024×1024

Full well (e)

180000

290000

350000

700000

450000

256000

o

TK1024B

Temperature ( C)

-45

-45

-40

-40

-40

-40

Dark current (e)

8

8

25

4

3

6

Readout Noise (e)

12

6

6

10

9

9

Quantum efficiency

40%

40%

40%

80%

35%

80%

Peak wavelength (nm)

700

670

670

650

670

650

The selection of an appropriate illumination source depends on the absorption spectrum of a luminescent paint and optical access of a specific facility. An illumination source must provide a sufficiently large number of photons in the wavelength band of absorption without saturating the luminescence and causing serious photodegradation. It is desirable for a source to generate a reasonably uniform illumination field over a surface such that the measurement uncertainty associated with model deformation can be reduced. A continuous illumination source should be stable and a flash source should be repeatable. A variety of illumination sources are commercially available. Pulsed and continuous-wave lasers with fiber-optic delivery systems were used in wind tunnel tests (Morris et al. 1993a, 1993b; Crites 1993; Bukov et al. 1992; Volan and Alati. 1991; Engler et al. 1991, 1992; Lyonnet et al. 1997). Lasers have obvious advantages in terms of providing narrow band intense illumination. Very stable blue LED arrays were developed for illuminating paints (Dale et al. 1999). LED arrays are attractive as an illumination source since they are light in weight and they produce little heat; they can be suitably distributed to form a fairly uniform illumination field. In addition, they can be easily controlled to generate either continuous or modulated illumination. Other light sources reported in the literature of PSP and TSP include xenon arc lamps with blue filters (McLachlan et al. 1993a), incandescent tungsten/halogen lamps with blue filters (Morris et al. 1993a; Dowgwillo et al. 1994) and fluorescent UV lamps (Liu et al. 1995a, 1995b). The spectral characteristics of illumination sources can be found in The Photonics Design and Applications Handbook (1999). Crites (1993) discussed some available light sources from a viewpoint of PSP application. Optical filters are used to separate the luminescent emission from the excitation light, or separate the luminescent emissions from different luminophores. There are two kinds of filters: interference filters and color glass filters. Interference filters select a band of light through a process of constructive and destructive interference. They consist of a substrate onto which chemical layers are vacuum deposited in such a fashion that the transmission of certain wavelengths is

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4. Radiative Energy Transport and Intensity-Based Methods

enhanced, while other wavelengths are either reflected or absorbed. Band-pass interference filters only transmit light in a spectral band; the peak wavelength and spectral width can be tightly controlled. Edge interference filters only transmit light above (long pass) or below (short pass) a certain wavelength. Color glass filters are used for applications that do not need precise control over wavelengths and transmission intensities. The ratio of transmission to blocking is a key filter characteristic. All filters are sensitive to the angle of incidence of the incoming light. For interference filters, the peak transmission wavelength decreases as the angle of incidence deviates from the normal, while the bandwidth and transmission characteristics generally remain unchanged. For color glass filters, an increase of the incident angle increases the transmission path, reducing the transmission efficiency. 4.4.2. Laser Scanning System A generic laser scanning system for PSP and TSP is shown in Fig. 1.5. A lowpower laser beam is focused to a small point and scanned over a model surface using a computer-controlled mirror to excite the paint on a model. The luminescent emission is detected using a low-noise photodetector (e.g. PMT); the photodetector signal is digitized with a high-resolution A/D converter in a PC and processed to calculate pressure or temperature based on the calibration relation for the paint. When the laser beam is modulated, a lock-in amplifier can be used to reduce the noise. Furthermore, the phase angle between the modulated excitation light and responding luminescence can be obtained using a lock-in amplifier for phase-based PSP and TSP measurements. The laser can be scanned continuously or in steps; it is synchronized to data acquisition such that the position of the laser spot on the model is known. In order to compensate for a laser power drift, the laser power variation is monitored using a photodiode. The laser scanning systems for PSP and TSP measurements were discussed by Hamner et al. (1994), Burns (1995), Torgerson et al (1996), and Torgerson (1997). Compared to a CCD camera system, a laser scanning system offers certain advantages. Since a low-noise PMT is used to measure the luminescent emission, before an analog output from the PMT is digitized, standard SNR enhancement techniques are available to improve the measurement accuracy. Amplification and band-limited filtering can be used to improve the SNR. The signal is then digitized with a high-resolution A/D converter (12 to 24 bits). Additional noise reduction can be accomplished using a lock-in amplifier when the laser beam is modulated. The laser scanning system is able to provide uniform illumination over a surface by scanning a single laser spot. The laser power is easily monitored and correction for the laser power drift can be made for each measurement point. The laser scanning system can be used for PSP and TSP measurements in a facility where optical access is so limited that a CCD camera system is difficult to use.

4.5. Basic Data Processing

75

4.5. Basic Data Processing The most basic processing procedure in the intensity-based method for PSP and TSP is taking a ratio between the wind-on image and the wind-off reference image to correct the effects of non-homogenous illumination, uneven paint thickness and non-uniform luminophore concentration. However, this ratioing procedure is complicated by model deformation induced by aerodynamic loads, which results in misalignment between the wind-on and wind-off images. Therefore, additional correction procedures are required to eliminate (or reduce) the error sources associated with model deformation, the temperature effect of PSP, selfillumination, and camera noises (dark current and fixed pattern noise). Figure 4.6 shows a generic data processing flowchart for intensity-based measurements of PSP and TSP with a CCD camera. A laser scanning system has similar data processing procedures for intensity-based measurements. The windon and wind-off images are acquired using a CCD camera. Usually, a sequence of acquired images is averaged to reduce the random noise like the photon shot noise. The dark current image and ambient lighting image are subtracted from data images to eliminate the dark current noise of the CCD camera and the contribution from the ambient light. The dark current image is usually acquired when the camera shutter is closed. In a wind tunnel environment, there is always weak ambient light that may cause a bias error in data images. The ambient lighting image is acquired when the shutter is open while all controllable light sources are turned off. The integration time for the dark current image and ambient lighting image should be the same as that for data images. The data images are then divided by the flat-field image to correct the fixed pattern noise. At a very high signal level, this correction is necessary since the fixed pattern noise may surpass the photon shot noise. Ideally, the flat-field image is acquired from a uniformly illuminated scene. A simple but less accurate approach is use of several diffuse scattering glasses mounted in the front of the lens of the camera to generate an approximately uniform illumination field. When a uniform illumination field cannot be achieved, a more complex noise-model-based approach can be used to obtain the fixed pattern noise field for a CCD camera (Healey and Kondepudy 1994). Normally, a scientific grade CCD camera has a good linear response of the camera output to the incident irradiance of light. However, conventional CCD video cameras often exhibit a non-linear response to the incident light intensity; in this case, a video camera should be radiometrically calibrated to correct the non-linearity. A simple but useful radiometric camera calibration technique is described in Chapter 5.

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4. Radiative Energy Transport and Intensity-Based Methods

Wind-On Image

Wind-Off Image

Correction for Dark Current, Ambient Light & Fixed Pattern Noise

Correction for Non-Linearity of Response of Detector Registered & Corrected Wind-On Image

Corrected Wind-Off Image

Image Registration

Ratio Image

Self-Illumination Correction

PSP/TSP Image

PSP/TSP Calibration

PSP/TSP Data on 3D Grid

Image Resection

Forces & Moments/Heat Transfer

Fig. 4.6. Generic data processing flowchart for intensity-based PSP and TSP measurements

In this stage, even though the noise-corrected wind-on and wind-off images are obtained, we cannot yet calculate a ratio of the wind-off image over the wind-on image, Vref / V , for conversion to a pressure or temperature image. This is because the wind-on image may not align with the wind-off image due to model deformation produced by aerodynamic loads. A ratio between those non-aligned images can lead to a considerable error in calculation of pressure or temperature using a calibration relation. Also, some distinct flow features such as shock, boundary layer transition and flow separation could be smeared. In order to correct the non-alignment problem, the image registration technique should be used to match the wind-on image to the wind-off image (Bell and McLachlan 1993, 1996; Donovan et al. 1993). The image registration technique is based on a mathematical transformation ( x' , y' )  ( x , y ) , which empirically maps the deformed wind-on image coordinate ( x' , y' ) onto the reference wind-off image coordinate ( x , y ) . For a small deformation, an image registration transformation is well described by polynomials m

( x, y ) = (

¦

i , j =0

m

aij x' i y' j ,

¦ b x' ij

i

y' j ) .

(4.28)

i , j =0

Geometrically, the constant terms, linear terms, non-linear terms in Eq. (4.28) represent translation, rotation and scaling, and higher-order deformation of a

4.5. Basic Data Processing

77

model in the image plane, respectively. In measurements of PSP and TSP, black fiducial targets are placed in the locations on a model where deformation is appreciable. The displacement of these marks in the image plane represents perspective projection of real model deformation in the 3D object space. From the corresponding centroids of the targets in the wind-on and wind-off images, the polynomial coefficients aij and bij in Eq. (4.28) can be determined using least-squares method. More targets will increase the statistical redundancy and improve the precision of least-squares estimation. For most wind tunnel tests, a second-order polynomial transformation (m = 2) is found to be sufficient. As a pure geometric correction method, however, the image registration technique fails to take into account a variation in illumination level on a model due to model movement in a non-homogenous illumination field. An estimate of this error requires the knowledge of the illumination field and the movement of the model relative to the light sources. Bell and McLachlan (1993, 1996) gave an analysis on this error in a simplified circumstance and found that this error was small if the illumination light field was nearly homogenous and model movement was small. Experiments showed that the image registration technique considerably improved the quality of PSP and TSP images (McLachlan and Bell 1995). Weaver et al. (1999) utilized spatial anomalies (dots formed from aerosol mists in spraying) in a basecoat and calculated a pixel shift vector field of a model using a spatial correlation technique similar to that used in particle image velocimetry (PIV). Based on the shift vector field, the wind-on image was registered. Le Sant et al. (1997) described an automatic scheme for target recognition and image alignment. A detailed discussion on the image registration technique is given in Chapter 5. After a ratio of the wind-off image over the registered wind-on image is taken, a pressure or temperature image can be obtained using the calibration relation (the Stern-Volmer relation for PSP or the Arrhenius relation for TSP). Compared to relatively straightforward conversion of an intensity ratio image to a temperature image, conversion to a pressure image is more difficult since the intensity ratio image of PSP is a function of not only pressure, but also temperature. The temperature effect of PSP often has a dominant contribution to the total uncertainty of PSP measurements if it is not corrected. When the Stern-Volmer coefficients A( T ) and B( T ) are determined in a priori laboratory PSP calibration and the temperature field on the surface are known, the pressure field can be, in principle, calculated from a ratio image. The need of temperature correction provoked the development of multiple-luminophore PSP and tandem use of PSP with TSP. The surface temperature distribution can be measured using TSP and infrared (IR) cameras. Also, the temperature field can be given by theoretical and numerical solutions to the motion and energy equations of flows. Unfortunately, experiments have shown that the use of a priori laboratory PSP calibration with a correction for the temperature effect still leads to a systematic error in the derived pressure distribution due to certain uncontrollable factors in wind tunnel environment. To correct this systematic error, pressure tap data at a number of locations are used to correlate the intensity ratio values to the pressure tap data; this procedure is referred to as in-situ calibration of PSP. In the worst

78

4. Radiative Energy Transport and Intensity-Based Methods

case where A( T ) and B( T ) are not known and the surface temperature field is not given, in-situ calibration is still able to give a pressure field. However, the accuracy of interpretation of PSP data between the pressure taps is not guaranteed especially when the gradients of the pressure and temperature fields between the taps are large. Obviously, the selection of the locations of the pressure taps is critical to assure the accuracy of in-situ calibration. The pressure tap data at the discrete locations for in-situ calibration should reasonably cover the pressure distribution on the surface. The in-situ calibration uncertainty of PSP is discussed in Chapter 7. PSP and TSP data in images have to be mapped onto a surface grid of a model in the 3D object space since the pressure and temperature fields on the surface grid are more useful for engineers and researchers. Further, this mapping is necessary for extraction of aerodynamic loads and heat transfer and for comparison with CFD results. In the literature of PSP and TSP, this mapping procedure is often called image resection. Note that the meaning of resection in the PSP and TSP literature is somewhat broader and looser than the strict one in photogrammetry. From the standpoint of photogrammetry, a key of this procedure is geometric camera calibration by solving the perspective collinearity equations to determine the camera interior and exterior orientation parameters, and lens distortion parameters. Once these parameters in the collinearity equations relating the 3D object space to the image plane are known, PSP and TSP data in images can be mapped onto a given surface grid in the 3D object space. A detailed discussion on analytical photogrammetric techniques is given in Chapter 5. In most PSP and TSP measurements conducted so far, data in images are mapped onto a rigid CFD or CAD surface grid of a model. However, when a model experiences a significant aeroelastic deformation in wind tunnel tests, mapping onto a rigid grid misrepresents the true pressure and temperature fields. Therefore, a deformed surface grid of a model should be generated for PSP and TSP mapping. Liu et al. (1999) discussed generation of a deformed surface grid based on videogrammetric model deformation measurements conducted along with PSP/TSP measurements (see Chapter 5). Finally, the integrated aerodynamic forces and moments can be calculated from the pressure distribution on the surface. For example, the lift is given by FL = ¦ p i ( n • l L ∆S )i , where n is the

unit normal vector of a panel on the surface, ∆S is the area of the panel, and l L is the unit vector of the lift. Similarly, the integrated quantities of heat transfer can be obtained from the surface temperature fields based on appropriate heat transfer models. The self-illumination correction is implemented after the luminescent intensity data are mapped on a surface grid in the 3D object space. The socalled self-illumination is a phenomenon that the luminescent emission from one part of a model surface illuminates another surface, thus increasing the observed luminescent intensity of the receiving surface and producing an additional error in calculation of pressure and temperature. This distorting effect often occurs on the surfaces of neighboring components such as wind/body junctures and concave surfaces. The self-illumination depends on the surface geometry, the luminescent field, and the reflecting properties of a paint layer. Assuming that a

4.5. Basic Data Processing

79

paint surface is Lambertian, Ruyten (1997a, 1997b, 2001a) developed an analytical model and a numerical scheme for correcting the self-illumination effect. The self-illumination correction scheme is discussed in Chapter 5. One of the original purposes of developing two-luminophore PSPs is to simplify the data processing for PSP. The dependency of a two-color intensity ratio I λ1 / I λ2 on pressure p and temperature T is generally expressed as

I λ1 / I λ2 = f ( p , T ) , where I λ1 and I λ2 are the luminescent intensities at the emission wavelengths λ1 and λ 2 , respectively. Ideally, a two-color intensity ratio can eliminate the effect of spatially non-uniform illumination on a surface. However, since two luminophores cannot be perfectly mixed, the simple twocolor intensity ratio I λ1 / I λ2 cannot completely compensate the effect of nonhomogenous dye concentration. In this case, a ratio of ratios ( I λ1 / I λ2 ) /( I λ1 / I λ2 )0 should be used to correct the effects of non-homogenous dye concentration and paint thickness variation, where the subscript 0 denotes the wind-off condition (McLean 1998). Since the wind-off images are required, the ratio-of-ratios method still needs image registration. The ratio-of-ratios approach was also applied to non-pressure-sensitive reference targets to compensate the effect of non-homogenous illumination on a moving model (Subramanian et al. 2002).

5. Image and Data Analysis Techniques

This Chapter describes image and data analysis techniques used in various processing steps for PSP and TSP. For quantitative PSP and TSP measurements, cameras should be geometrically calibrated to establish the accurate relationship between the image plane and the 3D object space and map data in images onto a surface grid in the object space. Analytical camera calibration techniques, especially the Direct Linear Transformation (DLT) and the optimization calibration method, are discussed. Since PSP and TSP are based on radiometric measurements, an ideal camera should have a linear response to the luminescent radiance. For a camera having a non-linear response, radiometric camera calibration is required to determine the radiometric response function of the camera for correcting the image intensity before taking a ratio between the wind-on and wind-off images. A simple but effective technique is described here for radiometric camera calibration. The self-illumination of PSP and TSP may cause a significant error near a conjuncture of surfaces when a strong exchange of the radiative energy occurs between neighboring surfaces. The numerical methods for correcting the selfillumination are generally described and the errors associated with the selfillumination are estimated for a typical case. The self-illumination correction is usually made on a surface grid in the object space since it highly depends on the surface geometry. A standard procedure in the intensity-based method for PSP and TSP is to take a ratio between the wind-on and wind-off images to eliminate the effects of non-homogenous illumination intensity, dye concentration, and paint thickness. However, since a model deforms due to aerodynamic loads, the wind-on image does not align with the wind-off image. The image registration technique based on a mathematical transformation between the wind-on and wind-off images is described to re-align these images. A crucial step for PSP is to accurately convert the luminescent intensity to pressure; cautious use of the calibration relations with a correction of the temperature effect of PSP is discussed. PSP measurements in low-speed flows are particularly difficult since a very small pressure change has to be sufficiently resolved by PSP. The pressure-correction method is described as an alternative to extrapolate the incompressible pressure coefficient from PSP measurements at suitably higher Mach numbers by removing the compressibility effect. The final processing step for PSP and TSP is to map results in images onto a model surface grid in the object space. When a model has a large deformation produced by aerodynamic loads, a deformed surface grid should be generated for more accurate PSP and TSP mapping. A methodology for generating a deformed wing grid is proposed based on

82

5. Image and Data Analysis Techniques

videogrammetric aeroelastic deformation simultaneously with PSP and TSP measurements.

measurements

conducted

5.1. Geometric Calibration of Camera

5.1.1. Collinearity Equations After the results of pressure and temperature are extracted from images of PSP and TSP, it is necessary to map the data onto a surface grid in the 3D object space (or physical space) to make the results more useful for design engineers and researchers. The collinearity equations in photogrammetry provide the perspective relationship between the 3D coordinates in the object space and corresponding 2D coordinates in the image plane (Wong 1980; McGlone 1989; Mikhail et al. 2001; Cooper and Robson 2001; Liu 2002). A key problem in quantitative image-based measurements is camera calibration to determine the camera interior and exterior orientation parameters, and lens distortion parameters in the collinearity equations. Simpler resection methods have often been used in PSP and TSP systems to determine the camera exterior orientation parameters under an assumption that the interior orientation and lens distortion parameters are known (Donovan et al. 1993; Le Sant and Merienne 1995). The standard Direct Linear Transformation (DLT) was also used to obtain the interior orientation parameters in addition to the exterior orientation parameters (Bell and McLachlan 1993, 1996). An optimization method for comprehensive camera calibration was developed by Liu et al. (2000), which can determine the exterior orientation, interior orientation and lens distortion parameters (as well as the pixel aspect ratio of a CCD array) from a single image of a 3D target field. The optimization method, combined with the DLT, allows automatic camera calibration without an initial guess of the orientation parameters; this feature particularly facilitates PSP and TSP measurements in wind tunnels. Besides the DLT, a closed-form resection solution given by Zeng and Wang (1992) is also useful for initial estimation of the exterior orientation parameters of a camera based on three known targets. Figure 5.1 illustrates the perspective relationship between the 3D coordinates ( X, Y, Z ) in the object space and the corresponding 2D coordinates (x, y) in the image plane. The lens of a camera is modeled by a single point known as the perspective center, the location of which in the object space is ( X c ,Yc ,Z c ) . Likewise, the orientation of the camera is characterized by three Euler orientation angles. The orientation angles and location of the perspective center are referred to in photogrammetry as the exterior orientation parameters. On the other hand, the relationship between the perspective center and the image coordinate system is defined by the camera interior orientation parameters, namely, the camera principal distance c and the photogrammetric principal-point location ( x p ,y p ) .

5.1. Geometric Calibration of Camera

83

The principal distance, which equals the camera focal length for a camera focused at infinity, is the perpendicular distance from the perspective center to the image plane, whereas the photogrammetric principal-point is where a perpendicular line from the perspective center intersects the image plane. Due to lens distortion, however, perturbation to the imaging process leads to departure from collinearity that can be represented by the shifts dx and dy of the image point from its ‘ideal’ position on the image plane. The shifts dx and dy are modeled and characterized by the lens distortion parameters. Z

Y Object point

X

O Object space

y Perturbed image point

Model

Perspective center c

dy

Principal point

dx yp

x

xp Ideal image point Image plane

Fig. 5.1. Camera imaging process and the interior orientation parameters

The perspective relationship is described by the collinearity equations

x − x p + d x =− c

m11 ( X − X c ) + m12 ( Y − Yc ) + m13 ( Z − Z c ) U = −c W m31 ( X − X c ) + m32 ( Y − Yc ) + m33 ( Z − Z c )

, y − y p + d y =− c

(5.1)

m21 ( X − X c ) + m22 ( Y − Yc ) + m23 ( Z − Z c ) V = −c W m31 ( X − X c ) + m32 ( Y − Yc ) + m33 ( Z − Z c )

where mij (i, j = 1, 2, 3) are the elements of the rotation matrix that are functions of the Euler orientation angles ( ω ,φ ,κ ) ,

84

5. Image and Data Analysis Techniques

m11 = cos φ cos κ m12 = sin ω sin φ cos κ + cos ω sin κ m13 = − cos ω sin φ cos κ + sin ω sin κ m21 = − cos φ sin κ m22 = − sin ω sin φ sin κ + cos ω cos κ

(5.2)

m23 = cos ω sin φ sin κ + sin ω cos κ m31 = sin φ m32 = − sin ω cos φ m33 = cos ω cos φ . The orientation angles ( ω ,φ ,κ ) are essentially the pitch, yaw, and roll angles of a camera in an established coordinate system. The terms dx and dy are the image coordinate shifts induced by lens distortion, which can be modeled by a sum of the radial distortion and decentering distortion (Fraser 1992; Fryer1989)

d x=d xr + d xd and d y =d y r + d y d ,

(5.3)

where

d x r = K 1 ( x' − x p ) r 2 + K 2 ( x' − x p ) r 4 , d y r = K 1 ( y' − y p ) r 2 + K 2 ( y' − y p ) r 4 , d x d = P1 [ r 2 + 2( x' − x p ) 2 ] + 2 P2 ( x' − x p )( y' − y p ) ,

(5.4)

d y d = P2 [ r 2 + 2( y' − y p ) 2 ] + 2 P1 ( x' − x p )( y' − y p ) , r 2 = ( x' − x p ) 2 + ( y' − y p )2 . Here, K1 and K2 are the radial distortion parameters, P1 and P2 are the decentering distortion parameters, and x’ and y’ are the undistorted coordinates in the image plane. When lens distortion is small, the unknown undistorted coordinates can be approximated by the known distorted coordinates, i.e., x' ≈ x and y' ≈ y . For large lens distortion, an iterative procedure can be employed to determine the appropriate undistorted coordinates to improve the accuracy of estimation. The following iterative relations can be used: ( x' )0 = x and ( y' )0 = y ,

( x' )k +1 = x + d x [( x' ) k ,( y' )k ] and ( y' )k +1 = y + d y [( x' ) k ,( y' ) k ] , where the superscripted iteration index is k = 0 , 1, 2  . The collinearity equations Eq. (5.1) contain a set of the camera parameters to be determined by camera calibration; the parameter sets ( ω ,φ ,κ , X c ,Yc , Z c ) , (c, x p , y p ) , and (K 1 , K 2 , P1 , P2 ) in Eq. (5.1) are the exterior orientation, interior orientation, and lens distortion parameters of a camera, respectively. Analytical camera calibration techniques have been used to solve the collinearity equations

5.1. Geometric Calibration of Camera

85

with the lens distortion model for the camera exterior and interior parameters (Rüther 1989; Tsai 1987). Since Eq. (5.1) is non-linear, iterative methods of leastsquares estimation have been used as a standard technique for the solution of the collinearity equations in photogrammetry (Wong 1980; McGlone 1989). However, direct recovery of the interior orientation parameters is often impeded by inversion of a nearly singular normal-equation-matrix in least-squares estimation. The singularity of the normal-equation-matrix mainly results from strong correlation between the exterior and interior orientation parameters. In order to reduce the correlation between these parameters and enhance the determinability of (c, x p , y p ) , Fraser (1992) suggested the use of multiple camera stations, varying image scales, different camera roll angles and a well-distributed target field in three dimensions. These schemes for selecting suitable calibration geometry improve the properties of the normal equation matrix. In general, iterative least-squares methods require a good initial guess to obtain a convergent solution. Mathematically, the singularity problem can be treated using the singular value decomposition that produces the best solution in a least-squares sense. Also, the Levenberg-Marquardt method can stay away to some extent from zero pivots (Marquardt 1963). Nevertheless, multiple-station, multiple-image methods for camera calibration are not easy to use in a wind tunnel environment where only a limited number of windows are available for cameras and the positions of cameras are fixed. Thus, it is highly desirable for PSP and TSP to have a single-image, easy-to-use calibration method devoid of the singularity problem and an initial guess. In the computer vision community, Tsai’s two-step method is particularly popular. Instead of directly solving the standard collinearity equations Eq. (5.1), Tsai (1987) used a radial alignment constraint to obtain a linear least-squares solution for a subset of the calibration parameters, whereas the rest of the parameters including the radial distortion parameter are estimated by an iterative scheme. Tsai’s method is fast, but less accurate than the standard photogrammetric methods. In addition, the radial alignment constraint prevents this method from incorporating a more general model of lens distortion. Here, we first discuss the DLT that can automatically provide initial values of the camera parameters and then describe an optimization method for more comprehensive calibration of a camera. 5.1.2. Direct Linear Transformation The Direct Linear Transformation (DLT), originally proposed by Abdel-Aziz and Karara (1971), can be very useful to determine approximate values of the camera parameters. Rearranging the terms in the collinearity equations leads to the DLT equations L1 X + L2 Y + L3 Z + L4 − ( x+ d x )( L9 X + L10 Y + L11 Z + 1 ) = 0 . (5.5) L5 X + L6 Y + L7 Z + L8 − ( y + d y )( L9 X + L10 Y + L11 Z + 1 ) = 0

86

5. Image and Data Analysis Techniques

The DLT parameters L1 ,  L11 are related to the camera exterior and interior orientation parameters ( ω ,φ ,κ , X c ,Yc , Z c ) and (c, x p , y p ) (McGlone 1989). Unlike the standard collinearity equations Eq. (5.1), Eq. (5.5) is linear for the DLT parameters when the lens distortion terms dx and dy are neglected. In fact, the DLT is a linear treatment of what is essentially a non-linear problem at the cost of introducing two additional parameters. The matrix form of the linear DLT L = ( L1 ,  L11 )T , equations for M targets is B L = C , where

C = ( x1 , y1 ,  x M , y M )T , and B is the 2M×11 configuration matrix that can be directly obtained from Eq. (5.5). A least-squares solution for L is formally given by L = (B T B) −1 B T C without using an initial guess. The camera parameters can be extracted from the DLT parameters from the following expressions

x p = ( L1 L9 + L2 L10 + L3 L11 )L2 , y p = ( L5 L9 + L6 L10 + L7 L11 )L2 , c = ( L21 + L22 + L23 )L2 − x 2p

,

φ = sin −1 ( L9 L ) , ω = tan −1 ( − L10 / L11 ) , κ = cos −1 ( m11 / cos( φ )) , m11 = L( x p L9 − L1 ) / c , L = − ( L29 + L210 + L211 )−1 / 2 ,

§ Xc · § L1 ¨ ¸ ¨ ¨ Yc ¸ = − ¨ L5 ¨ ¸ ¨ © Zc ¹ © L9

L2 L6 L10

L3 · § L4 · ¸¨ ¸ L7 ¸ ¨ L8 ¸ . ¸¨ ¸ L11 ¹ © 1 ¹

Because of its simplicity, the DLT is widely used in both non-topographic photogrammetry and computer vision. When dx and dy cannot be ignored, however, iterative solution methods are still needed and the DLT loses its simplicity. In general, the DLT can be used to obtain fairly good values of the exterior orientation parameter and the principal distance, although it gives a poor estimate for the principal-point location (x p ,y p ) (Cattafesta and Moore 1996). Therefore, the DLT is valuable since it can provide initial approximations for more accurate methods like the optimization method discussed below for comprehensive camera calibration.

5.1. Geometric Calibration of Camera

87

5.1.3. Optimization Method In order to develop a simple and robust method for comprehensive camera calibration, the singularity problem must be dealt with to solve the collinearity equations. Liu et al. (2000) proposed an optimization method based on the following insight. Strong correlation between the interior and exterior orientation parameters leads to the singularity of the normal-equation-matrix in least-squares estimation for a complete set of the camera parameters. Therefore, to eliminate the singularity, least-squares estimation is used for the exterior orientation parameters only, while the interior orientation and lens distortion parameters are calculated separately using an optimization scheme. This optimization method contains two separate, but interacting procedures: resection for the exterior orientation parameters and optimization for the interior orientation and lens distortion parameters. When the image coordinates (x, y) are given in pixels, we express the collinearity equations Eq. (5.1) as

f 1 = S h x n − x p + d x + cU / W = 0 f 2 = S v y n − y p + d y + cV / W = 0

,

(5.6)

where Sh and Sv are the horizontal and vertical pixel spacings (mm/pixel) of a CCD array, respectively. In general, the vertical pixel spacing is fixed and known for a CCD camera, but the effective horizontal spacing may be variable. Thus, an additional parameter, the pixel-spacing-aspect-ratio S h / S v , is introduced. We define Ȇ ex = ( Ȧ, ij, ț, X c , Yc , Z c )T for the exterior orientation parameters and

Ȇ in = (c, x p , y p , K 1 , K 2 , P1 , P2 , S h / S v )T for the interior orientation and lens distortion parameters in addition to the pixel-spacing-aspect-ratio. For given values of Ȇ in , and a set of known points (targets) pn =(x n ,y n )T and Pn =(X n ,Yn ,Z n )T , a solution for Ȇ ex in Eq. (5.6) can be found using an iterative least-squares method, referred to as resection in photogrammetry. The linearized collinearity equations for targets (n = 1, 2, , M) are written as V = ǹ ( ǻȆ ex ) − l , where ǻȆ ex is the correction term for the exterior orientation parameters, V is the 2M×1 residual vector, A is the 2M×6 configuration matrix, and l is the 2M×1 observation vector. The configuration matrix A and observation vector l in the linearized collinearity equations are § (∂ f 1 / ∂ Ȇ ex )1 · § ( f 1 )1 · ¸ ¨ ¸ ¨ ¨ (∂ f 2 / ∂ Ȇ ex )1 ¸ ¨ ( f 2 )1 ¸ ¸ ¨ ¸ ¨ (5.7) A=¨  ¸ and l = − ¨  ¸ , ¨ (∂ f 1 / ∂ Ȇ ex ) ¸ ¨ ( f 1 )M ¸ M ¸ ¨ ¸ ¨ ¨( f2 ) ¸ ¨ (∂ f 2 / ∂ Ȇ ex ) ¸ M ¹ M ¹ © ©

88

5. Image and Data Analysis Techniques

where

the

operator

∂ /∂ Ȇ ex

is

defined

as

( ∂ /∂ Ȧ,∂ /∂ ij,∂ /∂ ț, ∂ /∂ X c , ∂ /∂Yc , ∂ /∂ Z c ) and the subscript denotes a target. The components of the vectors ∂ f 1 / ∂ Ȇ ex and ∂ f 2 / ∂ Ȇ ex are

∂ f1 c U = { m12 ( Z − Z c ) − m13 ( Y − Yc ) − [ m32 ( Z − Z c ) − m33 ( Y − Yc )]} , ∂Ȧ W W ∂ f1 c U [ −cos ț W − (cos ț U − sin ț V)] , = ∂ij W W ∂ f 1 cV = W ∂ț

,

∂ f1 U c = ( −m11 + m31 ) , ∂Xc W W ∂ f1 c U = ( −m12 + m32 ) , ∂Yc W W ∂ f1 c U = ( −m13 + m33 ) , ∂Zc W W ∂ f2 c V = { m22 ( Z − Z c ) − m23 ( Y − Yc ) − [ m32 ( Z − Z c ) − m33 ( Y − Yc )]} , ∂Ȧ W W ∂ f2 c V [sin ț W − (cos ț U − sin ț V)] , = ∂ij W W ∂ f2 cU =− W ∂ț

,

∂ f2 c V = ( −m 21 + m31 ) , W ∂Xc W ∂ f2 c V = ( −m22 + m32 ) , ∂Yc W W ∂ f2 c V = ( −m23 + m33 ) . ∂Zc W W A least-squares solution to minimize the residuals V for the correction term is ǻȆ ex =(AT A) −1 AT l . In general, the 6 × 6 normal-equation-matrix ( AT A ) can be inverted without any singularity problem since the interior orientation and lens

5.1. Geometric Calibration of Camera

89

distortion parameters are not included in least-squares estimation. To obtain such Ȇ ex that the correction term ǻȆ ex becomes zero, the Newton-Raphson iterative method is used for solving the non-linear equation (AT A) −1 AT l = 0 for Ȇ ex . This approach converges over a considerable range of the initial values of Ȇ ex . Therefore, for given Ȇ in , the corresponding exterior orientation parameter Ȇ ex can be obtained, which are symbolically expressed as Ȇ ex = RESECTION( Ȇ in ) . At this stage, the exterior orientation parameters Ȇ ex are not necessarily correct unless the given interior orientation and lens distortion parameters Ȇ in are accurate. Obviously, an extra condition is needed to obtain correct Ȇ in and the determination of Ȇ in is coupled with the resection for Ȇ ex . An optimization scheme to obtain the correct Ȇ in is described as follows. We notice that the correct values of Ȇ in are intrinsic constants for a camera/lens system, and they are independent of the target locations p n = (x n ,y n )T in the image plane and Pn = (X n ,Yn ,Z n )T in the object space. Mathematically, Ȇ in is an invariant under a transformation ( p n ,Pn )  ( p m ,Pm ) ( m≠n ). Therefore, for the correct values of Ȇ in , the parameters (c, x p , y p ) are invariant under the transformation ( p n ,Pn )  ( p m ,Pm ) ( m≠n ). In other words, for the correct values of Ȇ in , the standard deviation of (c, x p , y p ) calculated over all the targets from the collinearity equations should be zero, i.e., M

std(x p ) = [ ¦ ( x p − < x p > ) 2 /( M − 1 ) ] 1/2 = 0 , where std denotes the standard n =1

deviation and < > denotes the mean value. Furthermore, since std ( x p ) ≥ 0 is always valid, the correct Ȇ in must correspond to the global minimum point of the function std ( x p ) . Hence, the determination of the correct Ȇ in becomes an optimization problem to seek such values of Ȇ in that the objective function

std ( x p ) is minimized, i.e., std ( x p ) → min . To solve this multiple-dimensional optimization problem, the sequential golden section search technique is used because of its robustness and simplicity. Since (c, x p , y p ) are estimated from Eq. (5.1) for given Ȇ ex , the optimization scheme for Ȇ in is coupled with the resection scheme for Ȇ ex . Other appropriate objective functions can also be used; an obvious choice is the root-mean-square (rms) deviation of the calculated object space coordinates of all the targets from the measured ones. In fact, it is found that std ( x p ) or std ( y p ) is equivalent to this rms deviation in the optimization problem. The quantities std ( x p ) and std ( y p ) for optimization have a simple topological structure near the global minimum point, exhibiting a single ‘valley’ structure in the parametric space (Liu et al. 2000). Generally, the topological

90

5. Image and Data Analysis Techniques

structure of std ( x p ) or std ( y p ) depends on three-dimensionality of a target field; stronger three-dimensionality of the target field produces a steeper ‘valley’ in topology, leading to faster convergence. The topological structure of std ( x p ) or std ( y p ) can also be affected by random disturbances on the targets. Larger noise in images leads to a slower convergence rate and produces a larger error in optimization computations. Although the simple ‘valley’ topological structure allows convergence of optimization computation over a considerable range of the initial values, appropriate initial values are still required to obtain a converged solution. The DLT can provide such initial values for the exterior orientation parameters ( Ȧ, ij, ț , X c , Yc , Z c ) and the principal distance c . Combined with the DLT, the optimization method allows rapid and comprehensive automatic camera calibration to obtain a total of 14 camera parameters from a single image without requiring a guess of the initial values.

Fig. 5.2. Step target plate for camera calibration

The optimization method was used for calibrating a Hitachi CCD camera with a Sony zoom lens (12.5 to 75 mm focal length) and an 8 mm Cosmicar television. As shown in Fig. 5.2, a three-step target plate with a 2-in step height provided a 3D target field for camera calibration, on which 54 circular retro-reflective targets of a 0.5-in diameter spaced out 2 inches apart are placed. Figure 5.3 shows the principal distance given by the optimization method versus zoom setting for the Sony zoom lens. Figures 5.4 and 5.5 show, respectively, the principal-point location and radial distortion coefficient K1 as a function of the principal distance for the Sony zoom lens. The results given by the optimization method are in reasonable agreement with measurements for the same lens using optical equipment in laboratory (Burner 1995). The optimization method was also used to calibrate the same Hitachi CCD camera with an 8 mm Cosmicar television lens. Table 5.1 lists the calibration results given by the optimization method compared well with those obtained using optical equipment. In order to determine accurately the interior orientation parameters, a target field should fill up an image for camera calibration. In large wind tunnels, however, a camera is often located far from a model such that the target field looks small in the image plane. In this case, a two-step approach is suggested that determines the interior and exterior orientation parameters separately. First, placing a target plate near a camera to produce a sufficiently large target field in the image plane, we can determine accurately the interior orientation parameters

5.1. Geometric Calibration of Camera

91

using the optimization method. Next, assuming that the determined interior orientation parameters are fixed for locked camera setting, we obtain the exterior orientation parameters using a resection scheme from the target field in a given wind-tunnel coordinate system. 90

Principal distance c (mm)

80

Optimization algorithm Linear fit

70 60 50 40 30 20 10 10

20

30

40

50

60

70

80

Zoom setting (mm)

Fig. 5.3. Principal distance vs. zoom setting for a Sony zoom lens. From Liu et al. (2000) 2 Optimization algorithm

xp or yp (mm)

Laser illumination technique

1

yp 0

xp -1 10

20

30

40

50

60

70

80

90

c (mm)

Fig. 5.4. Principal-point location as a function of the principal distance for a Sony zoom lens connected to a Hitachi camera. From Liu et al. (2000)

92

5. Image and Data Analysis Techniques 0.0010 Optimization algorithm Burner (1995)

0.0005

K1 (mm-2)

0.0000

-0.0005

-0.0010

-0.0015

-0.0020 10

20

30

40

50

60

70

80

90

c (mm)

Fig. 5.5. The radial distortion coefficient as a function of the principal distance for a Sony zoom lens connected to a Hitachi camera. From Liu et al. (2000)

Table 5.1. Calibration for Hitachi CCD camera with 8 mm Cosmicar TV lens -2

-4

Interior orientation

c (mm) xp (mm)

yp (mm)

Sh /Sv

K1 (mm )

K2 (mm )

Optimization

8.133

0.2014

0.99238

0.0026

3.3×10

Optical techniques

8.137

-0.156 -0.168

0.2010

0.99244

0.0027

-1

-1

P1 (mm )

P2 (mm )

-5

1.8×10-4

3×10-5

-5

-4

4.5×10

1.7×10

7×10

-5

5.2. Radiometric Calibration of Camera Since PSP and TSP are based on radiometric measurements, a CCD camera used for measurements should have a good linear response of the electrical output to the scene radiance. However, there are many stages of image acquisition that may introduce non-linearity; for example, video cameras often include some form of ‘gamma’ mapping. When the radiometric response function of a camera is known, the non-linearity can be corrected. Here, a simple algorithm is described to determine the radiometric response function of a camera from a scene image taken in different exposures. First, we define I ( x ) as a linear radiometric response to the scene radiance and m [ I ( x )] as the measurement of I ( x ) by camera electronic circuitry that may produce a non-linear electrical output. Actually, the measurement m [ I ( x )] is the brightness or gray level of an image, where x is the image coordinates. The non-dimensional response function relating I ( x ) to m [ I ( x )] is defined by

5.2. Radiometric Calibration of Camera

I ( x ) / I max = f [ ξ ( x )] ,

93

(5.8)

where ξ ( x ) = m[ I ( x )] / m( I max ) is the non-dimensional measurement of I ( x ) normalized by the maximum value and I max corresponds to the maximum

radiance in the scene. Recovery of f (ξ ) is the task of the radiometric calibration of a camera. Two images of a scene are taken in two different exposures. According to the camera formula (Holst 1998), I ( x ) is proportional to the integration time t INT and inversely proportional to the square of the f-number F. Thus, we have the following functional equation for f (ξ ) ,

f (ξ 1 ) / f ( ξ 2 ) = R12 ,

(5.9)

where the subscripts 1 and 2 denote the image 1 and image 2, and the factor R12 is defined as I ( t / F 2 )1 R12 = max 2 INT . (5.10) I max 1 ( t INT / F 2 )2 Since m( I max ) corresponds to I max , the boundary condition for

f (ξ = 1) = 1 . We assume that f (ξ ) can be expanded as f (ξ ) =

f (ξ ) is

N

¦ c φ (ξ ) , n

(5.11)

n

n =0

where the base functions φ n ( ξ ) are the Chebyshev functions although other orthogonal functions and non-orthogonal functions like polynomials can also be used. Substitution of Eq. (5.11) to Eq. (5.9) leads to the following equations for the coefficients c n N

¦ c [φ ( ξ n

n

1

) − R12 φ n ( ξ 2 )] = 0 ,

n =0 N

¦ c φ (1) = 1 .

(5.12)

n n

n =0

For selected M pixels in a scene image, Eq. (5.12) constitutes a system of M+1 equations for the N+1 unknowns c n ( M ≥ N ). For a given R12 , a least-squares solution for c n can be found. In practice, since the factor R12 is not exactly known a priori, we use an approximate value of R12

R12 ≈

m( I max 2 ) ( t INT / F 2 )1 m( I max 1 ) ( t INT / F 2 )2

.

An iteration scheme can be used to give an improved value of R12 . Figure 5.6 shows two images taken by a Cannon digital still camera (EOS D30) at two

94

5. Image and Data Analysis Techniques

different f-numbers of F = 4.0 and F = 5.6, where Ansel Adams’ photograph of Mirror Lake of Yosemite was used as a test scene providing a broad range of the gray levels for radiometric calibration. Figure 5.7 shows the radiometric response function of the camera retrieved from the two images, where six terms of the Chebyshev functions in Eq. (5.11) were used. The response function of the Cannon digital still camera exhibits a non-linear behavior; it is also different for the red, green and blue (RGB) color channels.

(a)

(b)

Fig. 5.6. Two images of Mirror Lake of Yosemite (Ansel Adams 1935) taken by a Cannon digital still camera (EOS D30) at different F-numbers (a) F = 4.0 and (b) F = 5.6 for radiometric calibration of the camera

Camera Response Function, I/Imax

1.0 R G B

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

m(I)/m(Imax)

Fig. 5.7. Response functions of a Cannon digital still camera (EOS D30) for the R, G, and B color channels obtained from radiometric calibration

5.3. Correction for Self-Illumination

95

5.3. Correction for Self-Illumination The self-illumination of PSP and TSP results from the luminescent contribution to a point on a surface from all visible neighboring points; it becomes appreciable near a conjuncture of two surfaces and on a concave surface (Ruyten 1997a, 1997b, 2001a; Ruyten and Fisher 2001; Le Sant 2001b). Although the selfillumination can be to certain extent suppressed by taking a ratio between a windon image and a wind-off image, it cannot be eliminated without considering an exchange of the radiative energy between neighboring surfaces, which may produce an error in data reduction of PSP and TSP. Therefore, we need to know how much the radiative energy leaves from an area element and travels toward another element. The geometric relations for this inter-surface process are known as view factors, configuration factors, shape factors, or angle factors (Modest 1993). We consider diffuse surfaces that absorb and emit diffusely, and also reflect the radiative energy diffusely. The view factor dFdA i − dA j between two infinitesimal surface elements dAi and dA j , as shown in Fig. 5.8, is defined as a ratio between the diffuse energy leaving dAi directly toward and intercepted by

dA j and the total diffuse energy leaving dAi , which is expressed as dFdA i − dA j =

cos θ i cos θ j

π | X ij |

2

dA j =

( ni • X ij )( n j • X ij )

π | X ij |4

dA j ,

(5.13)

where n i (or n j ) is the unit normal vector of dAi (or dA j ), X ij is the position vector directing from dAi toward dA j , and θ i (or θ j ) is the angle between the position vector X ij and the normal n i (or n j ). The view factors leaving dAi directly toward the total surface A j or leaving A j toward dAi , or leaving A j toward Ai can be similarly defined by integrating dFdA i − dA j (Modest 1993). The law of reciprocity dAi dFdA i −dA j = dA j dFdA j −dA i is valid for these view factors. The view factor is a function of the geometric parameters. Methods for evaluating the view factors were discussed by Modest (1993) and a large collection of the view factors for simple geometric configurations was complied by Howell (1982). For partially specular surfaces, the determination of the view factors is more complicated since the bidirectional reflectance distribution function (BRDF) of the paint must be known (Nicodemus et al. 1977; Asmail 1991).

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5. Image and Data Analysis Techniques

Xj nj

θj

dAj

Xij

ni θi

Xi

dAi

Fig. 5.8. Radiative exchange between two surface elements

The self-illumination correction is applied to an image intensity (or brightness intensity) field denoted by I in this sub-section after it is mapped onto a model surface grid in the object space. Because the image intensity is proportional to the luminescent energy flow rate, the image intensity I i at an area element dAi is a sum of the local intrinsic intensity I i( 0 ) and an integration of the contributions from all the neighboring elements, i.e.,

I i = I i( 0 ) + ρ λwp2

N

¦I j =1

j

dFdA j − dAi dA j

,

(5.14)

where ρ λwp2 is the reflectivity of the wall-paint interface at the luminescent wavelength. In simulations, given a set of the intrinsic intensities I i( 0 ) , the image intensity I i affected by the self-illumination can be obtained using a simple iteration scheme

I i( n+1 ) = I i( 0 ) + ρ λwp2

N

¦I j =1

(n) j

dFdA j −dA i dA j

.

(5.15)

The more efficient Gauss-Seidel iteration scheme was used by Ruyten and Fisher (2001). In measurements, since the image intensity I i is known in PSP and TSP images, an explicit relation is used to correct the self-illumination and recover the intrinsic intensity I i( 0 ) , i.e.,

I i( 0 ) = I i − ρ λwp2

N

¦I j =1

j

dFdA j −dA i dA j

.

(5.16)

5.3. Correction for Self-Illumination

97

The steps for correcting the self-illumination are: (1) measuring the reflectivity ρ λwp2 ; (2) defining a surface grid consisting of N surface elements dAi ; (3) evaluating the view factors dFdA j −dA i ; (4) mapping the image intensity I i onto the surface grid; (5) calculating the intrinsic (corrected) intensity I i( 0 ) using Eq. (5.16); and (6) calculating a ratio of the intrinsic (self-illumination-corrected) intensities and converting it to pressure or temperature. Ruyten and Fisher (2001) conducted a numerical simulation of correcting the self-illumination for a PSP test of the Alpha jet and found that the error associated with the self-illumination in PSP measurements could reach several percents of actual pressure.

plate 2

plate 1 dA1

α

Fig. 5.9. Wedge-shaped conjunction of two plates

Here, we consider a simple but representative geometric configuration, a wedge-shaped conjunction of two infinitely large plates, as shown in Fig. 5.9; this case allows an analytical estimate of the error induced by the self-illumination. The image intensity at a location on the plate 1 is

I 1 = I 1( 0 ) + ρ λwp2

³I

2

dFdA2 −dA1 dA2 .

(5.17)

plate 2

Assuming that the image intensity at the plate 2 is homogenous, by integrating the view factor for this configuration (Modest 1993), we obtain the image intensity at the plate1 affected by the plate 2

I 1 ≈ I 1( 0 ) + ε 1 I 2 ,

(5.18)

where the parameter ε 1 = ρ λwp2 ( 1 + cos α ) / 2 represents the combined effect of the angle α between the plates and reflectivity. Clearly, the self-illumination decreases from the maximum value at α = 0 o to zero at α = 180 o . A reciprocal relation gives the image intensity at the plate 2

98

5. Image and Data Analysis Techniques

I 2 ≈ I 2( 0 ) + ε 1 I 1 .

(5.19)

When the parameter ε 1 is small, the image intensity ratio at the plate 1 is 0) I 1 ref / I 1 ≈ ( I 1( ref / I 1( 0 ) ) ( 1 + ε 1 ε 2 ) .

(5.20)

0) − I 2( 0 ) / I 1( 0 ) reflects the difference of the relative The parameter ε 2 = I 2( 0ref) / I 1( ref influence of the plate 2 on the pate 1 between the wind-off reference and wind-on conditions. Using the Stern-Volmer relation for PSP, we obtain an estimate for the pressure error associated with the self-illumination for the wedge configuration

§ A p( 0 ) | p − p( 0 ) | ≈ ε1 | ε 2 | ¨ + (0 ) (0 ) ¨ B p ref p ref ©

· ¸, ¸ ¹

(5.21)

(0 ) where A and B are the Stern-Volmer coefficients, and p ( 0 ) and p ref are, respectively, the intrinsic PSP-derived pressures in the wind-on and wind-off reference conditions that are not affected by the self-illumination. Similarly, using the Arrhenius relation for TSP, we have an estimate for the temperature error associated with the self-illumination for the wedge configuration

|T − T (0 ) | R ≈ ε1 | ε2 | E nr T (0 )

,

(5.22)

where R is the universal gas constant, E nr is the activity energy of TSP, and

T ( 0 ) is the intrinsic TSP-derived temperature that is not affected by the selfillumination. The above discussion is based on an assumption that the luminescent paint surface is a diffuse surface or Lambertian surface. Nevertheless, a real paint surface is neither Lambertian nor specular. To characterize reflection on a general surface, the bidirectional reflectance distribution function (BRDF) was introduced by Nicodemus et al. (1977). As shown in Fig. 5.10, the incident radiance is generally a function of the incident direction defined by the incident polar angle and azimuthal angle ( θ i ,φ i ) , i.e., Li = Li ( θ i ,φ i ) .

(5.23)

The reflection radiance Lr ( θ i ,φ i ;θ r ,φ r ) is quantitatively characterized by the BRDF

f r ( θ i ,φ i ;θ r ,φ r ) = dLr ( θ i ,φ i ;θ r ,φ r ) / dEi ( θ i ,φ i ) .

(5.24)

where ( θ r ,φ r ) defines the direction of reflection and the infinitesimal incident irradiance dE i ( θ i ,φ i ) over a solid angle element dω i is

5.3. Correction for Self-Illumination

dEi ( θ i ,φ i ) = Li ( θ i ,φ i ) cosθ i dω i .

99

(5.25)

-1

The BRDF has a unit of steradian . Here, the conventional radiometric notations L and E are used for radiance and irradiance, which are also applicable to the luminescent emission. The BRDF depends on a surface roughness distribution. For a perfectly diffuse surface where the reflection radiance is isotropic, i.e., Lr = const . , the BRDF is

f r = 1 / π (Horn and Sjoberg 1979). For a general surface, the BRDF can be derived based on either the wave equation for electromagnetic waves or geometrical optics models (Beckmann and Spizzichino 1963; Torrance and Sparrow 1967; Nayar et al. 1991). Asmail (1991) gave a bibliographical review on the BRDF. From a viewpoint of application, empirical expressions for the scattered radiance from a rough surface are very useful due to their simplicity (Cook and Torrance 1981; Haussecker 1999). An empirical model for a single light source is Lr ( X ) = ρ a E a ( X ) + ρ d Els ( X )( N T Ls ) + ρ s Els ( X ) p( R T V ) ,

(5.26)

where the first, second and third terms are, respectively, the contributions from the ambient reflection, diffuse reflection, and specular reflection. In Eq. (5.26), ρ a ,

ρ d , and ρ s , are the empirical reflection coefficients for the ambient reflection, diffuse reflection, and specular reflection. As shown in Fig. 5.10, the vectors N , Ls , R , and V are, respectively, the unit normal vector of a surface, the unit vector directing the light source from the surface, the unit main directional vector of the specular reflection, and the unit viewing vector. E a ( X ) and Els ( X ) are the irradiances for the ambient environment and light sources, respectively. The function p( R T V ) is the directional distribution of the specular reflection, describing the spreading of scattered light. Phong (1975) gave a power function p( R T V ) = ( R T V )n . In general, the main directional vector of the specular reflection, R , is a function of the incident direction of light − Ls . Although there are certain theories for predicting the specular direction R (Torrance and Sparrow 1967), R is not known for a general surface. The unknowns in Eq. (5.26), such as R , the reflection coefficients, and the parameters in p( R T V ) , have to be determined experimentally by calibration. Le Sant (2001b) measured the BRDF for the B1 PSP paint with talc using a BRDF calibration rig. As illustrated in Fig. 5.11, the BRDF calibration rig included a lamp for illumination and a spectrometer to measure the reflected light from a sample. The lamp emitted white light, enabling the calibration of the o o BRDF in the visible range; the lamp moved from 0 at the vertical position to 60 . o o The zenith (or polar) angle of the spectrometer moved from 0 to 60 and the o o o azimuth angle moved from 0 to 180 , where 180 was in the opposite direction of the emission. Figure 5.12 shows the measured BRDF for the B1 paint, which was nearly Lambertian when the zenith (or polar) angle of illumination was 10°, while specular reflection occurred when the zenith angle increased further. The

100

5. Image and Data Analysis Techniques

maximum value was always achieved in the specular direction. The low value obtained at the azimuth angle of 0° was incorrect since the spectrometer was in the front of the lamp and thus the PSP sample was no longer illuminated. The measured BRDF showed a superposition of diffuse reflection and specular reflection. A specular peak was observed at the zenith angle of 60° as well as a secondary peak at the azimuth angle of 90°. The value of the diffuse reflection factor depended on the zenith angle of illumination. Le Sant (2001b) was able achieve a good fit to the measured BRDF using the modified Phong model (Phong 1975), as shown in Fig. 5.13, and the modeled BRDF captured the main features of the measured BRDF except the secondary specular peaks.

LI

θE

V

N

R

θH

φH

Fig. 5.10. Vectors of incident, reflecting, and viewing directions zenith (spectrometer) zenith (illumination)

0°

180° 90°

90°

azimuth 0°

Fig. 5.11. The zenith (or polar) and azimuth angles in the BRDF calibration rig. From Le Sant (2001b)

5.3. Correction for Self-Illumination BRDF 0.5

BRDF 0.5

20

0.4

30

0.4

0.2

0.1

0.1 60

0

60

0

160

zenith

180

40

140

20

zenith

80

20

40

20

azimuth

0

0

40

0.6

50

0.4

60

0.4

0.2

0.2

60

0

0.1 60

0

180

160

40

140

10

160

zenith

180

40

140

30

120 100 80

0

50

50

10

40

zenith

160

azimuth

80

10

40 20

20 0

0

azimuth

60

40

20

120 100

20

80 60

180

140

30

120 100

20

azimuth

60

0.4

0.3

0.1

50

20

0.5

0.3

0.1 60

BRDF 0.5

0.3

30

0

0.6

BRDF 0.5

40

azimuth

40 20 0

0.6

0

80 60

10

40

BRDF

zenith

120 100

20 0

180

160 140

30

zenith

80 60

10

40 20 0

180

120 100

azimuth

60 10

160 140

30

120 100

0

50

50

50

30

0.4

0.3

0.2

0.2

0.1

40

0.5

0.3

0.3

60

0.6

0.6

0.6

BRDF

10

101

0

0

0

Fig. 5.12. The measured BRDF of the B1 paint at the illumination zenith angles of 10, 20, 30, 40, 50 and 60 degrees. From Le Sant (2001b)

BRDF

BRDF 0.5

10

20

0.4

0.4

0.2

0.2

0.2

0.1

0.1 60

0

60

0

160

180

40

140

30 20

zenith

80

40

BRDF

0 160 140

30

120 100

20

80 60

10

40 20

azimuth

180

60

0.4

0.4

0.3

0.2

0.2

40

0.5

0.3

0.2

0.1

0.1

50

0.6

0.6

0.3

60

0

0.5

50

0.4

azimuth

40 20 0

BRDF

40

0

80 60

10

0.6

0

20

azimuth

40

0

0.5

180

120 100

20 0

0

BRDF

zenith

160 140

30

zenith

80 60

10

40 20 0

180

120 100

20

azimuth

60 10

160 140

30

120 100

0

50

50

50 40

0.4

0.3

0.3

0.1

zenith

0.5

30

0.5

0.3

60

0.6

0.6

0.6

BRDF

60

0

50 40

160

180

0

50 40

160

140

30

zenith

0.1 60

120 100

20

80 60

10

40

azimuth

120 100

20

80 60

10

0

azimuth

40

20 0

180

140

30

zenith

20 0

0

Fig. 5.13. The modeled BRDF of the B1 paint at the illumination zenith angles of 10, 20, 30, 40, 50 and 60 degrees using the modified Phong model. From Le Sant (2001b)

Le Sant (2001b) also studied the self-illumination in a corner to validate a correction algorithm. The corner was painted with a Pyrene-based paint providing an image significantly affected by the self-illumination near the junction of the two plates, as shown in Fig. 5.14. Then, the left plate was covered with a black sheet, removing the effect of the self-illumination on the right plate, as shown in

102

5. Image and Data Analysis Techniques

the right image in Fig. 5.14. Figure 5.15 shows results before and after correcting the self-illumination based on the diffuse surface model and the Phong model. The self-illumination correction was effective; the self-illumination effect was reduced to 15% from about 40% near the junction. This paint behaved mostly like a diffuse paint such that the Phong model did not exhibit a significant improvement. Although the Phong model might improve the accuracy of correction for a surface with strong specular reflection, the computation time for the Phong model was much longer than that for the diffuse surface model.

Fig. 5.14. Self-illumination in a corner coated with PSP. From Le Sant (2001b) 0.16

with SI effect

0.15 0.14 0.13

diffuse model

0.12 0.11

Phong model

0.10 0

10

20

30

without SI 40 X (mm) 50

Fig. 5.15. Self-illumination correction using the diffuse and Phong models. From Le Sant (2001b)

5.4. Image Registration The intensity-based method for PSP and TSP requires a ratio between the wind-on and wind-off images of a painted model. Since a model deforms due to aerodynamic loads, the wind-on image does not align with the wind-off image; therefore these images have to be re-aligned before taking a ratio between the images. The image registration technique, developed by Bell and McLachlan (1993, 1996) and Donovan et al. (1993), is based on an ad-hoc transformation that

5.4. Image Registration

103

maps the deformed wind-on image coordinates ( xon , yon ) onto the reference wind-off image coordinates ( xoff , yoff ) . In order to register the images, some black fiducial targets are placed on a model. When the correspondence between the targets in the wind-off and wind-on images is established, a transformation between the wind-off and wind-on image coordinates of the targets can be expressed as x off = aij φ i ( x on )φ j ( y on ) . (5.27) y off = bij φ i ( xon )φ j ( y on )

¦ ¦

The base functions φ i ( ξ ) are either the orthogonal functions like the Chebyshev functions or the non-orthogonal power functions φ i ( x ) = x i used by Bell and McLachlan (1993, 1996) and Donovan et al. (1993). Given the image coordinates of the targets placed on a model, the unknown coefficients aij and bij can be determined using least-squares method to match the targets between the wind-on and wind-off images. For image warping, one can also use a 2D perspective transform (Jähne 1999)

x off =

a 11 x on + a 12 y on + a 13 a 31 x on + a 32 y on + 1

. (5.28) a 21 x on + a 22 y on + a 23 y off = a 31 x on + a 32 y on + 1 Although the perspective transform is non-linear, it can be reduced to a linear transform using the homogeneous coordinates. The perspective transform is collinear that maps a line into another line and a rectangle into a quadrilateral. Therefore, Eq. (5.28) is more restricted than Eq. (5.27) for PSP and TSP applications. Before the image registration technique is applied, the targets must be identified and their centroid locations in images must be determined. The target centroid ( xc , y c ) is defined as

¦¦ x I ( x , y ) / ¦¦ I ( x , y ) , = ¦¦ y I ( x , y ) / ¦¦ I ( x , y )

xc =

i

i

i

i

i

yc

i

i

i

i

i

(5.29)

where I ( xi , y i ) is the gray level on an image. When a target contains only a few pixels and the target contrast is not high, the centroid calculation using the definition Eq. (5.29) may not be accurate. Another method for determining the target location is to maximize the correlation between a template f ( x , y ) and the target scene I ( x , y ) (Rosenfeld and Kak 1982). The correlation coefficient C fI is defined as

104

5. Image and Data Analysis Techniques

C fI =

³³ f ( x + x , y + y )I ( x , y )dxdy ³³ f ( x , y )dxdy ³³ I ( x , y )dxdy 0

2

0

.

(5.30)

2

For the continuous functions f ( x , y ) and I ( x , y ) , one can determine the location

( x0 , y0 ) of the target by maximizing C fI . However, it is found that for small targets in images, sub-pixel misalignment between the template and the scene can significantly reduce the value of C fI even when the scene contains a perfect replica of the template. To enhance the robustness of a localization scheme, Ruyten (2001b) proposed an augmented template f ( x , y ) = f 0 ( x , y ) + f x ∆ x + f y ∆ y , where f 0 ( x , y ) represented a conventional template and f x and f y are the partial derivatives of f ( x , y ) . The additional shift parameters ( ∆ x , ∆ y ) allowed more robust and accurate determination of the target locations. In PSP and TSP measurements, operators can manually select the targets and determine the correspondence between the wind-off and wind-on images. However, PSP and TSP measurements with multiple cameras in production wind tunnels may produce hundreds or thousands of images in a given test; thus, image registration becomes very labor-intensive and time-consuming. It is non-trivial to automatically establish the point-correspondence between images taken by cameras at different viewing angles and positions. This problem is generally related to the epipolar geometry in which a point on an image corresponds to a line on another image (Faugeras 1993). Ruyten (1999) discussed the methodologies for automatic image registration including searching targets, labeling targets and rejecting false targets. Unlike ad-hoc techniques, the searching technique based on photogrammetric mapping is more rigorous. Once cameras are calibrated and the position and attitude of a tested model are approximately given by other techniques (such as accelerators and videogrammetric techniques), the targets in the images can be found using photogrammetric mapping from the 3D object space to the image plane (see Section 5.1). The aforementioned methods of using a single transformation for the whole image is a global approach for image registration. A local approach proposed by Shanmugasundaram and Samareh-Abolhassani (1995) divides an image domain into triangles connecting a set of targets based on the Delaunay triangulation (de Berg et al. 1998). For a triangle defined by the vertex vectors R1 , R2 and R2 , a point in the plane of the triangle can be described by a vector u1 R1 + u 2 R2 + u 3 R3 , where (u 1 , u 2 , u 3 ) are referred to as the parametric (barocentric) coordinates and a constraint u1 + u 2 + u 3 = 1 is imposed. When a wind-on pixel is identified inside a triangle and its parametric coordinates is given, the corresponding wind-off pixel can be determined by using the same parametric coordinates in the vertex vectors of the corresponding triangle in the wind-off image. Finally, the image intensity at that pixel is mapped from the wind-on

5.5. Conversion to Pressure

105

image to the wind-off image. This approach is basically a linear interpolation assuming that the relative position of a point inside a triangle to the vertices is invariant under a transformation from the wind-on image to the wind-off image. Weaver et al. (1999) proposed a so-called Quantum Pixel Energy Distribution (QPED) algorithm that utilizes local surface features to calculate a pixel shift vector using a spatial correlation method. The local surface features could be targets, pressure taps, and dots formed from aerosol mists in spraying on a basecoat. Similar to particle image velocimetry (PIV), the QPED algorithm can give a field of the displacement vectors when the registration marks or features are dense enough. Based on the shift vector field, the wind-on image can be registered. Although the QPED algorithm is computationally intensive, it can provide the local displacement vectors at certain locations to complement the global image registration techniques. A comparative study of different image registration techniques was made by Venkatakrishnan (2003).

5.5. Conversion to Pressure In PSP measurements, conversion of the luminescent intensity to pressure is complicated by the temperature effect of PSP especially when the surface temperature distribution is not known. Empirically, a priori calibration relation between air pressure and the relative luminescent intensity is expressed by a polynomial 2

§ I ref · I ref p ¸ . = C1 ( T ) + C2 ( T ) + C3 ( T ) ¨¨ ¸ I pref © I ¹

(5.31)

The experimentally determined coefficients C1 , C2 and C3 in Eq. (5.31) can be expressed as a polynomial function of temperature. If a distribution of the surface temperature is not given and the thermal conditions in a priori laboratory calibrations are different from those in wind tunnel tests, a priori relation Eq. (5.31) cannot be directly applied to conversion to pressure. To deal with this problem, a short-cut approach is in-situ PSP calibration that directly correlates the luminescent intensity to pressure data from taps distributed on a model surface. In this case, the constant coefficients C1 , C2 and C3 in Eq. (5.31) are determined using least-squares method to achieve the best fit to the pressure tap data over a certain range of pressures. Through in-situ calibration, the effect of a non-uniform surface temperature distribution is actually absorbed into a precision error of leastsquares estimation. When the temperature effect of PSP overwhelms a change of the luminescent intensity produced by pressure, in-situ calibration has a large precision error. In addition, when the pressure tap data do not cover the full range of pressure on a surface, in-situ PSP conversion may lead to a large bias error in data extrapolation outside the calibration range of pressures.

106

5. Image and Data Analysis Techniques

A hybrid method between in-situ and a priori methods is the so-called K-fit method originally suggested by M. Morris and recapitulated by Woodmansee and Dutton (1998). Eq. (5.31) is re-written as 2 ª § I off · § I off · º p ¸ + C3 ( T ) ¨ K I ¸ », = K P «C1 ( T ) + C2 ( T )¨¨ K I ¨ « poff I ¸¹ I ¸¹ » © © ¬ ¼

(5.32)

where I off = I ( poff ,Toff ) is the luminescent intensity in the wind-off conditions, and K I = I ref / I off and K P = pref / poff are called the K-factors. The reference conditions under which a priori calibration is made in a laboratory are generally different from the wind-off conditions in a wind tunnel. While the factor K P = pref / poff is known, the factor K I = I ref / I off is generally not known and has to be determined since illumination conditions and photodetectors used in laboratory may be different from those in wind tunnel. Given the coefficients C1 ,

C2 and C3 at a known temperature on an isothermal surface, K I can be determined using a single data point from pressure taps. When the surface temperature data near a number of pressure taps are provided by other techniques like TSP and IR camera, a more accurate value of K I can be obtained using leastsquares method with larger statistical redundancy. In the worst case where the surface temperature distribution is totally unknown, assuming an average temperature over the surface, we still able to estimate K I by fitting the pressure tap data. Similar to in-situ calibration, the effect of a non-uniform temperature distribution is absorbed into a precision error of least-squares estimation for K I . Bencic (1999) used a similarity variable of the luminescent intensity to scale the temperature effect of certain PSP

I ref I

= g( T ) corr

I ref I

,

(5.33)

where g( T ) was a function of temperature to be determined by a priori calibration. Under this similarity transformation, the calibration curves for the paint at different temperatures collapsed onto a single curve with the temperatureindependent coefficients, i.e.,

I ref p = C1 + C2 I pref

corr

§ I ref + C3 ¨ ¨ I ©

2

· ¸ . ¸ corr ¹

(5.34)

In this case, instead of using a 2D calibration surface in the parametric space, only a single one-parameter relation Eq. (5.34) was used to convert the luminescent intensity ratio to pressure. Bencic (1999) found that this similarity was valid for a

5.6. Pressure Correction for Extrapolation to Low-Speed Data

107

Ruthenium-based PSP used at NASA Glenn. In fact, as pointed out in Section 3.6, this similarity is a property of the so-called ‘ideal’ PSP that obeys the following relations (Puklin et al. 1998; Coyle et al. 1999)

I ( p,T ) / I ( p,Tref ) = g( T ) , I ref ( pref ,Tref ) I ( p,Tref )

= g( T )

I ref ( pref ,Tref ) I ( p,T )

.

(5.35)

Puklin et al. (1998) found that PtTFPP in FIB polymer was an ‘ideal’ PSP over a certain range of temperatures. Note that this similarity (or invariance) is not the universal property of a general PSP.

5.6. Pressure Correction for Extrapolation to Low-Speed Data PSP is particularly effective in high subsonic, transonic and supersonic flow regimes. However, in low-speed flows where the Mach number is typically less than 0.3, PSP measurement is a challenging problem since a very small pressure change may not be sufficiently resolved by PSP. The major error sources, notably the temperature effect, image misalignment and CCD camera noise, must be minimized to obtain acceptable quantitative pressure results at low speeds. The resolution of PSP measurements is eventually limited by the photon shot noise of a CCD camera. Liu (2003) proposed a pressure-correction method as an alterative to extrapolate low-speed pressure data without directly attacking the intrinsic difficulty of PSP instrumentation for low-speed flows. This method is able to obtain the incompressible pressure coefficient from PSP measurements at suitably higher Mach numbers (typically Mach 0.3-0.6) by removing the compressibility effect. It is noticed that there is a significant difference between the responses of the absolute pressure p and the pressure coefficient C p to the freestream Mach number M ∞ .

The sensitivity of C p to the Mach number for M ∞2 2

and

cov (ȗ i ȗ j ) = < ǻȗ i ǻȗ j > are the variance and covariance, respectively, and the notation < > denotes the statistical ensemble average. Here, the variables {ȗ i , i = 1 M} denote a set of the parameters Dt (ǻt ) , D x (ǻx ) , Dq0 (ǻt ) , V , Vref , Ȇ c /Ȇ c ref , Ȇ f /Ȇ f ref , h / href , c / c ref , q0 /q0 ref , p ref , T , A, and B in Eq. (7.2). The sensitivity coefficients S i are defined as S i = ( ȗ i /p )( ∂ p / ∂ ȗ i ) . Eq. (7.3) becomes particularly simple when the cross-correlation coefficients between the variables vanish ( ȡi j = 0 , i ≠ j ). Table 7.1 lists the sensitivity coefficients, the elemental errors and their physical origins. Many sensitivity coefficients are proportional to a factor ϕ = 1 + [A(T) /B(T)] / (p / p ref ) . For Bath Ruth + silica-gel in GE RTV 118, Figure 7.1 shows the factor 1 + [A(T) /B(T)] / (p / p ref ) as a function of p / p ref for different temperatures, which is only slightly changed by temperature. The temperature sensitivity coefficient is S T = −T [B' (T) + A' (T) p ref /p]/ B ( T ) , where the prime denotes differentiation respect with temperature. Figure 7.2 shows the absolute value of S T as a function of p / p ref at different temperatures. After the elemental errors in Table 7.1 are evaluated, the total uncertainty in pressure can be readily calculated using Eq. (7.3). The major elemental error sources are discussed below.

140

7. Uncertainty

Table 7.1. Sensitivity coefficients, elemental errors, and total uncertainty of PSP Variable

Sensi. Coef.

ȗi

Elemental Variance

Physical Origin

var( ȗ i )

Si

1

D t (ǻt )

ij

2

D x (ǻx )

ij

3

D q0 (ǻt )

4 5 6

Vref V Ȇ c /Ȇ c ref

7

Ȇ f /Ȇ f ref

8

h / href

ij ij -ij ij ij ij

9

c / c ref

ij

−2 [ (∂ c /∂ x ) ı x2 + (∂ c /∂ y ) ı 2y ] c ref

10 q 0 /q 0 ref

ij

( q0 ref )−2 ( ∇q0 ) • ( ǻX )

p ref

1

var( p)

11

12 T 13 A 14 B 15 Pressure

mapping

[( ∂V / ∂ t )( ǻt)/V ] 2 [ (∂V/∂x ) ı x2 + (∂V /∂ y ) ı 2y ] V − 2 2

2

[( ∂ q 0 / ∂ t )( ǻt)/q 0 ref ] 2 V ref G ! Ȟ B d

Photodetector noise

V G !Ȟ Bd

Photodetector noise

[ R2 /(R1 + R2 )] (ǻR1 /R1 ) 2

2

var(Ȇ f /Ȇ f ref )

−2 [ (∂ h /∂ x ) ı x2 + (∂ h /∂ y ) ı 2y ] href 2

2

2

2

2

ST

var( T) 1 − ij var( A) -1 var( B) 1 (∂ p/∂x )2 ı x2 + (∂p /∂y )2 ı 2y and ( ∇ p )surf • ( ǻX )surf

Total Uncertainty in Pressure

Temporal variation in luminescence due to photodegradation and surface contamination Image registration errors for correcting luminescence variation due to model motion Temporal variation in illumination

2

var (p)/ p 2 =

Change in camera performance parameters due to model motion Illumination spectral variability and filter spectral leakage Image registration errors for correcting thickness variation due to model motion Image registration errors for correcting concentration variation due to model motion Illumination variation on model surface due to model motion Error in measurement of reference pressure Temperature effect of PSP Paint calibration error Paint calibration error Errors in camera calibration and pressure mapping on a surface of a presumed rigid body M

¦S

2 i

var( ȗ i )/ ȗ i

2

i =1

Note: (1) ı x and ı y are the standard deviations of least-squares estimation in the image registration or camera calibration. (2) The factors for the sensitivity coefficient are defined as ij = 1 + [ A( T ) / B( T )]( p ref / p ) and S T = − [ T / B ( T )] [B' (T) + A' (T)(p ref / p ) ] .

7.1. Pressure Uncertainty of Intensity-Based Methods

141

3.0 T = 293 K T = 313 K T = 333 K

1 + (A(T)/B(T))(Pref/P)

2.5

2.0

1.5

1.0

0.5 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

P/Pref

Fig. 7.1. The sensitivity factor 1 + [A(T) /B(T)] / (p / p ref ) as a function of p / p ref at different temperatures for Bath Ruth + silica-gel in GE RTV 118. From Liu et al. (2001a) 10 9

T = 293 K T = 313 K T = 333 K

|(dP/dT)(T/P)|

8 7 6 5 4 3 2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

P/Pref

Fig. 7.2. The temperature sensitivity coefficient as a function of p / p ref at different temperatures for Bath Ruth + silica-gel in GE RTV 118. From Liu et al. (2001a)

7.1.3. Photodetector Noise and Limiting Pressure Resolution The uncertainties in the outputs V and Vref from a photodetector (e.g. camera) are contributed from a number of noise sources in the detector such as the photon shot noise, dark current shot noise, amplifier noise, quantization noise, and pattern

142

7. Uncertainty

noise. When the dark current and pattern noise are subtracted and the noise floor is negligible, the detector is photon-shot-noise-limited. In this case, the signal-tonoise ratio (SNR) of the detector is SNR = ( V / G ! ȞBd )1 / 2 , where ! is the Planck’s constant, ν is the frequency, Bd is the electrical bandwidth of the detection electronics, G is the system’s gain, and V is the detector output. The uncertainties in the outputs are expressed by the variances var(V) = V G ! Ȟ Bd and

var(Vref ) = Vref G ! Ȟ Bd . In the photon-shot-noise-limited case in which the error propagation equation contains only two terms related to V and Vref , the pressure uncertainty is ǻ p §¨ G Bd ! Ȟ ·¸ = ¨ V ref ¸ p ¹ ©

1/ 2

ª A( T ) p ref «1 + B( T ) p «¬

ºª p º » » «1 + A( T ) + B( T ) p ref ¼» »¼ ¬«

1/ 2

,

(7.4)

which holds for both CCD cameras and non-imaging detectors. For a CCD camera, the first factor in the right-hand side of Eq. (7.4) can be simply expressed by the total number of photoelectrons collected over the integration time ( ∝ 1 / Bd ), n pe = V /( G ! ȞBd ) . When the full-well capacity of the CCD camera is achieved, we obtain the minimum pressure difference that PSP can measure from a single frame of image

( ǻp)min 1 = p (n pe ref )max

ª A(T) p ref º ª p º » «1 + » «1 + A(T) + B(T) B(T) p »¼ ¬« p ref »¼ ¬«

1/2

,

(7.5)

where (n pe ref )max is the full-well capacity of the camera in reference conditions. When N images are averaged, the limiting pressure difference (7.5) is further 1/2 reduced by a factor N . Eq. (7.5) provides an estimate for the noise-equivalent pressure resolution for a CCD camera. When (n pe ref )max is 500,000 electrons for a CCD camera and Bath Ruth + silica-gel in GE RTV 118 is used, the minimum pressure uncertainty ( ǻp)min / p is shown in Fig. 7.3 as a function of p / p ref for different temperatures, indicating that an increased temperature degrades the limiting pressure resolution.

Figure 7.4 shows

(n pe ref )max ( ǻp)min / p

as a

function of p / p ref for different values of the Stern-Volmer coefficient B(T) . Clearly, a larger B(T) leads to a smaller limiting pressure uncertainty ( ǻp)min / p . Figure 7.5 shows

(n pe ref )max ( ǻp)min / p

as a function of the Stern-Volmer

coefficient B(T) for different values of p / p ref . There is no optimal value of B in this case.

Minimum pressure uncertainty (%)

7.1. Pressure Uncertainty of Intensity-Based Methods

143

0.40

0.36

0.32 T = 333 K T = 313 K

0.28 T = 293 K 0.24

0.20 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

P/Pref

Fig. 7.3. The minimum pressure uncertainty ( ǻp)min / p as a function of p / p ref at different temperatures for Bath Ruth + silica-gel in GE RTV 118. From Liu et al. (2001a)

7

(∆P)min/P [(npe ref)max]

1/2

6

5 B = 0.5 4 0.6 3

0.7 0.8

2

0.9

1 0.0

0.5

1.0

1.5

2.0

P/Pref

Fig. 7.4. The normalized minimum pressure uncertainty

(n pe ref )max ( ǻp)min / p as a

function of p / p ref for different values of the Stern-Volmer coefficient B(T) . From Liu et al. (2001a)

144

7. Uncertainty

20 P/Pref = 0.2

(∆P)min/P [(npe ref)max]

1/2

0.5 15

1.0

1.5

10

2.0 5

0.0

0.2

0.4

0.6

0.8

1.0

B

Fig. 7.5. The normalized minimum pressure uncertainty

(n pe ref )max ( ǻp)min / p as a

function of the Stern-Volmer coefficient B for different values of p / p ref . From Liu et al. (2001a)

7.1.4. Errors Induced by Model Deformation Model deformation generated by aerodynamic loads causes a displacement ǻx = x' − x of the wind-on image relative to the wind-off image. This displacement leads to the deviations of the quantities D x (ǻx ) , h / href , c / c ref , and q0 /q0 ref in Eq. (7.2) from unity because the distributions of the luminescent intensity, paint thickness, dye concentration and illumination level are not spatially homogeneous on a surface. After the image registration technique is applied to re-align the wind-on and wind-off images, the estimated variances of these quantities are var[Dx (ǻx )] ≈ W ( V ) / V 2 , var( h / href ) ≈ W ( h ) /( href )2 , and

var( c / c ref ) ≈ W ( c ) /( c ref ) 2 .

The

operator

W( • )

is defined

as

W( • ) = (∂ /∂ x ) ı + (∂ /∂ y ) ı , where ı x and ı y are the standard deviations of least-squares estimation for image registration. The uncertainty in q0 ( X )/q0 ref ( X' ) is caused by a change in the illumination intensity on a model surface after the model moves with respect to the light sources. When a point on the model surface travels along the displacement vector ǻX = X' − X in the object space, the variance of q0 /q0 ref is estimated by 2

2 x

2

2 y

2

var[q0 ( X ) / q0 ref ( X' )] ≈ ( q0 ref )−2 ( ∇q0 ) • ( ǻX ) . Consider a point light source

7.1. Pressure Uncertainty of Intensity-Based Methods

with unit strength that has a light flux distribution q0 ( X − X s ) = X − X s

145 −n

,

where n is an exponent (normally n = 2) and X − X s is the distance between the point X on the model surface and the light source location X s . Thus, the variance of q0 /q0 ref for a single point light source is var[q0 ( X ) / q0 ref ( X' )]

= n2 X − X s

−4

2

( X − X s ) • ( ǻX ) .

The variance for multiple point light

sources can be obtained based on the principle of superposition. In addition, model deformation leads to a small change in the distance between the model surface and the camera lens. The uncertainty in the camera performance parameters due to this change is var(Ȇ c /Ȇ c ref ) ≈ [ R2 /(R1 + R2 )] 2 (ǻR1 /R1 )2 , where R1 is the distance between the lens and model surface and R2 is the distance between the lens and sensor. For R1 >> R2 , this error is very small.

7.1.5. Temperature Effect Since the luminescent intensity of PSP is intrinsically temperature-dependent, a temperature change on a model during a wind tunnel run results in a significant bias error in PSP measurements if the temperature effect is not corrected. In addition, temperature influences the total uncertainty of PSP measurements through the sensitivity coefficients of the variables in the error propagation equation. Hence, the surface temperature on a model must be known in order to correct the temperature effect of PSP. In general, the surface temperature distribution can be measured experimentally using TSP or IR camera and determined numerically by solving the motion and energy equations of flows coupled with the heat conduction equation for a model. For a compressible boundary layer on an adiabatic wall, the adiabatic wall temperature Taw can be estimated using a simple relation Taw / T0 = [ 1 + r( γ - 1)M 2 / 2 ] [ 1 + ( γ - 1)M 2 / 2 ] −1 , where r is the recovery factor for the boundary layer, T0 is the total temperature, M is the local Mach number, and γ is the specific heat ratio.

7.1.6. Calibration Errors The uncertainties in determining the Stern-Volmer coefficients A(T) and B(T) are calibration errors. In a priori PSP calibration in a pressure chamber, the uncertainty is represented by the standard deviation of data collected in replication tests. Because tests in a pressure chamber are well controlled, a priori calibration results usually show a small precision error. However, a significant bias error is found when a priori calibration results are directly used for data reduction in wind tunnel tests due to unknown surface temperature distribution and uncontrollable

146

7. Uncertainty

testing environmental factors. In contrast, in-situ calibration utilizes pressure tap data over a model surface to determine the Stern-Volmer coefficients. Because in-situ calibration correlates the local luminescent intensity with the pressure tap data, it can reduce the bias errors associated with the temperature effect and other sources, achieving a better agreement with the pressure tap data. The in-situ calibration uncertainty, which is usually represented as a fitting error, will be specially discussed in Section 7.3. 7.1.7. Temporal Variations in Luminescence and Illumination For PSP measurements in steady flows, a temporal change in the luminescent intensity mainly results from photodegradation and sedimentation of dusts and oil droplets on a model surface. The photodegradation of PSP may occur when there is a considerable exposure of PSP to the strong excitation light between the windoff and wind-on measurements. Dusts and oil droplets in air sediment on a model surface during wind-tunnel runs; the resulting dust/oil layer absorbs both the excitation light and luminescent emission on the surface and thus causes a decrease of the luminescent intensity. The uncertainty in Dt (ǻt ) due to the photodegradation and sedimentation can be collectively characterized by the variance var[Dt ( ǻt )] ≈ [( ∂V / ∂ t )( ǻt)/V ] 2 . Similarly, the uncertainty in

Dq0 (ǻt ) , which is produced by an unstable excitation light source, is described by var[Dq0 ( ǻt )] ≈ [( ∂q 0 / ∂ t )( ǻt)/q 0 ref ] 2 .

7.1.8. Spectral Variability and Filter Leakage The uncertainty in Ȇ f /Ȇ f ref is mainly attributed to the spectral variability of illumination lights and spectral leaking of optical filters. Possolo and Maier (1998) observed the spectral variability between flashes of a xenon lamp; the uncertainties in the absolute pressure and pressure coefficient due to the flash spectral variability were 0.05 psi and 0.01, respectively. If optical filters are not selected appropriately, a small portion of photons from the excitation light and ambient light may reach a detector through the filters, producing an additional output to the luminescent signal. 7.1.9. Pressure Mapping Errors The uncertainty in pressure mapping is related to the data reduction procedure in which PSP data in images are mapped onto a surface grid of a model in the object space. It is contributed from the errors in camera resection/calibration and mapping onto a surface grid of a presumed rigid body. The camera resection/calibration error is represented by the standard deviations ı x and ı y of the calculated target coordinates from the measured target coordinates in the

7.1. Pressure Uncertainty of Intensity-Based Methods

147

image plane. Typically, a good camera resection/calibration method gives the standard deviation of about 0.04 pixels in the image plane. For a given PSP image, the pressure variance induced by the camera resection/calibration error is 2 2 var(p) ≈ (∂ p/∂ x ) ı x2 + (∂p /∂ y ) ı 2y . The pressure mapping onto a presumably non-deformed model surface grid leads to another deformation-related error because a model may undergo a considerable deformation generated by aerodynamic loads in wind tunnel tests. When a point on a model surface moves by ǻX = X' − X in the object space, the pressure variance induced by mapping onto a presumed rigid body grid without 2

correcting the model deformation is var(p ) = ( ∇ p )surf • ( ǻX )surf , where (∇ p )surf is the pressure gradient on the surface and ( ǻX )surf is the component of the displacement vector ǻX projected on the surface in the object space. To eliminate this error, a deformed surface grid should be generated for PSP mapping based on optical model deformation measurements under the same testing conditions (Liu et al. 1999).

7.1.10. Paint Intrusiveness A thin PSP coating may slightly modify the overall shape of a model and produces local surface roughness and topological patterns. These unwanted changes in model geometry may alter flows over a model and affect the integrated aerodynamic forces (Engler et al. 1991; Sellers 1998a). Hence, this paint intrusiveness to flow should be considered as an error source in PSP measurements. The effects of a paint coating on pressure and skin friction are directly associated with locally changed flow structures and propagation of the induced perturbations in flow; these local effects may collectively alter the integrated aerodynamic forces. When a local paint thickness variation is much smaller than the boundary layer displacement thickness, a thin coating does not alter the inviscid outer flow. Instead of directly altering the outer flow, a rough coating may indirectly result in a local pressure change by thickening the boundary layer; coating roughness may reduce the momentum of the boundary layer to cause early flow separation at certain positions. Therefore, the effective aerodynamic shape of a model is changed and as a result the pressure distribution on the model is modified; this effect is mostly appreciable near the trailing edge due to the substantial development of the boundary layer on the surface. Vanhoutte et al. (2000) observed an increment in the trailing edge pressure coefficient relative to the unpainted model, which was consistent with an increase in the boundary layer thickness at the trailing edge. For certain models such as high-lift models, a coating may change the gap between the main wing and slat or flap when the gap is small; thus, the pressure distribution on the model is locally influenced. In addition, a coating may influence laminar separation bubbles near the leading edge at low Reynolds numbers and high angles-of-attack. The perturbations induced by a rough coating near the leading edge may enhance

148

7. Uncertainty

mixing that entrains the high-momentum fluid from the outer flow into the separated region. The perturbations could be amplified by several hydrodynamic instability mechanisms such as the Kelvin-Helmholtz instability in the shear layer between the outer flow and separated region and the cross-flow instability near the attachment line on a swept wing. Consequently, the coating causes the laminar separation bubbles to be suppressed. Vanhoutte et al. (2000) reported this effect that led to a reduction in drag. Schairer et al. (1998a, 2002) observed that a rough coating on the slats slightly decreased the stall angle of a high-lift wing. Also, they found that the empirical criteria for ‘hydraulic smoothness’ and ‘admissible roughness’ based on 2D data by Schichting (1979) were not sufficient to provide a satisfactory explanation for their observation. Indeed, in 3D complex flows on the high-lift model, the effect of the coating on the cross-flow instability and its interactions with the boundary layer and other shear layers such wakes and jets are not well understood. Schairer et al. (1998a, 2002) and Mebarki et al. (1999) found that a rough coating moved a shock wave upstream and the pressure distribution was shifted near the shock location. This change might be caused by an interaction between the shock and the incoming boundary layer affected by the coating. In an attached flow at high Reynolds numbers, a rough coating increases skin friction by triggering premature laminar-turbulent transition and increasing the turbulent intensity in a turbulent boundary layer (Mebarki et al. 1999; Vanhoutte et al. 2000). An increase in drag due to a rough coating was observed in airfoil tests in high subsonic flows (Vanhoutte et al. 2000). In fact, premature transition by coating roughness has been often observed in TSP transition detection experiments (see Chapter 10). Amer et al. (2001, 2003) reported that a very smooth coating on the upper surface of a delta wing model at Mach 0.2 and a semi-span arrow-wing model at Mach 2.4 did not significantly change the drag coefficients of these models. Generally speaking, the effect of a coating on aerodynamic forces highly depends on flows over a specific model configuration; there is no universal conclusion on this effect.

7.1.11. Other Error Sources and Limitations Other error sources include the self-illumination and induction effect; there are limitations in the time response and spatial resolution of PSP. The selfillumination is a phenomenon that the luminescent emission from one part of a model surface reflects to another surface, thus distorting the observed luminescent intensity at a point by superposing all the rays reflected from other points. It often occurs on surfaces of neighbor components of a complex model (Ruyten 1997a, 1997b, 2001a; Le Sant 2001b). The self-illumination effect on calculation of pressure and temperature are discussed in Section 5.3. Another problem is the ‘induction effect’ observed as an increase in the luminescent emission during the first few minutes of illumination for certain paints; the photochemical process behind it was explained by Uibel et al. (1993) and Gouterman (1997). In PSP measurements in unsteady flows, the limiting time response of PSP, which is

7.1. Pressure Uncertainty of Intensity-Based Methods

149

mainly determined by oxygen diffusion process across a PSP layer (see Chapter 8), imposes an additional restriction on the accuracy of PSP measurements. The spatial resolution of PSP is limited by oxygen diffusion in the lateral direction along a paint surface. Considering a pressure jump across a point on a surface (a normal shock wave), Mosharov et al (1997) gave a solution of the diffusion equation describing a distribution of the oxygen concentration in a PSP layer near the pressure jump point. According to this solution, the limiting spatial resolution is about five times of the paint layer thickness. 7.1.12. Allowable Upper Bounds of Elemental Errors In the design of PSP experiments, we need to give the allowable upper bounds of the elemental errors for the required pressure accuracy. This is an optimization problem subject to certain constraints. In matrix notations, Eq. (7.3) is expressed as ı P2 = ı T A ı , where the notations are defined as ı P2 = var (p)/p 2 ,

A ij = S i S j ȡi j , and ı i = [ var( ȗ i ) ] 1/2 / ȗ i . For required pressure uncertainty ı P , we look for a vector ı up to maximize an objective function H = W T ı , where

W is the weighting vector. The vector ı up gives the upper bounds of the elemental errors for a given pressure uncertainty ı P . The use of the Lagrange multiplier method requires H = W T ı + Ȝ ( ı P2 − ı T A ı ) to be maximal, where λ is the Lagrange multiplier. The solution to this optimization problem gives the upper bounds A −1 W (7.6) ı up = ıP . ( W T A −1 W )1/2 For the uncorrelated variables with ȡi j = 0 (i ≠ j) , Eq. (7.6) reduces to −1/2

· § (7.7) (ı i )up = S i− 2 Wi ı P ¨¨ S −k 2 Wk2 ¸¸ . © k ¹ When the weighting factors Wi equal the absolute values of the sensitivity coefficients | S i | , the upper bounds can be expressed in a very simple form

¦

(ı i )up / ı P = N V−1 / 2 S i

−1

, ( i = 1, 2 , , N V )

(7.8)

where N V is the total number of the variables or the elemental error sources. The relation Eq. (7.8) clearly indicates that the allowable upper bounds of the elemental uncertainties are inversely proportional to the sensitivity coefficients and the square root of the total number of the elemental error sources. Figure 7.6 shows a distribution of the upper bounds of 15 variables for PSP Bath Ruth + silica-gel in GE RTV 118 at p / p ref = 0.8 and T = 293 K. Clearly, the allowable upper bound for temperature is much lower than others, and therefore the temperature effct of PSP must be tightly controlled to achieve the required pressure accuracy.

150

7. Uncertainty

1.8 1.6 1.4 1.2

(ı i )up ıP

1.0

1

Dt (ǻt )

9

c / c ref

2

Dx (ǻx )

10

q0 /q0 ref

3

Dq0 (ǻt )

11

Pref

4

V Vref

12 13

T A

B

5

0.8 0.6

6

Ȇ c /Ȇ c ref

14

7

Ȇ f /Ȇ f ref

15 Pressure mapping

8

h / href

0.4 0.2 0.0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Variable Index

Fig. 7.6. Allowable upper bounds of 15 variables for Bath Ruth + silica-gel in GE RTV 118 when p / p ref = 0.8 , and T = 293 K. From Liu et al. (2001a)

7.1.13. Uncertainties of Integrated Forces and Moments The uncertainties of the integrated aerodynamic forces and moments can be estimated based on their definitions. For example, the uncertainty in the lift is

³³ ( ¦¦ ( n • l

∆FL / FL = FL−1 [ ∆ ≈F

−1 L

1/ 2

p( n • l L )dS ] 2 L

∆S )i ( n • l L ∆S ) j < ∆ p i ∆ p j >

)

,

(7.9)

1/ 2

where n is the unit normal vector of a surface panel, ∆S is the area of the surface panel, and l L is the unit vector of the lift. The correlation between the pressure differences at the panel ‘i’ and panel ‘j’ is simply modeled by < ∆ pi ∆ p j >= δ ij < ∆ pi >< ∆ p j > , where the Kronecker delta is δ ij = 1 for i = j and δ ij = 0 for i ≠ j . Thus, the uncertainty in the lift can be estimated based on the PSP uncertainty at all the surface panels, i.e.,

§ ǻF L / F L ≈ ¨ ¨ ©

N

¦ (n•l i =1

L

ǻS ) ( p i / FL ) ( ǻp / p ) 2 i

2

2 PSP i

· ¸ ¸ ¹

1/ 2

(7.10)

Similarly, the uncertainties in the pressure-induced drag and pichting moment are estimated by

7.2. Pressure Uncertainty Analysis for Subsonic Airfoil Flows

§ ǻFD / FD ≈ ¨ ¨ © § ǻM c / M c ≈ ¨ ¨ ©

N

¦ (n•l

D

ǻS ) ( pi / FD ) ( ǻp / p ) 2 i

2

2 PSP i

i =1

N

¦ [ n×( X − X

mc

· ¸ ¸ ¹

1/ 2

,

) ǻS ] ( p i / M c ) ( ǻp / p ) 2 i

151

2

2 PSP i

i =1

(7.11)

· ¸ ¸ ¹

1/ 2

, (7.12)

where l D is the unit vector of the drag and X mc is the assigned moment center.

7.2. Pressure Uncertainty Analysis for Subsonic Airfoil Flows PSP measurements on a Joukowsky airfoil in subsonic flows are simulated in order to illustrate how to estimate the elemental errors and the total uncertainty using the techniques described above. The airfoil and incompressible potential flows around it are generated using the Joukowsky transform; the pressure coefficients Cp on the airfoil in the corresponding subsonic compressible flows are obtained using the Karman-Tsien rule. Figure 7.7 shows typical distributions of the pressure coefficient and adiabatic wall temperature on a Joukowsky airfoil at o Mach 0.4 and AoA = 5 .

-8

-3

Cp Temperature -6

-2

-4

0

-2

Cp

-1

Taw - Tref (deg C)

AoA = 5 deg, Mach = 0.4

0

1

Joukowski Airfoil 2

2 0.0

0.2

0.4

0.6

0.8

1.0

x/c

Fig. 7.7. Typical distributions of the pressure coefficient and adiabatic wall temperature on o a Joukowsky airfoil at Mach 0.4, AoA = 5 , and Tref = 293 K

152

7. Uncertainty

Presumably, PSP, Bath Ruth + silica-gel in GE RTV 118, is used, which has the Stern-Volmer coefficients A(T) ≈ 0.13 [ 1 + 2.82( T − Tref ) / Tref ] and

B(T) ≈ 0.87 [ 1 + 4.32( T − Tref ) / Tref ] over a temperature range of 293-333 K. The uncertainties in a priori PSP calibration are ǻA/A = ǻB/B = 1% . We assume that the spatial changes of the paint thickness and dye concentration in the image plane are 0.5%/pixel and 0.1%/pixel, respectively. The rate of photodegradation of the paint is 0.5%/hour for a given excitation level and the exposure time of the paint is 60 seconds between the wind-off and wind-on images. The rate of reduction of the luminescent intensity due to dust/oil sedimentation on the surface is assumed to be 0.5%/hour. In an object-space coordinate system whose origin is located at the leading edge of the airfoil, four light sources for illuminating PSP are placed at the locations X s1 = ( − c , 3c ) , Xs2 = ( 2c, 3c ) , X s3 = ( − c , − 3c ) , and Xs4 = ( 2c, − 3c ) , where c is the chord of the airfoil. For the light sources with unit strength, the illumination flux distributions on the upper and lower surfaces are, respectively,

(q0 )up = X up - X s1

−2

+ X up - X s2

−2

and (q0 )low = X low - X s3

−2

+ X low - X s4

−2

,

where X up and X low are the coordinates of the upper and lower surfaces of the airfoil, respectively. The temporal variation of irradiance of these lights is assumed to be 1%/hour. It is also assumed that the spectral leakage of optical filters for the lights and cameras is 0.3%. Two cameras, viewing the upper surface and lower surface respectively, are located at ( c/2, 4 c ) and ( c/2, − 4c ) . The pressure uncertainty associated with the photon shot noise can be estimated by using Eq. (7.5). Assume that the full-well capacity of ( n pe )max = 350,000 electrons of a CCD camera is utilized. The numbers of photoelectrons collected in a CCD camera are mainly proportional to the distribution of the illumination field on the model surface. Thus, the photoelectrons on the upper and lower surfaces are estimated by ( n pe )up = ( n pe )max ( q0 )up / max[( q0 )up ] and

( n pe )low = ( n pe )max ( q0 )low / max[( q0 )low ] . Combination of these estimates with Eq. (7.5) gives the shot-noise-generated pressure uncertainty distributions on the surfaces. Movement of the airfoil produced by aerodynamic loads is expressed by a superposition of a local rotation (twist) and translation. A transformation between the non-moved and moved surface coordinates X = ( X ,Y )T and X' = ( X ' ,Y ' )T is X' = R( θ twist ) X + T , where R( θ twist ) is the rotation matrix, θ twist is the local wing twist, and T is the translation vector.

Here, for θ twist = −1o and

T = ( 0.001c , 0.01 c )T , the uncertainty in q0 ( X )/q0 ref ( X' ) is estimated by 2

var[q0 ( X ) / q0 ref ( X' )] ≈ ( q0 ref )−2 ( ∇q0 ) • ( ǻX ) , where the displacement vector is ǻX = X' − X . The pressure variance associated with mapping PSP data onto a rigid body grid without correcting the model deformation is estimated by

7.2. Pressure Uncertainty Analysis for Subsonic Airfoil Flows

153

2

var(p ) = ( ∇ p )surf • ( ǻX )surf , where (∇ p )surf is the pressure gradient on the surface and ( ǻX)surf = ( X' − X)surf is the component of the displacement vector projected on the surface. To estimate the temperature effect of PSP, an adiabatic model is considered at which the wall temperature Taw is given by

Taw / T0 = [ 1 + r( γ - 1)M 2 / 2 ] [ 1 + ( γ - 1)M 2 / 2 ] −1 , where the recovery factor is r = 0.843 for a laminar boundary layer. Assuming that the reference temperature Tref equals to the total temperature T0 = 293 K, we can calculate a temperature difference ∆T = Taw − Tref between the wind-on and wind-off cases. The adiabatic wall is the most severe case for PSP measurements since the surface temperature on a metallic model is much lower than the adiabatic wall temperature due to heat conduction to the model. The total uncertainty in pressure is estimated by substituting all the estimated elemental errors into Eq. (7.3). Figure 7.8 shows the pressure uncertainty distributions on the upper and lower surfaces of the airfoil for different freestream Mach numbers. It is indicated that the temperature effect of PSP dominates the uncertainty of PSP measurements on an adiabatic wall. The uncertainty becomes larger and larger as the Mach number increases because the adiabatic wall temperature increases. The local pressure uncertainty on the upper surface is as high as 50% at one location for Mach 0.7, which is caused by a local surface o temperature change of about 6 C. In order to compare the PSP uncertainty with the pressure variation on the airfoil, a maximum relative pressure variation on the airfoil is defined as

max ǻp

surf

/ p∞ = 0.5 γ M ∞2 max ∆C p . Figure 7.9 shows the maximum relative

pressure variation

max ǻp

surf

/ p∞

along with the chord-averaged PSP

uncertainty < ( ǻp/p)PSP > aw on the adiabatic airfoil at the Mach numbers of 0.050.7. The uncertainty < ( ǻp/p)PSP > ∆T =0 without the temperature effect is also plotted in Fig. 7.10, which is mainly dominated by the a priori PSP calibration error ǻB/B = 1% in this case. The curves max ǻp surf / p∞ , < ( ǻp/p)PSP > aw and < ( ǻp/p)PSP > ∆T =0 intersect near Mach 0.1. When the PSP uncertainty exceeds the maximum pressure variation on the airfoil, the pressure distribution on the airfoil cannot be quantitatively measured by PSP. As shown in Fig. 7.9, because a temperature change on a non-adiabatic wall is smaller, the PSP uncertainty for a real wind tunnel model generally falls into the shadowed region confined by < ( ǻp/p)PSP > aw and < ( ǻp/p)PSP > ∆T =0 . The PSP uncertainty associated with the photon shot noise < ( ǻp/p)PSP > ShotNoise is also plotted in Fig. 7.9. The intersection between max ǻp

surf

/ p∞ and < ( ǻp/p)PSP > ShotNoise gives the limiting low Mach number

154

7. Uncertainty

( ~ 0.06 ) for PSP measurements in this case. The uncertainties in the lift ( FL ) and pitching moment ( M c ) are also calculated from the PSP uncertainty distribution on the surface. Figure 7.10 shows the uncertainties in the lift and pitching moment relative to the leading edge for the Joukowsky airfoil over a o range of the Mach numbers when the angle of attack is 4 . The uncertainties in the lift and moment decrease monotonously as the Mach number increases since the absolute values of the lift and moment rapidly increase with the Mach number.

0.6 Upper Surface

0.5

Uncertainty in P

M = 0.7 0.4 0.3 0.2 M = 0.5

0.1

M = 0.3 0.0

M = 0.1 0.0

0.2

0.4

0.6

0.8

1.0

x/c

(a) 0.06

Lower Surface M = 0.7

Uncertainty in P

0.05

0.04

M = 0.5

0.03

0.02 M = 0.3 0.01

M = 0.1 0.0

(b)

0.2

0.4

0.6

0.8

1.0

x/c

Fig. 7.8. PSP uncertainty distributions for different freestream Mach numbers on (a) the upper surface and (b) lower surface of a Joukowsky airfoil. From Liu et al. (2001a)

7.2. Pressure Uncertainty Analysis for Subsonic Airfoil Flows

155

1

Relative Error or Variation

Upper Surface

< ( ǻP/P)PSP > aw max ǻP 0.1

surf

/ P∞

0.01

< ( ǻP/P)PSP > ∆T =0 < ( ǻP/P)PSP > ShotNoise 0.001 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Freestream Mach number

(a) 1

Relative Error or Variation

Lower Surface

max ǻP

0.1

surf

/ P∞

< ( ǻP/P)PSP > aw

0.01

< ( ǻP/P)PSP > ∆T =0 < ( ǻP/P)PSP > ShotNoise 0.001 0.0

(b)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Freestream Mach number

Fig. 7.9. The maximum relative pressure change and chord-averaged PSP uncertainties as a function of the freestream Mach number on (a) the upper surface and (b) the lower surface of a Joukowsky airfoil. From Liu et al. (2001a)

156

7. Uncertainty

0.7 0.6

Lift Pitching Moment

Uncertainty

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Freestream Mach Number

Fig. 7.10. Uncertainties in the lift and pitching moment of a Joukowsky airfoil as a function of the freestream Mach number. From Liu et al. (2001a)

7.3. In-Situ Calibration Uncertainty

7.3.1. Experiments

As pointed out before, the use of a priori PSP calibration in large wind tunnels often leads to a considerable systematic error since the surface temperature distribution is not known and the illumination change on a surface due to model deformation cannot be corrected by the image registration technique. The systematic error is also related to uncontrollable environmental testing factors. Therefore, in actual PSP measurements, experimental aerodynamicists are forced to calibrate PSP in situ by fitting (or correlating) the luminescent intensity to pressure tap data at a number of suitably distributed locations. In a sense, in-situ PSP calibration eliminates the systematic error associated with the temperature effect and the illumination change by absorbing it into an overall fitting error. Kammeyer et al. (2002a, 2002b) assessed the accuracy of the Boeing production PSP system by statistical analysis of comparison between PSP and pressure transducers over a large numbers of data points. The Boeing PSP system is a typical intensity-based system that uses eight CCD (1024×1024 or 512×512) cameras for imaging, thirty lamps for illumination, and two IR cameras measuring the surface temperature for correcting the temperature effect of PSP. The test article was a 1/12th-scale model of a Cessna Citation that was instrumented with a total of 225 pressure taps. The tests were conducted in the DNW/NLR HST wind tunnel, a variable-density, closed circuit, continuous tunnel with slotted top and bottom test section walls (12% open). The test section was 6.56 ft wide and was

7.3. In-Situ Calibration Uncertainty

157

configured to be 5.25 ft high. The cameras and lamps were mounted in the floor and ceiling. A run consisted of a lift polar at each of several Mach numbers from 0.22 to 0.82. Two Reynolds numbers, 4.5 and 8.3 millions, were run. Fourteen o angles of attack were from –4 to 10 . Over 8300 visual images and over 2000 IR images were obtained for 676 test points. The wind-off reference images were acquired after the run when the fan had stopped in order to reduce the effect of the model temperature distribution. Figure 7.11 shows a typical pressure distribution on the model obtained by PSP.

Fig. 7.11. Typical pressure distribution obtained from PSP on a Cessna Citation model. From Kammeyer et al. (2002a)

In-situ PSP calibrations were performed by utilizing 78 of 225 pressure taps for each of the cameras. Figure 7.12 shows the variation of the in-situ calibration slope (i.e. the Stern-Volmer coefficient B) as a function of test point throughout the tests, where no temperature correction was applied. The variation does not show an overall trend; the repeating pattern mirrors the pattern of the test conditions, wherein sequential angles of attack were run for sequentially increasing Mach numbers. The mean value of the slope is close to one, which is approximately consistent with the paint characteristics given by a priori calibration. The scatter is attributed to a number of factors, including the nonhomogeneous temperature distributions, temperature differences between the wind-off and wind-on conditions, lamp intensity drift, and image registration error. The accuracy of the PSP system was directly assessed by comparing the pressure value measured by a transducer/tap combination with that obtained from PSP at the same tap location. After some problematic pressure data were excluded, 130,391 comparisons from 221 taps and 676 wind-on test points were used as an overall set of realizations for statistical analysis. The PSP data

158

7. Uncertainty

processing included in-situ calibration, but did not exercise the explicit temperature correction. When examining the comparisons, the 78 taps were used for in-situ calibration to provide residual comparisons, while other taps provided truly independent comparisons. Figure 7.13 shows a histogram for the over set of comparisons, where a Gaussian distribution with the same mean and standard deviation is superimposed for comparison. Clearly, the distribution is nonGaussian. A robust estimate of the 68% confidence level gives an estimate of the standard uncertainty of 0.29 psi, which corresponds to 0.0065 in Cp. Figure 7.14 shows the standard uncertainty as a function of the angle of attack for the right wing. The behavior of the dependency of the uncertainty on the angle of attack corresponds to wing deformation. This indicates that the error is associated with the movement of the model in the non-homogenous illumination field, which cannot be corrected by the image registration technique. Kammeyer et al. (2002a, 2002b) also studied temperature correction using the IR cameras. Two sets of PSP data obtained before and after temperature correction were used to assess the effectiveness of the temperature correction. Figure 7.14 shows the standard uncertainty after the temperature correction as a function of the angle of attack. The temperature correction was increasingly effective when the angle of attack o was larger than 2 ; it removed the spatial biases associated with the temperature distribution on the model. Overall, the standard uncertainty, priori to the temperature correction, was in the range 0.16-0.45 psi (0.04-0.1Cp); with the temperature correction, it was in the range 0.17-0.35 psi (0.04-0.09Cp). The significance of the work of Kammeyer et al. (2002a, 2002b) is that it identifies the functional dependency of in-situ PSP calibration uncertainty on the testing parameters such as the angle of attack and Mach number.

Fig. 7.12. Variation of PSP in-situ calibration slope throughout the tests on a Cessna Citation model. From Kammeyer et al. (2002a)

7.3. In-Situ Calibration Uncertainty

159

Fig. 7.13. Histogram of the overall set of PSP errors compared with a Gaussian distribution of the equivalent mean and standard deviation. From Kammeyer et al. (2002a)

Fig. 7.14. Standard uncertainty of PSP on the right wing of a Cessna Citation model as a function of the angle of attack. From Kammeyer et al. (2002a)

7.3.2. Simulation

Inspired by the experimental study of Kammeyer et al. (2002a, 2002b), Liu and Sullivan (2003) studied in-situ calibration uncertainty of PSP through a simulation of PSP measurements in subsonic Joukowsky airfoil flows. It is assumed that insitu calibration uncertainty is mainly attributed to the temperature effect of PSP and illumination change on a surface due to model deformation. The Joukowsky airfoil and subsonic flows around it are generated using the Joukowsky transform plus the Karman-Tsien rule as described in Section 7.2. An adiabatic model is

160

7. Uncertainty

considered that is coated with Bath Ruth + silica-gel in GE RTV 118. Four point light sources for illuminating PSP and two cameras for imaging are placed at the same locations as described in Section 7.2. The twist θ twist and bending T y of the airfoil are a function of the angle of attack (AoA or α ) for a given Mach number and Reynolds number. Based on previous wing deformation measurements (Burner and Liu 2001), the typical linear relations θ twist = −0.113α (deg) and T y = 0.022 α ( in ) are used over a certain range of AoA at a certain spanwise location of a wing. Thus, a change of the illumination radiance on the airfoil surface due to the deformation is estimated using a transformation of rotation and translation for the airfoil moving in the given illumination field. In simulation, the measured luminescent intensity (I) distribution of PSP in the wind-on case (deformation case) is generated by I I ref 0

§ I ref =¨ ¨ I ref 0 ©

·§ · ¸ ¨ A( T ) + B( T ) p ¸ ¸¨ p ref ¸¹ ¹©

−1

§ L = ¨¨ © L0

−1

·§ p ·¸ ¸ ¨ A( T ) + B( T ) , ¸¨ p ref ¸¹ ¹©

where I ref 0 and I ref are the reference luminescent intensities (without wind) on the non-deformed airfoil and deformed airfoil, respectively. It is assumed that I ref 0 and I ref are proportional to the corresponding illumination radiance levels L0 and L on the non-deformed airfoil and deformed airfoil, respectively. The surface temperature T is substituted by the adiabatic wall temperature distribution Taw , and the pressure distribution is given by the Joukowsky transform plus the Karman-Tsien rule for subsonic flows. Therefore, the resulting luminescent intensity distribution contains the effects of both the illumination change and temperature variation on the surface. Assuming that the wind-on image (I) is already re-aligned with the wind-off image I ref 0 on the non-deformed airfoil by the image registration technique, in-

situ PSP calibration is made to correlate I ref 0 / I to p / p ref using the SternVolmer relation based on 104 virtual pressure taps on each of the upper and lower surfaces. For a given AoA and Mach number, the histogram of in-situ calibration error ∆p / p ref = ( p − pin − situ ) / p ref is found to be a near-Gaussian distribution, where ∆p is a difference between the true pressure from the theoretical distribution and the pressure converted from the luminescent intensity using insitu calibration. The standard deviation (std) of the probability density function is dependent on AoA and Mach number. Figures 7.15 and 7.16 show the std of the in-situ calibration error as a function of AoA for Mach 0.4 and as a function of the o Mach number for AoA = 5 , respectively. Figures 7.15 and 7.16 also show the isolated effects of the temperature and illumination change on the std. The behavior of the calculated std as a function of AoA is very similar to the experimental results shown in Fig. 7.14. The concavity of the std as a function of AOA in Fig. 7.15 is mainly attributed to the movement of the airfoil.

7.3. In-Situ Calibration Uncertainty

161

Figure 7.17 shows the simulated histogram for an overall sample set of

∆p / p ref (a total of 10920 samples) over the whole range of AoA and Mach

numbers, duplicating the experimental non-Gaussian distribution in Fig. 7.13 given by Kammeyer et al. (2002a, 2002b). The Gaussian distribution with the same std is also plotted in Fig. 7.17 as a reference. In fact, for a union of sample sets having near-Gaussian distributions with different the std values at different AoA and Mach numbers, the distribution becomes non-Gaussian because more and more samples accumulate near zero when forming a union of the sample sets. The probability density function of a union of the N sample sets should be given by a sum of the Gaussian distributions rather than the Gaussian distribution, i.e., N

N −1

¦ exp( − x

2

/ 2σ i2 ) / 2π σ i .

i =1

As shown in Fig. 7.17, this distribution correctly describes the simulated histogram. Note that we should not confuse this case with the central limit theorem that deals with a sum of independent random variables. Although the simulation is made for an airfoil section of a wing, the in-situ calibration error for a wing can be estimated by averaging the local results over the full wingspan; therefore, the behavior of the error for a wing should be similar to that for an airfoil. 0.012 Illumination change only with constant temperature Temperature effect only without illumination change Both temperature effect and illumination change

std[(p - pin-situ)/pref]

0.010

0.008

0.006

0.004

0.002

0.000

Mach = 0.4

-5

0

5

10

15

AoA (deg)

Fig. 7.15. In-situ PSP calibration error as a function of the angle-of-attack (AoA) for Mach 0.4 in Joukowsky airfoil flows. From Liu and Sullivan (2003)

162

7. Uncertainty 0.004 Illumination change only with constant temperature Temperature effect only without illumination change Both temperature effect and illumination change

std[(p - pin-situ)/pref]

0.003

0.002

0.001

0.000 AoA = 5 deg

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

Mach Number o

Fig. 7.16. In-situ PSP calibration error as a function of the Mach number for AoA = 5 in Joukowsky airfoil flows. From Liu and Sullivan (2003)

900 800

Sample Number

700

Sum of Gaussian Distributions

600 500 400

Gaussian Distribution

300 200 100 0 -0.01 -0.008 -0.006 -0.004 -0.002

0

0.002 0.004 0.006 0.008

0.01

∆p/pref

Fig. 7.17. Histogram of the overall set of in-situ PSP calibration errors in the whole ranges of AoA and Mach numbers in Joukowsky airfoil flows. From Liu and Sullivan (2003)

7.4. Pressure Uncertainty of Lifetime-Based Methods

163

7.4. Pressure Uncertainty of Lifetime-Based Methods

7.4.1. Phase Method

The phase method for PSP measurements, as described in Chapter 6, determines pressure by § · −1 ¨ Ȧ IJ 0 ¸ p = K SV (7.13) ¨ tan ij − 1¸ , © ¹ where tan ij = Ȧ IJ = − Vc / Vs is uniquely related to the lifetime for a fixed modulation frequency, and Vc = − Am IJ H M eff sin (ij ) and Vs = Am IJ H M eff cos (ij ) are the DC components from the low-pass filters. The error propagation equation gives the relative variance of pressure var( K SV ) var( IJ 0 ) var (p) var( T ) = S T2 + S K2 SV + S IJ20 2 2 2 2 p T K SV IJ0 . (7.14) 2 var( Vc ) 2 var( V s ) + S Vs + SVc Vs2 Vc2 The first term is the uncertainty related to temperature, the second is the uncertainty in PSP calibration, the third is the error in the given reference lifetime, and the last two terms are the uncertainties associated with the measurement system composed of a photodetector and lock-in amplifier. The sensitivity coefficients in Eq. (7.14) are T ∂p T ∂K SV 1 + K SV p T ∂τ 0 ST = , + =− p ∂T K SV ∂T K SV p τ 0 ∂T

S K SV =

K SV ∂p = −1 , p ∂K SV

S IJ0 =

IJ0 ∂ p = 1 + 1 /( K SV p ) , p ∂ IJ0

SVs =

Vs ∂ p = 1 + 1 /( K SV p ) , p ∂ Vs

SVc =

Vc ∂ p = − SVs . p ∂ Vc

Compared to the intensity-based method discussed in Chapter 4, many error sources associated with model deformation do not exist, which reflects the advantage of the lifetime-based method. When the photon shot noise of the detector dominates, the pressure uncertainty is mainly contributed by the last two terms in Eq. (7.14). In the photon-shot-noise-limited case, the uncertainties in the outputs of the detector and lock-in amplifier are var(Vs ) = | Vs |G ! Ȟ Bd and

164

7. Uncertainty

var(Vc ) = | Vc | G ! Ȟ Bd , where G is the system gain, Bd is the bandwidth of the system, and ! is the Planck’s constant. Therefore, the photon-shot-noise-limited pressure uncertainty is given by

∆p p

§ GBd !ν · ¸ ¸ © | Vs | ¹

= ¨¨

1/ 2

§ · § 1 + K SV p · ¨1 + 1 ¸ ¨1 + ¸ ¨ ȦIJ 0 ¸¹ K SV p ¸¹ ¨© ©

1/ 2

.

(7.15)

The estimate Eq. (7.15) for the phase method is similar to Eq. (7.4) for the intensity-based CCD camera system. The behavior of the pressure uncertainty as a function of pressure and the Stern-Volmer coefficient B is similar to that shown in Figs. 7.4 and 7.5. 7.4.2. Amplitude Demodulation Method

When the amplitude demodulation method is used, as indicated in Chapter 6, pressure is given by · Ȧ IJ0 1 § ¨ (7.16) − 1 ¸¸ , p= 2 2 1/ 2 ¨ K SV © [ H ( Vmean / Vstd ) / 2 − 1 ] ¹ where Vmean and Vstd are the mean and standard deviation of the photodetector output, respectively. Thus, the error propagation equation gives the relative variance of pressure var( K SV ) var( IJ 0 ) var (p) var( T ) = S T2 + S K2 SV + S IJ20 2 2 2 2 T K SV p IJ0 . (7.17) var( Vstd ) var( Vmean ) 2 S + SV2mean + Vstd 2 Vmean Vstd2 The first term is the uncertainty related to temperature, the second is the uncertainty in PSP calibration, the third is the error in the given reference lifetime, and the last two terms are the uncertainties associated with the photodetector. The sensitivity coefficients in Eq. (7.17) are T ∂p T ∂K SV 1 + K SV p T ∂τ 0 , ST = + =− p ∂T K SV ∂T K SV p τ 0 ∂T

S K SV = S IJ0 =

IJ0 ∂ p = 1 + 1 /( K SV p ) , p ∂ IJ0

§ 1 + K SV p ( 1 + K SV p )3 Vmean ∂p = − ¨¨ + p ∂( Vmean ) K SV p( ȦIJ 0 ) 2 © K SV p V ∂p = std = − SVmean . p ∂( Vstd )

SVmean = SVstd

K SV ∂p = −1 , p ∂K SV

· ¸, ¸ ¹

7.4. Pressure Uncertainty of Lifetime-Based Methods

165

In the photon-shot-noise-limited case, the uncertainties in the detector outputs are var(Vmean ) = Vmean G ! Ȟ Bd and var(Vstd ) = Vstd G ! Ȟ Bd . Thus the photon-shotnoise-limited pressure uncertainty is

∆p p

§ GB d !ν · ¸ ¸ © V mean ¹

1/ 2

= ¨¨

ª 1 + K SV p ( 1 + K SV p ) 3 º + « » K SV p( ȦIJ 0 ) 2 ¼» ¬« K SV p

1/ 2 ­ ( ȦIJ 0 ) 2 º ½° 2ª ° × ®1 + «1 + » ¾ H «¬ ( 1 + K SV p ) 2 ¼» ° °¯ ¿

.

1/ 2

(7.18)

Figure 7.18(a) shows the normalized pressure uncertainty 1/ 2 (ǻp/p)( Vmean / G ! Ȟ Bd ) as a function of p/p ref at different values of the SternVolmer coefficient B for Ȧ IJ 0 = 10 and H = 1 . Here, for a fixed temperature

T = Tref ,

we

use

the

following

= ( B / A )( p / p ref ) and A + B = 1 .

relations

K SV p = K SV p ref ( p / p ref )

Figure 7.18(b) shows the normalized

pressure uncertainty (ǻp/p)( Vmean / G ! Ȟ Bd )1 / 2 as a function of B at different values of p/p ref for Ȧ IJ 0 = 10 and H = 1 . Interestingly, in this case, there is an optimal value of the Stern-Volmer coefficient B at which 1/ 2 (ǻp/p)( Vmean / G ! Ȟ Bd ) is minimal. The optimal value of the Stern-Volmer coefficient B varies between 0.7 and 0.9, depending on the value of pressure. 20

50 P/Pref= 0.2

40

15

10

(∆P/P)(Vmean/GBdhν)1/2

(∆P/P)(Vmean/GBdhν)1/2

45

B = 0.5 0.6 0.7 0.8 0.9

5

35

0.5

30 25 20

1.0

15

1.5

10

2.0

5 0

0 0.0

0.5

1.0

1.5

2.0

0.0

P/Pref

(a)

0.2

0.4

0.6

0.8

1.0

B

(b)

Fig. 7.18. The normalized pressure uncertainty (ǻp/p)( Vmean / G ! Ȟ Bd )1 / 2 in the amplitude demodulation method with Ȧ IJ 0 = 10 and H = 1 as a function of p/p ref for different values of the Stern-Volmer coefficient B, and a function of B for different values of p/p ref

166

7. Uncertainty

7.4.3. Gated Intensity Ratio Method

In the gated intensity ratio method for the sinusoidally modulated excitation light, pressure can be expressed as a function of the gated detector output ratio V2 / V1 −1 / 2

· Ȧ IJ 0 § 2 H 1 + V2 / V1 1 ¨ p= − − 1 ¸¸ . (7.19) ¨ K SV © ʌ 1 − V2 / V1 K SV ¹ Therefore, the error propagation equation is var( IJ 0 ) var( K SV ) var( V2 ) var( V1 ) var (p) var( T ) = S T2 + S K2 SV + S IJ20 + SV21 + SV22 2 2 2 p2 K SV T2 V22 V IJ0 1 (7.20) where the sensitivity coefficients are T ∂p T ∂K SV 1 + K SV p T ∂τ 0 ST = + , =− p ∂T K SV ∂T K SV p τ 0 ∂T S K SV =

K SV ∂p = −1 , p ∂K SV

S IJ0 =

IJ0 ∂ p = 1 + 1 /( K SV p ) , p ∂ IJ0

SV1 =

V1 ∂ p { π [1 + Ȧ 2 IJ 02 ( 1 + K SV p )−2 ] − 2 H }( 1 + K SV p )3 , = p ∂V1 2ʌ Ȧ 2 IJ 02 K SV p

V2 ∂ p = − SV1 . p ∂V2 In the photon-shot-noise-limited case, the uncertainties in the detector outputs are var(V1 ) = V1 G ! Ȟ Bd and var(V2 ) = V2 G ! Ȟ Bd . Thus, the photon-shot-noiselimited pressure uncertainty for the gated intensity ratio method is SV2 =

§ GB d !ν = ¨¨ p © V1

∆p

· ¸ ¸ ¹

1/ 2

2π { π [1 + Ȧ 2 IJ 02 ( 1 + K SV p ) − 2 ] − 2 H } 1 / 2 2ʌ Ȧ 2 IJ 02

[1 + Ȧ 2 IJ 02 ( 1 + K SV p ) − 2 ] 1 / 2 ( 1 + K SV p ) 3 × K SV p

.

(7.21)

Figure 7.19(a) shows the normalized pressure uncertainty (ǻp/p)( V1 / G ! Ȟ Bd )1 / 2 as a function of p/p ref at different values of the Stern-Volmer coefficient B for

Ȧ IJ 0 = 10 and H = 1 . Figure 7.19(b) shows the normalized pressure uncertainty

(ǻp/p)( Vmean / G ! Ȟ Bd )1 / 2 as a function of B at different values of p/p ref for Ȧ IJ 0 = 10 and H = 1 . Similar to the amplitude demodulation method, there is an optimal value of B (around 0.8) to achieve the minimal value of (ǻp/p)( Vmean / G ! Ȟ Bd )1 / 2 . In general, to reduce the noise, the gated intensity ratio method has to collect sufficient photons over a large number of cycles. For

7.4. Pressure Uncertainty of Lifetime-Based Methods

167

example, compared to a standard CCD camera system with an integration time of 1 second, a gated CCD camera with a modulation frequency of 50 kHz needs to accumulate photons over 100,000 cycles to achieve the equivalently small uncertainty. The accumulation of photons can be done automatically in a phase sensitive camera. 15

(∆P/P)(V1/GBdhν)1/2

(∆P/P)(V1/GBdhν)1/2

4

3

B = 0.5

2

10

P/Pref= 0.2

0.5 5

0.6

1.0 1.5 2.0

0.7 0.8 0.9

0

1 0.0

0.5

1.0

P/Pref

(a)

1.5

2.0

0.0

0.2

0.4

0.6

0.8

1.0

B

(b)

Fig. 7.19. The normalized pressure uncertainty ( ∆p/p)( V1 / G ! Ȟ Bd )1 / 2 for the gated intensity method using a sinusoid modulation with Ȧ IJ 0 = 10 and H = 1 as a function of

p/p ref for different values of the Stern-Volmer coefficient B, and a function of B for different values of p/p ref

When the gated intensity ratio method is applied to the pulse excitation light, pressure can be expressed as a function of the gated detector output ratio V2 / V1

p=

IJ0 V2 / V1 1 , ln − t g K SV 1 + V2 / V1 K SV

(7.22)

where the time t g divides the two gating intervals [ 0 , t g ] and [ t g , ∞ ] . Thus, we have the pressure uncertainty var( IJ 0 ) var( K SV ) var( V2 ) var( V1 ) var (p) var( T ) = S T2 + S K2 SV + S IJ20 + SV21 + SV22 2 2 2 2 2 p K SV T V22 V1 IJ0 (7.23) where the sensitivity coefficients are ∂K SV ∂τ 0 T ∂p T ª 1 + K SV p § ¨¨ K SV −τ0 ST = = 2 « p ∂T K SV p «¬ τ 0 ∂T ∂T ©

· ∂K SV ¸¸ + ∂T ¹

º », »¼

168

7. Uncertainty

S K SV =

K SV ∂p = −1 , p ∂K SV

S IJ0 =

IJ0 ∂ p = 1 + 1 /( K SV p ) , p ∂ IJ0

SV1 =

τ0 V1 ∂ p =− 1 − exp[ −( 1 + K SV p )( t g / τ 0 )] , p ∂V1 t g K SV p

SV2 =

V2 ∂ p = − SV1 . p ∂V2

{

}

In the photon-shot-noise-limited case, only the terms associated with V1 and

V2 remain in Eq. (7.23) and the uncertainties of the system outputs are var(V1 ) = V1 G ! Ȟ Bd and var(V2 ) = V2 G ! Ȟ Bd . The photon-shot-noise-limited pressure uncertainty for the time-resolved multiple-gate method is

§ GBd !ν = ¨¨ p © V1

∆p

· ¸ ¸ ¹

1/ 2

1

1 − exp[ − ( 1 + K SV p )( t g / τ 0 )]

K SV p ( t g / τ 0 ) exp[ − 0.5( 1 + K SV p )( t g / τ 0 )]

. (7.24)

The factor V1 / G ! Ȟ Bd equals to the number of photoelectrons collected in the first gating interval [ 0 , t g ] .

Figure 7.20(a) shows the normalized pressure

uncertainty (ǻp/p)( V1 / G ! Ȟ Bd )1 / 2 as a function of p/p ref at different values of the Stern-Volmer coefficient B for a fixed gating time t g /IJ 0 = 0.2 , where the relations K SV p = ( B / A )( p / p ref ) and A + B = 1 are imposed. Figure 7.20(b) shows the normalized pressure uncertainty (ǻp/p)( Vmean / G ! Ȟ Bd )1 / 2 as a function of B at different values of p/p ref for t g /IJ 0 = 0.2 . The optimal value of the Stern-Volmer coefficient B is about 0.8-0.9. For t g /IJ 0 < 0.5 , the pressure uncertainty ǻp/p remains small, but ǻp/p rapidly increases as t g /IJ 0 approaches one.

7.5. Uncertainty of Temperature Sensitive Paint

169

15

4

(∆P/P)(V1/GBdhν)1/2

(∆P/P)(V1/GBdhν)

1/2

P/Pref= 0.2

3 B = 0.5

0.6

2

0.7

10 0.5

1.0

5

1.5 2.0

0.8 0.9

1

0 0.0

0.5

1.0

1.5

2.0

0.0

0.2

0.4

P/Pref

0.6

0.8

1.0

B

(b)

(a)

Fig. 7.20. The normalized pressure uncertainty ( ∆p/p)( V1 / G ! Ȟ Bd )1 / 2 for the gated intensity method with a pulse excitation and t g /IJ 0 = 0.2 as (a) a function of p/p ref for different values of the Stern-Volmer coefficient B, and (b) a function of B for different values of p/p ref

7.5. Uncertainty of Temperature Sensitive Paint

7.5.1. Error Propagation and Limiting Temperature Resolution

In principle, the above uncertainty analysis for PSP can be adapted for TSP since many error sources of TSP are the same as those of PSP. For simplicity, instead of the general Arrhenius relation, we use an empirical relation between the luminescent intensity (or the photodetector output) and temperature T for a TSP uncertainty analysis (Cattafesta and Moore 1995; Cattafesta et al. 1998) T − Tref = K T ln( I ref / I ) = K T ln( U 2 V ref / V ) ,

(7.25)

where K T is a TSP calibration constant with a temperature unit and U 2 is the factor defined previously in Eq. (7.2) for the PSP uncertainty analysis. Without model deformation and temporal illumination variation, the factor U 2 equals to one. Eq. (7.25) can be used to fit TSP calibration data over a certain range of temperature. The error propagation equation for TSP is M var( ȗ i ) var( K T ) K T2 var (T) = + , (7.26) 2 2 ȗ i2 K K2 (T − Tref ) (T − Tref ) i = 1

¦

170

7. Uncertainty

where the variables {ȗ i , i = 1 M} denote a set of the parameters Dt (ǻt ) ,

D x (ǻx ) , Dq0 (ǻt ) , V , Vref , Ȇ c /Ȇ c ref , Ȇ f /Ȇ f ref , h / href , c / c ref , and q0 /q0 ref as defined in Section 7.1. The summation term in the right-hand side of Eq. (7.26) include the errors associated with model deformation, unstable illumination, photodegradation, filter leakage, and luminescent intensity measurements. The last term in Eq. (7.26) is the TSP calibration error. Similar to the uncertainty analysis for PSP, in the photon-shot-noise-limited case without any model deformation, we are able to obtain the minimum temperature difference that TSP can measure from a single frame of image ª § T − Tref ( ǻT)min = «1 + exp¨¨ (n pe ref )max «¬ © KT )max is the full-well capacity of a CCD KT

where (n pe ref

1/2

·º ¸» , (7.27) ¸» ¹¼ camera in the reference

conditions. The minimum resolvable temperature difference ( ǻT)min is inversely proportional to the square-root of the number of collected photoelectrons, and approximately proportional to the calibration constant K T . When (n pe ref )max is o

500,000 electrons, for a typical Ruthenium-based TSP having K T = 37.7 C, the minimum resolvable temperature difference ( ǻT)min is shown in Fig. 7.21 as a o

function of T at a reference temperature Tref = 20 C. When N images are averaged, the limiting temperature resolution given by Eq. (5.27) should be 1/2 divided by a factor N . 7.5.2. Elemental Error Sources

The elemental error sources of TSP have been discussed by Cattafesta et al. (1998) and Liu et al. (1995c). Table 7.2 lists the elemental error sources, sensitivity coefficients, and total uncertainty of TSP. The sensitivity coefficients for many variables are related to ij = KT /(T − Tref ) . The elemental errors in the variables Dt (ǻt ) , D x (ǻx ) , Dq0 (ǻt ) , V , Vref , Ȇ c /Ȇ c ref , Ȇ f /Ȇ f ref , h / href ,

c / c ref , and q0 /q0 ref can be estimated using the same expressions given in the uncertainty analysis for PSP, which represent the error sources associated with model deformation, unstable illumination, photodegradation, filter leakage, and luminescence measurements. The camera calibration error and temperature mapping error can be also estimated using the similar expressions to those for PSP, i.e., var(T) ≈ (∂T/∂ x ) ı x2 + (∂T /∂ y ) ı 2y and var(T ) = ( ∇T )surf • ( ǻX )surf , 2

2

2

where ı x and ı y are the standard deviations of least-squares estimation in image registration or camera calibration. In order to estimate the TSP calibration errors, the temperature dependency of TSP was repeatedly measured using a calibration set-up over days for several TSP formulations (Liu et al. 1995c). Temperature measured by TSP was compared to accurate temperature values measured by a

7.5. Uncertainty of Temperature Sensitive Paint

171

Minimum Temperature Difference (deg. C)

standard thermometer. Figure 7.22 shows histograms of the temperature calibration error for EuTTA-dope and Ru(bpy)-Shellac TSPs, which exhibit a near-Gaussian distribution. The standard deviation for EuTTA-dope TSP is about o o 0.8 C over a temperature range of 15-70 C. For Ru(bpy)-Shellac TSP, the o histogram has a broader error distribution having the deviation of about 2 C over a o temperature range of 20-100 C. The temperature hysteresis introduces an additional error source for TSP, which was reported in calibration experiments for a Rhodamine(B)-based coating (Romano et al. 1989). The temperature hysteresis is related to the polymer structural transformation from a hard and relatively brittle state to a soft and rubbery one when temperature exceeds the glass temperature of a polymer. Since the thermal quenching of luminescence in a brittle condition is different from that in a rubbery state, the temperature dependency is changed after it is heated beyond the glass temperature. To reduce the temperature hysteresis, TSP should be preheated to a certain temperature above the glass temperature before it is used as an optical temperature sensor for quantitative measurements. It was found that for both pre-heated EuTTA-dope and Ru(bpy)-Shellac paints the temperature hysteresis was minimized such that the temperature dependency remained almost unchanged in repeated tests over several days (Liu et al. 1995c).

0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 -20

0

20

40

60

80

100

Temperature (deg. C)

Fig. 7.21. The minimum resolvable temperature difference as a function of temperature for o o a Ruthenium-based TSP for (n pe ref )max = 500,000e, K T = 37.7 C , and Tref = 20 C

172

7. Uncertainty 0.4

1.0 EuTTA - dope paint

Ru(bpy) - Shellac paint

Gaussian with σ = 0.8 C 0

0.3 Frequency

Frequency

0.8

0.6

0.4

Gaussian with σ = 2 0C

0.2

0.1

0.2 0.0

0.0 -4

-3

-2 -1 0 1 2 3 Temperature error (deg. C)

(a)

4

-10 -8 -6 -4 -2 0 2 4 6 Temperature Error (deg. C)

8 10

(b)

Fig. 7.22. Temperature calibration error distributions for (a) EuTTA-dope TSP and (b) Ru(bpy)-Shellac TSP, where σ is the standard deviation. From Liu et al. (1997b)

7.5. Uncertainty of Temperature Sensitive Paint

173

Table 7.2. Sensitivity coefficients, elemental errors, and total uncertainty of TSP Variable

Sensi. Coef.

ζi

Si

Physical Origin

Elemental Variance

var( ȗ i )

1

D t (ǻt )

ij

[( ∂V / ∂ t )( ǻt)/V ] 2

2

D x (ǻx )

ij

[ (∂V/∂x ) ı x2 + (∂V /∂ y ) ı 2y ] V − 2

3

D q0 (ǻt )

ij

[( ∂ q 0 / ∂ t )( ǻt)/q 0 ref ] 2

4 5 6

Vref V Ȇ c /Ȇ c ref

7

Ȇ f /Ȇ f ref

2

2

ij

V ref G ! Ȟ B d

Photodetector noise

-ij ij

V G !Ȟ Bd

Photodetector noise

ij

[ R 2 /(R 1 + R 2 )] 2 (ǻR1 /R1 ) 2

var(Ȇ f /Ȇ f ref )

Change in camera performance parameters due to model motion Illumination spectral variability and filter spectral leakage Image registration errors for correcting thickness variation due to model motion Image registration errors for correcting concentration variation due to model motion Illumination variation on model surface due to model motion Paint calibration error

h / href

ij

−2 [ (∂ h /∂ x ) ı x2 + (∂ h /∂ y ) ı 2y ] href

9

c / c ref

ij

−2 [ (∂ c /∂ x ) ı x2 + (∂ c /∂ y ) ı 2y ] c ref

10

q0 /q0 ref

ij

( q0 ref )−2 ( ∇q0 ) • ( ǻX )

11

KT

1

var( K T )

12

Temperature mapping

1

8

Temporal variation in luminescence due to photodegradation and surface contamination Image registration errors for correcting luminescence variation due to model motion Temporal variation in illumination

2

2

2

2

2

(∂T/∂x )2 ı x2 + (∂T /∂y )2 ı 2y and ( ∇T )surf • ( ǻX )surf

Total Uncertainty in Temperature

2

var (T)/ (T - Tref ) 2 =

Errors in camera calibration and temperature mapping on a surface of a presumed rigid body M

¦S

2 i

var( ȗ i )/ ȗ i

2

i =1

Note: (1) ı x and ı y are the standard deviations of least-squares estimation in the image registration or camera calibration. (2) The factor for the sensitivity coefficient is defined as ij = KT /(T − Tref ) .

8. Time Response

8.1. Time Response of Conventional Pressure Sensitive Paint

8.1.1. Solutions of Diffusion Equation The fast time response of PSP is required for measurements in unsteady flows, which is related to two characteristic timescales of PSP. One is the luminescent lifetime of PSP that represents an intrinsic physical limit for an achievable temporal resolution of PSP. Another is the timescale of oxygen diffusion across a PSP layer. Because the timescale of oxygen diffusion across a homogenous polymer layer is usually much larger than the luminescent lifetime, the time response of PSP is mainly determined by oxygen diffusion. In a thin homogenous polymer layer, when diffusion is Fickian, the oxygen concentration [O2] can be described by the one-dimension diffusion equation ∂ [O2 ] ∂ 2 [ O2 ] , (8.1) = Dm ∂z2 ∂t where Dm is the diffusivity of oxygen mass transfer, t is time, and z is the coordinate directing from the wall to the polymer layer. The boundary conditions at the solid wall and the air-paint interface for Eq. (8.1) are ∂ [O2 ] / ∂ z = 0 at z = 0 ,

[O2 ] = [O2 ]0 f ( t ) at z = h ,

(8.2)

where the non-dimensional function f ( t ) describes a temporal change of the oxygen concentration at the air-paint interface, [O2 ]0 is a constant concentration of oxygen, and h is the paint layer thickness. The initial condition for Eq. (8.1) is [O2 ] = [O2 ]0 f ( 0 ) at t = 0 . (8.3) Introducing the non-dimensional variables (8.4) n(t' , z' ) = [O2 ] / [O2 ]0 − f ( 0 ) , z' = z / h , t' = tDm / h 2 , we have the non-dimensional diffusion equation

176

8. Time Response

∂n ∂ 2 n = ∂ t' ∂ z' 2

,

(8.5)

with the boundary and initial conditions ∂n / ∂ z' = 0 at z = 0 , n = g( t' ) at z' = 1 , n = 0 at t = 0 ,

(8.6)

where the function g( t' ) is defined as g ( t' ) = f ( t' ) − f ( 0 ) that satisfies the initial condition g ( 0 ) = 0 . Applying the Laplace transform to Eq. (8.5) and the boundary and initial conditions Eq. (8.6), we obtain a general convolution-type solution for the normalized oxygen concentration n(t' , z' )

n( t' , z' ) =

³

t'

g t ( t' −u ) W ( u , z' ) du .

0

(8.7)

In Eq. (8.7), the function g t ( t ) = d g( t ) / dt = df ( t ) / dt is the differentiation of

g(t) with respect to t and the function W ( t , z ) is defined as W( t,z ) =

∞

¦

( −1 )k erfc(

1 + 2k − z 2 t

k =0

)+

∞

¦ ( −1 ) erfc( k

1 + 2k + z 2 t

k =0

).

(8.8)

The derivation of Eq. (8.7) uses the following expansion in negative exponentials

[ 1 + exp( −2 s )] −1 =

∞

¦ ( −1 )

n

exp( −2n s ) , where s is the complex variable of

n =0

the Laplace transform. In particular, for a step change of the oxygen concentration at the air-paint interface, after g t ( t ) = δ ( t ) is substituted into Eq. (8.7), the oxygen concentration distribution in a paint layer is simply n( t' , z' ) = W ( t' , z' ) , a classical solution given by Crank (1995) and Carslaw and Jaeger (2000). Instead of using the Laplace transform, Winslow et al. (2001) studied the solution of the diffusion equation using an approach of linear system dynamics. The special solutions for a step change and a sinusoidal change of oxygen were used for PSP dynamical analysis by a number of researchers (Winslow et al. 1996, 2001; Carroll et al. 1995, 1996; Mosharov et al. 1997; Fonov et al. 1998). The trigonometrical-series-type solution for a step change of oxygen given by Carroll et al. (1996) is

[O2 ]( t , z ) − [O2 ] min = 1− [O2 ] max − [O2 ] min

∞

¦ [ A cos( λ z ) exp( −λ D k

k

2 k

m

t )] ,

(8.9)

k =1

where Ak = −2( −1 )k /( hλk ) , λk = ( 2k − 1 )π /( 2h ) , [O2 ] max = [O2 ]( t , h ) , and

[O2 ] min = [O2 ]( 0 , z ) . Similarly, Winslow et al. (1996) used the trigonometricalseries-type solution for a sinusoidal change of oxygen

8.1. Time Response of Conventional Pressure Sensitive Paint

177

[O 2 ]( t , z ) − [O 2 ] 0 = [O2 ] 1

∞

π¦ 4

k =1

( 2k − 1 )π z ( −1 ) k − 1 cos[ ] sin( ω t − β k ) cos( β k ) ( 2k − 1 ) 2h

where

β k = tan −1 [

(8.10)

4 h 2ω ]. π 2 ( 2 k − 1 ) 2 Dm

The constants [O2 ]0 and [O2 ]1 are given in the initial and boundary conditions

[O2 ]( 0 , z ) = [O2 ]0 and [O2 ]( t , h ) = [O2 ]0 + [O2 ]1 sin( ω t ) . Mosharov et al. (1997) also presented the trigonometrical-series-type solution of the diffusion equation in a similar form to Eq. (8.9) for a step change at a surface. Note that they defined a coordinate system in such a way that the airpaint interface was at z = 0 and the wall was at z = h . For a sinusoidal change of oxygen [O2 ]( t ,0 ) = [O2 ]0 + [O2 ]1 sin( ω t ) at the air-paint interface, they gave a solution composed of two harmonic terms, i.e., [O2 ]( t , z ) = [O2 ]0 + [O2 ]1 [ X ( γ , z' ) sin( ω t ) + Y ( γ , z' ) cos( ω t )] ,

(8.11)

where γ = ( ω h 2 / Dm )1 / 2 is a non-dimensional frequency and z' = z / h is a nondimensional coordinate normal to the wall. The coefficients in Eq. (8.11) are X ( γ , z' ) = cosh[ 2 γ ( 1 − z' / 2 )] cos( γ z' /

2 ) + cos[ 2 γ ( 1 − z' / 2 )] cosh( γ z' /

2)

cosh( 2 γ ) + cos( 2 γ )

Y ( γ , z' ) = sinh[ 2 γ ( 1 − z' / 2 )] sin( γ z' / 2 ) + sin[ 2 γ ( 1 − z' / 2 )] sinh( γ z' / 2 )

.

cosh( 2 γ ) + cos( 2 γ ) (8.12) These trigonometrical-series-type solutions, which are often obtained using the method of separation of variables, should be equivalent to the general convolution-type solution Eq. (8.7) that is reduced in these special cases. The solutions of the diffusion equation give a classical square-law estimate for the diffusion timescale τdiff through a homogenous PSP layer,

τ diff ∝ h 2 / Dm .

(8.13)

The square-law estimate is actually a phenomenological manifestation of the statistical theory of the Brownian motion. Interestingly, this estimate is still valid even when the diffusivity of a homogeneous polymer is concentration-dependent. The 1D diffusion equation with the concentration-dependent diffusivity can be

178

8. Time Response

reduced to an ordinary differential equation by using the Boltzmann’s transformation ξ = z /( 2t 1 / 2 ) ; hence, the solution for the concentration distribution can be expressed by this similarity variable (Crank 1995). Clearly, the Boltzmann’s scaling indicates that the timescale for any point to reach a given concentration is proportional to the square of the distance (or thickness). Using the solution of the diffusion equation for a step change of pressure, Carroll et al. (1997) estimated the mass diffusivity Dm for oxygen in a typical silicon polymer binder and gave D m = 1.23 − 1.88 × 10 −9 m 2 /s over a temperature

range of 9.9-40.2 C. The values of D m = 3.55 × 10 −9 m 2 /s for the pure polymer o

Poly(dimethyl Siloxane) (PDMS) and D m = 1.2 × 10 −9 m 2 /s for PDMS with 10% fillers were also reported (Cox and Dunn 1986; Pualy 1989). For a 10 µm thick polymer layer having the diffusivity D m = 10 −10 − 10 −9 m 2 /s , the diffusion timescale is in the order of 0.1-1 s. Therefore, a conventional non-porous polymer PSP has slow time response, and it is not suitable to unsteady pressure measurements. 8.1.2. Pressure Response and Optimum Thickness

Schairer (2002) studied the pressure response of PSP based on the solution Eq. (8.11) of the diffusion equation given by Mosharov et al. (1997). In a simpler notation, the luminescent intensity integrated over a paint layer is expressed as

I(t) = C

³

h 0

exp( − β z ) dz , a + k [ O 2 ]( t , z )

(8.14)

where β is the extinction coefficient for the excitation light, C is a proportional constant, and a and k are the coefficients. In the quasi-steady case, the indicated pressure by PSP is p PSP ( t ) = [ I ref / I ( t ) − A ] / B , (8.15) where the Stern-Volmer coefficients are determined from steady-state calibration of PSP. As shown in Eq. (8.15) coupled with Eqs. (8.11), (8.12) and (8.14), the indicated pressure p PSP ( t ) is a non-linear function of the true pressure that sinusoidally varies with time, p( t ) = p0 + p1 sin( ω t ) , although the diffusion equation is linear. However, if the amplitude of the unsteady pressure is small compared to the mean pressure ( p1