Experimental Fluid MechanicsR. J. Adrian' M.Gharib- W. Merzkirch

D. Rockwell - J. H. Whitelaw

Springer-Verlag Berlin Heidelberg GmbH

Engineering ONLINE L1BRARY

http://www.springer.de/engine/

G. S. Settles

Schlieren and Shadowgraph Techniques Visualizing Phenomena in Transparent Media

With 208 Figures and 48 Color Plates

, Springer

Professor G. S. SETTLES

Penn State University Gas Dynamics Laboratory 301 D Reber Building University Park, PA 16802 USA e-mail: gss2@psu.edu

ISBN 978-3-642-63034-7

Iibrary of Congress Cataloging-in-Publication Data

Settles, G.S. (Gary S.), 1949-Schlieren und shadowgraph techniques: visualizing phenomena in transparent media / G. S. Settles. p. cm. - (Experimental fluid mechanics) Includes bibliographical references and index. ISBN 978-3-642-63034-7 ISBN 978-3-642-56640-0 (eBook) DOI 10.1007/978-3-642-56640-0 1. Schlieren methods (Optics) 2. Photography, Shadowgraph. I Title. 11 Series. QC 373.S3 S47 2001 535'.2-dc21 2001042034

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 ofSeptember 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law.

http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Originally published by Springer-Verlag Berlin Heidelberg New York in 2001 Softcover reprint of the hardcover Ist edition 200 I

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Series Editors

Prof. R. J. AdrianUniversity of Illinois at Urbana-ChampaignDept. of Theoretical and Applied Mechanics216 Talbot Laboratory104 South Wright StreetUrbana, IL 61801USA

Prof. M. GharibCalifornia Institute of TechnologyGraduate Aeronautical Laboratories1200 E. California Blvd.MC 205-45Pasadena, CA 91125USA

Prof. Dr. Wolfgang MerzkirchUniversitat EssenLehrstuhl fur StromungslehreSchiitzenbahn 7045141 Essen

Prof. Dr. D. RockwellLehigh UniversityDept. of Mechanical Engineering and MechanicsPackard Lab.19 Memorial Drive WestBethlehem, PA 18015-3085USA

Prof. J. H. WhitelawImperial CollegeDept. of Mechanical EngineeringExhibition RoadLondon SW7 2BXUK

Preface

It makes the most sense in French. I Between microskopie and telescopie comestrioscopie and ombroscopie - schlieren and shadowgraph techniques in English.All of a close family, these optical methods have common parents, grew up to­gether, and are made of the same optical stuff. Though there are hundreds ofbooks on microscopes and telescopes, however, there is no modem book onschlierenand shadowgraph methods, thus the principal justificationfor this book.

Unlike the microscopes and telescopes, schlieren and shadowgraph instrumentsare not seeing aids based on size or distance. Instead, they allow us to see the in­visible: the optical inhomogeneities in transparent media like air, water, and glassthat otherwise cause onlyghostly distortions of our normal vision.

These techniques are discussed briefly in many books and papers, but the bestand most complete treatments so far are dated and little-known. I might thereforeparaphrase Sir James Jeans by claiming that much of my book is merely Schardin[1,2] rewritten and broughtup to date.'

This book is intended more to be a practical guide and a thinking tool than anexhaustive theoretical treatment of the topic (which would require a separate vol­ume). In the western world few if any schlieren instruments can be bought out­right; unlike telescopes and microscopes, schlieren and shadowgraph systemsmust usually be assembled from components. Though this is actually not verydifficult, it can still be frustrating for the inexperienced. Thus Chap. 7 is devotedto practical issues in the selection of components, setup, and adjustment, andChap. 8 suggests a modestschlieren systemanyone can build.

This is a very visual topic. Just in fluid dynamics alone, one clearly sees thenonlinearity and pattern diversity of the governing equations in the wealthof opti­cal images that spill forth. Some of these (e.g. Toepler [3], Mach and Salcher [4],Sabine [5], Cranz and Schardin [6] and Brown and Roshko [7]) have spawnedentire fields of study. It is absurd to present such a visual subject in words alone,

I Originally from the Latin striatus (streaked) and umbra (shadow), and theGreek tele (far),micros (small), and skopein (tolook at).2 This is especially true in that Springer-Verlag has granted permission to reproduce imagesfrom Schardin's classic 1942 monograph onschlieren techniques and their applications [2],and I have followed Schardin's lead onseveral issues. His scholarship is hard to match, buthisGerman wartime publication isobscure today.

VIII

so I made every effort to illustrate the text with the best schlieren images andshadowgrams I could find. The courtesy of colleagues and some good luck al­lowed me to include a number of classic visualizations, and I thank all those whocontributed images. But even Sir Kenneth Clark [8] had to admit that some pic­tures he greatly admired were inaccessible to him.

For such time-honored techniques there remains a curious lack of familiarityand a misunderstanding of principles that I hope to rectify here. People sometimesconfuse these techniques with thermography and paranormal stuff like Kirlianphotography and the "human aura." Following recent coverage of some of myschlieren imagery on German public TV, I received a flurry of German inquiriesabout this "new" technique. One even asked for the English equivalent of theword "Schlieren." These misconceptions are dealt with throughout the text.

I thus hope to fulfill the needs of the novice for basic information, as well asthose of the professional for a thorough treatment of the topic. Other goals includemaking connections among several diverse fields and covering important recentdevelopments. At the end I make an attempt to predict where this topic is headed.Along the way, I try to convey a little of what Ernst Mach called "the charm andthe poetry of research" [9].

Consequently this book fails a primary goal if it turns out not very readable.Acronyms, for example, would save some space but the cost is too great: technol­ogy must not become the Tower ofBabel.

Almost every book requires some difficult choices of scope. Thus the gray areabetween schlieren and interferometry is explored only briefly here. Further,though several useful instruments embody schlieren optics without actually form­ing an image, present coverage is limited solely to imaging methods. No attemptis made to cite all of the thousands of literature references in which the subjecttechniques are mentioned or used. Quantitative methods are discussed in Chap.10, but the emphasis remains on the qualitative and semi-quantitative visualizationof natural phenomena, at which schlieren and shadowgraphy excel.

My audience is the user group of schlieren and shadowgraph methods. I hopethey find here a comprehensive resource. On a different level I also try to appealto all enthusiasts of seeing the invisible. The implications of this field - the opticsof inhomogeneous media - are so broad as to span most of science and technol­ogy. I try to do justice to this despite my obvious background in fluid dynamics.

The book assumes familiarity with geometric optics, optical components, andcalculus, though I do not stress the mathematical background. A few more­advanced topics such as aberrations, diffraction theory, and Fourier optics are alsoinvoked. No attempt is made to review this background in any detail, since it isreadily available in many fine textbooks. A few key optical concepts and a deriva­tion are covered in App. A in support of the main text.

Though introduced to schlieren techniques as a teenager in the 1960's, I firsthad the notion to write this book in 1983. I mentioned this to long-time friend andcolleague Wolfgang Merzkirch and it eventually led, with his encouragement, tomy writing the actual book.

I am privileged to acknowledge the assistance and friendship of three col­leagues, Leonard Weinstein, Horst Herbrich and Peter Krehl, who share my obses-

IX

sion with the optics of inhomogeneous media. Weinstein's inventive genius onthis topic is withoutparallel in the latter 20th Century. Herbrich's approach to ap­plications inspired me in that direction, and Krehl provided the example of ascientistwitha determined flairfor history.

I am further indebted to Anatol Roshko for pointingout, with characteristic in­sight, that a book on schlieren techniques shouldinclude shadowgraphy too.

Jack North and Felix Weinberg are the only masters of schlieren imaging fromthe previous generation that I was privileged to know personally. Jack passedaway during the writing of this book, but his widow, Nella North, and his col­leagues HerbertPearceyand EricRogers havebeen veryhelpful.

HeatherFerree, RossanaQuinones, and ThomasC. Hanson helped illustrate thebook with manyfine photos shot usingthe example schlieren systemof Chap. 8. Ialso thank Lori Dodson and J.D. Miller - staff members of the Penn State GasDynamics Lab- for their supportin this and all our other endeavors.

Mrs. E. Raufelder and Dr. H. Riedesel of Springer-Verlag Heidelberg assistedthis project over moreyears thanexpected, and I thankthemfor their patience.

Finally, special thanks are due to the following individuals who assisted the ef­fort in a wide variety of ways: M. Abramowitz, R. C. Benson, P. Bradshaw, Y. D.Chashechkin, A. Davidhazy, J. M. Dewey, I. V. Ershov, H. Kleine,1. Kozlik, H. F.Lehr, W. H. McCallum, 1. R. Meyer-Arendt, M. Raffel, 1. Rienitz, P. Rheinberg,M. T. Settles, 1. Srulijes, P. Steehouwer, J. K. Vandiver, A. A. Zheltovodov, andmy parents,CharlesH. and StellaM. Settles.

Bellefonte, PA, February 200I Gary S. Settles

Left-to-right: the author, Leonard Weinstein, and Horst Herbrich in Sorrento, Italy, Sept.1998: the self-proclaimed ''world's experts on schlieren photography". (No one else ap­plied for thetitle.) Photo byLiselotte Herbrich.

x

About the Author

Gary Settles was born in 1949 and grew up on a farm at the edge of the GreatSmoky Mountains in East Tennessee. His interest in fluid dynamics, optics, andexperiments began as a teenager with the construction of small subsonic and su­personic wind tunnels, winning awards at the 1967 International Science Fair. Hegained industrial and laboratory experience through summer jobs at the US NavalOrdnance Lab, NASA's Ames Research Center, and the Boeing Company, wherehe worked on both 747 and SST aerodynamics. He studied at the University ofTennessee under an Alcoa Foundation scholarship, receiving the Bachelor of Sci­ence Degree in Aerospace Engineering in 1971.

Supported by an NSF Traineeship at Princeton University, he studied shockwave/boundary-layer interactions under Prof Seymour Bogdonoff, earning hisPh.D. degree in 1976. For three years he was a Research Scientist in the PrincetonCombustion Lab Division of Flow Research, Inc., where he led a project to definea national program in energy-efficient pump utilization. He then joined the Re­search Staff of Princeton University's Mechanical and Aerospace EngineeringDept., where he conducted research and managed the Gas Dynamics Laboratory.

In 1983 Settles was appointed Associate Professor of Mechanical Engineeringat Penn State University, where he established the Penn State Gas Dynamics Lab.He was promoted to full professor in 1989. The Penn State Gas Dynamics Labhas become well-known for its work in high-speed viscous-inviscid interactionsand optical flow diagnostics. Research by Settles and his students on these topicswas honored by the Penn State Engineering Society with its 1986 Award for Out­standing Research and its 1992 Premiere Researcher Award.

The strong visual and artistic nature of the optical flow visualization practicedby Settles has been used in many film and television documentaries, museum ex­hibits, books, encyclopedias and magazines around the world. Examples includethe award-winning film "Search for Solutions," the CBS science series "Universe",the NBC "Today Show," CNN's "Science News," the series "Scientific Imagery"on German Public Television (NDR), the Learning Channel's series "Body Atlas,"and the 3-D IMAX film "The Hidden Dimension." Settles has also presentedmany invited lectures and seminars in the US, Europe, Russia, China, and Japan.

Beginning in 1992, the Penn State Gas Dynamics Lab redirected its research,emphasizing applications of gas dynamics, nozzles, and flow visualization to"non-traditional" problems in areas ranging from surface engineering and materi­als processing to rheology and aviation security. This has since resulted in severalinventions, the world's largest indoor optical flow visualization system, and a newaviation security screening portal based on the human thermal plume.

Fourteen Ph.D.'s and many Masters-degree and undergraduate scholars havebeen educated by Prof. Settles at Penn State. He currently teaches a 2-semesterlecture sequence on compressible flows, and takes recourse in oil painting andclassical music as antidotes to the rigors of an academic career.

The Settle/Suttle family immigrated from Yorkshire to Virginia in 1657 [10]

Table of Contents

List of Nomenclature XV

1 Historical Background I1. I The 1i h Century I1.2 The 18th Century .41.3 The 19th Century 61.4 The 20th Century 15

2 Basic Concepts 252.1 Light Propagation Through Inhomogeneous Media 252.2 Definition of a Schliere 282.3 Distinction Between Schlieren and Shadowgraph Methods 292.4 Direct Shadowgraphy 302.5 Simple Lens-Type Schlieren System 32

2.5.1 Point Light Source 322.5.2 Extended Light Source 34

2.6 On the Aspect of a Schlieren Image 37

3 Toepler's Schlieren Technique 393.1 Lens- and Mirror-Type Systems 39

3.1.1 Lens Systems .403.1.2 Mirror Systems .42

3.2 Sensitivity 483.2.1 Definition and Geometrical Theory .483.2.2 Sensitivity Examples 523.2.3 The Limits of Sensitivity 543.2.4 Sensitivity Enhancement by Post-Processing 57

3.3 Measuring Range 603.3.1 Definition of Measuring Range 603.3.2 Adjustment of Measuring Range 61

3.4 Estimating the Sensitivity and Range Required 633.5 Resolving Power 653.6 Diffraction Effects 66

3.6.1 Diffraction Halos Due to Opaque Edges in the Test Area 66

xn Table of Contents

3.6.2 Diffraction at the Knife-Edge 683.7 Magnification and Depth of Field 73

3.7.1 Image Magnification and the Focusing Lens 733.7.2 Depth ofField 74

4 Large-Field and Focusing Schlieren Methods 774.1 Large Single- and Double-Mirror Systems 77

4.1.1 Availability of Large Schlieren Mirrors 774.1.2 Examples of Large-Mirror Systems 784.1.3 Penn State 's l-Meter Coincident Schlieren System 79

4.2 Traditional Schlieren Systems with Large Light Sources 8I4.3 Lens-and-Grid Techniques 84

4.3.1 Simple Background Distortion 844.3.2 Background Grid Distortion 864.3.3 Large Colored Grid Background 874.3.4 The Modern Focusing/Large-Field Schlieren System 884.3.5 Penn State's Full-Scale Schlieren System 98

4.4 Large-Field Scanning Schlieren Systems 1044.4.1 Scanning Schlieren Systems for Moving Objects 1044.4.2 Schlieren Systems with Scanning Light Source and Cutoff I06

4.5 Moire-Fringe Methods 1084.6 Holographic and Tomographic Schlieren 109

5 Specialized Schlieren Techniques l l l5.1 Special Schlieren Cutoffs 111

5.1.1 Graded Filters 1125.1.2 Exponential Cutoffs and Source Filters I 155.1.3 Matched Spatial Filters at Source and Cutoff 1165.1.4 Phase Contrast 1195.1.5 Photochromic and Photorefractive Cutoffs 121

5.2 Color Schlieren Methods 1225.2.1 Reasons for Introducing Color 1225.2.2 Conversion from Monochrome to Color Schlieren 1235.2.3 Classification of Color Schlieren Techniques 1235.2.4 Recent Developments 128

5.3 Stereoscopic Schlieren 1305.4 Schlieren Interferometry 132

5.4.1 The Wollaston-Prism Shearing (Differential) Interferometer 1325.4.2 Diffraction-Based Schlieren Interferometers 134

5.5 Computer-Simulated Schlieren 1365.6 Various Specialized Techniques 137

5.6.1 Resonant Refractivity and the Visualization of Sound 1385.6.2 Anamorphic Schlieren Systems 1385.6.3 Schlieren Observation of Tracers 1385.6.4 Two-View Schlieren 140

Tableof Contents xm

5.6.5 Immersion Methods 1405.6.6 Infrared Schlieren 140

6 Shadowgraph Techniques 1436.1 Background 143

6.1.1 Historical Development... 1436.1.2 The Role of Shadowgraphy 1446.1.3 Advantages and Limitations 145

6.2 Direct Shadowgraphy 1476.2.1 Direct Shadowgraphy in Diverging Light 1486.2.2 Direct Shadowgraphy in Parallel Light 152

6.3 "Focused" Shadowgraphy 1556.3.1 Principle of Operation 1556.3.2 History and Terminology 1566.3.3 Advantages and Limitations 1576.3.4 Magnification , Illuminance, and the Virtual Shadow Effect 1586.3.5 "Focused" Shadowgraphy in Ballistic Ranges 158

6.4 Specialized Shadowgraph Techniques 1596.4.1 Large-Scale Shadowgraphy 1596.4.2 Microscopic, Stereoscopic, and Holographic Shadowgraphy 1616.4.3 Computed Shadowgraphy 1626.4.4 Conical Shadowgraphy 162

7 Practical Issues 1657.1 Optical Components 165

7.1.1 Light Sources 1657.1.2 Mirrors 1707.1.3 Schlieren Cutoffs and Source Filters 1717.1.4 Condensers and Source Slits 1727.1.5 The Required Optical Quality 174

7.2 Equipment Fabrication, Alignment, and Operation 1767.2.1 Schlieren System Design Using Ray Tracing Codes 1767.2.2 Fabricat ion of Apparatus 1777.2.3 Setup, Alignment, and Adjustment 1787.2.4 Vibration and Mechanical Stability 1817.2.5 Stray Light, Self-Luminous Events, and Secondary Images 1837.2.6 Interference from Ambient Airflows 184

7.3 Capturing Schlieren Images and Shadowgrams 1847.3.1 Photography and Cinematography 1857.3.2 Videography 1887.3.3 High-Speed imaging 1897.3.4 Front-Lighting 193

7.4 Commercial and Portable Schlieren Instruments 1957.4.1 Soviet Instruments 1957.4.2 Western Instruments 1987.4.3 Portable Schlieren Apparatus 198

XIV Table of Contents

8 Setting Up Your Own Simple Schlieren and Shadowgraph System 2018.I Designing the Schlieren System 20 I8.2 Determining the Cost. 2038.3 Choosing a Setup Location 2048.4 Aligning the Optics 2058.5 Troubleshooting 2068.6 Recording the Schlieren Image or Shadowgram 2088.7 Conclusion 209

9 Applications 2119.I Phenomena in Solids 2I I

9.1.1 GlassTechnology 21 I9.1.2 Polymer-Film Characterization 2 I39.1.3 Fracture Mechanics andTerminal Ballistics 2149.1.4 Specular Reflection from Surfaces 215

9.2 Phenomena in Liquids 2159.2.1 Convective Heatand Mass Transfer 2159.2.2 Liquid Surface Waves 2179.2.3 Liquid Atomization and Sprays 2199.2.4 Ultrasonics 2199.2.5 WaterTunnelTesting andTerminal BaIIistics 220

9.3 Phenomena in Gases 2219.3.1 Agricultural Airflows 2219.3.2 Aero-Optics 2229.3.3 Architectural Acoustics 2239.3.4 Boundary Layers 2249.3.5 Convective HeatandMassTransfer. 2259.3.6 Heating, Ventilation, andAir-Conditioning 2269.3.7 GasLeakDetection 2289.3.8 Electrical Breakdown andDischarge 2299.3.9 Explosions, Blasts, Shock Waves, and Shock Tubes 2309.3. 10 Ballistics 23 I9.3. I I Gas Dynamics and High-Speed WindTunnelTesting 2339.3.12 Supersonic Jets andJet Noise 2359.3.13 Turbomachinery and Rotorcraft 236

9.4 OtherApplications 2379.4.1 Art andmusic 2379.4.2 Biomedical Applications 2409.4.3 Combustion 2449.4.4 Geophysics 2459.4.5 Industrial Applications 2469.4.6 Materials Processing 2479.4.7 Microscopy 2499.4.8 Optical Processing 2529.4.9 Optical ShopTesting 2539.4.10 Outdoor Schlieren and Shadowgraphy 254

Table of Contents XV

9.4.11 Plasma Dynamics 2599.4.12 Television Light Valve Projection 2609.4.13 Turbulence 261

10 Quantitative Evaluation 26310.1 Quantitative Schlieren Evaluationby Photometry 264

10.1.1 Absolute Photometric Methods 26510.1.2 Standard Photometric Methods 266

10.2 Grid-Cutoff Methods 26810.2.1 FocalGrids 26810.2.2 Defocused Grids 27110.2.3 Defocused Filament Cutoff 272

10.3 Quantitative Image Velocimetry 27310.3.1 Background 27310.3.2 Multiple-Exposure Eddy and Shock Velocimetry 27410.3.3 Schlieren Image CorrelationVelocimetry 27410.3.4 Focusing Schlieren Deflectometry 27510.3.5 The Background-Oriented Schlieren System 276

10.4 Quantitative Shadowgraphy 27710.4.1 Double Integration of a2nJa! 27710.4.2 Turbulence Research 27710.4.3 Shock-Wave Strength Quantitation 27710.4.4 Grid Shadowgraphy Methods 277

11 Summary and Outlook 27911.1 Summary 279

11.1.1 PerceptionsOutside the Scientific Community 27911 .1.2 OtherLessons Leamed 28011.1.3 Further Comments on Historical Development... 28111 .1.4 FurtherComments on Images andVisualization 28111 .1.5 Renewed Vitality 283

11.2 Outlook: Issues for the Future 28411 .2.1 Predictions 28411 .2.2 Opportunities 28611 .2.3 Recommendations 288

11 .3 Closing Remarks 289

References 291

Appendix A Optical Fundamentals 333A.l Radiometry andPhotometry 333A.2 Refraction Angle e 334

A.2.1 Small Optical Angles and Paraxial Space 334A.2.2 Huygens' Principle and Refraction 334

A.3 Optical Components and Devices 335A.3.l Conjugate Optical Planes 335

XVI Table of Contents

A.3.2 Lens f/number 335A3.3 TheThin-Lens Approximation 335A.3A Viewing Screens andGround Glass 336A.3.5 Optical Density 336

AA Optical Aberrations 336A5 Lightand the Human Eye 337A6 Geometric Theory of LightRefraction by a Schliere 338

Appendix B The Schlieren System as a Fourier Optical Processor 341B.l The BasicFourier Processor withno Schlieren Present 3418.2 The Addition of a Schlieren TestObject 344B.3 The Schlieren Cutoff 345BA OtherSpatial Filters 3478.5 Partially-Coherent andPolychromatic Illumination 350

Appendix C Parts List for a Simple Schlieren/ Shadowgraph System 353C.l Optics 353C.2 Illunrination 354C.3 Miscellaneous Components 354C.4 Optical Mounts 354

Appendix D Suppliers of Schlieren Systems andComponents 355D.I Complete Schlieren Systems 355D.2 Schlieren FieldMirrors 356D.3 LightSources 357DA Components .358D.5 Focusing Schlieren Lenses 359D.6 Miscellaneous 359

Index 361

ColorPlates .367

List of Nomenclature

Man: 'Hello, myboy. Andwhatis yourdog's name?'Boy: '1 don't know. WecallhimRover.'

StaffordBeer

Metric units are used throughout this book. Distances units m, cm, mm, and umare used at various points for convenience. In the special case of photometricunits, see Sect. A.l for more detail. Finally, an attempt is made to maintain con­sistent notation, as far as possible, with the key precedent works [1,2,98,102,110].

a grid constant in cutoffplane, also acoustic speed, mlsa unobstructed height of source image in cutoff planeM change in a due to refractionA amplitude function; also clear aperture of schlieren objectiveB Luminance, candela/m'; also grid width constantb, h breadth and height of light source slitc constant; also denotes the speed of lightC contrastd width of schliere in test area; also diameter of circle of confusionD schlieren system aperture (test area diameter); also light source diametere base of natural logarithms; also grid width constantf focal lengthfino . the focal ratio of a lens or mirror, i.e. f/Df', f2 focal length of second lens or mirrorF Fourier transform; also focal distance coordinateg,h lengths; g also denotes gravitational acceleration constantE, I illuminance on viewing screen, lux~E change of illuminance on screen, luxE* indirect or unwanted illuminance component, luxj imaginary number V-Ik constant; also Gladstone-Dale constant

1, e length

L length oflight path or of schlieren object in test area, measured along z­axis; also designates a lens

m image magnification

XVIII

Mnpr,e,cpR

R.,d,ph

SsTtu,vVwx,y,z

List of Nomenclature

Machnumberrefractive indexstaticpressure, kPa; also denotespolarizerspherical coordinates (r =radius)rangeof schlieren systemin mmof ray deflection or arcseconds of angle;also specific gas constantand inverse oflens aperture angleresolution limitsdue to aberrations, diffraction, and photographic mediaSensitivity; also denotesschlieren objectdistancecoordinatetemperature, K; also transmittance functiontime, seconds; also usedas a length coordinate, e.g. thicknesscoordinates normal to opticalaxis in Fourierplanespeed, mlswidthcoordinate; also wallcondition, Wollaston prismRight-handed Cartesian coordinate systemin which the lightpropagationalongthe opticalaxis is in the +z-direction

Symbolsa apertureangle, radians~ light-loss factor~T coefficient of thermal expansion, K 1

y slopeof characteristic curveof photographic filmo surface inclination angle, degrees; also diameter of a small feature of a

schlieren object,and blur diameter~ represents an increment or changein the quantitywhichfollowse light ray deflection angleproduced by schlieren, radiansor arcseconds</>, cp phase function; also numberof line pairs contributing to an image point

in focusing schlieren, and an angularcoordinateA. wavelength of light, usually givenin um~ fluid viscosity, N·sec/m2

8 azimuth angle in plane perpendicular to optical axis, degrees; also rota­tion angleand axis offsetangle in a z-type schlierensystem

p density, kg/nr'1; constant

Subscriptso background or baselinevalueI first schlierenlens or mirror2 secondschlieren lens or mirror3 focusing lensb.u blur, unsharpnessmax maximummill mmimumref reference value