Particle image velocimetry: a practical guide

Markus Raffel · Christian E. Willert Steve T. Wereley · Jürgen Kompenhans Particle Image Velocimetry Markus Raffel · ...

Author: Markus Raffel | Christian E. Willert | Steven T. Wereley | Jürgen Kompenhans

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Markus Raffel · Christian E. Willert Steve T. Wereley · Jürgen Kompenhans Particle Image Velocimetry

Markus Raffel · Christian E. Willert Steve T. Wereley · Jürgen Kompenhans

Particle Image Velocimetry A Practical Guide Second Edition

With 288 Figures and 42 Tables

Markus Raffel DLR, Institut für Aerodynamik und Strömungstechnik Bunsenstraße 10 37073 Göttingen Germany [email protected]

Christian E. Willert DLR, Institut für Antriebstechnik 51170 Köln Germany [email protected]

Steve T. Wereley Purdue University Purdue Mall 585 47907-2088 West Lafayette USA [email protected]

Jürgen Kompenhans DLR, Institut für Aerodynamik und Strömungstechnik Bunsenstraße 10 37073 Göttingen Germany [email protected]

Library of Congress Control Number: 2007928306

ISBN

978-3-540-72307-3 Second Edition Springer Berlin Heidelberg New York

ISBN

978-3-540-63683-0 Springer Berlin Heidelberg New York

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 any other way, 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. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 1998 2007 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. Production: Integra Software Services Pvt. Ltd., Puducherry, India Cover design: WMX Design, Heidelberg Printed on acid-free paper SPIN: 11689782 42/3100/Integra 5 4 3 2 1 0

Preface

The development of Particle Image Velocimetry (PIV), a measurement technique, which allows for capturing velocity information of whole ﬂow ﬁelds in fractions of a second, has begun in the eighties of the last century. In 1998, when this book has been published ﬁrstly, the PIV technique emerged from laboratories to applications in fundamental and industrial research, in parallel to the transition from photo-graphical to video recording techniques. Thus this book, whose objective was and is to serve as a practical guide to the PIV technique, found strong interest within the increasing group of users. The early progress made with the PIV technique might best be characterized by the experience gained during our aerodynamic research at DLR (Deutsches Zentrum f¨ ur Luft- und Raumfahrt) at that time. The ﬁrst applications of PIV outside the laboratory, in wind tunnels, as performed in the mid-eighties were characterized by the following time scales: time required to set up the system and to obtain well focused photo-graphical PIV recordings was 2 to 3 days, time required to process the ﬁlm was 0.5 to 1 day, time required to evaluate a single photo-graphical PIV recording by means of optical evaluation methods was 24 to 48 hours. When the ﬁrst edition of this book was published in 1998, with electronic cameras and computers, it was possible to focus on-line, to capture several recordings per second, and to evaluate a digital recording within seconds. During a typical measurement campaign about 100 GigaByte of raw data could easily be collected. The time since then was again characterized by a rapid development of hard- and software for the PIV technique. Improved cameras, lasers, optics and software lead to a signiﬁcant increase in performance. We therefore updated the chapters on the principles only moderately and included information on microscopic, high-speed and three component measurements as well as a description of advanced evaluation techniques. Steve Wereley with his knowledge completed the group of authors of the ﬁrst edition and his participation as an author was the key for that update. But also the range of possible applications increased drastically over the years. PIV is nowadays used in very diﬀerent areas from aerodynamics to

VI

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biology, from fundamental turbulence research to applications in the space shuttle, from combustion to two phase pipe ﬂows and very intensively in micro devices and systems. Due to this wide range of possible applications of PIV, the number of research groups employing the PIV technique world wide has increased from a handful at the beginning of the eighties to far beyond a thousand today. Due to the variety of diﬀerent applications of PIV and the large number of diﬀerent possibilities to illuminate, to record and to evaluate, many diﬀerent technical modiﬁcations of the PIV technique have been developed. Much of the material covered in this book has already been published in conference proceedings or in scientiﬁc journals. However, this information is widespread and cannot easily be found by someone who wants to start employing the PIV technique for his special problems. Moreover, most publications illuminate the problems only from a speciﬁc point of view. We therefore felt that it was again timely to compile the knowledge about the basic principles of PIV and the main guidelines for its implementation in practice.

Organization of the Book The intention of this book is to present in a more general context mainly those aspects of the PIV technique relevant to applications. This strategy is supported by the experience gained during the work on the development of PIV for more than two decades and a large number of more than 100 diﬀerent applications of DLR’s mobile PIV systems in aerodynamics and related areas ranging from subsonic to transonic ﬂows, from turbulence research to the investigation of reacting ﬂows, and from small test facilities to large industrial wind tunnels. This strategy is similar to that of the annual course on Particle Image Velocimetry held since 1993 at the laboratories of DLR. The presentation of the material in this book takes into account the feedback from the participants of these courses as well. The fundamentals for the application of PIV to micro ﬂows being somewhat diﬀerent from that for large-scale ﬂows at DLR a fourth author, being a leading scientist in this ﬁeld, could be enlisted to contribute with his knowledge to this book. Furthermore the chapter describing representative applications of the PIV technique has been expanded considerably. Many ideas about problems associated with special applications of PIV and their solution can be found in this chapter, to which many PIV experts worldwide contributed with their expertise gained in the most diﬀerent areas of ﬂow research. This practical guide to PIV provides in a condensed form most of the information relevant for the planning, performance and understanding of experiments employing the PIV technique. It is mainly intended for engineers, scientists and students, who already have some basic knowledge of ﬂuid mechanics and non-intrusive optical measurement techniques. For many researchers and engineers, planning to utilize PIV for their special industrial or scientiﬁc ap-

Preface

VII

plications, PIV is just an attractive tool with unique features which may help them to gain new insights in problems of ﬂuid mechanics. These people are usually not interested in becoming specialists in this ﬁeld before starting their investigations. On the other side some of the basic properties of PIV must be well understood before a correct interpretation of the results is possible. Our hope is that this book will serve this purpose by providing an easy transfer of the know-how gathered by the authors during many years to the readers. It may help the readers to avoid beginners’ errors and bring them to a position to obtain high quality results when employing PIV right from the beginning of their work. For those, already working in the ﬁeld of PIV, this book may serve as a reference to further publications containing more details. As with all publications, the amount of information covered in this book is limited, unfortunately rendering completeness impossible. Nevertheless, it is our hope that we have been able to collect the relevant information for practical work with PIV.

About the Authors Markus Raﬀel received his degree in Mechanical Engineering in 1990 from the Technical University of Karlsruhe, his doctorate in in Engineering in 1993 from the University of Hannover and his lecturer qualiﬁcation (Habilitation) from the Technical University Clausthal, in 2001. He started working on PIV at the German Aerospace Establishment (DLR) in 1991 with emphasis on the development of PIV recording techniques in high-speed ﬂows. In this process he applied the method to a number of aerodynamic problems mainly in the context of rotorcraft investigations. He is professor at the University of Hannover and head of the Department of Technical Flows at DLR’s Institute of Aerodynamics and Flow Technology in G¨ ottingen. Christian Willert received his Bachelor of Science in Applied Science from the University of California at San Diego (UCSD) in 1987. Subsequent graduate work in experimental ﬂuid mechanics at UCSD lead to the development of several non-intrusive measurement techniques for application in water (particle tracing, 3-D particle tracking, digital PIV). After receiving his Ph.D. in Engineering Sciences in 1992, he assumed post-doctoral positions ﬁrst at the Institute for Nonlinear Science (INLS) at UCSD, then at the Graduate Aeronautical Laboratories at the California Institute of Technology (Caltech). In 1994 he joined DLR G¨ottingen’s measurement sciences group as part of an exchange program between Caltech and DLR. Since 1997 he has been working in the development and application of planar velocimetry techniques (PIV and Doppler Global Velocimetry (DGV)) at the Institute of Propulsion Technology of DLR and now is heading the Department of Engine Measurement Techniques there.

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Steven T. Wereley received both his Bachelor of Arts in Physics from Lawrence University at Appleton, Wisconsin, and his Bachelor of Science in Mechanical Engineering from Washington University at St. Louis in 1990. He received his Master of Science and Ph.D. degrees from Northwestern University in Evanston, Illinois, in 1992 and 1997, respectively. Subsequently, he spent two years at the Mechanical and Environmental Engineering Department at the University of California in Santa Barbara developing particle image velocimetry algorithms for micro-domain investigations. Since 1999 he has been a professor at Purdue University in the School of Mechanical Engineering–as an Assistant Professor from 1999 to 2005 and an Associate Professor since then. Professor Wereley’s research is largely concerned with micro-particle image velocimetry techniques and microelectromechanical systems with applications in bio-physics and bio-engineering. J¨ urgen Kompenhans received his doctoral degree in physics in 1976 from the Georg-August University of G¨ ottingen. Since 30 years he has been working for DLR in G¨ ottingen, Germany, mainly developing and applying nonintrusive measurement techniques for aerodynamic research, starting with the PIV technique back in 1985. At present, he is head of the Department of Experimental Methods of DLR’ s Institute of Aerodynamics and Flow Technology. Within this department, image based methods such as Pressure Sensitive Paint, Temperature Sensitive Paint, Particle Image Velocimetry, model deformation measurement techniques, density measurement techniques, acoustic ﬁeld measurement techniques etc. are developed for application as mobile systems in large industrial wind tunnels. As coordinator of several European networks J¨ urgen Kompenhans has contributed to promote the use of image based measurement techniques for industrial research.

Acknowledgments Steve Wereley would like to acknowledge the partial support of the Alexander von Humboldt Society that enabled the face-to-face meetings among the author without which completion of the manuscript would have been much more diﬃcult. During the past decades, a number of colleagues, some of them being members of our group only for a limited time, have contributed to the success of the ﬁrst edition of this book: technicians, students and scientists. Among these we especially want to acknowledge the contributions of KarlAloys B¨ uteﬁsch, Klaus Hinsch, Hans H¨ ofer, Christian K¨ ahler, Hugues Richard, Olaf Ronneberger, Andreas Schr¨ oder, and Andreas Vogt. Our work has been ﬁnancially supported by DLR, as well as by other national and international institutions. Stimulating discussions on correlation techniques with Jerry Westerweel, Mory Gharib and Thomas Roesgen, are greatly appreciated.

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IX

PIV wind tunnel measurements, discussed in the ﬁrst edition of this book, have been performed at the facilities of diﬀerent research organizations such as DLR, the German–Dutch Wind Tunnel (DNW) with the special support of its former director, Hans Ulrich Meier, and the Institut Franco–Allemand de Recherches de Saint-Louis (ISL) and others. With a view to the second edition, we would like to set apart Professor R. I. Sujith (Dept. of Aerospace Engineering, IIT Madras) for carefully reviewing the manuscript and Kolja Kindler (DLR) for organizing our work. Both made many valuable suggestions for the improvement of text and ﬁgures. Moreover, we gratefully acknowledge, that without Professor R. I. Sujith encouragement we had not started working on the second edition and without Kolja’s help it wouldn’t yet be ﬁnished. Many thanks to Fulvio Scarano, Bernhard Wieneke and Gerrit Elsinga for their description of the emerging tomographic evaluation technique. The second edition is, even more than the ﬁrst, based on the supplementary work of many scientists. We are very much indebted to a great number of researchers worldwide who have provided contributions on special topics related to the PIV technique, which allowed covering a broad ﬁeld of applications. These contributions shall be acknowledged next, giving the name of the author and institute where the research has been performed. In case authors are still active in the ﬁeld of PIV their most recent address has been given. Contributions to the second edition have been provided by: C. B¨ohm

Work performed at: ZARM University of Bremen Am Fallturm, 28359 Bremen, Germany

J. Bosbach

Institute of Aerodynamics and Flow Technology German Aerospace Center Bunsenstraße 10, 37073 G¨ottingen, Germany

A. Delgado

Institute of Fluid Mechanics, Technical Faculty University Erlangen-N¨ urnberg Cauerstraße 4, 91058 Erlangen, Germany

U. Dirksheide

LaVision GmbH Anna-Vandenhoeck-Ring 19, D-37081 G¨ ottingen, Germany

R. du Puits

Department of Mechanical Engineering Ilmenau University of Technology P.O. Box 100565, 98684 Ilmenau, Germany

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G.E. Elsinga

Department of Aerospace Engineering Delft University of Technology Kluyverweg 1, 2629 HS, Delft, The Netherlands

D. Favier

Laboratoire d’A´erodynamique et de Biom´ecanique du Mouvement CNRS, LABM, 163 Avenue de Luminy, CP 918, 13288 Marseille Cedex 09, France

C. Gharib

Aeronautics and Bioengineering California Institute of Technology 1200 East California Boulevard, 205-45, Passadena CA 91125, USA

E. G¨ ottlich

Institut for Thermal Turbomachinery and Machine Dynamics Technical University of Graz Inﬀeldgasse 25A, 8010 Graz, Austria

R. Hain

Institute of Fluid Mechanics Technical University of Braunschweig Bienroder Weg 3, 38106 Braunschweig, Germany

J.T. Heineck

Experimental Physics Branch NASA Ames Research Center Moﬀet Field, CA 94035, USA

M. Herr

Institute of Aerodynamics and Flow Technology German Aerospace Center Lilienthalplatz 7, 38108 Braunschweig, Germany

R.A. Humble

Department of Aerospace Engineering Delft University of Technology Kluyverweg 1, 2629 HS, Delft, The Netherlands

C.J. K¨ ahler

Institute of Fluid Mechanics Technical University of Braunschweig Bienroder Weg 3, 38106 Braunschweig, Germany

H. Kinoshita

Work performed at: Institute of Industrial Sciences University of Tokyo 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

Preface

XI

K. Kindler

Institute of Aerodynamics and Flow Technology German Aerospace Center Bunsenstraße 10, 37073 G¨ottingen, Germany

C. Klein

Institute of Aerodynamics and Flow Technology German Aerospace Center Bunsenstraße 10, 37073 G¨ottingen, Germany

R. Konrath

Institute of Aerodynamics and Flow Technology German Aerospace Center Bunsenstraße 10, 37073 G¨ottingen, Germany

W. Kowalczyk

Institute of Fluid Mechanics, Technical Faculty University of Erlangen-N¨ urnberg Cauerstraße 4, 91058 Erlangen, Germany

H. Lang

Work performed at: Institute for Thermal Turbomachinery and Machine Dynamics Technical University of Graz Inﬀeldgasse 25, 8010 Graz, Austria

T. Lauke

Institute of Aerodynamics and Flow Technology German Aerospace Center Lilienthalplatz 7, 38108 Braunschweig, Germany

C.D. Meinhart

Dept. of Mechanical and Environmental Engineering University of California - Santa Barbara Santa Barbara, CA 93106, USA

M. Oshima

Institute of Industrial Science University of Tokyo 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

H. Petermeier

Information Technology Weihenstephan Technical University of Munich Am Forum 1, 85354 Freising, Germany

P. Rambaud

Environmental and Applied Fluids Department Von Karman Institute for Fluid Dynamics 72 Chaussee de Waterloo 1640 Rhode St. Gen`ese, Belgium

C. Resagk

Department of Mechanical Engineering Ilmenau University of Technology P.O. Box 100565, 98684 Ilmenau, Germany

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H. Richard

Institute of Aerodynamics and Flow Technology German Aerospace Center Bunsenstrasse 10, 37073 G¨ottingen, Germany

M. Riethmuller

Environmental and Applied Fluids Department Von Karman Institute for Fluid Dynamics 72 Chaussee de Waterloo 1640 Rhode St. Gen`ese, Belgium

C. Rondot

Laboratoire d’A´erodynamique et de Biom´echanique du Movement CNRS, LABM, 163 Avenue de Luminy, CP 918, 13288 Marseille Cedex 09, France

O. Ronneberger

Work performed at: Institute of Aerodynamics and Flow Technology German Aerospace Center Bunsenstrasse 10, 37073 G¨ottingen, Germany

F. Scarano

Department of Aerospace Engineering Delft University of technology Kluyverweg 1, 2629 HS, Delft, The Netherlands

C. Schram

Work performed at: Environmental and Applied Fluids Department Von Karman Institute for Fluid Dynamics 72 Chaussee de Waterloo 1640 Rhode St. Gen`ese, Belgium

E.T. Schairer

Experimental Physics Branch NASA Ames Research Center Moﬀet Field, CA 94035, USA

A. Schr¨ oder

Institute of Aerodynamics and Flow Technology German Aerospace Center Bunsenstr. 10, 37073 G¨ottingen, Germany

A. Thess

Department of Mechanical Engineering Ilmenau University of Technology P.O. Box 100565, 98684 Ilmenau, Germany

B. van der Wall

Institute of Flight Systems German Aerospace Center Lilienthalplatz 7, 38108 Braunschweig, Germany

Preface

XIII

B.W. van Oudheusden Department of Aerospace Engineering Delft University of Technology Kluyverweg 1, 2629 HS, Delft, The Netherlands M. Voges

Institute of Propulsion Technology German Aerospace Center Linder H¨ ohe, 51147 K¨oln, Germany

C. Wagner

Institute of Aerodynamics and Flow Technology German Aerospace Center Bunsenstr. 10, 37073 G¨ottingen, Germany

S.M. Walker

Work performed at: Experimental Physics Branch NASA Ames Research Center Moﬀet Field, CA 94035, USA

B. Wieneke

LaVision GmbH Anna-Vandenhoeck-Ring 19, D-37081 G¨ ottingen, Germany

J. Woisetschl¨ ager

Institute for Thermal Turbomachinery and Machine Dynamics Technical University of Graz Inﬀeldgasse 25, 8010 Graz, Austria

M. Yoda

George W. Woodruﬀ School of Mechanical Engineering Georgia Institute of Technology 771 Ferst Drive, Atlanta, Georgia 30332-0405, USA

We are deeply indebted to all friends and colleagues of the worldwide PIV community who helped us to better understand the diﬀerent aspects of the PIV technique during recent years by their work, their publications and conference contributions, and by personal discussions. The authors, G¨ ottingen, March 2007 Markus Raﬀel Chris Willert Steve T. Wereley J¨ urgen Kompenhans

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Principle of Particle Image Velocimetry (PIV) . . . . . . . . . . . . . . . 3 1.3 Development of PIV During the Last Two Decades . . . . . . . . . . 8 1.3.1 PIV in Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2 Major Technological Milestones of PIV . . . . . . . . . . . . . . . 11

2

Physical and Technical Background . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Tracer Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fluid Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Light Scattering Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Particle Generation and Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Seeding of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Seeding of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Features and Components of PIV Lasers . . . . . . . . . . . . . 2.3.3 White Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Light Sheet Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Volume Illumination of the Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Imaging of Small Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Diﬀraction Limited Imaging . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Lens Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Perspective Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Discussion of the Perspective Error . . . . . . . . . . . . . . . . . . 2.6.5 Basics of Microscopic Imaging . . . . . . . . . . . . . . . . . . . . . . 2.6.6 In-Plane Spatial Resolution of Microscopic Imaging . . . . 2.6.7 Microscopes Typically Used in Micro-PIV . . . . . . . . . . . . 2.6.8 Confocal Microscopic Imaging . . . . . . . . . . . . . . . . . . . . . . .

15 15 15 18 21 21 22 28 28 35 41 43 46 48 48 52 55 57 59 62 62 65

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2.7 Photographic Recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 A Brief Description of the Chemical Processes . . . . . . . . 2.7.2 Introduction to Performance Diagrams . . . . . . . . . . . . . . . 2.8 Digital Image Recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Characteristics of CCD Sensors . . . . . . . . . . . . . . . . . . . . . 2.8.2 Characteristics of CMOS Sensors . . . . . . . . . . . . . . . . . . . . 2.8.3 Sources of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 Linearity and Dynamic Range . . . . . . . . . . . . . . . . . . . . . . 2.9 Standard Video and PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 The Video Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

Mathematical Background of Statistical PIV Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Particle Image Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Image Intensity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Mean Value, Autocorrelation and Variance of a Single Exposure Recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Cross-Correlation of a Pair of Two Singly Exposed Recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Correlation of a Doubly Exposed Recording . . . . . . . . . . . . . . . . . 3.6 Expected Value of Displacement Correlation . . . . . . . . . . . . . . . . 3.7 Optimization of Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 66 66 69 69 71 72 73 73 75 75 79 79 81 83 86 88 91 92

PIV Recording Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1 Film Cameras for PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1.1 Example of a PIV Film Camera . . . . . . . . . . . . . . . . . . . . . 100 4.1.2 High-Speed Film Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 Digital Cameras for PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2.1 Full-Frame CCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2.2 Frame Transfer CCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.3 Interline Transfer CCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2.4 Full-Frame Interline Transfer CCD . . . . . . . . . . . . . . . . . . 107 4.2.5 Active Pixel CMOS Sensor . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2.6 High-Speed CCD Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.7 High-Speed CMOS Cameras for PIV Recording . . . . . . . 110 4.3 Single Frame/Multi-Exposure Recording . . . . . . . . . . . . . . . . . . . 110 4.3.1 General Aspects of Image Shifting . . . . . . . . . . . . . . . . . . . 111 4.3.2 Optimization of PIV Recording for Autocorrelation Analysis by Image Shifting . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.3.3 Realizations of Image Shifting . . . . . . . . . . . . . . . . . . . . . . . 112 4.3.4 Layout of a Rotating Mirror System . . . . . . . . . . . . . . . . . 113 4.3.5 Calculation of the Mirror Image Shift . . . . . . . . . . . . . . . . 115 4.3.6 Experimental Determination of the Mirror Image Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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XVII

4.4 Multi-Frame PIV Recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.4.1 Video-Based Implementation of Double Frame/Single Exposure PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5

Image Evaluation Methods for PIV . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1 Correlation and Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.1.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.1.2 Optical Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.1.3 Digital Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2 Summary of PIV Evaluation Methods . . . . . . . . . . . . . . . . . . . . . . 127 5.3 Optical PIV Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.3.1 Young’s Fringes Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.4 Digital PIV Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.4.1 Digital Spatial Correlation in PIV Evaluation . . . . . . . . . 132 5.4.2 Correlation Signal Enhancement . . . . . . . . . . . . . . . . . . . . 139 5.4.3 Autocorrelation of Doubly Exposed PIV Images . . . . . . . 143 5.4.4 Advanced Digital Interrogation Techniques . . . . . . . . . . . 146 5.4.5 Peak Detection and Displacement Estimation . . . . . . . . . 158 5.5 Measurement Noise and Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.5.1 Synthetic Particle Image Generation . . . . . . . . . . . . . . . . . 165 5.5.2 Optimization of Particle Image Diameter . . . . . . . . . . . . . 166 5.5.3 Optimization of Particle Image Shift . . . . . . . . . . . . . . . . . 169 5.5.4 Eﬀect of Particle Image Density . . . . . . . . . . . . . . . . . . . . 170 5.5.5 Variation of Image Quantization Levels . . . . . . . . . . . . . . 172 5.5.6 Eﬀect of Background Noise . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.5.7 Eﬀect of Displacement Gradients . . . . . . . . . . . . . . . . . . . . 175 5.5.8 Eﬀect of Out-of-Plane Motion . . . . . . . . . . . . . . . . . . . . . . . 176

6

Post-Processing of PIV Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.1 Data Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.1.1 Global Histogram Operator . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.1.2 Dynamic Mean Value Operator . . . . . . . . . . . . . . . . . . . . . 183 6.1.3 Vector Diﬀerence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.1.4 Median Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.1.5 Normalized Median Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.1.6 Other Validation Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.1.7 Implementation of Data Validation Algorithms . . . . . . . . 188 6.2 Replacement Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.3 Vector Field Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.4 Estimation of Diﬀerential Quantities . . . . . . . . . . . . . . . . . . . . . . . 190 6.4.1 Standard Diﬀerentiation Schemes . . . . . . . . . . . . . . . . . . . . 191 6.4.2 Alternative Diﬀerentiation Schemes . . . . . . . . . . . . . . . . . . 194 6.4.3 Uncertainties and Errors in Diﬀerential Estimation . . . . 198 6.5 Estimation of Integral Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.5.1 Path Integrals – Circulation . . . . . . . . . . . . . . . . . . . . . . . . 200 6.5.2 Path Integrals – Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . . 201

XVIII Contents

6.5.3 Area Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.5.4 Pressure and Forces from PIV Data . . . . . . . . . . . . . . . . . . 205 6.6 Vortex Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7

Three-Component PIV Measurements . . . . . . . . . . . . . . . . . . . . . 209 7.1 Stereo PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.1.1 Reconstruction Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.1.2 Stereo Viewing Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.2 Dual-Plane PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.2.1 Mode of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.2.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.3 Three Component PIV Measurements in a Volume . . . . . . . . . . 231 7.3.1 Principles of Tomographic-PIV . . . . . . . . . . . . . . . . . . . . . . 234

8

Micro-PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.2 Overview of Micro-PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 8.2.1 μPIV Seeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.2.2 Special Processing Methods for μPIV Recordings . . . . . . 255 8.2.3 μPIV Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

9

Examples of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 9.1 Liquid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 9.1.1 Vortex-Free-Surface Interaction . . . . . . . . . . . . . . . . . . . . . 259 9.1.2 Study of Thermal Convection and Couette Flows . . . . . . 260 9.2 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 9.2.1 Boundary Layer Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 266 9.2.2 Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . 268 9.3 Transonic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.3.1 Cascade Blade with Cooling Air Ejection . . . . . . . . . . . . . 272 9.3.2 Transonic Flow Above an Airfoil . . . . . . . . . . . . . . . . . . . . 273 9.3.3 Shock Wave/Turbulent Boundary Layer Interaction . . . . 276 9.4 Stereo PIV Applied to a Vortex Ring Flow . . . . . . . . . . . . . . . . . 280 9.4.1 Imaging Conﬁguration and Hardware . . . . . . . . . . . . . . . . 281 9.4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 9.5 Dual-plane PIV Applied to a Vortex Ring Flow . . . . . . . . . . . . . 285 9.5.1 Imaging Conﬁguration and Hardware . . . . . . . . . . . . . . . . 285 9.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 9.6 Large Scale Rayleigh-B´enard Convection . . . . . . . . . . . . . . . . . . . 292 9.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.6.2 Stereo PIV in the Barrel of Ilmenau . . . . . . . . . . . . . . . . . 293 9.6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 9.7 Analysis of PIV Image Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 297 9.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 9.7.2 Evaluation of a Simulated PIV Image Sequence . . . . . . . 299 9.7.3 Investigation of Separation on a SD7003 Airfoil . . . . . . . 300

Contents

XIX

9.8 Velocity and Pressure Maps Above a Transonic Delta Wing . . . 301 9.9 Coherent Structure Detection in a Backward-Facing Step Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 9.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 9.9.2 Vortex Detection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 305 9.9.3 Application to the Backward-Facing Step Flow . . . . . . . . 306 9.9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 9.10 Quantitative Study of Vortex Pairing in a Circular Air Jet . . . . 310 9.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 9.10.2 Acoustically Excited Jet Facility . . . . . . . . . . . . . . . . . . . . 310 9.10.3 PIV Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.10.4 PIV Uncertainty: Random Errors . . . . . . . . . . . . . . . . . . . . 312 9.10.5 Particle Centrifugation in the Vortex Cores: Bias Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 9.10.6 Post-Processing: Automatic Vortex Tracking . . . . . . . . . . 315 9.10.7 Acoustical Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 9.10.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 9.11 Stereo and Volume Approaches to Helicopter Aerodynamics . . . 317 9.11.1 Rotor Flow Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 9.11.2 Wind Tunnel Measurements of Rotor Blade Vortices . . . 318 9.11.3 Measurement of Rotor Blade Vortices in Hover . . . . . . . . 321 9.12 Stereo PIV Applied to a Transonic Turbine . . . . . . . . . . . . . . . . . 327 9.12.1 Optical Conﬁguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.12.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 9.13 PIV Applied to a Transonic Centrifugal Compressor . . . . . . . . . 332 9.14 PIV in Reacting Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 9.15 A High-Speed PIV Study on Trailing-Edge Noise Sources . . . . . 344 9.15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 9.15.2 Setup, Measurements and Procedure . . . . . . . . . . . . . . . . . 344 9.15.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.16 Volume PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.17 Supersonic PIV Measurements on a Space Shuttle Model . . . . . 350 9.18 Multiplane Stereo PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 9.18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 9.18.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 9.18.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 9.19 Microscale PIV Wind Tunnel Investigations . . . . . . . . . . . . . . . . . 358 9.19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 9.19.2 The Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 9.19.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 9.19.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 9.20 Micro-PIV I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 9.20.1 Application of PIV to Microscopic Flow . . . . . . . . . . . . . . 363 9.20.2 Examples of Micro-PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

XX

Contents

9.20.3 Diﬀerences from Macroscale PIV . . . . . . . . . . . . . . . . . . . . 365 9.20.4 Advanced Technique: Confocal Micro-PIV . . . . . . . . . . . . 366 9.21 Micro-PIV II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 9.21.1 Flow in a Microchannel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 9.21.2 Flow in a Micronozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 9.21.3 Flow Around a Blood Cell . . . . . . . . . . . . . . . . . . . . . . . . . . 374 9.21.4 Flow in Microﬂuidic Biochip . . . . . . . . . . . . . . . . . . . . . . . . 377 9.22 Nano-PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 9.22.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 9.22.2 Nano-PIV Studies of Microscale Electroosmotic Flow . . 380 9.23 Micro-PIV in Life Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 9.23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 9.23.2 Biocompatible μPIV/μPTV . . . . . . . . . . . . . . . . . . . . . . . . 386 9.23.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 10 Related Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 10.1 Deformation Measurement by Digital Image Correlation (DIC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 10.1.1 Deformation Measurement in a High-Pressure Facility . . 391 10.2 Background Oriented Schlieren Technique (BOS) . . . . . . . . . . . . 393 10.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 10.2.2 Principle of the BOS Technique . . . . . . . . . . . . . . . . . . . . . 394 10.2.3 Application of the BOS to Compressible Vortices . . . . . . 396 10.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 A.1 Convolution with the Dirac Delta Distribution . . . . . . . . . . . . . . 437 A.2 Particle images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 A.3 Convolution of Gaussian Image Intensity Distributions . . . . . . . 437 A.4 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

1 Introduction

1.1 Historical Background Human beings are extremely interested in the observation of nature, as this was and still is of utmost importance for their survival. Human senses are especially well adapted to recognize moving objects as in many cases they mean eventual danger. One can easily imagine how the observation of moving objects has stimulated ﬁrst simple experiments with setups and tools easily available in nature. Today the same primitive behavior becomes obvious, when small children throw little pieces of wood down from a bridge in a river and observe them ﬂoating downstream. Even this simple experimental arrangement allows them to make a rough estimate of the velocity of the running water and to detect structures in the ﬂow such as swirls, wakes behind obstacles in the river, water shoots, etc. However, with such experimental tools the description of the properties of the ﬂow is restricted to qualitative statements. Nevertheless, being an artist with excellent skills and an educated observer of nature at the same time, Leonardo da Vinci, was able to prepare very detailed drawings of the structures within a water ﬂow by mere observation. A great step forward in the investigation of ﬂows was made after it was possible to replace such passive observations of nature by experiments carefully planned to extract information about the ﬂow utilizing visualization techniques. A well known promoter of such a procedure was Ludwig Prandtl, one of the most prominent representatives of ﬂuid mechanics, who designed and utilized ﬂow visualization techniques in a water tunnel to study aspects of unsteady separated ﬂows behind wings and other objects. Figure 1.1 shows Ludwig Prandtl in 1904 in front of his tunnel, driving the ﬂow manually by rotating a blade wheel [26]. The tunnel comprises an upper and lower section separated by a horizontal wall. The water recirculates from the upper open channel, where the ﬂow may be observed, back through the lower closed duct. Two-dimensional models such as cylinders, prisms and

2

1 Introduction

Fig. 1.1. Ludwig Prandtl in front of his water tunnel for ﬂow visualization in 1904.

wings can be easily mounted vertically in the upper channel, thereby extending above the level of the water surface. The ﬂow is visualized by distributing a suspension of mica particles on the surface of the water. Ludwig Prandtl studied the structures of the ﬂow in steady as well as unsteady ﬂow (at the onset of ﬂow) with this arrangement [65]. Being able to change a number of parameters of the experiment (model, angle of incidence, ﬂow velocity, steady–unsteady ﬂow) Prandtl gained insight into many basic features of unsteady ﬂow phenomena. However, at that time only a qualitative description of the ﬂow ﬁeld was possible. No quantitative data about ﬂow velocity, etc., could be achieved.

Fig. 1.2. Separated ﬂow behind wing, visualized with modern equipment in a replica of Ludwig Prandtl’s tunnel.

1.2 Principle of Particle Image Velocimetry (PIV)

3

10 pixel

0

5

0

5

0

5

0

0

5

10

15

20

25

30

35

40

Fig. 1.3. Vector map of instantaneous velocity ﬁeld corresponding to ﬁgure 1.2.

Today, a century after Ludwig Prandtl’s experiments, it is easily possible also to extract quantitative information about the instantaneous ﬂow velocity ﬁeld exactly from the same kind of images as were available to Prandtl. This is illustrated in ﬁgure 1.2. A replica of Ludwig Prandtl’s water tunnel together with a ﬂash lamp for illumination and a video camera have been employed to obtain visualization of the ﬂow by means of aluminum particles distributed on the water surface. Evaluation of this recording by methods which will be described later resulted in a vector map of the instantaneous velocity ﬁeld shown in ﬁgure 1.3. This means that the basic principles underlying the quantitative visualization technique which is the subject of this book have already been known for a long time. However, the scientiﬁc and technical progress achieved in the last 20 years in optics, lasers, electronics, video and computer techniques was necessary to further develop a technique for qualitative ﬂow visualization to such a stage that it can be employed for quantitative measurement of complex instantaneous velocity ﬁelds.

1.2 Principle of Particle Image Velocimetry (PIV) In the following, the basic features of this measurement technique, most widely named “particle image velocimetry” or “PIV”, will be described brieﬂy1 . 1

In earlier years other names such as laser speckle velocimetry, particle image displacement velocimetry etc. have been used as well.

4

1 Introduction

Light sheet optics

Mirror

Laser Light sheet

Flow with tracer particles

Illuminated particles

First light pulse at t Second light pulse at t y

Imaging optics Flow direction

x

t

Image plane t

Fig. 1.4. Experimental arrangement for particle image velocimetry in a wind tunnel.

The experimental setup of a PIV system typically consists of several subsystems. In most applications tracer particles have to be added to the ﬂow. These particles have to be illuminated in a plane of the ﬂow at least twice within a short time interval. The light scattered by the particles has to be recorded either on a single frame or on a sequence of frames. The displacement of the particle images between the light pulses has to be determined through evaluation of the PIV recordings. In order to be able to handle the great amount of data which can be collected employing the PIV technique, sophisticated post-processing is required. Figure 1.4 brieﬂy sketches a typical setup for PIV recording in a wind tunnel. Small tracer particles are added to the ﬂow. A plane (light sheet) within the ﬂow is illuminated twice by means of a laser (the time delay between pulses depending on the mean ﬂow velocity and the magniﬁcation at imaging). It is assumed that the tracer particles move with local ﬂow velocity between the two illuminations. The light scattered by the tracer particles is recorded via a high quality lens either on a single frame (e.g. on a high-resolution digital or ﬁlm camera) or on two separate frames on special cross-correlation digital cameras. After development the photo-graphical PIV recording is digitized by means of a scanner. The output of the digital sensor is transferred to the memory of a computer directly. For evaluation the digital PIV recording is divided in small subareas called “interrogation areas”. The local displacement vector for the images of the tracer particles of the ﬁrst and second illumination is determined for each in-

1.2 Principle of Particle Image Velocimetry (PIV)

5

terrogation area by means of statistical methods (auto- and cross-correlation). It is assumed that all particles within one interrogation area have moved homogeneously between the two illuminations. The projection of the vector of the local ﬂow velocity into the plane of the light sheet (two-component velocity vector) is calculated taking into account the time delay between the two illuminations and the magniﬁcation at imaging. The process of interrogation is repeated for all interrogation areas of the PIV recording. With modern charge coupled device (CCD) cameras (1000×1000 sensor elements and more) it is possible to capture more than 100 PIV recordings per minute. High-speed recording on complementary metaloxide semiconductor (CMOS) sensors even allows for acquisition in the kHzrange. The evaluation of one digital PIV recording with several thousand instantaneous velocity vectors (depending on the size of the recording, the interrogation area and processing algorithm) is of the order of a second with standard computers. If data is required at even faster rates for online monitoring of the ﬂow, dedicated software algorithms which perform evaluations of reduced precision within fractions of a second are commercially available. Before going into the details of the PIV technique, some general aspects have to be discussed in order to facilitate the understanding of certain technical solutions later on. Non-intrusive velocity measurement. In contrast to techniques for the measurement of ﬂow velocities employing probes such as pressure tubes or hot wires, the PIV technique being an optical technique works non-intrusively. This allows the application of PIV even in high-speed ﬂows with shocks or in boundary layers close to the wall, where the ﬂow may be disturbed by the presence of the probes. Indirect velocity measurement. In the same way as with laser Doppler velocimetry the PIV technique measures the velocity of a ﬂuid element indirectly by means of the measurement of the velocity of tracer particles within the ﬂow, which – in most applications – have been added to the ﬂow before the experiment starts. In two phase ﬂows, particles are already present in the ﬂow. In such a case it will be possible to measure the velocity of the particles themselves as well as the velocity of the ﬂuid (to be additionally seeded with small tracer particles). Whole ﬁeld technique. PIV is a technique which allows to record images of large parts of ﬂow ﬁelds in a variety of applications in gaseous and liquid media and to extract the velocity information out of these images. This feature is unique to the PIV technique. Aside from Doppler Global Velocimetry (DGV, also known as Planar Doppler Velocimetry) [46, 468, 470, 471], which is a new technique particularly appropriate for medium to high-speed air ﬂows, and Molecular Tagging Velocimetry (MTV) [469] all other techniques for velocity measurements only allow the measurement of the velocity of the ﬂow at a single point, however in most cases with a high temporal resolution. The spatial resolution of PIV is large, whereas the temporal resolution (frame rate

6

1 Introduction

of recording PIV images) is limited due to current technological restrictions. These features must be kept in mind when comparing results obtained by PIV with those obtained with traditional techniques. Instantaneous image capture and high spatial resolution of PIV allow the detection of spatial structures even in unsteady ﬂow ﬁelds. Velocity lag. The need to employ tracer particles for the measurement of the ﬂow velocity requires us to check carefully for each experiment whether the particles will faithfully follow the motion of the ﬂuid elements, at least to that extent required by the objectives of the investigations. Small particles will follow the ﬂow better. Illumination. For applications in gas ﬂows a high power light source for illumination of the tiny tracer particles is required in order to well expose the photographic ﬁlm or the video sensor by scattered light. However, the need to utilize larger particles because of their better light scattering eﬃciency is in contradiction to the demand to have as small particles as possible in order to follow the ﬂow faithfully. In most applications a compromise has to be found. In liquid ﬂows larger particles can usually be accepted which scatter much more light. Thus, light sources of considerably lower peak power can be used here. Duration of illumination pulse. The duration of the illumination light pulse must be short enough to “freeze” the motion of the particles during the pulse exposure in order to avoid blurring of the image (“no streaks”). Time delay between illumination pulses. The time delay between the illumination pulses must be long enough to be able to determine the displacement between the images of the tracer particles with suﬃcient resolution and short enough to avoid particles with an out-of-plane velocity component leaving the light sheet between subsequent illuminations. Distribution of tracer particles in the ﬂow. At qualitative ﬂow visualization certain areas of the ﬂow are made visible by marking a stream tube in the ﬂow with tracer particles (smoke, dye). According to the location of the seeding device the tracers will be entrained in speciﬁc areas of the ﬂow (boundary layers, wakes behind models, etc.). The structure and the temporal evolution of theses structures can be studied by means of qualitative ﬂow visualization. For PIV the situation is diﬀerent: a homogeneous distribution of medium density is desired for high quality PIV recordings in order to obtain optimal evaluation. No structures of the ﬂow ﬁeld can be detected on a PIV recording of high quality. Density of tracer particle images. Qualitatively three diﬀerent types of image density can be distinguished [31], which is illustrated in ﬁgure 1.5. In the case of low image density (ﬁgure 1.5a), the images of individual particles can be detected and images corresponding to the same particle originating from diﬀerent illuminations can be identiﬁed. Low image density requires tracking methods for evaluation. Therefore, this situation is referred to as “Particle Tracking Velocimetry”, abbreviated “PTV”. In the case of medium image density (ﬁgure 1.5b), the images of individual particles can be detected as well.

1.2 Principle of Particle Image Velocimetry (PIV)

7

Fig. 1.5. The three modes of particle image density: (a) low (PTV), (b) medium (PIV), and (c) high image density (LSV).

However, it is no longer possible to identify image pairs by visual inspection of the recording. Medium image density is required to apply the standard statistical PIV evaluation techniques. In the case of high image density (ﬁgure 1.5c), it is not even possible to detect individual images as they overlap in most cases and form speckles. This situation is called “Laser Speckle Velocimetry” (LSV), a term which has been used at the beginning of the nineteen-eighties for the medium image density case as well, as the (optical) evaluation techniques were quite similar for both situations. Number of illuminations per recording. For both photographic and digital techniques, we have to distinguish whether it is possible to store images of the tracer particles on diﬀerent frames for each illumination or whether all particle images due to the diﬀerent illuminations are stored on a single frame. Number of components of the velocity vector. Due to the planar illumination of the ﬂow ﬁeld only two (in plane) components of the velocity vector can be determined in standard two-component PIV (2C-PIV). Methods are available to extract the third component of the velocity vector as well (stereo techniques, dual-plane PIV and holographic recording [38, 39], which itself is three-dimensional). This would be labeled 3C-PIV. Both methods work in planar domains of the ﬂow ﬁeld (2D-PIV). Extension of the observation volume In the most general way an extension of the observation volume is possible by means of holographic techniques (3D-PIV) [210]. Other methods such as establishing several parallel light sheets in a volume [38] or scanning a volume in a temporal sequence [87, 225] would be referred to as 2+1D-PIV. Temporal resolution. Most PIV systems allow to record with high spatial resolution, but at relative low frame rates. However, the recent development of high-speed lasers and cameras allows time resolved measurements of most liquid and low-speed aerodynamic ﬂows.

8

1 Introduction

Spatial resolution. The size of the interrogation areas during evaluation must be small enough for the velocity gradients not to have signiﬁcant inﬂuence on the results. Furthermore, it determines the number of independent velocity vectors and therefore the maximum spatial resolution of the velocity map which can be obtained at a given spatial resolution of the sensor employed for recording. Repeatability of evaluation. In PIV full information about the ﬂow velocity ﬁeld (except the time delay between pulses and magniﬁcation at imaging) is stored at recording time at a very early stage of data reduction. This results in the interesting feature that PIV recordings can easily be exchanged for evaluation and post-processing with others employing diﬀerent techniques. The information about the ﬂow velocity ﬁeld completely contained in the PIV recording can be exploited later on in quite a diﬀerent way from that for which it had originally been planned without the need to repeat the experiment. In this section the main features of the PIV technique have been described brieﬂy to support a general understanding of its unique features. PIV oﬀers new insights in ﬂuid mechanics especially in unsteady ﬂows as it allows for capturing the whole velocity ﬁelds instantaneously. However, other quantitative ﬂow visualization techniques [19] giving information about other important physical quantities of a ﬂuid such as density, temperature, concentration, etc., are already well known and widely used, and new optical methods for the measurement of quantities on the surface of a model such as pressure or deformation are developed. Hence, a more complete experimental description of a complex ﬂow ﬁeld will be possible and available for comparison with the results of numerical calculations in future by combination of diﬀerent techniques.

1.3 Development of PIV During the Last Two Decades The development of particle image velocimetry during the past 20 years is characterized by the fact that analog recording and evaluation techniques have been replaced by digital techniques. Though these analog methods have widely contributed to the success of the PIV technique in its initial stage of development the discussion of these techniques will not be one of the main objectives of this handbook on PIV. We will rather concentrate on the description of the present state of the art of PIV. A number of sources describing the basic principles of PIV in the context of its historical development are readily available. Thus, for further information the reader is referred to the SPIE Milestone Series 99, edited by Grant in 1994 [35]. This volume comprises more than 70 original papers, ﬁrst published between 1932 and 1993. The majority of them originate from the eighties, including contributions describing the roots ofmodern PIV (i.e.

1.3 Development of PIV During the Last Two Decades

9

speckle interferometry), the early work of Meynart [42], the development of low and high image density PIV, optical correlation techniques, etc. Review articles by Lauterborn and Vogel (1984) [41] and by Adrian (1986, 1991) [30, 31], some of which are also reprinted in the Milestone Series on PIV [35], demonstrate the fast development and compilation of know-how about PIV within a decade. PIV seen from the side of optics is described in the chapter “Particle Image Velocimetry” written by Hinsch in 1993 [37], included in a book on “Speckle metrology”. This contribution is especially useful for the understanding of the optical aspects of PIV. It includes 104 references to other literature on PIV. At that time strong competition with respect to the better performance of optical and digital methods in the evaluation of PIV recordings took place. Details of the theoretical fundamentals of digital particle velocimetry can be found in the Ph.D. thesis Digital particle image velocimetry – Theory and practice published also in 1993 by Westerweel [51]. This book includes more than 100 references. A review paper “Particle image velocimetry: a review” by Grant appeared in 1997 [36] gives a summary of diﬀerent implementations of PIV illumination, recording and evaluation techniques. Many of those are not covered in this book. The paper includes 188 references. As indicated, all four publications mentioned previously include a detailed bibliography of the literature on PIV, which the reader should refer to if further details are required on special aspects of PIV exceeding the framework of this book. A further bibliography on PIV with nearly 1200 references was compiled by Adrian [32] and is available commercially. The large number of references listed in the review articles demonstrate that Particle Image Velocimetry is nowadays a well accepted tool for the investigation of velocity ﬁelds in many diﬀerent areas. This also means that a number of special implementations of the PIV technique had to be developed for such diﬀerent applications as, for example, in biology or turbomachinery [47, 49]. Another comprehensive source of information on PIV development and applications is provided through the von Karman Institute which has which has oﬀered a number of lecture series and associated monographs dedicated to PIV since 1991 [43, 44, 45]. At present, PIV is widely used in ﬂuid mechanics in the investigation of air and water ﬂows. The progress made in the recent past has brought PIV to a state that it is routinely applied in aerodynamic research and related ﬁelds. New areas of application are continuously reported. This means that today a nearly complete and nearly stable picture of the technical aspects of PIV can best be given if looking at the demands of applications in aerodynamics or in water ﬂows, where the technical problems are similar but usually less severe than in air ﬂows.

10

1 Introduction

Most of the technical problems in the application of PIV encountered in this special ﬁeld appear in other PIV applications as well. Many of the basic considerations can easily be transferred to other applications. 1.3.1 PIV in Aerodynamics The use of the PIV technique is very attractive in modern aerodynamics, because it helps to understand unsteady ﬂow phenomena as, for example, in separated ﬂows over models at high angle of attack. PIV enables spatially resolved measurements of the instantaneous ﬂow velocity ﬁeld within a very short time and allows the detection of large and small scale spatial structures in the ﬂow velocity ﬁeld. Another need of modern aerodynamics is that the increasing number and increasing quality of numerical calculations of ﬂow ﬁelds require adequate experimental data for validation of the numerical codes in order to decide whether the physics of the problem has been modeled correctly. Towards this purpose carefully designed experiments have to be performed in close cooperation with those scientists doing the numerical calculations. The experimental data of the ﬂow ﬁeld must possess high resolution in time and space in order to be able to compare them with high density numerical data ﬁelds. The PIV technique is an appropriate experimental tool for this task, especially if information about the instantaneous velocity ﬁeld is required. A PIV system for the investigation of air ﬂows in wind tunnels must be serviceable in low-speed ﬂows (e.g. ﬂow velocities of less than 1 m/s in boundary layers) as well as in high-speed ﬂows (ﬂow velocities up to 600 m/s in supersonic ﬂows with shocks). Flow ﬁelds over solid, moving or deforming models have to be investigated. The application of the PIV technique in large, industrial wind tunnels poses a number of special problems: large observation area, long distances between the observation area and the light source and the recording camera, restricted time for the measurement, and high operational costs of the wind tunnel. The description of the problems as given above leads to the deﬁnition of requirements which should be fulﬁlled when PIV is applied in aerodynamics. First of all, a high spatial resolution of the data ﬁeld is necessary in order to resolve large scale as well as small scale structures in the ﬂow. This condition directly inﬂuences the choice of the recording equipment. A second important condition is that a high density of experimental data is required for a meaningful comparison with the results of numerical calculations. Thus, the tracer particle image density (i.e. number of particle images per interrogation area) must be high. A powerful seeding generator (high concentration of tracer particles in the measuring volume in the ﬂow even at high ﬂow velocities) is needed for this purpose. As the ﬂow velocity is measured indirectly by means of the measurement of the velocity of tracer particles added to the ﬂow, the tracer particles must follow the ﬂow faithfully. This requires the use of very small tracer particles. However, small particles scatter very little light. This fact results in a third important condition for the application of PIV in aero-

1.3 Development of PIV During the Last Two Decades

11

dynamics: a powerful pulse laser is required for the illumination of the ﬂow ﬁeld. 1.3.2 Major Technological Milestones of PIV Earlier in this section some references to papers describing the general historical development of PIV have been given. In a handbook more devoted to the technical aspects of PIV it might be of even greater interest to outline the development of PIV towards its applicability in complex ﬂows in terms of the achievement of major technical milestones. The understanding of some of the technical restrictions in the application of PIV in the past and their conquest may be useful for new users of the PIV technique, in order to assess the discussion in some older publications sometimes dealing with – nowadays – “strange” looking eﬀorts to solve technical problems which no longer exist today. The selection of these milestones was done according to the technical progress in the past as experienced by the authors in their own work. Thus, the choice is a subjective one. Feasibility of modern PIV. The feasibility of employing the particle image velocimetry technique for the measurement of ﬂow velocity ﬁelds in water and even in air was demonstrated in the early eighties at the von Karman Institute in Brussels, mainly by R. Meynart [42]. At that time the evaluation methods were based on the work done in the ﬁeld of speckle interferometry (see references in [35]). Reliable high power light sources for application in air. The use of double oscillator Nd:YAG lasers (two resonators; frequency doubled, to achieve a wavelength of λ = 532 nm in visible light) allowed for the ﬁrst time the illumination of a plane in the ﬂow with laser pulses of the same, constant energy at any time delay between the two pulses as required by the experiment at repetition rates of the order of 10 Hz [59]. Alignment of the light sheet optics and image acquisition was thus facilitated considerably. Ambiguity removal. Especially with photographic recordings it was not possible in most cases to store the images of the tracer particles due to ﬁrst and second illumination on two diﬀerent recordings. Thus, the temporal sequence of the images of the tracer particles could not be distinguished. Methods to remove the ambiguity of the sign of the velocity vector had to be developed (see references in [35]). The most widely used technique was image shifting, which could be successfully applied later on even in high-speed ﬂows. By enabling the investigation of complex, unsteady 3D ﬂow ﬁelds, this development contributed considerably to the increasing interest in PIV from the side of wind tunnel users and industry. Generation and distribution of tracer particles in the ﬂow. The development of powerful aerosol generators and the know-how to distribute tracer particles of a well deﬁned size within the ﬂow homogeneously improved the particle image density and the quality of the PIV recordings considerably.

12

1 Introduction

Computer hardware. The improvement of computer hardware with respect to processor speed and larger memory still continues according to Moore’s law. Diﬃcult only a decade ago, the handling and processing of numerous mega-pixel sized PIV images has become trivial on today’s personal computers. Processing of these images is possible in fractions of a second. Improved peak detection. The impact of digital PIV was aﬀected by the limited size and resolution of the digital sensors as compared to that of photographic recording. The use of sub-pixel peak position estimation by means of the Gaussian function allowed for the determination of the displacement with further improved accuracy [174]. Thus, smaller interrogation windows could be utilized, leading to an increase of spatial resolution (number of vectors) in digital PIV. Cross correlation video camera. Today progressive scan video cameras allow users to store a pair of images of the tracer particles on separate frames for each illumination with interframing times of less than 1 μs [109]. This feature immediately solves the problem of ambiguity removal even for highspeed ﬂows. Sensor sizes of 2000 × 2000 pixels and more together with the application of cross correlation methods with superior signal-to-noise ratio at evaluation yield velocity vector ﬁelds of the same quality as was possible only with photographic recording in the past. High-speed CMOS camera. Another very recent technical improvement for PIV applications was the development of CMOS sensors with the active pixel sensor (APS) technology in which, in addition to the photodiode, a readout ampliﬁer is incorporated into each pixel. This converts the charge accumulated by the photodiode into a voltage which is ampliﬁed inside the pixel and then transferred in sequential rows and columns to further signal processing circuits. This, together with highly parallel readout electronics storage devices, allows for the recording and handling of up to a few thousand frames per second at acceptable noise levels. An additional advantage of most CMOS sensors is there ability to record images of high contrast without blooming. High-repetition rate Nd:YAG and Nd:YLF lasers. The introduction of kilo Hertz CMOS cameras has generated a need for laser light sources that can operate at frame rates of 1000’s of frames per second. As a consequence thereof, diode pumped lasers were adapted to high-speed PIV. These lasers were originally developed for materials processing and as pump sources for more complex scientiﬁc lasers. Specialized, high repetition rate, double oscillator lasers, designed speciﬁcally for high speed PIV were recently introduced. Such laser systems allow accommodating for pulse separation times shorter than the temporal distance between two successive pulses, while having acceptable beam proﬁles and stability. This allows high frame rate PIV experiments to be designed for a wide range of air and water ﬂow velocities. Microscopic PIV recording. During the past ﬁve years, signiﬁcant progress has been made in the development and application of micronresolution Particle Image Velocimetry (μPIV). Developments of the technique

1.3 Development of PIV During the Last Two Decades

13

have extended typical spatial resolutions of PIV from the order of 1 mm to the order of 1 μm. These advances have been obtained as a result of novel improvements in instrument hardware and post processing software. The utility of μPIV has been demonstrated by applying it to ﬂows in microchannels, micronozzles, BioMEMS, and ﬂow around cells. While the technique was initially developed for micro-scale velocity measurements, it has been extended to measure wall positions with tens of nanometers resolution, the deformation of hydrogels, micro-particle thermometry and infrared-PIV. Adaptive evaluation algorithms. Standard evaluation algorithms provide reliable velocity vectors. However, their accuracy is limited due to the loss of particle image pairs in complex ﬂow regions. As a straight forward approach to tackle this problem, the second interrogation window can be displaced with respect to the window in the ﬁrst image or diﬀerent window sizes can be used in combination with a direct correlation scheme [84]. More sophisticated algorithms evaluate the image repeatedly, ﬁrst with larger interrogation windows in order to ﬁnd the local mean displacements, then with smaller windows and increasingly higher spatial resolution. Many PIV algorithms additionally use the displacement information of a previous pass in order to determine the actual deformation of the interrogation volume between the illuminations and deform the secondly recorded images accordingly [159]. Theoretical understanding of PIV. At the beginning of the development of PIV the understanding of the technique was a more intuitive one. Progress was often made just by trial and error. In the past few years the theoretical understanding of the basic principles of the PIV technique has been improved considerably. Such theoretical considerations as well as simulations of the recording and evaluation process give useful information on many parameters important for the layout of an experiment utilizing PIV. In this chapter, a brief introduction to the basic principles of PIV and some of its problems and technical constraints to be kept in mind has been presented. Next, the diﬀerent topics will be described in more detail. We will start with providing the background of the most important physical principles. In the following, the mathematical background of PIV evaluation will be discussed. With this knowledge the path has been prepared for the understanding of the recording, evaluation and post-processing methods applied in PIV. Furthermore the present state of the technical development of stereo and dualplane PIV will be described – methods allowing access to the third component of the velocity vector in planar domains. In the ﬁnal chapter examples of the application of PIV will be presented, thereby explaining the speciﬁc problems experienced at each measurement due to the diﬀerent properties of the ﬂow under investigation.

2 Physical and Technical Background

2.1 Tracer Particles It is clear from the principle of PIV that this technique – in contrast to hot wire or pressure probe techniques – is based on the direct determination of the two fundamental dimensions of the velocity: length and time. On the other hand, the measurement technique is indirect as it determines the particle velocity instead of the ﬂuid velocity. Therefore, ﬂuid mechanical properties of the particles have to be examined in order to avoid signiﬁcant discrepancies between ﬂuid and particle motion. 2.1.1 Fluid Mechanical Properties A primary source of error is the inﬂuence of gravitational forces if the densities of the ﬂuid ρ and the tracer particles ρp do not match. Even though it can be neglected in many practical situations, we will derive the gravitationally induced velocity Ug from Stokes’ drag law in order to introduce the particle’s behavior under acceleration. Therefore, we assume spherical particles in a viscous ﬂuid at a very low Reynolds number. This yields: Ug = d2p

(ρp − ρ) g 18μ

(2.1)

where g is the acceleration due to gravity, μ the dynamic viscosity of the ﬂuid and dp is the diameter of the particle. In analogy to equation (2.1), we can derive an estimate for the velocity lag of a particle in a continuously accelerating ﬂuid: Us = Up − U = d2p

(ρp − ρ) a 18μ

(2.2)

where Up is the particle velocity. The step response of Up typically follows an exponential law if the density of the particle is much greater than the ﬂuid density:

16

2 Physical and Technical Background

t Up (t) = U 1 − exp − τs

(2.3)

with the relaxation time τs given by: τs = d2p

ρp 18μ

d2 . ν

(2.4)

If the ﬂuid acceleration is not constant or Stokes drag does not apply (e.g. at higher ﬂow velocities) the equations of the particle motion become more diﬃcult to solve, and the solution is no longer a simple exponential decay of the velocity. Nevertheless, τs remains a convenient measure for the tendency of particles to attain velocity equilibrium with the ﬂuid. The result of equation (2.3) is illustrated in ﬁgure 2.1, where the time response of particles with diﬀerent diameters is shown for a strong deceleration in an air ﬂow. When applying PIV to liquid ﬂows the problems of identifying particles with matching densities are usually not severe, and solid particles with adequate ﬂuid mechanical properties can often be found. Usually their size can easily be determined before suspension into the liquid and will not change afterwards. Some of the commonly used tracers are listed in table 2.1 and 2.3. However, early applications of PIV have already demonstrated that diﬃculties arise in providing high quality seeding in gas ﬂows compared to applications in liquid ﬂows [55, 58, 64, 68, 70]. The problems are similar to those faced by Laser Doppler Velocimetry. From equation (2.2) it can be seen that due to the diﬀerence in density between the ﬂuid and the tracer particles, the diameter of the particles should be very small in order to ensure good tracking of the ﬂuid motion. On the other hand, the particle diameter should not be too small as light scattering properties have also to be taken into account as will be shown in the following section. Therefore, it is clear that a compromise has to be found. This problem is discussed extensively in the literature Umax 1.0

10 μm 5 μm 1 μm

0.8

0.6

0.4

0.2

Umin

0.0

0

0.0002

0.0004

Time [sec]

0.0006

Fig. 2.1. Time response of oil particles with diﬀerent diameters in a decelerating air ﬂow.

2.1 Tracer Particles

17

U [m/s] U= 359 m/s

360

350

340

Interrogation volume (in flow field)

330

320 U= 317 m/s -30

-20

-10

0

10

X [mm]

Fig. 2.2. Comparison of the experimental (PIV) and theoretical (dashed line for a particle diameter: 1.7 μm) result for the change of the U -component of the instantaneous velocity vector along a line in the ﬂow ﬁeld about a bluﬀ cylinder when crossing a shock.

[71, 72, 74, 75, 76]. The most commonly used seeding particles for PIV investigations of gaseous ﬂows are listed in table 2.3. For most of our applications in aerodynamics, we used oil or DEHS particles which were generated by means of a Laskin nozzle (see ﬁgure 2.10), with a mean diameter of the aerosol droplets around 1 μm. It is well known from LDV measurements in gas ﬂows that the size and the distribution of the tracer particles may change during the travel from the aerosol generator to the test section, where the measurements take place. It is therefore advisable to gain information about the particles and especially about the velocity lag directly from the observation area [55, 60, 70]. The result of one of these examinations of this problem is presented in ﬁgure 2.2. The U -component of the instantaneous ﬂow velocity along one line of a PIV recording of a transonic ﬂow ﬁeld is plotted. The measured ﬂow velocity U drops from 359 m/s in front of a shock to 317 m/s within a distance of 8 mm. The real extent of the shock (≈ 10−3 mm) cannot be resolved due to the ﬁnite size of the interrogation areas (≈ 2−3 mm diameter, when projected back into the ﬂow ﬁeld, indicated as circles in ﬁgure 2.2). However, a calculation of the velocity lag of particles with a diameter of 1.7 μm, carried out according to a theoretical approach, considering the eﬀects of compressibility, deformation of droplets and high Reynolds numbers [67] yields a similar relation between velocity and distance as measured with PIV (compare the dashed line in ﬁgure 2.2). Hence, it does not make much sense to utilize tracer particles with a diameter much smaller than 1 μm for this experiment, because of the fact that during the evaluation of the PIV recording the velocities are averaged within

18

2 Physical and Technical Background

each interrogation area. Smaller tracer particles would only be necessary if a higher spatial resolution in the vicinity of the shock were required. Besides the bias error which might be introduced, large particle sizes can cause data drop-out in critical areas of the ﬂow ﬁeld as for example in vortex cores, shear ﬂows or boundary layers. In some cases the velocity lag due to centrifugal forces in a vortex leads to only small errors of measurement but due to its integration from vortex generation to the light sheet the particle density sometimes becomes too low for adequate measurements. 2.1.2 Light Scattering Behavior In this section, some of the most important characteristics of light scattered by tracer particles will be summarized. Since the obtained particle image intensity and therefore the contrast of the PIV recordings is directly proportional to the scattered light power, it is often more eﬀective and economical to increase the image intensity by properly choosing the scattering particles than by increasing the laser power. In general it can be said that the light scattered by small particles is a function of the ratio of the refractive index of the particles to that of the surrounding medium, the particles’ size, their shape and orientation. Furthermore, the light scattering also depends on polarization and observation angle. For spherical particles with diameters, dp , larger than the wavelength of the incident light λ, Mie’s scattering theory can be applied. A detailed description and discussion is given in the literature [13]. Figures 2.3 and 2.4 show the polar distribution of the scattered light intensity for oil particles of diﬀerent diameters in air with a wavelength λ of 532 nm according to Mie’s theory. The intensity scales are in logarithmic scale and are plotted so that the intensity for neighboring circles diﬀer by a factor of 100. The Mie scattering can be characterized by the normalized diameter, q, deﬁned by: πdp . q= λ If q is larger than unity, approximately q local maxima appear in the angular distribution over the range from 0◦ to 180◦. For increasing q the ratio of

Light

o

0

10

3

10

10

5

7

10

o

180

Fig. 2.3. Light scattering by a 1 μm oil particle in air.

2.1 Tracer Particles

Light

o

0

19

180o

Fig. 2.4. Light scattering by a 10 μm oil particle in air. Intensity scales as in ﬁgure 2.3.

forward to backward scattering intensity will increase rapidly. Hence, it would be advantageous to record in forward scatter, but, due to the limited depth of ﬁeld, recording at 90◦ is most often used. In general, the light scattered paraxially (i.e. at 0◦ or 180◦ ) from a linearly polarized incident wave is linearly polarized in the same direction and the scattering eﬃciency is independent of polarization. In contrast, the scattering eﬃciency for most other observation angles strongly depends on the polarization of the incident light. Furthermore, for observation angles in the range from 0◦ to 180◦ the polarization direction can be partially turned. This is particularly important if image separation or image shifting depending on polarization of the scattered light has to be applied. Therefore, such a technique works well only for certain particles, for example 1 μm diameter oil particles in air. There is a clear tendency for the scattered light intensity to increase with increasing particle diameter. However, if we recall that the number of local maxima and minima is proportional to q, it becomes clear, that the function of the light intensity versus particle diameter is characterized by rapid oscillations if only one certain observation angle is taken into account. One implication is that particle images of high intensity do not always mean that the particle crossed the center of the measurement volume. Hence, a determination of the out-of-plane particle displacement by analyzing particle positions in a light sheet with known intensity proﬁle by the image intensity is usually not feasible. When averaging over a range of observation angles, which is determined by the observation distance and the recording lens aperture, the intensity curve is smoothed considerably. The average intensity roughly increases with q 2 , and as already mentioned above, the scattering eﬃciency strongly depends on the ratio of the refractive index of the particles to that of the ﬂuid. Since the refractive index of water is considerably larger than that of air, the scattering of particles in air is at least one order of magnitude more powerful compared to particles of the same size in water. Therefore, much larger particles have to be used for water ﬂow experiments, which can mostly be accepted since the density matching of particles and ﬂuid is usually better. In ﬁgure 2.5 to 2.7 the normalized scattered intensity of diﬀerent diameter glass particles in water according to the Mie theory are shown at λ = 532 nm.

20

2 Physical and Technical Background

Fig. 2.5. Light scattering by a 1 μm glass particle in water.

Fig. 2.6. Light scattering by a 10 μm glass particle in water.

Fig. 2.7. Light scattering by a 30 μm glass particle in water.

It can be seen from all Mie scattering diagrams, that the light is not blocked by the small particles but spread in all directions. Therefore, for a large number of particles inside the light sheet massive multiscattering occurs. Then the light which is imaged by the recording lens is not only due to direct illumination but also due to fractions of light, which have been scattered by more than one particle. In the case of heavily seeded ﬂows this considerably increases the intensity of individual particle images, because the intensity of directly recorded light – at 90◦ to the incident illumination – is orders of magnitude smaller than that scattered in the forward scatter range. One interesting implication is that not only larger particles can be used to increase the scattering eﬃciency but also a larger number density of the particles. However, two problems limit this eﬀect from being intensively used. First, the background noise and therefore the noise on the recordings will increase

2.2 Particle Generation and Supply

21

signiﬁcantly as well. Second, if – as is usually the case – polydisperse particles (i.e. particles of diﬀerent sizes) are used, it is ﬁnally not certain whether the number of visible particles has been increased by simply increasing the number of very large particles. Since images of larger particles clearly dominate the results of PIV evaluation, it would be diﬃcult to give reliable estimates on the eﬀective particle size and the corresponding velocity lag.

2.2 Particle Generation and Supply 2.2.1 Seeding of Liquids Descriptions of seeding particles and their characteristics have been given in many scientiﬁc publications. In contrast to that, little information can be found in the literature on how to practically supply the particles into the ﬂow under investigation. Sometimes seeding can be done very easily or does not even have to be done. The use of natural seeding is sometimes acceptable, if enough visible particles are naturally present to act as tracers for PIV. In almost all other work it is desirable to add tracers in order to achieve suﬃcient image contrast and to control particle size. For most liquid ﬂows, seeding can easily be done by suspending solid particles into the ﬂuid and mixing them in order to ensure a homogeneous distribution. A number of diﬀerent particles which can be used for ﬂow visualization, LDV and PIV are listed in table 2.1 for liquid and in table 2.3 for gas ﬂows. For our experiments in oil and water ﬂows we used hollow coated glass spheres of approximately 10 μm diameter as shown in ﬁgure 2.8 for two diﬀerent magniﬁcations. They oﬀer good scattering eﬃciency and a suﬃciently small velocity lag. Tracer particles small compared to λ. When decreasing the observation ﬁeld size and increasing the optical resolution of the investigation, the tracer particle diameters have obviously to be decreased also. In the Rayleigh scattering regime, where the particle diameter dp is much smaller than the wavelength of light, dp λ, the amount of light scattered by a particle varies as d−6 [3]. Since the diameter of the ﬂow-tracing particles must be small p Table 2.1. Seeding materials for liquid ﬂows. Type

Material

Mean diameter in μm

Solid

Polystyrene Aluminum ﬂakes Hollow glass spheres Granules for synthetic coatings Diﬀerent oils Oxygen bubbles

10 2 10 10 50 50

Liquid Gaseous

– – – – – –

100 7 100 500 500 1000

22

2 Physical and Technical Background

Fig. 2.8. Micrographs of silver coated hollow glass spheres: ×500 and ×5000.

enough for the particles not to interact with the ﬂow being measured, they can frequently be of the order of 50–100 nm. Their diameters are then 1/10 to 1/5 the wavelength of green light, λgreen = 532 nm, and are therefore approaching the Rayleigh scattering criteria. This places signiﬁcant constraints on the image recording optics, making it extremely diﬃcult to record particle images. One solution to this imaging problem is to use epi-ﬂuorescence imaging to record light emitted from ﬂuorescently-labeled particles in which an optical wavelength-speciﬁc long-pass ﬁlter is used to remove the background light, leaving only the light ﬂuoresced by the particles. This technique has been used successfully in liquid ﬂows many times to record images of 200 to 300 nmdiameter ﬂuorescent particles [32, 406]. While ﬂuorescently-labeled particles are well-suited for micro-PIV studies in liquid ﬂows, they are not readily applicable to gaseous ﬂows for several reasons. First, commercially available ﬂuorescently-labeled particles are generally available as aqueous suspensions. A few manufacturers do have dry ﬂuorescent particles available, but only in larger sizes, > 7 μm. In principle, the particle-laden aqueous suspensions can be dried, and the particles subsequently suspended in a gas ﬂow but this often proves problematic because the electrical surface charge that the particles easily acquire allows them to stick to the ﬂow boundaries and to each other. Successful results have been reported using theatrical fog and smoke – both non-ﬂuorescent [285]. Furthermore, the emission decay time of many ﬂuorescent molecules is on the order of several nanoseconds, which may cause streaking of the particle images for high-speed ﬂows. 2.2.2 Seeding of Gases In gas ﬂows, the increased diﬀerence in density between the gaseous bulk ﬂuid and the particles can result in a signiﬁcant velocity lag. Health considerations are also more important since the experimentalists may inhale seeded air, for example in wind tunnels with an open test section. The particles which are often used are not easy to handle because many liquid droplets tend to evaporate rather quickly. Solid particles are diﬃcult to disperse and tend to

2.2 Particle Generation and Supply

23

Table 2.2. Seeding materials for gas ﬂows. Type

Material

Solid

Polystyrene Alumina Al2 O3 Titania TiO2 Glass micro-spheres Glass micro-balloons Granules for synthetic coatings Dioctylphathalate Smoke Diﬀerent oils Di-ethyl-hexyl-sebacate (DEHS) Helium-ﬁlled soap bubbles

Liquid

Mean diameter in μm 0.5 0.2 0.1 0.2 30 10 1

– – – – – – – < 0.5 – 0.5 – 1000 –

10 5 5 3 100 50 10 1 10 1.5 3000

agglomerate. Very often the particles must be injected into the ﬂow shortly before the gaseous medium enters the test section. The injection has to be done without signiﬁcantly disturbing the ﬂow, but in a way and at a location that ensures homogeneous distribution of the tracers. Since the existing turbulence in many test setups is not strong enough to mix the ﬂuid and particles suﬃciently, the particles have to be supplied from a large number of openings. Distributors, like rakes consisting of many small pipes with a large number of tiny holes, are often used. Therefore, particles which can be transported inside small pipes are required. A number of techniques are used to generate and supply particles for seeding gas ﬂows [71, 72, 73, 74, 75, 76]. Dry powders can be dispersed in ﬂuidized beds or by air jets. Liquids can be evaporated and afterwards precipitated in condensation generators, or liquid droplets can directly be generated in atomizers. Atomizers can also be used to disperse solid particles suspended in evaporating liquids [77], or to generate tiny droplets of high vapor pressure liquids (e.g. oil) that have been mixed with low vapor pressure liquids (e.g. alcohol) which evaporate prior to entry in the test section. For seeding wind tunnel ﬂows condensation generators, smoke generators and monodisperse polystyrene or latex particles injected with water-ethanol are most often used for ﬂow visualization and LDV. Oil droplet seeding of air ﬂows. For most of the PIV measurements in air ﬂows, Laskin nozzle generators and oil have been used. These particles oﬀer the advantage of not being toxic, they stay in air at rest for hours and do not change in size signiﬁcantly under various conditions. In recirculating wind tunnels they can be used for a global seeding of the complete tunnel volume or for a local seeding of a stream tube by a seeding rake with a few hundred tiny holes. A technical description of such an atomizer is given below. The aerosol generator consists of a closed cylindrical container with two air inlets and one aerosol outlet (ﬁgure 2.9). Four air supply pipes – mounted at

24

2 Physical and Technical Background

Pressurized air (separate pipe) Pressurized air

(max.1bar)

Valves Particles

Pressure adjustment

Impactor plate Laskin nozzles

Olive oil

Fig. 2.9. Oil seeding generator.

Fig. 2.10. Sketch of a Laskin nozzle.

the top – dip into vegetable oil or similar liquid inside the container. They are connected to one air inlet by a tube and each has a valve. The pipes are closed at their lower ends (see ﬁgure 2.10). Four Laskin nozzles, 1 mm in diameter, are equally spaced on the circumference of each pipe [69]. A horizontal circular impactor plate is placed inside the container, so that a small gap of about 2 mm is formed by the plate and the inner wall of the container. The second air inlet and the aerosol outlet are connected directly to the top. Two gauges measure the pressure on the inlet of the nozzles and inside the container, respectively. Compressed air with 0.5 to 1.5 bar pressure diﬀerence with respect to the outlet pressure is applied to the Laskin nozzles and creates air bubbles within the liquid. Due to the shear stress induced by the tiny sonic jets, small droplets are generated and carried inside the bubbles towards the oil surface. Big particles are retained by the impactor plate; small particles escape through the gap and reach the aerosol outlet. The number of particles can be controlled by the four valves at the nozzle inlets. The particle concentration can be decreased by an additional air supply via the second air inlet. The mean size of the particles generally depends on the type of liquids being atomized, but is only slightly dependent on the operating pressure of the nozzles. Vegetable oil is the most commonly used liquid, since oil droplets are believed to be less unhealthy than many other particles. However, any kind of seeding particles which are toxic or cannot be dissolved in water should not be inhaled. Most vegetable oils (except cholesterol-free oils) lead to polydisperse distributions with mean diameters of approximately 1 μm [67].

2.2 Particle Generation and Supply

25

Powder-based seeding of air ﬂows. In cases where the stability of the seeding material cannot be guaranteed due to increased temperatures or reactive environments, droplet-based seeding is no longer feasible. In these cases, seeding based on solids must be used. Metal oxide powders are especially well suited for this purpose due to their inertness, high melting point and rather low cost. Table 2.3 lists titanium dioxide, alumina and silica powders, with corresponding micrographs shown in ﬁgure 2.11 and ﬁgure 2.12. A controlled dispersion of these powders is more challenging than for liquid materials since the powders have a strong tendency to form agglomerates especially for small grain sizes in the sub-micron range. Therefore the seeding device has to either break up the agglomerates or remove them from the aerosol prior to delivery into the facility such as through the use of a cyclone separator [313, 314]. Another approach to de-agglomerate the bulk seed material was proposed by Wernet & Wernet [77]: the inter-particle forces responsible for the

Fig. 2.11. Micrographs of 800 nm mono-disperse SiO2 spheres (left), porous SiO2 spheres (right).

Fig. 2.12. Left: Ground Al2 O3 powder for seeding reactive ﬂows [377], right: TiO2 powder for seeding supersonic ﬂows [313].

26

2 Physical and Technical Background B Helium Air

Soap

Air

Screw thread H Cap l

h

d D

Fig. 2.13. Schematic drawing of an oriﬁce type nozzle for the generation of He-ﬁlled soap bubbles [63].

agglomeration can be directly inﬂuenced by controlling the acidity of liquid suspensions of the seed material. They suggest the use of alumina/water or alumina/ethanol dispersions with a pH value of 1 which can be dispersed using liquid atomization. The solid seed material remains after evaporation of the carrier liquid. Depending on the relative mass-ﬂow rates between seeded and unseeded ﬂows, the use of liquid-particle suspensions is not always feasible especially when the ﬂow is disturbed due to evaporation cooling and/or changes in the reactive chemistry. In this case the aerosol has to be created directly from the dry powder. A common approach is to aerate the powder inside a vertical tube from below resulting in a ﬂuidized bed. The ﬂow rate through the seeder is chosen just large enough to ﬂuidize the bed of particles carrying smaller particles into the region above the bed (known as freeboard) towards the exit oriﬁce and from there into the facility under investigation. Figure 2.14 shows a simple ﬂuidized bed seeding device for use in elevated pressure applications [377, 378]. This generator has two noteworthy features: the strong shear ﬂow present in a sonic oriﬁce at the exit serves to break up larger agglomerates. The size of this oriﬁce is chosen to ensure suﬃcient ﬂow rate to aerate the powder. The second feature is a switchable by-pass line which maintains constant mass ﬂow rates into a test facility even when no seeding is required. The following gives some recommendations for the successful operation of ﬂuidized bed seeders: • The seeding powder should be kept dry, preferably by heating the material to remove excess moisture before ﬁlling the seeding device. Dry air or nitrogen should be used to operate the seeder.

2.2 Particle Generation and Supply

27

Fig. 2.14. Fluidized bed seeding device for high pressure applications.

• Short supply lines between seeder and facility should be used to prevent the formation of agglomerates. If possible, additional carrier air should be used to reduce the relative seeding concentrations. • Frequent agitation of seeding system reduces the chance of channel formation within the ﬂuidized bed. • The mechanical interaction of the seed material with small brass spheres (100–500 μm) added to the ﬂuidized bed also helps to break up agglomerates. This conﬁguration is referred to as two-phase ﬂuidized bed. Soap bubble seeding for air ﬂows1 . The ﬁnite scattering eﬃciency of any tracer particle is usually the limiting factor when increasing the ﬁeld of view (FOV) in an actual PIV measurement. Oil droplets with diameters of d = 1 μm for example restrict the FOV to areas around 1000 × 750 mm2 when standard Nd:YAG lasers with pulse energies up to 300 mJ are used. One possibility to overcome this limitation is to use larger tracer particles. Here, however, the mass of the particles is critical, since for PIV ideally neutrally buoyant particles are required. A well established method to provide neutrally buoyant particles with dimensions around 1–3 mm is the generation 1

This description of the soap bubble seeding is based on the work of Bosbach et al. section 9.6.

28

2 Physical and Technical Background

Soap pressure tank

Fine regulation valves

Safety valve Shut off valve with venting Pressurized helium Helium

Pressurized air Shut off valve

Air

Fig. 2.15. Scheme of the air, helium and soap supply for a bubble generator[63].

of helium-ﬁlled soap bubbles, where a helium ﬁlling of the particles compensates for their gravity. Diﬀerent possibilities to generate such bubbles are discussed in detail in [63]. For the large-scale Rayleigh-B´enard experiments presented in section 9.6, an oriﬁce type nozzle has been used, as depicted in ﬁgure 2.13. Basically, gaseous helium is blown through a central pipe, which is mounted coaxially into a second pipe. Through this second pipe the bubble ﬂuid solution (BFS) is driven at a predeﬁned ﬂow rate. Thus, small bubbles are generated, their separation from the tubes being driven by the air ﬂow through the additional air-pipes. Generation of neutrally buoyant bubbles relies on a proper adjustment of helium pressure, air-pressure, BFS ﬂow rate, and the distance between nozzle cap and tubes, h (see ﬁgure 2.13). The BFS typically consists of a mixture of water, glycerin and soap. For providing the nozzle with the gases and the BFS, both a commercially available generator, see ﬁgure 2.15, as well as a home-built generator were used. Their function is basically the storage of the BFS, and the regulation of the BFS ﬂow as well as the helium and air-pressure.

2.3 Light Sources 2.3.1 Lasers Lasers are widely used in PIV, because of their ability to emit monochromatic light with high energy density, which can easily be bundled into thin light sheets for illuminating and recording the tracer particles without chromatic aberrations. In ﬁgure 2.16 a typical conﬁguration of a laser is shown. Generally speaking, as shown in ﬁgure 2.16, every laser consists of three main components. The laser material consists of an atomic or molecular gas, semiconductor or solid material.

2.3 Light Sources

29

L Laser

Laser material

M

beam

P

Fig. 2.16. Schematic diagram of a laser.

Pump energy

The pump source excites the laser material by the introduction of electromagnetical or chemical energy. The mirror arrangement, i.e., the resonator allows an oscillation within the laser material. In the following we will describe the principle of gas lasers and give an overview of the lasers used in PIV. It is well known from quantum mechanics that each atom can be brought into various energy states by three elementary kinds of interaction with electromagnetic radiation. This can be illustrated in an energy level diagram, as shown in ﬁgure 2.17 for a hypothetical atom with only two possible energy states. An excited atom at level E2 usually drops back to the state E1 after a very short, but not exactly deﬁned period of time and emits the energy E2 − E1 = h ν in the form of a randomly directed photon (h and ν are Plank’s constant and the frequency respectively). This process is called spontaneous emission. However, if, on the other hand, a photon with “appropriate” frequency ν impinges on an atom, two eﬀects are possible. Either – in the case of absorption – an atom in the state E1 can receive the energy h ν, that is it becomes raised to E2 ; or the incident photon can stimulate an atom in the excited E2 state into a speciﬁc, non-spontaneous, transition to E1 . Then, in addition to the incident photon, a second one is emitted in phase with the former. The impinging wave therefore is coherently ampliﬁed (stimulated emission).

t

t+τ

t

t+τ

t+τ

t

E2

E2

E2

E1

E1

E1

Absorption E 2 -E 1=h ν

Spontaneous emission hν

Stimulated emission 2 hν

Fig. 2.17. Elementary kinds of interactions between atoms and electromagnetic radiation.

30

2 Physical and Technical Background

For large numbers of atoms, one of the two processes – absorption or stimulated emission – predominates. In case of population density inversion, i.e., N2 > N1 [atoms/m3 ], stimulated emission predominates, otherwise, N1 > N2 , absorption is favored. Since the laser can only operate if a population inversion is forced to take place (N2 > N1 ), external energy has to be transferred to the laser material because atoms usually exist in their ground state. This is achieved by diﬀerent pump mechanisms depending on the kind of laser material. Solid laser materials are generally pumped by electromagnetic radiation, semiconductor lasers by electronic current, and gas lasers by collision of the atoms or molecules with electrons and ions. It should be noted that in a system which consists of only two energy states, as described so far, no population inversion can be achieved, because when the number of atoms N2 in level E2 equals the number N1 in level E1 , absorption and stimulated emission are equally likely and the material will become transparent at the frequency ν = (E2 − E1 )/h. In other words, the number of transitions from the upper level E2 to the lower level E1 and vice versa are on average the same. Hence, at least three energy levels of the laser medium are essential to achieve population inversion. However, a three level system is not very eﬃcient because a fraction of more than 50% of the atoms of the system has to be excited in order to amplify an impinging photon. This means that the energy needed for the excitation of this fraction is lost for the ampliﬁcation. In the case of a four level laser the lower laser level E2 does not coincide with the basic level E1 and therefore remains unoccupied at room temperature. In this way it is easier to achieve the population inversion and a four level laser requires substantially less pumping power. This is illustrated in ﬁgure 2.18. If for instance state E4 is achieved by optical pumping at frequency ν according to hν = E4 − E1 , then a rapid non-radiative transition to the upper laser level E3 occurs. The atoms remain in this metastable state E3 for a relatively long interim period before they drop down to the unoccupied lower laser level E2 . As a consequence of population inversion through energy transfer by the pump mechanism spontaneous emission occurs in all directions which causes excitation of further neighboring atoms. This initiates a rapid increase of stimulated emission and therefore of radiation in a chain reaction. In the case of a cylindrical shape of laser material, the rapid increase of radiation occurs in a deﬁned direction because the ampliﬁcation increases with increasing length of the laser medium. Within an optical resonator (mirror arrangement) the laser material can be arranged to form an oscillator. The simplest way to achieve this is to place the material between two exactly aligned mirrors. In this case, a photon which impinges randomly on one of the mirror surfaces is reﬂected and ampliﬁed in the laser material again. This process will be repeated and generates an avalanche of light which increases exponentially with the number of reﬂections, ﬁnally resulting in a stationary

2.3 Light Sources E4

nonradiative transition

pump transition E1

E3

upper laser level

E2

pump level nonradiative transition

pump level

E3

31

upper laser level

laser action

laser action E2 pump transition

ground state

E1

lower laser level nonradiative transition ground state

Fig. 2.18. Level diagrams of three (left) and four (right) level lasers.

process. In other words, standing waves are produced for a resonator length corresponding to the condition mλ (2.5) 2n where n is the refractive index, m an integer number, and L the resonator length. Since the frequency ν according to the transition ν h = E2 − E1 does not correspond to exactly one wavelength, but rather to a spectrum of a certain band width Δν depending on the transition time τ of the process, these conditions can be fulﬁlled by diﬀerent wavelengths λ or frequencies ν and the resonator can oscillate in many axial modes with distinct frequencies. Consecutive modes are separated by a constant diﬀerence Δν = c/(2 L n), wherein c is the speed of light. Moreover, the cross-section of the laser beam can be divided into several ranges oscillating out of phase with intermediate node lines, that is, diﬀerent transverse modes can be sustained as well (see ﬁgure 2.19). Their occurrence depends on the resonator design and alignment. The lowest order transverse mode TEM00 (TEM = transverse electric mode; index = node in X- and Y -direction) is most commonly used, because it produces a beam with uniform phase and a Gaussian intensity distribution along the beam cross-section. There are various types of resonators with diﬀerent mirror curvatures. The confocal resonator, shown in ﬁgure 2.16, is particularly stable and easy to adjust. Hemispherical resonators use one planar and one concave mirror, and critical resonators use two planar mirrors. Critical resonators oﬀer the advantage of having no beam waist inside the laser rod and therefore using its whole volume for ampliﬁcation. However, they are sensitive for thermal lens eﬀects and misalignment. L=

32

00

2 Physical and Technical Background

01

10

11

02

21

Fig. 2.19. Examples of diﬀerent transverse modes.

In gas lasers, which are usually used for continuous operations (CW = continuous wave), free electrons are accelerated by an electrical ﬁeld resulting in an excitation of the gaseous medium. The plasma tube in which the excitation takes place is closed by Brewster windows (plates tilted at the polarization angle). Therefore, these lasers emit linearly polarized light. In the case of optically pumped solid-state lasers, the light of the rod-shaped ﬂash lamp is concentrated on the laser rod by a cylindrical mirror with an elliptical cross-section. In lasers, which basically consist of luminescence diodes, two polished, parallel surfaces work as resonator mirrors. The laser light perpendicular to the p-n junction is more divergent due to the lower aperture width. Some of the popular lasers for PIV are brieﬂy described below in order to provide an overview of the abilities and limitations of the diﬀerent systems. Helium-neon lasers (He-Ne lasers λ = 633 nm) are the most common as well as the most eﬃcient lasers in the visible range. He-Ne lasers have mostly been used for the optical evaluation of photographic images in PIV rather than for illuminating the ﬂow. The power of commercial models ranges from less than 1 mW to more than 10 mW. The laser transitions take place in the neon atom. During the electrical discharge within a gaseous mixture of helium and neon, the helium atoms are excited by collisions with electrons. In the second step the upper laser levels of the Ne atoms are populated through collisions with metastable helium atoms. This is the main pump mechanism. The major properties of the He-Ne laser that are important for the evaluation of PIV images are the coherence of the laser beam and the Gaussian distribution of the laser beam intensity (TEM00 ). If necessary, the beam quality can further be improved by means of spatial ﬁlters. Copper-vapor lasers (Cu lasers λ = 510 nm, 578 nm) are the most important neutral metal vapor lasers for PIV. The wavelengths of Cu lasers are within the yellow and green spectrum (table 2.3). These lasers are characterized by their high average power (typically 1 – 30 W) and an eﬃciency of up to one percent. Continuous wave operations are not possible due to the long life-span of the lower laser level. During pulse operations, repetition rates within the kHz-range can be achieved. This type of laser has been intensively developed during the past decade because the coper-vapor laser is an important pump source for dye lasers. Whereas most lasers are cooled, metal vapor lasers need thermal insulation since the operating temperature for vaporizing the metal can typically reach 1500◦C. Two electrodes are located at the ends of a thermally insulated

2.3 Light Sources

33

Table 2.3. Properties of a commercial Cu laser. Wavelength: Average power: Pulse energy: Pulse duration: Peak power: Pulse frequency: Beam diameter: Beam divergence:

510.6 nm and 578.2 nm 50 W 10 mJ 15 ns – 60 ns < 300 kW 5 kHz – 15 kHz 40 mm 0.6 · 10−3 rad

ceramic tube with a pulsed charge burning in between them. For improving discharge quality, neon is added as the buﬀer gas at a pressure of around 3000 Pa. Table 2.3 lists some of the key properties of a typical Cu laser. Argon-ion lasers ( Ar+ lasers λ = 514 nm, 488 nm) are gas lasers, similar to the He-Ne lasers described above. In argon lasers, very high currents have to be achieved for ionization and excitation. This is technologically much more complicated compared to He-Ne lasers. Typically the eﬃciency of these lasers is of the order of a tenth of a percent. Large versions of these lasers can supply over 100 W in the blue-green range and 60 W in the near ultraviolet range. Emission is produced at several wavelengths through the use of broadband laser mirrors. Individual wavelengths can be selected by means of Brewster prisms in the laser resonator. The individual wavelengths can be adjusted by rotating the prism. The most important wavelengths are 514.5 and 488.0 nm. Nearly all conventional inert gas ion lasers supply TEM00 . Despite the extreme load on the tubes resulting from the high currents, product lives of several thousand operating hours can be achieved. Since argon lasers are frequently used for LDV measurements, they are often found in ﬂuid mechanics laboratories. In PIV they can easily be used for low-speed water investigations. Semiconductor lasers oﬀer the advantage of being very compact. The laser material is typically 1 cm long and has a diameter of the order of 1 mm. The total eﬃciency of commercial diode laser pumped Nd:YAG and ND:YLF systems is around 7%. Since heating is considerably reduced, these types of pumped lasers supply a very good beam quality of over 100 mW in the TEM00 mode during continuous operations. Diode lasers are of interest for PIV because of two reasons: The high eﬃciency allows the production of Nd:YLF lasers with high average and peak power needed for high-speed applications. Due to their ability to generate beams with excellent quality, semiconductor lasers can also be used as seed lasers for an improvement of the coherence length of ﬂash lamp pumped Nd:YAG lasers for use in holographic PIV. A particularly interesting variant is the combination of a diode-pumped laser oscillator and a ﬂash lamp pumped ampliﬁer. Together with other optical components, like vacuum-pinholes and phase-conjugated mirrors, this concept oﬀers very good beam properties, but at the same time, its initial purchase costs are high.

34

2 Physical and Technical Background

Ruby lasers ( Cr3+ lasers λ = 694 nm), historically the very ﬁrst lasers, use ruby crystal rods containing Cr3+ ions as the active medium. They are pumped optically by means of ﬂash lamps. As already mentioned above, the ruby laser is a three level system which has the disadvantage that approximately 50% of the atoms must be excited before population inversion takes place. The high pumping energy needed can usually only be achieved during pulse mode operations. The wavelength of the ruby laser is 694.3 nm. Like other solid-state lasers, the ruby laser can also be operated normally or in Qswitched mode. (For details about Q-switched mode see the next paragraph.) The ruby laser is of particularly interest for PIV because it delivers very high pulse energies and its beam is well suited for holographic imaging because of its good coherence. Its disadvantage is that the low repetitive rates hamper the optical alignment and its light is emitted at the edge of the visible spectrum. Photographic ﬁlms are usually not sensitive to red light while modern CCD cameras are usually optimized for smaller wavelengths. Neodym-YAG lasers (Nd:YAG lasers λ = 1064 nm and λ = 532 nm) are the most important solid-state laser for PIV in which the beam is generated by Nd3+ ions. The Nd3+ ion can be incorporated into various host materials. For laser applications, YAG crystals (yttrium-aluminum-garnet) are commonly used. Nd:YAG lasers have a high ampliﬁcation and good mechanical and thermal properties. Excitation is achieved by optical pumping in broad energy bands and non-radiative transitions into the upper laser level. Solid-state lasers can be pumped with white light as a result of the arrangement of the atoms which form a lattice. The periodic arrangement leads to energy bands formed by the upper energy levels of the single atoms. Therefore, the upper energy levels of the system are not discrete as in the case of single atoms, but are continuous. As already mentioned, the Nd:YAG laser is a four-level system which has the advantage of a comparably low laser threshold. At standard operating temperatures, the Nd:YAG laser only emits the strongest wavelength, 1064 nm. In the relaxation mode the population inversion takes place as soon as the threshold is reached, with this threshold value depending on the design of the laser cavity. In this way, many successive laser pulses can be obtained during the pump pulse of the ﬂash lamp. By including a quality switch (Q-switch) inside the cavity the laser can be operated in a triggered mode. The Q-switch has the eﬀect of altering the resonance characteristics of the optical cavity. If the Q-switch is operated, allowing the cavity to resonate at the most energetic point during the ﬂash lamp cycle, a very powerful laser pulse, the giant pulse, can be achieved. Q-switches normally consist of a polarizer and a Pockels cell, which change the quality of the optical resonator depending on the Pockels cell voltage. The Q-switched mode in general is of higher interest and is usually used in PIV. Although Q-switches can be used to generate more than one giant pulse out of one resonator, PIV lasers are mostly designed as double oscillator systems. This enables the user to adjust the separation time between the two illuminations of the tracer particles independently of the pulse strength. The

2.3 Light Sources

35

beam of Q-switch lasers is linearly polarized. For PIV, and many other applications, the fundamental wavelength of 1064 nm is frequency-doubled using special crystals. (For details about these KD*P crystals see the next section.) After separation of the frequency-doubled portion, approximately one third of the original light energy is available at 532 nm. Nd:YAG lasers are usually driven in a repetitive mode. Since the optical properties of the laser cavity change with changing temperature, good and constant beam properties will only be obtained at nominal repetition rates and ﬂash lamp voltage. Due to thermal lensing, the beam quality which is very often poor compared to other laser types, decreases signiﬁcantly when, for example, single pulses are used. This is not that critical for telescopic resonator arrangements but very important for modern critical resonator systems. The coherence length of pulsed Nd:YAG lasers is normally of the order of a few centimeters. For holographic recording, lasers with a narrow spectral bandwidth have to be used. This is usually done by injection from a smaller semiconductor laser into the cavity by a partially reﬂecting mirror. Then, the laser pulse builds up from this small seeding pulse of narrow bandwidth, resulting in coherence lengths of 1 or 2 meters. However, very precise laser timing and temperature control for the primary cooling circuit are required for this purpose. Neodym-YLF lasers (Nd:YLF lasers λ = 1053 nm and λ = 526 nm) are used for an increasing number of applications, including high-speed PIV techniques, which require a reliable high average-power laser source that enables eﬃcient frequency conversion to visible wavelengths. For this and other applications, several variants of the diode-pumped solid state lasers have been developed, and of these, the Nd:YLF (neodymium: yttrium lithium ﬂuoride) laser produces the highest pulse energy and average power, with repetition rates ranging from single pulse up to approximately 10 kHz. They can be operated at diﬀerent fundamental wave lengths. The fundamental wavelength most frequently used in PIV is λ = 1053 nm, which is turned into the visible range at λ = 526 nm by frequency doublers in a similar way as in case of the Nd:YAG lasers. A fundamental wavelength of λ = 1047 nm can be obtained by rotating a polarizer inside the resonator. However, the advantage of the λ = 1053 nm wavelength is its ability of an ampliﬁcation by neodymphosphate glass. 2.3.2 Features and Components of PIV Lasers Commercially available Nd:YAG laser rods are up to 150 mm long and have diameters up to 10 mm. Typically pulse energies of 400 mJ or more can be achieved out of one oscillator. In this case, more than one ﬂash lamp and critical resonators with plane mirror surfaces have to be used. The price paid for the high output power that can be achieved with those resonators is that the beam proﬁle tends to be very poor: hot spots and diﬀerent ring modes can often be found. In order to improve the beam proﬁle, output mirrors with a reﬂectivity that varies with the radius are frequently used. However, even

36

2 Physical and Technical Background

with these mirrors the beam proﬁle is sometimes very poor, even though it is speciﬁed to be 80% Gaussian in the near and 95% in the far ﬁeld. Two laser systems of the same manufacturer often have diﬀerent beam properties depending on the laser rod properties and the alignment of the laser. Since a good beam proﬁle is essential for PIV (see chapter 3) it must be speciﬁed not only in the near and in the far ﬁeld – as most manufacturers do – but also in the mid-ﬁeld at a distance of 2–10 m from the laser. The description of the beam intensity distribution should not only be based on a good ﬁt to a Gaussian distribution, but also on the minimum and maximum energy in order to ensure a hole-free intensity distribution without hot spots. In ﬁgure 2.20, the intensity proﬁles along the light sheet thickness measured at four diﬀerent distances from the laser are shown. The light sheet optics used for this experiment are shown in ﬁgure 2.28. The peak value of the distribution has been adjusted close to full scale for each position (1.8 m, 3.3 m, 4.3 m, 5.8 m). It can be seen that the thickness of the light sheet increases slowly with the distance from the laser. A small side peak is visible at every position but seems to disappear at 5.8 m. The ﬂuctuations in the distribution are minimized at position three (4.3 m). When assessing these light sheet proﬁles it has to be taken into account that the loss of correlation during the evaluation of PIV recordings is mainly inﬂuenced by the light sheet intensity distribution at recording (see chapter 3). The light energy contained in the side peak will be lost in most situations, because a very small ﬂow component in the lateral direction would displace particles from bright towards dark areas and would therefore lead to only one illumination of the tracer particles. For ﬂow ﬁelds without any signiﬁcant out-of-plane velocity component the light sheet can be focused more precisely and a better, more Gaussian-like, intensity proﬁle can be obtained in the out-of-plane direction. However, even in those cases the intensity proﬁle along the light sheet height strongly depends on the laser beam properties. If data is lost in some regions of the observation area due to insuﬃcient illumination, the result of the whole measurement may become questionable. In other words, pulse energy is one value required for the speciﬁcation of light sheet illumination, but deﬁnitely not the most important one. In table 2.4, some critical parameters are listed which should be speciﬁed when assessing a double oscillator Nd:YAG laser. The laser system and all the speciﬁcations are made with respect to a wavelength of 532 nm and a repetition rate of 10 Hz of the two pulses, unless otherwise stated. All trigger signals of the laser should be TTL compatible. In ﬁgure 2.21, a laser system with telescopic resonators is shown. It oﬀers only 2 × 70 mJ pulse energy but has the advantage of a good and stable beam proﬁle. Based on the authors’ experience it can be said that the beam proﬁle of this laser has been stable during more than ten years of application. In ﬁgure 2.22 a laser system with critical resonators is shown. These systems typically have 150–450 mJ per pulse. The disadvantages of critical resonators have already been described above. In general it has to be mentioned

2.3 Light Sources

37

250

Grey levels

200

Distance 1.8 m

Distance 3.3 m

Distance 4.3 m

Distance

150

100

50

0 250

Grey levels

200

5.8 m

150

100

50

0

0

200

400

Pixels

600 0

200

400

600

Pixels

Fig. 2.20. Evolution of the light sheet proﬁle with increasing distance from the laser.

that beam properties very much depend on manufacturers’ know-how and the tuning of each individual laser. In the following section a short description of basic Nd:YAG and Nd:YLF laser components is given. The ﬂash lamp pumping chamber contains the laser crystal rod and a linear ﬂash lamp which are sealed into their respective mountings with O-rings. These two components are surrounded by ceramic reﬂectors which provide eﬃcient optical pumping of the laser rod. Glass ﬁlter plates absorb the ultraviolet radiation from the ﬂash lamp. Flash lamp pumped Nd:YAG laser are still the most common for conventional PIV applications where high pulse energies at moderate repetition rates are required. Diode-pumped lasers, on the other hand, oﬀer better stability and higher reliability than ﬂash lamp pumped laser systems. The diode pumping chambers can be divided into two major categories: end pumping and side pumping conﬁgurations. End pumped designs oﬀer the advantage of reaching best quality by reshaping an astigmatic diode-laser beam

38

2 Physical and Technical Background

Table 2.4. Properties and speciﬁcations of modern Nd:YAG PIV-laser systems. Repetition ratea Pulse energy for each of two pulses Roundness at 8 m from laser outputb Roundness at 0.5 m from laser outputc Spatial intensity distribution at 8 m from laser outputd Spatial intensity distribution at 0.5 m from laser outputd Linewidth Power drift over 8 hourse Energy stabilityf Beam pointing stabilityg Deviation from collinearity of laser beams Beam diameter at laser output Divergenceh Jitter between two following laser pulses Delay between two laser pulses Resolution Working temperatures Cooling wateri Power requirements

10 Hz 320 mJ 75% 75% < 0.2 < 0.2 1.4 cm−1 < 5% < 5% 100 μrad < 0.1 mm/m 9 mm 0.5 mrad 2 ns 0 to 10 ms 5 ps 15◦ –35◦ C 10◦ C – 25◦ C 220–240V , 50 Hz

a

and integral fractions of 10 Hz, eg. 5 Hz, 2.5 Hz etc. ratio between two perpendicular axis (major and minor axis). c if laser beam is elliptical, major axis of both oscillators must be parallel. d |(Imax − Imin )|/|(Imax + Imin )|, with I being the peak intensity in the spatial distribution limited by the diameter at half maximum for both oscillators. e without readjustment of phase-matching for ambient temperatures of 18◦ C< T < 35◦ C. f shot to shot, peak to peak, 100% of shots. g RMS, on 200 alternating pulses at the focal plane of a 2 m lens. h full angle on 200 pulses at e−2 of the peak, 85% of total energy. i secondary circuit, 10 l/min pressure, 1.5 to 3 bar. b

into a beam with a circular symmetry. Their disadvantage is the greater complexity and the fact that they can not easily be scaled to high average-power output. Main advantages of the side pumped conﬁgurations are simplicity, reliability and physical and thermal tolerance. Most modern high-speed lasers used for PIV contain side pumped designs which are producing high average power and good beam quality needed for the eﬃcient generation of the green wavelengths at λ = 526 nm. The output mirror has a plane surface in most cases with a partially reﬂecting coating facing into the cavity. The opposite plane surface has an

2.3 Light Sources Front mirror

Pump cavity

Pockels cell

Telescope

39

Back mirror

Output beam aperture

Tube dump

Polarizer

45 -Rotator

Crystal doubler housing

Wavelength separator

Fig. 2.21. Double oscillator laser system with telescopic resonators.

antireﬂection coating. In some cases, the output mirror has a curved surface with a variable reﬂectivity coating decreasing from the center to the edges. The back mirror has a highly reﬂective surface facing towards the cavity. Usually, this mirror has a slightly curved surface. The decay of energy inside a resonator can be described by introducing a quality factor or Q-factor. This factor can be changed with the help of the Q-Switch. The Q-switch normally consists of a polarizer plate, a temperature stabilized Pockels cell crystal and a beam path correcting prism. It is driven by high voltages and is used to release the energy stored in the laser rod in a giant pulse by rapidly changing the resonance conditions. Its principle is as follows: during the beginning of the ﬂash lamp pulse the (voltage dependent) birefringence of the Pockels cell makes it act as a quarterwave plate. The polarization of the light that passes the polarizer is rotated by 90◦ on its way through the crystal towards the mirror and back towards the polarizer. The reﬂected light is then rejected by the polarizer. Therefore, no laser oscillation and no light ampliﬁcation will take place. When the energy stored in the laser rod reaches a maximum the Pockels cell voltage is altered and the polarization of passing light will no longer be changed. As a result, the laser oscillation begins immediately and the energy stored is extracted in a pulse of only a few nanoseconds duration. Since the birefringence of the Pockels cell is also temperature dependent, the Pockels cell is generally temperature stabilized. An intra-cavity telescope is used to avoid high order modes within the cavity. It also compensates for thermal lensing eﬀects inside the laser rod. A second harmonic generator (SHG) is a nonlinear crystal used for the frequency doubling of the Nd:YAG laser emission. Simply speaking, it converts infrared light of a wavelength of 1064 nm into visible green light of 532 nm. The

40

2 Physical and Technical Background

Back mirror

Pockels Pump cell Glan cavity polarizer

Variable reflectivity output mirror 90 -Rotator

532nm beam shutter

Mirror

45 -Rotator

Output beam aperture

Crystal doubler Phase angle housing adjustment

Beam dump with heat sink Dielectric Beam Dichroic polarizer dump mirror

Fig. 2.22. Double oscillator laser system with critical resonators.

process of frequency doubling takes place only when the crystal is oriented such that the direction of propagation of the pump beam is at a speciﬁc angle to the crystal axis. This condition is known as phase matching. Therefore, the crystal can usually be angle tuned by the user. Since the refractive index and therefore the actual phase matching changes with temperature, the crystal has to be temperature stabilized to ensure stable conversion eﬃciencies. As most crystals used are hygroscopic the heating of the crystals should not be switched oﬀ in order to protect its surface from moisture. The crystal most commonly used is called KD*P. It can be cut in two diﬀerent orientations (Type I or Type II) and has to be chosen depending on the ﬁnal conﬁguration of the laser. For Type I crystals, the incident laser light has to be linearly, typically vertically or horizontally, polarized. The frequency doubled light emerges with a polarization which is orthogonal to that of the pump radiation. This type of doubler is used for PIV laser systems with two polarization directions as, for example, shown in ﬁgure 2.22. In order to generate green light of identical polarization one Type II crystal is generally used. Therefore, the infrared laser light must have two polarization components. The second harmonic will then have one polarization direction parallel to one of both original components depending on the orientation of the crystal. In order to provide two components of the incident laser light, its linear polarization is turned by an angle of 45◦ using a polarization rotator. A polarization rotator is a crystal which continuously rotates the polarization angle of linearly polarized light as it propagates through it. The rate of rotation is dependent on the material, its thickness and the wavelength. A 45◦ rotator will be used when a Type II doubling crystal is used. A 90◦ rotator might be used in front of the beam combination optics, if two oscillators of identical orientation are used (see ﬁgure 2.22). A prism harmonic separator can be used to separate the second harmonic wave by deﬂecting it into an energy dump. Two energy dumps are provided;

2.3 Light Sources

41

Intracavity shutter Pump cavity

Crystal doubler housing

Pockels cells

Fig. 2.23. Double oscillator high-speed PIV laser system with intra cavity doubler.

one for the fundamental and one for the third harmonic wave. These separators are most eﬃcient when used with only one polarization direction, as the reﬂection losses at the prism surfaces are lower for one polarization (see ﬁgure 2.21). A dichroic mirror has maximum reﬂectivity for one given wavelength. The fundamental and any unwanted harmonic waves pass through such a mirror and can therefore be steered into an external energy dump. Figure 2.23 shows the sophisticated optical layout of a modern high-speed PIV laser system, in which the two infrared beams are combined. This principle is patent pending by one of the leading PIV laser manufactures and oﬀers an eﬀective mechanism, the intra-cavity doubling. The working principle is as follows: Both infrared beams are reﬂected by the output mirror that is coated to reﬂect infrared and transmit green laser light. Both infrared beams are reﬂected back towards the second harmonic generator (SHG) by this same output mirror. They pass through the second harmonic generator. The beams are then reﬂected by a back mirror through the second harmonic generator again and then are transmitted through the output mirror out of the laser head. Therefore, the beams are always co-linear since they are combined within the combined resonators. This doubling mechanism provides a high conversion from infrared to green. The two separate cavities produce two pulses with the same pulse width given by identical resonator lengths. The properties and speciﬁcations of a high-speed Nd:YLF PIV-laser system are listed in table 2.5. Pulse width and pulse energies vary with the chosen repetition rate. Typical values of pulse energy and pulse width are shown in ﬁgure 2.24 and ﬁgure 2.25 respectively. 2.3.3 White Light Sources Even though most PIV investigations are performed using laser light sheets, white light sources might also be used. Due to the ﬁnite extension of these

42

2 Physical and Technical Background

Table 2.5. Properties and speciﬁcations of a high-speed Nd:YLF PIV-laser system. Repetition ratea Pulse energy for each of two pulses Roundness at 4 m from laser outputb Roundness at 0.5 m from laser outputc Spatial intensity distribution at 4 m from laser outputd Spatial intensity distribution at 0.5 m from laser outputd Pulse width at 1 kHz Power drift over 8 hours Energy stabilitye Deviation from co-linearity of laser beams Beam diameter at laser output Divergencef Spatial mode Cooling water

0.01 − 10 kHz 15–3 mJ 75% 75% < 0.2 < 0.2 < 180ns < 5% < 1% < 0.1 mm/m 2 mm < 3 mrad multi-mode, M2 < 6 10 – 25◦ C

a

repetition rate per cavity ratio between two perpendicular axis (major and minor axis). c if laser beam is elliptical, major axis of both oscillators is parallel. d |(Imax − Imin )|/|(Imax + Imin )|, with I being the peak intensity in the spatial distribution limited by the diameter at half maximum for both oscillators. e RMS after 10 minutes warm up at 2 kHz. f full angle at e−2 of the peak, 85% of total energy. b

sources and since white light cannot be collimated as well as monochromatic light, they clearly have some disadvantages. On the other hand, the spectral output of sources like Xenon lamps is well suited for use with CCD cameras because of their similar spectral sensitivity. Systems are commercially available which can easily be triggered and oﬀer a repetition rate that matches the video rate. Two ﬂash lamps can be linked by optical ﬁber bundles in order to

Energy vs. repetittion rate

Energy per pulse, mJ

15 10

5 0 0

1000

2000 3000 4000 Repetition rate, Hz

5000

6000

Fig. 2.24. Pulse energy versus repetition rate of a modern double oscillator highspeed PIV laser system.

2.4 Light Sheet Optics

43

Pulse width, ns

250 200 150 100 50 0 0 200 500

2000 4000 6000 Repetition rate, Hz

8000

10000

Fig. 2.25. Pulse width versus repetition rate of a modern double oscillator highspeed PIV laser system.

achieve short pulse separation times. If the outputs of the ﬁbers are arranged in line, the generation of a light sheet is considerably simpliﬁed. The main advantage of these white light sources is – aside from reduced cost – that their application is not hampered by laser safety rules.

2.4 Light Sheet Optics This section treats the optics for the illumination of the particles by a thin light sheet. Therefore, we describe three diﬀerent lens conﬁgurations which have been used during various experiments. Rules for the calculation of the light sheet intensity distribution are not given herein. The reason for this is that geometric optics rules are already suﬃcient for a general layout of the chosen lens conﬁguration. They do not require a special description and can readily be found in any book on optics [11]. On the other hand, more sophisticated calculations based on Gaussian optics usually require some assumptions, which are valid only for exceptional cases. Computer programs can be used in order to predict further parameters such as the light sheet thickness at the beam waist where the theoretical (geometrical) thickness is zero, but their description is beyond the scope of this book. Optical ﬁbers for beam delivery can be used to improve the handling of the system or for experimental situations where mirror systems would not be feasible. For CW lasers, a variety of systems are available and for their use we refer to the manufacturers. Since pulsed lasers and white light sources have only limited repetition rates, ﬁber bundles can be used for the combination of two sources for shorter pulse separation times. New developments can already deliver more than 10 mJ per pulse and further improvements can be expected [374]. However, the use of optical ﬁbers will always be associated with a certain loss in intensity.

44

2 Physical and Technical Background Side view Light sheet

-50 mm 200 mm

500 mm Top view Thickness

-50 mm 200 mm

500 mm

Fig. 2.26. Light sheet optics using three cylindrical lenses (one of them with negative focal length).

Higher pulse energies can be delivered by an articulated mirror arm allowing controlled laser illumination and encapsulated beams. This beam delivery is particularly eﬀective in cases when the laser needs to be kept away from the experiment due to space constraints or hostile conditions. The addition of compact light sheet optics results in a versatile and self-contained illumination system that can safely deliver high power laser pulses. In most cases it allows traversing the light sheet with six degrees of freedom. The essential element for the generation of a light sheet is a cylindrical lens. When using lasers with a suﬃciently small beam diameter and divergence – like for example argon-ion lasers – one cylindrical lens can be suﬃcient to generate a light sheet of appropriate shape. For other light sources – like for example Nd:YAG lasers – a combination of diﬀerent lenses is usually required in order to generate thin light sheets of high intensity. At least one additional lens has then to be used for focusing the light to an appropriate thickness. Such a conﬁguration is shown in ﬁgure 2.26, where also a third cylindrical lens has been added in order to generate a light sheet of constant height. The reason why a diverging lens has been used ﬁrst is that focal lines should be avoided. In high power pulse lasers focal points have to be avoided, as otherwise the air close to the focal point will be ionized. Focal lines usually do not ionize the air but dust particles might be burnt if the area in the vicinity of the line is not covered or evacuated. In both cases acoustic radiation will occur and the beam properties will change signiﬁcantly. For the light sheet shown in ﬁgure 2.26, the position of minimum thickness is given by the beam divergence of the light source and the focal length of the cylindrical lens on the right hand side, for example at a distance of 500 mm from the last lens for the conditions illustrated in ﬁgure 2.26. The combination of a cylindrical lens together with two telescope lenses makes the system more versatile. This is shown in ﬁgure 2.27 where spherical lenses have been used, because they are in general easier to manufacture especially if short focal length lenses are required. The height of the light sheet shown in ﬁgure 2.27 is mainly given by the focal length of the cylindrical lens

2.4 Light Sheet Optics

45

Side view Light sheet -100 mm 200 mm 350 mm Top view Thickness

-100 mm 200 mm 350 mm

Fig. 2.27. Light sheet optics using two spherical lenses (one of them with negative focal length) and one cylindrical lens.

in the middle. A diverging lens – negative focal length – could also be used, however, since the focal line has a relatively large extension this conﬁguration can also be used for pulsed lasers. The adaptation of the light sheet height has to be done by changing the cylindrical lens. The adjustment of the thickness can easily be done by shifting the spherical lenses with respect to each other. The use of spherical lenses in general does not allow light sheet height and thickness to be changed independently. This can be done by the conﬁguration shown in ﬁgure 2.28. Additionally this setup allows for the generation of light sheets which are thinner than the beam diameter at every location. It therefore enables the generation of light sheets which are already thin shortly after the last lens. With this arrangement the thickness can be held constantly small. However, the energy per unit area of these conﬁgurations is high. Therefore, the critical region close to the focal line has to be covered in order to avoid reﬂections by dust or seeding particles. Using a diverging cylindrical lens ﬁrst would solve those problems, but the combination shown in ﬁgure 2.28 has the advantage of imaging of the beam proﬁle from a certain position in front of the lens to the observation area while keeping its properties constant. Simple geometric considerations can be used for these lens combinations to determine from which position the laser beam has been imaged and, if the development of the beam proﬁle of the laser is known, this information can be used to optimize the light sheet intensity distribution. For lasers with a critical beam proﬁle this can improve the valid data yield, because the light sheet intensity distribution especially in the out-of-plane direction is essential for the quality of the measurement (see section 3). In ﬁgure 2.20 on page 37 the evolution of a light sheet proﬁle generated by a lens conﬁguration similar to that shown in ﬁgure 2.28 has been shown as a function of the distance from the laser. A few general rules should also be given here. Uncoated lens surfaces in 2 air exhibit a slight reﬂectivity of [(n − 1)/(n + 1)] . Since this value is of the order of 4% for common lenses the losses due to the reﬂection can usually be accepted. However, these reﬂections can cause damage if they are focused close to other optical components. In most cases this can easily be avoided by the right orientation of the lenses as demonstrated in ﬁgure 2.29.

46

2 Physical and Technical Background

Side view Light sheet 100 mm 200 mm 60 mm Top view Thickness Fig. 2.28. Light sheet optics using three cylindrical lenses.

100 mm 200 mm 60 mm

Furthermore cases c and d in ﬁgure 2.29 should be used in order to minimize aberrations. For other situations it might be possible to tilt the lens slightly in order to avoid reﬂections on to other lenses or towards the laser or even into the resonator.

2.5 Volume Illumination of the Flow Whereas most conventional PIV investigations utilize light sheet illumination, they are typically not a practical source of illumination for micro-ﬂows, due to a lack of optical access along with signiﬁcant diﬀraction in light sheet forming optics. Consequently, the ﬂow must be volume illuminated, leaving two choices for the visualization of the seed particles – with an optical system whose depth of ﬁeld exceeds the depth of the ﬂow being measured or with an optical system whose depth of ﬁeld is small compared to that of the ﬂow. Both of these techniques have been used in various implementations of μPIV. Cummings [281] uses a large depth of ﬁeld imaging system to explore electrokinetic and pressure-driven ﬂows. The advantage of the large depth of ﬁeld optical system is that all particles in the ﬁeld of view of the optical system are well-focused Not recommended a)

b)

Recommended c)

d)

Fig. 2.29. General considerations on the orientation of lenses inside the light sheet optics.

2.5 Volume Illumination of the Flow

47

and contribute to the correlation function comparably. The disadvantage of this scheme is that all knowledge of the depth of each particle is lost, resulting velocity ﬁelds that are completely depth-averaged. For example in a pressuredriven ﬂow where the velocity proﬁle is expected to be parabolic with depth, the fast moving particles near the center of the channel will be focused at the same time as the slow moving particles near the wall. The measured velocity will be a weighted-average of the velocities of all the particles imaged. Cummings [281] addresses this problem with advanced processing techniques that will not be covered here. The second choice of imaging systems is one whose depth of ﬁeld is smaller than that of the ﬂow domain, as shown in Figure 2.30. The optical system will then sharply focus those particles that are within the depth of ﬁeld δ of the imaging system while the remaining particles will be unfocused – to greater or lesser degrees – and contribute to the background noise level. Since the optical system is being used to deﬁne thickness of the measurement domain, it is important to characterize exactly how thick the depth of ﬁeld, or more appropriately, the depth of correlation zcorr , is. The distinction between depth of ﬁeld and depth of correlation is an important although subtle one. The depth of ﬁeld refers to distance a point source of light may be displaced from the focal plane and still produce an acceptably focused image whereas the depth of correlation refers to how far from the focal plane a particle will contribute signiﬁcantly to the correlation function. The depth of correlation can be calculated starting from the basic principles of how small particles are imaged [300, 408].

Fig. 2.30. Schematic showing the geometry for volume illumination particle image velocimetry. The particles carried by the ﬂow are illuminated by light coming out of the objective lens (i.e. upward).

48

2 Physical and Technical Background

2.6 Imaging of Small Particles 2.6.1 Diﬀraction Limited Imaging This section provides a description of diﬀraction limited imaging, which is an eﬀect of practical signiﬁcance in optical instrumentation, and of particular interest for PIV recording. In the following we will restrict our description of imaging by considering only one-dimensional functions. If plane light waves impinge on an opaque screen containing a circular aperture they generate a far-ﬁeld diﬀraction pattern on a distant observing screen. By using a lens – for example an objective in a camera – the far ﬁeld pattern can be imaged on an image sensor close to the aperture without changes. However, the image of a distant point source (e.g. a small scattering particle inside the light sheet) does not appear as a point in the image plane but forms a Fraunhofer diﬀraction pattern even if it is imaged by a perfectly aberration-free lens [11]. A circular pattern, which is known as the Airy disk, will be obtained for a low exposure. Surrounding Airy rings can be observed for a very high exposure. Using an approximation (the Fraunhofer approximation) for the far ﬁeld it can be shown that the intensity of the Airy pattern represents the Fourier transform of the aperture’s transmissivity distribution [10, 18]. Taking into account the scaling theorem of the Fourier transform, it becomes clear that large aperture diameters correspond to small Airy disks and small apertures to large disks as can be seen in ﬁgure 2.31. The Airy function is equivalent to the square of the ﬁrst order Bessel function. Therefore, the ﬁrst dark ring, which deﬁnes the extension of the Airy disk, corresponds to the ﬁrst zero of the ﬁrst order Bessel function shown in ﬁgure 2.32. The Airy function represents the impulse response – the so-called point spread function – of an aberration-free lens. We will now determine the diameter of the Airy disk ddiﬀ , because it represents the smallest particle image that can be obtained for a given imaging conﬁguration. In ﬁgure 2.32 the value of the radius of the ring and therefore of the Airy disk can be found for a given aperture diameter Da and wavelength λ:

Fig. 2.31. Airy patterns for a small (left hand side) and a larger aperture diameter (right hand side).

2.6 Imaging of Small Particles

49

1.0

0.8

Airy function Gaussian approximation

0.6 I/ Imax 0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

x/x0

I(x) =0 Imax

⇒

Fig. 2.32. Normalized intensity distribution of the Airy pattern and its approximation by a Gaussian bell curve.

ddiﬀ = 1.22 2x0

with x0 =

λ . Da

If we consider imaging of objects in air – the same media on both sides of the imaging lens – the focus criterion is given by (see ﬁgure 2.33): 1 1 1 + = z0 Z0 f

(2.6)

where z0 is the distance between the image plane and lens and Z0 the distance between the lens and the object plane. Together with the deﬁnition of the magniﬁcation factor M=

z0 Z0

the following formula for the diﬀraction limited minimum image diameter can be obtained: (2.7) ddiﬀ = 2.44 f#(M + 1) λ where f# is the f-number, deﬁned as the ratio between the focal length f and the aperture diameter Da [10]. In PIV, this minimum image diameter ddiﬀ will only be obtained when recording small particles – of the order of a few microns – at small magniﬁcations. For larger particles and/or larger magniﬁcations, the inﬂuence of geometric imaging becomes more and more dominant. The image of a ﬁnite-diameter particle is given by the convolution of the point spread function with the geometric image of the particle. If lens aberrations can be neglected and the point spread function can be approximated by the

50

2 Physical and Technical Background

Y Image plane f

Object Z plane

f y

Z0

Z0

Fig. 2.33. Geometric image reconstruction.

Airy function, the following formula can be used for an estimate of the particle image diameter [53]: dτ = (M dp )2 + d2diﬀ . (2.8) This expression is dominated by diﬀraction eﬀects and reaches a constant value of ddiﬀ when the size of the particle’s geometric image M dp is considerably smaller than ddiﬀ . It is dominated by the geometric image size for geometric image sizes considerably larger than ddiﬀ where dτ ≈ M dp In practice the point spread function is often approximated by a normalized Gaussian curve also shown in ﬁgure 2.32 and deﬁned by: I(x) x2 = exp − 2 (2.9) Imax 2σ √ where the parameter σ must be set to σ = f# (1 + M )λ 2 /π, in order to approximate diﬀraction limited imaging. This approximation is particularly useful because it allows a considerable simpliﬁcation of the mathematics encountered in the derivation of modulation transfer functions, which also includes other kinds of optical aberrations of the imaging lens as will be described later. In practice there are two good reasons for optimizing the particle image diameter: First, an analysis of PIV evaluation shows that the error in velocity measurements strongly depends on the particle image diameter (see e.g. section 5.5.2). For most practical situations, the error is minimized by minimizing both the image diameter dτ and the uncertainty in locating the image centroid or correlation peak centroid respectively. Second, sharp and small particle images are particularly essential in order to obtain a high particle image intensity Imax , since at constant light energy scattered by the tracer particle the light energy per unit area increases quadratically with decreasing image areas (Imax ∼ 1/d2τ ). This fact also explains why increasing the particle diameter not always compensates for insuﬃcient laser power.

2.6 Imaging of Small Particles

51

Equation (2.8) shows that for a range of particle diameters greater than the wavelength of the scattered light (dτ λ), the diﬀraction limit becomes less important and the image diameter increases nearly linearly with increasing particle diameter. Since the average energy of the scattered light increases with (dp /λ)2 for particles with a diameter greater than the wavelength (Mie’s theory), the image intensity becomes independent of the particle diameter, as both the scattered light and the image area increase with d2p . As can be seen in ﬁgure 2.34, a point in the object plane generates a sharp image at only one deﬁned position in the image space, where the rays transmitted by diﬀerent parts of the lens intersect. This point of intersection and therefore of best focusing has been determined in equation (2.6). If the distance between the lens and the image sensor is not perfectly adjusted, the geometric image is blurred and its diameter can also be determined by geometric optics. This blur of images due to misalignment of the lens does not depend on diﬀraction or lens aberrations. However, the minimum image diameter ddiﬀ which can be obtained due to diﬀraction is commonly used also as the acceptable diameter of the geometric image (Δdg in ﬁgure 2.34). Therefore, the particle image diameter obtained by equation (2.8) can be used to estimate the depth of ﬁeld δZ using the following formula [28]: δZ = 2f# ddiﬀ (M + 1)/M 2 .

(2.10)

Some theoretical values for the diﬀraction limited imaging of small particles (dp ≈ 1 μm) are shown in table 2.6 (calculated with a wavelength of λ = 532 nm and a magniﬁcation of M = 1/4). It can be seen that a large aperture diameter is needed to get suﬃcient light from each individual particle within the light sheet, and to get sharp particle images, because – as already shown in ﬁgure 2.31 – the size of the diﬀraction pattern can be decreased by increasing the aperture diameter. Unfortunately, a big aperture diameter Object plane

Image plane

Z0

z0

Δ dg Da

Zb

Δ dg Zf

δz

Fig. 2.34. Out-offocus imaging.

52

2 Physical and Technical Background

Table 2.6. Theoretical values for diﬀraction limited imaging of small particles (λ = 532 nm, M = 1/4, dp = 1 μm). f# = f /Da dτ [ μm] δZ [ mm] 2.8 4.0 5.6 8.0 11 16 22

4.7 6.6 9.1 13.0 17.8 26.0 35.7

0.5 1.1 2.0 4.2 7.8 16.6 31.4

yields a small focal depth which is a signiﬁcant problem when imaging small tracer particles. Since lens aberrations become more and more important for an increasing aperture, they will be considered next. 2.6.2 Lens Aberrations In analogy to linear system analysis the performance of an optical system can be described by its impulse response – the point spread function – or by the highest spatial frequency that can be transferred with suﬃcient contrast. This upper frequency – the resolution limit – can be obtained by the reciprocal value of the characteristic width of the impulse response. According to this, the physical dimension is the reciprocal of a length; typically it is interpreted as the number of line pairs per millimeter ( lps/mm) that can be resolved. The traditional means of determining the quality of a lens was to evaluate its limit of resolution according to the Rayleigh criterion: two point sources were said to be “barely resolved” when the center of one Airy disk falls on the ﬁrst minimum of the Airy pattern of the other point source. This means that the theoretical resolution limit ρm is the reciprocal value of the radius of the Airy disk [11]: ρm =

2 ddiﬀ

=

1 . 1.22 f#(M + 1)λ

(2.11)

Another useful parameter in evaluating the performance of an optical system is the contrast or image modulation deﬁned by the following equation: Mod =

Imax − Imin . Imax + Imin

(2.12)

The measurement of the ratio of image modulation Mod for varying spatial frequencies yields the modulation transfer function (MTF). The modulation transfer function has become a widely used means of specifying the performance of lens systems and photographic ﬁlms.

2.6 Imaging of Small Particles

53

In practice, an approximation of the modulation transfer function can be obtained by an inverse Fourier transformation of the point spread function. The Gaussian approximation given by equation (2.9) and shown in ﬁgure 2.32 greatly simpliﬁes this transformation. In the following we continue to simplify the description of imaging by considering only one-dimensional functions. The Fourier transformation FT of a one-dimensional Gaussian function is given by: √ x2 1 FT √ exp − ⇐⇒ σ 2π exp −2π 2 σ 2 r2 (2.13) 2σ σ 2π where σ determines the width of the Gaussian curve and r represents the variable for the spatial frequency. Figure 2.35 shows a plot of image modulation versus spatial frequency for three apertures of a hypothetical lens system without spherical aberration as obtained by the inverse transformation of the Gaussian approximation of the Airy function. As shown in table 2.6, the minimum image diameter decreases with decreasing f-numbers. As a consequence, high spatial frequencies can only be recorded at small f-numbers and for a given spatial frequency r, small f-numbers yield better contrast compared to large f-numbers. Although the shape of these curves only roughly approximates the shape of MTFs of real lens systems, the qualitative behavior can clearly be seen. However, taking lens aberrations into account results in major changes of the MTFs, especially when using small f-numbers. This can be seen in ﬁgure 2.36 where the measured values of a high quality 100 mm lens are presented together with a Gaussian curve ﬁtted to the value measured at the highest frequency. These values of the modulation transfer are often given in the data sheets of a lens system for diﬀerent f-numbers and magniﬁcations or can easily be estimated when studying diagrams of the modulation versus the image height. From experience these values can be used for a rough estimation of the image diameters to be expected, regardless of the fact that they were 1.0

Modulation transfer

0.8 f#=2.8 f#=4 f#=5.6

0.6

0.4

0.2

0.0 0.0

100

200 300 400 Spatial frequency [ line pairs/ mm]

500

Fig. 2.35. Image modulation versus spatial frequency for a hypothetical lens systems without spherical aberrations at three diﬀerent f-numbers (Gaussian approximations).

54

2 Physical and Technical Background 1.0

Modulation transfer

0.9

0,92 0,9

0.8

0,78

0.7

0,7

0.6

f#=2.8 f#=5.6

0,55

0.5 0,45 0.4 0.3 0.2 0.1 0.0 0.0

100

200

300

400

Spatial frequency ( line pairs/mm )

500

Fig. 2.36. Modulation transfer data of a high quality 100 mm lens for two diﬀerent f-numbers at three spatial frequencies and a Gaussian ﬁt through the data measured at the highest frequency.

originally measured using white light and therefore consider also chromatic aberrations, which do not have to be taken into account for monochromatic laser light. According to the previous discussion, we assume that the MTF can be described by the inverse Fourier transform of the Gaussian approximation of the normalized image intensity distribution I(x)/Imax , which can be written in normalized form as: ˜ TF = exp −2π 2 σ 2 r2 . M (2.14) Taking a characteristic value r for the spatial frequency (r ≈ dτ ) and the ˜ TF into account, equation (2.9) can be used in corresponding MTF value M order to determine σ and therefore a function through this point in the MTF ˜ TF (r ) = 0.55 at r = 40 line pairs/mm from ﬁgure 2.36): (e.g. M ˜ TF (r )] ln[M . (2.15) σ= − 2π 2 r2 Now, equation (2.9) can be used to approximate the normalized intensity distribution of the image. In contrast to the Bessel function the Gaussian approximation has no zero-crossings (see ﬁgure 2.32). An image diameter can therefore not be determined by taking the x-value of the ﬁrst zero. This requires some kind of threshold level. In photographic PIV this threshold level represents the start of the nonlinear behavior of the ﬁlm material used for recording and contact copy (see ﬁgure 2.44). A two step photographic process is normally used to reduce the eﬀects of the ﬁlm fog and other background noise for subsequent optical evaluation. In digital PIV a certain threshold is normally used because of electronic background noise. In the following we assume that the lower 20% of the intensity of an image will not be used for evaluation. This assumption and equation (2.9) yield the following formula for the radius of an image of a circular object:

2.6 Imaging of Small Particles

x = −2σ 2 ln 0.2

55

(2.16)

When substituting equation (2.15) we obtain an approximation of the image diameter d that corresponds to a circular object which has an extension (2r )−1 : d ≈ 0.8 −

˜ TF (r )] ln [M . r2

(2.17)

For the estimation of the image diameters of smaller objects the same approximation as in equation (2.8) can be used to obtain: ˜ TF (r )] M 2 ln [M − . (2.18) dτ ≈ −0.64 r2 2r The deﬁnition of a MTF is a practical approach to describe the performance of optical systems. Since the underlying optical processes are much more complex, the description would be more complete when considering also the relative phase shift. However, phase shifts in optical systems occur only oﬀ axis and are of less interest than the MTF [11]. For many practical applications, the MTF of a complex optical system can be assumed to be simply the product of the MTFs of the individual components. For photographic recordings, the value of the ﬁlm MTF (e.g. shown in ﬁgure 2.45) at a certain spatial frequency r must be multiplied by the lens MTF value for the same spatial frequency and equation (2.18) can be used to estimate the image diameter for a given recording condition. Practical experience has shown that the resolution of a ﬁlm with high sensitivity (3200 ASA) can be used to perform high quality PIV measurements even in the case of diﬃcult recording conditions [60]. The image diameter estimated by equation (2.18) is approximately 20 μm (M = 1/4, f# = 2.8) and is in good correspondence with image diameters found during experiments. However, image recording by video sensors cannot be described by those models adequately. The sampling due to the regular arrangement of picture elements and its inﬂuence on the PIV evaluation has to be modeled in a diﬀerent manner in order to adequately describe the important eﬀects. 2.6.3 Perspective Projection In order to fully explain the inﬂuence of a velocity component perpendicular to the light sheet on the location of the image points in the coordinate system x, y, z (ﬁgure 2.37), the imaging through the lens must be taken into account. Ideal imaging conditions are assumed for this calculation. Image distortions resulting from non-ideal lenses can be taken into account by means of an extended model which is beyond the scope of this book (see [9]). The underlying perspective projection can either be modeled by deﬁning a homogeneous coordinate system in a more general way or in this simple case by trigonometric considerations (see ﬁgure 2.37).

56

2 Physical and Technical Background

i

α

i

0

0

Fig. 2.37. Imaging of a particle within the light sheet on the recording plane.

D deﬁnes the particle displacement between the light pulses by Dx , Dy , Dz . The following relation between the location of image points on the recording plane due to the imaging of a particle at position xi and xi is obtained: tan(α) =

xi z0

(2.19)

The image displacement d = xi − xi corresponding to a certain particle displacement D can be obtained: xi − xi = −M (DX + DZ xi /z0 )

(2.20)

yi − yi = −M (DY + DZ yi /z0 ) .

(2.21)

Assuming a particle displacement only in the X, and Y directions (DZ ≈ 0) would simplify equation (2.20) and equation (2.21) considerably. Then, the in-plane particle displacement could easily be determined by multiplying the image displacement by (−M ). In this particular case, the only uncertainty of the velocity measurement would be introduced by the uncertainty in determining the image displacement and the geometric parameters. However, in practical cases a ﬂow ﬁeld is never strictly two-dimensional over the whole observation ﬁeld. Moreover, conventional PIV, which was – at the beginning – developed for measurements of ﬂow ﬁelds with weak out-of-plane components only, has been adapted also for use in highly three-dimensional ﬂows over the last decade. It can be seen in equation (2.20) and equation (2.21), that a particle displacement in the Z-direction inﬂuences the particle image displacement, especially for large magnitudes of X i and Y i at the edges of the observation ﬁeld [62, 66]. This eﬀect introduces an uncertainty in measuring the in-plane velocity components, because it cannot be separated from

2.6 Imaging of Small Particles

57

the in-plane components. This uncertainty will turn into a systematic error if it is assumed that PIV determines just the in-plane components even for larger viewing angles. It will be shown by a simple analysis of the recording of a hypothetical three-dimensional ﬂow ﬁeld that this systematic measurement error can increase up to more than 15% of the mean ﬂow velocity. 2.6.4 Discussion of the Perspective Error The following discussion is based on the assumption of a ﬂow with a strong out-of-plane component. In order to describe the inﬂuence of the perspective projection qualitatively and quantitatively an example will be given: a potential vortex is assumed, similar to vortices occurring in natural ﬂows (hurricane) and ﬂows in technical devices (cyclone). In the case of an isoenergetic ﬂow the Bernoulli equation yields for the velocity components V and W :

˜ |Umax | X V 2 + W2 = √ , 2 X +Y2 ˜ is the distance where Umax is the maximum velocity of the vortex ﬂow, and X from the point where it occurs to the vortex axis. The X-axis of the observed plane of the ﬂow is parallel to the vortex axis, and is located outside the vortex core as illustrated in ﬁgure 2.38. The V and W components can be calculated according to the formula given above, while the U -component of the velocity is assumed to have 20% of the magnitude of

Fig. 2.38. 3D representation of the coordinate system, the observation ﬁeld and a streamline (α0 < π/4 for better representation) of a potential vortex.

58

2 Physical and Technical Background

Fig. 2.39. Velocity vector map of the U and V components of a simulated potential vortex within the observation ﬁeld shown in ﬁgure 2.38.

the other velocity components at point (0, 0, ΔZ). This leads to the following equations for the velocity components as functions of the X, Y, Z-coordinates: 1 2 0.2 Y ΔY V = −|Umax | cos arctan + α0 √ ΔY ΔY 2 + Y 2 Y ΔY W = −|Umax | sin arctan . + α0 √ ΔY ΔY 2 + Y 2 U = |Umax | √

In the case of the following numerical simulation of the PIV recording process α0 = π/4 is assumed, so that V = 0 at the upper left corner of the observation area and W = 0 at the lower left corner. In ﬁgure 2.39 we plot the U and V components of the velocity ﬁeld, which are in general obtained by a PIV measurement. However, since PIV determines the perspective projection of the threedimensional velocities, we come to the following representation of the measured velocity components. In ﬁgure 2.40 the values that would have been obtained by a PIV measurement of the 3D velocity vectors in the XY plane are presented, computed by means of the algorithms for the transformation described above. The computation was based on the following parameters: Magniﬁcation M = 1/4, ﬁlm size 35 × 24 mm2 , focal length of the objective lens f = 60 mm. These parameters are realistic values for many practical applications. The diﬀerence between the real U V data and the calculated “PIV recording” data is up to

2.6 Imaging of Small Particles

59

Fig. 2.40. Vector map of two velocity components after “PIV recording” (M = 1/4, 35 mm ﬁlm, f = 60 mm) calculated by means of perspective transformation from the same velocity vector ﬁeld as presented in ﬁgure 2.38.

16.6% of the mean ﬂow velocity in this case of idealized conditions. In order to achieve accurate data, this deviation has clearly to be assessed as a considerable error, but its systematic inﬂuence does not hamper the interpretation of the instantaneous ﬂow ﬁeld when looking for structures of the ﬂow ﬁeld as would be the case for random errors. In order to illustrate this statement, an error of the same size but with random angular distribution was superimposed on the known velocity data (ﬁgure 2.41). The fact that the error introduced by the perspective projection is not that visible in the results explains why it is commonly neglected. However, the only way to avoid this error or uncertainty is – at least for highly threedimensional ﬂows – to measure all three components of the velocity vectors, for example by means of stereoscopic techniques (see section 7.1). 2.6.5 Basics of Microscopic Imaging To date, micron-resolution PIV is of increasing interest with a view to BioMEMS and micro-ﬂuidics in general. Both might be considered increasingly popular and spreading ﬁelds of research since the beginning of the century. Therefore, the basics of microscopic imaging are depicted in this section. Microscopes usually consist of two magnifying stages, the objective and the eyepiece (sometimes referred to as the ocular). The objective is composed of several lenses that together form the intermediate image of the object under investigation. The intermediate image is further magniﬁed by the eyepiece. The total magniﬁcation of a microscope is given by the product of the individual magniﬁcations of the objective and eyepiece.

60

2 Physical and Technical Background

Fig. 2.41. Vector map of two velocity components with a random error of the same order as contained in the “PIV recording”, shown in ﬁgure 2.40.

The object illumination is one of the critical aspects in optical microscopy. In conventional microscopy the illumination is performed by a white light source and a condenser, which is a lens arrangement generating a homogeneous illumination of the object. It is usually based on transmission, reﬂection or ﬂuorescence. Regardless of the imaging mode used, image brightness is governed by the illumination light intensity and the numerical aperture. The brightness of the microscope illumination is determined by the square of the condenser working numerical aperture, whereas the brightness of the image is proportional to the square of the objective numerical aperture. A microscope objective is the most important component of an optical microscope since it determines the image quality. Standard bright-ﬁeld objectives are the most common for investigations utilizing traditional illumination techniques. More complex methods frequently require a detector close to the rear focal plane. Plan apochromat or ﬂuorite objectives with a high numerical aperture allow for the investigation of thinly cut ﬁxed tissues adhered to glass substrates and usually produce high-resolution images. However, attempts to image details at micrometer distances from the cover glass in a ﬂuid often suﬀer from severe spherical aberration. The use of water instead of oil reduces those aberration problems. Most microscope objectives are designed to be used with a cover glass. Thicknesses of 0.17 millimeters are satisfactory when the objective numerical aperture is 0.4 or less. However, when using a higher numerical aperture, cover glass thickness variations of only a few micrometers result in dramatic image degradation due to aberration, which grows with increasing cover glass thickness. The correction collars allowing the adjustment of the central lens group position to coincide with ﬂuctuations in cover glass thickness can be used in order to compensate for this error.

2.6 Imaging of Small Particles

61

In modern wide-ﬁeld ﬂuorescence and the later described laser scanning confocal microscopy, the collection and measurement of secondary emission gathered by the objective can be accomplished by photomultipliers, photodiodes and CCD or CMOS sensors. In analogy to the imaging of conventional objective lenses described in section 2.6.2, microscopes might typically suﬀer from ﬁve common aberrations: spherical, chromatic, curvature of ﬁeld, comatic and astigmatic. In stereoscopic microscopy geometrical distortion has to be considered additionally. All optical microscopes, including conventional wide-ﬁeld microscopes and the later described confocal microscopes are limited in the resolution that can be achieved. As described in section 2.6.1, in a perfect optical system, resolution and therefore contrast is limited by numerical aperture of the optical components and by the wavelength of the light. In a real ﬂuorescence microscope, contrast is determined by the dynamic range of the signal, optical aberrations of the imaging system, number of photons collected from the ﬂuorophore (if ﬂuorescence is used) and the number of picture elements (pixels2 ) per unit area. The numerical aperture of a microscope objective determines the ability to gather light and resolve ﬁne details at a certain object distance. However, the total resolution of a microscope system additionally depends on the numerical aperture of the substage condenser. The eyepieces in combination with microscope objectives further magniﬁes the intermediate image. Literature usually distinguishes two types of eyepieces depending on the lens and aperture diaphragm arrangement: the negative eyepieces with an internal aperture diaphragm and positive eyepieces with an aperture diaphragm below the lenses of the eyepiece. The eyepiece is usually designed to work together with objectives to eliminate chromatic aberration. The substage condenser gathers light from the microscope light source and concentrates it into a cone of light that illuminates the object with uniform intensity over the entire ﬁeld of view. The performance of the condenser is one of the most important factors in obtaining high quality images in the microscope. In reﬂected light microscopy oblique or epi-illumination (illumination from above) is utilized for the study of objects that are opaque, including semiconductors, ceramics, metals, polymers and many others. Beside reﬂection, ﬂuorescence can be used in order to emit light from parts of the object for imaging. The lasers commonly employed in micro PIV need larger pulse widths compared to PIV applications where scattered light is recorded. More details on modern microscopy can be found in [14].

2

Pixel is an acronym derived from picture element which is a single cell within a digital image or on a digital sensor. In an image, each pixel is associated with a numerical intensity value describing the local gray value or color.

62

2 Physical and Technical Background

2.6.6 In-Plane Spatial Resolution of Microscopic Imaging For inﬁnity-corrected microscope objective lenses, Meinhart and Wereley [289] showed that 12 1 n 2 −1 . (2.22) f# = 2 NA The numerical aperture is deﬁned as NA ≡ n sin θ, where n is the index of refraction of the recording medium and θ is the half-angle subtended by the aperture of the recording lens. Numerical aperture is a more convenient expression to use in microscopy because of the diﬀerent immersion media used. In photography, generally air is the only immersion medium used and hence f# is suﬃcient. To avoid confusion when reading the μPIV literature, it must be noted here that Equation 2.22 reduces to f# ≈

1 2N A

(2.23)

for the immersion medium being air (nair ≈ 1.0) and small numerical apertures. This is a small angle approximation that is accurate to within 10% for NA ≤ 0.25 but approaches 100% error for NA ≥ 1.2 [289]. This approximation is used, for example, by [294, 297]. Combining equation (2.7) and 2.22 yields the expression 12 n 2 −1 (2.24) ddiﬀ = 1.22M λ NA for the diﬀraction-limited spot size in terms of the numerical aperture directly. As mentioned above, the actual recorded image can be estimated as the convolution of the point-spread function with the geometric image (see equation (2.8)). 2.6.7 Microscopes Typically Used in Micro-PIV The most common microscope objective lenses range from diﬀraction-limited oil-immersion lenses with M = 60, N A = 1.4 to low magniﬁcation airimmersion lenses with M = 10, N A = 0.1. Table 2.7 gives eﬀective particle diameters recorded through a circular aperture and then projected back into the ﬂow, dτ /M . Using conventional microscope optics, particle image resolutions of dτ /M ∼ 0.3 μm can be obtained using oil-immersion lenses with numerical apertures of NA = 1.4 and particle diameters dp < 0.2 μm. For particle diameters dp > 0.3 μm, the geometric component of the image decreases the resolution of the particle image. The low magniﬁcation air-immersion lens with M = 10, N A = 0.25 is diﬀraction-limited for particle diameters dp < 1.0 μm. Eﬀective numerical aperture. In experiments where the highest possible spatial resolution is desired, researchers often choose high numerical aperture oil-immersion lenses. These lenses are quite complicated and designed to conduct as much light as possible out of a sample (see Figure 2.42) by not

2.6 Imaging of Small Particles

63

Table 2.7. Eﬀective particle image diameters when projected back into the ﬂow, dτ /M ( μm) [300]. M NA n

60 1.40 1.515

40 0.75 1.00

40 0.60 1.00

20 0.50 1.00

10 0.25 1.00

dp (μm) Eﬀ. particle image dia. dτ /M (μm) 0.01 0.10 0.20 0.30 0.50 0.70 1.00 3.00

0.29 0.30 0.35 0.42 0.58 0.76 1.04 3.01

0.62 0.63 0.65 0.69 0.79 0.93 1.18 3.06

0.93 0.94 0.95 0.98 1.06 1.17 1.37 3.14

1.24 1.25 1.26 1.28 1.34 1.43 1.59 3.25

2.91 2.91 2.92 2.93 2.95 2.99 3.08 4.18

allowing the light to pass into a medium with a refractive index as low as that of air (nair ≈ 1.0). When the index of refraction of the working ﬂuid is lower than that of the immersion medium, the eﬀective numerical aperture that an objective lens can deliver is decreased from that speciﬁed by the manufacturer because of total internal reﬂection. Meinhart and Wereley [289] have analyzed this numerical aperture reduction through a ray tracing procedure. As a speciﬁc example, assume that a 60× magniﬁcation, numerical aperture N AD = 1.4 oil-immersion lens optimized for use with a 170 μm coverslip and a maximum working distance wd of 200 μm is used with immersion oil having an index of refraction no matching that of the coverslip and a working ﬂuid (water) with a lower refractive index (nw ). This is a common situation in μPIV and is described in ref [406], among many others. The following

Fig. 2.42. Geometry of a high numerical aperture oil-immersion lens, immersion oil, coverslip, and water as the working ﬂuid. A point source of light emanating from a depth in the water, lw , appears to be at a distance wd from the lens entrance. After [289].

64

2 Physical and Technical Background

analysis is not restricted to these speciﬁc parameters and is easily generalizable to any arbitrary immersion medium and working ﬂuid as long as the immersion medium refractive index exceeds that of the working ﬂuid. Using a complicated ray-tracing procedure, Meinhart and Wereley [289] derived an implicit expression relating the depth into the ﬂow at which the focal plane is located, called the imaging depth lw , to the eﬀective numerical aperture N Aef f . The expression is

wd

no N AD

−1

12

=

lw

nw N Aef f

−1

12 +

wd − no N Aef f

no nw lw

−1

12

(2.25)

where no and nw are the refractive indices of the oil and water and wd is the lens’s working distance. Since no closed-form analytical solution is possible for Equation 2.25, it is solved numerically for the imaging depth lw in terms of N Aef f . Figure 2.43 shows the numerical solution of Equation 2.25 using no = 1.515 and nw = 1.33. When imaging at the coverslip boundary, i.e. only slightly into the water, the eﬀective numerical aperture is approximately equal to the refractive index of the water, 1.33. The eﬀective numerical aperture decreases with increasing imaging depth. At the maximum imaging depth the eﬀective numerical aperture is reduced to approximately 1.21. The eﬀective numerical aperture and the diﬀraction-limited spot size are given as a function of the imaging medium and imaging depth in Table 2.8. The refractive index change at the water/glass interface signiﬁcantly reduces the eﬀective numerical aperture of lens, even when the focal plane is right at the surface of the glass. This in turn increases the diﬀraction-limited spot size which reduces the spatial resolution of the μPIV technique.

Fig. 2.43. Eﬀective numerical aperture of an oil-immersion lens imaging into water as a function of the dimensionless imaging depth. After [289].

2.6 Imaging of Small Particles

65

Table 2.8. Estimates of the diﬀraction-limited spot size ddif f for various imaging media as a function of imaging depth lw [289]. Imaging Depth lw Imaging Medium N Aef f ddif f (μm) All Depths 0 μm (min) 200 μm (max)

Oil Water Water

1.40 1.33 1.21

18.8 24.8 34.2

In experiments where the working ﬂuid is water, a similar diﬀractionlimited spot size can be achieved using a N A=1.0 water-immersion lens, ddif f =39.8 μm, compared to an oil-immersion lens, which may only achieve an eﬀective numerical aperture, N Aef f ≈ 1.21 where ddif f =34.2 μm. A waterimmersion lens with N AD =1.2 will exhibit better performance than the oilimmersion lens, having ddif f =21.7 μm. Further, water-immersion lenses, are designed to image into water and will produce superior images to the oil immersion lens when being used to image a water ﬂow. 2.6.8 Confocal Microscopic Imaging The laser scanning confocal microscope (LSCM) techniques oﬀer some advantages over conventional optical microscopy. This kind of microscopes feature a controllable depth of ﬁeld, the elimination of image degrading out-of-focus information and have the ability to image serial optical sections of relatively thick objects. Its basic concept has been developed by Marvin Minsky and was patented in 1957 [292]. As described earlier, in a conventional wide-ﬁeld microscope, the object is illuminated mostly by a white light source. In contrast to that, the method of image formation in a laser confocal microscope is fundamentally diﬀerent. Illumination is achieved by a scanning laser beam through the object. In its classical form, the light of this beam is focussed by the objective lens and the sequences of points of light from the object are detected by a photomultiplier tube through a pinhole. The output is built into an image and displayed by the computer. Digital image processing methods applied to sequences of images allow the representation image series of diﬀerent depth and three-dimensional representation of specimens, as well as the time-sequence presentation of 3D data as four-dimensional imaging. The reﬂected light within the object can as well be used for imaging as ﬂuorescence. The latest generation of confocal instruments has tunable ﬁlters for the control of excitation wavelength ranges and intensity. They allow the control of the intensity on a pixel-by-pixel basis while maintaining a high scanning rate.

66

2 Physical and Technical Background

2.7 Photographic Recording 2.7.1 A Brief Description of the Chemical Processes Over the past decades photographic ﬁlms were widely used in optical systems to detect and store optical information. In some cases, they are still in use because of their high spatial resolution and possibly their availability. In the following, a short and simpliﬁed description of the chemical processes from exposure to ﬁxing will be given, together with an introduction to the diagrams normally used to characterize ﬁlm performance. An unexposed photographic emulsion mainly consists of tiny silver halide grains in a gelatine support. The interaction of an impinging photon with the photographic emulsion consists of random events. The probability that a photon strikes a silver halide grain depends on the density of the grains. Those grains that have absorbed a suﬃcient number of photons are found to contain tiny patches of metallic silver. These patches are referred to as “development centers”. During the developing process the chemicals used are diﬀusing from the surface through the gelatine. The existence of a single tiny development center precipitates the change from silver halide to silver. The unexposed grains are left unchanged and are removed during the “ﬁxing”. 2.7.2 Introduction to Performance Diagrams The locally varying intensity transmittance of the emulsion after development is deﬁned by [10]: Itrans (x, y) . (2.26) T (x, y) = local average Iinc Iinc is the intensity of a light source illuminating the recording after development, Itrans is the intensity of the light locally transmitted through the recording. The area of the local average is large with respect to the size of a ﬁlm grain, but small compared with an area within which the transmitted intensity changes signiﬁcantly. The local variance of T (x, y) is – besides other process parameters – caused by the local variance of the prior exposure E, which is the integral of the intensity per unit area over the exposure time Δt: I dt . (2.27) E= Δt

Hurter & Driffield demonstrated that the logarithm of the intensity transmittance log[1/T (x, y)], the photographic density Dphoto , is proportional to the silver mass per unit area of a developed transparency. Plots of the photographic density versus the logarithm of exposure log(E) are therefore commonly referred to as Hurter–Driﬃeld curves. They are often given in data sheets of photographic ﬁlms. The following ﬁgures display diagrams for two

2.7 Photographic Recording

67

3

Density

Shoulder

2

3200 ASA

γ 100 ASA

1 Gross fog

0 -3.0

-2.0

-1.0 log (E)

0.0

1.0

Fig. 2.44. Hurter– Driﬃeld curves of the photographic density for two standard black and white ﬁlms.

ﬁlms of diﬀerent sensitivity, which are widely used in black and white photography. It can be seen in ﬁgure 2.44 that the density is equal to a minimum value referred to as “gross fog” when the exposure is below a certain level. The density increases with increasing exposure at the so-called “toe”. For further increase in exposure it is linearly proportional to the logarithm of exposure (see ﬁgure 2.44). The slope of the linear region is referred to as the photographic gamma γ. For further increase in exposures the curve saturates at a constant level. This region is called the “shoulder” of the Hurter–Driﬃeld curve. The linear region of the curve is the portion generally used for PIV recording. Films can be classiﬁed as high contrast ﬁlm for a photographic gamma of two or more or low contrast ﬁlms for γ < 1. The photographic gamma can further be varied by varying the development time. This eﬀect and the nonlinear features of photographic emulsions described by their Hurter– Driﬃeld curves can be used in an additional photographic process – a contact copy3 – which can be performed after evaluation and which reduces the noise in optical PIV evaluation as described in section 5.3. Up to now it has been assumed that any spatial variation of the incident light during recording will be transferred into corresponding spatial density variations on the ﬁlm. However, this is not valid if the spatial scale of exposure variations is too small. Generally speaking, a given photographic emulsion possesses only a limited spatial frequency response. Therefore, the highest spatial frequency that can be recorded with a suﬃcient contrast is a further important ﬁlm speciﬁcation. However, for many applications the evaluation of ﬁlm resolution properties only on the basis of an upper limit of spatial frequency response is not suﬃcient. Especially for PIV, where high resolution is required, the response of a ﬁlm over the entire operating frequency range is more useful. In ﬁgure 2.45, the spatial frequency response of black and white ﬁlms with sensitivity of 100 ASA and 3200 ASA respectively is shown. 3

A contact copy is a high contrast negative of the original photographic PIV recording used to improve the signal-to-noise ratio in the optical evaluation process by means of a probing laser.

Modulation transfer [%]

68

2 Physical and Technical Background

100

100 ASA 3200 ASA

10

1

10 Spatial frequency [line pairs/mm]

Fig. 2.45. Spatial frequency response (MTF) of two standard black and white ﬁlms.

100

It can be seen that the ﬁlm with the higher sensitivity oﬀers lower frequency resolution. Such frequency response diagrams (MTF’s) have become a widely used means of specifying the performance of ﬁlms and lens systems. The concept of MTFs and the analysis of their eﬀect on the obtainable particle image diameter has been described in more detail in section 2.6. Another important diagram for selecting the proper ﬁlm for PIV recording is the curve of the spectral sensitivity. The sensitivity is deﬁned as the reciprocal value of the exposure E, which is necessary to obtain a certain photographic density. Figure 2.46 shows such curves for two commonly used ﬁlms. The curves are given for a density that is the density of the ﬁlm fog plus one, in other words, they represent log(1/E) which is needed to obtain a ﬁlm transmittance that is a tenth of the transmittance of the ﬁlm fog. It can be seen that the 3200 ASA ﬁlm shows superior sensitivity especially in the green. This was one of the reasons why frequency doubled Nd:YAG lasers with a wavelength of 532 nm (vertical line in ﬁgure 2.45) became fairly popular in PIV. 2.0

Log (sensitivity)

3200 ASA 1.0 100 ASA 0.0

-1.0

300

400

500

Wave length [nm]

600

700

Fig. 2.46. Spectral sensitivity of two standard black and white ﬁlms.

2.8 Digital Image Recording

69

2.8 Digital Image Recording Recent advances in electronic imaging have forced the photographic methods mostly aside. Immediate image availability and thus feedback during recording as well as a complete avoidance of photochemical processing are a few of the apparent advantages brought about with electronic imaging. The present trends suggest that electronic recording will even be able to substitute large format ﬁlm cameras and holographic plates in near future. For this reason more attention is placed on the description of electronic recording although it should be noted that the development in this area is very rapid. However, the potential uses of sensors introduced in the future should be assessable given a basic understanding of the interdependence between current sensor architecture and their possible application to PIV. Since the optical and electronic characteristics of sensors have a direct inﬂuence on the technical possibilities in PIV recording and the accompanying error sources, this section will be devoted to describing the operation and characteristics of these electronic sensors. Their potential application to PIV recording will be described in section 4.2 in the context of the existing variety of CCD and CMOS architectures. A performance comparison of CCD- and CMOS-based PIV cameras can be found in [94]. There is a variety of electronic image sensors available today, but only solid state sensors will be described here. Although electronic imaging based on vidicon tubes has reached a high state of development since their introduction more than 50 years ago, their importance to typical imaging applications has decreased dramatically in favor of solid state sensors. The most common are charge coupled devices, or CCD, charge injection devices (CID) and CMOS devices. Over the past two decades, the CCD has found the most widespread use. However, the rapid development of chip technology in the early 90s of the last century allowed the manufacturing of CMOS sensors with an improved signal-to-noise ratio and resolution. Since a couple of years they are more and more frequently used for digital photography, machine vision and, last but not least, for high-speed PIV. 2.8.1 Characteristics of CCD Sensors In general a CCD is an electronic sensor that can convert light (i.e. photons) into electric charge (i.e. electrons). Speaking of a CCD sensor, we generally refer to an array of many individual CCDs, either in the form of a line (e.g. in a line scan camera), or arranged in a rectangular array (of course other specialized forms also exist). The individual CCD element in the sensor is called a pixel. Its size is generally of the order of 10×10 μm2 , or 100 pixel/mm. The operation of these pixels is best described by referring to the schematic cross-section shown in ﬁgure 2.47. The CCD is built on a semiconducting substrate, typically silicon, with metal conductors on the surface, an insulating oxide layer, an n-layer (anode) and a p-layer (cathode) below that. A small

70

2 Physical and Technical Background

Light (Photon) hν Metal conductors

n-Layer

- e- e- ee

p-Layer

ePhotoelectric effect

Oxide layer (Insulator)

Electric field

Fig. 2.47. Simpliﬁed model of a (CCD) pixel.

voltage applied between the metal conductors and the p-layer generates an electric ﬁeld within the semiconductor. The local minimum in the electric ﬁeld that is formed below the center of the pixel is associated with a lack of electrons and is known as a potential well. In essence the potential well is equivalent to a capacitor allowing it to store charge, that is, electrons. When a photon of proper wavelength enters the p-n junction of the semiconductor an electron-hole pair is generated. In physics this eﬀect is known as the photoelectric eﬀect. While the hole, considered as a carrier of positive charge, is absorbed in the p-layer, the generated electron (or charge) migrates along the gradient of the electric ﬁeld toward its minimum (i.e. potential well) where it is stored. Electrons continue to accumulate for the duration of the pixel’s exposure to light. However, the pixel’s storage capacity is limited, described by its full-well capacity which is measured in electrons per pixel. Typical CCD sensors have a full-well capacity of the order of 10 000 to 100 000 electrons per pixel. When this number is exceeded during exposure (overexposure) the additional electrons migrate to the neighboring pixels which leads to image blooming. This eﬀect is signiﬁcantly reduced through specialized antiblooming architectures incorporated in modern CCD sensors: the overﬂowing charge is captured by conductors as it migrates toward the neighboring CCD cells. Another characteristic of a pixel is its ﬁll factor or aperture which is deﬁned as the ratio of its optically sensitive area and its entire area. This value can reach 100% for special, scientiﬁc-grade, back-illuminated sensors or may be as low as 15% for complex interline-transfer sensors which will be described in a later section. The primary reason for the limited aperture of most pixels is the presence of opaque areas on the surface of the sensors, either metal conductors used to form the potential wells and facilitate the transport of the accumulated charge to the readout port(s), or areas which are masked oﬀ to locally store charge before it is read out. Two methods exist for improving the ﬁll factor: back-thinning is a costly process which removes the back of the substrate to a few tens of microns such that the sensor may be exposed from the back.

2.8 Digital Image Recording

71

Back-thinned CCDs are custom built and are frequently applied in astronomy and spectroscopy. Also the process cannot be applied to all CCD architectures because opaque regions are frequently needed to temporarily store collected charge. An alternative and more economical approach to enhance the ﬁll factor is to deposit an array of microlenses on to the sensor allowing each pixel to collect more of the incoming light. The light sensitivity of each pixel, as well of CCD as of CMOS sensors, may then be improved up to a factor of three. 2.8.2 Characteristics of CMOS Sensors

Line selector

In most of the CMOS sensors the underlying electro-optical principle of each pixel is the photodiode as described in section 2.8.1. Their main advantage compared to other techniques like photogates or phototransistors is their high sensitivity and relatively low noise. But in contrast to CCD pixels, the photodiodes in CMOS sensors can be controlled separately by MOS-FET transistors. Since the beginning of the new century, the CMOS technology has drastically improved, and oﬀers some interesting advantages with respect to CCD technology. The breakthrough on the high-speed sensor market came with the progress in lithography that allowed the very large scale integration (VLSI) technique to be applied for CMOS sensors. Their speciﬁc architecture allows the combination of the electro-optical process with further electronic processing of the signal directly on chip. The fundamental principle of a CMOS sensor is shown in ﬁgure 2.48. Each individual pixel contains an electronic circuit. This active pixel architecture together with the individual access to each pixel oﬀer some major advantages

One pixel

Amplifier

Row selector

Fig. 2.48. Simpliﬁed model of a CMOS sensor.

72

2 Physical and Technical Background

and allows to integrate fundamental camera function like ampliﬁcation, nonlinear signal transformations and AD-conversion on chip. Furthermore, the number of pixels to be active can be chosen by the deﬁnition of a sub-domain, the region of interest (ROI). This allows to come to higher framing rates in trade of image resolution. The major drawback that still hinders the CMOS sensors to completely replace CCDs are the capacities of the relatively long electrical lines for each row and column. This leads to a signiﬁcant electronic noise since the noise of the MOS-FET transistors used for readout increases with the capacitance connected to their gate. This yields especially for larger sensors which are usually needed for PIV applications. The way out of this problem is the the active pixel sensor (APS) which contains a separate electronic ampliﬁer for each pixel. This type of sensor is described in more detail in section 4.2.5. 2.8.3 Sources of Noise As with any electronic device, the digital image sensors are subject to electronic noise. For many electronic imaging applications the issue of noise only plays a secondary role in that it corrupts the visual perception of the image. In the case of PIV, the light scattered from small particles is ideally captured on an otherwise black background. Due to the limited light scattering eﬃciency of the tiny tracer particles, the recorded signal will sometimes only barely exceed the background noise level of the sensor as the observation area and observation distance is increased. A major source of this noise is due to thermal eﬀects which also generate electron-hole pairs that cannot be separated from those generated by the photoelectric eﬀect: as a result weak particle images can no longer be distinguished from noise. Since the production rate of the electron-hole pairs is constant at a given temperature and exposure, this dark current or dark count can be accounted for by subtracting a constant bias voltage at the output of the charge-to-voltage converter. However, the dark current has a tendency to ﬂuctuate over time giving rise to noise, better known as dark current noise or dark noise which also increases with temperature and has a value of approximately the square root of the dark current. The rate of generation doubles for every 6–7◦C increase in temperature, which is the primary motivation for the use of cooled sensors in scientiﬁc imaging. Cryogenically cooled CCD sensors as applied in astronomy may generate less than one electron per second in each pixel. Another source of noise is read noise or shot noise which is a direct consequence of the charge-to-voltage conversion during the readout sequence. In general, the read noise increases with the readout frequency which is why many scientiﬁc applications require slow-scan cameras. Under standard operating conditions a normal CCD camera will have a noise level of several hundred electrons generated in each pixel for the period of integration (1/25 or 1/30 s). A careful optimization of the conversion electronics, a reduced

2.8 Digital Image Recording

73

readout frequency as well as cooling of the sensor may limit the read noise to a few electrons RMS per pixel. Up to now the prohibitive cost of these specialized cameras has made their use for PIV recording unfeasible. Nevertheless cameras based on Peltier-cooled (i.e. electrically cooled) CCD sensors are increasingly used for PIV applications. Images from cameras which have multiple ampliﬁers like the later described active pixel CMOS cameras and high-speed CCD cameras usually contain ﬁxed pattern noise (FPN). Fixed pattern noise includes the formerly described shot noise and dark current noise. In CMOS sensors additionally spike noise appears and contributes to the FPN. Spike noise is a switching noise occurring on the video line via the drain to gate capacitance of the MOS-FET transistor when an address pulse is input. A large fraction of this noise is constant and can therefore be subtracted from each pixel value before PIV evaluation (ﬂatﬁeld correction). 2.8.4 Spectral Characteristics Similar to photographic ﬁlm, the digital sensor has a sensitivity and spectral response. A pixel’s sensitivity or quantum eﬃciency, QE, is deﬁned as the ratio between the number of collected photoelectrons and the number of incident photons per pixel and is measured in collected charge over light intensity Cb/( J · cm2 ). Alternatively, units of current, I = Q/Δt, over incident power, P = E/(Δt·Area), are used: A/( W·cm2 ). To a large extent this value depends on the pixel’s architecture, that is, its aperture (i.e. ﬁll factor), material and thickness of the optically sensitive area. Due to the width and position of the frequency-dependent band-gap of silicon, the sensors substrate material, photons of diﬀerent frequencies will penetrate the sensor diﬀerently resulting in a wavelength dependent quantum eﬃciency of the sensor. Examples for the spectral response of several sensors are given in ﬁgure 2.49. To reduce the susceptibility to infrared light, many commercially available CCD and CMOS cameras come equipped with an infrared ﬁlter in front of the sensor. Other ﬁlters may also be used to match the spectral characteristics to that of the human eye. The responsivity of a sensor element expresses the ratio of useful signal voltage to exposure for a given illumination. This quantity depends on both the quantum eﬃciency and the on-chip charge-to-voltage conversion. 2.8.5 Linearity and Dynamic Range Since each electron captured in the potential well adds linearly to the cumulative collected charge, the output signal voltage for the individual pixel is practically directly proportional to the collected charge. Nonlinearities of CCD images are usually due to overexposure or poorly designed output ampliﬁers. In contrast to CCDs, CMOS sensors allow for a non-linear ampliﬁcation and conversion of the signals on chip. However, for most of the cameras used for

74

2 Physical and Technical Background I

−I

G

A

n Io

r−

A

N

d

−Y

e

N

e−

H

by

Ru

Quantum efficiency [%]

100 Back−thinned CCD #1 Back−thinned CCD #2 Standard CCD #1 Standard CCD #2 Standard IT CCD NI−IT CCD NI−IT CCD, High Res.

80 60 40 20 0 300

400

500

600 700 800 Wavelength [nm]

900

1000

Fig. 2.49. Quantum eﬃciencies for various CCD sensors. The principle frequencies for the most common laser sources are shown as vertical lines (IT = interline transfer sensor architecture, NI-IT = noninterlaced, interline transfer sensor architecture).

PIV, the signal is linearly ampliﬁed and encoded. With adequate design, linearities with deviations of less than 1% are possible. Linearity is of importance in PIV recording when small particle images are to be located with accuracies below half a pixel. Any nonlinear behavior during recording jeopardizes the capability of measuring the particle image displacement in the sub-pixel regime4 . Especially if the particles themselves are to be located and tracked as in PTV, a linear dependency between recorded signal and scattered light is of importance. The sensors’ dynamic range is deﬁned as the ratio between the full-well capacity and the dark current noise. Since the dark current noise is temperature dependent, the dynamic range of a CCD increases as the temperature is lowered. Standard video sensors operating at room temperature typically have a dynamic range of 100–200 gray levels which exceeds that of human perception. Once digitized the useful signal is 7–8 bits in depth. With additional cooling and careful camera design a dynamic range exceeding 65 000 gray levels (16 bits/pixel) is possible. For the application of electronic imaging in digital PIV recording a dynamic range of 6–8 bits allows the use of small interrogation windows (322 pixel) with a reasonable measurement uncertainty of less than 0.1 pixel (see section 5.5). 4

A commonly used expression signifying a length scale below the spatial resolution limit of a digital image, i.e. a pixel, is called sub-pixel. For instance the intensity distribution of a particle image may be used to estimate its position with sub-pixel accuracy.

2.10 The Video Standard

75

2.9 Standard Video and PIV A question which frequently arises is whether consumer-grade video equipment adhering to the standard analog video formats (i.e. NTSC or PAL) can be used for PIV. The answer is neither “yes” nor “no” since it inadequately addresses the issue. The potential user of PIV ﬁrst has to clarify for himself in which context PIV is to be used. The following questions need to be addressed: Is spatial resolution important? Using equally sized interrogation windows of 32 × 32 pixel, PIV images recorded with standard (consumergrade) video equipment can provide only up to 22 by 16 discrete vectors; a 1000 × 1000 pixel sensor provides more than 30 by 30 discrete vectors, while the utilization of photographic techniques can yield even more. Is measurement accuracy important? Is the PIV system to be used as a quantitative visualization tool or is it intended to get accurate estimates of vorticity or circulation? Due to the analog nature of the standard video signal a small frame-to-frame jitter during the digitization process can cause pixels to be slightly misaligned which in turn increases the measurement uncertainty in the displacement data (i.e. velocity data). The problem typically worsens when standard (analog) video recorders are used. Is temporal resolution important? The primary advantage of standard video equipment is that it can be used to record image sequences at 25 Hz (PAL) or 30 Hz (NTSC) using standard equipment. If the ﬂow under investigation is slow enough that it can be resolved temporally at this frame rate, PIV at this lower resolution may be of interest. If, in addition, the PIV images can be stored digitally as they are being recorded, a good measurement accuracy in the PIV data may be achieved at the same time.

2.10 The Video Standard Video is a general term describing the television type of imaging and is typically associated with the analog transmission of images whose information is coded in time, line-by-line. The video standards in use today are derivatives of standards such as NTSC-1 which was established in 1948 by the National Television Standards Committee (NTSC). Frame rates, line counts and other characteristics were determined at that time according to the then available technology (see table 2.9). To this date, the standard (television) video signal is transmitted in an interlaced format in which alternating ﬁelds of either odd or even lines of the image are transmitted at twice the video frame rate. This procedure provides the viewer with a uniform image appearance without the need for increasing the video signal’s bandwidth. The transmission scheme shown in

76

2 Physical and Technical Background Table 2.9. Characteristics of black & white video standards. Main area of use

Europe

North America/Japan

B& W Color

CCIR PAL/SECAM

RS-170 (EIA) NTSC

Format

4:3 aspect ratio 2 interlaced ﬁelds per frame 25 frames/s 50 ﬁelds/s 625 lines 574 lines visible 15.625 kHz ⇔ 64 μs 52.48 μs active 11.52 μs retrace 5.5 MHz

4:3 aspect ratio 2 interlaced ﬁelds per frame 29.97 frames/s 59.94 ﬁelds/s 525 lines 484 lines visible 15.734 kHz ⇔ 63.55 μs 52.80 μs active 10.75 μs retrace 4.2 MHz

Frame rate Resolution Row scan time

Bandwidth

ﬁgure 2.50 applies to the image display on the monitor as well as to the readout of the recording camera or any other device employing video signal transmission. When a video image is digitized, for instance, the odd ﬁeld (i.e. odd lines) is digitized, top-to-bottom before the ﬁrst even line of the second, even ﬁeld is sampled (ﬁgure 2.50, right). As a result adjacent vertical pixels will always be sampled with a time-separation of one ﬁeld, that is 1/50th (PAL) or 1/60th (NTSC) of a second. Most common video cameras provide the signal in a similar manner such that the ﬁnal digitized image actually will consist of two images at half the vertical resolution with one ﬁeld of time delay between them. On an analog display monitor the image will be perceived to ﬂicker. Even the electronic shutters on most video-format, CCD Start of odd field

Analog interlaced video trace Start of even field

Odd line Start at t0

Digitized video image

Pixel

Odd line Even line Start at t0 +Dt N/2 N lines

Even line

End of odd field

End of even field

Line scan period = Dt

Fig. 2.50. Images in standard video equipment are interlaced, consisting of two separate ﬁelds (left). Digitization preserves the interlaced nature of the analog image (right).

2.10 The Video Standard

77

sensors operate on a ﬁeld-to-ﬁeld basis. In essence, interlaced video makes the implementation of single-exposure double-frame PIV more diﬃcult than it needs to be. Nevertheless, there are some methods of successfully using video for PIV recording as given in the next section. The introduction of new video standards (HDTV or digital television) may alleviate many of the problems brought about by the old standards, but may well bring up others (e.g. compression artifacts).

3 Mathematical Background of Statistical PIV Evaluation

A detailed mathematical description of statistical PIV evaluation has been given by Adrian [78]. This early work from 1988 concentrated on autocorrelation methods and was later expanded to cross-correlation analysis [84]. Most of the characteristics and limitations of the statistical PIV evaluation have been described therein. To date the most complete and careful mathematical description of digital PIV has been given by Westerweel [51]. In this chapter, a simpliﬁed mathematical model of the recording and subsequent statistical evaluation of PIV images will be presented. For this purpose the two-dimensional spatial estimator for the correlation will be referred to as the correlation. First, we analyze the cross-correlation of two frames of singly exposed recordings. Then we expand the theory for the evaluation of doubly exposed recordings. The motivation behind employing auto- and crosscorrelation methods are employed in PIV evaluation will be given in chapter 5.

3.1 Particle Image Locations Typically, PIV recordings are subdivided into interrogation areas during evaluation. These areas are called interrogation spots – in the case of optical interrogation – or interrogation windows1 when digital recordings are considered. Due to reasons stated afterwards, for cross-correlation analysis those interrogation areas need not necessarily be located at the same position of the PIV recording. Their geometrical back-projection into the light sheet will be referred to as interrogation volumes in the following (see ﬁgure 3.1). Two interrogation volumes used for statistical evaluation together deﬁne the measurement volume. Now, a single exposure recording is considered. It consists 1

The local sample of a PIV image from which a velocity vector is determined is referred to as the interrogation spot or window. Its size determines to what degree the recovered velocity ﬁeld is spatially smoothed. In optical interrogation systems it is deﬁned by the diameter of the probing laser beam; in digital systems the rectangular pixel grid imposes a rectangular window.

80

3 Mathematical Background of Statistical PIV Evaluation

of a random distribution of particle images, which correspond to the following pattern of N tracer particles inside the ﬂow: ⎛ ⎞ X1 ⎜ X2 ⎟ ⎛ ⎞ ⎜ ⎟ Xi ⎜ · ⎟ ⎟ with Xi = ⎝ Yi ⎠ Γ=⎜ ⎜ · ⎟ ⎜ ⎟ Zi ⎝ · ⎠ XN being the position of a tracer particle in a 3N -dimensional space. Γ describes the state of the ensemble at a given time t. Xi is the position vector of the particle i at time t. For more details about the mathematical description of the tracer ensemble, see [51]. The lower case letters refer to the coordinates in the image plane (ﬁgure 3.1) such that x x= y is the image position vector in this plane. In the remainder of this section we will assume that the particle position and the image position are related by a constant magniﬁcation factor M for simplicity, such that: Xi = xi /M

and Yi = yi /M .

As already described in section 2.6.3, a more complex model of imaging geometry has to be used to take the eﬀect of perspective projection into account.

Fig. 3.1. Schematic representation of geometric imaging.

3.2 Image Intensity Field

81

3.2 Image Intensity Field In this section a mathematical representation of the intensity distribution in the image plane is given. It is assumed that the image can best be described by a convolution of the geometric image and the impulse response of the imaging system, the point spread function. For inﬁnite small particles and perfectly aberration-free, well focused lenses the amplitude of the point spread function can mathematically be described by the square of the ﬁrst order Bessel function also known as Airy function (see section 2.6). A more complex model of imaging has to include imperfections of lenses and photographic ﬁlms or sensors. For lenses and photographic ﬁlms an estimation of the main eﬀects besides diﬀraction can be obtained by analyzing their modulation transfer functions (MTF’s) (see section 2.6.2). For CCD sensors, a careful analysis requires more complex models which have not yet been described in the PIV literature suﬃciently. The description of digital imaging of very small objects is especially important, because the systematic arrangement of sensor elements can cause signiﬁcant bias errors in statistical particle image displacement estimation (q.v. peak locking; in section 5.5). In the following analysis we assume the point spread function of the imaging lens τ (x) to be Gaussian versus x and y (see appendix A.2), which is a common practice in literature and a good approximation for the point spread function of real lens systems [51, 78]. The convolution product of τ (x) with the geometric image of the tracer particle at the position xi therefore describes the image of a single particle located at position Xi . Furthermore, we restrict the analysis to inﬁnitely small geometric particle images which would be the case for small particles imaged at small magniﬁcations. Therefore, we use the Dirac delta-function shifted to position xi to describe the geometric part of the particle image. As schematically illustrated in Fig 3.2, the image intensity ﬁeld of the ﬁrst exposure may be expressed by: I = I(x, Γ) = τ (x) ∗

N

V0 (Xi ) δ(x − xi )

(3.1)

i=1

where V0 (Xi ) is the transfer function giving the light energy of the image of an individual particle i inside the interrogation volume VI and its conversion into an electronic signal or optical transmissivity2 . τ (x) is considered to be identical for every particle position. The visibility of a particle depends on many parameters as for example the scattering properties of the particle, the light intensity at the particle position, the sensitivity of the recording optics and the sensor or ﬁlm at the corresponding image position. However, we adopt that the particles at every position have the same scattering properties and 2

Strictly speaking equation (3.1) is valid only for incoherent light. For coherent light a term considering the interference of overlapping particle images has to be included [51]. In most practical situations the particle images do not overlap. Therefore, we use equation (3.1) also for coherent illumination.

82

3 Mathematical Background of Statistical PIV Evaluation

the recording optics and media have a constant sensitivity over the image plane. In many situations diﬀerent weight is assigned to diﬀerent locations inside the interrogation area. This can be done by a multiplication of the recorded image intensity with weight kernels in the case of digital evaluation or implicitly due to the spatial intensity distribution of the interrogating laser beam in the case of optical evaluation. Further on, we presume that Z is the viewing direction, the light intensity inside the interrogation volume is only a function of Z and the image intensity ﬁnally analyzed depends on X and Y only due to the weight function. Therefore, V0 (X) just describes the shape, extension and location of the actual interrogation volume: V0 (X) = W0 (X, Y ) I0 (Z)

(3.2)

where I0 (Z) is the intensity proﬁle of the laser light sheet in the Z direction and W0 (X, Y ) is the interrogation window function geometrically backprojected into the light sheet. This is mathematically not correct, because it does not consider the convolution with the point spread function. For rectangular interrogation windows this means that in our mathematical description we neglect the eﬀects of partially cropped images at the edges of the interrogation area. However, we will use this simple model of the interrogation volumes in the ﬂow, because it also simpliﬁes the description of PIV evaluation: (Z − Z0 )2 I0 (Z) = IZ exp −8 ΔZ02 might be used to describe the Gaussian intensity proﬁle of the laser light sheet, where ΔZ0 is the thickness of the light sheet measured at the e−2 points and IZ is the maximum intensity of the light sheet. W0 (X, Y ) can be described in a similar way if a Gaussian window function with a maximum weighting WXY at position X0 , Y0 has to be considered: (X − X0 )2 (Y − Y0 )2 W0 (X, Y ) = WXY exp −8 − 8 . ΔX02 ΔY02 Since many pulsed lasers used for PIV have an intensity distribution which is closer to a top-hat function than to a Gaussian function and since digitized recordings are commonly interrogated with rectangular windows, V0 (X) can also be deﬁned as a rectangular box: IZ if |Z − Z0 | ≤ ΔZ0 /2 I0 (Z) = (3.3) 0 elsewhere W0 (X, Y ) =

WXY if |X − X0 | ≤ ΔX0 /2 and 0 elsewhere.

|Y − Y0 | ≤ ΔY0 /2 (3.4)

3.3 Mean Value, Autocorrelation and Variance of a Single Exposure Recording

83

I x1 x2 x3 0,0

Δ x0

Fig. 3.2. Example of an intensity ﬁeld I (single exposure).

The factor I0 (Zi ) represents the amount of light received from the particle i inside the ﬂow, and located at distance |Zi − Z0 | from the center plane of the laser light sheet. ΔZ0 is the light sheet thickness and therefore the extension of the interrogation volume in the Z direction. ΔX0 = Δx0 /M and ΔY0 = Δy0 /M are the extension of the interrogation volume in the X- and Y -direction respectively. With τ (x − xi ) = τ (x) ∗ δ(x − xi ) (see appendix A.1) and the assumption that the particle images under consideration do not overlap, equation (3.1) can alternatively be written as: I(x, Γ) =

N

V0 (Xi ) τ (x − xi ) . (see appendix A)

(3.5)

i=1

This expression for the image intensity ﬁeld will intensively be used in the following sections. In the following we will illustrate diﬀerent representations of the intensity ﬁeld and their correlation by giving an example for the recording of three arbitrarily located particles.

3.3 Mean Value, Autocorrelation and Variance of a Single Exposure Recording In this section we will determine spatial estimators for the mean value and the variance of the image intensity ﬁeld, because these quantities will be used for the normalization of the cross-correlation. Furthermore, autocorrelation and auto-covariance of a single exposure intensity ﬁeld will be introduced. The main equations used in the following are taken from Papoulis [20, 21]. The spatial average is deﬁned as: 1 I(x, Γ) dx I(x, Γ) = aI aI where aI is the interrogation area. Employing equation (3.5) yields: 1 I(x, Γ) = aI

N

aI i=1

V0 (Xi ) τ (x − xi ) dx .

84

3 Mathematical Background of Statistical PIV Evaluation

The mean value of the intensity ﬁeld can be approximated by: μI = I(x, Γ) =

N 1 V0 (Xi ) τ (x − xi ) dx . aI i=1 aI

We can now derive the autocorrelation of the single exposure intensity ﬁeld in a similar way: RI (s, Γ) = I(x, Γ)I(x + s, Γ) N N 1 = V0 (Xi ) τ (x − xi ) V0 (Xj ) τ (x − xj + s) dx aI aI i=1 j=1 where s is the separation vector in the correlation plane. By distinguishing the i = j terms which represent the correlation of diﬀerent particle images and therefore randomly distributed noise in the correlation plane, and the i = j terms which represent the correlation of each particle image with itself, we come to the following representation:

RI (s, Γ) =

N 1 V0 (Xi ) V0 (Xj ) τ (x − xi ) τ (x − xj + s) dx aI aI i=j

+

N 1 2 V0 (Xi ) τ (x − xi ) τ (x − xj + s) dx . aI i=j aI

Following the decomposition proposed by Adrian, we can write: RI (s, Γ) = RC (s, Γ) + RF (s, Γ) + RP (s, Γ) where RC (s, Γ) is the convolution of the mean intensities of I and RF (s, Γ) is the ﬂuctuating noise component both resulting from the i = j terms. RP (s, Γ) ﬁnally is the self-correlation peak located at position (0, 0) in the correlation plane. It results from the components that correspond to the correlation of each particle image with itself (i = j terms). The autocorrelation of actual particle image data is provided in Fig 3.3 and clearly shows a strong central self-correlation peak surrounded by a noise ﬂoor. We will now concentrate on this central peak in order to evaluate its features. For a Gaussian particle image intensity distribution 8 |x|2 τ (x) = K exp − 2 dτ it can be shown that √ the autocorrelation Rτ (s) is again a Gaussian function with a width that is 2 dτ (see appendix A.3). Consequently RP (s, Γ) may be rewritten as follows:

3.3 Mean Value, Autocorrelation and Variance of a Single Exposure Recording

85

RP

RC +R F

sx

sy

0

0

RP (s, Γ) =

N

2

V0 (Xi ) exp

i=1

2

−8|s| √ 2 ( 2 dτ )

1 aI

Fig. 3.3. Composition of peaks in the autocorrelation function.

s dx . τ 2 x − xi + 2 aI

In the remainder of this book we will always use the representation: Rτ (s) = exp

2

−8|s| √ 2 ( 2 dτ )

1 aI

s dx τ 2 x − xi + 2 aI

taking into account that its features are mainly the same also for non-Gaussian τ (x): the maximum of Rτ (s) is located at |s| = 0 and the characteristics of its shape is given by the particle images shape. Therefore, we will write RP as RP (s, Γ) = Rτ (s)

N

V0 2 (Xi ) .

i=1

In ﬁgure 3.4 the schematic of the autocorrelation of the example intensity ﬁeld I is given. The correlation peaks (RP and RF ) occur at locations which are given by the vectorial diﬀerences between particle image locations. Their strength is proportional to the number of all possible diﬀerences which result

RI x1 - x3 x1 - x2 x3 - x2

0,0

x3 - x1

x2 - x3 x1 - x1 = x2 - x2 = x3 - x3 = 0 x2 - x1 Fig. 3.4. Schematic representation of the autocorrelation of the intensity ﬁeld I given in ﬁgure 3.2.

86

3 Mathematical Background of Statistical PIV Evaluation

in that location. For intensity ﬁelds with zero mean value the autocorrelation equals the auto-covariance. For nonzero mean values of the intensity ﬁeld the auto-covariance CI (s) can be obtained by [20]: CI (s) = RI (s) − μI 2 . An estimator of the variance of the intensity ﬁeld can be obtained by: σI2 = CI (0, Γ) = RI (0, Γ) − μI 2 = RP (0, Γ) − μI 2 .

3.4 Cross-Correlation of a Pair of Two Singly Exposed Recordings As already mentioned before, PIV recordings are most often evaluated by locally cross-correlating two frames of single exposures of the tracer ensemble. The mathematical background of this technique will be described in the following. In the remainder of this section, a constant displacement D of all particles inside the interrogation volume is assumed, so that the particle locations during the second exposure at time t = t + Δt are given by: ⎞ ⎛ Xi + DX Xi = Xi + D = ⎝ Yi + DY ⎠ . Zi + DZ Furthermore, we assume that the particle image displacements are given by: M DX d= M DY which is a simpliﬁcation of the perspective projection that is valid only for particles located in the vicinity of the optical axis (see section 2.6.3). We come to the following representation of the image intensity ﬁeld for the time of the second exposure (see equation 3.5): I (x, Γ) =

N

V0 (Xj + D) τ (x − xj − d)

j=1

where V0 (X) deﬁnes the interrogation volume during the second exposure. If we ﬁrst consider identical light sheet and windowing characteristics, the cross-correlation function of the two interrogation areas can be written as: 1 V0 (Xi ) V0 (Xj + D) τ (x − xi ) τ (x − xj + s − d) dx RII (s, Γ, D) = aI i,j aI

3.4 Cross-Correlation of a Pair of Two Singly Exposed Recordings

I

I

x1

x1 x2

x2

x3 0,0

87

x3

Fig. 3.5. The intensity ﬁeld I recorded at time t and the intensity ﬁeld I recorded after a time delay of Δt at t .

0,0

where s is the separation vector in the correlation plane. Analogous to the procedure used in the previous section we arrive at: V0 (Xi ) V0 (Xj + D) Rτ (xi − xj + s − d) . RII (s, Γ, D) = i,j

By distinguishing the i = j terms which represent the correlation of diﬀerent randomly distributed particles and therefore mainly noise in the correlation plane and the i = j terms, which contain the displacement information desired, we obtain: RII (s, Γ, D) =

V0 (Xi ) V0 (Xj + D) Rτ (xi − xj + s − d)

i=j

+Rτ (s − d)

N

V0 (Xi ) V0 (Xi + D) .

i=1

Again, we can decompose the correlation into three parts: RII (s, Γ, D) = RC (s, Γ, D) + RF (s, Γ, D) + RD (s, Γ, D) where RD (s, Γ, D) represents the component of the cross-correlation function that corresponds to the correlation of images of particles obtained from the

RD

RC + R F

sx sy

0

0

Fig. 3.6. Composition of peaks in the crosscorrelation function.

88

3 Mathematical Background of Statistical PIV Evaluation

RII x1 - x3 x1 - x2 x1 - x1= x3 - x3 = d 0,0 x3 - x2 x3 - x1

Fig. 3.7. Schematic representation of the cross-correlation of the intensity ﬁelds I and I given in ﬁgure 3.5.

ﬁrst exposure with images of identical particles obtained from the second exposure (i = j terms): RD (s, Γ, D) = Rτ (s − d)

N

V0 (Xi ) V0 (Xi + D) .

(3.6)

i=1

Hence, for a given distribution of particles inside the ﬂow, the displacement correlation peak reaches a maximum for s = d. Therefore, as already anticipated, the location of this maximum yields the average in-plane displacement, and thus the U and V components of the velocity inside the ﬂow. In ﬁgure 3.7 the schematic of the cross-correlation of the example intensity ﬁelds I and I is given. Nearly the same correlation peaks occur as in the autocorrelation shown in ﬁgure 3.4, but at locations which are displaced by d. Correlations of x2 do not appear here, because this image is located outside the interrogation window (see ﬁgure 3.5). It can be seen from equation (3.6) that the displacement correlation is a function of the random variables (Xi )i=1···N . Consequently it is a random variable itself and for diﬀerent realizations at the same overall conditions we will obtain diﬀerent qualities of the displacement estimation depending on the state of the tracer ensemble. In order to derive rules for a general optimization of the displacement estimation, we will determine the expected value of the displacement correlation in section 3.6.

3.5 Correlation of a Doubly Exposed Recording The correlation function of a doubly (or multiply) exposed recording can be derived by analogy to the correlation for single exposed recordings. Instead of cross-correlating I with I , we will consider the correlation of the intensity ﬁeld I + = I + I with itself. Assuming identical light sheets and windowing characteristics, the intensity ﬁeld of both exposures I + can be written as:

3.5 Correlation of a Doubly Exposed Recording

89

I+I x1 x3

x1 x2

x2

x3

0,0

Fig. 3.8. The sum of the intensity ﬁelds I and I (see ﬁgure 3.5) as obtained by a recording of the tracer ensemble at t and t on the same frame.

I + (x, Γ) = I(x, Γ) + I (x, Γ) =

N

(V0 (Xi ) τ (x − xi ) + V0 (Xi + D) τ (x − xi − d)) .

i=1

It can be shown that the autocorrelation of I + consists of four terms:

RI+ (s, Γ, D) = RI (s, Γ) + RI (s, Γ) + RII (s, Γ, D) + RII (−s, Γ, D) . It is therefore appropriate to decompose the estimator into the following terms:

RI+ (s, Γ, D) = RC (s, Γ, D) + RF (s, Γ, D) + RP (s, Γ) +RD+ (s, Γ, D) + RD− (s, Γ, D)

(3.7)

where RC (s, Γ, D) is the convolution of the mean intensity of I + and RF (s, Γ, D) is the ﬂuctuating noise component. RP (s, Γ) is the self-correlation peak located at the center of the correlation plane. It results from the components that correspond to the correlation of each particle image with itself. RD+ (s, Γ, D) and RD− (s, Γ, D) represent the components of the correlation function which correspond to the correlation of images of particles obtained

RP

RD-

RD+

RC + R F

sy

sx 0

0

Fig. 3.9. Components of the autocorrelation function.

90

3 Mathematical Background of Statistical PIV Evaluation

from the ﬁrst exposure with that of images of identical particles obtained from the second exposure and vice versa. When comparing the correlation of a doubly exposed recording with the correlation of a pair of two singly exposed recordings, the following statements can be made: RI+ is symmetric with respect to its central peak RP . Two identical displacement peaks RD+ and RD− appear and as a consequence the sign of the displacement cannot be determined. Therefore, the correlation of a doubly exposed recording is not conclusive if the displacement ﬁeld of the whole recording is not unidirectional. Another problem appears if the ﬁeld contains displacements close to zero, which would lead to an overlap between the displacement peaks with the central peak. However, these problems have to be solved during recording. Precautions have to be made so that the images of identical particles due to the diﬀerent exposures do not overlap and the sign of their displacement is determined. If the ﬂow ﬁeld under investigation contains areas of reverse ﬂow or of relative slow velocities image shifting has to be used (see section 4.3). It can be seen from ﬁgure 3.10 that the correlation of doubly exposed recordings contains more than twice the number of randomly distributed noise peaks. The example given in ﬁgure 3.10 shows that in situations for which the crosscorrelation of single exposure yields good results, the correlation of doubly exposed recordings contains noise peaks of the strength of the displacement peak. Hence, the evaluation of multiply exposed recordings has to be performed with more particle image pairs in order to get the same performance as that of single exposure evaluation. This can be done by diﬀerent methods: the seeding density, the number of exposures or the light sheet thickness can be increased. Besides other problems related to these methods their application is restricted due to the limited number of particle images that can be stored on the sensor without a signiﬁcant overlap. Therefore, in most cases the size of the interrogation areas has to be increased compared to the evaluation of single exposures resulting in a lower spatial resolution of the measurement at the same sensor size.

RI+I x1 - x3 = x1 - x3 (R ) F

0,0

x1 - x1 = x3 - x3 (R +) D x1 - x1 = x1 - x1 = x2 - x2 = x3 - x3 = x3 - x3 = 0 x3 - x3 = x1 - x1 (RD-) x3 - x1 = x3 - x1 (R ) F

Fig. 3.10. Schematic representation of the autocorrelation of the intensity ﬁeld I + I given in ﬁgure 3.8.

3.6 Expected Value of Displacement Correlation

91

3.6 Expected Value of Displacement Correlation In order to derive rules for a general optimization of the displacement estimation we will determine the expected value of the displacement correlation E{RD } for all realizations of Γ. More concretely: we want to calculate the mean correlation function of all possible “patterns” that can be realized with N particles. From equation (3.6), it follows that E{RD } = E

Rτ (s − d)

N

V0 (Xi )V0 (Xi + D)

i=1

= Rτ (s − d) E

N

V0 (Xi )V0 (Xi + D)

i=1

Deﬁning fl (X) = V0 (X) V0 (X + D) yields: E{RD } = Rτ (s − d) E

N

fl (Xi )

.

(3.8)

fl (X) dX .

(3.9)

i=1

We prove in appendix A.4 that: N N E fl (Xi ) = fl (X) dX VF VF i=1 where V fl (X) dX is the volume integral F fl (X, Y, Z) dXdYdZ . Thus: E{RD } =

N Rτ (s − d) VF

VF

Since we deﬁned N to be the number of all particles of the ensemble, VF has to be interpreted as the whole volume of ﬂuid that has been seeded with particles. According to the above deﬁnition of fl (X) we can say in a more practical sense that the integration has to be performed over the volume which contained all particles that were inside the interrogation volumes during the ﬁrst or second exposure. We can rewrite the integral over fl (X) as:

fl (X) dX =

I0 (Z) I0 (Z + DZ ) dZ

VF

×

W0 (X, Y ) W0 (X + DX , Y + DY ) dXdY V02 (X) dX · FO (DZ ) FI (DX , DY )

= VF

92

3 Mathematical Background of Statistical PIV Evaluation

with FI (DX , DY ) =

W0 (X, Y ) W0 (X + DX , Y + DY ) dXdY W02 (X, Y ) dXdY

(3.10)

and FO (DZ ) =

I0 (Z) I0 (Z + DZ ) dZ . I02 (Z) dZ

(3.11)

Keane & Adrian [82, 83, 84] have deﬁned FI as a factor expressing the in-plane loss-of-pairs, and FO as a factor expressing the out-of-plane loss-ofpairs. When no in-plane or out-of-plane loss-of-pairs are present the latter two are unity. Finally equation (3.9) yields: E{RD (s, D)} = CR Rτ (s − d) FO (DZ ) FI (DX , DY )

(3.12)

where the constant CR is deﬁned as: N CR = V 2 (X) dX . VF VF 0

3.7 Optimization of Correlation The ﬁrst parameter that has to be optimized during a PIV measurement is the pulse separation time between the successive light pulses. Besides technical limitations some general eﬀects have to be considered. According to the principle of PIV the measured velocity is determined by the ratio of two components of the measured particle displacement between successive light pulses DX and DY respectively, and the pulse separation time Δt. Since the particle displacement – which is considered to be a function of Δt in the following – is determined by the particle image displacement with DX (Δt) = dx (Δt)/M and DY (Δt) = dy (Δt)/M respectively, and the measured image displacements contain certain residual errors, εresid , we can deﬁne the following equation for the magnitude of the locally measured velocity: εresid |d(Δt)| + . (3.13) M Δt M Δt Since the particle image displacement for a given recording conﬁguration reduces linearly with the pulse separation time, the ﬁrst term of the above equation stays constant for vanishing pulse separations: |U| =

3.7 Optimization of Correlation

93

|d(Δt)| = |U| . Δt→0 M Δt In contrast to that, the residual error contained in the measured image displacement will not be reduced below a certain limit by a reduction of the pulse separation, because the uncertainty in determining the particle image positions will be unaﬀected. Therefore, the second term of equation (3.13) – which states that the measurement error is weighted with 1/Δt – increases rapidly with decreasing pulse separation: lim

εresid =∞. Δt→0 M Δt From these considerations it can be seen that the accuracy of PIV measurements can be increased by increasing the separation time between the exposures at least within certain limits. However, for high values of Δt the measurement noise increases. This becomes clear when looking at the expectation of the displacement correlation given in equation (3.12). It can be seen that the average signal strength is weighted with the loss of pairs due to the particle displacement D(Δt). For a very large separation time the particle displacement, which increases linearly with Δt, will exceed the extent of the interrogation volume. Then, no particle will be illuminated twice and no image correlation would be obtained. What can be done to improve the situation? First of all the pulse separation time can be reduced. This directly reduces the particle displacement and the loss of pairs. In ﬁgure 3.11 we have tried to illustrate the two aspects of the choice of Δt on the quality of the PIV data: the dotted line, curve g, represents the eﬀect of the weighting of the residual error with Δt, the solid line, curve f , represents the inﬂuence of the loss of pairs. The optimum Δt could therefore be found lim

QPIV [%] f (NI ,F0 (Δ t), Fi (Δ t))

100

g (εrms/ Δ t)

QPIV(Δ t)

0

0

Δ t opt

Δt

Fig. 3.11. Schematic representation for the optimization of the pulse delay time.

94

3 Mathematical Background of Statistical PIV Evaluation

by determining the maximum of a quality function QPIV , for example the product of curves f and g which is represented by the dashed line. However, the shape of curve f has been chosen arbitrarily, since a general value for the quality of a measurement is diﬃcult to deﬁne. When using digital equipment, which allows immediate feedback during the measurement, the optimum can be found interactively by slowly increasing the pulse separation until the number of obvious outliers3 within the vector map increases. However, the number of valid data yield is only one parameter of the obtained quality, but not an exact measure of it. Another parameter, which can be used for optimization, if it is made available from the evaluation software, is the normalized strength of the displacement correlation. The cross-correlation coeﬃcient given by: CII RII − μI μI = . σI σI σI σI While using photographic recording the choice of all recording parameters merely depends on the experiences of the experimentalist, because the valid data yield, the cross-correlation coeﬃcient, or, in the case of optical evaluation, the visibility of the Young’s fringes, can be assessed only after several hours. Another way to reduce the loss of pairs is to change the size of the interrogation volumes and/or to displace them slightly with respect to each other in order to compensate for the mean particle displacement. The extension of the interrogation volume in the out-of-plane direction is given by the light sheet thickness. This parameter can be increased only if adequate laser power is available. If one of the two possible out-of-plane directions is predominant, the light sheet can be displaced between the successive illuminations towards the mean ﬂow. While using double oscillator systems, this can be achieved by a slight “misalignment” of the beam combining optics. In the case of CW lasers a displacement requires additional equipment (see e.g. section 9.5). The extension and location of the interrogation volumes in the in-plane directions is given by the size of the interrogation areas during evaluation and the magniﬁcation during recording. In the case of cross-correlation analysis the location of the interrogation windows with respect to each other can be changed. This is one main advantage of cross-correlation and the reason why it is frequently applied also for the evaluation of single frame recordings instead of autocorrelation. The eﬀects of the interrogation volume locations during the ﬁrst and second exposure X0 = (X0 , Y0 , Z0 ) and X0 = (X0 , Y0 , Z0 ) respectively, can best be described by presenting equation (3.12) in a more generalized form: cII =

3

Outlier is a common term describing data whose validity is questionable on the basis of a certain acceptance criterion.

3.7 Optimization of Correlation

95

E{RD (s, D, X0 − X0 )} = CR Rτ [s − d − (x0 − x0 )]

×FO [DZ − (Z0 − Z0 )] ×FI [DX − (X0 − X0 ), DY − (Y0 − Y0 )] .

From this equation the eﬀect of an interrogation window oﬀset x0 − x0 can clearly be seen: the peak location has changed by the amount of the oﬀset, and the inﬂuence of the in-plane loss of correlation on to the peak strength has changed. The signiﬁcance of the loss of correlation also depends on the absolute extension of the interrogation volume. This is implied in the equations for FO and FI as given in the previous section (equation (3.10) and equation (3.11)), but shall be illustrated by the following equation which has been derived for a top-hat light sheet proﬁle (equation (3.3)) and rectangular interrogation windows (equation (3.4)): E{RD (s, D, X0 − X0 )} = CR Rτ [s − d − (x0 − x0 )] DX − (X0 − X0 ) · 1− ΔX0 DY − (Y0 − Y0 ) · 1− ΔY0 DZ − (Z0 − Z0 ) · 1− . ΔZ0 Generally speaking, a stronger peak results in a better peak detection probability and in reduced inﬂuence of noise components on the determination of the peak location. In many cases prior knowledge of the main displacement due to mean ﬂow or image shifting can be used in order to improve the result of the evaluation. In other cases more sophisticated algorithms are required in order to take advantage of this eﬀect. The software can either work in a multiple path scheme or with diﬀerent sizes of the interrogation windows to be correlated. In both cases the resolution and accuracy of the measurement can be increased considerably at the cost of more computing time. The diﬀerent aspects of choosing the right interrogation window sizes and locations for the evaluation by cross-correlation are described in section 5.4 in detail. According to our experience it is advisable to apply a quick-look evaluation during recording for the optimization of the experimental parameters, and also to store the original recordings in order to be able to optimize the evaluation after the experiment.

4 PIV Recording Techniques

In this chapter diﬀerent approaches to PIV recording are introduced. It is important to realize that the various recording methods are not necessarily deﬁned by the recording medium. The same approach may for instance be applied using either photography or digital recording. The PIV recording modes can be classiﬁed into two main categories: (1) methods which capture the illuminated ﬂow on to a single frame and (2) methods which provide a single illuminated image for each illumination pulse. These branches are referred to as single frame/multi-exposure PIV (ﬁgure 4.1) and multi-frame/single exposure PIV (ﬁgure 4.2), respectively [31]. The principal distinction between the two categories is that the former method, without additional eﬀort, does not retain information on their temporal order of the illumination pulse giving rise to a directional ambiguity in the recovered displacement vector. This necessitated the introduction of a wide variety of schemes to account for the directional ambiguity, such as displacement biasing, the so-called image shifting (i.e. using a rotating mirror or birefringent crystal), pulse tagging or color coding1 [57, 86, 91, 93, 98, 99]. In contrast, multi-frame/single exposure PIV recording inherently preserves the temporal order of the particle images and hence is the method of choice if the technological requirements can be met. Also in terms of evaluation this approach is much easier to handle. Historically single frame/multi-exposure PIV recording was ﬁrst utilized in conjunction with photography. Although multiple frame/single exposure PIV recording is possible using high-speed motion cameras, other problems such as interframe registration arise. Continual development over the past decade in the area of electronic imaging has made multi-frame/single exposure PIV recording possible at ﬂow velocities extending into the hypersonic domain.

1

Strictly speaking color coding is a form of multi-frame/single exposure PIV: the color recording can be separated into diﬀerent color channels containing single exposed particle images.

98

4 PIV Recording Techniques

Single Frame/Single Exposure

t

t’

Time

Single Frame/Double Exposure

t t’

Time

Single Frame/Multi- Exposure

t t’ t’’

Time

Fig. 4.1. Single frame techniques.

This chapter on PIV recording is organized as follows: after an introduction to the cameras most frequently used, the advantages and associated problems of single frame recording and image shifting will be discussed. The second part will introduce multiple frame PIV recording mainly in the context of digital imaging. A description of all possible recording techniques that can be used for PIV cannot be given here. A variety of diﬀerent sophisticated ideas related to this problem has been reported in the literature over the past few years, most of them with special advantages for individual applications. It is clear that a decision on which method is best cannot be made without taking the individual needs of each application into account. Therefore, the recording techniques described here are not complete and not necessarily the best, but the most common. In summary it can be said that the design of an experimental setup for PIV is based on a decision of which of the following goals have priority: • • • •

high spatial and/or temporal resolution of the ﬂow ﬁeld under investigation, the required resolution of velocity ﬂuctuations, the time interval between the individual PIV measurements, and which components are already available in the laboratory or can be obtained at adequate costs.

4.1 Film Cameras for PIV

99

Double Frame/Single Exposure

t

t’

t

t’

Time

Multi-Frame/Single Exposure

t’’

Time

Multi-Frame/Double-Exposure

t t’

t’’....

Time

Fig. 4.2. Multiple frame techniques (open circles indicate the particles’ positions in previous frames).

Depending on the choice of priority an appropriate system for recording can be conﬁgured. However, it must be kept in mind that not every requirement can be fulﬁlled, which is mainly due to technical limitations such as the available laser power, pulse repetition rates, camera frame rate, etc. The selection of the recording system also inﬂuences the method for directional ambiguity removal and, hence, the evaluation technique to be used. Today, photographic recording and mechanical image shifting – which have been used for a long time because of the obtained quality of the recordings – might not be considered the ﬁrst choice anymore. Video recording oﬀers so many advantages by allowing for immediate feedback and quality optimization during the course of the experiment. This is of high interest for most applications, especially with a view to operational costs.

4.1 Film Cameras for PIV The current level of technology allows digital recording to achieve a high spatial resolution that can be compared with that of 35 mm ﬁlm cameras. However, especially if using large format ﬁlms, resolutions can be obtained which are more than one order of magnitude larger than that of digital cameras. The

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4 PIV Recording Techniques

photographic technique is a method of choice for PIV applications requiring high spatial resolution without peak-locking. One major disadvantage of the photographic technique is that it is diﬃcult to record the images of the tracer particles on to diﬀerent frames, especially in the case of high-speed investigations where pulse separation times, of the order of a few microseconds are required. This indicates that the problem of directional ambiguity removal has to be solved for photographic PIV in a reliable and ﬂexible manner using a technique such as image shifting which will be described below. 4.1.1 Example of a PIV Film Camera Initial PIV experiments have shown that a high quality and reliable focusing device is necessary, in order to save time in the alignment of the system. For this purpose a photographic camera should be equipped with a device for fast focusing. A small area in the ﬁlm plane can for example be observed by means of a CCD camera looking through an oriﬁce in the back wall of the camera in order to control focusing[58]. This focusing device worked well and helped to reduce the time necessary for alignment considerably. A diﬀerent solution which possesses some technical advantages has been suggested in the literature [60, 68]. A low cost standard CCD sensor mounted in the viewﬁnder of a singlelens reﬂex (SLR) photo camera can be used for fast and reliable focusing (ﬁgure 4.3). The position of the CCD sensor has to be carefully aligned in such a way that the distance between the lens and the CCD sensor via the mirror is exactly the same as that from the lens to the ﬁlm plane. The distance between the light sheet and the ﬁlm plane can be changed by moving the complete camera system by means of a traversing table, thereby observing for minimum particle image diameters on a monitor.

Fig. 4.3. Photo camera with CCD sensor for fast focusing.

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101

4.1.2 High-Speed Film Cameras The ﬁrst high-speed cameras have already been developed shortly before and more intensively during the second world war [96]. First developments consisted of standard components to which a motorized fast drive had been coupled, which rapidly moved the ﬁlm from one roll to the other. Rotating prisms generated a synchronized periodic shift of the projected image in order to allow for longer exposure times. The limiting factors for those solutions are the acceleration forces and the mechanical ﬁlm properties. The maximum frame rates of those cameras are approximately 1000 frames/s. Much higher frame rates can be obtained by drum cameras. They consist of a rotating mirror on which the image is projected by the primary object lens as shown in ﬁgure 4.4. The mirror has an inclination of 45◦ with respect to the axis of rotation. A couple of secondary lenses around the rotating mirror projects the image onto the photographic ﬁlm, which is attached to the inner wall of the cylinder around the mirror. Some major technical problems had to be solved for this type of camera in order to reach the current state of technology. First, a motor had to be coupled which oﬀers a high acceleration before the recording and a high and constant speed during recording. Second, resonances causing inner vibrations and high structural loads during the acceleration and constant revolution had to be avoided. And third, highly complex electro-mechanical or electro-optical shutters were required. Even if high-speed drum cameras have not been developed much further over the past decade, their frame rates of up to 107 frames/s together with the high spatial resolution of photographic material makes them superior with respect to modern high-speed digital cameras. Only the costs of multiple oscillator lasers that would allow such high pulse rates and their complicated handling prevent them to be used in many applications.

4.2 Digital Cameras for PIV Over the last decade, CCD based digital cameras, as described in the following sections, became the work horses for nearly all technical and scientiﬁc PIV applications that required only moderate or no temporal resolution. Flash lamp pumped double oscillator Nd:YAG-lasers oﬀer high pulse energies and

Image

Film

Secondary lenses Object

Rotating mirror Mirror image Main lens

Fig. 4.4. Sketch of a high-speed photographic drum camera.

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4 PIV Recording Techniques

repetition rates that matched with the frame rates of most of the commercially available CCD cameras. The CCD cameras oﬀer two important advantages, one being increased spatial resolution, the second the electronic architecture that permits two PIV recordings, temporally spaced by microseconds, by the same camera (see 4.2.4). Therefore, the CCD sensors will be described in considerable detail in the following. In section 2.8 the CCD as an imaging sensor was described. Next the various types of CCDs are introduced in the context of application to PIV recording. Figure 4.5 schematically describes the layout of a CCD sensor. The individual pixels are typically grouped into a rectangular array to form a light sensitive area (linear, circular or hexagonal formats also exist). It should be pointed out that, in contrast to most CMOS sensors, in CCD sensors the pixels in the array cannot be randomly addressed the way memory can be addressed in a computer. Rather, the array has to be read out sequentially in a two-step process: after exposing the sensor the accumulated charge (i.e. electrons) is shifted vertically, one row at a time, into a maskedoﬀ analog shift register on the lower edge of the sensor’s active area. Each row in the analog shift register is then clocked, pixel by pixel, through a charge-to-voltage converter and thereby provides one voltage for each pixel. The stream of pixel voltages along with a variety of synchronization pulses compose the actual (analog) video signal. Depending on the employed image transmission format the read-out of the sensor can either be sequential (also known as progressive scan) or interlaced, in which ﬁrst all odd rows are read out before the even rows are accessed. The latter is the common format for standard video equipment (see also section 2.10). Since the progressive scan approach preserves the image integrity, it is more useful for PIV recording as well as for other imaging applications such as machine vision. In the following four sections we will concentrate on the operation of the various types of CCD sensors and how these may be utilized in PIV recording. Section 4.2.5 deals with the recently developed active pixel CMOS sensors, which became the state of the art design of sensors used for high-speed PIV. Sections 4.2.6 and 4.2.7 describe camera types that can be used for high-speed recording, whereas sections 2.9 and 4.4.1 should be consulted in regard to the utilization of standard (consumer) video equipment for PIV recording. A short summary of the features of digital cameras that are considered to be essential for PIV is given in section 2.8.1. 4.2.1 Full-Frame CCD The full-frame CCD sensor represents the CCD in its classical form (ﬁgure 4.5): a photosensitive area of pixels that is ﬁrst exposed to light and then read out sequentially (progressive scan) on a row-by-row basis without separating the image into two separate interlaced ﬁelds such as in standard

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103

Column Pixel Array Pixel

Row transfer direction

Row

Charge-to-voltage converter

Analog shift register

Fig. 4.5. Typical CCD sensor geometry.

video. This sensor has been in use in scientiﬁc imaging such as astronomy, spectroscopy and remote sensing ever since its introduction in the 1960’s. It is characterized by large ﬁll factors which can even reach 100% for special back-thinned, back-illuminated sensors2 . With adequate cooling and slow read-out speeds, imaging at very low noise levels with high dynamic range (up to 16 bits) is possible. The most striking advantage is that these sensors are available as very large arrays with pixel counts exceeding tens of millions (7000 × 5000 pixel). The use of these sensors does however have some major drawbacks. To achieve the low read-out noise and high dynamic range, the pixel read-out rate has to be kept low. Even at standard video characteristics the data rate is limited to 10 − 20 MHz which results in a decreasing frame rate as the number of pixels increases. Frame rates of less than 1 Hz are not uncommon for larger sensors. For this reason multiple read-out ports are sometimes used, which brings about the problem of calibrating the respective charge-to-voltage converters with respect to one another. Another drawback is that the sensor stays active during read-out. Unless a shutter is placed in front of it, light falling on to the sensor will also be captured resulting in a vertical smear in the ﬁnal image. Because of its high spatial resolution, the full-frame sensor can be used as a direct replacement of photographic ﬁlm. These sensors are frequently incorporated into 35 mm SLR camera bodies. As for their use in PIV, single images containing multiple exposed particle images (nexp ≥ 2) can be recorded analogous to the photographic method. The same ambiguity removal schemes as in photographic PIV recording (rotating mirror, birefringent crystal) can be employed. If the ﬂow under investigation is suﬃciently slow in comparison to the frame rate of a camera based on this sensor, then single exposed PIV 2

In case of a back-illuminated CCD, the photo-active parts are illuminated from “behind” through the silicone-substrate. Therefore, the back of the device is thinned down to O[10] μm and coated to avoid reﬂexions.

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4 PIV Recording Techniques

recordings can be obtained. In this case the ambiguity removal schemes are not needed. The timing charts given in ﬁgure 4.9(a) and (b) summarize how the particle illumination pulses have to be placed to produce single exposed or multiple exposed PIV images. 4.2.2 Frame Transfer CCD

Masked storage area

Row transfer direction

Light sensitive area

The pixel architecture of the frame transfer CCD sensor (ﬁgure 4.6) is essentially equivalent to that of the full-frame CCD sensor with the diﬀerence that the lower half of its rows are masked oﬀ and cannot be exposed by incoming light. Once exposed, the rows of accumulated charge are rapidly shifted down into the masked-oﬀ area at rates as fast as Δtrow-shift = 1 μs per row. The entire image can thereby be shielded from further exposure within Δttransfer = 0.5 − 1 ms depending on the vertical clocking speed and vertical image size. However, the sensor does stay active during the vertical transfer time such that vertical smear is possible. Charge stored within the masked area prior to the shift is lost however. Once the shift has been completed, the sequential read-out is equivalent to that of a full-frame CCD. The frame transfer CCD sensor oﬀers two application possibilities in PIV recording. The fast transfer of the accumulated charge into the storage area allows two single exposed PIV images to be captured at a time delay, Δt, slightly longer than the transfer time, for example Δt ≥ Δttransfer . To achieve this, the illumination pulses are placed such that the ﬁrst pulse occurs immediately before the frame transfer event (i.e. on frame n), while the second pulse occurs immediately thereafter (i.e. on frame n + 1, see ﬁgure 4.9(c)).

Charge-to-voltage converter

Analog shift register

Fig. 4.6. Frame transfer CCD sensor layout.

4.2 Digital Cameras for PIV Vertical shift registers (masked)

Vertical shift register

Row transfer direction

Masked storage area

105

Pixel layout Odd-line pixel

Even-line pixel Charge barrier

Light sensitive area Charge-to-voltage converter

Horizontal shift register

Fig. 4.7. Interline transfer CCD layout.

This placement of the illumination pulses with respect to the CCD sensor’s periodic exposure cycles is sometimes referred to as frame straddling. At standard video resolution and a ﬁeld of view of 20 cm the measurement of ﬂow velocities up to the order of 5 m/s is possible. In this case the PIV frame rate is half the camera frame rate (e.g. 15 Hz for NTSC video, 12.5 Hz for PAL video). The frame transfer CCD sensor can alternatively be used to impose an image shift in order to remove the displacement bias associated with double exposure single frame PIV recording. This is achieved by placing the ﬁrst illumination pulse just prior to the start or at the beginning of the vertical transfer period (ﬁgure 4.9(d)). The second light pulse is placed such that it occurs while the collected charge of the ﬁrst exposure is transferred into the masked area. For example, at a transfer rate of Δtrow-shift = 1 μs per row, a pulse delay of Δt = 10 μs would produce a maximum of 10 pixel image shifts. In this mode of operation the PIV frame rate is equal to the camera’s frame rate. In this context it should be mentioned that standard CCD sensors are also capable of performing this type of displacement biasing [113]. 4.2.3 Interline Transfer CCD The interline transfer CCD sensor takes its name from additional vertical transfer registers located between the active pixels. Typically two vertically adjacent pixels share a common storage site in the vertical shift registers as shown on the right of ﬁgure 4.7. Charge accumulated in the active area of the pixel can be rapidly transferred into the storage area (Δttransfer < 1 μs). This fast charge dumping feature also opens the possibility for full electronic shuttering on the sensor level. This type of sensor is among the most common in consumer video

106

4 PIV Recording Techniques

products and hence readily available at standard video resolutions (see also table 2.9) although high resolutions are also in use. Contrary to the previously described framing CCD sensors, the interline transfer CCD sensor tends to be more sensitive in the blue-green region of the light spectrum (see ﬁgure 2.49). The major drawback of these sensors is their reduced ﬁll factor due to the additional storage sites next to each light-sensitive area. Additional microlenses on the face of the sensor improve their light gathering capability. The alternative back-thinning approach is not possible with this sensor due to the additional storage sites which have to stay shielded against light exposure. Since the sensor only provides half as many charge storage sites as there are active pixels, an image can only be stored at half the vertical resolution. This storage mode is an artifact of standard video transmission which separates a full image frame into distinct ﬁelds containing only even or odd lines (see section 2.10). Thus, the sensor can oﬀer only half the vertical resolution in the shuttered mode of operation. For example if the odd lines of captured image data are read out from the sensor, the even lines are accumulating charge and vice versa. As a result, the odd and even lines are active during diﬀerent periods of time, resulting in the capture of image data that is staggered by the period of one ﬁeld for adjoining video lines – captured images of moving objects seem to ﬂicker back and forth. Cameras based on the interline transfer CCD have two possible applications in electronic PIV recording. The electronic shutter can be used to shutter the light of a continuous wave laser such as an argon-ion laser (ﬁgure 4.9(e)). This electronic shutter is implemented by means of a clamping voltage on each pixel which inhibits the photon-to-charge conversion of the CCD for most of the framing interval; leaving a short period for photon collection just prior to the charge transfer event. Since the temporal position of the light-sensitive period is ﬁxed relative to the camera’s ﬁeld rate, the eﬀective pulse delay, Δt, will be equivalent to the ﬁeld rate (e.g. Δt = 20 ms for CCIR, Δt = 16.7 ms for NTSC). This limits the application to low-speed phenomena which can however be resolved in time. In the second mode of operation the electronic shutter has to be completely deactivated making the sensor active at all times except for the brief charge transfer event. The illumination is provided by a pulsed laser (ﬁgure 4.9(f)). In this case the frame straddling method is applied in which the ﬁrst of the two illumination pulses is placed before the transfer event and the second right afterward (see also section 4.2.2). Thereby two single exposed images with half the vertical resolution (i.e. ﬁelds) can be recorded with a very short eﬀective pulse delay which may be as short as the 1–2 μs duration of the charge transfer. The technique was ﬁrst applied by Wernet [110] to measure a free jet in the 100 m/s range using a CW-laser and an intensiﬁed, interlinetransfer CCD camera. Further implementations and applications are reported in [97, 100, 106, 111]. Since two ﬁelds comprise a frame the eﬀective PIV image frame rate is equivalent to the camera frame rate (e.g. 25 Hz for CCIR, 30 Hz for NTSC), given that the pulse laser can provide pulse pairs at this

4.2 Digital Cameras for PIV

107

frequency. In this context it should be observed that PIV recording based on interlaced images is only reliable when the particle image diameter is large enough such that particle images will not disappear in the inactive scan-lines of the second exposure, and vice versa. 4.2.4 Full-Frame Interline Transfer CCD This sensor is a derivative of the interline transfer CCD described before with the diﬀerence that each active pixel has its own storage site (ﬁgure 4.8, right side). Introduced in the ﬁrst half of the 1990’s cameras based on these progressive scan sensors rapidly gained popularity in the ﬁeld of machine vision as they removed all the artifacts associated with interlaced video imaging. The electronic shutter can be applied to the entire image rather than to one of its ﬁelds as for standard interline transfer CCDs. Here also microlenses above each pixel help raise the eﬀective ﬁll factor from 20% to up to 60%. The fast transfer of the entire exposed image into the adjoining storage sites within a few microseconds in conjunction with higher resolution formats, in a departure from the standard video resolutions, has extended the application of single exposure double frame PIV images into the transonic ﬂow velocity domain. Here the maximum PIV image frame rate is half the camera’s frame rate with pulse delays as low as Δt = 1 μs [109, 112]. As these cameras also frequently have asynchronous reset possibilities, their range of application is the most ﬂexible of the CCD systems described in this chapter. A timing diagram for PIV recording based on this sensor is given in ﬁgure 4.9(g).

Vertical shift registers (masked)

Vertical shift register

Pixel layout

Odd-line pixel

Row transfer direction

Masked storage areas

Even-line pixel Charge barrier Light sensitive area Charge-to-voltage converter

Horizontal shift register

Fig. 4.8. Progressive scan, interline transfer CCD layout.

108

4 PIV Recording Techniques Time

Standard CCD Camera frame rate

(a) Mode 1

On Off

Exposure n-1

n

n+1

n+2

n-1

n

n+1

n+2

Read-out

Laser light pulses

Camera frame rate

(b) Mode 2 Laser light pulses Charge transfer period

Frame-transfer CCD Camera frame rate

n-1

n

n+1

n+2

n-1

n

n+1

n+2

(c) Mode 1 Laser light pulses

Camera frame rate

(d) Mode 2 Laser light pulses

Interline transfer CCD (interlaced) Camera frame rate

(e) Mode 1

Camera field rate

n-1 Even

Odd

Even

Odd

Even

Odd

Even

Odd

n

n+1

n+2

n

n+1

n+2

Electronic shutter

(f)

Mode 2

Camera field rate Laser light pulses

Full-frame interline transfer CCD Camera frame rate (g)

n-1

Laser light pulses

Fig. 4.9. Timing diagrams for PIV recording based on various types of CCD sensors.

4.2.5 Active Pixel CMOS Sensor The most relevant CMOS sensors for PIV applications are based on the active pixel sensor (APS) technology in which, in addition to the photodiode, a readout ampliﬁer is incorporated into each pixel. This converts the charge accumulated by the photodiode into a voltage which is ampliﬁed inside the pixel and then transferred in sequential rows and columns to further signal processing circuits as described in section 2.8.2. As can be seen in ﬁgure 4.10 each pixel contains a photodiode, a triad of transistors that converts accumulated electron charge to a measurable voltage, resets the photodiode and transfers the voltage to a vertical column bus. In addition to that, some CMOS sensors contain shutter transistors for each pixel. The ampliﬁer transistor represents the input device of what is generally termed a source follower. It converts the charge generated by the photodiode into a voltage that is output to the column bus. The reset transistor controls integration time, and a row-select transistor connects the pixel output to the column bus for readout. During the operation of the sensor, ﬁrst the reset transistor is initialized in order to drain the charge from the photosensitive region. Then, the inte-

4.2 Digital Cameras for PIV

109

Uref Pixel reset Source follower

Row selector

Fig. 4.10. APS-CMOS pixel layout with integrated ampliﬁer (source follower).

gration period begins and the electrons from the photo diode are stored in the potential well lying beneath the surface. After the integration period, the row-select transistor connects the ampliﬁer transistor in the selected pixel to its load to form a source follower and thus converts the charge of the photodiode into a voltage on the column bus. The cyclic repetition of this process to read out every row thereby forms the image. One problem that frequently occurs when recording PIV images close to model surfaces with CCD cameras is blooming (see section 2.8.1). In this case, the high intensity of the light scattered from the surface leads to a migration of electrons to neighboring pixels, that makes the recording of particle images in those areas impossible. One of the main advantages of most CMOS sensors is there ability to record images with high contrast without blooming. 4.2.6 High-Speed CCD Cameras In section 4.2 the diﬀerent types of CCD sensors were introduced. These sensors, especially the full-frame interline transfer CCD, is chosen for most of the conventional PIV applications with moderate frame rates. However, special developments based on the CCD technology can also be used for high-speed PIV. High frame rates, together with the relatively high spatial resolution needed for most PIV applications result in large amount of data that has to be transferred from the chip into the storage. This required high clock speeds and, as a consequence thereof, a high bandwidth of the read out electronics. The high bandwidth of the sensor increases the noise while the eﬃciency decreases. Those problems resulted in sensor designs, in which the sensor is divided into smaller segments, which are read out in parallel. The required read out speed could therefore be reduced by the number of separate channels [108]. In addition to that, most of the CCD based high-speed cameras contain a so-called split-frame storage into which half of the image is read-out from the top of the chip and the other half from the bottom. In spite of all the eﬀorts to increase the combined pixel readout rates by parallel transfer and storage, the rates are considerably higher in comparison to conventional cameras and the read out electronics need to be carefully optimized with respect to noise.

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Todays commercially available high-speed CCD cameras oﬀer frame rates of about 1000 frames/s at moderate resolutions in the 512 × 512 pixel range. 4.2.7 High-Speed CMOS Cameras for PIV Recording The most advanced high-speed cameras suitable for PIV have CMOS sensors. The parallel structure of the CMOS sensor, which has been described previously, allows more channels than were practical with CCDs. CMOS cameras used for high-speed PIV application frequently have 32 or more output channels. For the clock speed CMOS cameras oﬀer higher pixel rates, since CMOS pixels can be read out with one clock pulse, while CCDs typically require two to four clock pulses per pixel read out. In contrast to most CCD sensors, highspeed CMOS sensors have electronic shuttering integrated in each pixel. As already mentioned, CMOS sensors are not prone to blooming. While saturation eﬀects can occur, high pixel intensities do not eﬀect the image in the way blooming in CCD sensors does. Windowing, the formerly described technique to read smaller sub windows of the CMOS sensor array, is a feature that allows to produce higher frame rates at reduced image resolutions. This feature is available in most of the high-speed CMOS cameras and frequently used for high-speed PIV recording, since it allows for using the very high repetition rates of most high-speed lasers. Some CMOS cameras allow an extremely ﬂexible read-out of the sensor. They may have hundreds of selectable resolutions, helping the user to obtain the desired resolution at maximum performance. Advanced CMOS-sensor designs use fewer components per pixel than earlier designs and take advantage of smaller size of the components. This leads to a better light sensitivity than that of most CCD based high-speed cameras. Additionally, the image quality of new CMOS high-speed cameras has signiﬁcantly been improved. And the fact that some leading manufactures of digital SLR cameras nowadays oﬀer CMOS sensors in these products, leads to the assumption that this trend is continuing. Today’s commercially available high-speed CMOS cameras oﬀer frame rates exceeding 3000 frames/s at full mega pixel resolution. At 512 × 512 pixel resolution they even reach 10, 000 frames per second. In order to save the enormous amount of data, that can be recorded within seconds with such a performance, many cameras have memory of up to 16 GB on board where data are stored prior to transfer to the computer.

4.3 Single Frame/Multi-Exposure Recording When using photographic ﬁlm or single frame digital cameras for PIV recording, two or more exposures of the same particles are stored on a single recording. Therefore, the sign of the direction of the particle motion within each interrogation window cannot be determined uniquely, since there is no way to

4.3 Single Frame/Multi-Exposure Recording

111

decide which image is due to the ﬁrst and which is due to the second illumination pulse. Although, for many applications the sign of the velocity vector can be derived from a priori knowledge of the ﬂow, other cases involving ﬂow reversals, such as in separated ﬂows, require a technique by which the sign of the displacement can be determined correctly. 4.3.1 General Aspects of Image Shifting The great interest in PIV measurements in many diﬀerent ﬁelds of research requires a ﬂexible technique for ambiguity removal that can be applied to a variety of experimental situations. Especially for aerodynamic investigations it is very important to be able to apply this technique to high-speed ﬂows, that is with short time intervals of the order of a few microseconds between the exposures. One method is the image shifting technique as described by various authors [86, 89, 98] that removes the directional ambiguity. Image shifting enforces a constant additional displacement on the image of all tracer particles at the time of their second illumination. In contrast to other methods for ambiguity removal, which require a special, or at least a specially adapted, method of evaluation, image shifting leaves the proven evaluation process employing statistical methods unchanged. Elimination of the ambiguity of direction: Figure 4.11 explains the removal of directional ambiguity of two tracer particles by means of image shifting, one of which is moving to the right and the other one is moving to the left (ﬂow reversal). Introducing an additional image shift, dshift , to the ﬂowinduced displacements of the particle images d1 and d2 , the situation changes. By a selection of the additional image shift, dshift , in such a manner that it is always greater than the maximum value of the reverse-ﬂow component (i.e. d2 ), it is guaranteed that the tracer images of the second exposure are always located in the “positive” direction with respect to the location of the ﬁrst exposure (ﬁgure 4.11). The elimination of the directional ambiguity does not depend on the direction within the observation plane where the shift takes place if the maximum of the corresponding reverse-ﬂow component is predicted accordingly. Thus, an unambiguous determination of the sign of the displacement vector is established. The value and correct sign for the displacement vectors d1 and d2 will be obtained by subtracting the “artiﬁcial” contribution dshift after the extraction of the displacement vectors for the PIV recording. 4.3.2 Optimization of PIV Recording for Autocorrelation Analysis by Image Shifting As already mentioned in chapter 3 the application of the cross-correlation technique for two subareas of a single frame/multi-exposure recording instead of performing an autocorrelation on a single subarea, increases the ﬂexibility of the PIV system. This evaluation approach cannot remove the directional ambiguity of the velocity vectors or handle situations where the image

112

4 PIV Recording Techniques d shift d 1 + dshift d1

d2

d1 d 2 + dshift d2

Fig. 4.11. Elimination of the ambiguity of direction of the displacement vector as observed in the recording plane.

displacements are of the order of the particle image diameter. However, the pulse separation time can be adapted in a wider range, because the size of the two interrogation windows and their displacement can be adapted later on during evaluation. If the evaluation system is not ﬂexible enough to allow cross-correlation of slightly displaced interrogation areas, the evaluation process of doubly exposed PIV recordings has to be analyzed by autocorrelation. This problem often occurred in the past when optical evaluation was used in order to improved the signal-to-noise ratio. Besides resolving the directional ambiguity, image shifting is required in order to optimize the recording for later autocorrelation evaluation. More details on the diﬀerent aspects of image shifting were described by Raffel & Kompenhans [104] 4.3.3 Realizations of Image Shifting The most widely used experimental technique for image shifting involves the use of a rotating mirror over which the observation area within the ﬂow is imaged on to the recording area in the camera. The magnitude of the additional displacement of the images of the tracer particles depends on the angular speed of the mirror, the distance between the light sheet plane and the mirror, the magniﬁcation of the imaging system and the time delay between the two illumination pulses [98]. Various experimental setups employing rotating mirrors were realized by diﬀerent authors and have shown good experimental results, such as the investigation of dynamic ﬂow separation (wake with ﬂow reversals) above proﬁles [61]. In order to achieve very high shift velocities, electro-optical methods employing diﬀerently polarized light for illumination have been proposed and applied by Landreth & Adrian [99], Lourenc ¸ o [101] and Molezzi & Dutton [64]. The constant shift of the particle images is obtained by means of birefringent crystals of appropriate thickness. Reuss describes the problems associated with this method, such as “depolarization eﬀects” [105]. Another scheme by which the directional ambiguity problem may be resolved has been presented by Wormell & Sopchak and involves a CCD camera in which the charge associated with the ﬁrst illumination is electronically moved by a known distance within the sensor during the time period between the ﬁrst and the second laser pulse [113] (see section 4.2.2). This

4.3 Single Frame/Multi-Exposure Recording

113

arrangement allows a minimum pulse separation of approximately 40 μs. At DLR many successful applications have been performed with a rotating mirror system for image shifting. A detailed description of such a system, is given below. A rotating mirror based image shifting system has the following advantages: it can easily be implemented into already existing apparatus for employing single frame/multi exposure recordings; it makes no additional demands on the scattering characteristics of the particles (light depolarization eﬀects can be neglected); the shift velocity can be adapted to the problem very easily; and a much higher shift velocity can be attained than by moving the entire camera. It should be mentioned here that the rotating mirror technique is limited by a maximum frame rate which is given by the product of the angular speed and number of mirror surfaces. In most aerodynamic applications the maximum framing rate is higher than the repetition rate of a standard pulse laser system. Nevertheless, the framing rate may become a limiting factor, for example, when ﬂow ﬁelds are observed at low ﬂow (and shift) velocities with high resolution in time. Furthermore, it is not possible to synchronize non-periodic ﬂow events with a mirror that is rotating at a constant speed. 4.3.4 Layout of a Rotating Mirror System Figure 4.12 shows the high-speed rotating mirror system for image shifting as used at DLR in G¨ ottingen. It allows shift velocities exceeding 500 m/s without any noticeable reduction of the optical quality of the images. The system shown in ﬁgure 4.12 consists of the following components: a shaft mounted in precision bearings, a mirror mount attached to one end of the shaft and an optical encoder connected to the other end. The mirror assembly is driven by a stepper motor with stable revolution frequencies ranging from 1 to 100 Hz. A toothed belt guarantees slip free transmission while a revolving mass attached to the rotating shaft compensates for the velocity ﬂuctuations of the motor drive. The optical encoder is a commercially available angle encoder that is coupled to the shaft using a twist free shaft clutch and supplies the signals for the laser triggering as well as angular frequency monitoring. This precisely machined setup ensures that the 90◦ angle required between objective lens and observation plane is exactly reproducible even in a noisy industrial environment. The adaption of the signal frequency from the angle encoder to the repetition rate of the pulse laser is performed by digital dividers, see ﬁgure 4.13. The angular position of the mirror at the time of image capture is kept constant by means of a digital controller and is phase-locked to the trigger for the laser pulses. The controller itself was designed in such a manner that it is able to handle severe electronic noise due to the stepper motors or the electric drive systems present in many wind tunnel environments. The procedure

114

4 PIV Recording Techniques

Encoder

Motor

35mm-Camera

1

2 3

Flywheel

Mirror

Fig. 4.12. Schematic diagram of the rotating mirror system.

of controlling the observation angle is as follows: Signal (a) is the increment signal from the encoder. The resolution of 1000 pulses per revolution as delivered by the encoder was suﬃcient. Signal (b) is the reference signal (recording position) from the encoder which usually is one inverted pulse per revolution. Signals (a) and (b) are combined by a logical AND gate in the angle controller. The resulting signal (c) and a signal obtained from the light pulse of the laser (d) are combined by a logical OR gate. This signal, (e), is used to control the laser and the camera for recording. If the reference signal (b) and the light

Fig. 4.13. Flow and impulse chart of the angle control.

4.3 Single Frame/Multi-Exposure Recording

115

pulse (dI ) do not coincide (Case I), diﬀerent pulse rates are obtained for laser (eI ) and mirror control (a). This leads to a phase shift in each revolution until the reference signal (b) and the light pulse (dII ) coincide temporally. In this case (Case II) the pulse rates of the increment signal of the encoder (a) and of the control signal for the laser (eII ) are the same and the control error equals zero. The digital dividers included in the signal chain allow a variation of the mirror revolutions and the framing rate. The main advantage of this image shifting setup is the easy handling and the ﬂexibility in adjusting the shift velocity, which can be chosen from a wide range in very small steps just by “pressing a button”. This section described how to build a rotating mirror system which is precisely phase locked with the laser light pulses. However, there exist additional error sources in rotating mirror systems. In particular the way in which the virtual image of the tracer particles is moved by the rotation of the mirror has to be taken into account. It will be shown in the following that the movement of the particle images due to the rotating mirror are not uniform over the recording, but vary locally. By means of the equations arising from the system’s optical geometry derived in the following, algorithms can be implemented, which allow a full compensation of these errors. 4.3.5 Calculation of the Mirror Image Shift In the literature, the motion of the virtual image due to the rotation of the mirror is derived from geometric relations in two dimensions with assumptions that are not valid in general [92]. Only the displacement of the depth-of-ﬁeld center from the light sheet was determined in order to estimate the eﬀect of this parameter on the defocusing of the images of the particles [103]. In most cases, the shift of the particle images dshift is assumed to be constant in the entire observation area: dshift = 2ωm Zm M Δt

(4.1)

where ωm is the angular velocity of the rotating mirror, Zm the distance between the light sheet and mirror-axis (Zm = const.), M the magniﬁcation (image size/object size) and Δt = t − t the time delay between the light pulses. The fact that the distance between the mirror axis and a point of the virtual light sheet is not constant but a function of the x-coordinate leads to an error of equation (4.1) of more than 1% of the mean image shift at typical experimental conditions. A further, additional error results from the direction of movement of the virtual image. Since the z component of the virtual particle image shift increases towards the edges of the virtual observation area, there is also an inﬂuence on the x and y components due to the perspective projection of the virtual particle image on to the recording plane. In order to fully describe the eﬀects of a shift component perpendicular to the virtual light sheet on the position of the image points in the coordinate system x, y, z, the imaging through the lens must also be considered.

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4 PIV Recording Techniques

Details on the mathematical procedure can be found in [104]. As a result the image displacement d = x − x due to mirror rotation can be obtained (see equation (4.2) and equation (4.3)). Equation 4.1, which is given in the literature for the shift of the tracer images due to a rotating mirror, can be derived from the exact solution by the following assumptions: •

Δt approaches zero;

•

the axis of rotation of the mirror lies on the optical axis;

•

the particle images are located close to the center of the image.

The ﬁrst assumption leads to a negligible error due to the fact that the shift angle 2ωm Δt is usually less than 0.1◦ . The relations sin(2ωm Δt) = 2ωm Δt and cos(2ωm Δt) = 1 lead to the following formulae for the calculation of the particle image shift, when the formulae 1/f = 1/z0 − 1/Z0 and M = z0 /Z0 (see section 2.6.1) are used: dx (x, y) =

x − M · Zm 2ωmΔt −x (x + Xm M ) 2ωm Δt f −1 (1 + M )−1 + 1

(4.2)

dy (x, y) =

y −y . (x + Xm M ) 2ωm Δt f −1 (1 + M )−1 + 1

(4.3)

These formulae are recommended for the practical use of image shifting by means of rotating mirror systems. In many cases, the distance Xm from the mirror axis to the optical axis of the lens can be adjusted to zero. However, asymmetric conﬁgurations with Xm greater than zero allow the use of a smaller mirror. The assumption that the particle images are located close to the center of the recording (x ∼ = 0; y ∼ = 0) leads to a systematic shift error in the measured displacement data. This error = (dx − dshift , dy ), which can be calculated when using equations (4.2) and (4.3), must be accounted for in the evaluation of PIV images. As an example, the diﬀerence between the local tracer image shift resulting from the mirror rotation, (dx , dy ), and the tracer image shift in the center of the observation ﬁeld, dshift , was calculated, given a set of typical experiment parameters: a magniﬁcation of M = 1 : 4, pulse separation, Δt = 12 μs, distance from the rotating mirror axis to the light sheet Zm = 512 mm, mirror speed ωm = 62.8 rad/s, focal length f = 100 mm. Figure 4.14 indicates that the image displacement increases to 200 μm at the edges of the recording with a shift of only 194 μm present at the center of the recording.

4.3 Single Frame/Multi-Exposure Recording

117

Reference Vector: 1 m/sec

0.08

0.06

y [m] 0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x [m]

Fig. 4.14. Map of the calculated tracer image shift (dx −dshift ,dy ) due to perspective transformation.

4.3.6 Experimental Determination of the Mirror Image Shift The error in the particle image shift on the PIV recording due to the use of a rotating mirror system for image shifting has also been determined experimentally. For this purpose PIV measurements were performed in air at rest. Tracer particles were injected into the air in the area of the observation ﬁeld. The experimental setup is deﬁned by the same parameters as given in the example calculation in the previous section. Particle motion resulting from convection and/or gravity eﬀects was minor and therefore is negligible compared to the shift velocity of 64 m/s. Figure 4.15 displays the diﬀerence between the local tracer image shift and the tracer image shift in the center of the observation ﬁeld. The scaling of the vector ﬁeld is identical to ﬁgure 4.14. The deviations between the experimental values and the theoretical values as calculated with equations (4.2) and (4.3) lie within the measurement resolution of our PIV evaluation system. Therefore we can safely assume that the rotating mirror-camera system was modeled correctly. The conventional practice – as described in the literature – of assuming that the value for the tracer image shift is constant over the entire image leads to systematic errors of 2–3% in the actual values of tracer image shift even for small observation angles. A virtual shift greater than the mean ﬂow velocity is chosen frequently. This results in errors of up to 10% of the mean ﬂow velocity. Equations (4.2) and (4.3) indicate that the shift of the particle images depends on the magniﬁcation, M ,

118

4 PIV Recording Techniques Reference Vector: 1 m/sec

0.08

0.06

y [m] 0.04

0.02

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x [m]

Fig. 4.15. Map of the experimentally determined tracer image shift (dx − dshift , dy ) of particles at rest on the recording. Same parameters were used for calculation and experiment.

the focal length of the lens, f , as well as the position of the mirror axis. These parameters stay constant during an experiment. Therefore, in order to be able to correct the measurement results appropriately, the locally varying tracer image shift has to be determined only once for a given recording conﬁguration by means of the equations given above. An alternative practical approach had also been implemented which avoids the need for an exact measurement of the parameters in equations (4.2) and (4.3). Usually after completion of the experiments, the ﬂuid is left to come to rest while the shifting mirror continues to operate at constant angular speed. A PIV recording of the quiescent ﬂuid (Uresidual Ushift ) only records the eﬀects of the image shift. By performing a second order least squares ﬁt to the obtained virtual ﬂow ﬁeld, the distortions brought about by the mirror can be very well estimated and can subsequently be used to correct the actual PIV data of the original experiment. Such a direct calibration with subsequent ﬁt to the expected function can also be used to analyze and compensate for the nonuniform image shift of other devices (e.g. nonuniform image shift by a birefringent crystal as reported by Reuss [105]). Since the availability of the progressive scan CCD technology section 4.2, most PIV recordings are obtained in double-frame mode. However, ambiguity removal by means of image shifting might still be of interest if the cameras do not allow for frame transfer within the pulse separation time required.

4.4 Multi-Frame PIV Recording

119

4.4 Multi-Frame PIV Recording In the following, a short summary of diﬀerent techniques for multi-frame PIV recording is given. A more detailed description of the most frequently used video-based implementations of double frame/single exposure recording is presented in a subsequent section 4.4.1. The main advantages of separating the light of the subsequent recordings on to diﬀerent frames have already been stated above: it solves the directional ambiguity, allows the adaptation of the pulse separation time in a wider range, and higher signal-to-noise ratios in the correlation plane are available at the same interrogation window size. In most cases, the improved signal-to-noise ratio will be used to compute the displacement within smaller interrogation windows and therefore increasing the spatial resolution at the same resolution of the recording. Distinguishing the diﬀerent illuminations could be done by coding the light sheets by diﬀerent polarizations [88, 95], or by image separation using a color video camera and a color coded light sheet [87, 90, 91]. Both methods seem to be feasible in some situations, but because of additional optical problems associated with these methods they are no general solution. Depolarization due to glass windows and model surface scattering has to be considered as well as depolarization due to particles of larger size. First tests of the recording of small tracer particles in air with a two-color pulse laser system and color ﬁlm already showed that the focal plane for green light can be displaced by up to 20 mm with respect to the focal plane for red light. The separation of diﬀerent exposures can also be performed by the timing of the image recording with respect to illumination. This can be done, for example, by means of high-speed ﬁlm cameras in combination with copper vapor lasers [376] or multiple oscillator Nd:YAG lasers [107]. However, these experimental setups are very diﬃcult to handle and are suitable only for special applications as for example for ﬂow investigations in piston engines. More general solutions based on the proper timing of video cameras will be described in the following. 4.4.1 Video-Based Implementation of Double Frame/Single Exposure PIV The low cost and frequent availability of standard video equipment and associated PC-based frame grabbers make their use for PIV especially attractive if spatial resolution is not of primary concern. In the following, three schemes capable of providing image pairs of single exposed particle image recordings are brieﬂy described. Mode 1: The ﬁrst approach can only be applied to rather slow ﬂows (Umax < 5 cm/s), which makes it useful only in water applications and is essentially equivalent to the original implementation of DPIV described by Willert & Gharib [174]. As shown in the timing diagram in the upper part of ﬁgure 4.16, the laser light is strobed exactly in phase with the frame

120

4 PIV Recording Techniques

rate of the camera. If a vertical synchronization pulse is not directly available from the camera, it may be obtained using a synchronization stripper. The duration of the light pulse from a continuous wave laser should not exceed one fourth the frame period (approximately 8 ms) to avoid excessive streaking of the particle images in the recording. Mode 2: The second method of PIV recording based on video imaging utilizes the electronic shutter frequently available in today’s video cameras. However, as mentioned before, these shutters generally work only on a ﬁeld-byﬁeld basis such that the recorded PIV images will have only half the vertical resolution. The interlacing nature of video can thus provide the user with video frames, each containing a PIV image pair. The odd video ﬁeld (i.e. all odd lines) comprise the ﬁrst PIV recording, while the even ﬁeld (i.e. all even lines) correspond to the second PIV recording. The time delay between the recordings and hence the light pulse delay, Δt, is equal to the ﬁeld interval of 1/50th or 1/60th of a second, which doubles the temporal resolution as well as the maximum recordable ﬂuid velocity compared to Mode 1 (ﬁgure 4.16, middle). Another advantage of this method is that the utilization of the electronic shutter allows continuous light sources to be used. The shutter time should be chosen to be long enough to allow the sensor to be exposed but short enough to avoid excessive streaking, typically less than one fourth of the ﬁeld rate (e.g. 1/250 s). To process the digitized video frame, the user (a) Mode 1

Time On

Camera frame rate

n-1

Off

n

n+1

n+2

n+1

n+2

Interline transfer events

Light sensitive period Odd field Even field On

Laser light pulses

Off

(b) Mode 2 n-1

Camera frame rate Camera field rate

Odd

n Even

Odd

Even

Odd

Even

Odd

Even

Interline transfer events

Electronic shutter w/ continuous light source Light sensitive period

(c) Mode 3 Camera frame rate

n-1

Camera field rate

Odd

n Even

Odd

n+1 Even

Odd

n+2 Even

Odd

Even

Interline transfer events

Laser light pulses

On Off

Fig. 4.16. Timing diagrams for PIV recording based on standard video equipment.

4.4 Multi-Frame PIV Recording

121

must ﬁrst separate the interlaced image into an image pair, optionally also interpolating the missing lines of each image. Mode 3: This approach is similar to the one above with the diﬀerence that no electronic shutter is used on the sensor and that the illumination pulses are provided asynchronously (i.e. frame straddling as described on page 106). This approach extends video-based PIV recording to provide images of highspeed ﬂows [97, 100, 106]. The associated timing diagram is shown on the bottom of ﬁgure 4.16.

5 Image Evaluation Methods for PIV

This chapter treats the fundamental techniques for statistical PIV evaluation. In spite of the fact that most realizations of PIV evaluation systems are quite similar – in nearly every case they are based on digitally performed Fourier algorithms – we will also consider optical techniques because they are still important for the classiﬁcation and understanding of existing setups. In order to extract the displacement information from a PIV recording some sort of interrogation scheme is required. Initially, this interrogation was performed manually on selected images with relatively sparse seeding which allowed the tracking of individual particles [125, 178]. With computers and image processing becoming more commonplace in the laboratory environment it became possible to automate the interrogation process of the particle track images [118, 185, 231, 246]. However, the application of tracking methods, that is to follow the images of an individual tracer particle from exposure to exposure, is only practicable in the low image density case, see ﬁgure 1.5(a). The low image density case often appears in strongly three-dimensional high-speed ﬂows (e.g. turbomachinery) where it is not possible to provide enough tracer particles or in two phase ﬂows, where the transport of the particles themselves is investigated. Additionally, the Lagrangian motion of a ﬂuid element can be determined by applying tracking methods [127, 162, 229]. In principle, however, a high data density is required on the PIV vector maps, especially for the comparison of experimental data with the results of numerical calculations. This demand requires a medium concentration of the images of the tracer particles in the PIV recording. (In particular, in air ﬂows it is not possible to achieve a high image density, because beyond a certain level the number of detectable images cannot be increased by further increasing the tracers density in the ﬂow [53].) Medium image concentration is characterized by the fact that matching pairs of particle images – due to subsequent illuminations – cannot be detected by visual inspection of the PIV recording, see ﬁgure 1.5(b). Hence statistical approaches, which will be described in the next sections, had to be developed. After a statistical evaluation has been performed ﬁrst, tracking algorithms can be applied additionally in order to

124

5 Image Evaluation Methods for PIV

achieve subwindow spatial resolution of the measurement, which is known as super resolution PIV [179]. However, since the extraction of the displacement information from individual particle images requires spatially well resolved recordings of the particle images, those techniques are more appropriate to increase the spatial resolution of photographic PIV recordings than that of digital recordings. Tracking algorithms have continuously been improved during the past decade. Methods like the application of neural networks [134, 250] seem to be very promising. Thus, for some applications particle tracking might be an interesting alternative to statistical PIV evaluation methods on which we focus in this book. Readers interested in obtaining a survey of tracking methods are referred to the survey paper by Grant [36], to the lecture notes on Threedimensional velocity and vorticity measuring and image analysis techniques, edited by Th. Dracos [34] and to the section on low image density PIV in the SPIE milestone series on PIV [35]. A comparison between cross-correlation methods and particle tracking techniques together with an assessment of their performance has recently been performed in the framework of the International PIV Challenge [48, 50].

5.1 Correlation and Fourier Transform 5.1.1 Correlation The main objective of the statistical evaluation of PIV recordings at medium image density is to determine the displacement between two patterns of particle images, which are stored as a 2D distribution of gray levels. Looking around in other areas of meteorology, it is common practice in signal analysis to determine, for example, the shift in time between two (nearly) identical time signals by means of correlation techniques. Details about the mathematical principles of the correlation technique, the basic relations for correlated and uncorrelated signals and the application of correlation techniques in the investigation of time signals can be found in many textbooks [2, 20]. The theory of correlation can be extended in a straight forward manner from the one dimensional (1D time signal) to the two-dimensional (2D gray value distribution) case [4]. In chapter 3 the use of auto- and cross-correlation techniques for statistical PIV evaluation has already been explained. Analogously to spectral time signal representations, a 2D spatial signal I(x, y) the power ˆ x , ry )|2 can be determined where rx , ry are spatial frequencies in spectrum |I(r orthogonal directions. The basic theorems for correlation and Fourier transform known from the theory of time signals are also valid for the 2D case (with appropriate modiﬁcations) [4]. For the calculation of the autocorrelation function two possibilities exist: either direct numerical calculation or indirectly (numerically or optically), using the Wiener-Kinchin theorem [2, 4]. This theorem states that

5.1 Correlation and Fourier Transform Interrogation window with image pairs

Autocorrelation

125

R I+(sx ,sy )

I (x, y)

1.FT

OFT

2.FT (FT-1 )

2

(?) Complex spectrum I (rx , r y)

Power spectrum 2 I (rx , r y)

Fig. 5.1. Sketch of relation between 2D correlation function and spatial spectrum by means of the Wiener–Khinchin theorem. FT – Fourier transform, FT−1 – inverse Fourier transform, OFT – optical Fourier transform.

the Fourier transform of the autocorrelation function RI and the power specˆ x , ry )|2 of an intensity ﬁeld I(x, y) are Fourier transforms of each trum |I(r other. Figure 5.1 illustrates that the autocorrelation function can either be determined directly in the spatial domain (upper half of the ﬁgure) or indirectly by Fourier transform FT (left hand side), multiplication, that is the calculation of the squared modulus, in the frequency plane (lower half of the ﬁgure), and by inverse Fourier transform FT−1 (right hand side). 5.1.2 Optical Fourier Transform As already mentioned in section 2.6 the far ﬁeld diﬀraction pattern of an aperture transmissivity distribution is represented by its Fourier transform [10, 18, 102]. A lens can be used to transfer the image from the far ﬁeld close to the aperture. For a mathematical derivation of this result some assumptions have to be made, which are described by the Fraunhofer approximation. These assumptions (large distance between object and image plane, phase factors) can be fulﬁlled in practical optical setups for Fourier transforms. Figure 5.2 shows two diﬀerent setups for such optical Fourier processors. In the arrangement on the left hand side the object, which would consist of a transparency to be Fourier transformed (e.g. the photographic PIV recording), is placed in front of the so-called Fourier lens. In the second setup (right hand side) the object is placed behind the lens. As derived in the book of Goodman [10] both arrangements diﬀer only by the phase factors of the complex spectrum and a scale factor. Light sensors (photographic plates as

126

5 Image Evaluation Methods for PIV

Plane wavefronts

Object

Lens

Spherical wavefronts

Focal plane

Lens

Object

Focal plane

Fig. 5.2. Optical Fourier processor, diﬀerent positions of object and Fourier lens.

well as CCD sensors) are only sensitive to the light intensity. The intensity corresponds to the squared modulus of the complex distribution of the electromagnetic ﬁeld; hence phase diﬀerences in the light wave cannot be detected. Therefore, both of the arrangements shown in ﬁgure 5.2 can be used for PIV evaluation. The result of the optical Fourier transform (OFT, dashed line in ﬁgure 5.1) directly is the power spectrum of the gray value distribution of the transparency. In the following this will be illustrated for the case of a pair of two particle images. White (transparent) images of a tracer particle on a black (opaque) background will form a double aperture on the photographic PIV recording. With good lens systems the diameter of an image of a tracer particle on the recording is of the order of 20 to 30 μm. The spacing between the two images of a tracer particle should be approximately 150–250 μm, in order to obtain optimum conditions for optical evaluation (compare section 4.3). Figure 5.3 shows a cross-sectional cut through the diﬀraction pattern of a double aperture (parameters are similar to those of the PIV experiment). The ﬁgure at the left side shows several peaks of the light intensity distribution under an envelope. The envelope represents the diﬀraction pattern of a single aperture with the same diameter (i.e. the Airy pattern, see section 2.6). The intensity distribution will extend in the 2D presentation in the vertical direction, thus forming a fringe pattern, that is the Young’s fringes. The fringes are oriented normal to the direction of the displacement of the apertures (tracer images). The displacement between the fringes is inversely proportional to the displacement of the apertures (tracer images). If the distance between the apertures (tracer images) is decreased, the distance between the fringes will increase inversely. This is illustrated in the center of ﬁgure 5.3, where the distance between the two apertures is only half that of the example on the left side. It can be seen that the distance between the fringes is increased by a factor of two. The same inverse relation, which is due to the scaling theorem of the Fourier transform, is valid for the envelope of the diﬀraction pattern: if the diameter of the aperture (particle images) decreases, the extension of the Airy pattern will increase inversely (see ﬁgure 5.3, right side). As a consequence, more fringes can be detected in those fringe patterns which are generated by

5.2 Summary of PIV Evaluation Methods a)

-1.0

b)

0.0

1.0 -1.0

Light

127

c)

0.0

1.0 -1.0

Light

0.0

1.0

Light

Fig. 5.3. Fraunhofer diﬀraction pattern of three diﬀerent double apertures, from left to right, ﬁrst the separation between the apertures has been decreased, then – on the right hand side – the diameter of the apertures has been decreased.

smaller apertures (particle images). This is one reason to explain why small and well focused particle images will increase the quality and detection probability in the evaluation of PIV recordings. Due to another property of the Fourier transform, that is the shift theorem, the characteristic shape of the intensity pattern does not change if the position of the particle image pairs is changed inside the interrogation spot. Increasing the number of particle image pairs also does not change the Young’s fringe pattern signiﬁcantly. Of course this is not true for the case of just two image pairs: two fringe systems of equal intensity will overlap, allowing no unambiguous evaluation. 5.1.3 Digital Fourier Transform The digital Fourier transform is the basic tool of modern signal and image processing. A number of textbooks describe the details [2, 4, 15, 29]. The breakthrough of the digital Fourier transform is due to the development of fast digital computers and to the development of eﬃcient algorithms for its calculation (Fast Fourier Transformation, FFT) [2, 4, 5, 29]. Those aspects of the digital Fourier transform relevant for the understanding of digital PIV evaluation will be described in section 5.4.

5.2 Summary of PIV Evaluation Methods In the following the diﬀerent methods for the evaluation of PIV recordings by means of correlation and Fourier techniques will be summarized.

128

5 Image Evaluation Methods for PIV

FT

I I*

FT

-1

t and t + Δ t

Fig. 5.4. Analysis of single frame/double exposure recordings: the fully digital autocorrelation method.

Figure 5.4 presents a ﬂow chart of the fully digital autocorrelation method, which can be implemented in a straight forward manner following the equations given in chapter 3. The PIV recording is sampled with comparatively small interrogation windows (typically 20 - 50 samples in each dimension). For each window the autocorrelation function is calculated and the position of the displacement peak is determined. The calculation of the autocorrelation function is carried out either in the spatial domain (upper part of ﬁgure 5.1) or – in most cases – via the bypass over the frequency plane through the use of FFT algorithms. If the PIV recording system allows the employment of the double frame/single exposure recording technique (see ﬁgure 4.2) the evaluation of the PIV recordings is performed by cross-correlation (ﬁgure 5.5). In this case, the crosscorrelation between two interrogation windows sampled from the two recordings is calculated. As will be explained later in section 5.4, it is advantageous to oﬀset both these samples according to the mean displacement of the tracer particles between the two illuminations. This reduces the in-plane loss of correlation and therefore increases the correlation peak strength. The calculation of the cross-correlation function is generally computed numerically by means of eﬃcient FFT algorithms. Single frame/double exposure recordings may also be evaluated by a crosscorrelation approach instead of autocorrelation (ﬁgure 5.6). In this case the interrogation windows can be chosen of diﬀerent size and/or slightly displaced with respect to each other in order to compensate for the in-plane loss of correlation due to the mean displacement of particle images. Depending on the diﬀerent parameters, autocorrelation peaks may also appear in the correlation plane in addition to the cross-correlation peak. This is illustrated in more detail in ﬁgure 5.6.

FT

t + Δt

I1 I2*

FT -1

t

Fig. 5.5. Analysis of double frame/single exposure recordings: the digital crosscorrelation method.

5.2 Summary of PIV Evaluation Methods

FT

I I*

FT

129

-1

t and t + Δ t

Fig. 5.6. Single frame/double exposure cross-correlation method ﬂow chart.

The counterpart of the fully digital evaluation by means of autocorrelation is a system employing optical Fourier transform (OFT) for evaluation. In order to obtain the autocorrelation function a setup with two optical Fourier processors has to be implemented, following the bypass through the frequency plane as outlined in ﬁgure 5.1. A spatial light modulator is required to store the output of the ﬁrst Fourier processor and to serve as input of the second Fourier processor. This is shown in ﬁgure 5.7. Up to now, no optical setups giving the 2D cross-correlation function for PIV evaluation have been described in literature.

Fig. 5.7. Analysis of single frame/double exposure recordings: the fully optical method.

Computer memory and computation speed being limited in the beginning of the eighties, PIV work was strongly promoted by the existence of optical evaluation methods. The most widely used method was the Young’s fringes method, which in fact is an optical-digital method, employing optical as well as digital Fourier transforms for the calculation of the correlation function. The ﬂow chart of this evaluation method is shown in ﬁgure 5.8. In the next section the fully optical method of PIV evaluation will be treated in order to give an introduction to the problems of this “old-fashioned” technique which still oﬀers some advantages compared to digital techniques.

Fig. 5.8. Analysis of single frame/double exposure recordings: the hybrid (optical/digital) method utilizing the Young’s fringes technique.

130

5 Image Evaluation Methods for PIV

The most commonly used and very ﬂexible digital evaluation methods will be discussed in the sections thereafter in more detail.

5.3 Optical PIV Evaluation In order to achieve high quality in optical PIV evaluation some pre-processing of the recordings is required. Due to the granular composition of the emulsion, photographic noise is contained in every photographic recording in addition to the particle images. This noise hampers the classical optical/digital evaluation of photographic PIV recordings. The noise is generated by scattering of the light from the illuminating laser beam at the ﬁlm grain (grain noise) and variations of the refractive index (phase noise). Principally, phase noise can be reduced by immersing the negative in an index matching liquid as reported by Pickering & Halliwell [122]. However, a much better improvement of the Young’s fringes visibility and, thus, of the probability of detecting valid velocity data during PIV evaluation can be achieved by a two step photographic process [123]. Interrogating the original PIV negative (park images of tracer particleson a bright background), the noise in the Fourier plane where the Young’s fringes are formed reaches a considerable level because the areas on the negative which have the highest transmittance (background) maintain the gross fog (ﬁgure 2.44). By preparing a contact copy from the negative a positive transparency can be obtained (i.e. bright images of tracer particles on a dark background) which reduces the bias transmittance. This process prevents noise being transferred to the contact copy, which will be employed for evaluation, by taking advantage of the nonlinear behavior of the ﬁlm used for copying. Thus, a much better signal-to-noise ratio can be obtained during PIV evaluation especially in regions of the PIV recording where the image density is low. 5.3.1 Young’s Fringes Method An experimental setup for the implementation of the Young’s fringes technique is shown in ﬁgure 5.9. In this setup only the ﬁrst Fourier transform (compare ﬁgure 5.8) is performed optically. In order to determine the local autocorrelation, the input to the ﬁrst optical Fourier processor is achieved by simply illuminating a small area (i.e. the interrogation spot) of the photographic negative of the PIV exposure with a He-Ne laser light beam. After optical Fourier transform by means of an arrangement already shown on the right side of ﬁgure 5.2, the Young’s fringe pattern is obtained in the Fourier plane. The light intensity distribution in the Fourier plane is recorded by means of a video camera. Its image is digitized and stored in a computer. As explained, the spacing of the fringes is

5.4 Digital PIV Evaluation

Image pairs

Laser

Spatial filter

Fourier lens

131

Young's fringes

Object

Fourier plane (Ground glass)

Camera

Fig. 5.9. Setup for PIV evaluation employing the Young’s fringes technique.

inversely proportional to the displacement of matched image pairs. The direction of the fringes is perpendicular to the direction of the displacement. Thus, by evaluating the distance between the fringes and their direction, the magnitude and velocity of the tracer particles in the ﬂow can be determined. The second better and more widely used method to evaluate the Young’s fringes pattern is to perform a second Fourier transform by means of the FFT algorithm and the peak is found numerically in the computer. The major advantage of this procedure was the increased speed (in the nineteen-eighties) and the higher accuracy of the optical-digital method as compared to digitaldigital methods. Setups like the one described here have been widely used in the ﬁrst ten years of the development of PIV. Details about the diﬀerent optical evaluation techniques for PIV and the Young’s fringes method can be found in [37]. The problem in the development of fully optical evaluation systems was to store the output of the ﬁrst optical processor (i.e. the Young’s fringes) in such a way that it can be used as input to a second optical Fourier processor. Only after the development of easy to use and cheap spatial light modulators (SLM’s) it was possible to set up fully operational optical PIV evaluation systems. For the details of fully optical PIV evaluation methods and their experimental realization, the interested reader might refer to [10, 114, 116, 117, 120, 121, 124, 168].

5.4 Digital PIV Evaluation The complexity of the optical assemblies for fully optical PIV evaluation not only implies a thorough understanding of Fourier optics but also requires the use of electro-mechanical parts (e.g. translation stages), digital imaging as well as computer interfaces. With computer speed and storage capacity increasing almost inversely proportional to its cost, it was merely a matter of time before the interrogation could be performed fully digitally. In the case

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of a photographic recording, a desktop slide scanner for its digitization and a personal computer for the subsequent analysis can completely replace the optical interrogation assembly. Recent advances in electronic imaging further allow for a replacement of the rather cumbersome photographic recording process. In the following we describe the necessary steps in the fully digital analysis of PIV recordings using statistical methods. Initially, the focus is on the analysis of single exposed image pairs, that is single exposure/double frame PIV, by means of cross-correlation. The analysis of multiple exposure/single frame PIV recordings can be viewed as a special case of the former. 5.4.1 Digital Spatial Correlation in PIV Evaluation Before introducing the cross-correlation method in the evaluation of a PIV image, the task at hand should be deﬁned from the point of view of linear signal or image processing. First of all let us assume we are given a pair of images containing particle images as recorded from a light sheet in a traditional PIV recording geometry. The particles are illuminated stroboscopically so that they do not produce streaks in the images. The second image is recorded a short time later during which the particles will have moved according to the underlying ﬂow (for the time being ignoring eﬀects such as particle lag, three-dimensionality, etc.). Given this pair of images, the most we can hope for is to measure the straight-line displacement of the particle images since the curvature information between the recording instances is lost. (Acceleration information also cannot be obtained from a single image pair.) Further, the seeding density is too homogeneous that it is diﬃcult to match up discrete particles. In some cases the spatial translation of groups of particles can be observed. The image pair can yield a ﬁeld of linear displacement vectors where each vector is formed by analyzing the movement of localized groups of particles. In practice, this is accomplished by extracting small samples or interrogation windows and analyzing them statistically (ﬁgure 5.10). From a signal (image) processing point of view, the ﬁrst image may be considered the input to a system whose output produces the second image of the pair (ﬁgure 5.11). The system’s transfer function, H, converts the input image I to the output image I and is comprised of the displacement function d and an additive noise process, N. The function of interest is a shift by the vector d as it is responsible for displacing the particle images from one image to the next. This function can be described, for instance, by a convolution with δ(x − d). The additive noise process, N, in ﬁgure 5.11 models eﬀects due to recording noise and three-dimensional ﬂow among other things. If both d and N are known, it should then be possible to use them as transfer functions for the input image I to produce the output image I . With both images I and I known the aim is to estimate the displacement ﬁeld d while excluding the eﬀects of the noise process N. The fact that the signals (i.e. images) are not continuous – the dark background cannot provide any displacement information – makes it

5.4 Digital PIV Evaluation Image 1

133

Image 2 Time = to+Δ t

Time = to Sample 1

Sample 2

I(m,n)

I (m,n)

Displacement estimate

d(m,n)

Field of estimated displacements

Fig. 5.10. Conceptual arrangement of frame-to-frame image sampling associated with double frame/single exposure Particle Image Velocimetry.

necessary to estimate the displacement function d using a statistical approach based on localized interrogation windows (or samples). One possible scheme to recover the local displacement function would be to deconvolve the image pair. In principle this can be accomplished by dividing the respective Fourier transforms by each other. This method works when the noise in the signals is insigniﬁcant. However, the noise associated with realistic recording conditions quickly degrades the data yield. Also the signal peak is generally too sharp to allow for a reliable subpixel estimation of the displacement. Rather than estimating the displacement function d analytically, the method of choice is to locally ﬁnd the best match between the images in a statistical sense. This is accomplished through the use of the discrete

Input image (Image 1)

I(m,n)

Image transfer function (Spatial shift)

Additive noise process

d(m,n)

Output image (Image 2)

I (m,n)

Noise N(m,n)

Fig. 5.11. Idealized linear digital signal processing model describing the functional relationship between two successively recorded particle image frames.

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5 Image Evaluation Methods for PIV

cross-correlation function, whose integral formulation was already described in section 3: RII (x, y) =

L K

I(i, j)I (i + x, j + y) .

(5.1)

i=−K j=−L

The variables I and I are the samples (e.g. intensity values) as extracted from the images where I is larger than the template I. Essentially the template I is linearly ‘shifted’ around in the sample I without extending over edges of I . For each choice of sample shift (x, y), the sum of the products of all overlapping pixel intensities produces one cross-correlation value RII (x, y). By applying this operation for a range of shifts (−M ≤ x ≤ +M, −N ≤ y ≤ +N ), a correlation plane the size of (2M + 1) × (2N + 1) is formed. This is shown graphically in ﬁgure 5.12. For shift values at which the samples’ particle images align with each other, the sum of the products of pixel intensities will be larger than elsewhere, resulting in a high cross-correlation value RII at this position (see also ﬁgure 5.13). Essentially the cross-correlation function statistically measures the degree of match between the two samples for a given shift. The highest value in the correlation plane can then be used as a direct estimate of the particle image displacement which will be discussed in detail in section 5.4.5. Upon examination of this direct implementation of the cross-correlation function two things are obvious: ﬁrst, the number of multiplications per Shift (x=-2, y=2)

Shift (x=0, y=0)

Shift (x=2, y=2)

Shift (x=-1, y=-1)

Shift (x=1, y=-2)

2 0

y -2 -2

0

2

x Cross-correlation plane

Fig. 5.12. Example of the formation of the correlation plane by direct crosscorrelation: here a 4 × 4 pixel template is correlated with a larger 8 × 8 pixel sample to produce a 5 × 5 pixel correlation plane.

5.4 Digital PIV Evaluation Sample 1 I(x,y)

135

Sample 2 I’(x,y) Cross-correlation plane RII (x,y) +16

0

1

-16

32

-16 1

0

+16

64

Fig. 5.13. The cross-correlation function RII (right) as computed from real data by correlating a smaller template I (32×32 pixel) with a larger sample I (64×64 pixel). The mean shift of the particle images is approximately 12 pixels to the right. The approximate location of best match of I within I is indicated as a white rectangle.

correlation value increases in proportion to the interrogation window (or sample) area, and second, the cross-correlation method inherently recovers linear shifts only. No rotations or deformations can be recovered by this ﬁrst order method. Therefore, the cross-correlation between two particle image samples will only yield the displacement vector to ﬁrst order, that is, the average linear shift of the particles within the interrogation window. This means that the interrogation window size should be chosen suﬃciently small such that the second order eﬀects (i.e. displacement gradients) can be neglected. In a later section this will be discussed in more detail. The ﬁrst observation concerning the quadratic increase in multiplications with sample size imposes a quite substantial computational eﬀort. In a typical PIV interrogation the sampling windows cover of the order of several thousand pixels while the dynamic range in the displacement may be as large as ±10 to ±20 pixels which would require up to one million multiplications and summations to form only one correlation plane. Clearly, taking into account that several thousand displacement vectors can be obtained from a single PIV recording, a more eﬃcient means of computing the correlation function is required. Frequency domain based correlation The alternative to calculating the cross-correlation directly using equation (5.1) is to take advantage of the correlation theorem which states that the crosscorrelation of two functions is equivalent to a complex conjugate multiplication of their Fourier transforms: RII ⇐⇒ Iˆ · Iˆ

∗

(5.2)

where Iˆ and Iˆ are the Fourier transforms of the functions I and I , respectively. In practice the Fourier transform is eﬃciently implemented for discrete

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5 Image Evaluation Methods for PIV

Input

Image sampling at position (i,j)

Image 1

(i,j)

Output

Real−to−complex FFT Complex− conjugate multiplication

Image 2

(i,j)

Complex−to−real inverse FFT

Cross− correlation data

Real−to−complex FFT

Fig. 5.14. Implementation of cross-correlation using fast Fourier transforms.

data using the fast Fourier transform or FFT which reduces the computation from O[N 2 ] operations to O[N log2 N ] operations [5, 15, 23]. The tedious twodimensional correlation process of equation (5.1) can be reduced to computing two two-dimensional FFT’s on equal sized samples of the image followed by a complex-conjugate multiplication of the resulting Fourier coeﬃcients. These are then inversely Fourier transformed to produce the actual cross-correlation plane which has the same spatial dimensions, N ×N , as the two input samples. Compared to O[N 4 ] for the direct computation of the two-dimensional correlation the process is reduced to O[N 2 log2 N ] operations. The computational efﬁciency of this implementation can be increased even further by observing the symmetry properties between real valued functions and their Fourier transform, namely the real part of the transform is symmetric: Re(Iˆi ) = Re(Iˆ−i ), while the imaginary part is antisymmetric: Im(Iˆi ) = −Im(Iˆ−i ). In practice two real-to-complex, two-dimensional FFTs and one complex-to-real inverse, two-dimensional FFT are needed, each of which require approximately half the computation time of standard FFTs (ﬁgure 5.14). A further increase in computation speed can of course be achieved by optimizing the FFT routines such as using lookup tables for the required data, reordering and weighting coeﬃcients and/or ﬁne tuning the machine level code [132, 133]. The use of two-dimensional FFT’s for the computation of the cross-correlation plane has a number of properties whose eﬀects have to be dealt with. Fixed sample sizes: The FFT’s computational eﬃciency is mainly derived by recursively implementing a symmetry property between the even and odd coeﬃcients of the discrete Fourier transform (the Danielson–Lanczos lemma [5, 23]). The most common FFT implementation requires the input data to have a base-2 dimension (i.e. 32 × 32 pixel or 64 × 64 pixel samples). For reasons explained below it generally is not possible to simply pad a sample with zeroes to make it a base-2 sized sample. Periodicity of data: By deﬁnition, the Fourier transform is an integral (or sum) over a domain extending from negative inﬁnity to positive inﬁnity. In practice however, the integrals (or sums) are computed over ﬁnite domains which is justiﬁed by assuming the data to be periodic, that is, the signal (i.e. image sample) continually repeats itself in all directions. While for spectral estimation there exist a variety of methods to deal with the associated artifacts,

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137

such as windowing, their use in the computation of the cross-correlation will introduce systematic errors or will even hide the correlation signal in noise. One of these methods, zero padding, which entails extending the sample size to four times the original size by ﬁlling in zeroes, will perform poorly because the data (i.e. image sample) generally consists of a nonzero (noisy) background on which the signal (i.e. particle images) is overlaid. The edge discontinuity brought about in the zero padding process contaminates the spectra of the data with high frequency noise which in turn deteriorates the cross-correlation signal. The slightly more advanced technique of FFT data windowing removes the eﬀects of the edge discontinuity, but leads to a nonuniform weighting of the data in the correlation plane and to a bias of the recovered displacement vector. The treatment of this systematic error is described in more detail below. Aliasing: Since the input data sets to the FFT-based correlation algorithm are assumed to be periodic, the correlation data itself is also periodic. If the data of length N contains a signal (i.e. displacements) exceeding half the sample size N/2, then the correlation peak will be folded back into the correlation plane to appear on the opposite side. For a displacement dx,true > N/2, the measured value will be dx,meas. = dx,true − N . In this case the sampling criterion (Nyquist theorem) has been violated causing the measurement to be aliased. The proper solution to this problem is to either increase the interrogation window size, or, if possible, reduce the laser pulse delay, Δt. Displacement range limitation: As mentioned before the sample size N limits the maximum recoverable displacement range to ±N/2. In practice however, the signal strength of the correlation peak will decrease with increasing displacements, due to the proportional decrease in possible particle matches. Earlier literature reports a value of N/3 to be an adequate limit for the recoverability of the displacement vector [174]. A more conservative, but widely adopted limit is N/4, sometimes referred to as the one-quarter rule [82]. Bias error: Another side eﬀect of the periodicity of the correlation data is that the correlation estimates are biased. With increasing shifts less data are actually correlated with each other since the periodically continued data of the correlation template makes no contribution to the actual correlation value. Values on the edge of the correlation plane are computed from only the overlapping half of the data and should be weighted accordingly. Unless the correlation values are weighted accordingly, the displacement estimate will be biased to a lower value (ﬁgure 5.15). The proper weighting function for this purpose will be described in section 5.4.5. If all of the above points are properly handled, an FFT-based interrogation algorithm as shown in ﬁgure 5.14 will reliably provide the necessary correlation data from which the displacement data can be retrieved. For the reasons given above, this implementation of the cross-correlation function is sometimes referred to as circular cross-correlation compared to the linear cross-correlation of equation (5.1).

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5 Image Evaluation Methods for PIV

1

True peak (with weighting accounted for)

Correlation

Measured peak

Weighting factor due to reduced interrogation window overlap

Fig. 5.15. Bias error introduced in the calculation of the cross-correlation using FFTs.

0 0

Shift

N

Bias error

Calculation of the correlation coeﬃcient For a number of cases it may be useful to quantify the degree of correlation between the two image samples. The standard cross-correlation function equation (5.1) will yield diﬀerent maximum correlation values for the same degree of matching because the function is not normalized. For instance, samples with many (or brighter) particle images will produce much higher correlation values than interrogation windows with fewer (or weaker) particle images. This makes a comparison of the degree of correlation between the individual interrogation windows impossible. The cross-correlation coeﬃcient function normalizes the cross-correlation function equation (5.1) properly: CII (x, y)

cII (x, y) =

σI (x, y) σI (x, y)

(5.3)

where CII (x, y) =

M N

[I(i, j) − μI ] [I (i + x, j + y) − μI (x, y) ]

(5.4)

i=0 j=0

σI (x, y) =

M N

[I(i, j) − μI ]

2

(5.5)

i=0 j=0

σI (x, y) =

M N

[I (i, j) − μI (x, y) ] . 2

(5.6)

i=0 j=0

The value μI is the average of the template and is computed only once while μI (x, y) is the average of I coincident with the template I at position (x, y). It has to be computed for every position (x, y). Equation (5.3) is

5.4 Digital PIV Evaluation

139

considerably more diﬃcult to implement using an FFT-based approach and is usually computed directly in the spatial domain. In spite of its computational complexity, the equation does permit the samples to be of unequal size which can be very useful in matching up small groups of particles. Nevertheless a ﬁrst order approximation to the proper normalization is possible if the interrogation windows are of equal size and are not zero-padded: Step 1: Sample the images at the desired locations and compute the mean and standard deviations of each. Step 2: Subtract the mean from each of the samples. Step 3: Compute the cross-correlation function using 2D-FFTs as described in ﬁgure 5.14. Step 4: Divide the cross-correlation values by the standard deviations of the original samples. Due to this normalization the resulting values will fall in the range −1 ≤ cII ≤ 1. Step 5: Proceed with the correlation peak detection taking into account all artifacts present in FFT-based cross-correlation. 5.4.2 Correlation Signal Enhancement Image pre-processing The correlation signal is strongly aﬀected by variations in image intensity. The correlation peak is dominated by brighter particle images with weaker particle images having a reduced inﬂuence. Also the non-uniform illumination of particle image intensity, due to light-sheet non-uniformities or pulse-topulse variations, as well as irregular particle shape, out-of-plane motion, etc. introduce noise in the correlation plane. For this reason image enhancement prior to processing the image is oftentimes advantageous. The main goal of the applied ﬁlters is to enhance particle image contrast and to bring particle image intensities to a similar signal level such that all particle images have a similar contribution in the correlation function [51, 161, 377]. Among the image enhancement methods, background subtraction from the PIV recordings reduces the eﬀects of laser ﬂare and other stationary image features. This background image can either be recorded in the absence of seeding, or, if this is not possible, through computation of an average or minimum intensity image from a suﬃciently large number of raw PIV recordings (at least 20-50). These images can also be used to extract areas to be masked. A ﬁlter-based approach to image enhancement is to high-pass ﬁlter the images such that low-frequency background variations are removed leaving the particle images unaﬀected. In practice this is realized by calculating a low-passed version of the original image and subtracting it from the original. Here the ﬁlter kernel width should be larger than the diameter of the particle images: ksmooth > dτ . Thresholding or image binarization, possibly in combination with prior high-pass ﬁltering, results in images where all particles have the same intensity

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5 Image Evaluation Methods for PIV

and thus have equal contribution to the correlation function. As will be described in section 5.5.5 this binarization results in an increase in measurement uncertainty. The application of a narrow-width, low-pass ﬁlter may be suitable to remove high-frequency noise (e.g. camera shot noise, pixel anomalies, digitization artifacts, etc.) from the images. It also results in widening of the correlation peaks, thus allowing a better performance of the sub-pixel peak ﬁtting algorithm (see section 5.5.2). In cases where images are under-sampled (dτ < 2) it reduces the so-called peak-locking eﬀect (page 167), but also increases the measurement uncertainty. Range clipping is another method of improving the data yield. The intensity capping technique [161], which was found to be both very eﬀective and easy to implement, relies on setting intensities exceeding a certain threshold to the threshold value. Although optimal threshold values may vary with the image content, it may be calculated for the entire image from the grayscale median image intensity, Imedian , and its standard deviation, σI : Iclip = Imedian +nσI . The scaling factor n is user deﬁned and generally positive in the range 0.5 < n < 2. A similar approach to intensity capping is to perform dynamic histogram stretching in which the intensity range of the output image is limited by upper and lower thresholds. These upper and lower thresholds are calculated from the image histograms by excluding a certain percentage of pixels from the either the upper or lower end of the histogram, respectively. While the previous two methods provide contrast normalization in a global sense, the min/max ﬁlter approach suggested by Westerweel [51] also adjusts to variations of image contrast throughout the image. The method relies on computing envelopes of the local minimum and maximum intensities using a given tile size. Each pixel intensity is then stretched (normalized) using the local values of the envelopes. In order not to aﬀect the statistics of the image the tile size should be larger than the particle image diameter, yet small enough to eliminate spatial variations in the background [51]. Sizes of 7 × 7 to 15 × 15 are generally suitable. When applying any of the previously describes contrast enhancement methods, it should be remembered that modiﬁcations of the image intensities may also aﬀect the image statistics which in turn can result in increased measurement uncertainties. This has to be balanced against the increase in data yield. Selective application of contrast enhancing ﬁlters in areas of low data would be the logical consequence. Phase-only correlation Further improvement of the correlation signal may be achieved through adequate ﬁlters in the spectral domain (ﬁgure 5.16). Since most PIV correlation implementations rely on FFT based processing, spectral ﬁltering is easily accomplished with very little computational overhead. A recent processing

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141

technique, proposed by Wernet [171], is called symmetric phase only ﬁltering (SPOF) and is based on phase-only ﬁltering techniques which are commonly found in optical correlator systems. SPOF has been shown to improve the signal-to-noise ratio in PIV cross-correlation. In eﬀect these ﬁlters also normalize the contribution of all sampled particle images and in eﬀect provide contrast normalization. In addition inﬂuences due to wall reﬂections (streaks or lines) and other undesired artifacts can be reduced. According to [161] SPOF yields more accurate results in the presence of DC-background noise, but is not as well suited as the intensity capping technique (page 140) in reducing the displacement bias inﬂuence of bright spots with high spatial frequencies. Correlation-based correction Another form of improving the signal-to-noise ratio in the correlation plane (e.g. displacement peak detection rate) was proposed by Hart [139, 141]. The technique involves the multiplication of at least two correlation planes calculated from samples located close-by - typically oﬀset by one quarter to half the correlation sample width. Provided that the displacement gradient between the samples is not signiﬁcant the multiplication of the correlation planes will enhance the main signal correlation peak while removing noise peaks that generally have a more random placement. Correlation plane averaging, that is summation of the of correlation planes instead of multiplication, produces similar results and is more robust when the number of combined correlation planes increases [290]. Ensemble correlation While the previous method is applied within a given PIV image it can also be applied to a sequence of images. This PIV processing approach, also known as ensemble correlation, was developed in the framework of micro-PIV applications in an eﬀort to reduce the inﬂuence of Brownian motion that introduces signiﬁcant noise in data obtained from a single PIV recording. Rather than obtaining displacement data for each individual image pair, the technique relies on averaging coincident correlation planes from a sequence of images. With increasing frame counts a single correlation peak will accumulate for each correlation plane reﬂecting the mean displacement of the ﬂow [290, 399, 407]. Although computationally very eﬃcient, the main drawback of this approach is that all information pertaining to the unsteadiness of the ﬂow is lost (e.g. no RMS values). Its use with conventional (macroscopic) PIV recording has been veriﬁed by the authors and seems appropriate for fast calculation of the mean ﬂow. Since this processing is very fast, it has potential as a quasi-online diagnostic tool. To demonstrate the eﬀectiveness of the ensemble correlation technique, Meinhart et al.compared the three diﬀerent averaging algorithms applied

142

5 Image Evaluation Methods for PIV

Input

Image sampling at position (i,j)

Image 1

Image 2

Output

(i,j)

Real−to−complex FFT

Complex− conjugate multiplication

(i,j)

Real−to−complex FFT

Spectral filtering (weighting)

Complex−to−real inverse FFT

Cross− correlation data

Fig. 5.16. Modiﬁcation of the cross-correlation processor of ﬁgure 5.14 to more accurately represent the matched spatial ﬁltering operation as proposed in [171].

to a series of images acquired from a steady Stokes ﬂow of water through a 30 μm × 300 μm glass microchannel [407]. Details of the same experiment but for a diﬀerent ﬂow rate are discussed by Meinhart et al. [406]. The signal-to-noise ratio for measurements generated from a single pair of images was relatively low, because there was an average of only 2.5 particle images located in each 16×64 pixel interrogation window. The velocity measurements are noisy and approximately 20% appear to be erroneous. The three diﬀerent types of averages compared are: • image averaging in which the images themselves averaged to produce an average ﬁrst image and an average second image that are then correlated; • correlation ﬁeld averaging in which the correlation function at each measurement point is averaged across all image pairs; • velocity ﬁeld averaging in which a velocity measurement is calculated at each measurement position in each image pair and then averaged across all image pairs. The relative performance of the three averaging algorithms was quantitatively compared by varying the number of image pairs used in each averaging technique from one to twenty. The fraction of valid measurements for each averaging technique was determined by identifying the number of velocity measurements in which the streamwise velocity component deviated by more than 10% from the known solution at each point. For this comparison, the known solution was the velocity vector ﬁeld estimated by applying the average correlation technique to 20 realizations, and then smoothing the ﬂow ﬁeld. Figure 5.17 shows the fraction of valid measurements for each of the three averaging algorithms as a function of the number of realizations used in the average. The average correlation method clearly performs better than the other two methods, and produces less than 0.5–1% erroneous measurements after averaging eight realizations. The average image method produces about 95% reliable velocity measurements, and reaches a maximum at four averages. Further increases in the number of realizations used to average the images decreases the signal to noise ratio of the average particle-image ﬁeld, and produces noise in the correlation plane due to random correlation between

5.4 Digital PIV Evaluation

143

Fig. 5.17. Comparison of the performance of the three averaging techniques: average velocity •, average image , and average correlation .

nonpaired particle images. The average velocity method reaches a maximum of 88% reliable measurements using two velocity averages. Further increases in the number of averages decreases the fraction of reliable measurements, due to an increase in the probability of an encountering an erroneous measurement. Single Pixel Evaluation One key observation from the ensemble correlation technique is that where previously signal-to-noise ratio of an interrogation could only be raised by increasing the number of particles in an interrogation region, now it can be raised by increasing the number of images acquired. In fact, the signal-to-noise ratio can be held constant at an acceptable value while decreasing the interrogation region size by increasing the number of image pairs in the ensemble. Thus, in situations where it is possible to acquire thousands of image sets in the region of interest, one can shrink the interrogation window from thousands of pixels (e.g. 32 × 32 pixel) to a single pixel while maintaining the same signal-to-noise ratio as in conventional spatial cross-correlation. Some studies have already demonstrated that this technique works using both actual and synthetic ﬂow images. In the case of micro-PIV, an ultimate in-plane resolution of 60 nm is predicted when a 60 nm-particle is imaged with an M = 100, NA = 1.4 objective lens and a CCD camera of 6 μm pixel size. There are a number of issues presently unexplored in this technique such as bias errors and peak locking. 5.4.3 Autocorrelation of Doubly Exposed PIV Images Although the current trends in technology suggest that the standard PIV recording method will be of the single exposure/multiple frame type,

144

5 Image Evaluation Methods for PIV

recordings with multiple exposed particle images may still be utilized. This is especially the case when photographic recording with high spatial resolution is used. In previous sections, optical techniques for extracting the displacement information from the photographs were described. However, desktop slide scanners now also make it possible to digitize the photographic negatives and thus enable a purely digital evaluation. Alternatively, high resolution, single frame CCD sensors can directly provide multiple exposed digital PIV recordings. Essentially the same algorithms utilized in the digital evaluation of PIV image pairs described before can be used with minor modiﬁcations to extract the displacement ﬁeld from multiple exposed recordings. The major diﬀerence between the evaluation modes arises from the fact that all information is contained on a single frame – in the trivial case a single sample is extracted from the image for the displacement estimation (ﬁgure 5.18, case I). From this sample, the autocorrelation function is computed by the same FFT method described earlier. In eﬀect, the autocorrelation can be considered to be a special case of the cross-correlation where both samples are identical. Unlike the cross-correlation function computed from diﬀerent samples the autocorrelation function will always have a self-correlation peak located at the origin (see also the mathematical description in section 3.5). Located symmetrically around the self-correlation peak two peaks with less than one fourth the intensity describe the mean displacement of the particle images in the interrogation area. The two peaks arise as a direct consequence of the directional ambiguity in the double (or multiple) exposed/single frame recording method. To extract the displacement information in the autocorrelation function the peak detection algorithm has to ignore the self-correlation peak, RP , located at the origin, and concentrate on the two displacement peaks, RD+ and RD− . If a preferential displacement direction exists, either from the nature of the ﬂow or through the application of displacement biasing methods (e.g. image shifting), then the general search area for the peak detection can be predeﬁned. Alternatively a given number of peak locations can be saved from which the correct displacement information can be extracted using a global histogram operator (section 6.1.1). The digital evaluation of multiple exposed PIV recordings can be signiﬁcantly improved by sampling the image at two positions which are oﬀset with respect to each other according to the mean displacement vector. This oﬀers the advantage of increasing the number of paired particle images while decreasing the number of unpaired particle images. This minimization of the in-plane loss-of-pairs increases the signal-to-noise ratio, and hence the detection of the principle displacement peak RD+ . However, the interrogation window oﬀset also shifts the location of the self-correlation peak, RP , away from the origin as illustrated in ﬁgure 5.18 (Case II–Case V). The use of FFTs for the calculation of the correlation plane introduces a few additional aliasing artifacts that have to be dealt with. As the oﬀset of the interrogation window is increased, ﬁrst the negative correlation peak,

5.4 Digital PIV Evaluation Sampling windows

Correlation plane

N

N

RP

dx I = I’

Case I

1

No offset Symmetric correlation

145

RD-

RD+

2

RD-

0

RP RD+

= pure autocorrelation I

Case II

RP

I’

RD+

RD-

Offset

0 < |Offset| < dx

RD- RP

I

Case III

0

RD+

RP

I’

RD+ RD-

dx < |Offset| < N/2 Aliasing of peak RD-

RD-

I

Case IV

RP

0

RD+ RD-

I’

RP

RD+

RP RD-

|Offset| = N/2 Split peak RP

RP

I

Case V

RD+

RD-

0

RP RD+

I’

N/2 < |Offset| < N Aliasing of peak RP

RP

RD+

RP 0

RP RD+

Case VI |Offset| > N No overlap No self-correlation peak RP

I

I’ 1

2 0

RD+

Shift = 0

Fig. 5.18. The eﬀect of interrogation window oﬀset on the position of the correlation peaks using FFT based cross-correlation on double exposed images. RD+ marks the displacement correlation peak of interest, RP is the self-correlation peak. In this case a horizontal shift is assumed.

RD− , and then the self-correlation peak, RP , will be aliased, that is, folded back into the correlation plane (ﬁgure 5.18 Case III – Case V). In practice, detection of the two strongest correlation peaks by the procedure described in section 5.4.5 is generally suﬃcient to recover both the positive displacement peak, RD+ , and the self-correlation, RP . The algorithm can be designed to

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automatically detect the self-correlation peak because it generally falls within a one pixel radius of the interrogation window oﬀset vector. 5.4.4 Advanced Digital Interrogation Techniques The transition of PIV from the analog (photographic) recording to digital imaging along with improved computing resources prompted signiﬁcant improvements of interrogation algorithms. The various schemes can roughly be split into ﬁve groups: • • • • •

single pass interrogation schemes such as presented in Willert & Gharib [174] multiple pass interrogation schemes with integer sampling window oﬀset [169, 172]. coarse-to-ﬁne interrogation schemes (resolution pyramid [140, 220]) or (ﬂow-)adaptive resolution schemes second-order schemes relying on the deformation of the interrogation samples according to the local velocity gradients [159] super-resolution schemes and single particle tracking [176, 179, 183]

Especially the combination of the grid reﬁning schemes in conjunction with image deformation have recently found widespread use as they combine signiﬁcantly improved data yield with higher accuracy compared to ﬁrst-order schemes. The following sections give brief overview of each of these schemes. Multiple pass interrogation The data yield in the interrogation process can be signiﬁcantly increased by using a window oﬀset equal to the local integer displacement in a second interrogation pass [172]. By oﬀsetting the interrogation windows according to the mean displacement, the fraction of matched particle images to unmatched particle images is increased, thereby increasing the signal-to-noise ratio of the correlation peak (see section 5.5.3). Also, the measurement noise or uncertainty in the displacement, , reduces signiﬁcantly when the particle image displacement is less than half a pixel (i.e. |d| < 0.5 pixel) where it scales proportional to the displacement [172]. The interrogation window oﬀset can be relatively easily implemented in an existing digital interrogation software for both single exposure/double frame PIV recordings or multiple exposure/single frame PIV recordings described in the previous section. The interrogation procedure could take the following form: Step 1: Perform a standard digital interrogation with an interrogation window oﬀset close to the mean displacement in the data.

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Step 2: Scan the data for outliers using a predeﬁned validation criterion as described in section 6.1. Replace outlier data by interpolating from the valid neighbors. Step 3: Use the displacement estimates to adjust the interrogation window oﬀset locally to the nearest integer. Step 4: Repeat the interrogation until the integer oﬀset vectors converge to ±1 pixel. Typically three passes are required. The speed of this multiple pass interrogation can be increased signiﬁcantly by comparing the new integer window oﬀset to the previous value allowing unnecessary correlation calculations to be skipped. The data yield can be further increased by limiting the correlation peak search area in the last interrogation pass. As pointed out by Wereley & Meinhart [169] a symmetric oﬀset of the interrogation samples with respect to the point of interrogation corresponds to a central diﬀerence interrogation which is second-order accurate in time in contrast to a simple forward diﬀerencing scheme that simply adds the oﬀset to the interrogation point (see ﬁgure 5.19). Grid reﬁning schemes The multiple pass interrogation algorithm can be further improved by using a hierarchical approach in which the sampling grid is continually reﬁned while the interrogation window size is reduced simultaneously. This procedure has the added capability of successfully utilizing interrogation window sizes smaller than the particle image displacement. This permits the dynamic spatial range1 to be increased by this procedure. This is especially useful in PIV recordings with both a high image density and a high dynamic range in the

Fig. 5.19. Sampling window shift using a forward diﬀerence scheme (left) and central diﬀerence scheme, right (from [159]). 1

The dynamic spatial range (DSR) is deﬁned as the ratio of the largest observable length scale to the smallest observable length scale (typically the interrogation window size). Thereof, the dynamic velocity range (DVR) is derived as the ratio of the maximum measurable velocity to the minimum resolvable velocity [54].

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displacements. In such cases standard evaluation schemes cannot use small interrogation windows without losing the correlation signal due to the larger displacements. However, a hierarchical grid reﬁnement algorithm is more diﬃcult to implement than a standard interrogation algorithm. Such an algorithm may look as follows: Step 1: Start with a large interrogation sample that is known to capture the full dynamic range of the displacements within the ﬁeld of view by observing the one-quarter rule (page 137). Step 2: Perform a standard interrogation using little or no interrogation window overlap. Step 3: Scan for outliers and replace by interpolation. As the recovered displacements serve only as estimates for the next higher resolution level, the outlier detection criterion can be more stringent than usual. Data smoothing may also be useful. Step 4: Project the estimated displacement data on to the next higher resolution level. Use this displacement data to oﬀset the interrogation windows with respect to each other. Step 5: Increment the resolution level and repeat steps 1 through 4 until the actual image resolution is reached. Step 6: Finally perform an interrogation at the desired interrogation window size and sampling distance (without outlier removal and smoothing). By limiting the search area for the correlation peak the ﬁnal data yield may be further increased. In the ﬁnal interrogation pass the window oﬀset vectors have generally converged to ±1 pixel of the measured displacement thereby guaranteeing that the PIV image was optimally evaluated. The choice for the ﬁnal interrogation window size depends on the particle image density. Below a certain number of matched pairs in the interrogation area (typically NI < 4) the detection rate will decrease rapidly (see section 5.5.4). Figure 5.20 shows the displacement data of each step of the grid and interrogation reﬁnement. The processing speed may be signiﬁcantly increased by downsampling the images during the coarser interrogation passes. This can be achieved by consolidating neighboring pixels by placing the sum of a block of N × N into a single pixel. This allows the use of smaller interrogation samples that are evaluated much faster. In eﬀect, a constant interrogation window size can be used regardless of the image resolution (e.g. a 4× downsampled image interrogated by a 32 × 32 pixel sampling window corresponds to a 128 × 128 pixel sample at the initial image resolution). Image deformation schemes The particle image pattern displacement is measured by cross-correlation under the assumption that the motion of the particles image within the interrogation window is approximately uniform. In practice this hypothesis is never

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a)

b)

c)

d)

149

Fig. 5.20. Iteration steps used in a multiple pass, multi-grid interrogation process. The gray squares in the lower left of each data set indicate the size of the utilized interrogation window. In the ﬁrst pass the original image was desampled by a factor of three.

strictly valid and in most ﬂows of interest the velocity ﬁeld exhibits signiﬁcant variations within the interrogation window. In these cases the cross-correlation peak produced by image pairs with a diﬀerent velocity becomes broader and in extreme cases it can split into multiple peaks due to large velocity diﬀerences across the window (ﬁgure 5.21). As a result, the measurement of velocity in presence of large velocity gradient is aﬀected by larger uncertainty and suﬀers from a higher vector rejection rate. The iterative window deformation technique is conceived for the compensation of the in-plane velocity gradient and the peak broadening eﬀect, which can be largely reduced when the two PIV recordings are iteratively deformed according to the velocity ﬁeld. The technique can be implemented within the

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0.4 0.3 0.2 0.1

Correlation Coefficient

150

0.0

Fig. 5.21. Discrete spatial correlation map in a shear ﬂow. Interrogation with 1-step correlation. The peak broadens and splits into several individual peaks.

multi-grid philosophy described before (see section 5.4.4) and its main advantage with respect to the discrete window shifting method is an increased robustness and accuracy over highly sheared ﬂows such as boundary layers, vortices and turbulent ﬂows in general. The basic principle is illustrated in ﬁgure 5.22, where a continuous image deformation progressively transforms the images towards two hypothetical recordings taken simultaneously at time t + Δt/2. In analogy with the discrete window shift technique, this method is referred to as window deformation; however, the eﬃcient implementation of the procedure is based on the deformation of the entire PIV recordings, which is sometimes referred to as image deformation (ﬁgure 5.23). The two approaches are therefore synonyms of the same concept. The image deformation technique can be summarized as follows:

Fig. 5.22. Principle of the window deformation technique. Left: tracer pattern in the ﬁrst exposure. Right: tracer pattern at the second exposure (solid circles represent the tracers correlated with the ﬁrst exposure in the interrogation window). In grey deformed window according to the displacement distribution estimated from a previous interrogation.

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Fig. 5.23. Graphical scheme of the image deformation technique with one multigrid step. Undeformed interrogation windows as solid lines and deformed windows as dashed lines.

Step 1: Standard digital interrogation with an interrogation window complying with the one quarter rule (page 137) Step 2: Low-pass ﬁltering of the velocity vector ﬁeld. A ﬁlter kernel equivalent to the window size is suﬃcient to smooth spurious ﬂuctuations and suppress ﬂuctuations at sub-window wavelength. Moving average ﬁlters or spatial regression with a 2nd order least squares regression are suitable choices [313]. Step 3: Deformation of the PIV recordings according to the ﬁltered velocity vector ﬁeld with a central diﬀerence scheme. The image resampling scheme inﬂuences the accuracy of the procedure [126]. For typical PIV images with particle image diameter of 2 to 3 pixel, high order schemes (cardinal interpolation, B-splines) yield better results than low order interpolators (bilinear interpolation)[163, 165]. Step 4: Additional digital interrogation pass now on the deformed images with an interrogation window complying with the 4-pairs rule. Step 5: Add the result of the correlation to the ﬁltered velocity ﬁeld. Step 6: Scan the velocity vector ﬁeld for outliers and replace these by interpolation. Step 7: Repeat steps 2 to 6 two or three times. With each iteration the incremental change in the displacement ﬁeld will decrease.

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In analogy to the standard cross-correlation function given in equation (5.1) the equation used for the spatial cross-correlation for deformed images reads as follows: RII (x, y) =

K L

˜ j)I˜ (i + x, j + y) . I(i,

(5.7)

i=−K j=−L

˜ j) and I˜ (i, j) are the image intensities reconstructed after deforwhere I(i, mation using the predicted deformation ﬁeld Δs(x) in a central diﬀerence scheme: Δs(x) ˜ I(x) = I x− 2 Δs(x) I˜ (x) = I x + 2

(5.8) (5.9)

The deformation ﬁeld Δs(x) is a spatial distribution which generally is not uniform and therefore requires interpolation at each pixel in the image. Here a truncation of the Taylor series at the ﬁrst order term is commonly suﬃcient for the reconstruction of the local displacement: Δs1 (x) = Δs(x0 ) + ∇[Δs(x0 )] · (x − x0 ) + ... + O(x − x0 )2

(5.10)

1.0 0.8 0.5 0.3

Correlation Coefficient

Here x0 denotes the position of the center of the interrogation window. Since the size of the interrogation window normally exceeds the spacing of the displacement vectors (overlap factor 50% to 75%) the displacement

0.0

Fig. 5.24. Correlation map of a shear ﬂow as in ﬁgure 5.21 with multi-step correlation and window deformation. A single peak can be clearly distinguished from the correlation noise with a correlation coeﬃcient of 99.5% compared to 27.3% for the undeformed image.

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Fig. 5.25. Displacement error as a function of the particle image displacement (32 × 32 pixels window size). 1-step cross-correlation; • window deformation.

distribution within the window is a piecewise linear function resulting in a higher order approximation of the ﬂow pattern within the window. When steps 2 to 6 are repeated a number of times, the distance between ˜ j) and I˜ (i, j) tend to coincide expect particle image pairs is minimized and I(i,

Fig. 5.26. Block diagram of the iterative image deformation interrogation method with ﬁltered predictor.

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5 Image Evaluation Methods for PIV

for out-of-plane particle motion and shot-to-shot image intensity variation. As a consequence, the correlation function returns a peak located at the center of the correlation plane. Aside from minimizing the in-plane loss of pairs due to velocity gradients, the procedure has additional beneﬁts: ﬁrstly, the peak broadening due to velocity gradients is reduced; secondly, the correlation peak is located at the origin of the correlation plane and is symmetric, reducing uncertainties due to distorted peak shape or inaccurate peak reconstruction (see ﬁgure 5.24 and 5.25). Another advantage is that the spatial resolution is approximately doubled with respect to a standard interrogation. However, as will be described later, the iterative interrogation tends to selectively amplify wavelengths smaller than the interrogation window, which needs to be compensated for by low-pass ﬁltering the intermediate result [313]. Image interpolation for PIV Due to the continuously varying deformation ﬁeld, the image intensity is has to be interpolated at non-integer pixel locations which increases the computational load. Depending on the choice of interpolator signiﬁcant bias errors may be introduced as shown in ﬁgure 5.28. The data was obtained from synthetic images with constant particle image displacement using the Monte Carlo methods described in section 5.5.1. Depending on the choice of image interpolator the deviation from the true displacement may reach one ﬁfth of a pixel. Since a symmetric image shifting is applied the both images of the image pair, the deviation reaches a maximum at ±1 pixel. This means that the polynomial interpolators have their poorest performance when interpolating image intensities at ±0.5 pixel and are obviously not particularly suited for this purpose. Image interpolation has been investigated extensively in the ﬁeld of image processing and a number of interpolation schemes have been developed. A review of the topic on the background of medical imaging along with a performance comparison is given by Th´ evenaz et al. [163]. They suggest the use of generalized interpolation with non-interpolating basis functions such as Bsplines in favor of more commonly used polynomial or bandlimited sinc-based interpolation. More detailed information on the theory and implementation of B-splines can be found in [165, 166, 167]. In contrast to many other imaging applications, a properly recorded PIV image generally contains almost discontinuous data with signiﬁcant signal strength in the shortest wavelengths close to the sampling limit (i.e. strong intensity gradients). Because of this the image interpolator should primarily be capable of properly reconstructing these steep intensity gradients. A concise comparison of various advanced image interpolators for use in PIV image deformation is provided by Astaria & Cardone [126]. In accordance to the ﬁndings reported by Th´ evenaz et al. [163], they suggest the use of B-splines for an optimum balance between computational cost and performance. The

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Fig. 5.27. Sine wave test: normalized amplitude response as a function of the normalized window size. Solid line: theoretical response (sinc); 1-step crosscorrelation; • window deformation with 2nd order least squares ﬁlter; ◦ window deformation without ﬁltering.

bias error for B-splines of third and ﬁfth order shown in ﬁgure 5.28 clearly supports this. If higher accuracy is required then sinc-based interpolation, such as Whittaker reconstruction [158], or FFT-based interpolation schemes [175] with a large number of points should be used. However processing time may increase an order of magnitude. Iterative PIV interrogation and its stability Multi-step analysis of PIV recordings can be seen as comprising of two procedures: 1) Multi-grid analysis where the interrogation window size is progressively decreased. This process allows to eliminate the one quarter rule constraint and it is usually terminated when the required window size (the smallest) is applied. 2) Iterative analysis at a ﬁxed sampling rate (grid spacing) and spatial resolution (window size). This process further improves the accuracy of the image deformation and enhances the spatial resolution. In essence the iterative analysis can be described by a predictor-corrector process governed by the following equation: Δsk+1 (x) = Δsk (x) + Δscorr (x)

(5.11)

where Δsk+1 indicates the result of the evaluation at the kth iteration. The correction term Δscorr can be viewed as a residual and is obtained by interrogating the deformed images as calculated by the central diﬀerence expression equation (5.8). The procedure can be repeated several times, however two to three iterations are already suﬃcient to achieve a converged result with most of the in-plane particle image motion compensated through the image deformation.

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5 Image Evaluation Methods for PIV Bilinear Poly 3rd Order Poly 5th Order B-Spline 3rd Order B-Spline 5th Order

Bias (Meas.-True) [pixel]

0.20

0.10

0.00

-0.10

-0.20 0.0

0.5

1.0

Displacement [pixel]

1.5

2.0

0.15 Bilinear Poly 3rd Order Poly 5th Order B-Spline 3rd Order B-Spline 5th Order

Bias (Meas.-True) [pixel]

0.10 0.05 0.00 -0.05 -0.10 -0.15 0.0

0.5

1.0

Displacement [pixel]

1.5

2.0

Fig. 5.28. Bias error in image deformation PIV processing for three image interpolation schemes. Particle image diameter: dτ = 2.0 (top), dτ = 4.0 (bottom).

The iterative scheme introduced above appears very logical and its simplicity makes it straightforward to implement, which probably explains why it has been so broadly adopted in the PIV community [131, 142, 157, 158, 169]. However, when such iterative interrogation is performed without any spatial ﬁltering of the velocity ﬁeld, the process tends to oscillate and may eventually diverge unless the image processing is interrupted at an early stage. The instability arises from the sign reversal in the sinc shape of the response function associated to the top-hat function of the interrogation window. For instance, taking two almost identical images except for artiﬁcial pixel noise, the displacement ﬁeld measurement after some iterations begins to oscillate at a spatial wavelength λunst ≈ 2/3DI and yields a wavy pattern. The above result is consistent with the fact that response function of a top-hat weighted interrogation window (the common option) is rs = sin(x/DI )/(x/DI ). Therefore wavelengths in the range with negative values of the sinc function are systematically ampliﬁed. The iterative process requires therefore a stabilization by means of a low-pass ﬁlter applied to the updated

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result (ﬁgure 5.26), which damps the growth of ﬂuctuations at wavelengths smaller than the window size. A moving average ﬁlter with a kernel size corresponding to that of the interrogation window is more than suﬃcient to stabilize the process. However, ﬁltering with a second order least-squares spatial regression allows to both maximize spatial resolution and minimize the noise. Other means of stabilization are based on interrogation window weighting techniques (e.g. Gaussian or LFC [150]). A numerical simulation using a sine-modulated shear ﬂow shows that the single-pass cross-correlation amplitude modulation (empty squares in ﬁgure 5.27) follows closely that of a sinc function with a 10% cut-oﬀ occurring when the window size is about one-quarter the spatial wavelength (DI /λ = 0.25). The iterative interrogation (full circles in ﬁgure 5.27) delays the cut oﬀ at DI /λ = 0.65. This implies that with a window size of 32 pixels the single-step cross-correlation is only capable of accurately recovering ﬂuctuations with a wavelength larger than 120 pixel, whereas the minimum wavelength reduces to 50 pixel. The higher response of the iterative interrogation without ﬁlter (empty circles in ﬁgure 5.27) is only hypothetical because the process is unstable and the error is dominated by the ampliﬁed wavy ﬂuctuations. From the above discussion, it can be concluded that the spatial resolution of the iterative analysis is about twice as high than that of the single-step or window-shift procedure. However, the increase in resolution is only eﬀective when the velocity ﬁeld is sampled spatially at a higher rate, that is, increasing the overlap factor to 75% between adjacent windows instead of 50%. Adaptive interrogation techniques The iterative multi-grid interrogation may help in increasing the spatial resolution by decreasing the ﬁnal window size. However, in several cases the ﬂow and the ﬂow seeding distribution are not homogeneous over the observed domain. In this case, the optimization rules for interrogation can only be satisﬁed in an average sense and local non-optimal conditions may occur such as poor correlation signal or too low ﬂow sampling rate. Moreover, the window ﬁltering eﬀect can be minimized when the ﬂow exhibits variations along a preferential direction. For instance, in case of interfaces an adaptive choice of the interrogation volume shape and orientation may contribute to achieve further improvements especially when dealing with shear layers or shock waves. Quite possibly, spatially adaptive algorithms could be oﬀered by image motion estimation techniques that do not rely on cross-correlation. Image motion estimation, also referred to as optical ﬂow, is a fundamental issue in low-level vision and has been the subject of substantial research for application in robot navigation, object tracking, image coding or structure reconstruction [186, 187]. These applications are commonly confronted with the problem of fragmented occlusion (i.e. looking through branches of a tree) or depth discontinuities in space, which are analogous to shocks within ﬂuid ﬂows.

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Optical ﬂow for the analysis of PIV images was ﬁrst reported by Quenot et al. who investigated a thermally driven ﬂow of water [188]. Further implementations of optical ﬂow adapted to the evaluation of PIV images have been reported by Ruhnau et al.[189, 190]. The potential of optical ﬂow for achieving high spatial resolution and accurate results in high-gradient regions was demonstrated in the scope of the ”International PIV-Challenge” [48, 50]. One known deﬁciency of established optical ﬂow techniques available in computer vision is their instability in the presence of out-of-plane motion of particles that is associated with a loss of image correspondence. Therefore additional constraints have to be implemented in the algorithms to successfully apply them to PIV recordings [189, 190]. Super-resolution and single particle tracking The spatial resolution in PIV evaluation can be even further increased by eventually tracking the individual particle images, a procedure referred to as super-resolution PIV by Keane et al.[179] who applied the technique to double-exposed images. A similar procedure was also implemented for image pairs by Cowen & Monismith [129] for the study of a ﬂat plate turbulent boundary layer. The common approach in particle tracking velocimetry (PTV) is to ﬁrst detect the location of individual particles within the images followed by a suitable particle pairing algorithm [181]. Although this approach works on a pair of images it generally is much more reliable when working with image sequences, because a predictor-corrector scheme for matching particle can be used in a addition [180, 181, 246]. PTV schemes applied to single image pairs can only rely on neighborhood information and additional constraints such as particle image intensity. Here the use of PIV to provide a predictor for subsequent tracking of individual particle images has gained increased popularity, partially also because of the widespread availability of PIV algorithms. While many implementations rely on detection and position estimation of particle images prior to matching [129, 176, 181, 183], other schemes prefer to use cross-correlation of small samples (typ. 8 × 8 pixel) centered on the detected individual particle image. The existence of a matched pair is conﬁrmed by applying the procedure in reverse by starting from the second particle image. The main advantage of this correlation-based approach is that it is much more robust in the presence of overlapping images, and thus more suited for high particle image density data. The recovered displacement estimates also tend to be more accurate [50] than for “pure” PTV methods. 5.4.5 Peak Detection and Displacement Estimation One of the most crucial, but not necessarily easily understood features of digital PIV evaluation, is that the position of the correlation peak can be

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estimated to subpixel accuracy. Estimation accuracies of the order of 1/10th to 1/20th of a pixel are realistic for 32 × 32 pixel samples from 8-bit digital images. Simulation data such as those presented in section 5.5 can be used to quantify the achievable accuracy for a given imaging setup. Since the input data itself is discretized, the correlation values exist only for integral shifts. The highest correlation value then permits the displacement to be determined with an uncertainty of ±1/2 pixel. However, with the cross-correlation function being a statistical measure of best match, the correlation values themselves also contain useful information. For example, if an interrogation sample contains ten particle image pairs with a shift of 2.5 pixel, then from a statistical point of view, ﬁve particle image pairs will contribute to the correlation value associated with a shift of 2 pixel, while the other ﬁve will indicate a shift of 3 pixel. As a result, the correlation values for the 2 pixel and 3 pixel shifts will have the same value. An average of the two shifts will yield an estimated shift of 2.5 pixel. Although this is a rather crude example, it illustrates that the information hidden in the correlation values can be eﬀectively used to estimate the mean particle image shift within the interrogation window. A variety of methods of estimating the location of the correlation peak have been utilized in the past. Centroiding, which is deﬁned by the ratio between the ﬁrst order moment and zeroth order moment, is frequently used, but requires a method of deﬁning the region that comprises a correlation peak. Generally, this is done by assigning some sort of threshold which separates the correlation peak from the background noise. The method works best with broad correlation peaks where many values contribute in the moment calculation. Nevertheless, separating the signal from the background noise is not always unambiguous. A more robust method is to ﬁt the correlation data to some function. Especially for narrow correlation peaks, the approach of using only three adjoining values to estimate a component of displacement has become wide-spread. The most common of these three-point estimators are listed in table 5.1, with the Gaussian peak ﬁt most frequently implemented. The reasonable explanation for this is that the particle images themselves, if properly focused, describe Airy intensity functions which are approximated very well by a Gaussian intensity distribution (see section 2.6.1). The correlation between two Gaussian functions can be shown also to result in a Gaussian function. The three-point estimators typically work best for rather narrow correlation peaks formed from particle images in the 2–3 pixel diameter range. Simulations such as those shown in ﬁgure 5.32 indicate that for larger particle images the achievable measurement uncertainty increases which can be explained by the fact that, while the noise level on each correlation value stays nearly the same, the diﬀerences between the three adjoining correlation values are too small to provide reliable shift estimates. In other words, the noise level becomes increasingly signiﬁcant while the diﬀerences between the neighboring correlation values decrease. In this case, a centroiding approach

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Table 5.1. Three-point estimators for determining the displacement from the correlation data at the subpixel level. Fitting function

Estimators

Peak centroid

x0 =

(i − 1)R(i−1,j) + iR(i,j) + (i + 1)R(i+1,j) R(i−1,j) + R(i,j) + R(i+1,j)

y0 =

(j − 1)R(i,j−1) + jR(i,j) + (j + 1)R(i,j+1) R(i,j−1) + R(i,j) + R(i,j+1)

f (x) = ﬁrst order moment zero order moment

R(i−1,j) − R(i+1,j) 2 R(i−1,j) − 4 R(i,j) + 2 R(i+1,j) R(i,j−1) − R(i,j+1) y0 = j + 2 R(i,j−1) − 4 R(i,j) + 2 R(i,j+1)

Parabolic peak ﬁt

x0 = i +

f (x) = Ax2 + Bx + C

Gaussian peak ﬁt

f (x) = C exp

ln R(i−1,j) − ln R(i+1,j) 2 ln R(i−1,j) − 4 ln R(i,j) + 2 ln R(i+1,j) ln R(i,j−1) − ln R(i,j+1) y0 = j + 2 ln R(i,j−1) − 4 ln R(i,j) + 2 ln R(i,j+1)

x0 = i + −(x0 −x)2 k

may be more adequate since it makes use of more values around the peak than a three-point estimator. If in turn, the particle images are too small (dτ < 1.5 pixel), the three-point estimators will also perform poorly, mainly because the values adjoining the peak are hidden in noise. In the remainder we describe the use and implementation of the three-point estimators, which were used for almost all the data sets presented as examples in this book. The following procedure can be used to detect a correlation peak and obtain a subpixel accurate displacement estimate of its location: Step 1: Scan the correlation plane R = RII for the maximum correlation value R(i,j) and store its integer coordinates (i, j). Step 2: Extract the adjoining four correlation values: R(i−1,j) , R(i+1,j) , R(i,j−1) and R(i,j+1) . Step 3: Use three points in each direction to apply the three point estimator, generally a Gaussian curve. The formulas for each function are given in table 5.1. Two alternative peak location estimators also deserve to be mentioned in this context as they provide even higher accuracy than the previously mentioned methods. First, the ﬁt to a two-dimensional Gaussian, as introduced by Ronneberger et al. [155], is capable of using more than the immediate values neighboring the correlation maximum and also recovers the aspect ratio

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and skew of the correlation peak. Therefore it is well suited in the position estimation of non-symmetric (e.g. elliptic) correlation peaks.

−(x − xo )2 (y − yo )2 kxy (x − xo )(y − yo ) f (x, y) = Io exp − 2 − dτ x dτ y (1/8) dτ x (1/8) dτ y 2

(5.12)

The expression given in equation (5.12) contains a total of six coeﬃcients that need to be solved for: dτ x and dτ y deﬁne the correlation peak widths along x and y respectively, while kxy describes the peak’s ellipticity. The correlation peak maximum is located at position coordinates xo and yo and has maximum peak height of Io . The solution of equation (5.12) can usually only be achieved by nonlinear regression methods, utilizing for instance a Levenberg-Marquardt least-squares minimization algorithm [23]. If only 3 × 3 points are used the coeﬃcients in equation (5.12) can also be solved for explicitly in a least squares sense [148]. The second estimator is based on signal reconstruction theory and is often referred to as Whittaker or cardinal reconstruction. [153, 261]. The underlying function is a superposition of shifted sinc functions whose zeroes coincide with the sample points. Values between the sample points (i.e. correlation values) are formed from the sum of the sinc functions. Since the reconstructed function is continuous, the position of the peak value between the sample points has to be determined iteratively using for instance Brent’s method [23]. In principle all correlation values of the correlation plane could be used for the estimation, but in practice it suﬃces to perform one-dimensional ﬁts on the row and column intersecting the maximum correlation value. Multiple peak detection To detect a given number of peaks, n, within the same correlation plane, a diﬀerent search algorithm is necessary which sorts out only the highest peaks. In this case it is necessary to extract local maxima based on some sort of neighborhood comparison. This procedure is especially useful for correlation data obtained from single frame/multiple exposed PIV recordings. Also, multiple peak information is useful in cases where the strongest peak is associated with an outlier vector. An easily implemented recipe based on looking at the adjoining ﬁve or nine (3 × 3) correlation values is given here: Step 1: Allocate a list to store the pixel coordinates and values of the n highest correlation peaks. Step 2: Scan through the correlation plane and look for values which deﬁne a local maximum based on the local neighborhood, that is, the adjoining 4 or 8 correlation values. Step 3: If a detected maximum can be placed into the list, reshuﬄe the list accordingly, such that the detected peaks are sorted in the order of

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intensity. Continue with Step 2 until the scan through the correlation plane has been completed. Step 4: Apply the desired three-point peak estimators of table 5.1 for each of the detected n highest correlation peaks, thereby providing n displacement estimates. Displacement peak estimation in FFT-based correlation data As already described in section 5.4.1 the assumption of periodicity of both the data samples and resulting correlation plane brings in a variety of artifacts that need to be dealt with properly. The most important of these is that the correlation plane, due to the method of calculation, does not contain unbiased correlation values, and results in the displacement to be biased to lower magnitudes (i.e. bias error, page 137). This displacement bias can be determined easily by convolving the sampling weighting functions, generally unity for the size of the interrogation windows, with each other. For example, the circular cross-correlation between two equal sized uniformly weighted interrogation windows results in a triangular weighting distribution in the correlation plane. This is illustrated in ﬁgure 5.29 for the one-dimensional case. The central correlation value will always have unity weight. For a shift value of N/2 only half the interrogation windows’ data actually contribute to the correlation value such that it carries only a weight of 1/2. When a threepoint estimator is applied to the data, the correlation value closer to the origin

a)

Sample weighting

Effective weighting of correlation values

1

f 0

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Fig. 5.29. Eﬀective correlation value weighting in FFT-based ‘circular’ crosscorrelation calculation: (a) for interrogation windows of equal size, and (b) for interrogation windows of unequal size (using zero-padding).

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163

is weighted more than the value further out and hence the magnitude of the estimated displacement will be too small. The solution to this problem is very straightforward: before applying the three-point estimator, the correlation values RII have to be adjusted by dividing out the corresponding weighting factors. The weight factors can be obtained by convolving the image sampling function with itself – generally a unity weight, rectangular function – as illustrated in ﬁgure 5.29(a). In the case where the two interrogation windows are of unequal size a convolution between these two sampling functions will yield a weighting function with unity weighting near the center (ﬁgure 5.29(b)). The extension of the method to nonuniform interrogation windows is of course also possible. On a related note it should be mentioned that many FFT implementations result in the output data to be shuﬄed. Often the DC-component is found at index (0) with increasing frequencies up to index (N/2 − 1). The next index, (N/2), is actually both the highest positive frequency and highest negative frequency. The following indices represent the negative frequencies in descending order such that index (N − 1) is the lowest negative frequency component. By periodicity, the DC component reappears at index (N ). In order to achieve a frequency spectrum with the DC component in the middle, the entire data set has to be rotated by (N/2) indices. As illustrated in ﬁgure 5.30 two-dimensional FFT-data has to be unfolded in a corresponding manner. For correlation planes calculated by means of a two-dimensional FFT, the zero-shift value (i.e. origin) would initially appear in the lower left corner which would make a similar unfolding of the resulting (periodic) correlation data necessary. Without unfolding, negative displacement peaks will actually appear on the opposite side. However, a careful implementation of the peak ﬁnding algorithm allows proper peak detection and shift estimation without having to unscramble to the correlation plane ﬁrst.

a)

b)

N-1

4

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1

3

4

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1

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Fig. 5.30. Spatially folded output from a two-dimensional FFT routine (a) requires unfolding to the place the origin back at the center of the correlation plance (b).

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5.5 Measurement Noise and Accuracy The overall measurement accuracy in PIV is a combination of a variety of aspects extending from the recording process all the way to the methods of evaluation. This section is devoted to analyzing the contributing factors in the digital evaluation of the PIV recording. The absolute measurement error in the estimation of a single displacement vector, tot , can be decomposed into a group of systematic errors, sys , and a group of residual errors, resid : tot = sys + resid

(5.13)

The systematic errors comprise all errors which arise due to the inadequacy of the statistical method of cross-correlation in the evaluation of a PIV recording, such as its application in gradient regions or the use of an inappropriate subpixel peak estimator. The nature of these errors is that they follow a consistent trend which makes them predictable. By choosing a diﬀerent analysis method or modifying an existing one to suit the speciﬁc PIV recording, the systematic errors can be reduced or even removed. The second type of errors, the residual errors, remain in the form of a measurement uncertainty even when all systematic errors have been removed. In practice, however, it is not always possible to completely separate the systematic errors, sys , from the residual errors, resid , such that we choose to express the total error as the sum of a bias error bias and a random error or measurement uncertainty, rms : tot = bias + rms

(5.14)

Each displacement vector is associated with a certain degree of over or under estimation, hence a bias error bias , and some degree of random error or measurement uncertainty ±rms . The measurement uncertainty and systematic errors in digital PIV evaluation can be assessed in a variety of ways. One approach is to use actual PIV recordings for which the displacement data is known reliably. For instance, PIV recordings obtained from a static (quiescent) ﬂow were used in determining the measurement uncertainty in the cross-correlation evaluation of single exposed particle image pairs [111, 174] as well as double exposed single images [124]. Although this approach is likely to provide the most realistic estimate for the measurement uncertainty, it only permits a limited study of how speciﬁc parameters, such as particle image diameter and background noise, inﬂuence the measurement precision. An alternative approach to assessing the measurement precision in PIV evaluation is based on numerical simulation which is an approach taken by a number of researchers [51, 82, 83, 84, 124, 129, 179]. By varying only a single parameter at a time, artiﬁcial particle image recordings of known content can be generated, evaluated and compared with the known result. Random positioning of particle images and a high number of simulations (O[1000]) per

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165

choice of parameters are crucial in providing reliable measurement precision estimates. The predictions of these Monte Carlo simulations can then be compared with either theory [51, 172] or existing data sets [124]. In the following sections the methodology for Monte Carlo simulation in the assessment of the measurement uncertainty of PIV along with a few important results will be explained. 5.5.1 Synthetic Particle Image Generation The core of the Monte Carlo based measurement error estimation in digital PIV evaluation lies in the generation of adequate particle image recordings. The particle image generator has to fulﬁl the requirement of providing artiﬁcial images with known characteristics: diameter, shape, dynamic range, spatial density and image depth, among others. For most of the simulations presented here, the individual particle images are described by a Gaussian intensity proﬁle: −(x − xo )2 − (y − yo )2 (5.15) I(x, y) = Io exp (1/8) d2τ where the center of the particle image is located at (xo , yo ) with a peak intensity of Io . For simplicity the magniﬁcation factor between object plane and image plane is chosen to be unity, such that (x, y) ≡ (X, Y ). The particle image diameter, dτ , is deﬁned by the e−2 intensity value of the Gaussian bell which by deﬁnition contains 95% of the scattered light. When the particle image diameter is reduced to zero, the particle images will be represented as delta functions. The factor Io is a function of the particle’s position, Zo , within the light sheet and the eﬃciency q with which the particle scatters the incident light. For a light sheet centered at Z = 0 with a Gaussian intensity proﬁle, typical of continuous wave argon-ion lasers, Io could be expressed as: Z2 Io (Z) = q exp − (5.16) (1/8) ΔZ02 where ΔZ0 is the thickness of the light sheet measured at the e−2 intensity waist points. Further on, it is assumed that the particle diameter is much smaller than the light sheet thickness, ΔZ0 . For a top-hat intensity proﬁle the expression for Io would take the following form: 1 if |Z| ≤ 12 ΔZ0 (5.17) Io (Z) = q · 0 otherwise . To generate a particle image, a random number generator speciﬁes the particle’s position (X1 , Y1 , Z1 ) within a three-dimensional slab containing the light sheet (ﬁgure 5.31). The peak intensity Io (Z1 ) is estimated using either equation (5.16), equation (5.17) or any other intensity proﬁle. This value is then substituted into equation (5.15) for the calculation of the light captured by

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5 Image Evaluation Methods for PIV

Y

Z

X

Light sheet intensity profile

Fig. 5.31. Three-dimensional volume containing a light sheet and particles used in the generation of artiﬁcial particle images.

each pixel. Here the integration of equation (5.15) across each pixel can be greatly simpliﬁed by computing the product of the error functions (closed form integral of the Gaussian function) along both X and Y . To generate a displacement, an artiﬁcial ﬂow then moves the particle location to a new position (X2 , Y2 , Z2 ) for which a new particle image intensity distribution is calculated. This operation is repeated until a desired particle image density N is reached. The image is then quantized to the desired image depth (i.e. bits per pixel) and noise may be added to simulate for instance the sensor’s shot noise. The next sections illustrate the use of Monte Carlo simulation to test which parameters aﬀect the measurement uncertainty, that is the random error, in digital PIV evaluation. The aim here is not to predict the measurement uncertainty or bias error for a speciﬁc set of parameters. Rather, the behavior of these errors with respect to the variation of a given parameter is to be illustrated. 5.5.2 Optimization of Particle Image Diameter Figure 5.32(a) and (b) predict the existence of an optimum particle image diameter for digital PIV evaluation employing three-point Gaussian peak approximators. For the cross-correlation between two images this diameter is slightly more than 2.0 pixel, while double exposed PIV recordings have an ideal particle image diameter of dτ ≈ 1.5 pixel. Although the same software modules for particle image generation and evaluation (FFTs, peak ﬁnder, etc.) were used for both Monte Carlo simulations, there is a discrepancy in the optimum particle image diameter for which no plausible explanation is known. When the particle images become too small, another eﬀect arises which can also be observed in the simulation data (ﬁgure 5.33): the displacements tend to be biased towards integral values. The eﬀect increases as the particle image

5.5 Measurement Noise and Accuracy

167

a)

RMS-Uncertainty [pixel]

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16 px window 2 32 px window 2 64 px window

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0.001

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Fig. 5.32. Measurement uncertainty (RMS random error) in digital crosscorrelation PIV evaluation with respect to varying particle image diameter: (a) single exposure/double frame PIV imaging, (b) double exposure/single frame PIV imaging. Simulation parameters: QL = 8 bits/pixel, no noise, optimum exposure, top-hat light sheet proﬁle, N = 1/64 pixel−1 .

diameter is reduced which is a clear indication that the chosen subpixel peak estimator – here a three-point Gaussian peak ﬁt – is unsuited at these particle image diameters. Other three-point ﬁts may perform even worse [51, 172]. In actual displacement data, the presence of this “peak-locking”2 eﬀect can be detected by plotting a displacement histogram such as given in ﬁgure 5.34. 2

The term peak locking is frequently used term to describe a displacement bias error that has a periodic pattern on pixel intervals. Mostly it is caused by improper sub-pixel displacement estimation or sensor artifacts.

168

5 Image Evaluation Methods for PIV 0.10

Bias error [pixel]

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Fig. 5.33. “Peak locking” is introduced when the particle image diameter is too small for the three-point estimator (simulation parameters equivalent to ﬁgure 5.32).

Number of vectors

Such a distorted histogram can serve as a good indicator that the systematic errors (due to e.g. the peak ﬁt algorithm) are larger than the random noise in the displacement estimates. Consequently, a smooth histogram can also be present when the random noise is larger than the systematic error, so care must be taken with regard to misinterpreting the histogram data. In the literature this “beating” eﬀect in the histogram is also referred to as a “bias error” [152]. The source of this eﬀect, however, is not only limited to the insuﬃcient particle image size, but can also arise due to a reduced ﬁll factor or possibly even a spatially varying illumination response over the extent of the individual pixel. A variety of solutions to decrease the eﬀect exist. First the particle image

2500 2000 1500 1000 500 0

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Displacement [pixel]

Fig. 5.34. Histograms of actual PIV displacement data obtained from a 10-image sequence of a turbulent boundary layer illustrating the “peak locking” associated with insuﬃcient particle image size (left). Image pre-processing can reduce this eﬀect (right). Histogram bin-width = 0.05 pixel.

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169

diameter dτ can be increased during the recording process by increasing the sampling rate or maybe even by defocusing the particle image. In this way the particle images are properly sampled (dτ > 2 pixel) and should be the method of choice, provided there is suﬃcient light to still expose the sensor. The second choice is to choose a diﬀerent peak estimator which is better suited for smaller particle image diameters. A third approach is to pre-process the images using ﬁlters which optimize the particle image diameter with respect to the peak estimator. Finally another technique was reported by Roth & Katz [156] applies an equalization transfer function to the peak-locked data. This transfer function is calculated from the histograms of the displacement data and requires a suﬃciently large range of displacements and displacement vector count. Advanced iterative processing schemes relying on image deformation such as those reported in section 5.4.4 can also be used to reduce the pixel locking eﬀect because the correlation peak will center itself on the correlation plane origin. Thus the biasing eﬀects of inadequate sub-pixel peak estimators are mitigated [128]. 5.5.3 Optimization of Particle Image Shift Figure 5.35 shows the simulation results for the measurement uncertainty (RMS random error) as a function of the displacement. For most of the displacements the uncertainty is nearly constant except for displacements less than 0.5 pixel where a linear dependency can be observed (see also ﬁgure 5.38). This behavior can also be observed in the experimentally obtained error estimates given in [174]. Theory may be used to explain this behavior [172].

RMS-Uncertainty [pixel]

2

dτ=4, 32 px window 2 dτ=4, 64 px window 2 dτ=2, 32 px window

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Fig. 5.35. Monte Carlo simulation results for the measurement uncertainty in digital cross-correlation PIV evaluation as a function of particle image displacement.

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5 Image Evaluation Methods for PIV

The drastic reduction in the measurement uncertainty for |d| < 0.5 pixel may be exploited by oﬀsetting the interrogation windows with respect to each other according to the mean displacement vector within the interrogation window. This oﬀset has the additional side-eﬀect of increasing the detectability of the correlation peak by increasing the number of particle matches [84]. The displacement bias arising due to the in-plane loss of pairs is shown in ﬁgure 5.36. The measured displacement will always be underestimated for reasons explained earlier in section 5.4.1. By dividing out the appropriate weighting function from the correlation values prior to applying the threepoint ﬁt, this displacement bias can be nearly completely removed which is also shown in ﬁgure 5.36. 5.5.4 Eﬀect of Particle Image Density The particle image density has two primary eﬀects in the evaluation of PIV images. First, the probability of a valid displacement detection increases when more particle image pairs enter in the correlation calculation. The number of image pairs captured in an interrogation area itself depends on three factors, namely, the overall particle image density, N , the amount of in-plane displacement and the amount of out-of-plane displacement. Keane & Adrian [82, 83, 84] have deﬁned these three quantities as the eﬀective particle image pair density within the interrogation spot, NI , a factor expressing the in-plane loss-of-pairs, Fi and a factor expressing the out-of-plane loss-of-pairs, Fo . When no in-plane or out-of-plane loss-of-pairs are present the latter two are unity.

Bias error [pixel]

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8

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Fig. 5.36. Simulation results showing the diﬀerence between actual and measured displacement as a function of the particle image displacement. Bias correction removes the displacement bias (simulation parameters: dτ = 2.0, no noise, top-hat intensity proﬁle, N = 1/64 pixel−1 ).

5.5 Measurement Noise and Accuracy

171

Valid Detection Probability [%]

100 P[n>1]

80 60 40

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20 0

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NIFiFo Fig. 5.37. Vector detection probability as a function of the product of image density NI , in-plane loss of pairs Fi and out-of-plane loss of pairs Fo . The solid line represents the probability for having at least a given number of particle images in the interrogation spot (see also ﬁgure 4 in Keane & Adrian [84]).

The product of the three quantities NI Fi Fo expresses the mean eﬀective number of particle image pairs in the interrogation spot. Monte Carlo simulations performed by Keane & Adrian showed that for NI Fi Fo > 8 the valid detection probability exceeds 95% in double exposure/single frame PIV, while triple-pulse single frame PIV require only NI Fi Fo > 4. In contrast, single exposure/double frame PIV requires that NI Fi Fo > 5, which is consistent with the data shown in ﬁgure 5.37. However, depending on the chosen method for validation, the probability curves may be shifted up or down. The theoretical Poisson distribution curves describing the presence of at least a given number of particle image pairs, P [n ≥ i], are also plotted in ﬁgure 5.37 and indicate that the presence of at least three particle image pairs in the interrogation spot matches the simulation data. In practice, the data yield can be easily optimized by ensuring the presence of at least three or four particle image pairs. Further optimization is possible by oﬀsetting the interrogation windows as described in the previous section (section 5.5.3) which minimizes the in-plane loss of pairs, that is Fi → 1. The second eﬀect the particle image density has for the evaluation of PIV images is its direct inﬂuence on the measurement uncertainty. In ﬁgure 5.38 the measurement uncertainty is plotted as a function of particle image displacement for various particle image densities, NI . The displacement range was limited to the one pixel range which can be ensured by an interrogation window oﬀset. For displacements less than 1/2 pixel the same linear trend as in ﬁgure 5.35 can be observed for all NI . For |d| > 0.5 pixel the uncertainty

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RMS-Uncertainty [pixel]

0.05 0.04

NI = 5.2 NI = 10.2 NI = 20.5 NI = 32

0.03 0.02 0.01 0.00 0.0

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0.2

0.3

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1.0

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Fig. 5.38. Measurement uncertainty for single exposure/double frame PIV as a function of particle image shift for various particle image densities NI . (Simulation parameters: dτ = 2.2 pixel, QL = 8 bits/pixel, 32 × 32 pixel, no noise, optimum exposure, top-hat light sheet proﬁle.).

remains approximately constant. The principle eﬀect of the particle image density, NI , is that it can reduce the measurement uncertainty substantially, which can be explained by the simple fact that more particle image pairs increase the signal strength of the correlation peak. Together the eﬀects described above indicate that if a ﬂow can be densely seeded then both a high valid detection rate as well as a low measurement uncertainty can be achieved using small interrogation windows, which in turn allows for a high spatial resolution. 5.5.5 Variation of Image Quantization Levels Monte Carlo simulations for double exposed single frame PIV described in Willert [111] already indicated that the image quantization (i.e. bits/pixel) has only little inﬂuence on the measurement uncertainty or displacement bias error. This is further conﬁrmed for single exposed double frame PIV as shown in ﬁgure 5.39. Interestingly, a reduction of quantization levels from QL = 8 bits/pixel to QL = 4 bits/pixel practically has no inﬂuence on the RMS error for the given particle image density. This eﬀect can be explained by the fact that the noise introduced by the FFT-based correlation calculation dominates, which also implies that a high number of image quantization levels does not necessarily warrant better PIV measurement accuracy unless the noise eﬀects due the correlation algorithm are removed at the same time (such as through a direct, linear correlation). At lower image quantization levels, QL < 4 bits/pixel, the measurement uncertainty does however increase up to ten-fold. The number of quantization levels is however of importance when single particle images are measured such as in particle tracking velocimetry or super-resolution PIV. In this case, the position of a particle image can be

RMS-Uncertainty [pixel]

5.5 Measurement Noise and Accuracy

173

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Fig. 5.39. Measurement uncertainty for single exposure/double frame PIV as a function of displacement and image quantization (simulation parameters: dτ = 2.2 pixel, NI = 10.2, 32 × 32 pixel, no noise, optimum exposure, top-hat light sheet proﬁle).

more accurately estimated when its pixel intensity values are better resolved. Another noteworthy observation with regard to the image quantization level is that suﬃcient particle image density (NI > 30) allows for low-noise measurements even for binary images (ﬁgure 5.40). This fact may be of interest when

RMS-Uncertainty [pixel]

0.20 NI = 5.2 NI = 10.2 NI = 20.5 NI = 32

0.15

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Fig. 5.40. Measurement uncertainty for binary image pairs (i.e. QL = 1 bits/pixel) as a function of displacement and particle image density, NI . The solid line indicates the measurement uncertainty at QL = 8 bits/pixel (simulation parameters: same as in ﬁgure 5.39 except NI ).

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storage space needs to be conserved because a binary image requires eight times less memory than a standard eight bit image. However, a binarization of the original image without sacriﬁcing too much of the information is not necessarily trivial (i.e. nonuniform background, varying particle image intensities and diameters, etc.). Once binary images or run-length-encoded images are available, very fast direct correlation algorithms can be implemented using bit-wise or integer operations common to most computer processors [124, 138]. 5.5.6 Eﬀect of Background Noise

RMS-Uncertainty [pixel]

Figure 5.41 illustrates the increase of measurement uncertainty due to background noise. In the simulations, normal-distributed (white) noise at a speciﬁed fraction of the image dynamic range was linearly added to each pixel. Further the noise for a given pixel was completely uncorrelated with its neighbors or with its counterpart on a diﬀerent image. Both of these are not always the case for actual image sensors. Although this simulation is not quite realistic, it shows that minor noise has little eﬀect on the measurement uncertainty. Again the noise contributions due to the FFT-based correlations dominate. For the chosen simulation parameters, 10% of noise, which corresponds to roughly QL = 4 bits/pixel, cause little deterioration in the measurements. This agrees with the observations made in ﬁgure 5.39 for the variation of the image quantization levels. In that case an image quantization of QL ≥ 4 bits/pixel shows little eﬀect on the RMS-error.

0.100

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Fig. 5.41. Measurement uncertainty as a function of displacement and various amounts of white background noise (simulation parameters: dτ = 2.2 pixel, NI = 10.2, 32 × 32 pixel, optimum exposure, top-hat light sheet proﬁle).

5.5 Measurement Noise and Accuracy

175

5.5.7 Eﬀect of Displacement Gradients

RMS−Uncertainty [pixel]

Since PIV is based on a statistical measurement of the displacement using the correlation between two interrogation windows, a displacement gradient across the window is likely to result in biased data. This is because not all of the particle images present in the ﬁrst interrogation window will also be present in the second interrogation window, even if the mean particle image displacement is accounted for. For interrogation windows without an oﬀset, the displacement will be biased to a lower value because particle images with small displacements will be present more frequently than those with higher displacements (e.g. in-plane loss-of-pairs, [84]). This measurement error does not arise in particle tracking methods because they measure the displacements of individual particle images [129]. In ﬁgure 5.42 the measurement uncertainty is plotted as a function of the displacement gradient for two diﬀerent interrogation window sizes. One interesting observation that can be concluded from this ﬁgure is that smaller interrogation windows can tolerate much higher displacement gradients. Even at the same normalized particle image density, N , where the larger window contains four times as many image pairs, this eﬀect cannot be compensated. The reason for this behavior lies in the wider spread of the correlation peak in the larger window: for the same displacement gradient the dynamic range of the displacements scales linearly with the dimension of the interrogation window, resulting in a proportional increase of the correlation peak width. As a consequence, smaller interrogation windows should be favored, provided the particle image density is suﬃciently high.

1.000

NI=5, 16x16 px NI=10, 16x16 px NI=5, 32x32 px NI=20, 32x32 px NI=5, 64x64 px NI=20, 64x64 px

0.100

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Fig. 5.42. Measurement uncertainty as a function of displacement gradient for various particle image densities and interrogation window sizes (simulation parameters: dτ = 2.0 pixel, QL = 8 bits/pixel, no noise, optimum exposure, top-hat light sheet proﬁle).

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The bias error due to displacement gradients can be accounted for by estimating the mean particle image location in each interrogation window and assigning the displacement estimate to this point. A subsequent bilinear interpolation scheme can then be used to estimate the local displacement vectors at the center of the interrogation windows [305]. In principle, the estimation of the mean particle image location requires that only paired particle images which contribute to the correlation are used. As this is diﬃcult to implement, it generally suﬃces to threshold the data in each sample and calculate the centroid of the remaining pixel intensities (e.g. particle images). It should be noted that the higher-order PIV processing algorithms presented in section 5.4.4 are capable of signiﬁcantly reducing the artifacts introduced in gradient regions. 5.5.8 Eﬀect of Out-of-Plane Motion Frequently the PIV method has to be applied in highly three-dimensional ﬂows, in some cases even with the mean ﬂow normal to the light sheet. Examples of these may be the study of wing-tip vortices or other structures aligned with the ﬂow. In this arrangement the out-of-plane loss of pairs is signiﬁcant such that the correlation peak signal strength diminishes. As a result, the possibility of valid peak detection reduces. Three methods exist to compensate for the out-of-plane motion. First, the pulse delay Δt between the recordings can be reduced which has the side eﬀect of reducing the dynamic range in the measurement. Second, the light sheet can be thickened to accommodate the out-of-plane motion for a given pulse delay. However, this is not always possible because the energy density in the light sheet is reduced proportionally to the increased thickness. Third, the mean out-of-plane ﬂow component can be accommodated with a parallel oﬀset of the light sheet between the illumination pulses in the direction of the ﬂow. This method works best when the mean out-of-plane ﬂow component is nearly constant across the ﬁeld of view. The best results can be achieved by combining all three of these approaches. Just as with the in-plane loss-of-pairs, the general guideline is to keep the out-of-plane loss-of-pairs small enough to still ensure the presence of a minimum number of particle image pairs within the interrogation window (typically NI ≥ 4, see section 5.5.4).

6 Post-Processing of PIV Data

The recording and evaluation of PIV images has been described in the previous two chapters. Investigations employing the PIV technique usually result in a great number of images which must be further processed. If looking for statistical quantities the recorded data can easily amount to some gigabytes, which is now possible with today’s computer hardware. Even more data per investigation are to be expected in future. Thus, it is quite obvious that a fast, reliable and fully automatic post-processing of the PIV data is essential. In principle, post-processing of PIV data is characterized by the following steps: Validation of the raw data. After automatic evaluation of the PIV recordings, a certain number of obviously incorrectly determined velocity vectors (outliers) can usually be found by visual inspection of the raw data. In order to detect these incorrect data, the raw ﬂow ﬁeld data have to be validated. For this purpose, special algorithms have to be developed, which must work automatically. Replacement of incorrect data. For most post-processing algorithms (e.g. calculation of vector operators) it is required to have complete data ﬁelds as is quite naturally the case for numerically obtained data. Such algorithms will not work if gaps (data drop-outs) are present in the experimental data. Thus, means to ﬁll the gaps in the experimental data must be developed. Data reduction. It is quite diﬃcult to inspect several hundred velocity vector maps and to describe their ﬂuid mechanical features. Usually techniques like averaging (in order to extract the information about the mean ﬂow and its ﬂuctuations), conditional sampling (in order to distinguish between periodic and non-periodic parts of the ﬂow) and vector ﬁeld operators (e.g. vorticity, divergence in order to detect structures in the ﬂow) are applied. Analysis of the information. At present this is the most challenging task for the user of the PIV, technique. PIV being the ﬁrst technique to oﬀer information about complete instantaneous velocity vector ﬁelds, allows new insights in old and new problems of ﬂuid mechanics. Tools for analysis such

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as proper orthogonal decomposition (POD) [276] or neural networks [250] are applied to PIV data. Presentation and animation of the information. A number of software packages – commercial as well as in-house developed ones – are available for the graphical presentation of the PIV ﬁeld data. It is also very important to support the easier understanding of a human observer of the main features of the ﬂow ﬁeld. This can be done by contour plotting, color coding, etc. Animation of the PIV data is very useful for better understanding in the case of time series of PIV recordings or 3D data. In the following sections those steps of post-processing with special requirements due to the PIV technique will be explained in greater detail.

6.1 Data Validation Some of the problems associated with PIV raw data after automatic evaluation can be seen in ﬁgure 6.1. It shows the instantaneous ﬂow ﬁeld above a NACA 0012 airfoil at a free stream Mach number Ma = 0.75. The vector of the average ﬂow velocity (344 m/s) as calculated for this PIV recording has been subtracted from each individual velocity vector in order to enhance details of the ﬂow ﬁeld. The supersonic ﬂow regime above the leading edge of the airfoil and the terminating shock with its strong velocity gradient can clearly be detected. For more details on the experiments see chapter 9. Typical features of incorrect velocity vectors, which can be detected in ﬁgure 6.1, are that: • their magnitude and direction diﬀer considerably from their surrounding neighbors,

0

0.1

0.2

0.3

0.4 Naca0012 airfoil

0.5

0.6

0.7

Fig. 6.1. Velocity vector map (raw data) of instantaneous ﬂow ﬁeld (U − U , V ) above NACA 0012 airfoil (Ma= 0.75, α = 5◦ , lc = 20 cm, τ = 4 μs, U = 344 m/s).

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179

• they very often appear at the edges of the data ﬁeld (near the surface of the model, at the edges of drop-out areas, at the edges of the illuminated area), • in most cases, they appear as single incorrect vectors. From this description we can conclude that it is most likely that during the evaluation procedure a correlation peak was detected which is due to noise or artifacts (model surface, noise of diﬀerent sources, etc.) and not due to the correlation of properly matched image pairs. These questionable or spurious data points are frequently deﬁned as outliers. In general an outlier may be deﬁned as an observation (data point) which is very diﬀerent from the rest of the data based on some measure. The human perceiver is very eﬃcient in detecting these outliers. For a small number of PIV recordings these erroneous velocity vectors may be treated interactively. This is no longer possible if a great number of recordings has to be evaluated. However, for the further processing of the ﬂow ﬁeld data it is absolutely necessary to eliminate all such erroneous data. All subsequent operations involving diﬀerential operators on the raw vector data still including such outliers would enhance and smear out these errors locally and could thus mask data of good quality. Diﬀerential operators are, for example, the divergence, the vorticity operator or the calculation of diﬀerences between numerical and experimental ﬂow ﬁeld data. In contrast to this, the application of operators utilizing averaging processes over a great number of data, mean value, variance, degree of turbulence, etc., are less severely aﬀected by a few incorrect data values. It follows that all PIV data should generally be checked for erroneous data. Because of the great amount of data this can only be performed by means of an automatic algorithm. The guiding principle for handling questionable data should be: • The algorithm must ensure with a high level of conﬁdence so that no questionable data are stored in the ﬁnal PIV data set. • Questionable data should be rejected, if it cannot be decided by application of the algorithm whether data are valid or not. As a consequence of the application of the validation algorithm, the number of PIV data obtained from a given recording will be reduced by approximately 0.1–1.5% depending on the quality of the PIV recording and the type of ﬂow to be investigated. The problem of ﬁlling gaps where no valid data were found on the ﬂow ﬁeld map (by means of interpolation or extrapolation) should only be performed after completing the data validation. It should again be emphasized that this procedure prevents information arising from incorrect data being spread into areas with data of good quality. For the same reason no smoothing of data should be carried out before validating the data. The challenge in data validation is to provide algorithms that strike a good balance between overdetection, that is the removal of valid data, and

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underdetection in which too many spurious vectors are accepted. Diﬀerent techniques for PIV data validation have been described in the literature [115, 119, 258, 273, 274]. However, to date no general solution can be oﬀered for the problem of data validation in PIV. However, some degree of generalized validation is possible through the use of the normalized median ﬁlter [275] (see page 185). Variable threshold approaches determine the detection threshold from ﬁltered versions of the unvalidated data set [264, 268]. In our applications we use several algorithms, which have been developed and tested utilizing real PIV recordings obtained in diﬀerent experimental situations and for diﬀerent types of ﬂows [266]. Some of these modules are alternatives, some of them may be applied successively. Two of these modules, the global histogram operator and the dynamic mean value operator, will be discussed in detail in the following. Further validation schemes are described as well. Some deﬁnitions will be made for the following discussion of the diﬀerent data validation algorithms. The instantaneous velocity vector ﬁeld (U, V ) has been sampled (“interrogated”) at positions which form a regular grid in the ﬂow ﬁeld. In our case the grid, a part of which is shown in ﬁgure 6.2, consists of I × J grid points in the X and Y directions with constant distance ΔXstep , ΔYstep between neighboring grid points in both directions. The two-dimensional velocity vector at the position i, j (i = 1 · · · I, j = 1 · · · J) is denoted U2D (i, j). In the following discussion the relation between the central velocity vector U2D (i, j) and one of its nearest neighbors U2D (n) is considered. The nearest neighbors are labeled by n, (n = 1, ..., N ). Usually, N is chosen to be eight. The distance d between the central velocity vector U2D (i, j) and its nearest neighbors U2D (n) is either ΔXstep , ΔYstep or

2 2 , depending on its position on the grid. The magnitude of ΔXstep + ΔYstep the vector diﬀerence between the central velocity vector U2D (i, j) and U2D (n) is |Udiﬀ,n | = |U2D (n) − U2D (i, j)|. For the demonstration of the performance of the diﬀerent algorithms for data validation, the ﬂow ﬁeld shown in ﬁgure 6.3 is utilized. It represents the lower left part of the ﬂow ﬁeld already shown in ﬁgure 6.1.

U(4) = U2D(i-1,j+1)

U(5) = U2D(i-1,j)

U(6) = U2D(i-1,j-1)

U(3) = U2D(i,j+1)

U2D(i,j)

U(7) = U2D(i,j-1)

U(2) = U2D(i+1,j+1)

U(1) = U2D(i+1,j)

U(8) = U2D(i+1,j-1)

Fig. 6.2. Sketch of data grid with notation of vectors.

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181

Reference Vector: 200 m/s

Fig. 6.3. Enlarged area (lower left part) of ﬁgure 6.1 for the demonstration of the data validation algorithm.

6.1.1 Global Histogram Operator Principle. It is assumed that for real ﬂow ﬁelds the vector diﬀerence between the neighboring velocity vectors is smaller than a certain threshold. This is certainly true as long as the length scale of the ﬂow is much greater than the distance d between the position of neighboring vectors. Consequently all correct velocity vectors must lie within a continuous area in the (u, v) plane for ﬂow ﬁelds without discontinuities. As a ﬁrst step this criterion is used to discriminate against incorrect data. Procedure. A 2D histogram of the displacement (or velocity) data is obtained by plotting the positions of all recovered correlation peaks in a common correlation plane. An equivalent presentation is given in ﬁgure 6.4 where a point has been plotted in the correlation plane at the location of the highest correlation peak for each interrogation window. Two separated areas of accumulated correlation peaks (velocity vectors) can be detected, one circular region (I) and a second region (II) with greater scattering of the peak’s locations (i.e. of the velocity vectors). Now a rectangular box can automatically be calculated, circumscribing the area with the highest accumulation of displacement peaks (velocity vectors). In the next step of the validation procedure the displacement peaks detected for each interrogation window will be checked individually. All displacement peaks (velocity vectors) lying outside the rectangle(s) or other suitable boundary will be marked and rejected. Discussion. Most outliers due to noise in the correlation plane can be rejected by application of this simple algorithm. Also, if autocorrelation is employed, data originating from the vicinity of the origin of the correlation plane (zero order central peak) can be eliminated by deﬁning another rectangle of rejection around the zero order peak (not shown in ﬁgure 6.4). Noise originating from the recording process, which is concentrated in certain spectral bands (noise from AC sources, imperfections of the optics for interrogation,

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II 15

Pixel y

10

I 5 0 -5 40

60

50

70

Pixel X x

Fig. 6.4. Location of correlation peaks in the correlation plane. Rectangles indicate areas of plausible data, area (I) Ma < 1, (II) Ma > 1.

structures of the CCD-camera, etc.) can be detected and eliminated because in most cases its response is located in areas of the correlation plane which are not connected to those areas where the correct velocity vectors accumulate. Figure 6.4 also demonstrates that there may be situations with two or more areas in the correlation (or velocity) plane, where the peaks (or velocity vectors) accumulate. This is the case if discontinuities are present in the ﬂow ﬁeld, for example for transonic ﬂow ﬁelds, if shocks are embedded in the ﬂow ﬁeld. In ﬁgure 6.4 the area marked (I) is due to the subsonic part of the ﬂow ﬁeld at the right side of ﬁgure 6.3, whereas the area (II) is due to the supersonic part of the ﬂow ﬁeld just above the leading edge of the airfoil. Summarizing, one can say that the global histogram operator employs physical arguments (upper and lower limit of possible ﬂow velocities) to remove all data, which physically cannot exist in the ﬂow ﬁeld. Moreover, the inspection of the global velocity histogram gives useful information about the quality of the PIV evaluation (number of incorrect data due to noise, dynamic range of the ﬂow ﬁeld, maximum utilization of the optimal range for PIV evaluation by selecting the proper time delay between the light pulses for illumination, etc.). If at this stage of the validation process a velocity vector is rejected, it will automatically be checked whether for this grid position the automatic evaluation procedure has detected more than one peak in the correlation plane (see section 5.4.5). If this should be the case, these data are checked as well, whether they fulﬁll the criteria explained above. It is obvious that the maximum number of peaks admitted during the automatic evaluation for a ﬁxed grid position should be limited to two or three, as otherwise – with a great number of peaks for selection – the chance would be rather high to pick up peaks due to noise, which, however, accidentally fulﬁll the criterion for selection.

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183

6.1.2 Dynamic Mean Value Operator Principle. Many PIV validation schemes described in the literature makes use of mean value tests. These algorithms check each velocity vector individually by comparing its magnitude |U2D (i, j)| with the average value over its nearest neighbors μU (i, j). Typically, a 3 × 3 neighborhood with eight nearest neighbors is selected. The velocity vector to be validated will be rejected if the absolute diﬀerence between its magnitude and the average over its neighbors is above a certain threshold thresh . The test can be modiﬁed by applying it not only to the magnitude but also to the U and V components of the vector, or by utilizing a larger number of neighbors for comparison. However, the application of this test to transonic ﬂows has shown some problems if shocks are present in the ﬂow ﬁeld (discontinuity of the ﬂow velocity along a line). Thus, the algorithm had to be improved for ﬂows with velocity gradients by locally varying the threshold level for validation. Procedure. The following expression is calculated for N = 8 closest neighbors: μU (i, j) =

N 1 U2D (n) . N n=1

(6.1)

The averaged magnitude of the vector diﬀerence between the average vector and its 8 neighbors is also calculated: 2 σU (i, j) =

N 1 2 (μU (i, j) − U2D (n)) . N n=1

(6.2)

The criterion for data validation is: |μU (i, j) − U2D (i, j)| < thresh

(6.3)

where thresh = C1 + C2 σU (i, j) with C1 , C2 = constants. Problems will arise in the case of drop-outs within the data ﬁeld or at the edges of the data ﬁeld, that is if less than N = 8 neighbors are available for comparison. Satisfying results were obtained when the drop-out areas were ﬁlled with the mean value of the whole data ﬁeld and lines and rows were added at each edge of the data ﬁeld by doubling the outermost lines and rows. However, these artiﬁcially generated data were only kept during the application of the local mean value test or other validation tests. Discussion. The eﬀect of the application of the dynamic mean value operator is demonstrated in ﬁgure 6.5, where all velocity vectors identiﬁed as “incorrect” are marked by thick vector symbols. Figure 6.5 clearly shows the power of the algorithm described above as those velocity data, which are incorrect – as they are obviously diﬀerent in magnitude and direction from their neighbors – have been consequently marked. However, the application of this algorithm does not lead to diﬃculties with the strong velocity gradients across

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6 Post-Processing of PIV Data Reference Vector: 100 m/s

0.2

0.15

0.1

0.05

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fig. 6.5. Application of the dynamic mean value operator on the velocity vector map of ﬁgure 6.3. Vectors identiﬁed as “incorrect” are marked by thick vector symbols.

the shock, that is no correct data are marked as “incorrect”. Thus, it has been shown that this algorithm is able to handle ﬂows with strong velocity gradients by the introduction of the locally varying threshold level for validation (see ﬁgure 6.6). The constants C1 , C2 have to be determined once experimentally and can then be utilized for the whole series of PIV recordings taken for the same type of ﬂow. 6.1.3 Vector Diﬀerence Test Similar to the previous ﬁlter the gradient ﬁlter or vector diﬀerence ﬁlter computes the magnitude of the vector diﬀerence of a particular vector in question

Naca 0012 airfoil

Fig. 6.6. Velocity vector map of the instantaneous ﬂow ﬁeld above a NACA 0012 airfoil of ﬁgure 6.1 after clean-up.

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185

U2D (i, j) to each of its four or eight neighbors U2D (n): |Udiﬀ,n | = |U2D (n) − U2D (i, j)| < thresh

with thresh > 0 .

(6.4)

Rather than averaging the vector diﬀerences as for the dynamic mean ﬁlter the idea is to count the number of instances for which the validation criterion |Udiﬀ,n | < thresh is violated. A displacement vector can then be classiﬁed as questionable when it is ‘conﬂicting’ with at least half its neighbors. 6.1.4 Median Test PIV data validation by means of median ﬁltering has been proposed by Westerweel [274]. While median ﬁltering is frequently utilized in image processing to remove spurious noise, it may also by used for the eﬃcient treatment of spurious velocity vectors. Median ﬁltering simply speaking means that all neighboring velocity vectors U2D (n) are sorted linearly either with respect to the magnitude of the velocity vector, or their U and V components. The central value in this order (i.e. either the fourth or ﬁfth of eight neighbors) is the median value. The velocity vector under inspection U2D (i, j) is considered valid if |U2D (med) − U2D (i, j)| < thresh . 6.1.5 Normalized Median Test A slight modiﬁcation of the median test results in a very powerful validation scheme for spurious velocity vectors. Westerweel & Scarano [275] demonstrated that a normalization of the standard median test given in equation (6.1.4) yields a rather universal probability density function for the residual such that a single threshold value can be applied to eﬀectively detect spurious vectors. The normalization requires that the residual ri , deﬁned as: ri = |Ui − Umed | is ﬁrst determined for each surrounding vector {Ui |i = 1, . . . , 8}. Next the median of these eight residuals, rmed , is determined and used to normalize the standard median test as follows: |U2D (med) − U2D (i, j)| < thresh rmed + 0 The additional term 0 is required to account for remaining ﬂuctuations obtained from correlation analysis of otherwise quiescent or homogeneous ﬂow. In practice this value should be set around 0.1 − 0.2 pixel, corresponding to the mean noise level of PIV data [85] (see also section 5.5). The eﬃciency of the normalized median test was demonstrated by Westerweel & Scarano [275] by applying it to a number of PIV experiments covering a wide range of Reynolds numbers. The probability density functions of both the standard and normalized median for these experiments is shown

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6 Post-Processing of PIV Data

(a) median test grid turb. (9.4%) grid turb. (4.3%) grid turb. (3.8%) jet 16 16 jet 32 32 pipe microchannel Karman wake Ma = 2 wake backw. fac. step

1

fraction [ ]

0.1 0.01 0.001 10-4 10-5 10-6 10-7 0

1

2

3

4

residual [px] (b) normalized median test 1

fraction [ ]

0.1 0.01 0.001 10-4 10-5 10-6 10-7 0

2

4

6

8

normalized residual [ ] Fig. 6.7. Histograms of the residuals using the conventional median (a) and the normalized median (b) for a wide variety of experimental data as presented by Westerweel & Scarano [275].

in ﬁgure 6.7. Integration of the histograms of the residual for the normalized median indicated that the 90-percentile occurs for rmed ≈ 2. This meant that in all experiments investigated a single detection threshold labeled the largest 10% of residuals. The detection eﬃciency is less stringent for thresholds rmed > 2 and vice versa. The universality of this detection scheme makes it especially well suited for iterative PIV interrogation schemes such as those presented in section 5.4.4 and should be very suited for self-optimizing PIV algorithms. 6.1.6 Other Validation Filters While ﬂuid mechanical information can be used for validation, it is commonly only used indirectly by assuming that the investigated ﬂow must observe a certain degree of continuity or coherency through the application of neighborhood

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187

operators. Another forms of data validation are possible through the use of redundant information that is available from time resolved, multi-frame PIV data [323] or from additional view points such as in stereo-PIV (see page 215). The following describes a few more validation methods which are of lesser importance for various reasons. Their performance in comparison to the previous methods is outlined in table 6.1. Minimum correlation ﬁlter As mentioned earlier, a low correlation coeﬃcient is indicative of a strong loss a particle match and may have a variety of causes. Thus, a validation ﬁlter may be very helpful in detecting problematic areas in the ﬁeld of view. However it is of lesser importance for the actual validation of PIV data, as low correlation values do not necessarily point to invalid displacement readings. Peak height ratio ﬁlter In this case the correlation peak representative of the displacement reading is compared to the ﬁrst noise peak in the correlation map. A low ratio of the peak heights may point to an inadequately seeded area and a higher likelihood that the measured displacement is questionable. In terms of validation it is less eﬀective because mismatched areas may have high correlation coeﬃcients especially when seeding levels are low. Signal-to-noise ﬁlter Here the signal-to-noise ratio in the correlation plane – deﬁned as the quotient of correlation peak height with respect to the mean correlation level – is used to validate the data. However its use is questionable because mismatched particle images or stationary background features can also produce high levels of correlation. Table 6.1. Various validation ﬁlters - outlier detection eﬃciency, number of required parameters and potential for self-optimization validation no. of ﬁlter params. magnitude 1 range 2-4 dynamic mean 2 diﬀerence 1 median 1 normalized median 1 minimum correlation 1 correlation peak ratio 1 correlation SNR 1 reconstruction residualsa 1 a only for stereo PIV data

detection eﬃciency poor medium medium high high high poor poor poor high

automated optimization simple simple diﬃcult possible possible simple possible diﬃcult simple simple

reference page page page page page page page page page

181 183 184 185 185 187 187 187 215

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6.1.7 Implementation of Data Validation Algorithms Since there is no unique validation ﬁlter suitable for all applications, the common approach has been to apply a combination of several diﬀerent ﬁlters in succession. By adjusting the validation parameters individually for each ﬁlter, high data validation rates can be achieved even if the individual ﬁlters are not operating at their optimum. This strategy is especially attractive when processing larger image quantities. A successful validation procedure should be to collect as much a priori information about the ﬂow ﬁeld to be investigated as possible and to express this knowledge in the form of ﬂuid mechanical or image processing operators. The ﬁrst simple ﬂuid mechanical operators have already been developed [88].

6.2 Replacement Schemes After having validated all PIV data it is possible to ﬁll in missing data using, for instance, bilinear interpolation. According to Westerweel [274] the probability that there is another spurious vector in the direct neighborhood of a spurious vector is given by a binominal distribution. For instance, if the data contains 5% spurious vectors, more than 80% of the data can be recovered by a straight bilinear interpolation from the four valid neighboring vectors. (Incidentally, the bilinear interpolation also fulﬁlls continuity.) The remaining missing data can be estimated by using some sort of weighted average of the surrounding data, such as the adaptive Gaussian window technique proposed ¨i & Jim´ by Agu enez [125]. Some post-processing methods also require smoothing of the data. The reason is that the experimental data is aﬀected by noise in contrast to numerical data. A simple convolution of the data with a 2 × 2, 3 × 3 or larger smoothing kernel (with equal weights) is generally suﬃcient for this purpose. By choosing the kernel size to have spatial dimensions smaller than the effective interrogation window size, additional lowpass ﬁltering of the velocity ﬁeld can be minimized. Median ﬁltering is another eﬀective means for spurious noise reduction. High-quality PIV data typically exhibits less than 1% of spurious vectors under regular conditions and less than 5% in rather challenging experimental situations. Replacement schemes should therefore not be used thoughtlessly if the amount of spurious vectors is (locally) larger than that.

6.3 Vector Field Operators In many ﬂuid mechanical applications the velocity information by itself is of secondary interest in the physical description, which is principally due to the lack of simultaneous pressure and density ﬁeld measurements. In general

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189

the pressure, density and velocity ﬁelds are required to completely recover all terms in the Navier–Stokes equation: ρ

DU = −∇p + μ∇2 U + F Dt

(6.5)

where F represents the contribution of the body forces such as gravity. Efforts to obtain some of these ﬁeld quantities in addition to the velocity ﬁeld is subject of current research, which is partially realized by the application of several methods in parallel (i.e. PSP + PIV, page 301) or through volume capturing methods such as tomographic PIV (page 347). Clearly, the task of obtaining all of these ﬁeld quantities simultaneously is a remaining challenge. By itself, the planar velocity ﬁeld obtained by PIV can already be used to estimate other ﬂuid mechanically relevant quantities by means of diﬀerentiation or integration which will be outlined in the following. Of the diﬀerential quantities the vorticity ﬁeld is of special interest because this quantity, unlike the velocity, is independent of the frame of reference. In particular, if it is resolved temporally, the vorticity ﬁeld can be much more useful in the study of ﬂow phenomena than the velocity ﬁeld by itself, especially in highly vortical ﬂows such as turbulent boundary layers, wake vortices and complex vortical ﬂows. For incompressible ﬂows (∇·U = 0) 1 the Navier– Stokes equation can actually be rewritten in terms of the vorticity, that is the vorticity equation: ∂ω + U · ∇ω = ω · ∇U + ν∇2 ω ∂t

(6.6)

which expresses the rate of change of vorticity of a ﬂuid element (for simplicity, F = 0). Although the pressure term has been eliminated from this expression, the estimation of the last term, ∇2 ω, is diﬃcult from actual PIV data. Because of its frequent use in ﬂuid mechanical descriptions, the estimation of vorticity from PIV data will serve as an example for the available diﬀerentiation schemes given in the following sections. Integral quantities can also be obtained from the velocity ﬁeld. The instantaneous velocity ﬁeld obtained by PIV can also be integrated yielding either single values through path integrals or another ﬁeld quantity such as the stream function. Analogous to the vorticity ﬁeld, the circulation which is obtained through path integration is also of special interest in the study of vortex dynamics, mainly because it is also independent of the reference frame. Other PIV applications may require the calculation of mass ﬂow rates in a control volume type of analysis. A crucial condition for the application of integral analysis is that the ﬁeld of view allows for an appropriate choice of the integration path. The later sections of this chapter will be devoted to the aspects of integration. 1

Incompressibility is a fairly stringent condition for equation (6.6). However, the pressure term vanishes and equation (6.6) holds, if ∇ρ ∇p, that is the ﬂuid is barotropic.

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6.4 Estimation of Diﬀerential Quantities Before addressing the actual calculation schemes available for the diﬀerentiation of the velocity ﬁeld data, it should be determined which terms can actually be calculated from the planar velocity ﬁeld. Standard PIV data provide only the two components2 of the three-dimensional vector ﬁeld while more advanced PIV methods like stereoscopic PIV provide three-component velocity data. Unless several light sheet planes are recorded simultaneously, the PIV method can only provide a single plane of velocity data thereby excluding all possibilities of calculating gradients normal to the light sheet. In order to see which diﬀerential terms actually can be calculated, the full velocity gradient tensor or deformation tensor, dU/dX, will be given ﬁrst: ⎡ ∂U ∂V ∂W ⎤ ∂X ∂X ∂X ∂V ∂W ∂Y ∂Y ∂U ∂V ∂W ∂Z ∂Z ∂Z

dU ⎢ = ⎢ ∂U dX ⎣ ∂Y

⎥ ⎥ ⎦

(6.7)

This deformation tensor can be decomposed into a symmetric part and an antisymmetric part: ⎡ 1 ∂W ⎤ ∂U ∂U ∂U 1 ∂V ∂X 2 ∂X + ∂Y 2 ∂X + ∂Z ⎥ dU ⎢ 1 ∂U ∂V ∂V 1 ∂W ⎥ =⎢ + ∂X + ∂V 2 ∂Y ∂Y 2 ∂Y ∂Z ⎣ ⎦ dX 1 ∂U ∂W 1 ∂V ∂W ∂W 2 ∂Z + ∂X 2 ∂Z + ∂Y ∂Z ⎡ ⎢ +⎢ ⎣

1 2 1 2

∂U ∂Y ∂U ∂Z

0 − −

∂V ∂X ∂W ∂X

1 2

1 2

∂V ∂X

∂V ∂Z

−

∂U ∂Y

0 −

∂W ∂Y

1 2 1 2

∂W ∂X ∂W ∂Y

− −

⎤

∂U ∂Z ⎥ ∂V ⎥ ∂Z ⎦

(6.8)

0

A substitution of the strain and vorticity components yields: ⎤ ⎤ ⎡ ⎡ 1 1 XX 12 XY 21 XZ 0 2 ωZ − 2 ωX dU ⎢ 1 ⎥ ⎥ ⎢ 1 = ⎣ 2 Y X Y Y 21 Y Z ⎦ + ⎣ − 12 ωZ 0 2 ωY ⎦ dX 1 1 − 12 ωX 12 ωY 0 2 ZX 2 ZY ZZ

(6.9)

Thus the symmetric tensor represents the strain tensor with the elongational strains on the diagonal and the shearing strains on the oﬀ-diagonal, whereas the antisymmetric part contains only the vorticity components. Given that conventional two-component PIV provides only the U and V velocity components and that this data can only be diﬀerentiated in the X and Y directions, only a few terms of the deformation tensor, dU/dX, can be estimated with PIV: 2

We ignore the fact that standard PIV only yields a two-dimensional projection of the three-dimensional vector.

6.4 Estimation of Diﬀerential Quantities

∂U ∂V − ∂X ∂Y ∂V ∂U + = ∂Y ∂X ∂V ∂U + = ∂X ∂Y

ωZ = XY η = XX + Y Y

191

(6.10) (6.11) (6.12)

Therefore, only the vorticity component normal to the light sheet can be determined, along with the in-plane shearing and extensional strains. In this regard it is interesting to note that the additional availability of the third velocity component, W , by stereoscopic PIV methods, does not yield any additional strains or vorticity components. Assuming incompressibility, that is, ∇ · U = 0, the sum of the in-plane extensional strains in equation (6.12) can be used to estimate the out-of-plane strain ZZ : ∂U ∂V ∂W =− − = −η (6.13) ZZ = ∂Z ∂X ∂Y However it should be kept in mind that the quantity η only indicates the presence of out-of-plane ﬂow; it does not recover the out-of-plane velocity, W , which should be retrieved directly using stereo PIV for instance. The recently introduced multiplane stereo PIV technique (page 353) can be used to estimate the full vorticity vector. The technique provides threecomponent velocity data at typically two slightly Z-displaced parallel planes, i from which the out-of-plane diﬀerentials ∂u ∂Z can be estimated through central diﬀerences. 6.4.1 Standard Diﬀerentiation Schemes Since PIV provides the velocity vector ﬁeld sampled on a two-dimensional, evenly spaced grid, ﬁnite diﬀerencing has to be employed in the estimation of the spatial derivatives of the velocity gradient tensor, dU/dX. Moreover, each of the velocity data, Ui , is disturbed by noise, that is, a measurement uncertainty, U . Although the error analysis used for the estimation of the uncertainty in the diﬀerentials assumes the measurement uncertainty of each quantity to be decoupled from its neighbors, this is not always the case. For instance, if the PIV image is oversampled, that is the interrogation interval (sample points) is smaller than the interrogation area dimensions (ΔX < ΔX0 and/or ΔY < ΔY0 ), the recovered velocity estimates are not independent because the neighboring interrogation areas partly sample the same particles. At low image densities, N , this problem worsens especially in regions of high displacement gradients (see also ﬁgure 5.42 in section 5.5.7). For simplicity the diﬀerentiation schemes described in the next section assume the measurement uncertainties to be independent of their neighbors. Table 6.2 lists a number of ﬁnite diﬀerence schemes to obtain estimates for the ﬁrst derivate, df /dx, of a function f (x) sampled at discrete locations fi = f (xi ). The “accuracy” in this table reﬂects the truncation

192

6 Post-Processing of PIV Data

Table 6.2. First order diﬀerential operators for data spaced at uniform ΔX intervals along the X-axis. Operator Forward diﬀerence Backward diﬀerence Central diﬀerence Richardson extrapol. Least squares

Implementation fi+1 − fi df ≈ dx i+1/2 ΔX fi − fi−1 df ≈ dx i−1/2 ΔX df fi+1 − fi−1 ≈ dx i 2ΔX fi−2 − 8fi−1 + 8fi+1 − fi+2 df ≈ dx i 12ΔX 2fi+2 + fi+1 − fi−1 − 2fi−2 df ≈ dx i 10ΔX

Accuracy Uncertainty O(ΔX)

≈ 1.41

U ΔX

O(ΔX)

≈ 1.41

U ΔX

O(ΔX 2 ) ≈ 0.7

U ΔX

O(ΔX 3 ) ≈ 0.95 O(ΔX 2 ) ≈ 1.0

U ΔX

U ΔX

error associated with derivation of each operator by means of Taylor series expansion. The actual uncertainty in the diﬀerential estimate due to the uncertainty in the velocity estimates U can be obtained using standard error propagation methods assuming the individual data to be independent of each other. The diﬀerence between the Richardson extrapolation scheme and the least squares approach is that the former is designed to minimize the truncation error while the latter attempts to reduce the eﬀect of the random errors, that is, the measurement uncertainty, U . The least squares approach therefore seems to be the most suitable method for PIV data. In particular, for oversampled velocity data where neighboring data are no longer uncorrelated, the Richardson extrapolation scheme along with the less sophisticated ﬁnite diﬀerence schemes will perform poorly with respect to the least-squares approach. On the other hand, the least-squares approach has a tendency to smooth the estimate of the diﬀerential because the outer data fi±2 are more weighted than the inner data fi±1 . The eﬀect of oversampling on the estimation of the diﬀerential quantities is demonstrated in ﬁgure 6.8 for vorticity ﬁelds computed from the same velocity data at diﬀerent mesh spacings. Since the data is taken from a laminar vortex pair, the vorticity contours are expected to be smooth (data from [305]). For a 50% interrogation window overlap all schemes produce reasonable results since neighboring data are only weakly correlated. The estimate obtained from the forward diﬀerence scheme is the most noisy because the data entering in the formula are correlated (by 50% overlap) which is not the case for the center diﬀerence scheme.

6.4 Estimation of Diﬀerential Quantities Forward differences

ωz, max = 33.7 s-1

ωz, min = -33.0 s-1

Richardson extrapolation

ωz, max = 30.9 s-1

ωz, min = -31.0 s-1

193

Center differences

ωz, max = 27.8 s-1

ωz, min = -28.1 s-1

Least-squares approach

ωz, max = 20.5 s-1

ωz, min = -20.9 s-1

Fig. 6.8. Vorticity ﬁeld estimates obtained from twice-oversampled PIV data, e.g. the interrogation window overlap is 50%. The vortex pair is known to be laminar and thus should have smooth vorticity contours.

By doubling the interrogation window overlap, such as in ﬁgure 6.9, much noisier vorticity ﬁelds are obtained which has two related causes: ﬁrst the grid spacing, ΔX, ΔY , is reduced by a factor of two while the measurement uncertainty for the velocity, U , stays the same. As a result the vorticity measurement uncertainty is doubled. Secondly, all or part of the data used in the diﬀerentiation scheme will be correlated because of the increased overlap. For instance velocity gradient induced bias errors will be similar in neighboring points which in turn results in a biased estimate of the vorticity. Thus, the estimation of diﬀerential quantities from the velocity ﬁeld has to be optimized with respect to the grid spacing. A coarser grid not only yields less noisy estimates of the gradient quantity, but also results in a reduced spatial resolution. The noise performance and frequency response of these as well as more advanced diﬀerentiation schemes have been investigated in further detail by Foucaut & Stanislas [252], Fouras & Soria [253] and Etebari & Vlachos [251]. In the following section, some alternative diﬀerentiation schemes are introduced.

194

6 Post-Processing of PIV Data Forward differences

ωz, max = 50.9 s-1

ωz, min = -41.1 s-1

Center differences

ωz, max = 41.5 s-1

Richardson extrapolation

ωz, max = 45.1 s-1

ωz, min = -38.3 s-1

ωz, min = -35.7 s-1

Least-squares approach

ωz, max = 32.8 s-1

ωz, min = -30.7 s-1

Fig. 6.9. Vorticity ﬁeld estimates obtained from four times oversampled PIV data, e.g. the interrogation window overlap is 75%

6.4.2 Alternative Diﬀerentiation Schemes The ﬁnite diﬀerencing formulae given in table 6.2 have been derived for functions of one variable, that is, they are applied in one dimension at a time. Almost by deﬁnition, the velocity data obtained by PIV is provided on a twodimensional grid which also holds for the diﬀerential quantities obtained from it. As a consequence, the use of one-dimensional ﬁnite diﬀerence schemes for the estimation of the two-dimensional diﬀerential ﬁeld quantities seems inadequate. Using the estimation of the out-of-plane vorticity component, ωZ , as an example several alternative approaches to the problem of diﬀerential estimation will be given. By deﬁnition the vorticity is related to the circulation by Stokes theorem % Γ = U · dl = (∇ × U) · dS = ω · dS (6.14) where l describes the path of integration around a surface S. The vorticity for a ﬂuid element is found by reducing the surface S, and with it the path l, to zero: % 1 U · dl (6.15) n ˆ·ω =n ˆ · ∇ × U = lim S→0 S where the unit vector n ˆ is normal to the surface S. Stokes theorem can also be applied to the (X, Y )-gridded PIV velocity data:

6.4 Estimation of Diﬀerential Quantities

(ω Z )i,j =

1 1 Γi,j = A A

195

% l(X,Y )

(U, V ) · dl

(6.16)

where (ω Z )i,j reﬂects the average vorticity within in the enclosed area. In practice equation (6.16) is implemented by choosing a small rectangular contour (ﬁgure 6.10, for instance two mesh points wide and two mesh points high) around which the circulation is calculated using a standard integration scheme such as the trapezoidal rule. The local circulation is then divided by the enclosed area to arrive at an average vorticity in this area. The following formula provides a vorticity estimate at point (i, j) based on a circulation estimate around the neighboring eight points: & (ωZ )i,j =

Γi,j 4ΔXΔY

(6.17)

with Γi,j =

1 ΔX(Ui−1,j−1 + 2Ui,j−1 + Ui+1,j−1 ) 2 1 + ΔY (Vi+1,j−1 + 2Vi+1,j + Vi+1,j+1 ) 2 1 − ΔX(Ui+1,j+1 + 2Ui,j+1 + Ui−1,j+1 ) 2 1 − ΔY (Vi−1,j+1 + 2Vi−1,j + Vi−1,j−1 ) . 2

(6.18)

Vorticity ﬁelds estimated by this expression are shown in ﬁgure 6.11. When compared to ﬁgures 6.8 and 6.9, this diﬀerentiation scheme clearly performs better, especially in the four-times oversampled data. The principle reason for this is that more data enter in each vorticity estimate. A closer inspection of equation (6.17) reveals that the expression is equivalent to applying the central diﬀerence scheme (table 6.2) to a smoothed (3 × 3 kernel) velocity ﬁeld [51]. While the vorticity estimation by one-dimensional ﬁnite diﬀerences (table 6.2) requires only 4 to 8 velocity data values this expression utilizes 12 data values. The uncertainty in the vorticity estimate, assuming uncorrelated velocity data, then reduces to ω ≈ 0.61U /ΔX compared to ω ≈ U /ΔX for center diﬀerences or ω ≈ 1.34U /ΔX for the Richardson extrapolation i-1

i

i+1 j+1

j Y j-1 X

Fig. 6.10. Contour for the circulation calculation used in the estimation of the vorticity at point (i, j).

196

6 Post-Processing of PIV Data 50% Overlap

ωz, max = 26.7 s-1

75% Overlap

ωz, min = -26.6 s-1

ωz, max = 39.0 s-1

ωz, min = -35.3 s-1

Fig. 6.11. Vorticity ﬁeld estimates obtained from PIV velocity ﬁelds by the circulation method: (left) the velocity ﬁeld is twice oversampled, (right) four times oversampled. The contours of this laminar vortex pair are known to be smooth such that the non-uniformities are due to measurement noise

method. Further eﬀects due to data oversampling are not as signiﬁcant as with some of the simpler one-dimensional diﬀerentiation schemes because no diﬀerences of directly adjoining data are used. A similar approach may be used in the estimation of the shear strain and the out-of-plane strain. ∂U ∂V Ui−1,j−1 + 2Ui,j−1 + Ui+1,j−1 + = & − (xy )i,j = ∂Y ∂X i,j 8ΔY Ui+1,j+1 + 2Ui,j+1 + Ui−1,j+1 8ΔY Vi−1,j+1 + 2Vi−1,j + Vi−1,j−1 − 8ΔX Vi+1,j−1 + 2Vi+1,j + Vi+1,j+1 + 8ΔX

+

−(zz )i,j =

∂U ∂V + ∂X ∂Y

= & i,j

(6.19)

Vi−1,j−1 + 2Vi,j−1 + Vi+1,j−1 8ΔY

Vi+1,j+1 + 2Vi,j+1 + Vi−1,j+1 8ΔY Ui+1,j−1 + 2Ui+1,j + Ui+1,j+1 + 8ΔX Ui−1,j+1 + 2Ui−1,j + Ui−1,j−1 − 8ΔX

−

(6.20)

For the out-of-plane or normal strain an analogy to the vorticity/circulation relation can be given: in place of the circulation the net ﬂow across the

6.4 Estimation of Diﬀerential Quantities

a)

b)

Vorticity

dV > 0 dX

c)

Shear strain

197

Normal strain

dV > 0 dX

dU > 0 dX dV > 0 dY dU > 0 dY

dU < 0 dY

ΔX ΔY (i,j)

(i,j)

( U δ s)

(i,j)

( U n ) δs

Fig. 6.12. Implementation of the three major diﬀerential quantities obtainable with planar PIV data. The deformation of the ﬂuid element is given on top while the bottom shows the path of integration.

boundaries of the contour is calculated. However, no such analogy exists for the shear strain, XY . Figure 6.12 graphically illustrates the three diﬀerential estimation schemes described in this section. Aside from the above mentioned techniques for diﬀerential estimation in PIV velocity data, the literature has suggested alternative methods. Earlier it was noted that the estimation uncertainty, Δ , for the same diﬀerentiation scheme is directly proportional to the grid spacing (ΔX, ΔY ), that is, Δ ≈ U /ΔX. Once the interrogation window overlap exceeds 50%, the velocity data entering in the diﬀerentiation are increasingly correlated (i.e. biased) and cause the diﬀerential estimates to be biased. For these reasons Lourenc ¸o & Krothapalli [261] suggested the use of an adaptive scheme for the computation of vorticity. The method is based on Richardson extrapolation and is aimed at minimizing the total error in the vorticity estimate by combining vorticity estimates at several diﬀerent grid spacings. Even better results can be obtained by also including a least squares second order polynomial approximation in the diﬀerentiation scheme. The extension of this principally one-dimensional approach to diﬀerential estimation would certainly also be possible. The task of diﬀerential estimation from velocity ﬁeld data can also be studied from a two-dimensional signal processing point of view as suggested by Lecuona et al. [259, 264]. Linear ﬁlter theory is used to derive and optimize a variety of one- and two-dimensional diﬀerentiating ﬁlters whose performance is

198

6 Post-Processing of PIV Data 50% Overlap

ωz, max = 27.8 s-1

ωz, min = -27.2 s-1

75% Overlap

ωz, max = 39.2 s-1

ωz, min = -34.2 s-1

Fig. 6.13. Vorticity ﬁeld estimates obtained from PIV velocity ﬁelds using a linear, two-dimensional ﬁlter: (left) the velocity ﬁeld is twice-oversampled, (right) four times oversampled.

tested on noisy PIV data. Vorticity estimates from one such ﬁlter (f) are shown in ﬁgure 6.13. Compared to the circulation method equation (6.17) this differential ﬁlter is more susceptible to the side-eﬀects of oversampling described before because it is designed to perform well at higher spatial frequencies. 6.4.3 Uncertainties and Errors in Diﬀerential Estimation As already noted in the previous section a variety of factors enter in the uncertainty of a diﬀerential estimate. Uncertainty in velocity: Each PIV velocity estimate Ui,j is associated with a measurement uncertainty U whose magnitude depends on a wide variety of aspects such as interrogation window size, particle image density, displacement gradients, etc. (see section 5.5). Since diﬀerential estimates from the velocity data require the computation of local diﬀerences on neighboring data the noise increases inversely proportional to the local diﬀerence, Ub − Ua , as the spacing between the data ΔX = |Xa − Xb | is reduced. That is, the estimation uncertainty in the diﬀerential, Δ , scales with U /ΔX. Oversampled velocity data: It is common practice to oversample a PIV recording during interrogation at least twice in order to bring out smallscale features in the ﬂow. Because of this oversampling, neighboring velocity data are estimated partially from the same particle images and therefore are correlated with each other. Because of this, neighboring data are likely to be biased to a similar degree, especially in regions containing high velocity gradients and/or low seeding densities (see section 5.5.7). This localized velocity bias then causes the diﬀerential estimate to be biased as well. The oversampling eﬀects can be observed very well by comparing ﬁgures 6.8 and 6.9. The eﬀect of oversampling is partially reduced through application of advanced processing schemes such as iterative image deformation techniques as introduced in section 5.4.4. To illustrate this the vortex pair data shown in

6.4 Estimation of Diﬀerential Quantities

ω z, max= 29.4 s-1

ω z, min= -29.0 s

-1

199

Fig. 6.14. Vorticity estimate calculated from 75% overlapped PIV data using equation (6.17). Compared to the earlier examples an iterative image deformation PIV algorithm was ﬁrst used to calculate the velocity data.

the previous section was processed by an image deforming algorithm at 75% overlap and diﬀerentiated using the circulation approach (equation (6.17)). In spite of the high oversampling the contours in ﬁgure 6.14 are smooth as expected from the laminar ﬂow. In part this is a direct result of the smoothing applied during the intermediate processing steps of the processing algorithm. Nonetheless, the advanced processing algorithms are capable of providing nearly bias-free data where the choice of subsequent diﬀerentiation scheme is less critical. Interrogation window size: The size of the interrogation window in the object plane (ΔX0 × ΔY0 ) deﬁnes the spatial resolution in the recovered velocity data. The spatial resolution in the velocity ﬁeld in turn limits the obtainable spatial resolution of the diﬀerential estimate. Depending on the utilized diﬀerentiation scheme the spatial resolution will be reduced to some degree due to smoothing eﬀects. The eﬀect of the interrogation window size on both the velocity as well as vorticity estimate is given in ﬁgure 6.15. Curvature eﬀects: The standard PIV method is only a ﬁrst order approximation to the true particle image displacement. Since it generally relies on only two illumination pulses, eﬀects due to acceleration and curvature are lost. In regions of rotating ﬂow this straight line approximation underestimates the actual particle image displacement and thereby the local velocity. Diﬀerential estimates will then have a tendency to be biased to lower magnitudes as well. By reducing the illumination pulse delay, Δt, this eﬀect can be reduced at the cost of increased noise in the diﬀerential estimate due to the velocity measurement uncertainty, U , itself. As pointed out by Wereley & Meinhart [169] the symmetric oﬀset of the interrogation windows during iterative processing provides second order accurate displacement estimates and thus reduces the eﬀects of curvature because the displacement vector is attached to the midpoint between the oﬀset samples.

200

6 Post-Processing of PIV Data

Actual velocity profile

Measured velocity profile

Measured velocity profile

(Large interrogation window)

(Small interrogation window)

Y

Y

Y U1

U1

U1

Y2 Window size

Y1 U1

U1

Actual vorticity profile

U1 U(Y)

U(Y)

U(Y)

Estimated vorticity profile

Y

Estimated vorticity profile

Y

Y

ω+

Y2

ω−

U2 Window size

U2

U2

ω+

ω+

Y1

ω−

ω−

ω (Y)

ω (Y)

ω (Y)

Fig. 6.15. Eﬀect of spatial resolution on vorticity estimation.

6.5 Estimation of Integral Quantities 6.5.1 Path Integrals – Circulation By deﬁnition, the vorticity integrated over an area A equals the circulation, Γ . Using Stokes theorem (equation 6.14) this operation reduces to a line integral of the dot product between the local velocity vector, U, and the incremental path element vector dX where the integration path is deﬁned by the boundary, C, of the enclosed area A: ω dA (6.21) Γ = %A = U · dX (6.22) C

For two-component velocity data constrained to the XY plane with U = (U, V ), the above equation reduces to:

6.5 Estimation of Integral Quantities

201

Γ = %

ωZ dX dY

(6.23)

U(X, Y) · dX

(6.24)

U dX + V dY

(6.25)

A(X,Y )

= C(X,Y )

% =

C(X,Y )

Given the path of integration, the evaluation of equation (6.25) is straightforward using integration schemes such as the trapezoidal approximation or Simpson’s rule. To determine the circulation of clearly deﬁned, nearly round vortical structures, a circular integration path centered at the position of maximum vorticity is generally suﬃcient. By plotting the circulation with respect to integration path radius, an asymptotic convergence towards the value of the structure’s circulation can be observed (provided no other vortices are included by the integration contour). This convergence coincides with a decay of vorticity away from the vortex core. For more complex vortical structures the assignment of a reasonable integration path is not very simple. For vortical structures, the ideal integration path would be deﬁned by a dividing stream line which separates it from other vortical structures. However, the computation of the required stream function from the unsteady velocity data is nontrivial and often nonunique (see section 6.5.3). Since the circulation actually is an area-integral of vorticity, an integration along a constant-vorticity contour near zero will evaluate to a value close to the vortex’s actual circulation. This approach was chosen, for instance, in the evaluation of the Karman vortex street data shown in ﬁgure 6.16. Although this approach is very robust, the diﬃculty often lies in retrieving the desired closed contour from the vorticity data. Once the contour is available, a bi-linear interpolation of the velocity on to the contour is suﬃcient for the evaluation of equation (6.25). 6.5.2 Path Integrals – Mass Flow In some applications the rate mass or volume across a control surface, CS, is of interest and is expressed as a surface integral: dm ˙ = ρ(U · n ˆ) dS (6.26) M= dt CS For two-dimensional data constrained to the xy plane, the surface reduces to a path integral similar to equation (6.25): % ˙ XY = dmXY = M ρ(U dY − V dX) (6.27) dt C ˙ XY are mass ﬂow per unit depth and if ρ ≡ 1 then equaThe units of M tion (6.27) represents a volume ﬂow rate per unit depth. With regard to its

202

6 Post-Processing of PIV Data

a)

[mm]

50 cm/s

b)

Vorticity Field 60.2

51.5

48.6

[mm]

54.0

Circulation [cm 2/s] 57.3 56.2 52.8

[mm]

Fig. 6.16. The contours of the vorticity ﬁeld are used as integration paths for the circulation whose magnitude is given above the individual vortices (data courtesy ¨ der [267]). of Schro

numerical implementation, similar integration schemes as for the estimation of the circulation (section 6.5.1) can be used. In cases where three-dimensional velocity data is available in a plane, the actual mass (or volume) ﬂow rate across this surface or portions thereof can be determined using an area integral: ˙ = W dX dY (6.28) M A(X,Y )

where W is the velocity component normal to the light sheet. In this case the approximation of the integral is more complicated than for the previously described line integrals. 6.5.3 Area Integrals The following integration schemes are based on the assumption that the integrand, that is, the ﬂow ﬁeld, is two-dimensional as well as incompressible. Further on, assuming the ﬂow to be irrotational, potential theory relates the

6.5 Estimation of Integral Quantities

203

velocity ﬁeld, U = (U (X, Y ), V (X, Y )), to the stream function, Ψ, and potential function, Φ: ∂Ψ ∂Φ = ∂Y ∂X ∂Φ ∂Ψ = V =− ∂X ∂Y U =

which can be integrated over the domain (i.e. XY plane) to: Ψ= U dY − V dX Y X Φ= U dX + V dY X

(6.29) (6.30)

(6.31) (6.32)

Y

Although these purely kinematic conditions will work reasonably well for the ﬂows studied with PIV, the inherent problem is that, depending on the chosen frame of reference, nonunique solutions to Ψ and Φ are obtained. This is due to the fact that equation (6.31) and equation (6.32) are a simpliﬁcation of the Poisson equation: (6.33) ∇2 Ψ = −ωZ to a Laplace equation, ∇2 Ψ = 0 using the condition of irrotationality (i.e. ωZ = 0). The integration of equation (6.33) is rather diﬃcult because the integrand can only be approximated from the velocity ﬁeld data (see section 6.4). Further, the boundary conditions along the edges of the ﬁeld of view need to be deﬁned prior to integrating equation (6.33). Figure 6.17 shows the result of integrating equation (6.31) on an actual ﬂow, a vortex pair, which can be assumed to be nearly two-dimensional, but

Fig. 6.17. Two-dimensional stream function computed from vortex pair velocity data in a laboratory-ﬁxed reference frame (left) and in a reference frame moving 20 mm/s upward with the vortex pair (right).

204

6 Post-Processing of PIV Data

not irrotational. Depending on the choice of the reference frame two entirely diﬀerent results are obtained. For instance, if the vortex propagation speed is subtracted, the bounding streamline of the Kelvin oval, that is, the body of ﬂuid moving with the vortex pair, can be approximated. This is not the case for streamlines computed in a laboratory-ﬁxed reference frame (ﬁgure 6.17 left). Since equation (6.31) and equation (6.32) are path-independent integrals, the numerical integration of the velocity ﬁeld can be freely chosen. In this case an integration scheme as presented by Imaichi & Ohmi [256] is used. The trapezoid approximation is used to integrate between two neighboring points. To start the integration a starting point, Po , is chosen, preferably near the middle of the velocity ﬁeld since errors in the individual velocity data are propagated through integration. As illustrated in ﬁgure 6.18 there are two principal integration methods: a column-major integration or a row-major integration. In the ﬁrst case, the integration proceeds in opposite horizontal directions away from the starting point, Po , producing new values of the integral for each node on the horizontal. These new estimates are then used as initial values for the integration in opposite directions along the vertical columns, producing estimates of the integral throughout the domain. A second estimate of the integral can then be obtained by reversing the order of the integration scheme, that is, by starting the integration oﬀ in opposite directions along the vertical line containing the starting point. The two results are then arithmetically averaged together. Since the described integration scheme tends to propagate disturbances due to noisy or erroneous data “down-stream” of its occurrence, more sophisticated integration schemes, such as a multigrid approach, could be used. In this case the integral is ﬁrst computed on highly smoothed and subsampled versions of the ﬂow ﬁeld and successively updated as the sampling mesh is reﬁned. Y

Po X

Fig. 6.18. Integration routes used to integrate two-dimensional stream and potential functions as well as pressure. Two paths of integration follow either the dashed or solid arrows (after [256]).

6.5 Estimation of Integral Quantities

205

6.5.4 Pressure and Forces from PIV Data If the ﬂow ﬁeld under investigation is nearly two-dimensional, steady (i.e. dU /dt = 0) as well as incompressible (i.e. dρ/dt = 0) the pressure ﬁeld can be estimated through the numerical integration of the steady Navier–Stokes equations in two-dimensional form [256]: 2 ∂ U ∂U ∂U 1 ∂p ∂2U U +V =− +ν + , (6.34) ∂X ∂Y ρ ∂X ∂X 2 ∂Y 2 2 ∂ V ∂V 1 ∂p ∂V ∂2V +V =− +ν U + . (6.35) ∂X ∂Y ρ ∂Y ∂X 2 ∂Y 2 To obtain the pressure ﬁeld, the pressure gradients ∂p/∂X and ∂p/∂Y are approximated using ﬁnite diﬀerence approximations of the velocity gradients. The pressure gradients are then integrated starting from a point near the center with an initial value of p0 using an integration scheme similar to the one illustrated in ﬁgure 6.18. An alternative method of pressure ﬁeld estimation from incompressible PIV velocity data is by means of integration of the Poisson pressure equation [254, 255]: ∇2 p = −ρ∇ · (U · ∇U)

(6.36)

with appropriate boundary conditions. On surfaces a Neumann boundary condition for pressure can be derived from the Navier-Stokes equation [254, 255, 269]: dU + ρ(U · ∇)U − μ∇2 U (6.37) dt If the outer ﬂow around an object is steady in the far ﬁeld, an outer boundary condition can be also prescribed using the Dirichlet condition p = 0 which then allows an integration of equation (6.36) across the ﬁeld of view. The feasibility of this approach has been demonstrated for constricting pipe ﬂow and impinging jet ﬂow [255] as well as for the ﬂow around an oscillating cylinder [254]. In all cases the acceleration term dU/dt in equation (6.37) was assumed to be zero which limits the applicability of the technique. A more rigorous discussion on the possibilities of pressure and force estimation from PIV data using a control-volume approach (ﬁgure 6.19) is provided by Noca et al. [263], who applied several techniques for the estimation of forces on a low Reynolds number ﬂow around a cylinder. The availability of time-resolved data allows the proper treatment of the acceleration term in equation (6.37) as demonstrated by Liu & Katz [260] on a 2D cavity turbulent ﬂow ﬁeld using a four frame PIV system. The availability of time-resolved data allows the proper treatment of the acceleration term in equation (6.37) as demonstrated by Liu & Katz [260] on a 2D cavity turbulent ﬂow ﬁeld using a four frame PIV system. The unsteady forces on a square cylinder along with the surrounding pressure ﬁelds were −∇p = ρ

206

6 Post-Processing of PIV Data

Fig. 6.19. Control-volume approach for determining integral forces from two dimensional ﬂow (after [263, 270]).

estimated from time-resolved 2C PIV obtained with a high speed PIV system operated at 1 kHz (Kurtulus et al. [257]). In this particular case the estimation of pressure gradients was based on potential ﬂow theory for the inviscid regions of the ﬂow while the Navier-Stokes equations were integrated between two boundary points from the edges of the viscous regions (i.e. cylinder wake). As suggested by van Oudheusden et al. [270, 271] equation (6.37) can be averaged to obtain the time-mean pressure gradient: −

∂ui uj ∂u ¯i ¯i ∂2u ∂ p¯ = ρ¯ uj +ρ −μ . ∂xi ∂xj ∂xj ∂xj ∂xj

(6.38)

Here ui are velocity components in tensor notation with averaged properties indicated by an overbar. As all terms on the right-hand side of equation (6.38) can be obtained from statistically averaged 2C PIV data, the time-mean pressure and body forces can thus be estimated through appropriate integration. van Oudheusden have also further extended their method to steady compressible ﬂow and demonstrated its viability by applying it to a two-dimensional Ma = 2 ﬂow around a bi-convex airfoil [271]. The previously described pressure and force estimation methods mostly assume two-dimensionality of the ﬂow and require the evaluation of momentum equation on the surface S (i.e. equation (6.37)) which considerably limits their applicability. One possible solution to this deﬁciency is oﬀered by the simultaneous measurement of velocity ﬁeld and surface pressure, the latter for example provided by means of pressure sensitive paint (PSP, see also section 9.8) . This provides a more reliable boundary condition for the integration of the pressure ﬁeld using equation (6.36) and is the only suitable approach when only a portion of the ﬂow around a body can be acquired. Clearly, the further incorporation of time- and volume resolved velocity data grants full access to all terms necessary for a complete recovery of the pressure distribution.

6.6 Vortex Detection

207

6.6 Vortex Detection The velocity ﬁeld obtained by PIV frequently is an intermediate result in the investigation of complex ﬂow phenomena. Further postprocessing is required to extract important ﬂuid mechanical properties. For validation of numerical tools and for aerodynamic studies in particular, a precise knowledge of the vortical ﬂow is desired, which is exempliﬁed by a number of applications provided in the later portion of this book. The following is intended to brieﬂy outline the potential of PIV in the analysis of vortical ﬂows. Though there seems to be a common understanding about what a vortex looks like, it is mostly deﬁned by empirical arguments or may be very subjective. In general, vortices are created due to conservation of angular momentum and not necessarily assume an easily detectible circular shape, especially if several vortices interact with each other. A vortex may be characterized by its location, circulation, core radius, drift velocity, peak vorticity, maximum circumferential velocity, for instance. The velocity ﬁeld generally has a tendency to hide vortices in the presence of convective ﬂow. Streamlines, if computable (see section 6.5.3), give a fairly good indication of vortical structures in the ﬂow ﬁeld as outlined in ﬁgure 6.20. Alternatively the vorticity ﬁeld obtained from the gradient tensor already indicate the presence of vortices regardless of the frame of reference. As a ﬁnite diﬀerence from noisy data the vorticity ﬁeld tends to be very noisy especially for the determination of the vortex centers. The squared vorticity ﬁeld or enstrophy further enhances the visibility of the vortices but is equally susceptible to noise. A more rigorous analysis of the available vortex characterization methods is given by Vollmers [272] as well as Schram et al. [340]. Among the available methods the rather useful λ2 -operator shall be brieﬂy described here.

Fig. 6.20. Behavior of autonomous ordinary diﬀerential equations of two degrees of freedom in diﬀerent regions dependent on trace and determinant of the velocity gradient tensor given in equation equation (6.39) (after Vollmers [272]).

208

6 Post-Processing of PIV Data

Following Vollmers [272] vortices appear in ﬁgure 6.20 for non-real eigenvalues of the gradient tensor ∂U ∂V dU ∂X ∂X = ∂U ∂V G= (6.39) dX ∂Y

∂Y

The discriminant λ2 of non-real eigenvalues of the velocity tensor G separates vortices from other patterns. It is obtained from 2

λ2 = (trace G) − 4 det(G) 2 ∂U ∂U ∂V ∂V ∂U ∂V + · − · = −4 ∂X ∂Y ∂X ∂Y ∂Y ∂X

(6.40) (6.41)

Regions with negative values of λ2 indicate vortices. Unlike vorticity the discriminant λ2 generally does not identify boundary layers and shear layers as vortices. This makes it very useful for the position estimation of vortices but unfortunately does not provide information on direction of rotation. This information can be obtained by analyzing the surrounding velocity ﬁeld. As the expression 6.40 is restricted to two-dimensional interpretations of the ﬂow, false detection may result for strongly three-dimensional ﬂows. If the full 3-C gradient tensor is available, the expression is readily extensible for three-dimensional vortex detection. A variant of equation (6.40) is also used for the detection of vortices in applications given in sections 9.9 and 9.10. These applications also describe the potential of wavelets for the detection and characterization of vortices.

7 Three-Component PIV Measurements

In spite of all its advantages, the PIV method contains some shortcomings that necessitate further developments on the basis of instrumentation. One of these disadvantages is the fact that the “classical” PIV method is only capable of recording the projection of the velocity vector into the plane of the light sheet; the out-of-plane velocity component is lost while the in-plane components are aﬀected by an unrecoverable error due to the perspective transformation as described in section 2.6.3. For highly three-dimensional ﬂows this can lead to substantial measurement errors of the local velocity vector. This error increases as the distance to the principal axis of the imaging optics increases. Thus it is often advantageous to select a large viewing distance in comparison to the imaged area to keep the projection error to a minimum. This is easily achieved using long focal length lenses. Nevertheless, an increasing number of PIV applications require the additional knowledge of the out-of-plane velocity component. A variety of approaches capable of recovering the complete set of velocity components have been described in the literature [38, 210]. The most straightforward, but not necessarily easily implemented, method is an additional PIV recording from a diﬀerent viewing direction using a second camera, which can be generally referred to as stereoscopic PIV recording [198, 204, 205, 217, 218, 220]. Reconstruction of the three-component velocity vector in eﬀect relies on the perspective distortion of a displacement vector viewed from diﬀerent directions. While most stereoscopic setups employ two cameras, stereoscopic viewing can also be achieved with a single camera by placing a set of mirrors in front of the recording lens [193]. Holographic PIV recording is another approach capable of recovering the third displacement component [213, 228]. If suﬃcient illumination power is present, the light sheet can be expanded into a thick slab such that three-dimensional PIV measurements are possible throughout a volume [224]. A completely diﬀerent approach, referred to as dual-plane PIV, is implemented by oﬀsetting the light sheet a small amount between the recordings to obtain a third PIV recording [208]. Based on measuring the change

210

7 Three-Component PIV Measurements

of the respective correlation peak height from one recording to the next, the out-of-plane displacement component can be estimated (see section 7.2). A similar approach was utilized for the analysis of PIV image sequences obtained in a scanning light sheet setup [87]. Of all the previously described three-dimensional/three-component PIV methods, holography is capable of the highest measurement precision but, at its present stage of development, is not well suited for experiments, where setup time, optical access and observation distances are important factors. For application to low-speed liquid ﬂows the “dual-plane” extension to standard PIV may be the most easily implemented, as it only requires an additional third illumination pulse and a slight out-of-plane displacement of the light sheet. The measurement precision of the third component does however depend on a continuous and known shape of the light sheet’s intensity proﬁle among other factors, many of which are still under investigation. Unless special measures are taken, the intensity proﬁles of the commonly utilized frequency doubled Nd:YAG lasers are not adequate for providing out-of-plane information with reasonable precision. Also, the recording of three separate images in short succession (microsecond range) is not easily tackled with existing PIV equipment; high-speed video or cinematography is necessary. Aside from a proof of concept on a vortex ring in air [209], the dual-plane technique, described in section 7.2, has not yet been applied to ﬂows of medium or high speed. To overcome these limitations the multiplane PIV technique, an extension of stereoscopic PIV technique, has been developed. Relying on four laser pulses and four cameras, it is capable of recovering velocity data in two adjacent image planes with the same high spatial resolution available for standard PIV [392, 393]. Thus it can be used to estimate the all quantities of the deformation tensor (equation (6.7)). More information on the multiplane PIV technique along with application examples can be found in section 9.18.1. A combination between stereo PIV and photogrammetric-based particle matching has also been demonstrated for the recovery of volume-resolved velocity data. The three-dimensional hybrid stereoscopic PIV technique [247] relies on the identiﬁcation and matching of tracer particle using redundant information available from a minimum of three cameras. Although the technique could be classiﬁed as a PTV approach it uses a three-dimensional crosscorrelation method to recover the displacement information as well as camera calibration methods, both of which also are essential building blocks for tomographic PIV described next. More recent eﬀorts have been directed toward the recovery of full volume data sets using multiple camera digital imaging along with tomographic reconstruction. The volume PIV technique, also known as tomographic PIV, simultaneously images a given volume with a least three high resolution cameras from diﬀerent directions. The over-determined nature of the image data allows for a tomographic reconstruction of the particle positions in space, such that a pair of these volume images recorded in short succession can be

7.1 Stereo PIV

211

used to obtain a three-dimensional distribution of three-component velocity data [230]. In practice the technique often makes use of four cameras which signiﬁcantly increases the data yield with respect to three cameras. The low illumination levels due to volume illumination require large camera lens apertures which again limits the depth of ﬁeld, such that a rather thick light sheet provides the best results. As shown in the application given in section 9.16, time resolved measurements of the volume resolved ﬂow even become possible through the use of high speed imaging. In the following section, the emphasis, however, will be on the recovery of a single plane of three-component velocity data using ‘only’ two cameras in a stereoscopic viewing arrangement. It has become a rather widespread technique for which much knowledge already has been accumulated but diﬀerent implementations in terms of calibration and vector recovery exist [193, 194, 202, 204, 206, 212, 217, 218, 219, 220].

7.1 Stereo PIV In the following a stereoscopic approach will be described, which has been developed for application in an industrial wind tunnel environment. The adaptation of this approach to applications in liquid ﬂows can easily be performed by changing the angles between the lens plane and sensor plane according to the refraction of the air-water interfaces. Detailed descriptions of the adaptation of stereo PIV to liquid ﬂows are given in [204, 205, 215, 216]. Since the measurement precision of the out-of-plane component increases as the opening angle between the two cameras reaches 90◦ , it is not always possible to mount the camera pair on a common base, when using large observation distances, and much less to provide a symmetric arrangement. Therefore a general description for asymmetric recording and associated calibration has been developed. Another problem that arises through the use of large focal length imaging lenses: their limited angular aperture restricts the distance between the lenses in a translation imaging approach (ﬁgure 7.1a, [201]). Designed for use with a ﬁxed format sensor centered on the optical axis of the lens, most lenses are not only limited in their optical aperture but also characterized by a strong decrease in the modulation transfer function (MTF) towards the edges of the ﬁeld of view. To adequately image small particles a good MTF at small f numbers (f# < 4) is a stringent requirement (section 2.6). Since lens systems with an oblique principal axis are practically nonexistent, a departure from the translation imaging method of ﬁgure 7.1a is unavoidable. As the best MTF is generally present near the lens principal axis, the alternative angular displacement method (ﬁgure 7.1b) aligns the lens with the principal viewing direction. The additional requirement for small f -numbers is associated with a very small depth of ﬁeld which only can be accommodated by additionally tilting the back plane according to the Scheimpﬂug criterion in which the

212

7 Three-Component PIV Measurements

a)

Object Plane

Lens Plane Camera #1

Camera #2 Image plane

b)

Object Plane

θ φ

ne

la sp

n

Im

ag

ep

lan

e

Le

Camera #1

Camera #2

Fig. 7.1. Basic stereoscopic imaging conﬁgurations: a) lens translation method, b) angular lens displacement with tilted back plane (Scheimpﬂug condition).

image plane, lens plane and object plane for each of the cameras intersect in a common line [38, 205, 211]. The oblique view of the scene is associated with a perspective distortion that is further increased by the Scheimpﬂug imaging arrangement. In essence, the perspective distortion results in a magniﬁcation factor that is no longer constant across the ﬁeld of view and requires an additional means of calibration to be described later. In the following sections the generalized, that is nonsymmetric, description for stereoscopic PIV imaging is given ﬁrst and is followed by a methodology for calibrating the perspective distortion. The feasibility of this approach is demonstrated in an experiment with an unsteady ﬂow ﬁeld, which is described in section 9.4. 7.1.1 Reconstruction Geometry This section describes the geometry necessary to reconstruct the threedimensional displacement ﬁeld from the two projected, planar displacement ﬁelds. Past descriptions of stereoscopic PIV imaging systems attempt to use a symmetric arrangement [66, 199, 204, 205, 217, 218]. In the present case

7.1 Stereo PIV

213

the two cameras may be placed in any desirable conﬁguration provided the viewing axes are not collinear. In section 2.6.3 we determined the basic equations - 2.20 and 2.21 - for particle image displacement assuming geometric imaging: xi − xi = −M DX + DZ z0 y i yi − yi = −M DY + DZ z0

xi

(7.1) (7.2)

In the following we will use the angle α in the XZ plane between the Z axis and the ray from the tracer particle through the lens center O to the recording plane as shown in ﬁgure 7.2. Correspondingly, β deﬁnes the angle within the Y Z plane. xi z0 y tan β = i z0

tan α =

The velocity components measured by the left camera are given by: xi − xi M Δt yi − yi V1 = − M Δt

U1 = −

The velocity components for the right camera U2 and V2 can be determined accordingly. Using the above equations, the three velocity components

V x

w

u

z α1 α2 u1 O Camera 1

u2 P Camera 2

Fig. 7.2. Stereo viewing geometry in the XZ-plane.

214

7 Three-Component PIV Measurements

(U, V, W ) can be reconstructed from the four measured values. For α, β ≥ 0 we obtain: U1 tan α2 + U2 tan α1 tan α1 + tan α2 V1 tan β2 + V2 tan β1 V = tan β1 + tan β2 U1 − U2 W = tan α1 + tan α2 V1 − V2 = tan β1 + tan β2 U =

(7.3) (7.4) (7.5) (7.6)

These formulae are general and apply to any imaging geometry. Note, that there are three unknowns and four known measured values, which results in an overdetermined system that can be solved in a least-squares sense (see below). Also, the denominators can approach zero as the viewing axes become collinear in either of their two-dimensional projections. For example, in the setup described in section 9.4 the cameras are approximately positioned in the same vertical position as the ﬁeld of view which makes the angles β1 , β2 and their tangents tan β1 and tan β2 very small. Clearly, component W can only be estimated with higher accuracy using equation (7.5), while V has to be rewritten using equation (7.5) which does not include tan β1 and tan β2 in the denominator: V1 + V2 W + (tan β1 − tan β2 ) 2 2 U1 − U2 tan β1 − tan β2 V1 + V2 + V = 2 2 tan α1 + tan α2 V =

(7.7) (7.8)

If tan β1 and tan β2 are very small, then V is given as the arithmetic mean of V1 and V2 with the out-of-plane component W having no eﬀect. As mentioned above the velocity components may also be solved for in a least squares sense as the system of equations is once overdetermined, that is, there are three unknown Cartesian displacement components but four known displacement components: ⎡ ⎤ 1 U1 ⎢ ⎢ V1 ⎥ ⎢ 0 ⎢ ⎥=⎢ ⎣ U2 ⎦ ⎢ 1 ⎣ V2 0 ⎡

x 0 −O Oz

⎤

⎡ ⎤ O ⎥ U 1 − Oyz ⎥ ⎥ ⎣V ⎦ Px ⎥ · 0 − Pz ⎦ W P 1 − Pyz

Umeas = A · V ⇒ V = (AT · A)−1 · AT · Umeas

(7.9)

(7.10) (7.11)

7.1 Stereo PIV

215

The residuals resid of this least squares ﬁt can be used as a measure of quality for the three component measurement result, since they should vanish in an ideal (noise-free) measurement. In practice residuals in the range of 0.1 − 0.5 pixel are common. Increased misalignment between the camera views results in signiﬁcant residuals, especially for ﬂows with high spatial variations. To use the above reconstruction, the displacement data set must ﬁrst be converted from the image plane to true displacements in the global coordinate system while taking into account all the magniﬁcation issues. Following Wieneke [219] the literature reports three main approaches: 1. The 2-C displacements for each view are computed on a regular grid in the raw image space. 3-C reconstruction is performed by projecting the vector maps onto a common grid using interpolation [206]. 2. The 2-C displacements are computed from the raw images at positions corresponding to the desired object space coordinates [194]. 3. The raw images are ﬁrst mapped onto a common image space before being analyzed at coincident object positions using 2-C vector processing [220]. While the ﬁrst approach is suited for rather fast processing time, the main drawback is that false or inaccurate vectors may result in unreliable reconstruction of the neighboring interpolated vectors. The second approach avoids the interpolation step but generally is associated with diﬀerent-sized interrogation windows in object space unless elaborate processing with spatially varying, non-rectangular interrogation samples is used. The alternative, also followed here, is to ﬁrst map both camera images onto a common image space as depicted in ﬁgure 7.3, such that subsequent PIV processing automatically makes use of common interrogation grid with constant sample size throughout. Regardless of the chosen approach, the projection onto a common grid in object space requires a priori calibration procedures which are described next. 7.1.2 Stereo Viewing Calibration In order to reconstruct the local displacement vector the viewing direction and magniﬁcation factor for each camera must be known at each point in the respective images. This correspondence between the image (x, y) and the object plane (X, Y ) may in fact be described through geometric optics; however, it requires exact knowledge of the imaging parameters such as the lens focal length, f , the angles between the various planes, θ, φ (see ﬁgure 7.1b), the actual position of the lens plane (which is not simple to determine) and the nominal magniﬁcation factor, Mo (the magniﬁcation along the principal optical axis): f x sin φ M0 sin θ(x sin φ + f M0 ) fy Y = x sin φ + f M0

X=

216

7 Three-Component PIV Measurements

These approximate expressions, given in [217], do not incorporate nonlinearities such as lens distortions and are sensitive to small variations in each of the parameters. A more robust approach is the second order image mapping approach employed by other researchers [22, 217]: X = a0 + a1 x + a2 y + a3 x2 + a4 xy + a5 y 2 + . . . Y = b0 + b1 x + b2 y + b3 x2 + b4 xy + b5 y 2 + . . .

(7.12) (7.13)

The above equations do not constitute a mapping based on the geometry at hand. Nevertheless the twelve unknown parameters can easily be determined using a least squares approach if at least six image-object point pairs are given. The advantage of this approach is that the imaging parameters such as focal length, magniﬁcation factor, etc., never need to be determined. Also lens distortions or other image nonlinearities can be accounted for by the higher order terms. For the reconstruction of the images, we implemented the projection equations based on perspective projection as put forth in [15, 203]. Using homogeneous coordinates the perspective projection is expressed by: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ a11 a12 a13 wi x wo X ⎣ wo Y ⎦ = ⎣ a21 a22 a23 ⎦ · ⎣ wi y ⎦ (7.14) a31 a32 a33 wo wi where wo and wi are constants and a33 = 1. When rewritten in standard coordinates the following two nonlinear expressions are obtained: a11 x + a12 y + a13 a31 x + a32 y + 1 a21 x + a22 y + a23 Y = a31 x + a32 y + 1

X=

(7.15) (7.16)

Mapping algorithm

Image area lost in mapping

Fig. 7.3. The back-projection algorithm has to map the recorded image on the left to the reconstructed image on the right.

7.1 Stereo PIV

217

The principal property of the perspective projection is that it maps a rectangle onto a general four-sided polygon. In other words, this mapping preserves only the straightness of lines. By setting a31 and a32 equal to zero the perspective transformation equation (7.14) reduces to the more frequently used aﬃne transformation which can only map a rectangle onto a parallelogram. To account for geometric distortions due to imperfect imaging optics (i.e. pin cushion and barrel distortion) equation (7.15) can be extended to a higher order: a11 x + a12 y + a13 + a14 x2 + a15 y 2 + a16 xy a31 x + a32 y + a33 + a34 x2 + a35 y 2 + a36 xy a21 x + a22 y + a23 + a24 x2 + a25 y 2 + a26 xy Y = a31 x + a32 y + a33 + a34 x2 + a35 y 2 + a36 xy a33 = 1 X=

(7.17) (7.18)

The determination of the unknowns in equation (7.15) and especially in equation (7.17) by means of a linear least squares method is not as simple as for the second order warping approach given in equation (7.12) because the equations no longer constitute linear polynomials but rather are ratios of two polynomials of the same order. Strongly deviating or erroneous point pairs cause a standard least squares method to rapidly diverge from the “true” best match. To ﬁnd the best match to the eight or seventeen unknowns, a nonlinear least squares method such as the Levenberg-Marquart method [23] is required. The Levenberg-Marquart method is implemented by ﬁrst solving for the unknowns in the ﬁrst order projection equations (7.15) and then using these as initial estimates for the solution of the higher order unknowns in equation (7.17). The described projection equations can be used to either map recovered 2-C displacement data or entire images onto an object space that is common to both camera views. Here the back-projection of the images is somewhat empirical because the operator needs to deﬁne a common image magniﬁcation factor for the reconstructed images (see ﬁgure 7.3). Due to the perspective distortion however the original raw image pixels can never be mapped at optimum sampling distances; oversampling, undersampling and aliasing may result. Here we suggest the use of a magniﬁcation factor that avoids loss of signal due to undersampling. The reconstruction of the image is performed by interpolating the image intensity in the raw image using the inverse versions of the mapping functions given before. Here an adequate choice of image interpolator has direct inﬂuence on the quality of the recovered displacement data. The image interpolation methods used for the iterative image deformation PIV algorithms (see section 5.4.4) are equally adequate here. In fact, one may combine image deformation and image back-projection in a single step [219].

218

7 Three-Component PIV Measurements

Camera calibration Up to now the described calibration procedures can provide adequate mappings from image to object space, but provide no attitude information for the cameras (i.e. viewing direction) themselves that may be used to reconstruct the three-component velocity vector. The pragmatic approach is to measure the position of each camera with respect to a known point in the calibration target by means of triangulation assuming that the camera more or less abides to pinhole viewing, that is, all imaging rays pass through a single point. In practice, the location of the pinhole is located along the optical axis within the camera lens and can be easily approximated for larger observation distances. Camera triangulation through distance measurement is not always possible especially in the presence of obstructions such as viewing windows or even air-to-water interfaces. To recover the camera positions with respect to the imaged plane, two primary calibration solutions have established themselves, one entirely empirical motivated [212], the other more or less relying on physical models [194, 214, 221]. A third approach, the so-called camera self-calibration originates from machine vision but is not described in detail here, in part because it attempts to ﬁnd adequate calibration parameters for an entire volume with large depth of ﬁeld – conditions that are not typical for PIV with large lens apertures imaging planar domains. Nonetheless selfcalibration has recently been applied in tomographic PIV [230], stereoscopic PIV [197, 219] and even in μPIV [402]. A common approach to calibrating a stereo PIV imaging setup relies on images of planar calibration targets which are placed coincident with the light sheet plane. These calibration targets typically consist of a precise grid of markers (dots, crosses, line grid, checker board) that are easily detected with simple image processing techniques [195, 220]. A single image of planar calibration marks is then suﬃcient to calculate adequate mappings between image space and object space as described in the previous section, but it generally does not provide information on the camera viewing angles that are essential for reconstruction of the three-component displacement vector. This important parameter can only be calculated from a set of image-to-object correspondence points that are not coplanar. Such a calibration data set can be rather easily generated by recording a set of images with the target slightly displaced at known positions in the direction normal to the light sheet plane. Another method is to use multi-level calibration targets that have reference markers at diﬀerent heights (see ﬁgure 7.5). Given the set of non-coplanar correspondence points, it is now possible to relate two-dimensional displacements on two diﬀerent imaging planes to threedimensional displacements in object space. The empirical approach mentioned earlier uses two-dimensional polynomials of second or third order to connect object volume coordinates with planar image coordinates thus simplifying the vector reconstruction in stereo-PIV [212]. One drawback of this approach is that the volume reconstruction makes use of a large number of polynomial

7.1 Stereo PIV

219

coeﬃcients not all of which are statistically relevant [219]. In fact insuﬃcient calibration data, especially near the edges, can result in undesired oscillations in the mapping functions. The more physically motivated camera calibration method for stereo-PIV originates from the ﬁeld of photogrammetry and image vision, which frequently use so-called camera models to describe the imaging geometry. The simplest camera model reduces the imaging process to a pinhole conﬁguration in which all rays passing from object to sensor must pass through a single point in space (ﬁgure 7.4). This model can be extended by additional parameters to account for radial distortions as suggested in the literature [214, 222]. The calibration procedure involves a nonlinear ﬁt of object-space calibrated feature points to a given functional. The ﬁtted camera parameters can thus be used to retrieve the local viewing angle for each point on the sensor and therefore is well suited for stereo PIV. A quantitative analysis of two these camera models on the background of PIV application is provided in [221]. In essence only a few parameters describing a physical imaging model are necessary to calibrate each viewing direction independently of the other. Typically 11-12 parameters are required to fully describe each view compared to a much larger number of parameters for polynomial-based reconstruction methods as proposed by Soloff et al. [212]. Higher order distortions can be corrected by adding additional distortion

Fig. 7.4. Simple pinhole imaging model used to describe oblique camera viewing. For clarity the projection onto the XZ plane is shown on the right (reprinted from [221]).

220

7 Three-Component PIV Measurements

Fig. 7.5. Precision-machined twin level calibration target with dot pattern for stereo PIV calibration. Levels are separated by 2 mm, dots are equally spaced on a 10 mm grid.

terms to the underlying models. For many practical purposes (standard lenses with narrow ﬁeld of view) the local viewing angle is generally suﬃciently well deﬁned (within 0.1◦ ) using the pinhole location estimate itself. As a consequence the image-reconstruction (back-projection) could use higher order functions to account for higher order distortions while the local viewing angle is suﬃciently well estimated by a simple imaging model. Additional correction terms on the camera model were even observed to lead to erroneous results as the camera model attempted to ﬁt the noise in the correspondence point data. A simple camera model based on an idealized (distortion-free) imaging system, shown in ﬁgure 7.4, is described here. The aim is to ﬁnd a mapping relating image-to-object space using correspondence points from a calibration target such that the position of the point X0 = (X0 , Y0 , Z0 ) can be estimated. These correspondence points consist of image coordinates xj = (xj , yj ) with associated object coordinates Xj = (Xj , Yj , Zj ). A direct linear transform (DLT) between object and image space, as ﬁrst proposed in [192], yields the homogeneous equation: ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ Xj wo xj wi a11 a12 a13 a14 ⎥ ⎢ ⎣ yj wi ⎦ = ⎣ a21 a22 a23 a24 ⎦ · ⎢ Yj wo ⎥ (7.19) ⎣ Zj wo ⎦ wi a31 a32 a33 a34 wo which is a generalized version of the projective mapping given in equation (7.14). The matrix A = a11 , . . . , a34 , also called homography, has 11 degrees of freedom (3 rotations, 3 translations, 5 intrinsic parameters). The coeﬃcients can be solved for by normalizing the weighting factors w = wi /wo and eliminating these resulting in the following projection equations: a11 Xj + a12 Yj + a13 Yj + a14 a31 Xj + a32 Yj + a33 Zj + a34 a21 Xj + a22 Yj + a23 Zj + a24 yj = a31 Xj + a32 Yj + a33 Zj + a34 with a34 = 1 xj =

(7.20) (7.21) (7.22)

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The coeﬃcient a34 behaves as an arbitrary scaling factor for the projection matrix and should be constrained by setting aij = aij /a34 to prevent a trivial solution with aij = 0. A standard nonlinear least-squares solver such as the Levenberg-Marquardt method [23] mentioned earlier can be used to solve for a11 , . . . , a33 . The pinhole position X0 shown in ﬁgure 7.4 provides an estimate for the local viewing angle (e.g. for 3-C vector reconstruction) and is located where the image coordinates x vanish: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ a11 a12 a13 X0 a14 ⎣ a21 a22 a23 ⎦ · ⎣ Y0 ⎦ = ⎣ a24 ⎦ (7.23) a31 a32 a33 Z0 −1 The coeﬃcient matrix A can be further decomposed to extract the camera’s intrinsic parameters I as well as extrinsic parameters E which is detailed in [196]: ⎡ ⎤ ⎡ t ⎤ −fx 0 xc r1 tx A = I · E = ⎣ 0 −fy yc ⎦ · ⎣ rt2 ty ⎦ (7.24) 0 0 1 rt3 tz Here (fx , fy ) is the projected focal length, (xc , yc ) the optical center on the image plane x. Variables rtk and tk relate to rotation and translation respectively. While the DLT assumes an idealized imaging system, real-world eﬀects such as radial distortions are included in more advanced camera models, such as those developed by Tsai [214] and Zhang [222]. A noteworthy aspect of model-based camera calibration is that oblique camera views may even be calibrated using a single set of coplanar calibration points (coplanar calibration, [214]). This is especially attractive in environments where the translation of a target is unfeasible due to diﬃcult access or non-acceptable eﬀort. In most cases the estimated camera positions are more reliable than a triangulation by hand, but requires viewing angles greater than 10◦ from normal. For near-normal viewing conditions the lack of perspective prevents a reliable numerical convergence of the camera models [221]. Disparity correction The previously described camera calibration provides the essential information needed for reconstruction of the 3-C vector from two separate 2-C PIV recordings obtained at the same instant but from diﬀerent directions. However this reconstruction approach assumes that the calibration target is perfectly aligned with the center of the light sheet plane. In practice this alignment is diﬃcult to achieve. Therefore a slight out-of-plane position and/or minor rotation of the target can introduce a signiﬁcant misalignment of the imaged light sheet volumes with respect to each other (see ﬁgure 7.6, [194, 200, 201, 219, 220]).

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Fig. 7.8. Tip vortex ﬂow after disparity correction (≈ 3.5 mm horizontal shift); top row: left camera view, right camera view, reconstructed ﬂow ﬁeld; bottom row: corresponding maps of out-of-plane vorticity.

Figure 7.8 shows the same set of data after application of a disparity correction scheme (explained later). Thereafter the vortex centers coincide and the reconstructed data shows a uniform vortex with a single round core that matches those of the individual camera views. Aside from the obvious mismatch of vortex centers, a further indicator of viewing disparity are the residuals of the vector reconstruction as described on page 215. The misalignment results in an increase of these residuals due to the mismatch of recovered vectors between the views. In the present case a correct alignment of the views reduces the residuals to levels below 0.2 pixel (ﬁgure 7.9). So in theory a minimization of the residuals could be used to align the camera views with each other (provided the ﬂow has notable variations the ﬁeld of view). In the previous example the camera view disparity was roughly 3.5 mm along the horizontal, which corresponds to an out-of-plane target placement of about 1.3 mm for a viewing geometry of ±55◦. Given a light sheet thickness of ≈ 2 mm the target is only slightly oﬀset (observation distance is about 1500 mm). While manual alignment of ﬂow structures or a minimization of the reconstruction residuals could be used to align the views, a much more eﬃcient and reliable method for disparity correction procedure is described in the following. The approach relies on the actual PIV recordings from both views. These images are then dewarped according to their projection coeﬃcients and a cross-correlation is performed between the views, that is, the ﬁrst image of

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view A is correlated with the ﬁrst image of view B. The thus recovered displacement ﬁeld in fact represents the disparity of the views with respect to each other. The displacement data can now be used to correct the disparity by modifying the mapping coeﬃcients accordingly. In practice the quality of the disparity map is improved by including the second image, or even an entire sequence of images if the cameras are stationary. In this case the average or ensemble correlation described in section 5.4.2 should be used. Closer inspection of disparity maps such as ﬁgure 7.11 indicates that a low order two-dimensional polynomial, obtained through a least-squares ﬁt to the data, is suﬃcient to describe it. Following ﬁgure 7.6, one displacement and two angles position the light sheet with respect to the calibration target. For optimum performance the positions of the cameras should be shifted and rotated using the recovered angles to match the coordinate system of the light sheet. However this is rarely done. Nonetheless the camera calibration described earlier can be included to iteratively recover the position of the light sheet [219].

Fig. 7.10. Image of 16 crosscorrelation planes sized 64 × 64 pixel obtained during disparity correction of image data corresponding to ﬁgure 7.7. A mean oﬀset of 30 pixel (≈ 3.5 mm) approximately centers the correlation peaks which measure about 20 × 4 pixel.

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Fig. 7.11. Misalignment between the two viewing directions due to a slight oﬀset and rotation of the calibration grid within the light sheet. A constant 6 pixel horizontal shift has already been removed.

Due to the ﬁnite thickness of the light sheet the images of the particles will never coincide between the views, even for perfect alignment. Hence, the crosscorrelation peak in the disparity correction procedure will be smeared into an oblong shape (see ﬁgure 7.10). The major axis of this widened correlation peak coincides with the plane spanned by the two camera viewing axes. Its width is directly related to a projection of the light sheet thickness [219]. The ﬁnite thickness of the light sheet also places restrictions on the obtainable spatial resolution: the oblique viewing arrangement reduces the true probe volume to the intersection of two larger sampling rhomboids (ﬁgure 7.12). As the sample size decreases, the eﬀective probe volume can decrease to questionable sizes. If a high spatial resolution is desired then either the light sheet thickness ΔZ0 should be reduced appropriately or a 3-D particle tracking scheme should be applied. As a rule of thumb the minimum PIV sample size should at least be the same size as the light sheet thickness: ΔX0 ≥ ΔZ0 . General recommendations for stereo PIV • Multi-level or translated targets provide suﬃciently good calibration results. Volumetric camera calibration commonly found in robot vision is not necessary and generally not possible due to the limited depth of ﬁeld for large aperture lenses used for PIV.

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Fig. 7.12. Eﬀective probe volume (hashed) for stereo PIV is smaller than probe volume for each individual view. Right: 3-C displacement data for ΔX0 < ΔZ0 are questionable.

• Calibration from a single image of a planar target is possible but should only be used if translation is not possible (i.e. due conﬁned space). • For optimum measurement accuracy the enclosed angle between the camera viewing axes should be close to 90◦ . • The unavoidable misalignment between calibration target and light sheet plane requires disparity correction or self-calibration schemes using the actual particle images. • Disparity correction can also be used to correct for movement of the camera(s) during the measurements (i.e. vibration). • The minimum PIV sample size should be at least the same size as the light sheet thickness: ΔX0 ≥ ΔZ0 . • The residuals resid of the 3-C vector reconstruction serve as a quality check for the SPIV measurement and should have values in the range 0.1 − 0.5 pixel. • For SPIV experiments in water the air-glass-water interface should be normal to the camera viewing axis to keep the astigmatism minimal. Oblique views are possible by attaching suitable water prisms to the test section [205, 216]. • Image reconstruction should make use of appropriate image interpolation schemes such as those reported in section 5.4.4. For best performance iterative image deformation and image back-projection can be combined in a single step [219].

7.2 Dual-Plane PIV In this section we will describe an approach to obtain information about the out-of-plane velocity components, which is based on the analysis of the height of the peak in the correlation plane. This value depends on the portion of paired particle images, which itself depends on the out-of-plane velocity component and on other parameters. To circumvent problems with other inﬂuences (e.g. background light, amount and size of images), images from an

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additional light sheet plane parallel to the ﬁrst one were also captured for peak height normalization. This concept is referred to as “dual-plane” PIV [208] or as the “spatial-correlation” technique [225]. Experimental results of this technique are shown in section 9.5. Two problems arise when measuring highly three-dimensional ﬂows with PIV. First, it is obvious from the description of the recording process that an appropriate pulse separation Δt can only be selected for ﬂows with a limited extent of the out-of-plane velocity component. This restriction is imposed because particles moving perpendicular to the light sheet will leave and enter the light sheet in between the two illumination pulses and thus will not correlate when evaluating the PIV recording. This leads to a decreased probability in detecting the particle image displacement, as has been shown by numerical simulation (see section 5.4). If the detection probability is increased by a strongly reduced Δt, an ampliﬁcation of the measurement noise will be obtained as described in section 3. Second, allowing a signiﬁcant velocity component perpendicular to the light sheet plane leads to an additional error because the camera lens reproduces the tracer particles by perspective projection and not by parallel projection. The only way to avoid this error when observing highly threedimensional ﬂows is to measure all three components of the velocity vectors (see section 2.6.3). The most commonly used technique to measure instantaneous three-component velocity ﬁelds in a plane is stereoscopic PIV, which has been described in the previous section. For this technique the achievable accuracy of the measurement of the out-of-plane velocity component is less than that of the in-plane component, but the out-of-plane measurement error can be reduced below 1% of full scale in best cases (see section 7.1). As already mentioned, specially developed calibration techniques and two recording cameras are necessary for the stereoscopic approach, which both need optical access to the test section. This makes the operation of stereoscopic PIV more diﬃcult than that of dual-plane PIV, when dealing with low-speed liquid ﬂows. 7.2.1 Mode of Operation A three-dimensional representation of the measurement volume is shown in ﬁgure 7.13. The ﬂuid element (left hand side) contains the particles, which were in the interrogation volume during the ﬁrst exposure at t = t0 . Due to the out-of-plane motion of this ﬂuid element it will partly leave the light sheet between t = t0 and t = t0 + Δt. Therefore, particles which have left the light sheet will not be imaged at t = t0 + Δt. In dual-plane PIV a slightly displaced light sheet has to be generated additionally. If the displacement of this light sheet corresponds to the displacement of the ﬂuid element the particles will be illuminated and imaged on a third frame at t = t0 + 2Δt. When dealing with a suﬃcient number of particles in the measurement volume, the number of particle image pairs inside the interrogation windows of two frames can be used to estimate the out-of-plane ﬂow component. This

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Fig. 7.13. Size and location of the interrogation volume and position of the particles, which were illuminated by the ﬁrst light pulse, represented at the time of the ﬁrst, second and third exposure at light sheet position (A), (A), and (B) respectively.

number is proportional to the number of particles within the interrogation volume at t, decreased by the number of second images lost due to out-ofplane motion and by the number lost by in-plane motion. Using evaluation methods with a constant size and ﬁxed location of the interrogation window, and assuming a constant particle density, the number of lost particle image pairs is proportional to the dark area shown in ﬁgure 7.13. Now, following the analysis for single-plane PIV (see section 3), we obtain the following results for the expectation of the image cross-correlations E{RII (s = d)} and E{RI I (s = d)} between the pairs of interrogation windows I(x, y),I (x, y) and I (x, y), I (x, y) respectively: E{RII (s = d)} = CR Rτ (0) F0 (DZ − Z + Z) Fi (DX , DY )

E{RI I (s = d)} = CR Rτ (0) F0 (DZ − Z + Z ) Fi (DX , DY )

(7.25) (7.26)

where CR is a constant factor (deﬁned in section 3), Rτ is the image selfcorrelation and Fi is interpreted as the in-plane loss of correlation. Z, Z and Z represent the position of the light sheet in the Z direction during the ﬁrst, second and third exposure respectively. F0 , the loss-of-correlation due to out-of-plane motion, is the relevant term for the description of “dual-plane” evaluation. It is given by:

7.2 Dual-Plane PIV

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In a practical situation, we have to estimate the expectation of the correlation by the ﬁnite spatial average. Therefore, we will replace the (mathematically correct) expectation by its estimator: the computed correlation RII and RI I respectively (for brevity of notation we omit s). Provided that Rτ and the light sheet intensity proﬁle IZ (Z) are known, it is in principle possible to determine |DZ − Z − Z| from RII only. However, in a practical situation the evaluation of |DZ − Z − Z| leaves the sign of the out-of-plane motion undetermined. Dual-plane PIV avoids these complications by taking the ratio of RII and RI I , which yields a general expression that relates DZ for an arbitrary light-sheet intensity proﬁle IZ (Z) and given shifts Z − Z and Z − Z: RII F0 (DZ − Z + Z) = RI I F0 (DZ − Z + Z )

(7.28)

It can be seen from the above equation, that taking the ratio of the correlation estimators also compensates for the loss-of-correlation due to in-planemotion. Obviously, the ability to solve DZ in equation (7.28) depends on the accurate knowledge of the light-sheet proﬁle IZ (Z). Most CW-lasers offer a suﬃcient beam-pointing stability, and their intensity proﬁles can easily be determined to ﬁnd the exact relation between the out-of-plane velocity and the cross-correlation values. When using solid-state pulsed laser systems – for example a Nd:YAG laser – a variation of the pulse-to-pulse beam-pointing of up to 20% of the beam diameter has to be taken into account. If the spatial beam proﬁle varies in time, the intensity proﬁle should be determined simultaneously, e.g. by using a beam splitter and a CCD sensor. In the remainder of this section we will deal with two particular solutions of equation (7.28), namely the solutions for (1) Z = Z < Z (which is the situation for the investigation given in section 9.5), and (2) for Z = Z < Z (which is the situation if ﬂows with bidirectional out-of-plane components have to be observed). To keep the subsequent analysis simple we consider a light sheet with a top-hat shaped intensity proﬁle and a width ΔZ0 that is IZ if |Z| ≤ ΔZ0 /2 I0 (Z) = 0 elsewhere and F0 (Z) =

1 − |Z| /ΔZ0 if |Z| ≤ ΔZ0 0 elsewhere .

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For Z = Z < Z , the solution of equation (7.28) for the out-of-plane velocity W = DZ /Δt, with F0 (Z) given by equation (7.29), yields:

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⎧ RI I − OZ RII ⎪ ⎪ ⎪ ⎪ RII − RI I ⎪ ⎪ RI I − OZ RII ΔZ ⎨ · W = RII + RI I Δt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ RI I + (2 − OZ )RII RII − RI I

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with OZ = 1 − (Z − Z )/ΔZ0 . For the experiments described in section 9.5 we only considered the solution for 0 < DZ < Z − Z . For a situation where the out-of-plane ﬂuid velocity can be both positive or negative, a reverse displacement of the light sheet positions between ﬁrst and second, and second and third exposure has to be taken into account. If the absolute value of displacement is the same for both directions equation (7.28) has to be solved for Z = Z < Z . The solution, with FO (Z) again given by equation (7.29), diﬀers slightly with respect to the solution given by equation (7.30), that is W =

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for −(Z − Z ) < DZ < Z − Z . Note that the range of DZ for this solution is symmetric with respect to DZ = 0. The following simpliﬁcations are implied in the above formulas: First, a top-hat intensity proﬁle of the light sheets in the Z direction has been assumed instead of a Gaussian distribution, which would be more adequate for CW lasers. This leads to the fact that F0 (Z), which is the normalized correlation of the intensity distributions in the Z direction of two successive pulsed light sheets and is therefore also a Gaussian function, is approximated by a triangle function. Second, the eﬀect of the variation of the displacement within the interrogated cell, and the fraction of second images lost by in-plane motion is assumed to be identical for both correlations. This is only a rough approximation as long as the second and third frames are not captured at the same time, which would require a more sophisticated image separation technique. Third, the ﬂuctuating noise component is neglected. Its eﬀect on the measurement accuracy can be reduced by averaging results over neighboring interrogation cells. However, this has to be balanced against a decrease in spatial resolution. 7.2.2 Conclusions In this section, we described the approach of using information rendered by the correlation function to estimate the out-of-plane components of velocity ﬁelds. Among other limitations, a larger number of particle images per interrogation window is required for this technique than for conventional PIV. The

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existence of a practically achievable maximum of the image density therefore results in a lower spatial resolution and/or accuracy compared to the in-plane measurement. However, we found the results of this approach and the ease of operation of the described technique to be encouraging. For investigations of low-speed liquid ﬂows, the operation of dual-plane PIV is easier than that of stereoscopic PIV, since the only calibration necessary is the determination of thickness and overlap of the light sheets. Furthermore, only one camera is needed. By changing the light sheet position after each exposure this technique can also be applied to ﬂows with out-of-plane components in the positive and negative Z direction. Further details on this technique may be found in [209].

7.3 Three Component PIV Measurements in a Volume The PIV technique in stereo arrangement as described in section 7.1 has become a standard for the measurement of all three velocity components (3C) in a plane of the ﬂow (2D). This is even true for complex applications in industrial test facilities (see sections 9.11, 9.12). In most applications the factor limiting the use of stereo PIV is not associated with PIV equipment or evaluation algorithms but with the limited optical access in such test facilities. This means that the temporal and spatial resolution achievable for stereo (3C) PIV is more or less the same as for ‘classical’ 2C PIV. However, there is still no universal method available to capture instantaneous velocity ﬁelds in a volume of the ﬂow (3D), in order to be able to investigate complex unsteady 3-dimensional ﬂow phenomena (vortices, boundary layers, separated ﬂows, etc.), which are of increasing importance in both fundamental as well as industrial research. The main reason is that the required laser light sources, cameras, evaluation algorithms, computing capacity, etc. are not yet available today due to lack of technical progress. Thus, diﬀerent PIV methods need to be applied today depending on the respective application. Holographic PIV methods (HPIV) are in principle suited to capture ﬂow velocity ﬁelds completely three-dimensional. The registration of the particle images on high resolution photographic material (photochemical holographic PIV (P-HPIV) allows a high spatial resolution in a medium sized volume without temporal resolution. Registration via CCD sensors (digital holographic PIV (D-HPIV) allows time-resolved measurements in a small volume. A new technique (volume PIV or tomographic PIV) achieves the information in depth of the volume via simultaneous observation with several cameras with diﬀerent viewing angles. As all these techniques are undergoing rapid development at present and their potential strongly depends on future technical developments, their characterization is beyond the scope of this book. Instead, a short overview of the present state-of-the art and possible applications of the diﬀerent methods shall be given.

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Multi-plane stereo PIV (details given in section 9.18) requires several planes of the ﬂow ﬁeld to be illuminated simultaneously. The particle images within each of such light sheets will be recorded with a separate stereo PIV system. The optical separation of these light sheets will be performed by polarization of diﬀerent wavelength of the light used for illumination. This method exhibits the same properties as stereo PIV and does not allow to capture a complete volume (see ﬁgure 7.14c). Information about the instantaneous velocity ﬁeld being available in diﬀerent planes, the determination of local velocity gradients and space-time correlation becomes possible. Drawback of this method is the high experimental eﬀort. For each plane a complete stereo PIV system has to be set up together with elements for the required optical separation (polarization, color ﬁlters). A further method consists in fast step-wise scanning of the volume by means of a conventional light sheet arrangement [87, 226]. However this method is only applicable for either time averaged measurements or for very low ﬂow velocities (typ. < 1 m/s, e.g. ﬂows in water), as the diﬀerent planes within the volume have to be recorded sequentially at diﬀerent instants in time. Furthermore, the measurement volume depth is limited by the depth of focus of the cameras. Photogrammetric methods [180, 227, 239, 246, 247] allow to capture a volume of the ﬂow instantaneously. The complete volume will be illuminated by one light pulse. The images of the tracer particles will be recorded from diﬀerent camera positions. The position of the individual particles will be identiﬁed

Fig. 7.14. Schematic overview over present variants of the PIV method with their typical spatial and temporal resolution. The symbols present the measurable three-component velocity vectors for each method in their spatial and temporal arrangement.

7.3 Three Component PIV Measurements in a Volume

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on the diﬀerent recordings and their position in space will be calculated by photogrammetric methods utilizing the diﬀerent views. Local velocities will be determined with particle tracking methods. The seeding concentration in the ﬂow needs to be limited in order to be able to identify individual particles uniquely. This in principle results in a low spatial resolution of this method. A further problem arises from the limited depth of focus, which leads to larger unfocused particle images, which may overlap. At present only holographic methods allow volumetric resolution without detection of particles [39], making use of the coherence of the illumination laser. The light scattered by the particles is superimposed with light of a reference wave on the photographic recording medium. The resulting interference pattern (hologram) stores intensity and phase of the light coming from the object. The photographic medium must possess high spatial resolution in order to correctly store the interference pattern. Photographic holographic plates typically allow resolving 3000 lines / mm. Volumes of 50 × 50 × 50 mm3 with a resolution of 1 mm3 , corresponding to 106 vectors, could be successfully captured with photochemical holographic PIV (P-HPIV) [223, 224, 240]. This method, involving wet chemical processing, has the disadvantage that only single recordings can be taken. With very high experimental eﬀort, a few recordings can be stored on the photographic plate by spatial multiplexing. After reconstruction of the hologram, the particle images are usually scanned by a CCD sensor for subsequent PIV evaluation, performed digitally by a computer. A further approach is to use electronic sensors to record the hologram. The spatial pixel size of such sensors (typ. 5 μm) is by far larger than the wavelength of light. This can be tolerated as far as certain restrictions are accepted. Only incident light within a certain restricted angle contributes to the information. Outside this angle the incident light only increases the noise level. For practical applications this means that the reference wave must be approximately normal to the sensor (“in-line”) and that the objects (particles) must be suﬃciently distant from it. In addition the reduced eﬀective aperture reduces the depth resolution. Digital holographic PIV (D-HPIV) typically captures a volume of 1 cm3 and allows resolving a few thousand vectors [236, 237]. In holography maximum spatial resolution and observation volume strongly vary depending on the properties of the image sensors and the photographic plates, respectively (ﬁgure 7.14d & e). In this context digital holography becomes more and more attractive due to the fast progress made at image sensors (CCD, CMOS) at present. There are further attempts to increase the capability of holographic systems, e.g. by improving the signal-to-noise ratio by holography utilizing short coherence length (“light in ﬂight”) [233] or improvement of depth resolution by observation from different directions [235, 244, 248]. Whereas 3C-2D stereo PIV having has become a standard method, no universal method (3C-3D) for capturing the instantaneous ﬂow ﬁeld with 3-D unsteady ﬂow phenomena is available at present. As many complex ﬂows still

234

7 Three-Component PIV Measurements

require experimental investigations due to insuﬃciency of present numerical methods, there is a strong need to further develop the PIV technique towards full 3C-3D capability, which allows better description and understanding of such ﬂows. Further, 3C-3D methods allow calculation of space-time correlations with high spatial and temporal resolution. However, further progress of the PIV technique in direction of 3C-3D capability will strongly depend on technological developments carried out in other areas for other customers, such as the consumer market. The following outlines a recently introduced 3D-3C method that relies on tomographic approaches. 7.3.1 Principles of Tomographic-PIV Introduction Tomographic Particle Image Velocimetry Tomographic-PIV1 is a measurement technique that allows resolving the particle motion within a three-dimensional measurement volume without the need to detect individual particles. This recently developed method (Elsinga et al. [230]) is based on the simultaneous view of the illuminated particles by digital cameras placed along several observation directions similarly to the stereoscopic PIV conﬁguration. The innovative element of the technique is the tomographic algorithm used to reconstruct the 3D particle ﬁeld from the individual images. The 3D light intensity distribution is discretized over an array of voxels and then analyzed by means of 3D cross-correlation interrogation returning the instantaneous three-component velocity vector ﬁeld over the measurement volume. Tomographic-PIV has been developed for research laboratory applications and its extension to industrial wind tunnel environments so far has not been performed. Despite its recent introduction, by early 2007 the technique has already been successfully applied in both air and water ﬂow measurements within about 10 experiments conducted over several European laboratories (TU Delft, LaVision, DLR G¨ ottingen, TU Braunschweig, Poitiers University). The requirement to implement time-resolved Tomographic-PIV is the same as for planar PIV, which has been already veriﬁed with both low repetition-rate Tomographic-PIV measurements in water ﬂows and a high-repetition rate boundary layer experiment in air ﬂow. The most important limiting factors are the extent of the measurement volume depth (typical depth-to-width ratio 0.25), the power required for volume illumination (typically 5 times higher than for a stereo-PIV experiment), the time required for digital evaluation of recordings (nowadays a few hours per snapshot) and the associated data storage (typically 10 times larger than for planar PIV). Last but not least a slightly extended optical access is required with respect to stereo-PIV because the viewing directions typically 1

The text on tomographic-PIV has been contributed by Fulvio Scarano, Gerrit E. Elsinga and Bernhard Wieneke

7.3 Three Component PIV Measurements in a Volume

235

cover a solid angle, although the linear conﬁguration is also a possible option if required. Working principle Tracer particles immersed in the ﬂow are illuminated by a pulsed light source within a three-dimensional region of space (ﬁgure 7.15). Particle images are recorded in focus from several viewing directions using CCD cameras. The Scheimpﬂug condition between the image plane, lens plane and the medianobject-plane has to be fulﬁlled, which is practically achieved by means of camera-lens tilt mechanisms with the rotation axis freely adjustable. The latter condition is not suﬃcient to ensure that particle images are in focus over the entire depth of the measurement volume. This requirement is satisﬁed with an appropriate depth of focus by selecting an appropriate lens aperture. The reconstruction of the 3D object from the digital images requires prior knowledge of the mapping function between the image planes and the physical space. This is achieved by means of a calibration procedure similar to that of stereoscopic PIV. However the procedure requires such mapping function to be deﬁned onto a volumetric domain as opposed to that used for planar

Fig. 7.15. Principle of Tomographic-PIV.

236

7 Three-Component PIV Measurements

stereo-PIV. The requirement for a precise alignment between calibration and measurement plane is not a restriction in this case since no such plane exists for the tomographic technique, which makes it easier to implement the experiment with respect to laser light positioning. However as shown in the remainder, the requirement for a precise relative position between cameras for the tomographic technique (one-ﬁfth of the particle image diameter) is much more stringent than that for planar stereo PIV. This is practically achieved by a-posteriori correction of the calibration mapping function (self-calibration technique, WIENEKE [219]) similar to the light sheet disparity correction technique used for stereo PIV (section 7.1.2). The 3D light scattering ﬁeld (the object) is reconstructed as a discretized 3D array of light intensity distribution in physical space E(X, Y, Z) from its projections on the CCD arrays. This reconstruction problem is of inverse nature and is in general underdetermined, meaning that a single set of projections (viz. images) can result from many diﬀerent 3D objects. Determining the most likely 3D distribution is the topic of tomography [232]. However for a sparse ﬁeld of emitters (or scatterers) the problem can be solved accurately provided that the information density is not too high. The relation between the intensity of the ith pixel with image coordinates (xi , yi ) and the light intensity distribution of the j th voxel in the 3D physical space is given by a linear equation describing the projection of the object ﬁeld onto the image: wi,j E(Xi , Yj , Zj ) = I(xi , yi ) (7.32) j∈Ni

Ni is the voxel’s neighborhood (typically a cylinder of 3 × 3 voxels crosssection) around the line of sight of the ith pixel (xi , yi ) through the volume. The weighting coeﬃcient wi,j describes the contribution of the j th voxel with intensity E(Xj , Yj , Zj ) to the pixel intensity I(xi , yi ) and is calculated as the volume fraction intersected by the line of sight with the considered voxel. A schematic of the projection conﬁguration is shown in ﬁgure 7.16. The described process requires an iterative implementation with a predictor-corrector technique since the left-hand side term of equation (7.32) (i.e. the integrand) is the unknown. The process requires a non-zero intensity E 0 (X, Y, Z) as initial condition (e.g. uniform) and the result is updated through the multiplicative correction equation: k+1

E(Xi , Yj , Zj )

k

= E(Xi , Yj , Zj )

I(xi , yi ) + k j∈Ni wi,j E(Xi , Yj , Zj )

μwi,j (7.33)

where μ ≤ 1 is a scalar relaxation parameter. When μ = 1 the fastest convergence rate is achieved. The magnitude of the update is determined by the ratio of the measured pixel intensity I with the projection of the current object E(Xi , Yj , Zj )k . The weighting coeﬃcient at the exponent ensures that

7.3 Three Component PIV Measurements in a Volume

237

Fig. 7.16. 3D space to image projection model.

only the elements in E(X, Y, Z) with non-zero projection onto the ith pixel are updated. Furthermore the multiplicative MART scheme requires that E and I are positive deﬁnite. The iterative process can be considered converged after 4 or 5 iterations, since further iterations do not yield appreciable diﬀerence in the measurements. The accuracy of the reconstruction process strongly depends upon several factors. The most important ones are the number of viewing cameras and number of particle images present in the images. A numerical evaluation of the MART algorithm shows that systems with two cameras are only adequate for very low seeding density (below 0.01 particles per pixel, ppp). Conversely the most adopted conﬁguration with four cameras is able to reconstruct a ﬁeld with particle image density of 0.05 ppp (i.e. 50,000 particles/Mpixel) with a reconstruction quality normalized coeﬃcient Q = 0.75 (ﬁgure 7.17). The accuracy of reconstruction is mildly dependent upon the angle between the viewing directions when this angle is in the range between 15 and 40 degrees. The evaluation of the reconstructed particle pattern displacement is performed by means of 3D spatial cross correlation analysis. The normalized cross-correlation function R(ΔX, ΔY, ΔZ) reads in this case as: +I,J,K i,j,k=1 E(i, j, k, t) · E(i − l, j − m, k − n, t + Δt)

(7.34) R(l, m, n) = cov(E(t)) · cov(E(t + Δt)) The analysis can be performed with an iterative technique based on the window deformation technique [158] extended to three dimensional intensity ﬁelds, where the interrogation boxes are displaced/deformed on the basis of the result from the previous interrogation. The intensity ﬁeld of the deformed volume at the k + 1th iteration is obtained from the original intensity and the predictor velocity ﬁeld according to the expression: 1 1 E k+1 (X, Y, Z, t) = E(X − ukd , Y − vdk , Z − 2 2 1 k 1 k+1 E (X, Y, Z, t + Δt) = E(X + ud , Y + vdk , Z + 2 2

1 k w , t) (7.35) 2 d 1 k w , t + Δt) 2 d

238

7 Three-Component PIV Measurements

Fig. 7.17. Cylinder wake at Re = 5500. Instantaneous vorticity iso-surfaces.

Where Vkd = (ukd , vdk , wdk ) represents the particle pattern deformation ﬁeld obtained at the k th interrogation. Summary of applications Although recently introduced the tomographic PIV technique has been applied to several ﬂow conditions from more academic conﬁgurations such as circular cylinder wakes to increasingly challenging topics such as low speed turbulent boundary layers and shock wave boundary layer interaction at Mach 2. The technique as also been implemented in conjunction with high repetition rate hardware for the time resolved measurement of boundary layer transition to turbulence described in details in the section of PIV applications. The main experimental parameters characterizing the mentioned experiments are given in table 7.1. For 3D measurements an estimator of the measurement precision can be based on mass conservation law. For the incompressible ﬂows regime the spatial distribution of velocity divergence ∇V = ∂u/∂x + ∂v/∂y + ∂w/∂z should virtually be null everywhere. The RMS ﬂuctuations of the divergence < V > constitutes a suitable norm of the error aﬀecting the diﬀerential quantities such as vorticity and rate of strain. The typical measurement returned by

7.3 Three Component PIV Measurements in a Volume

239

Table 7.1. Applications overview with some experimental parameters Cylinder wake Time-resolved Shock wave- boundary Time-resolved (air ﬂow) cylinder wake layer interaction boundary layer (air ﬂow) (air ﬂow) (air ﬂow) [241] [238] [234] [243] 5 0.02 ÷ 0.10 500 7 1376 × 1040 2048 × 1440 2048 × 1100 800 × 768 1 10 2 5000 37 × 36 × 8 88 × 59 × 16 70 × 38 × 8 34 × 30 × 19

Reference Flow velocity [m/s] Image size [pixels] Recording rate [Hz] Measurement volume L × H × D [ mm3 ] Spatial resolution 18.2 23.6 [voxels/mm] Particle concentration 2.1 1.2 [particles/ mm3 ] (total) (23,000) (98,000) Number of vectorsa 77 × 79 × 15 174 × 117 × 32 (total) (91,000) (651,000) a 75% overlap factor between interrogation volumes

30

24

3

0.94

(65,000) 140 × 76 × 18 (191,000)

(18,000) 46 × 41 × 24 (45,000)

a tomographic PIV experiment is a velocity vector ﬁeld distribution over a volume. A clear representation of the velocity or vorticity ﬁeld is not straightforward and requires the use of 3D computer graphics as shown in ﬁgure 7.17 with the iso-surfaces of vorticity magnitude describing the vortex organization in the wake.

8 Micro-PIV

8.1 Introduction There are many areas in science and engineering where it is important to determine the ﬂow ﬁeld at the micron scale. Industrial applications of microfabricated ﬂuidic devices are present in the aerospace, computer, automotive, and biomedical industries. In the aerospace industry, for instance, micron-scale supersonic nozzles measuring approximately 35 μm are being designed for JPL/NASA to be used as microthrusters on micro-satellites and for AFOSR/DARPA as ﬂow control devices for palm-size micro-aircraft [277]. In the computer industry, inkjet printers, which consist of an array of nozzles with exit oriﬁces on the order of tens of microns in diameter, account for 65% of the computer printer market [277]. The biomedical industry is currently developing and using microfabricated ﬂuidic devices for patient diagnosis, patient monitoring, and drug delivery. The i-STAT device (i-STAT, Inc.) is the ﬁrst microfabricated ﬂuidic device that has seen routine use in the medical community for blood analysis. Other examples of microﬂuidic devices for biomedical research include microscale ﬂow cytometers for cancer cell detection, micromachined electrophoretic channels for DNA fractionation, and polymerase chain reaction (PCR) chambers for DNA ampliﬁcation [293]. The details of the ﬂuid motion through these small channels, coupled with nonlinear interactions between macromolecules, cells, and the surface-dominated physics of the channels create very complicated phenomena, which can be diﬃcult to simulate numerically. A wide range of diagnostic techniques have been developed for experimental microﬂuidic research. Some of these techniques have been designed to obtain the highest spatial resolution and velocity resolution possible, while other techniques have been designed for application in nonideal situations where optical access is limited [287], or in the presence of highly scattering media [280]. Several of the common macroscopic full-ﬁeld measurement techniques have been extended to microscopic length scales. These are Scalar

242

8 Micro-PIV

Image Velocimetry [467], Molecular Tagging Velocimetry [469], as well as PIV, which at small length scales has become known as micro-PIV or μPIV. In 1998 Santiago et al. [297] demonstrated the ﬁrst μPIV system – a PIV system with a spatial resolution suﬃciently small enough to be able to make measurements in microscopic systems. Since then the technique has grown in importance at a tremendous rate. As of 2007 there are well over 350 μPIV journal articles. Because of this large amount of activity, well over an order of magnitude more than the previously described techniques combined, the remainder of this chapter will concentrate on μPIV, its applications and extensions. The ﬁrst μPIV system was demonstrated measuring slow ﬂows – velocities on the order of hundreds of microns per second – with a spatial resolution of 6.9×6.9×1.5 μm3 [297]. The system used an epi-ﬂuorescent microscope and an intensiﬁed CCD camera to record 300 nm-diameter polystyrene ﬂow-tracing particles. The particles are illuminated using a continuous Hg-arc lamp. The continuous Hg-arc lamp is chosen for situations that require low levels of illumination light (e.g., ﬂows containing living biological specimens) and where the velocity is suﬃciently small enough so that the particle motion can be frozen by the CCD camera’s electronic shutter (ﬁgure 8.1). Koutsiaris et al. [286] demonstrated a system suitable for slow ﬂows that used 10 μm glass spheres for tracer particles and a low spatial resolution, high-speed video system to record the particle images yielding a spatial resolution of 26.2 μm. They measured the ﬂow of water inside 236 μm round glass capillaries and found agreement between the measurements and the analytical solution within the measurement uncertainty.

Microfluidic device

Immersion fluid Microscope lens

λ = 532 nm Epi - fluorescent Nd:YAG Laser

Prism/Filter Cube Beam E x pa n d e r

λ =560nm Lens (Intensified) CCD Camera

Fig. 8.1. Schematic of a μPIV system. A pulsed Nd:YAG laser is used to illuminate ﬂuorescent 200 nm ﬂow-tracing particles, and a cooled CCD camera is used to record the particle images.

8.1 Introduction

243

Later applications of the μPIV technique moved steadily toward faster ﬂows more typical of aerospace applications. The Hg-arc lamp was replaced with a NewWave two-headed Nd:YAG laser that allowed cross-correlation analysis of singly exposed image pairs acquired with sub-microsecond time steps between images. At macroscopic length scales this short time step would allow analysis of supersonic ﬂows. However, because of the high magniﬁcation, the maximum velocity measurable with this time step is on the order of meters per second. Meinhart et al. [406] applied μPIV to measure the ﬂow ﬁeld Table 8.1. Comparison of High-Resolution Velocimetry Techniques [288]. Technique

Author

Flow Tracer Spatial Resolution Observation ( μm)

LDA

Tieu et al. (1995) Chen et al. (1997)

—

Optical Doppler tomography (ODT) Optical ﬂow using video microscopy Optical ﬂow using X-ray imaging Uncaged ﬂuorescent dyes Particle streak velocimetry PIV Superresolution PIV μPIV

5 × 5 × 10

1.7 μm 5 × 15 polystyrene beads

4–8 fringes limits velocity resolution Can image through highly scattering media

Hitt et al. (1996)

5 μm blood 20 × 20 × 20 cells

In vivo study of blood ﬂow

Lanzillotto et al. (1996)

1 – 20 μm emulsion droplets Molecular Dye

Paul et al. (1997)

∼ 20 – 40

Can image without optical access

100 × 20 × 20

Resolution limited by molecular diﬀusion

0.9 μm ∼ 10 polystyrene beads Urushihara et al. 1 μm oil 280 × 280 × 200 (1993) droplets Keane et al. 1 μm oil 50 × 50 × 200 (1995) droplets Brody et al. (1996)

Santiago et al. (1998)

μPIV

Meinhart et al. (1999)

μPIV

Westerweel et al. (2004)

300 nm 6.9 × 6.9 × 1.5 polystyrene particles 200 nm 5.0 × 1.3 × 2.8 polystyrene particles 500 nm 0.5 × 0.5 × 2.0 polystyrene particles

Particle streak velocimetry Turbulent ﬂows Particle tracking velocimetry Hele-Shaw Flow

Microchannel ﬂow

Silicon microchannel ﬂow

244

8 Micro-PIV

in a 30 μm × 300 μm (hight×width) rectangular channel, with a ﬂow rate of 50μl/h, equivalent to a centerline velocity of 10 mm/s. The experimental apparatus, shown in ﬁgure 8.1, images the ﬂow with a 60×, NA = 1.4, oilimmersion lens. The 200 nm-diameter polystyrene ﬂow-tracing particles were chosen small enough so that they faithfully followed the ﬂow and were 150 times smaller than the smallest channel dimension. A subsequent investigation by Meinhart & Zhang [291] of the ﬂow inside a microfabricated inkjet printer head yielded the highest speed measurements made with μPIV. Using a slightly lower magniﬁcation (40×) and consequently lower spatial resolution, measurements of velocities as high as 8 m/s were made. Here, we will give an overview of μPIV techniques and provide several application examples in section 9.

8.2 Overview of Micro-PIV Three fundamental problems diﬀerentiate μPIV from conventional macroscopic PIV: The particles are small compared to the wavelength of the illuminating light; the illumination source is typically not a light sheet but rather an illuminated volume of the ﬂow; and the particles are small enough that the eﬀects of Brownian motion must be considered. Three-Dimensional Diﬀraction Pattern. Following Born & Wolf [3], the intensity distribution of the three-dimensional diﬀraction pattern of a point source imaged through a circular aperture of radius a can be written in terms of the dimensionless diﬀraction variables (u, v): 2 , 2 2 U1 (u, v) + U22 (u, v) I0 (8.1) I(u, v) = u 2 2 1 v2 2 2 I(u, v) = 1 + V0 (u, v) + V1 (u, v) − 2V0 (u, v) cos u+ n 2 u 1 v2 (8.2) − 2V1 (u, v) sin u+ I0 2 u where Un (u, v) and Vn (u, v) are called Lommel functions, which may be expressed as an inﬁnite series of Bessel functions of the ﬁrst kind: n+2s +∞ J (v) Un (u, v) = s=0 (−1)s uv n+2s n+2s +∞ Jn+2s (v) Vn (u, v) = s=0 (−1)s uv

(8.3)

8.2 Overview of Micro-PIV

dp

245

θ

r

z Geometric

2a

Shadow

f Particle

Lens

Fig. 8.2. Geometry of a particle with a diameter dp , being imaged through a circular aperture of radius a, by a lens of focal length f (after [408]).

The dimensionless diﬀraction variables are deﬁned as: 2 u = 2π λz fa , 2 v = 2π λr fa

(8.4)

where f is the radius of the spherical wave as it approaches the aperture (which can be approximated as the focal length of the lens), λ is the wavelength of light, and r and z are the in-plane radius and the out-of-plane coordinate, respectively, with the origin located at the point source (Figure 8.2). Although both, equation (8.2) and (8.2) are valid in the region near the point of focus, it is computationally convenient to use equation (8.2) within the geometric shadow, where |u/v| < 1, and to use equation (8.2) outside the geometric shadow, where |u/v| > 1 [3]. Within the focal plane, the intensity distribution reduces to the expected result 2 2J1 (v) I0 (8.5) I(0, v) = v which is the Airy function for Fraunhofer diﬀraction through a circular aperture. Along the optical axis, the intensity distribution reduces to 2 sin u/4 I0 . (8.6) I(u, 0) = u/4 The three-dimensional intensity distribution calculated from equation (8.2) and (8.2) is shown in ﬁgure 8.3. The focal point is located at the origin, the optical axis is located at v = 0, and the focal plane is located at u = 0. The maximum intensity, I0 , occurs at the focal point. Along the optical axis, the intensity distribution reduces to zero at u = ±4π, ±8π, while a local maximum occurs at u = ±6π. Depth of ﬁeld. The depth of ﬁeld of a standard microscope objective lens is given by Inou´e and Spring [14] as: δz =

ne nλ0 2 + NA · M NA

(8.7)

246

8 Micro-PIV

Fig. 8.3. Three-dimensional intensity distribution pattern expressed in diﬀraction units (u, v), following Born & Wolf [3]. The focal point is located at the origin, the optical axis is located along v = 0, and the focal plane is located along u = 0 (after [3]).

where n is the refractive index of the ﬂuid between the microﬂuidic device and the objective lens, λ0 is the wavelength of light in a vacuum being imaged by the optical system, NA is the numerical aperture of the objective lens, M is the total magniﬁcation of the system, and e is the smallest distance that can be resolved by a detector located in the image plane of the microscope (for the case of a CCD sensor, e is the spacing between pixels). Equation (8.7) is the summation of the depths of ﬁeld resulting from diﬀraction (ﬁrst term on the right-hand side) and geometric eﬀects (second term on the right-hand side). The cutoﬀ for the depth of ﬁeld due to diﬀraction (ﬁrst term on the righthand side of equation (8.7)) is chosen by convention to be one-quarter of the out-of-plane distance between the ﬁrst two minima in the three-dimensional point spread function, that is, u = ±π in ﬁgure 8.3 and equation (8.2) and 8.2. Substituting NA = n sin θ = n · a/f , and λ0 = nλ yields the ﬁrst term on the right-hand side of equation (8.7). If a CCD sensor is used to record particle images, the geometric term in equation (8.7) can be derived by projecting the CCD array into the ﬂow ﬁeld, and then, considering the out-of-plane distance, the CCD sensor can be moved before the geometric shadow of the point source occupies more than a single pixel. This derivation is valid for small light collection angles, where tan θ ∼ sin θ = NA/n. Depth of correlation. The depth of correlation is deﬁned as twice the distance that a particle can be positioned from the object plane so that the intensity along the optical axis is an arbitrarily speciﬁed fraction of its focused

8.2 Overview of Micro-PIV

247

intensity, denoted by ε. Beyond this distance, the particle’s intensity is suﬃciently low that it will not inﬂuence the velocity measurement. While the depth of correlation is related to the depth of ﬁeld of the optical system, it is important to distinguish between them. The depth of ﬁeld is deﬁned as twice the distance from the object plane in which the object is considered unfocused in terms of image quality. In the case of volumeilluminated μPIV, the depth of ﬁeld does not deﬁne precisely the thickness of the measurement plane. The theoretical contribution of an unfocused particle to the correlation function is estimated by considering (1) the eﬀect due to diﬀraction, (2) the eﬀect due to geometric optics, and (3) the ﬁnite size of the particle. For the current discussion, we shall choose the cutoﬀ for the on-axis image intensity, ε, to be arbitrarily one-tenth of the in-focus intensity. The reason for this choice is that the correlation function varies like the intensity squared so a particle image that with one-tenth the intensity of a focused image can be expected to contribute less than 1% to the correlation function. The eﬀect of diﬀraction can be evaluated by considering the intensity of the point spread function along the optical axis in equation (8.6). If = 0.1, then the intensity cutoﬀ will occur at u ≈ ±3π. Using equation (8.4), substituting δz = 2z, and using the deﬁnition of numerical aperture, NA ≡ n sin θ = n·a/f , one can estimate the depth of correlation due to diﬀraction as: δcg =

3nλ0 NA2

(8.8)

The eﬀect of geometric optics upon the depth of correlation can be estimated by considering the distance from the object plane in which the intensity along the optical axis of a particle with a diameter, dp , decreases an amount, ε = 0.1, due to the spread in the geometric shadow, that is, the lens’ collection cone. If the light ﬂux within the geometric shadow remains constant, the intensity along the optical axis will vary as ∼ z −2 . From ﬁgure 8.3, if the geometric particle image is suﬃciently resolved by the CCD array, the depth of correlation due to geometric optics can be written for an arbitrary value of as: √ (1 − )dp e for dp > (8.9) δzcd = √ tan θ M Following the analysis of Olsen & Adrian [294, 295] and using equation (2.22), the eﬀective image diameter of a particle displaced a distance z from the objective plane can be approximated by combining the eﬀective image diameter dτ with a geometric approximation to account for the particle image spreading due to displacement from the focal plane to yield 2 12 M Da z n 2 2 2 2 2 dτ = M dp + 1.49 (M + 1) λ −1 + (8.10) NA so + z where so is the object distance and Da is the diameter of the recording lens aperture.

248

8 Micro-PIV

The relative contribution ε of a particle displaced a distance z from the focal plane, compared to a similar particle located at the focal plane can be expressed in terms of the ratio of the eﬀective particle image diameters raised to the fourth power 4 dτ (0) (8.11) ε= dτ (zcorr ) Approximating Da2 /(so + z)2 ≈ Da2 /s2o = 4[(n/NA)2 − 1]−1 , combining equation (8.10) and (8.11), and solving for zcorr yields an expression for the depth of correlation √ 2 dp [(n/NA)2 −1] 1− √ ε zcorr = 4 ε . 12 (8.12) 2 1.49(M+1)2 λ2 [(n/NA)2 −1] + 4M 2 From equation (8.12) it is evident that the depth of correlation zcorr is strongly dependent on numerical aperture NA and particle size dp and is weakly dependent upon magniﬁcation M . Table 8.2 gives the thickness of the measurement plane, 2zcorr , for various microscope objective lenses and particle sizes. The highest out of plane resolution for these parameters is 2zcorr = 0.36 μm for a NA = 1.4, M = 60 oil-immersion lens and particle sizes dp < 0.1 μm. For these calculations, it is important to note that the eﬀective numerical aperture of an oil-immersion lens is reduced according to Equation 2.25 when imaging particles suspended in ﬂuids such as water, where the refractive index is less than that of the immersion oil. One important implication of volume illumination that aﬀects both large and small depth of focus imaging systems is thatall particles in the illuminated Table 8.2. Thickness of the measurement plane for typical experimental parameters, 2zcorr [ μm] [300] M NA n

60 1.40 1.515

40 0.60 1.00

20 0.50 1.00

10 0.25 1.00

Meas. plane thickness 2zcorr [ μm]

dp [ μm] 0.01 0.10 0.20 0.30 0.50 0.70 1.00 3.00

40 0.75 1.00

0.36 0.38 0.43 0.52 0.72 0.94 1.3 3.7

1.6 1.6 1.7 1.8 2.1 2.5 3.1 8.1

3.7 3.8 3.8 3.9 4.2 4.7 5.5 13

6.5 6.5 6.5 6.6 7.0 7.4 8.3 17

34 34 34 34 34 35 36 49

8.2 Overview of Micro-PIV

249

volume will contribute to the recorded image. This implies that the particle concentrations will have to be minimized for deep ﬂows and lead to the use of low image density images as described below. Olsen & Adrian [294] used a small angle approximation to derive the depth of correlation as 12 √ 2 2 #4 1− ε 5.95 (M + 1) λ f 2 2 √ f# dp + , (8.13) zcorr = ε M2 where all the variables are as given above. Because it is given in terms of f# instead of NA, it is only applicable for air-immersion lenses and not oilor water-immersion lenses. This model for the depth of correlation has been indirectly experimentally conﬁrmed for low magniﬁcation (M ≤ 20×) and low numerical aperture (NA ≤ 0.4) air-immersion lenses. The depth of correlation was not evaluated explicitly but rather a weighting function used in the model for how the particle intensity varies with distance from the object plane was conﬁrmed with experimental measurements [279]. Particle visibility. The quality of μPIV velocity measurements strongly depends upon the quality of the recorded particle images from which those data are calculated. In macroscopic PIV experiments, it is customary to use a sheet of light to illuminate only those particles that are within the depth of ﬁeld of the recording lens. The light sheet has two important eﬀects – it minimizes the background noise from out of focus particles and ensures that every particle visible to the camera is well-focused. However, in μPIV, the microscopic length scales and limited optical access necessitate using volume illumination. Experiments using the μPIV technique must be designed so that focused particle images can be observed even in the presence of background light from unfocused particles and test section surfaces. The background light scattered from test section surfaces can be removed by using ﬂuorescence techniques to ﬁlter out elastically scattered light (at the same wavelength as the illumination) while leaving the ﬂuoresced light (at a longer wavelength) virtually unattenuated [297]. Background light ﬂuoresced from unfocused tracer particles is not so easily removed because it occurs at the same wavelength as the signal, the focused particle images, but it can be lowered to acceptable levels by choosing proper experimental parameters. Olsen & Adrian [294] present a theory to estimate particle visibility, deﬁned as the ratio of the intensity of a focused particle image to the average intensity of the background light produced by the unfocused particles. The analysis in this section refers to the dimensions labeled on ﬁgure 2.42 and 2.30. Assuming light is emitted uniformly from the particle, the light from a single particle reaching the image plane can be written as J(z) =

Jp Da2 2

16 (so + z)

(8.14)

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8 Micro-PIV

where Jp is total light ﬂux emitted by a single particle. We approximate the intensity of a focused particle image as Gaussian, −4β 2 r2 I(r) = I0 exp (8.15) d2τ where the unspeciﬁed parameter β is chosen to determine the cutoﬀ level that deﬁnes the edge of the particle image. Approximating the Airy distribution by a Gaussian distribution, with the area of the two axisymmetric functions being equal, the ﬁrst zero in the Airy distribution corresponds to [53] I = exp −β 2 ≈ exp(−3.67). I0

(8.16)

Integrating equation (8.15) over an entire particle image and equating that result to equation (8.14) allows I0 to be evaluated and equation (8.15) to be written as Jp Da2 β 2 −4β 2 r2 I(r, z) = exp (8.17) 2 d2τ 4πd2τ (so + z) Making the simplifying assumption that particles located outside a distance |z| > δ/2 from the object plane as being completely unfocused and contributing uniformly to background intensity, while particles located within a distance |z| < δ/2 as being completely focused, the total ﬂux of background light JB can be approximated by δ −2

JB = Av C

L−a

J(z)dz + −a

δ 2

J(z)dz

,

(8.18)

where C is the number of particles per unit volume of ﬂuid, L is the depth of the device, and Av is the average cross sectional area contained within the ﬁeld of view. Combining equation (8.14) and (8.18), correcting for the eﬀect of magniﬁcation, and assuming so δ/2, the intensity of the background glow can be expressed as [294] IB =

CJP LDa2 . 16M 2 (so − a) (so − a + L)

(8.19)

Following Olsen & Adrian [294], the visibility V of a focused particle can be obtained by combining equation (8.10) and (8.17), dividing by equation (8.19), and setting r = 0 and z = 0, V =

4M 2 β 2 (so − a) (so − a + L) I(0, 0) /

0 . = 2 n 2 IB πCLs2o M 2 d2p + 1.49 (M + 1) λ2 NA −1

(8.20)

From this expression it is clear that for a given recording optics conﬁguration, particle visibility V can be increased by decreasing particle concentration C

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251

or by decreasing test section thickness L. For a ﬁxed particle concentration, the visibility can be increased by decreasing the particle diameter dp or by increasing the numerical aperture NA of the recording lens. Visibility depends only weakly on magniﬁcation and object distance so . An expression for the volume fraction Vf r of particles in solution that produce a speciﬁc particle visibility can be obtained by rearranging equation (8.20) and multiplying by the volume occupied by a spherical particle to get 2d3 M 2 β 2 (so − a) (so − a + L) / p

0 . Vf r = (8.21) 2 n 2 3V Ls2o M 2 d2p + 1.49 (M + 1) λ2 NA −1 Reasonably high quality velocity measurements require visibilities in excess of 1.5. Although this is an arbitrary threshold, it works well in practice. To see this formula in practice, assume that we are interested in measuring the ﬂow at the centerline (a = L/2) of a microﬂuidic device with a characteristic depth of L = 100 μm. Table 8.3 shows for various experimental parameters the maximum volume fraction of particles that can be seeded into the ﬂuid while maintaining a focused particle visibility greater than 1.5. Here, the object distance so is estimated by adding the working distance of the lens to the designed coverslip thickness. Meinhart, et al. [408] veriﬁed these trends with a series of imaging experiments using known particle concentrations and ﬂow depths. The particle visibility V was estimated from a series of particle images taken of four diﬀerent particle concentrations and four diﬀerent device depths. A particle solution was prepared by diluting dp = 200 nm diameter polystyrene particles in deionized water. Test sections were formed using two feeler gauges of known Table 8.3. Maximum volume fraction of particles Vf r , expressed in percent, necessary to maintain a visibility V greater than 1.5 when imaging the center of an L = 100 μm deep device [300]. M NA n so [ mm]

60 1.40 1.515 0.38

40 0.60 1.00 3

20 0.50 1.00 7

10 0.25 1.00 10.5

Volume Fraction (%)

dp [ μm] 0.01 0.10 0.20 0.30 0.50 0.70 1.00 3.00

40 0.75 1.00 0.89

2.0E-5 1.7E-2 1.1E-1 2.5E-1 6.0E-1 9.6E-1 1.5E+0 4.8E+0

4.3E-6 4.2E-3 3.1E-2 9.3E-2 3.2E-1 6.4E-1 1.2E+0 4.7E+0

1.9E-6 1.9E-3 1.4E-2 4.6E-2 1.8E-1 4.1E-1 8.7E-1 4.5E+0

1.1E-6 1.1E-3 8.2E-3 2.7E-2 1.1E-1 2.8E-1 6.4E-1 4.2E+0

1.9E-7 1.9E-4 1.5E-3 5.1E-3 2.3E-2 6.2E-2 1.7E-1 2.5E+0

252

8 Micro-PIV

Table 8.4. Experimental assessment of particle visibility as a function of depth and particle concentration [406].

Depth[ μm] Particle Concentration (by Volume) 0.01% 0.02% 0.04% 0.08% 25 50 125 170

2.2 1.9 1.5 1.3

2.1 1.7 1.4 1.2

2.0 1.4 1.2 1.1

1.9 1.2 1.1 1.0

thickness sandwiched between a glass microscope slide and a coverslip. The images were recorded with an oil-immersion M = 60×, NA = 1.4 objective lens. The remainder of the μPIV system was as described above. The measured visibility is shown in table 8.4. As expected, the results indicate that, for a given particle concentration, a higher visibility is obtained by imaging a ﬂow in a thinner device. This occurs because decreasing the thickness of the test section decreases the number of unfocused particles, while the number of focused particles remains constant. Also, increasing the particle concentration decreases the visibility, as expected. In general, thinner test sections allow higher particle concentrations to be used, which can be analyzed using smaller interrogation regions. Consequently, the seed particle concentration must be chosen judiciously so that the desired spatial resolution can be obtained, while maintaining adequate image quality (i.e. particle visibility). 8.2.1 μPIV Seeding When the seed particle size becomes small, the collective eﬀect of collisions between the particles and a moderate number of ﬂuid molecules is unbalanced, preventing the particle from following the ﬂow to some degree. This phenomenon, commonly called Brownian motion, has two potential implications for μPIV: One is to cause an error in the measurement of the ﬂow velocity; the other is to cause an uncertainty in the location of the ﬂow tracing particles. In order to fully consider the eﬀect of Brownian motion, it is ﬁrst necessary to establish how particles suspended in ﬂows behave. Flow/Particle Dynamics. In stark contrast to many macroscale ﬂuid mechanics experiments, the hydrodynamic size of a particle (a measure of its ability to follow the ﬂow based on the ratio of inertial to drag forces) is usually not a concern in microﬂuidic applications because of the large surface to volume ratios at small length scales. As described in section 2.1.1, a simple model for the response time of a particle subjected to a step change in local ﬂuid velocity can be used to gauge particle behavior. Based on a simple

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253

ﬁrst-order inertial response to a constant ﬂow acceleration (assuming Stokes ﬂow for the particle drag), the response time τp of a particle is: τp =

d2p ρp 18μ

(8.22)

where dp and ρp are the diameter and density of the particle, respectively, and μ is the dynamic viscosity of the ﬂuid. Considering typical μPIV experimental parameters of 300 nm-diameter polystyrene latex spheres immersed in water, the particle response time would be 10−9 s. This response time is much smaller than the time scales of any realistic liquid or low-speed gas ﬂow ﬁeld. In the case of high-speed gas ﬂows, the particle response time may be an important consideration when designing a system for microﬂow measurements. For example, a 400 nm particle seeded into an air micronozzle that expands from sonic at the throat to Mach 2 over a 1 mm distance may experience a particle-to-gas relative ﬂow velocity of more than 5% (assuming a constant acceleration and a stagnation temperature of 300 K). Particle response to ﬂow through a normal shock would be signiﬁcantly worse. Another consideration in gas microchannels is the breakdown of the no-slip and continuum assumption as the particle Knudsen number Knp , deﬁned as the ratio of the mean free path of the gas to the particle diameter, approaches (and exceeds) one. For the case of the slip-ﬂow regime (10−3 < Knp < 0.1), it is possible to use corrections to the Stokes drag relation to quantify particle dynamics [278]. For example, a correction oﬀered by MELLING [74] suggests the following relation for the particle response time: τp = (1 + 2.76Knp )

d2p ρp 18μ

(8.23)

Velocity Errors. Santiago et al. [297] brieﬂy considered the eﬀect of the Brownian motion on the accuracy of μPIV measurements. A more indepth consideration of the phenomenon of Brownian motion is necessary to completely explain its eﬀects in μPIV. Brownian motion is the random thermal motion of a particle suspended in a ﬂuid [296]. The motion results from collisions between ﬂuid molecules and suspended microparticles. The velocity spectrum of a particle due to the Brownian motion consists of frequencies too high to be resolved fully and is commonly modeled as Gaussian white noise [298]. A quantity more readily characterized is the particle’s average displacement after many velocity ﬂuctuations. For time intervals Δt much larger than the particle inertial response time, the dynamics of Brownian displacement are independent of inertial parameters such as particle and ﬂuid density, and the mean square distance of diﬀusion is proportional to DΔt, where D is the diﬀusion coeﬃcient of the particle. For a spherical particle subject to Stokes drag law, the diﬀusion coeﬃcient D was ﬁrst given by Einstein [282] as: D=

KTa 3πμdp

(8.24)

254

8 Micro-PIV

where dp is the particle diameter, K is Boltzmann’s constant, Ta is the absolute temperature of the ﬂuid, and μ is the dynamic viscosity of the ﬂuid. The random Brownian displacements cause particle trajectories to ﬂuctuate about the deterministic pathlines of the ﬂuid ﬂow ﬁeld. Assuming the ﬂow ﬁeld is steady over the time of measurement and the local velocity gradient is small, the imaged Brownian particle motion can be considered a ﬂuctuation about a streamline that passes through the particle’s initial location. An ideal, nonBrownian (i.e., deterministic) particle following a particular streamline for a time period Δt has x- and y-displacements of: Δx = uΔt Δy = vΔt where u and v are the x- and y- components of the time-averaged, local ﬂuid velocity, respectively. The relative errors, εx and εy , incurred as a result of imaging the Brownian particle displacements in a two-dimensional measurement of the x- and y-components of particle velocity, are given as: σx εx = Δx = u1 2D Δt σy 1 εy = Δy = u 2D Δt This Brownian error establishes a lower limit on the measurement time interval Δt since, for shorter times, the measurements are dominated by uncorrelated Brownian motion. These quantities (ratios of the root mean square (rms) ﬂuctuation-to-average velocity) describe the relative magnitudes of the Brownian motion and will be referred to here as Brownian intensities. The errors estimated by equation (8.25)a and (8.25)b show that the relative Brownian intensity error decreases as the time of measurement increases. Larger time intervals produce ﬂow displacements proportional to Δt while the rms of the Brownian particle displacements grows as Δt1/2 . In practice, Brownian motion is an important consideration when tracing 50–500 nm particles in ﬂow ﬁeld experiments with ﬂow velocities of less than about 1 mm/s. For a velocity on the order of 0.5 mm/s and a 500 nm seed particle, the lower limit for the time spacing is approximately 100 μs for a 20% error due to Brownian motion. This error can be reduced by both averaging over several particles in a single interrogation spot and by ensemble√averaging over several realizations. The diﬀusive uncertainty decreases as 1/ N , where N is the total number of particles in the average [2]. Equation (8.25) demonstrates that the eﬀect of the Brownian motion is relatively less important for faster ﬂows. However, for a given measurement, when u increases, Δt will generally decrease. Equation (8.25)a and (8.25)b also demonstrate that when all conditions but Δt are ﬁxed, going to a larger Δt will decrease the relative error introduced by the Brownian motion. Unfortunately, a longer Δt will decrease the accuracy of the results because the PIV measurements are based on a ﬁrst-order accurate approximation to the

8.2 Overview of Micro-PIV

255

velocity. Using a second-order accurate technique (the central diﬀerence interrogation (CDI)) allows for a longer Δt to be used without increasing this error. Particle Position Error. In addition to the ﬂow velocity measurement error associated with particle displacement measurements, the Brownian motion incurred during the exposure time texp may also be important in determining the particle location, especially for slow ﬂows with long exposures and small tracer particles. For example, a 50 nm particle in water at room temperature will have an rms displacement of 300 nm if imaged with a 10 ms exposure time. For this particle image, the Brownian displacements projected into the image plane during the time of exposure are on the order of the image size estimated by equation (8.23) (given the best available far-ﬁeld, visual optics with a numerical aperture of 1.4). This random displacement during image exposure can increase the uncertainty associated with estimating the particle location. For low velocity gradients, the centroid of this particle image is an estimate of the average location of the particle during the exposure. This particle location uncertainty is typically negligible for exposure times where the typical Brownian displacement in the image plane is small compared to the particle image diameter or a value of the diﬀusion time d2τ /(4DM 2 ) much less than the exposure time. For the experimental parameters mentioned above, d2τ /(4DM 2 ) is 300 ms and the exposure time is 5 ns for a typical Nd:YAG laser. 8.2.2 Special Processing Methods for μPIV Recordings When evaluating digital PIV recordings with conventional correlation-based algorithms or image-pattern tracking algorithms, a suﬃcient number of particle images are required in the interrogation window or the tracked image pattern to ensure reliable and accurate measurement results. However, in many cases, especially in μPIV measurements, the particle image density in the PIV recordings is usually not high enough (ﬁgure 8.4a). These PIV recordings are called low image density (LID) recordings and are usually evaluated with particle-tracking algorithms. When using particle-tracking algorithms, the velocity vector is determined with only one particle, and hence the reliability and accuracy of the technique are limited. In addition, interpolation procedures are usually necessary to obtain velocity vectors on the desired regular grid points from the random distributed particle-tracking results (ﬁgure 8.5a), and therefore, additional uncertainties are added to the ﬁnal results. Fortunately, some special processing methods can be used to evaluate the μPIV recordings, so that the errors resulting from the low-image density can be avoided [299]. In this section two methods are introduced to improve measurement accuracy of μPIV: by using a digital image processing technique and by improving the evaluation algorithm. Overlapping of LID-PIV recordings. In the early days of PIV, multiple exposure imaging techniques were used to increase the particle image

256

8 Micro-PIV

Fig. 8.4. Example of image overlapping: (a) one of the LID-PIV recordings; and (b) result of overlapping 9 LID-PIV recordings. Image size: 256 × 256 pixel [299]. Copyright 2002, AIAA. Reprinted with permission.

(a)

(b)

Fig. 8.5. Eﬀect of image overlapping: (a) results for a single LID-PIV recording pair with a particle-tracking algorithm; and (b) results for the overlapped PIV recording pair with a correlation-based algorithm [299]. Copyright 2002, AIAA. Reprinted with permission.

numbers in PIV recordings. Similar to multiply exposing a single frame, highimage-density (HID) PIV recordings can be generated by computationally overlapping a number of LID-PIV recordings with: g0 (x, y) = max[gk (x, y), k = 1, 2, 3, ..., N ]

(8.25)

wherein gk (x, y) is the gray value distribution of the LID-PIV recordings with a total number N, and g0 (x, y) is the overlapped recording. Note that in equation (8.25) the particle images are positive (i.e., with bright particles and dark background); otherwise, the images should be inverted or the minimum function used. An example of the image overlapping can be seen in ﬁgure 8.4 for overlapping nine LID-PIV recordings. The size of the PIV recordings in ﬁgure 8.5 is 256 × 256 pixel, and the corresponding measurement

8.2 Overview of Micro-PIV Fk(m,n)

Fens(m,n)

m

(a)

257

m

n

(b)

n

Fig. 8.6. Eﬀect of ensemble correlation: (a) results with conventional correlation for one of the PIV recording pairs; and (b) results with ensemble correlation for 101 PIV recording pairs [299]. Copyright 2002, AIAA. Reprinted with permission.

area is 2.5 × 2.5 mm2 . The eﬀect of the image overlapping is shown in ﬁgure 8.5. Figure 8.6a has the evaluation results for one of the LID-PIV recording pairs with a particle-tracking algorithm [135]. Figure 8.5b has the results for the overlapped PIV recording pair (out of nine LID-PIV recording pairs) with a correlation-based algorithm. The results in ﬁgure 8.5b are more reliable, more dense, and more regularly spaced than those in ﬁgure 8.5a. The image overlapping method is based on the fact that ﬂows in microdomains typically have very low Reynolds numbers, so that the ﬂow can be considered as laminar and steady in the data acquisition period. Note that this method cannot be extended to measurements of turbulent or unsteady ﬂows, and it may not work very well with overlapping HID-PIV recordings or too many LID-PIV recordings because with large numbers of particle images, interference between particle images will occur [407]. Further study of this technique will be necessary to quantify these limitations but the promise of the technique is obvious. 8.2.3 μPIV Summary Currently, applying the advanced techniques outlined here, the maximum spatial resolution of the technique stands at approximately 1 μm. By using smaller seed particles that ﬂuoresce at shorter wavelengths, this limit could be reduced by a factor of 2 to 4. Still higher spatial resolutions could be obtained by adding a particle tracking step after the correlation-based PIV. Spatial resolutions of an order of magnitude smaller could then reasonably be reached. The various μPIV apparatus and algorithms described in this chapter have been demonstrated to allow measurements at length scales on the order of 1 μm, signiﬁcantly below the typical Kolmogorov length scale. These spatial resolutions are indispensable when analyzing ﬂows in microdomains or the smallest scales of turbulence. The most signiﬁcant problem standing in the

258

8 Micro-PIV

way of extending μPIV to gas phase ﬂows is seeding. With adequate seeding, the results presented here can be extended to gas ﬂows. The signiﬁcant issues associated with extending the results presented here to gas phase ﬂows are further explored by Meinhart et al. [408].

9 Examples of Application

In this chapter a number of applications of the PIV technique will be described, contributed by leading PIV experts from diﬀerent research establishments and universities worldwide. A complete list of authors and their aﬃliation is given in the acknowledgement at the beginning of this book. Primarily, the objective of presenting these applications is to show how the PIV technique has spread out to the most diﬀerent research areas. However, it is of even higher importance to gain the reader access to a wide variety of ideas for PIV measurements by presenting many diﬀerent applications in fundamental or industrial research. For each experiment the most important parameters of the object under investigation, of the illumination and recording setup, etc. will be given. These data together with the hints and tricks brieﬂy described and the references to further, more detailed, literature may be useful for the reader when trying to solve problems of his own application.

9.1 Liquid Flows 9.1.1 Vortex-Free-Surface Interaction Contributed by: C. Willert, M. Gharib

The present example was chosen to illustrate the possibility of PIV in providing time-resolved measurements in low-speed ﬂows, which is of importance in many ﬂuid mechanical investigations. Time resolution is possible when the image frame rate exceeds the time scales present in the ﬂow. For this investigation the frame rate of the utilized video equipment was 30 Hz (RS-170) resulting in an image-pair rate of 15 Hz using the frame–straddling approach, whereas the time-scales in the ﬂow were longer than 1/10th of a second. The PIV parameters used for this investigation are listed in table 9.1.

260

9 Examples of Application Table 9.1. PIV recording parameters for vortex–free-surface interaction. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Pulse duration Seeding material

Nearly two-dimensional ﬂow aligned with the light sheet Umax ≈ 10 cm/s 103 × 97 mm2 6.4 × 6.4 × 1.5 mm3 (H × W × D) DSR ≈ 16 : 1 DVR ≈ 100 : 1 z0 ≈ 1.5 m (through glass/water) double frame/single exposure frame separation (frame-straddling) frame transfer CCD (512 × 480 pixel) f = 50 mm, f# = 1.8 5 W CW argon-ion laser, mechanical shutter Δt = 10 ms 2 ms silver-coated, glass spheres (dp ≈ 10 μm)

The vortex-pair ﬂow under investigation is motivated in the context of understanding the fundamentals of the interaction of vortical structures with a free surface [306]. The vortex pair was generated by a pair of counter-rotating ﬂaps whose sharp tips were located at y ≈ −10 cm. Once the ﬂaps were closed the separation vortex from each tip formed a symmetric vortex pair which propagated towards the surface within two seconds. The interesting interaction events typically took place in the following 2–5 seconds. This means that of the order of O[10 s] worth of PIV data were needed to resolve a single interaction process. This translated to 300 separate PIV recordings, which were acquired using a computer-based, real-time digitization and hard-disk array. Figure 9.1 shows four selected instantaneous PIV velocity and corresponding vorticity ﬁelds from a 150 image-pair sequence. Since the ﬂow was repeatable, it was imaged in several diﬀerent planes in order to reconstruct the entire ﬂow ﬁeld. Using the circulation measurement schemes described in section 6.5.1 time-resolved circulation measurements of the vortex structures could be obtained from the velocity maps to study the vortex dynamics at the surface, namely, vortex reconnection and dissipation. Further details on this as well as related experiments can be found in Willert [305] and Weigand & Gharib [304]. 9.1.2 Study of Thermal Convection and Couette Flows Contributed by: C. B¨ ohm, C. Willert, H. Richard

These experimental investigations of ﬂows by means of PIV have been carried out by DLR in cooperation with the Center of Applied Space Technology

9.1 Liquid Flows

261

a) t = 1.56s

b) t = 2.23s

c) t = 2.90s

d) t = 3.90s

Fig. 9.1. Time resolved PIV measurements of the right core of a vortex pair impinging on a contaminated free surface (at y = 0). Shown in the right column are vorticity estimates computed from the velocity ﬁelds to the left.

and Microgravity (ZARM), University of Bremen, in order to complete their numerical simulations and LDV measurements [301]. The experimental setup is shown in ﬁgure 9.2. The PIV parameters used for this investigation are listed in table 9.2. A ﬂuid (silicone oils M20 and M3) seeded with 10 μm diameter glass 3 particles with a volumetric mass near 1.05 g/cm and a refraction index of n = 1.55, is ﬁlled in the gap between two concentric spheres. The outer sphere is composed of two transparent acrylic glass hemispheres (refraction index of n = 1.491), with a radius of 40.0 mm, and the inner sphere is made out of aluminum with a radius of 26.7 mm. To minimize optical distortions because of the curvature of the model, the outer sphere is included in a rectangular

262

9 Examples of Application Table 9.2. PIV recording parameters for thermal convection. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material

U∞ = 0.5 cm/s parallel to light sheet Umax ≈ 0.5 cm/s 50 × 40 mm2 1.6 × 1.6 × 2 mm3 (H × W × D) DSR ≈ 24 : 1 DVR ≈ 200 : 1 z0 ≈ 1.5 m dual frame/single exposure frame separation full frame interline transfer CCD (782 × 582 pixel) f = 100 mm f# = 2.8 to 22 continuous argon-Ion laser, 1 watt, internal shutter of camera Δt = 40 ms glass particles (dp ≈ 10 μm)

cavity ﬁlled with silicone oil to provide a plane liquid–air interface and reduce optical refraction. To study the thermal convection ﬂows, the inner sphere is heated homogeneously up to 45◦ C whereas the outer sphere is held at constant temperature. Six temperature sensors are installed on both spheres as indicated in ﬁgure 9.2. A 25 Hz CCD camera with an internal shutter (40 ms between each frame) was used in combination with a continuous argon-ion laser. This was possible because of the low velocity ﬂow studied (≈ 0.5 cm/s). A 100 mm Zeiss Makro Planar objective lens was used during the ﬂow measurements with a f# number of 2.8. For a magniﬁcation between 1/2 and 1/4, and a f-number of f# = 11, the particle image diameters are in the range between 22 and 18 μm,

Fig. 9.2. The experimental apparatus to study the thermal convection and the Taylor ﬂow.

9.1 Liquid Flows Outer sphere:To

263

Outer sphere:To

Siliconeoil

Light sheet Inner sphere Ti

Inner sphere Ti

Light sheet

1cm

Fig. 9.3. The light sheet position.

that is to say between 2 and 3 pixels which give the lowest measurement uncertainty. For the particles utilized in this experiment, the gravitational velocity is found to be: vg = 2.9 · 10−7 m/s with M20 oil and vg = 3 · 10−6 m/s with M3 oil, which are disturbances that can be disregarded. The investigations were carried out in a meridional light sheet (ﬁgure 9.3) through the sphere’s center. For small ΔT between the two spheres we have the laminar convective state, and the ﬂow structures of the PIV measurement (see ﬁgure 9.4) are in good agreement with the streamlines computed numerically by Garg [302]. We have an upward ﬂow of 0.1 cm/s at the inner sphere and a downward ﬂow of 0.05 cm/s at the outer sphere, and a ratio of 2 to 1 which has also been predicted theoretically by Mack & Hardee [303], and at the north pole we have a radial outward ﬂow of 0.2 cm/s whereas in the equatorial region we have an area of zero velocity in the middle of the sphere as in the model of Garg [302]. By increasing the ΔT a time dependent pulsating ring vortex sets in at the north pole near the outer sphere (see ﬁgure 9.4). The maximal velocities increase up to 1 cm/s in the polar region and we have an upward ﬂow of about 0.35 cm/s at the inner sphere and a downward ﬂow of 0.1 cm/s at the outer sphere. The convective motion is dominant at the boundary regions and near to the pole in contrast to the vanishing velocities in a wide range of radial positions. Additionally there are small radial inward ﬂows at the outer sphere boundary which is in agreement with the numerical simulations. The Couette ﬂow study required the use of another setup: the velocity being contained between 5 cm/s and 10 cm/s and therefore the previous delay used between each frame was too large. As a consequence, a pulsed Nd:YAG laser synchronized with a large format video camera was used allowing us to select appropriate pulse delays. The frame-straddling technique was employed for directional ambiguity removal. The study has been performed at 0.4 cm above the pole region (ﬁgure 9.5). The rotation of the inner sphere was 250 revolutions per minute for the experiment and ΔT = 0. The velocity vector maps are presented in ﬁgure 9.6. The PIV parameters used for this investigation are listed in table 9.3.

264

9 Examples of Application

Fig. 9.4. Thermal convection velocity ﬁelds and ﬂow picture with the two exposures. Table 9.3. PIV recording parameters for Couette ﬂow. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material a

frequency doubled

U∞ = 10 cm/s parallel to light sheet Umax ≈ 10 cm/s 50 × 50 mm2 1.6 × 1.6 × 2 mm3 (H × W × D) DSR ≈ 31 : 1 DVR ≈ 200 : 1 z0 ≈ 0.5 m dual frame/single exposure frame separation (frame-straddling) full frame interline transfer CCD (1008 × 1018 pixel) f = 60 mm, f# = 2.8 to 22 Nd:YAG lasera , 70 mJ/pulse Δt = 20 ms glass particles (dp ≈ 10 μm)

9.2 Boundary Layers Outer sphere

265

Outer sphere

Silicone oil

Light sheet

Inner sphere

Inner sphere

Light sheet

1cm

Fig. 9.5. The light sheet position: 0.4 cm up to the pole region. Reference Vector: 5 cm/sec

3.5

3

2.5

Y [cm]

2

1.5

1

0.5

0.5

1

1.5

2

2.5

3

3.5

X [cm]

Fig. 9.6. Couette ﬂow velocity ﬁeld and picture with the two exposures.

9.2 Boundary Layers Contributed by: C. K¨ ahler, J. Kompenhans

The following two experiments have been performed in the DLR low turbulence wind tunnel (TUG), which is of an Eiﬀel type. Screens in the settling chamber and a high contraction ratio of 15:1 lead to a low turbulence level in the test section (cross section 0.3 × 1.5 m2 ). The basic turbulence level in the test section of the TUG of T u = 0.06% (measured by means of a hot wire) allows the investigation of acoustically exited transition from laminar to turbulent ﬂow as well as turbulent boundary layers that develop in the relatively long test section. The ﬂow was seeded in the settling chamber upstream of the screens used to reduce the turbulence of the wind tunnel ﬂow.

266

9 Examples of Application

9.2.1 Boundary Layer Instabilities In the case of periodic ﬂows, the conditional sampling technique can be utilized in order to record instantaneous velocity vector maps always at the same phase angle. The excitation of the periodic process and the recording sequence must be phase locked. As an example for the application of conditional sampling, the investigation of instabilities in a boundary layer will be described. The transitional process in a boundary layer is determined by a mechanism of generation and interaction of various instabilities. Small oscillations may cause primary instability – two-dimensional waves, the Tollmien-Schlichting (TS) waves. The growth of such TS waves leads to a streamwise periodic modulation of the basic ﬂow, which gets sensitive to three-dimensional, spanwise periodic disturbances. These disturbances are ampliﬁed and lead to a three-dimensional distortion of the TS waves and farther downstream to the generation of three-dimensional Λ vortices. The extension of the knowledge about this mechanism enables the prediction and control of transition as required for applications in ﬂuid mechanical engineering. In order to study the behavior of instabilities, quantitative data of velocity ﬁelds with known initial conditions have been acquired in a ﬂat plate boundary layer, in the TUG wind tunnel (see ﬁgure 9.7). In order to get reproducible and constant conditions for the development of the instabilities it is necessary to know the initial amplitude of the velocity ﬂuctuations at the be¨hler & Wiegel ginning of the observation area [276]. In the experiment of Ka this is achieved by introducing controlled disturbances by means of a device for acoustic excitation which consists of a single spanwise slot for the controlled input of two-dimensional disturbances and 40 separate slots (positioned

1.5 6

7

18.65 6.25 2.3 1

2

3

4

0.3 5.0 5

Units [m] Fig. 9.7. Low turbulence wind tunnel.

9.2 Boundary Layers

267

Table 9.4. PIV recording parameters for boundary layer instabilities. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material a

parallel to light sheet and plate Umax ≈ 12 m/s 70 × 70 mm2 1.9 × 1.9 × 0.5 mm3 (H × W × D) DSR ≈ 31 : 1 DVR ≈ 137 : 1 Z0 ≈ 0.6 m dual frame/single exposure frame separation (frame-straddling) full frame interline transfer CCD f = 60 mm, f# = 2.8 Nd:YAG lasera 320 mJ/pulse Δt = 80 μs oil droplets (dp ≈ 1 μm)

frequency doubled

spanwise as well) for the input of controlled three-dimensional disturbances. The velocity at the outer edge of the boundary layer was about U = 12 m/s. The average free stream turbulence level was T u = 0.065%. The light sheet (thickness δZ = 0.5 mm in the observation area) was oriented parallel to the plate. Its height above the plate could be varied but was usually 0.5 mm in the experiment. The observation area was 70 × 70 mm2 . The PIV parameters used for this investigation are listed in table 9.4. By applying diﬀerent input signals to the acoustic excitation it was possible to excite diﬀerent transition types. We mention here the fundamental type, the subharmonic type and the oblique type. Figure 9.8 presents the phase

Fig. 9.8. Field of instantaneous velocity ﬂuctuations of boundary layer instabilities above a ﬂat plate for two diﬀerent amplitudes of the input signal.

268

9 Examples of Application

locked ﬁeld of the instantaneous velocity ﬂuctuations (U − Umean , V ) obtained by exciting the oblique type for two diﬀerent disturbances. The Λ-vortices exhibit in an aligned pattern. The spanwise wavelength of these Λ-vortices (here ≈ 20 mm) matches with the wavelength of the controlled input of the 3D-waves. The direction of the ﬂow is from left to right. The mean velocity Umean (calculated by averaging over all velocity vectors in the recording) has been subtracted from all velocity vectors in order to show the ﬂuctuating components of the velocity vector ﬁeld. 9.2.2 Turbulent Boundary Layer The following PIV application in a turbulent boundary layer at the wall of a ﬂat plate illustrates two problems: obtaining PIV data close to a wall and recovering PIV data even in a ﬂow with gradients (due to the velocity proﬁle of the boundary layer). In the present series of experiments the measurement position was 2.3 m downstream of a tripping region in the low-turbulence wind tunnel (see ﬁgure 9.7) at the DLR-G¨ ottingen research center [309]. At this position the turbulent boundary layer thickness δ was of the order of 5 cm, of which the lower 3 cm was imaged. At free stream velocities of 10.3, 14.9 and 19.8 m/s between 90 and 100 PIV image pairs were recorded. By removing a constant velocity proﬁle of Uref = 8 m/s from the PIV data set, the small scale strucTable 9.5. PIV recording parameters for turbulent boundary layer over a ﬂat plate with zero pressure gradient. Flow geometry Maximum in-plane velocity Field of view Interrogation volume

Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material a

frequency doubled

parallel to light sheet U∞ = 10.3, 14.9, 19.8 m/s 30 × 30 mm2 2.0 × 2.0 × 1.0 mm3 (H × W × D) 2.0 × 1.0 × 1.0 mm3 (H × W × D) 2.0 × 0.5 × 1.0 mm3 (H × W × D) 1.0 × 1.0 × 1.0 mm3 (H × W × D) DSR ≈ 31 : 1 DVR ≈ 44 : 1 z0 ≈ 1.5 m dual frame/single exposure frame separation (frame-straddling) full frame interline transfer CCD f = 180 mm, f# = 2.8 Nd:YAG lasera , 70 mJ/pulse Δt = 7 − 20 μs oil droplets (dp ≈ 1 μm)

9.2 Boundary Layers

269

Reference Vector: 5 m/s

25

20

Y [mm]

15

10

5

5

10

15

X [mm]

20

25

30

Fig. 9.9. Field of instantaneous velocity ﬂuctuations in a fully turbulent boundary layer, (U − Uref , V ). Position of the wall at Y = 0.

tures in the boundary layer are highlighted as can be seen in ﬁgure 9.9. It is remarkable how close to the wall, the velocity data could be recovered. The PIV parameters used for this investigation are listed in table 9.5. In the ﬁrst part of the evaluation the boundary layer proﬁle and the RMS components of the velocity ﬂuctuations were calculated as an average over all PIV recordings. These averaged quantities agree very well with the results from theory and pointwise velocity measurements as carried out by means of a hot wire. The nondimensional velocity proﬁles given in ﬁgure 9.10 start near the outer edge of the viscous sublayer (y + ≈ 10) and extend well into the region where the large scale structures in the boundary layer cause a departure from the logarithmic proﬁle (y + ≈ 200). As already mentioned, the strong velocity gradients within the interrogation areas close to the wall have mainly two eﬀects. First, due to the inhomogeneous displacement of paired particle images, the amplitude of the signal peak RD+ is diminished. In addition, the diameter of the peak is broadened in the direction of shear. Therefore the velocity variation in the near wall region will decrease the likelihood of detection of the displacement peak. Second, besides these experimental diﬃculties it has to be carefully checked whether the velocity vector assigned to the center of the interrogation window really represents the ﬂow velocity at this location also in the presence of velocity gradients, as has been obtained by averaging over the interrogation window (see ﬁgure 6.15). To investigate the eﬀect of diﬀerent interrogation area size on the number of outliers all PIV recordings were interrogated four times. The result can be

270

9 Examples of Application

25

U

+

20

15

U∞ = 10.3 U∞ = 14.9 U∞ = 19.8 U∞ = 1.6

10

5

10

100 + Y

Fig. 9.10. Mean velocity proﬁles, scaled with inner variables (averaged over 100 PIV recordings).

1000

seen in table 9.6 (for more details see [309]). The number of outliers in the 64 × 32 window is smaller compared to the other cases, because the number of particles is two times larger. The fraction of outliers is only of the order of 1% in the worst case, which clearly shows the reliability of the measurement technique. Figure 9.11 represents the semilogarithmic mean velocity proﬁles as a function of the distance from the wall. For wall distances y ≥ 2 mm the mean velocity is independent of the size of the interrogation window for three diﬀerent measurements (U∞ = 10.3, 14.9 and 19.8 m/s). However, for 0 < y < 2 mm the curves do not coincide due to the diﬀerent averaging. The extension of the interrogation windows in the y direction is mainly responsible for this. Rectangular windows (extending parallel to the wall) show better performance as compared to square windows. Windows deformation techniques for PIV evaluation as developed during the past few years have considerably contributed to the improvement of the data quality in boundarylayers and shear ﬂows. However, it should be stressed Table 9.6. Number of outliers as a function of the interrogation area size, shape and free stream velocities. Δx0 × Δy0 [ pixel]

outliers [%] [10.3 m/s]

outliers [%] [14.9 m/s]

outliers [%] [19.8 m/s]

32 × 32 64 × 16 64 × 32 64 × 64

1.07 0.72 0.20 0.14

1.00 0.58 0.21 0.19

1.26 1.03 0.30 0.17

9.3 Transonic Flows

271

14

12

⎯U [m/s]

10

32 × 32 64 × 16 64 × 32 64 × 64

8

6

4

2 0.1

1.0 Y [mm]

10.0

Fig. 9.11. Spatial resolution eﬀects in the near wall region (averaged over 100 PIV recordings).

that a test on scale sensitivity (size and shape of interrogation window) of the velocity data and the number of outliers should be carried out in order to assess the data quality.

9.3 Transonic Flows Contributed by: M. Raﬀel, J. Kompenhans

Common problems appearing in the application of PIV at high ﬂow velocities in wind tunnels are limited optical access and problems of focusing the images of the tracer particles due to vibrations and density gradients in the ﬂow [60]. Nevertheless, the instantaneous ﬂow ﬁelds above a helicopter blade proﬁle and in the wake of a model of a cascade blade have been investigated successfully at transonic ﬂow velocities by means of the photographic PIV technique already a decade ago [310]. Today, PIV can be applied to transonic ﬂows in industrial wind tunnels, such as the DNW-TWG with a cross-section of 1 × 1 m2 , even on a routine basis. Modern model deformation measurement techniques allow for the determination of the exact model location and deformation under load in parallel with the measurement of the instantaneous ﬂow ﬁelds. Furthermore, the simultaneous application of Pressure Sensitive Paints (PSP) and PIV improves our understanding of the complex ﬂow ﬁeld, for example over delta wings in the transonic regime. Thus, PIV – in combination with PSP or deformation measurements – provides high quality data for the validation of numerical codes (see section 9.8).

272

9 Examples of Application Table 9.7. PIV recording parameters for cascade ﬂow. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material

a

M a = 1.27 parallel to light sheet Umax ≈ 400 m/s 150 × 100 mm2 2.8 × 2.8 × 1 mm3 (H × W × D) DSR ≈ 57 : 1 DVR ≈ 100 : 1 z0 ≈ 1 m single frame/double exposure image shifting/rotating mirror 35 mm ﬁlm, ASA 3200, 100 lps/mm f = 100 mm, f# = 2.8 Nd:YAG lasera 70 mJ/pulse Δt = 2 − 4 μs oil droplets (dp ≈ 1 μm)

frequency doubled

The ﬁrst two experiments were carried out in the DLR high-speed blowdown wind tunnel (HKG). Transonic ﬂow velocities are obtained by sucking air from an atmospheric intake into a large vacuum tank. A quick-acting valve, located downstream of the test section, is rapidly opened to start the ﬂow. Ambient air, which is dried before entering the test chamber, ﬂows for a maximum of 20 s through a test section with 725 mm spanwise extension. Grids in the settling chamber and a high contraction ratio lead to a low turbulence level in the test section. 9.3.1 Cascade Blade with Cooling Air Ejection The aim of this investigation was to study the eﬀect of the ejection of cooling air on the wake behind a model of a cascade blade. Due to a specially adapted wind tunnel wall above and below the model and an adjustable tailboard above the model, the ﬂow ﬁeld of a real turbine blade could be simulated in a realistic manner. The PIV recordings were taken with the photo-graphical PIV recording system utilizing the high-speed rotating mirror for image shifting at a time delay between the two laser pulses of 2−4 μs. The PIV parameters used for this investigation are listed in table 9.7. Figure 9.12 presents the instantaneous ﬂow velocity ﬁeld at the trailing edge of the plate (thickness 2 cm) for a cooling mass ﬂow rate of 1.4% at a free stream Mach number of M a = 1.27. Expansion waves and terminating shocks can be easily seen. No data were obtained in the area above the model as the laser light was blocked oﬀ by the model. Data drop-out was also found in the area directly downstream of the model. The reason is mainly that the size of the interrogation area could not be further decreased at evaluation. This would have been necessary in order to satisfactorily resolve the strong velocity gradients close to the trailing edge

9.3 Transonic Flows (u −365m/s, v +0m/s)

273

Reference Vector: 200 m/s

3

2

1

Y/d 0

−1

−2

−2

−1

0

1

2

3

4

5

4

5

6

7

X/d

3

2

1

Y/d 0

−1

−2

−2

−1

0

1

2

3

6

7

X/d

Fig. 9.12. Flow velocity (top) and vorticity ﬁeld (bottom) behind a cascade blade at M a = 1.27 and a cooling mass ﬂow rate of 1.4%.

of the model. In addition, strong density gradients in this part of the ﬂow ﬁeld caused much broader particle images. Without ejection of cooling air, the wake behind the plate can be characterized as a vortex street [310]. With ejection of air this is no longer true: two separate thin shear layers can be detected in the presentation of the instantaneous vorticity shown in ﬁgure 9.12. 9.3.2 Transonic Flow Above an Airfoil The application of PIV in high-speed ﬂows yields two additional problems: the resulting behavior of the tracer particles and the presence of strong velocity gradients. For the proper understanding of the velocity maps it is important to know how far behind a shock will the tracer particles again move with the velocity of the surrounding ﬂuid. Experience shows that a good compromise between

274

9 Examples of Application

Table 9.8. PIV recording parameters for transonic ﬂow above a NACA0012 airfoil. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material a

M a = 0.75 parallel to light sheet Umax ≈ 520 m/s 300 × 200 mm2 5.6 × 5.6 × 1 mm3 (H × W × D) DSR ≈ 57 : 1 DVR ≈ 150 : 1 z0 ≈ 1 m single frame/double exposure image shifting/rotating mirror 35 mm ﬁlm, ASA 3200, 100 lps/mm f = 100 mm, f# = 2.8 Nd:YAG lasera 70 mJ/pulse Δt = 3 μs oil droplets (dp ≈ 1 μm)

frequency doubled

particle behavior and light scattering can be found if this distance is allowed to be of the order of one or two interrogation areas. Strong velocity gradients in the ﬂow will lead to a variation of the displacement of the images of the tracer particles within the interrogation area. This inﬂuence can be reduced by application of image shifting, that is by decreasing the temporal separation between the two illumination pulses and increasing the displacement between the images of the tracer particles by image shifting to the optimum for evaluation. This is especially important if autocorrelation and optical evaluation methods are applied as in this case it is required to be able to adjust the displacement of the tracer particles to the range for optimal evaluation (i.e. ≈ 200 μm). The PIV parameters used for this investigation are listed in table 9.8. In the case of optical evaluation methods image shifting helps also to solve the problem of large variations of the displacements of the tracer particle images within the PIV recording. A successful evaluation is achieved for a range of particle image displacements of 150 μm ≤ dopt ≤ 250 μm. The upper and lower limits for this optimal particle image displacement is determined by the ﬂow to be investigated and can be adapted to the optimal range of displacement on the recording medium by applying the image shifting technique, and adding an additional shift in the direction of the mean ﬂow. Less data drop-out can be expected by this means. Strong velocity gradients are present in ﬂow ﬁelds containing shocks as are present in transonic wind tunnels. Figure 9.13 shows such an instantaneous ﬂow ﬁeld – again above a NACA 0012 airfoil with a chord length of Cl = 20 cm – at M a∞ = 0.75 [103]. By subtracting the speed of sound from all velocity vectors the supersonic ﬂow regime and the shock are clearly detectable. Due

9.3 Transonic Flows

275

Naca 0012 airfoil

Fig. 9.13. Instantaneous ﬂow ﬁeld over a NACA 0012 airfoil at α = 5◦ and M a∞ = 0.75, Ushift = 174 m/s, Cl = 20 cm.

to the application of image shifting (Ushift = 174 m/s) the requirement for the ﬂuctuations to be less or equal to the particle image diameter could be fulﬁlled with an optimum interrogation spot diameter of 0.7 mm even at the location of the shock. No data drop-out is found even in interrogation spots located in front and behind the shock (ﬂow velocities from U = 280 m/s to 520 m/s). The associated parameters for the PIV recording are presented in table 9.9. The previous two examples should demonstrate, that already a decade ago the physical problems associated with the application of PIV in transonic ﬂows could be tackled with some experimental eﬀort even for photographic recording. To date, many of these problems have been solved in a more general way, for example using frame straddling enabling pulse delays of the laser of much less than 1 μs (optimal displacement of particle images). Furthermore, sophisticated evaluation algorithms providing high local resolution even in presence of strong displacement gradients and much stronger pulse lasers providing much higher intensities allow for either larger observation areas or a smaller Table 9.9. Image recording parameters associated with the instantaneous ﬂow ﬁeld of ﬁgure 9.13. N ≈ 15

M = 1 : 6.7

without I.S. with I.S.

Umin [ m/s] 200

Umax [ m/s] 520

Δt [ μs] 5 3

Ushift [ m/s] 0 174

ΔXshift [ μm] 0 78

ΔXmin [ μm] 149 167

ΔXmax [ μm] 388 311

276

9 Examples of Application

aperture leading to sharp particle images even in case of strong vibrations of the wind tunnel and the PIV setup. 9.3.3 Shock Wave/Turbulent Boundary Layer Interaction Contributed by: F. Scarano, R.A. Humble, B.W. van Oudheusden

The interaction between an oblique shock wave and a turbulent boundary layer (SWTBLI) creates a series of complicated ﬂow phenomena, such as unsteady ﬂow separation and shock/turbulence interaction that present unique experimental challenges [307], [313] and [314]. The application of PIV in the supersonic ﬂow regime presents the speciﬁc challenges of describing the high-speed ﬂow in the presence of shock waves with suﬃcient accuracy, necessitating the quantitative evaluation of the tracer particle’s dynamic behavior [312]. The SWTBLI problem additionally requires the large velocity gradient close to the wall and the high-frequency turbulent ﬂuctuations to be resolved. The PIV parameters used for this investigation are listed in table 9.10. The particle tracer relaxation time/length is a crucial parameter dictating the spatio-temporal resolution of the measurement. It is directly evaluated by measuring the particle velocity proﬁle across a planar steady shock wave. Figure 9.14 shows the normal velocity proﬁle against the shock-normal abscissa s, and returns a particle relaxation time of τp = 2.1 μs. The Table 9.10. PIV recording parameters for shock wave/turbulent boundary layer interaction on a ﬂat plate (second set denotes parameters for boundary layer study.) Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material

parallel to light sheet Umax ≈ 500 m/s (Ma = 2.1) 124 × 39 mm2 (16 × 5 mm2 ) (W × H) 1.9 × 1.9 × 1.5 mm3 (0.7 × 0.08 × 1.5 mm3 ) DSR ≈ 136 : 1 DVR ≈ 400 : 1 z0 ≈ 600 mm (z0 ≈ 150 mm) dual frame/single exposure frame separation (frame-straddling) full frame interline transfer CCD 1376 × 1040 (432 active) pixels f = 60 mm, f# = 8 (f = 105 mm, f# = 8) Freq. doubled Nd:YAG laser, 400 mJ/pulse at 532 nm Δt = 2 μs (0.6 μs) TiO2 (dp ≈ 400 nm)

9.3 Transonic Flows

277

Fig. 9.14. Normal velocity proﬁle against shock-normal abscissa s.

corresponding frequency response is fp ≈ 0.5 MHz. The Stokes number St = τp /τﬂow (τﬂow = δ/U∞ ) expresses the ﬁdelity of particle tracers in the speciﬁc ﬂow experiment [311]. For the present case St = 0.06, yielding an RMS tracking error below 1%. The ﬂow statistical properties are evaluated on the basis of 500 PIV recordings acquired at 10 Hz in a supersonic wind tunnel of 270(H) × 280(W ) mm2 test section. The upstream mean boundary layer proﬁle (δ99 = 20 mm, Reθ = 3.36 × 104 ) scaled with inner variables is shown in ﬁgure 9.15. The experimental data agree with the composite formula down to y + ≈ 200. When the evaluation is carried out using high-aspect ratio interrogation windows (61 × 7 pixels) the wall-normal spatial resolution is improved extending the agreement to the overlap region (y + ≈ 80, y < 0.2 mm).

Fig. 9.15. Upstream mean boundary layer velocity proﬁle.

278

9 Examples of Application

Fig. 9.16. Single PIV recording with mean velocity proﬁle.

The instantaneous recording depicted in ﬁgure 9.16 shows a non-uniform seeding particle concentration due to the density variation in the ﬂow. The incident and reﬂected shock waves can be visualized by the increase in tracer particle density, whilst the boundary layer is highlighted by a comparatively lower seeding level. Laser light reﬂections from the wall were minimized during the experiment by illuminating almost tangent to the wall. An instantaneous stream-wise velocity distribution is shown in ﬁgure 9.17 (top). The incoming boundary layer has a clear intermittent nature. The global structure of the interaction is formed by the impinging shock penetrating the boundary layer, turning and weakening until it vanishes at the sonic line. The adverse pressure gradient generated by the shock causes a dilation of the subsonic layer, which causes a second compression wave system to emanate upstream of the impinging shock. The irregular shape of the separated region exhibits turbulent coherent structures, mostly originating from the separated shear layer instability. Downstream of the interaction these structures enhance the momentum mixing, which drives the boundary layer recovery. The mean ﬂow behavior is described by the averaged velocity ﬁeld in ﬁgure 9.17 (middle). The incident and reﬂected shock waves are visible as a sharp ﬂow deceleration and change of direction for the ﬁrst, whereas the reﬂected shock exhibits a smoother spatial variation of the velocity due to its unsteady nature and the averaging eﬀect. An inﬂection point prior to separation is visible in the boundary layer proﬁle. However, from the mean velocity vector proﬁles no reverse ﬂow can be inferred. After reattachment, the distorted boundary layer has approximately doubled its thickness and develops downstream with a relatively low rate of recovery. The spatial distribution of the turbulence intensity magnitude (u2 + v 2 )1/2 /U∞ is depicted in ﬁgure 9.17 (bottom) and shows the turbulent properties of the incoming boundary layer, the increased level of ﬂuctuations throughout the interaction region and its redevelopment downstream. The higher level of ﬂuctuations associated to the impinging shock (approximately 4%) is typically encountered in these experimental conditions and is ascribed to the combined eﬀect of the decreased measurement precision and

9.3 Transonic Flows

279

Fig. 9.17. Instantaneous stream-wise velocity distribution (top), mean velocity vector ﬁeld (middle) and turbulence intensity magnitude (bottom).

280

9 Examples of Application

to small ﬂuctuations of the shock position. The increased level of ﬂuctuations associated with the impinging shock penetrating the boundary layer is due to its interaction with turbulent coherent structures convected in this region. The reﬂected shock exhibits a clear unsteady behavior and relatively high levels of ﬂuctuation, which in this case should not be regarded as turbulence. Two weak features downstream of the reﬂected shock (one parallel and the other roughly perpendicular to it) are due to optical aberration eﬀects introduced by the inhomogeneous index of refraction ﬁeld of this compressible ﬂow [308].

9.4 Stereo PIV Applied to a Vortex Ring Flow Contributed by: C. Willert

The various methods of image reconstruction and calibration as described in section 7.1 were applied in the measurement of the unsteady vortex ring ﬂow ﬁeld. Figure 9.18 outlines a vortex ring generator having a simple construction with very reproducible ﬂow characteristics. The vortex ring is generated by discharging a bank of electrolytic capacitors (60 000 μF) through a pair of loudspeakers which are mounted facing inward on to two sides of a wooden box. By forcing the loudspeaker membranes inward, air is impulsively forced out of a cylindrical, sharpened nozzle (inner diameter= 34.7 mm) on the top of the box. The shear layer formed at the tip of the nozzle then rolls up into Vortex ring Nozzle

Charging Vo voltage

Feedpipe for seeding Bassloudspeaker 60W, 4Ω

TTL-triggered relais

+

-

-

+

Capacitor bank

(60000 μF)

Fig. 9.18. Schematic of the vortex ring generator used to obtain an unsteady, yet reproducible ﬂow ﬁeld.

9.4 Stereo PIV Applied to a Vortex Ring Flow

281

a vortex ring and separates from the nozzle as the membranes move back to their equilibrium positions due to the decay in supply voltage. As long as the charging voltage is kept constant, the formation of the vortex ring will be very reproducible. The generator also has a seeding pipe with a check valve allowing the interior of the box and ultimately the core of the vortex ring to be seeded. 9.4.1 Imaging Conﬁguration and Hardware A noteworthy feature of the imaging conﬁguration outlined in ﬁgure 9.19 – which has also been used for the error estimation given in section 7.1 – is that the cameras are positioned on both sides of the light sheet. This arrangement allows both cameras to make use of the much higher forward scattering properties of the small (1 μm) oil droplets used for seeding. The principal viewing axes are both around 35◦ from the light sheet normal such that the combined opening angle is approximately 70◦ near the center of the image. A pair of 100 mm, f# 2.8 objective lenses constitute the recording optics and are connected to the CCD cameras using specially built tilt-adapters (see ﬁgure 9.20). Using a pair of set screws on each of the adapters, the angle between the lens and the sensor (image plane) can be easily and precisely

Camera #2 100mm/f2.8

Vortex ring generator 1110 mm

Position 2

Vortex ring x z

Light sheet

Position 1

1100 mm

Imaged area

Camera #1 100mm/f2.8

770 mm

Fig. 9.19. Stereoscopic imaging conﬁguration in forward scattering mode for both cameras.

282

9 Examples of Application

Fig. 9.20. A specially built tilt-adapter between the lens and the sensor allows adjustment according to the Scheimpﬂug criterion.

adjusted to meet the Scheimpﬂug imaging criterion (see section 7.1). A live display of the particle images at large f -numbers permits an accurate adjustment within minutes. These ﬁrst prototype Scheimpﬂug adapters had the disadvantage that the ﬁeld of view changed when the Scheimpﬂug angle was adjusted. The following generation of adapters – now commonly available – keep the lens ﬁxed on the desired ﬁeld of view while the sensor plane (CCD) is rotated within the plane of focus. In the imaging arrangement shown in ﬁgure 9.19 the Scheimpﬂug angle (φ in ﬁgure 7.1b) was measured to be approximately 2.7◦ . The ﬁeld of view covered about 145 mm horizontally by 115 mm vertically across the center of the image. The edge loss due to the Scheimpﬂug imaging arrangement was about 5 mm vertical from side to side, but since both cameras were positioned nearly symmetrically, the ﬁeld of view could be matched very well, thereby allowing three-dimensional PIV measurements across the entire sensor area. This is an advantage over the “classical” stereoscopic arrangement in which both cameras view from the same side of the light sheet because the nonoverlapping areas are of no use in the three-dimensional reconstruction. The cameras used for this experiment are based on a full frame interline transfer CCD sensor with a 1008H by 1018V pixel resolution. The light sheet was generated by a frequency doubled, double oscillator Nd:YAG laser with more than 300 mJ per pulse. Synchronization between the cameras and the laser was achieved by means of a multiple channel sequencer. Since one of the cameras was not capable of operating in a triggered mode it provided the master timing of the entire PIV recording system. The second was operated in an asynchronously triggered mode. Two separate personal computers with interface cards captured the image pairs from the cameras at a common image pair rate of 5 Hz. (In principle the use of a common PC for both cameras would have also been possible.) One of the computers provided the trigger pulse for the vortex generator as soon as the image acquisition was started. By adding

9.4 Stereo PIV Applied to a Vortex Ring Flow

283

a time delay (or by moving the vortex generator back and forth) the position of the vortex ring within the PIV recording could be adjusted. The light sheet thickness was set at approximately 2.5 mm, while the pulse delay was varied within 300 ≤ Δt ≤ 500 μs with the vortex ring propagating in-line with the light sheet (position 1 in ﬁgure 9.19), and Δt = 200 μs while propagating normal to the light sheet (position 2 in ﬁgure 9.19). With maximum velocities of 3.5 m/s this translated to maximum displacements of 0.7 mm for the vortex ring passage normal to the light sheet. Eﬀectively, the loss of pairs was kept to less than 30% thereby ensuring a high data yield even in regions of high out-of-plane motion. The f -number was set to f# = 2.8. 9.4.2 Experimental Results Initially the nozzle of the vortex ring generator was placed collinearly with the light sheet to provide cross-sectional cuts through the vortex ring (ﬁgure 9.21). This provided reference data as well as information on the ring’s circulation and stability. In the second conﬁguration, that is, position 2 in ﬁgure 9.19, the generator was placed normal to the light sheet. Figure 9.22 shows a pair of twocomponent velocity ﬁelds prior to their combination into a three-component data set. The stereoscopic view is clearly visible. Stereoscopic reconstruction using equation (7.3), equation (7.5) and equation (7.8) then produces the desired three-component data set of ﬁgure 9.23. In terms of processing, the image back projection was chosen such that the magniﬁcation factor was constant at 10 pixel mm−1 in all images after recon-

Fig. 9.21. PIV velocity (left) and vorticity data (right) in the symmetry plane of the vortex ring. A velocity of U = 1.5 m/s, V = 0.25 m/s has been removed to enhance the visibility of the ﬂow’s features. The propagation of the ring is left to right and slightly upward (the nozzle was inclined with the horizontal). The vorticity contours are spaced in intervals of 100 s−1 excluding 0.

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9 Examples of Application

Fig. 9.22. Two-component PIV velocity data of the vortex ring propagating normal to the light sheet as viewed by camera 1 (left) and camera 2 (right).

struction. The ﬁnal image size of 1450 horizontal by 1200 vertical pixel is about 70% larger than the original images. An interrogation area of 32 × 32 pixel with an overlap (oversampling) of 66% was chosen although only every fourth vector is shown in the plots (i.e. 33% overlap). In physical space the interrogation window covers 3.2 × 3.2 mm2 while the grid spacing is 1.0 × 1.0 mm2 . The particle image density was high enough to achieve valid data rates exceeding 99% over the entire ﬁeld of view. It was also found that image preconditioning

Fig. 9.23. Reconstructed three-component PIV velocity data obtained by combining the data sets shown in ﬁgure 9.22. The out-of-plane velocity component w is shown as a contour plot on the right (contour levels at 0.25 m/s).

9.5 Dual-plane PIV Applied to a Vortex Ring Flow

285

– adaptive background subtraction using a 7×7 pixel kernel highpass ﬁlter and subsequent binarization – signiﬁcantly improved the data yield by bringing most particle images to the same intensity level. After the displacement estimation automated outlier detection found of the order of 100 outliers per 16 600 vector data set. Most of these were found on the edges of the original image domain outside of which particle images do not exist. Only very few outliers (< 10) were detected in the central 95% of the ﬁeld of view, especially in regions of high gradients. The detected outliers were then linearly re-interpolated to allow a subsequent three-dimensional reconstruction.

9.5 Dual-plane PIV Applied to a Vortex Ring Flow Contributed by: M. Raﬀel, O. Ronneberger

In the experiments described in the following we observed a low-speed vortex ring ﬂow in water. Glass spheres with a diameter of 10 μm were mixed with the water in a Plexiglas tank. The vortex rings were generated by a 30 mm piston that pushes water out of a sharp-edged cylindrical nozzle into the surrounding ﬂuid. The piston was driven by a linear traversing mechanism and a computer controlled stepper motor. The ﬂow generated by this setup is well suited for three-dimensional measurements since its properties have been documented and tested during various previous experiments [304]. As already mentioned in the previous section a vortex ring experiment oﬀers a good challenge for three-component measurement techniques, since the ﬂow ﬁeld is suﬃciently complex and reproduces reasonably. 9.5.1 Imaging Conﬁguration and Hardware Figure 9.24 shows the main components of the setup except the light sheet shaping optics and the electronic equipment. The arrangement of the optical and the electro-mechanical components are shown in ﬁgure 9.25 and are described below. An argon-ion laser produced a continuous beam of about 6 W output power. An electro-mechanical shutter controlled by a timer box generated light pulses with a pulse length of te = 5 ms and a pulse separation time of Δt = 33 ms. The shutter was phase locked with the video camera which had a frame-transfer time of tf = 2 ms. The aperture of the shutter was of a size that cuts oﬀ the outer area of the laser beam of lower intensity. A computer controlled micro stepper motor with a mirror mounted to one end of the shaft was used as a scanner, which, together with the cylindrical scanner lens (see

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9 Examples of Application

CCD-Camera

Vortex ring generator y

Light sheet (shifted) Light sheet

x z

Fig. 9.24. Sketch of the main components of the setup.

ﬁgure 9.25), generated a parallel displacement of the light sheet. An additional cylindrical lens in front of the scanner mirror focused the light on to the mirror. Thus one compensates for the scanner lens’ focusing eﬀect, which would result in a variation of the light sheet thickness. The light sheet shaping lens had a focal length small enough to generate a light sheet height that was twice as large as the height of the observation ﬁeld. This minimized the variation in light intensity with respect to the observed ﬁeld. The scanner was phased locked to the video signal of the recording camera and alternated the light sheet location after each second capture of a complete video frame (see ﬁgure 9.26). Synchronized with the motion of the piston three subsequent video frames were captured. Two frames contain images of tracer particles within the same light sheet oriented perpendicular to the vortex ring axis (intensity ﬁelds I and I captured at t = t0 and t = t0 +Δt, respectively). The third frame contains images of tracer particles within a light sheet parallel to the ﬁrst one (I captured at t = t0 + 2Δt). The shift of the light sheet was (Z − Z ) = 2.5 mm resulting in an overlap of OZ = 17% of the light sheet thickness (ΔZ0 = 3 mm).

Scanner Laser

Shutter Cyl.lens Cyl. scanner lens

First light sheet position Second light sheet position

Sheet shaping cyl. lens Light sheet thickness ΔZ 0

Fig. 9.25. Sketch of the optical components.

9.5 Dual-plane PIV Applied to a Vortex Ring Flow

287

tf

Z

Frame transfer time t f

2 ΔZ0 te ΔZ0

f

f’’

Exposure time t e

f’

0

Z’’ - Z’ 0

Δt

2 Δt

t

Fig. 9.26. Timing diagram of image capture and light sheet position.

9.5.2 Experimental Results In order to obtain more information about the ﬂow ﬁeld generated by the setup described above, we ﬁrst acquired PIV data along the centerline of the vortex ring (see ﬁgure 9.27). The axial components of the velocity vectors along the indicated line give information on the out-of-plane velocity component we had to expect when observing the ﬂow ﬁeld in a plane perpendicular to the vortex ring axis. The magnitude of this velocity component parallel to the axis is plotted in ﬁgure 9.28. Following the described method, we then captured images of particles within two parallel light sheets on to three diﬀerent frames. Both light sheetplanes were orientated perpendicular to the vortex ring axis as shown

Fig. 9.27. Flow ﬁeld in an intersection on the vortex ring axis.

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9 Examples of Application

Axial velocity component (cm/s)

10 8 6 4 2 0 -2 0

100

200 300 y-position (pixels)

400

Fig. 9.28. Velocity component parallel to the vortex ring axis along the line shown in ﬁgure 9.27.

in ﬁgure 9.24. The frames were evaluated by correlating interrogation windows I with I and I with I , detecting the location of the stronger peak, and storing the normalized intensities of both correlation planes at location d for each interrogation area. The size of the interrogation windows was 32 × 32 pixels and the interrogation step-width in both the x and y directions was 16 pixels. The results of the evaluation of the interrogation windows I and I containing images of particles within the same light sheet show outliers in a ring close to the center of the ﬂow ﬁeld (see ﬁgure 9.29). This area of low detection probability is caused by the decreased seeding density near the center of the vortex ring and by the strong out-of-plane motion in the center of the observed ﬁeld. The normalized heights of the tallest peaks in the cross-correlation planes RII’ are shown in ﬁgure 9.30. They clearly show the inﬂuence of the out-ofplane velocity component (i.e. low correlation peak heights in the center of the vortex ring). The results of the evaluation of the interrogation areas I and I show outliers in a ring further outward (see ﬁgure 9.31 ). The normalized heights of the correlation planes RI I are shown in ﬁgure 9.32. In this case, out-of-plane velocity components increase the correlation peak heights. The following evaluation procedure was used to take advantage of the images captured in diﬀerent planes. The correlations RII between the interrogation windows I and I and the correlations RI I between the interrogation windows I and I were computed and normalized in order to obtain cII and cI I . The distribution containing the highest peak of the interrogated cell was then used to determine the particle image displacement. This procedure reduces the number of outliers (see ﬁgure 9.33) and therefore shows that compared to conventional PIV a larger out-of-plane

9.5 Dual-plane PIV Applied to a Vortex Ring Flow Reference vector:

289

5 cm/sec

30

25

y-position [interrogation steps]

20

15

10

5

0 0

5

10

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25

30

35

40

45

x-position [interrogation steps]

Fig. 9.29. Velocity vector map obtained by images of particles illuminated by the same light sheet.

component can be tolerated for identical Δt. The peak positions found by this procedure were used to ﬁnd the correct and identical locations in both cross-correlation planes for intensity analysis. Figure 9.34 shows the plot of the out-of-plane velocity distribution computed from the intensities found in

Fig. 9.30. Cross-correlation coeﬃcients cII for images of particles illuminated by the same light sheet (smoothed by a spatial averaging (3 × 3 kernel) for this representation).

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9 Examples of Application pixel Reference vector: 514cm/sec

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25

y-position [interrogation steps]

20

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5

0 0

5

10

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20

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x-position [interrogation steps]

Fig. 9.31. Velocity vector map obtained by images of particles illuminated by the two parallel light sheets.

the procedure described in section 7.2 and according to equation (7.30). In contrast to the results obtained by evaluating only two frames (see ﬁgure 9.30 and ﬁgure 9.32) the expected structures of the ﬂow can now clearly be seen in ﬁgure 9.34.

Fig. 9.32. Cross-correlation coeﬃcients cI I for images of particles illuminated by the two parallel light sheets (smoothed by a spatial averaging (3 × 3 kernel) for this representation).

9.5 Dual-plane PIV Applied to a Vortex Ring Flow

291

Fig. 9.33. In-plane velocity vector map obtained by considering the strongest peak of both correlations.

The maximum value of the out-of-plane velocity obtained by the dualplane correlation technique, of 2.9 mm/33 ms = 8.79 cm/s is in good correspondence with the maximum shown in ﬁgure 9.28. Minimum values of the measured velocity distribution are approximately zero in both cases. The ﬁnal result is shown in ﬁgure 9.35 in a three-dimensional representation.

Fig. 9.34. Out-ofplane velocity distribution obtained by analyzing the results of the correlation coeﬃcients cII and cI I for each interrogation cell (smoothed by a spatial averaging (3 × 3 kernel) for this representation).

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9 Examples of Application

Fig. 9.35. Threedimensional representation of the velocity vectors of the observed plane (raw data without any smoothing, data validation or interpolation).

9.6 Large Scale Rayleigh-B´ enard Convection Contributed by: J. Bosbach, C. Wagner, C. Resagk, R. du Puits, and A. Thess

Table 9.11. PIV recording parameters for large scale Rayleigh-B´enard convection. Flow geometry Maximum velocity Field of view Interrogation volume Observation distance Recording method Recording medium Recording lens Illumination Pulse delay Seeding material

central crosscut, one half of the convection cell 0.4 m/s 12.3 m2 96 × 96 × 40 mm3 3.4 m - 5.9 m double frame/single exposure CCD-camera f = 8 mm, f# = 1.3 Nd:YAG laser 160 mJ/pulse Δt = 30 ms helium-ﬁlled soap bubbles (dp ≈ 1 . . . 3 mm)

9.6.1 Introduction The application of PIV to measurement of ﬂows on large scales is a challenging necessity especially for investigations of convective air ﬂows. By combination of helium ﬁlled soap bubbles as tracer particles with high power quality switched solid state lasers as light sources stereoscopic PIV on scales of more than 10 m2 becomes possible. The technique has been applied to nat-

9.6 Large Scale Rayleigh-B´enard Convection

293

ural convection in a large scale Rayleigh-B´enard facility for the detection of characteristic convection patterns as described in the following. Turbulent convective air ﬂows are essential not only for many technical applications like for example climatization of vehicle compartments [316, 317, 319, 320] or residential buildings, but also for the warming of the atmosphere and mixing of the oceans [318]. Especially in combination with forced convection, such ﬂows are usually diﬃcult to scale. Therefore large scale PIV at full model size is highly desirable. 9.6.2 Stereo PIV in the Barrel of Ilmenau One of the best-known experiments to investigate natural thermal convection is the Rayleigh-B´enard experiment, where a ﬂuid is heated at the bottom and cooled at the ceiling of its conﬁnements. Stereoscopic PIV measurements of large-scale Rayleigh-B´enard convection were conducted in the Barrel of Ilmenau, a large scale Rayleigh-B´enard facility. The Barrel of Ilmenau is a cylindrical container with an inner diameter of D = 7.15 m and a variable height H of 0 m < H < 6.3 m. It is equipped with a heating plate at the bottom and a cooling plate at the top and nearly adiabatic side walls. As a ﬂuid, ambient air is used. The heating plate at the bottom consists of a heating wire in a concrete layer similar to an electrical under ﬂoor heating. In order to minimize heat losses through the bottom a thermal isolation layer with a thickness of 300 mm is placed directly below the heating layer. The surface of the heating plate is coated with aluminum foil in order to prevent radiation exchange with the cooling plate and the sidewalls. The maximum surface temperature which can be realized at the ﬂoor amounts to 75◦ C. The cooling plate consists of 16 separate water-cooled segments. They are made of two aluminum-plates with an interconnecting cooling coil and measure 40 mm in thickness. Together with a cooling system and a big water reservoir an accurate regularity of the temperature at the surface of the cooling plate was realized. The maximum temperature deviation over the whole surface is less than 1 K. The inner sidewall is made of a ﬁberglass-epoxy compound with an embedded thermal isolation layer with a thickness of 160 mm. In order to minimize heat losses it is covered with a heating system for active compensation of heat losses and a further isolation with a thickness of 140 mm. The measurements discussed in the following were conducted at a Rayleigh number of Ra = 1.2 · 1011 . The ﬂoor temperature was set to 60◦ C and the ceiling temperature amounted to 20◦ C. The aspect ratio of the cell was set to D/H = 2. The neutrally buoyant helium ﬁlled soap bubbles used as tracer particles for the PIV measurements were generated with two bubble generators and had diameters between 1 and 3 mm. They were injected into the cell through two small holes in the ceiling. For illumination of the tracer particles a quality switched air-cooled double cavity Nd:YAG laser with a pulse energy of 160 mJ was used. The particles were detected with two CCD cameras (Sensicam QE,

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9 Examples of Application

Fig. 9.36. Sketch of the stereo setup for PIV of natural convection in the Barrel of Ilmenau, top view.

PCO) with a Peltier-cooled CCD-chip and a resolution of 1376 × 1040 pixel in a stereo PIV setup, see ﬁgure 9.36 and 9.37. The double frames were recorded at a repetition rate of 2.5 Hz. For evaluation of the velocity vector ﬁelds, interrogation windows of 48 × 48 pixel were used leading to an interrogation window size of 96 × 96 mm2 . The thickness of the light sheet is thus of the order of the interrogation window size as it is appropriate for highly 3D convective air ﬂow. Multiple pass interrogation was used for calculation of the correlation between the interrogation windows of subsequent images and the correlation maximum was determined with sub pixel accuracy by a 3-point Gauss ﬁt to the correlation peak. In order to further reduce noise, double

Fig. 9.37. Sketch of the stereo setup for PIV of natural convection in the Barrel of Ilmenau, side view.

9.6 Large Scale Rayleigh-B´enard Convection

295

correlation was applied. The interrogation window overlap was set to 50%. Since thermal convection in a Rayleigh-B´enard conﬁguration is very sensitive to deteriorations in the symmetry, the optical access to the Barrel of Ilmenau is kept quite limited. In order to couple the laser light into the convection cell, a small viewport has been drilled into the wall and covered with the cylindrical lens, which was used in order to generate a light sheet, see ﬁgure 9.36 and 9.37. The thickness of the light sheet was adjusted with a telescope to amount between 3 and 5 cm. Due to the optical restrictions, as already mentioned, the CCD cameras had to be mounted inside of the convection cell. In order to cover one half of the convection cell lenses with a focal length as small as f = 8 mm have been used. The angle between the cameras was set to 88◦ , see ﬁgure 9.36, the cameras looking in a backward scattering conﬁguration on the measurement plane, which covered slightly more than one half of the Rayleigh-B´enard cell, see ﬁgure 9.37. Here the very high backward scattering eﬃciency of the helium ﬁlled soap bubbles comes into play handy. The overall FOV in this measurement was as large as 12.3 m2 . Usually in a stereo PIV setup, the CCD chips being tilted with respect to the light sheet plane, a Scheimpﬂug adapter has to be used in order to focus on the particles in the whole image plane. Estimating the depth of ﬁeld δz for our conﬁguration, however, with a magniﬁcation of M = 1.5 × 10−3 , a particle diameter of dp = 1 mm, and a F-number of f# = 1.3 yields δz = 2.6 m. Consequently in this setup there was no need to use a Scheimpﬂug adapter in order to focus on the particles. Since at aspect ratio D/H = 2 a transition of a steady two dimensional ﬂow structure like a single roll into a three dimensional structure of several rolls or toroids with varying directions is expected to occur, the structures changing periodically [321, 322], the aim of our measurements was to capture such structures with our particle images. In order to identify these typical ﬂow structures instantaneous 3C velocity maps as well as averages of 114 velocity ﬁelds have been evaluated. As an example ﬁgure 9.38 and ﬁgure 9.39 depict the instantaneous and averaged velocity ﬁeld, respectively, of a single convection roll in the Barrel of Ilmenau. It can be seen clearly that the ﬂuid moves to the right at the bottom towards the barrel wall where it rises up in order to move back to the barrel center at the ceiling. Comparing the homogeneous ﬂow structures in the time averaged vector plots to the instantaneous velocity map reveals thermal plumes, that is, spatially varying local jets from the hot bottom. In addition to toroidal structures, which are dominated by velocity components in the light sheet plane, structures with dominating out of plane velocity components have been detected as well [315]. 9.6.3 Conclusions Combination of neutrally buoyant helium ﬁlled soap bubbles as tracer particles with nanosecond laser pulses as light source in a PIV experiment has

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9 Examples of Application

3500

3000

z [mm]

2500

2000

1500

1000

500

0 0

0.4 m/s

1000

2000

3000

x [mm]

Fig. 9.38. Instantaneous velocity map of a convection roll in the Barrel of Ilmenau at aspect ratio D/H = 2. The barrel center is located at x = 0. Spatial velocity ﬂuctuations near the hot bottom indicate the existence of thermal plumes. 3500

3000

z [mm]

2500

2000

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500

0 0 0.3 m/s

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2000

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x [mm]

Fig. 9.39. Average of 114 velocity ﬁelds of a convection roll in the Barrel of Ilmenau at aspect ratio D/H = 2. The barrel center is located at x = 0.

9.7 Analysis of PIV Image Sequences

297

been demonstrated to be a very promising approach for measurement of air ﬂow on large scales. The feasibility of the method has been demonstrated by stereoscopic PIV of natural convection in a large scale Rayleigh-B´enard experiment on a ﬁeld of view as large as 12.3 m2 . In addition to the exceptionally large scattering eﬃciency the high backward scattering eﬃciency of the tracer particles comes into play as an advantage of the helium ﬁlled soap bubbles in the presented stereo setup. Measurement of natural convection in the Barrel of Ilmenau by large scale PIV made possible identiﬁcation of characteristic ﬂow patterns in RB convection. At aspect ratio D/H = 2, as expected, diﬀerent kinds of ﬂow structures could be detected. Along with toroidal structures like single or dual convection rolls, structures with dominating out of plane velocity components have been observed as well. For veriﬁcation or falsiﬁcation of model predictions of Rayleigh-B´enard convection in the Barrel of Ilmenau by PIV, however, the cameras should be installed preferably inside the insulating walls in order neither to deteriorate the ﬂow nor to break the thermal shielding of the convection cell. Even though discussed for application to basic research facilities here, large scale PIV involving helium ﬁlled soap bubbles could be a mighty tool for many technical applications involving air ﬂow on large scales as well, like e.g. ventilation of cars, trains, aircraft cabins [315], clean rooms, or even wind tunnel testing.

9.7 Analysis of PIV Image Sequences Contributed by: R. Hain, C.J. K¨ ahler 9.7.1 Introduction The principle drawback of all conventional double pulse PIV systems is the missing temporal ﬂow information and the relatively low measurement precision in regions where the particle image displacement is small [79]. While the ﬁrst problem can be solved nowadays by using fast CMOS cameras in combination with high repetition rate diode-pumped lasers the second problem requires a completely new evaluation and recording approach because the particle image displacement has to be adapted locally to reduce the relative measurement error. ¨hler & For this purpose an evaluation strategy was proposed by Ka Kompenhans [200] that allows to increase the dynamic range, and thus the measurement precision, by taking into account the temporal ﬂux of information. By using this evaluation approach it becomes possible to study ﬂows with a very large dynamic velocity range such as laminar separation bubbles (see ﬁg. 9.40) or boundary layer ﬂows, according to [324]. The ﬂow velocity inside a laminar separation bubble for instance is typically 40-times

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9 Examples of Application

Table 9.12. Recording parameters for time–resolved PIV investigations on a SD7003 airfoil. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Recording method Recording medium Recording lens Illumination Recording frequency Seeding material

parallel to light sheet 0.1 m/s 27.35 × 21.9 mm2 0.68 × 0.68 × 0.5 mm3 ≈ 40 ≈ 1000 (Multiframe evaluation) single frame / single exposure, equidistant time intervals CMOS-camera (pco.1200 hs) f = 108 mm, f# = 2.8 cw Ar+ laser (8 W) f = 636 Hz Glass hollow spheres (dp ≈ 30 μm)

smaller than the outer ﬂow velocity. The principle idea behind the so called Multiframe PIV evaluation is the local optimization of the particle image displacement, see [324]. This is achieved by calculating the local displacement with second order accuracy in space and time whereby the temporal separation between a pair of image-templates (interrogation windows) is the deciding optimization parameter. This requires a properly sampled ﬂow ﬁeld. In order to obtain the largest possible particle image separation and thus measurement accuracy a conventional multi-pass interrogation with standard window-shifting and window-deformation technique is performed at ﬁrst to estimate the local displacement with standard accuracy. In the next iteration this displacement ﬁeld determines the temporal separation between the interrogation windows for each vector position by using properly selected criteria such as the desired particle image displacement, the signal strength and the signal to noise ratio whereby gradient, curvature and acceleration eﬀects have to be taken into account [323, 324]. In a next pass the evaluation is repeated at each location with the optimized temporal separation whereby the spatial

Fig. 9.40. Laminar separation bubble (averaged velocity ﬁeld).

9.7 Analysis of PIV Image Sequences

299

templates required for the calculation of the correlation are selected symmetrically around the given time instant. As the temporal separation aﬀects the signal strength of the correlation peak, because of out-of-plane loss-of-pairs and gradient eﬀects for example, the desired dynamic range is automatically reduced when the validation criteria are not satisﬁed or increased when the accuracy can be further enhanced. This approach results ﬁnally in an optimized displacement ﬁeld with the best possible accuracy according to predetermined criteria. After this evaluation at time t the procedure is performed at times t+ n·Δt whereby n has to be selected such that the temporal ﬂow phenomena are properly sampled. 9.7.2 Evaluation of a Simulated PIV Image Sequence To demonstrate the capability of the analysis method quantitatively a synthetic image sequence was generated [323] based on the solution of a direct numerical simulation (DNS) of a laminar separation bubble with transition and turbulent reattachment [325]. The strong diﬀerences between the ﬂow velocities inside and outside the bubble can clearly be seen in ﬁgure 9.41. Figure 9.42 shows the relative deviations to the exact solution of the DNS 1000

y [px]

|U| [m/s] 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

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Fig. 9.41. Numerical simulation. Top: Velocities inside and outside the laminar separation bubble (only every 4th vector displayed). Bottom: Enlargement of lower left corner (every vector displayed, vector length 30 times higher than in the upper ﬁgure).

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9 Examples of Application

Fig. 9.42. Standard and multi-frame PIV evaluation based on DNS data. Top: Relative vector lengths deviations of the conventional evaluation to the exact solution. Bottom: Relative vector lengths deviations of the multiframe PIV evaluation to the exact solution.

for the conventional evaluation (top) and the multiframe evaluation approach (bottom). In case of the conventional evaluation the measurement error exceeds up to 40% inside the bubble and in the transition region. This error becomes about 15-times smaller on average when the multiframe evaluation approach is applied. Only a few vectors could not be improved because of the strong out-of-plane motions in this regions (total loss of information). 9.7.3 Investigation of Separation on a SD7003 Airfoil To demonstrate the performance of the evaluation strategy experimentally, the laminar separation on the suction side of a SD7003 airfoil at a Reynoldsnumber of Re = 2 × 104 was examined in the water-tunnel of the Institute of Fluid Mechanics of the Technical University of Braunschweig, see ﬁgure 9.43 (top). The PIV recording parameters for this application are given in table 9.12. Due to the multi-frame evaluation method an extremely large dynamic velocity range of about DVR ≈ 1000 could be realized! The velocity inside the separation bubble is about 50-times smaller than outside of it and it can be seen in ﬁgure 9.43 (bottom) that the multi-frame evaluation works very well even in practical situations.

9.8 Velocity and Pressure Maps Above a Transonic Delta Wing

301

Fig. 9.43. Top: Airfoil SD7003. Bottom: Flow ﬁeld in the ﬁeld-of-view as derived from multi-frame evaluation of PIV data.

These results demonstrate clearly the enhancement of the dynamic velocity range, and thus the measurement accuracy, that can be gained by using the multiframe PIV evaluation strategy. For this reason this method is in particular suited for the validation of numerical ﬂow simulations and turbulence models because in that case a very high precision is required to decide between computational concepts. For further details see [323].

9.8 Velocity and Pressure Maps Above a Transonic Delta Wing Contributed by: R. Konrath, C. Klein

The parallel application of experimental diagnostics methods allows a more complete description of complex ﬂow phenomena and provides high quality data for validation of numerical calculations. As an example the parallel application of the Pressure Sensitive Paint (PSP) technique and PIV shall be presented. Such investigations have been performed in the frame of the Inter-

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9 Examples of Application

Table 9.13. PIV recording parameters for combined velocity and pressure measurements on a delta wing. Flow condition Maximum in-plane velocity Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Recording medium Recording lens Illumination Pulse delay Seeding material a

M a = 0.4/ReM AC = 3 · 106 Umax = 200 m/s 146 × 60 × 294 mm−3 (H × W × D) DSR ≈ 30 DVR ≈ 80 z0 = 0.6 m multi frame, single exposure PCO SensiCam QE, 1376 × 1024 pixel f = 60.0, f# = 5.6 Nd:YAG lasera , 300 mJ/pulse Δt = 4 μs DEHS droplets (dp = 1μ)

frequency doubled

national Vortex Flow Experiment 2 (VFE-2), where wind tunnel tests were carried out on a 65◦ delta wing at sub- and transonic speeds applying both techniques. Since 2003 the VFE-2 is being carried out within the framework of the task group AVT-113 of RTO (NATO’s Research and Technology Organization). The objectives of this working group are to perform new wind-tunnel tests on a delta wing by using modern measurement techniques and to compare these data with results of numerical state-of-the-art codes [326]. For the tests, the delta wing model, provided by NASA Langley, was equipped with sharp as well as with rounded leading edges. With PSP the pressure distributions on the model surface measurements were determined, which serve as “pathﬁnder” tests. Their results gave ﬁrst information of the ﬂow topology over the delta-wing for a large range of angles of attack. The PIV measurements were performed in a second test campaign for which speciﬁc angles of attack and locations of the measurement planes above the delta wing were selected on the basis of a ﬁrst analysis of the PSP results. The measured velocity ﬁelds provide detailed information of the instantaneous and time averaged ﬂow ﬁelds. The stereo-PIV setup (see ﬁgure 9.44) allows for ﬂow velocity measurements above the delta wing within planes perpendicular to the model axis at diﬀerent chord stations. The light-sheet and the cameras can be translated along the model axis during wind tunnel operation. The arrangement also incorporates rotary plates in order to adjust quickly the setup for diﬀerent angles of attack. For the current delta wing conﬁguration, a speciﬁc ﬂow topology occurs in the rounded leading edge case. In addition to the well known outer primary vortex another inner primary vortex develops which was ﬁrst evidenced within the VFE-2 group for M = 0.4, RMAC = 3 · 106 (w.r.t. mean aerodynamic chord) and an angle of attack of 13◦ by a ﬂow computation of FRITZ (EADS-Munich, see [326]). This computation was invoked by the PSP results

9.8 Velocity and Pressure Maps Above a Transonic Delta Wing X-Y translation stage

303

Rotary plate Camera 2

Camera 1 Mirror

Window Delta wing

U? Light sheet Test section

Window

Linear translation stage Rotary plate Plenum, p0

Light sheet optics

Rotation axis of model

Light arm Laser beam

Fig. 9.44. Stereoscopic PIV arrangement inside the perforated test section of the transonic wind tunnel DNW-TWG showing the coated delta-wing and light sheet.

and used the measured pressure distributions to set up simulation parameters. The PIV results obtained later agree with the computed ﬂow topology. The ﬂow topology can be seen in ﬁgure 9.45 showing the measured pressured distribution on the model surface together with the measured velocity and vorticity distributions in planes of the diﬀerent chord stations. In this case the ﬂow separates at the leading edge ﬁrst at x/cr = 0.5 and the primary vortex is formed, which produces a strong suction peak in the pressure distributions. However, another weaker suction peak can be detected more inboard with a highest peak height just downstream the origin of the outer primary vortex. The velocity distributions at x/cr = 0.6 reveal that this suction peak is produced by another inner vortex co-rotating to the outer one. This vortex develops from a thin vortex structure which occurs more upstream close to the surface, i.e. x/cr = 0.4. Instantaneous PIV results show [327] that this vortex structure consists of several small co-rotating vortices spreading in the spanwise direction. Between the outer primary vortex and the inner vortical structure the ﬂow re-attaches to the surface and separates again so that vortices of the inner vortical structure detach from the surface, i.e. x/cr = 0.5. These vortices merge and a circular inner vortex is formed such that two corotating vortices of approximately the same size can be observed at x/cr = 0.6. Further downstream vorticity is fed only into the outer primary vortex and the strength of the inner vortex gradually decreases, whereas the inner and the outer vortex remain separated and do not merge.

304

9 Examples of Application

Fig. 9.45. Time averaged pressure, velocity and vorticity distributions above the delta wing with rounded leading edges for α = 13.3◦ , M a = 0.4 and RM AC = 3 · 106 . The in-plane velocity vectors are plotted in diﬀerent planes perpendicular to the delta wing axis. The gray levels of the vectors correspond to the out-of-plane vorticity. The gray levels at the surface are related to the local pressure coeﬃcient.

9.9 Coherent Structure Detection in a Backward-Facing Step Flow Contributed by: C. Schram, P. Rambaud, M. L. Riethmuller

9.9.1 Introduction An algorithm has been developed to automatically detect and characterize coherent structures from a general two-dimensional, two-components (2D-2C)

9.9 Coherent Structure Detection in a Backward-Facing Step Flow

305

velocity ﬁeld. The method is based on the continuous wavelet transform [329], whose characteristic selectivity in both space and frequency domains allows to automatically locate and determine the core size of the coherent structures contained in the velocity ﬁeld. Coherent structures are usually regarded as patches of vortical (swirling) ﬂuid that exhibits some coherence over a certain spatial extent and lifetime, and are individually more energetic than the background turbulence ([329, 331, 335, 336]). The present method has been developed to take advantage of the enstrophy (squared vorticity) ﬁelds calculated from the PIV data. 9.9.2 Vortex Detection Algorithm The robustness of the detection algorithm is a key issue in the present application. It should be immune to contaminations of the vortical ﬁeld by experimental noise, and discriminate between shearing and swirling motion. The ﬁltering properties of the continuous wavelet transform are combined with a topological criterion to achieve the highest robustness. We assume that the distribution of vorticity over a cross-sectional area of a vortex can be approximated by a Gaussian distribution (Oseen vortex). The two-dimensional Marr wavelet, also called Mexican Hat wavelet, has accordingly been selected as a mother wavelet: , (9.1) Ψ (x, y) = 2 − x2 − y 2 exp −(x2 + y 2 )/2 The vortex detection algorithm consists in ﬁnding the local maxima of the continuous wavelet transform of the instantaneous squared vorticity ﬁeld ω 2 (x, y) or enstrophy : 12 (l, x , y ) = ω ω 2 (x, y) Ψl,x ,y (x, y) dxdy (9.2) R2

where Ψl,x ,y (x, y) is a daughter wavelet obtained by translation and proper scaling of the mother wavelet (9.1). The position of the local maxima in 12 (l, x , y ) map yields the location the three-dimensional wavelet transform ω (x , y ) and the corresponding scale l of the candidate vortices. The proportionality factor between the wavelet scale l and the vortex core diameter σc has been obtained analytically in [339]. Shearing and swirling motion A topological criterion is used at each position in order to assess whether the ﬂow pattern is dominated by shearing or swirling motion [272, 328, 330]. The two-dimensional simpliﬁcation of the criterion based on the the λ2 -parameter is used [332]: 2 (9.3) λ2 = (∂u/∂x) + (∂v/∂x) (∂u/∂y) .

306

9 Examples of Application

λ2 has a negative value in regions of swirling ﬂuid motion. Please note that the expression (9.3) is only relevant if the ﬂow is locally two-dimensional and aligned with the measurement plane. The λ2 < 0 criterion might otherwise erroneously indicate the presence of a vortex (or its absence) because of the missing components of the full velocity gradient tensor. When no assumption can be made on the two-dimensionality of the ﬂow, another criterion is used to disregard shear layers. It is based on the isotropy of the candidate vortex. A coeﬃcient translating the correlation between the candidate vortex pattern and an isotropic Gaussian pattern is calculated: α(l, x , y ) ≡

12 (l, x , y )max ω 12 (l, x , y )th ω

(9.4)

12 (l, x , y )max is the maximum wavelet coeﬃcient at the location of the where ω 12 (l, x , y )th is the theoretical value that the wavelet candidate vortex, and ω transform would have if the detected structure was an Oseen vortex with a core radius corresponding to the wavelet scale l. To summarize, the vortex detection algorithm accepts the vortical struc12 (l, x , y ), tures that correspond to a local maximum of the wavelet transform ω and for which the conditions 12 (l, x , y ) ≥ ω 12 (l, x , y )thresh ω

;

λ2 (x , y ) ≤ λ2,thresh

; α(l, x , y ) ≥ αthresh

are simultaneously satisﬁed. 9.9.3 Application to the Backward-Facing Step Flow Experimental setup and measurements The experimental investigation is performed in a low-speed wind tunnel sketched in ﬁgure 9.46(a). The test section is divided in two parts by a horizontal Plexiglas plate, forming a ∼ (0.2 × 0.08) m2 duct section upstream of the step (ﬁgure 9.46(a)). The step height is h = 0.02 m. The expansion ratio,

4

1

y/h

3

2

3

2

4

1 0 -1

5 0

1

2

3

4

5

6

7

8

x/h

(a) Test section.

(b) Measurement zones.

Fig. 9.46. Backward-facing step experimental conﬁguration.

9

9.9 Coherent Structure Detection in a Backward-Facing Step Flow

307

deﬁned as the ratio of the test section height past the step Ly to the height upstream, is equal to 1.25. The value of the free stream inlet velocity, upstream of the step is U0 = 3.8 m/s, giving a Reynolds number Reh = U0 h/ν 5100. A double pulsed Nd:YAG laser providing up to 200 mJ per pulse (pulse duration 5 ns) is used to illuminate a thin vertical sheet of the ﬂow in the mid-plane of the test section. The ﬂow is seeded at the entrance of the centrifugal blower by 1 μm oil particles. The particle images are recorded using a 12-bit fan-cooled digital PCO camera, having a full frame resolution of 1280 × 1024 pixel2 . A 50 mm Nikon objective was used, with an f-number of f# = 4. Series of 32 pairs of images are grabbed by an acquisition board mounted on a PC with the PCO software at a frame rate of 4.1 Hz. Five campaigns of PIV measurements have been performed, covering the ﬁeld of view indicated in ﬁgure 9.46(b). The zones 1, 2 and 3 have been considered for the vortex detection. For each series, 1024 pairs of images have been acquired and processed using Window Distortion Iterative Multigrid (WIDIM), a PIV cross-correlation software developed at the von Karman Institute for Fluid Dynamics. Details on this software are given in [337]. Velocity ﬁelds containing 212 × 169 vectors have been obtained using a 50% window overlap. The spatial resolution is h/50 = 0.4 mm. Statistical results The mean velocity ﬁeld shown in ﬁgure 9.47 is obtained by ensemble-averaging the 1024 ﬁelds acquired one for each of the zones 1, 2 and 3. Mean velocity proﬁles, turbulence intensities and Reynolds stresses have been extracted in [340] and show an excellent agreement with DNS data obtained by Le et al. in [334]. Vortex detection Figure 9.49 shows a typical result of the vortex detection procedure. The circles superimposed on the vorticity ﬁeld indicate the dimension of the vortex

3 U0

y/h

2

1

0

0

1

2

3

4

5

6

7

x/h

Fig. 9.47. Ensemble-averaged velocity ﬁeld.

8

9

308

9 Examples of Application

Fig. 9.48. Turbulence spectrum calculated on the basis of the detected vortices characteristics.

core at each detection location. The thumbnails illustrate the swirling motion by subtracting from the velocity ﬁeld the average velocity computed over the core region. The core circulation Γc and diameter Dc of the detected coherent structures are combined on ﬁgure 9.48 to mimic the classical power spectrum representation in the Fourier domain. The normalized energy based on the circulation Γc is plotted versus a normalized wave number built as the inverse of the diameter Dc . The averaged spectrum represented by the line in ﬁgure 9.48 shows a slope about −3. This slope has also been obtained by Le & Moin [333] for the spectrum of the streamwise velocity component Euu at the position (x = 5 h ; y = 2.03 h). Further statistical results on the coherent structures of this ﬂow have been reported in [338]. It has been observed that these results are fairly robust with respect to a variation of the thresholds. 9.9.4 Conclusions A coherent structure detection algorithm has been developed, based on the properties of selectivity in space and scale of the continuous wavelet transform. For application to experimental data, a special attention was given to the robustness of the procedure. Moreover, with regard to the large amount of data that is often needed to obtain statistical convergence, the analysis has to be automatized. The statistical properties of the coherent structures have been derived from the results obtained with the detection algorithm. A pseudoturbulence energy spectrum has been generated on the basis of the detected vortex sizes and circulations. The decay of this spectrum has a slope similar to numerical results previously obtained in the literature.

9.9 Coherent Structure Detection in a Backward-Facing Step Flow

309

Fig. 9.49. Example of vortex detection using thresholds ωthresh h/U0 = 2.695, λ2,thresh h2 /U02 = −4.543 and αthresh = 0.5. The circles indicate the size of the vortex cores. The detailed velocity ﬁelds that are shown for some vortices are obtained by subtraction of the average velocity over the vortex core from the instantaneous velocity ﬁeld.

310

9 Examples of Application

9.10 Quantitative Study of Vortex Pairing in a Circular Air Jet Contributed by: C. Schram, M.L. Riethmuller

9.10.1 Introduction The purpose of the present work is the prediction of the aerodynamic sound produced by vortex pairing in a low Mach number circular air jet. The acoustical far ﬁeld is obtained using an aeroacoustical analogy known as Vortex Sound Theory, on the basis of a description of the ﬂow ﬁeld provided through PIV. The use of experimental data as input of an aeroacoustical prediction method constitutes the originality of the present work. The accuracy of the measurements is of crucial importance due to the high sensitivity of the aeroacoustic prediction with respect to errors in the ﬂow data [342, 345]. A synchronization technique has been devised in order to reconstruct a pseudo-temporal evolution of the vortex pairing using a conventional, low frequency PIV acquisition system. The technique involves the excitation of the jet instability in order to obtain a repetitive vortex pairing stabilized in space and phase. The acquisition of the PIV series is synchronized with respect to the excitation signal in such a way that pseudo-time resolved series are obtained [346]. The remainder of this section is focused on the technical details of our PIV measurements and the assessment of the random and bias errors. The developments related to the aeroacoustical theory are given in full length in [345, 347], only the ﬁnal comparison between the PIV-based sound prediction and acoustical measurements is given in section 9.10.7. 9.10.2 Acoustically Excited Jet Facility The experimental facility is sketched in ﬁgure 9.50. Air enters into the nozzle pipe through a honeycomb and turbulence is further reduced by grids placed just upstream of the nozzle contraction. The boundary layer is laminar at the downstream end of the contraction. The outlet diameter is D = 0.041 m, and the jet discharges into an anechoic room with a velocity U0 = 34.2 m/s. The walls of the settling chamber are covered with a 10 cm thick layer of acoustical absorbing foam to reduce acoustical resonances of the settling chamber. The set of images in ﬁgure 9.51 shows one pairing sequence obtained by smoke visualization. The leading and trailing vortex rings prior to pairing are respectively indicated by L and T. They are coplanar in the central image, at the instant that is known to correspond to the peak acoustical emission [343]. The vortex ring resulting from the merging of the leading and trailing rings is indicated by M.

9.10 Quantitative Study of Vortex Pairing in a Circular Air Jet

311

Fig. 9.50. Acoustically excited subsonic jet facility.

9.10.3 PIV Measurements A seeding generator provides 1 μm particles obtained by evaporation of oil on a heated plate followed by re-condensation. In order to obtain a homogeneous seeding across the shear layer, seeding is spread over the volume of the whole test room. Pairs of short pulses (duration below 5 ns) with a pulse separation equal to 10 μs are ﬁred by double-pulse Nd:YAG laser. The light beams are reshaped in light sheets using two spherical lenses and a cylindrical lens, to illuminate a meridian plane of the jet ﬂow in the nozzle outlet region. Series of 32 pairs of particle images are obtained using a fan-cooled 12-bit PCO CCD camera having a 1280 × 1024 pixel2 non-interlaced sensor chipset and a PC-mounted frame grabber controlled by the PCO software. A pulse delay generator is used to synchronize the camera and the laser. The laser is trig-

Fig. 9.51. Smoke visualization of vortex pairing.

312

9 Examples of Application

gered at a frequency equal to 8.012 Hz, and the CCD camera captures pairs of images at half that frequency, i.e. 4.006 Hz. A lens with 50 mm focal length was used to minimize optical distortion. The synchronization of the PIV system with the acoustic excitation signal yields pseudo-time resolved sequences of 32 images, spanning over two excitation periods to cover at least one complete pairing event. A total of 100 sequences, randomly triggered, was recorded in order to sample the vortex pairing with a good time resolution. The particle images are processed using a PIV software, Window Distortion Iterative Multigrid (WIDIM [157]), developed at the von Karman Institute for Fluid Dynamics. The data analysis is based on the cross-correlation of the pairs of particle images. The position of the correlation peak is obtained within sub-pixel accuracy using a Gaussian interpolation scheme. A ﬁne spatial resolution is obtained thanks to a multigrid predictor-corrector approach. A window overlap of 50% was used to further increase the spatial resolution. Finally, a ﬁrst order deformation is applied to the interrogation windows to cope with the strong velocity gradients encountered in the jet shear layer. These features allowed for obtaining a spatial resolution of 0.33 mm while keeping the validation rate above 96%. A vector is validated when the amplitude of the correlation peak is at least 50% larger than the amplitude of the second highest peak. The processing parameters are summarized in table 9.14. The spatial resolution is expressed as a fraction of the jet diameter D and vortex core diameter σc . The vortex core is here deﬁned as the region of the vortex where the tangential velocity increases for increasing distance from its center [339]. 9.10.4 PIV Uncertainty: Random Errors The velocity uncertainty is assessed on the basis of a comprehensive evaluation of the performance of the PIV software used in this work, made by Scarano & Riethmuller [158]. The error on the velocity is related to two factors: • The random error on the determination of the position of the crosscorrelation peak within sub-pixel accuracy. The evolution of the uncertainty on the particles displacement as a function of the local displacement gradient is given in [158]. • The errors due to the ﬁnite dimension of the interrogation windows. This leads to two important limitations bearing on i) the maximum amplitude of the displacement gradient that can be measured, and ii) the minimum measurable spatial wavelength, related to the spatial integration performed by the cross-correlation. The ﬁrst order window distortion applied by the present PIV algorithm allows to measure spatial gradients up to 0.5 pixel/ pixel. The spatial ﬁltering eﬀect has been quantiﬁed in [158] as a function of the ratio of the wavelength Λ of a reference sinusoidal velocity ﬁeld, divided by the interrogation window size Ws .

9.10 Quantitative Study of Vortex Pairing in a Circular Air Jet

313

Fig. 9.52. Evaluation of the uncertainty on the velocity measurements. For clarity, every third vector is shown. Table 9.14. Parameters of the PIV cross-correlation image interrogation algorithm and uncertainty analysis. Interrogation algorithm Initial window size Number of reﬁnement steps Final window size Ws Window overlap Spatial resolution Δx = Δy Validation rate

20 × 20 pixel 1 10 × 10 pixel 50% 0.33 mm D/125 σc /4 96%

Uncertainty analysis (∂u/∂y)max ; (∂v/∂x)max εu ; εv εu /δu ; εv /δv ε∂u/∂y ; ε∂v/∂x ε∂u/∂y / (∂u/∂y)max ; ε∂v/∂x / (∂v/∂x)max

0.25 pixel/ pixel; 0.16 pixel/ pixel 0.1 pixel ; 0.06 pixel 1.4% ; 1.5% 0.02 pixel/ pixel; 0.011 pixel/ pixel 8% ; 6.9%

In order to evaluate the accuracy of the measurements for the most unfavorable case, we consider the local velocity ﬁeld in the core of the vortices.

314

9 Examples of Application

Figure 9.52 shows a PIV particle displacement ﬁeld, together with axial and radial proﬁles obtained through the core of one of the vortex rings that undergo pairing. It can be seen that the maximum displacement gradient obtained in this case does not exceed 0.25 pixel/ pixel, which is well below the maximum measurable gradient. Hence no gradient truncation is expected. The same velocity proﬁles can be used to infer a minimum wavelength of the velocity ﬂuctuations in the vortex core region. It has been veriﬁed in [339] that the ratio Ws /Λ is about 0.1. This yields a truncation of 3% of the peak velocity found around the vortex core [158]. The absolute uncertainties on the displacement εu and εv (in pixels) are given as functions of the displacement gradients ∂v/∂x and ∂u/∂y in [158]. The corresponding values for our case are indicated in table 9.14. The corresponding relative uncertainties are calculated by dividing the absolute values by the variations δu and δv of the corresponding velocity component inside the core. The uncertainty on the vorticity has been obtained using the expressions given in section 6.4.1 for the 3rd order accurate Richardson scheme. We obtain an uncertainty on the vorticity in the core of about 8%. A maximum absolute uncertainty on the measured displacement of about 0.1 pixel is found. Being given the maximum particle displacement of about 10 pixel, a dynamical range of about 100, i.e. a range of displacements extending over 2 decades, can therefore be measured. 9.10.5 Particle Centrifugation in the Vortex Cores: Bias Error The particles motion is supposed to be identical to the motion of the surrounding ﬂuid. This is achieved in liquid ﬂows using neutrally buoyant particles. In air ﬂows, the Lagrangian accelerations must not be too severe. For the vortical ﬂows investigated in this study, the radial acceleration that is encountered in intense vortices induces a gradual rarefaction of the particles in the vortex core if their mass density is higher than the ﬂuid density. An example of such a rarefaction by centrifugation is illustrated in ﬁgure 9.53, that shows a doubly-exposed image of the particles inside the core of vortices measured in our jet ﬂow. We clearly observe a particle rarefaction in the core of the vortex. The centrifugation of particles within the vortex core depends on their density compared to the ﬂuid density, their diameter and the size of the vortex core, amongst other parameters [344]. We use a seeding generator that provides a poly-dispersed distribution of oil particles with a mean diameter Dp of the order of 1 μm and density equal to 850 kg/m3. The ﬂow ﬁeld inside the vortex and the vortex core radius σc have been determined using the focused, high-resolution PIV acquisitions described in [339]. For these conditions the particles have a negligible lag with respect to the ﬂow in the azimuthal direction, the error concerns mainly the radial component of the velocity. This is an important result since the radial velocity component does not contribute

9.10 Quantitative Study of Vortex Pairing in a Circular Air Jet

315

to the curl of the ﬂow ﬁeld. The centrifugation of the particles in the vortex core does therefore not aﬀect the measurement of the vorticity. The particle radial velocity vp is shown on the right side of ﬁgure 9.53, and reaches 1.5% for our case. In spite of this small value, we observe in the same ﬁgure on the left depletion in the vortex core. This is due the fact that smaller particles remain longer in the core, combined with the quadratic law relating the light intensity scattered by a particle to diameter [13]. A more intense illumination and the use of a camera with a large intensity dynamic range allows however to obtain a reliable correlation inside the vortex core. 9.10.6 Post-Processing: Automatic Vortex Tracking An important issue, when integrating the acoustical source term deﬁned by the vortex sound theory, is the deﬁnition of the integration domain, centered on the pairing vortices. As many realizations of the vortex pairing are required to achieve statistical convergence of the acoustical source term, an automatic and robust vortex detection algorithm was needed. The method is based on the vortex detection algorithm described in section 9.9. This allows to automatically process the PIV series and deﬁne an integration window centered on the pairing vortices as they evolve in the ﬁeld of view.

Fig. 9.53. Centrifugation of oil particles inside of the vortex cores; raw image data (left) and the radial velocity (right).

316

9 Examples of Application

1.2

X/D

1

Uw = 0.57Uo

0.8 0.6 Δx / D = 0.3 0.4 0.2 –12.56

– 6.28

0 ϕ (rad)

6.28

12.56

Fig. 9.54. Detection of the axial position of the vortex cores as a function of the phase of vortex pairing, and deﬁnition of the integration domain.

Figure 9.54 shows the axial position of the vortex rings’ cores during their interaction, obtained using the vortex detection algorithm. The phase used as abscissa is based on the excitation signal period, and is arbitrarily set to zero at the moment when the leading and trailing rings are coplanar. A linear ﬁt of these data gives the trajectory of the integration window, which follows the two vortices at a constant speed Uw = 0.57 U0 where U0 is the jet outlet velocity. The axial extent of the integration window is Δx = 0.3 D. Pseudotime resolved series of the acoustical source term were computed for the 100 sequences of 32 PIV ﬁelds. Details on the sound prediction method are given in [339]. 9.10.7 Acoustical Prediction Figure 9.55 shows the acoustical spectrum predicted using the PIV acoustical source data (symbols), superimposed to the acoustical measurements (line). We observe a good quantitative agreement between the predicted amplitude and the measured one for the frequencies not contaminated by the acoustical excitation (2.5 kHz): 1.25 kHz and 3.75 kHz. The predictions at 6.25 kHz and higher frequencies are below the background noise of the measurements. The measured spectrum shows nevertheless signiﬁcant peaks above the background noise at frequencies that can be attributed to the pairing: 6.25 kHz and 8.75 kHz. These peaks are largely underestimated in the prediction, presumably due to the smoothing process involved in the derivation of the PIV data [347]. Such agreement between the prediction and the measurements is nevertheless remarkable for the 1.25 kHz and 3.75 kHz components. The frequency of 1.25 kHz is the one of leapfrogging 1 of the trailing ring inside the leading one. 1

The axis-symmetric leapfrogging of two vortex rings with equal vorticity sign results from their mutual Biot-Savart interaction, leading to one or more slipthrough of the trailing ring inside the leading one. This interaction would be periodic if the two vortices are modelled by vorticity ﬁlaments, while it eventually

9.11 Stereo and Volume Approaches to Helicopter Aerodynamics

317

80

SPL (dB Re 20μPa)

70 60 50 40 30 20 10 0

0

2.5

5

7.5

10

12.5

f (kHz)

Fig. 9.55. Comparison of predicted (symbols) and measured (line) sound pressure level spectrum, at a distance of 0.9 m from the nozzle outlet and 90◦ from the jet axis.

9.10.8 Conclusions The originality of the present work is the prediction of aerodynamical sound on the basis of an experimental description of the ﬂow ﬁeld. This approach is problematic because of the large sensitivity of the aeroacoustical formulations to small errors in the source description [341]. Yet, the results shown above indicate that the processing of high-resolution, accurate PIV data with an ad-hoc vortex detection algorithm provides a good prediction of the sound produced by our vortex pairing.

9.11 Stereo and Volume Approaches to Helicopter Aerodynamics Contributed by: H. Richard, B.G. van der Wall, M. Raﬀel

Rotor Nomenclature a∞ c CT MH Nb R r rc

speed of sound ( m/s) chord ( m) rotor thrust coeﬃcient (CT = T /ρπΩ 2 R4 ) hover tip Mach number (MH = ΩR/a∞ ) number of blades blade radius ( m) radial distance ( m) vortex core radius ( m)

results in merging of the two rings if physical models based on ﬁnite size and deformable cores are applied.

318

T σ ψ Ω

9 Examples of Application

thrust ( N) solidity (σ = Nb c/πR) blade azimuthal position (deg) rotational rotor speed ( rad/s)

9.11.1 Rotor Flow Investigation With increasing use of civil helicopters the problem of noise radiation has become increasingly important within the last decades. Blade vortex interactions (BVI) have been identiﬁed as a major source of impulsive noise. As BVI-noise is governed by the induced velocities of tip vortices, it depends on vortex strength and miss-distance, which itself depends on vortex location, orientation, and convection speed relative to the path of the advancing blade. Blade vortex interaction can occur at diﬀerent locations inside the rotor plane depending on ﬂight velocity and orientation of the blade tip path plane. Within the last decade a large number of experimental investigations were performed in order to better understand and to model the development of rotor blade tip vortices [349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 363]. Most of these studies were done in hovering condition, because the ﬂow ﬁeld is azimuthally axisymmetric under this condition, the vortices are convected below the rotor plane and are isolated in early stage in comparison to forward or descent ﬂight where the vortices are entrained downstream and might interact with blade wake, other vortices and with the following blades. While earlier velocity measurements were obtained using intrusive techniques such as hot-wires, more recent ﬂow measurements rely exclusively on optical techniques, mainly LDV [348, 356] and PIV [351, 361]. 9.11.2 Wind Tunnel Measurements of Rotor Blade Vortices This wind tunnel experiment has been performed on a rotor model of 4 m diameter in the 6 m × 8 m open test section of the Large Low-speed Facility (LLF) of the German Dutch Wind Tunnel (DNW) operated at 33 m/s. The helicopter rotor model used was a model of the MBB Bo 105 of the German Aerospace Center (DLR) Institute of Flight Systems in Braunschweig. The PIV parameters used for this investigation are listed in table 9.15. The rotor consists of four hingeless blades with 0.121 m chord length, rectangular blade planform, and −8◦ linear twist. The airfoil was a NACA 23012 with tabbed trailing edge. During the test, the model was mainly operated in descent ﬂight conditions where blade vortex interaction dominates the noise radiation. The rotor rotational frequency was 17.35 Hz leading to a tip speed of 218 m/s. The PIV system consisted of ﬁve digital cameras and three double pulse Nd:YAG lasers with 2 × 320 mJ each, which were mounted, as sketched on Fig 9.56, onto a common traversing system in order to keep the distance between the cameras and the light sheet constant while scanning the rotor

9.11 Stereo and Volume Approaches to Helicopter Aerodynamics

319

Table 9.15. PIV recording parameters for HART II. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material a

U∞ = 70 m/s Umax ≈ 70 m/s 450 × 380 mm2 and 150 × 130 mm2 3.1 × 3.1 × 2 mm3 (H × W × D) DSR ≈ 31 : 1 DVR ≈ 40 : 1 z0 ≈ 5.6 m double frame/single exposure frame separation (frame-straddling) full frame interline transfer CCD (1280 × 1024 px) f = 100 mm and 300 mm, f = 2.8 3×Nd:YAG lasera 320 mJ/pulse t = 7 − 20 μs DEHS droplets (dp ≈ 1 μm)

Frequency doubled

wake. The length of the traversing system was in the order of 10 m, the height approximately 15 m. Two stereo systems were used simultaneously. One system was equipped with 300 mm lens in order to record a small observation area: 0.15 m × 0.13 m, centered to the blade tip vortex and the second system was equipped with 100 mm resulting of a ﬁeld of view of 0.45 m × 0.38 m. The large ﬁeld of view was intended for an overview of the vortex and of the surrounding ﬂow whereas the small ﬁeld of view was intended for vortex analysis.

Fig. 9.56. HART II stereoscopic PIV setup.

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For each position and rotor condition, 100 instantaneous PIV images were recorded. More than half a Tera Byte (650 GB) of PIV raw data had been recorded at various positions on the advancing and the retreating side. The measurements have been performed in the frame of the HART II program [364]. 3C-PIV data from rotating tip vortices experiments are challenging from an analysis point of view. Conditional averaging is mandatory for proper analysis of vortex properties in order to take into account the model motion, vortex wander, aperiodic phenomena and other disturbing eﬀects. In most cases the vortices are not measured perfectly perpendicular to their axis and the measurement plane must be re-oriented into the vortex axis system. Several methods to perform such post-processing are described in [361]. Conditionally averaged velocity and vorticity maps obtained by each stereo system are displayed in ﬁgure 9.57. The detection of the vortex needed to perform the conditional average was done using the wavelet detection method described in section 9.9. The results on the left were obtained with the large ﬁeld of view stereo system. The blade tip vortex is clearly visible as well as the wake of the previous blade whereas on the small ﬁeld of view on the right only the vortex can be seen but with a high spatial resolution.

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Fig. 9.57. Averaged velocity (top) and vorticity (bottom) maps of a blade tip vortex; large ﬁeld of view (left) and close-up view (right).

9.11 Stereo and Volume Approaches to Helicopter Aerodynamics

321

9.11.3 Measurement of Rotor Blade Vortices in Hover In order to investigate the development of blade tip vortices under diﬀerent rotor conditions like thrust and rotational speed, both two- and threecomponent PIV measurements were performed on the same 40% Mach scaled Bo105 model rotor in hover condition. The vortices were traced from their creation at the trailing edge of the blade up to half a revolution behind the blade with azimuth steps between 1◦ to 10◦ and diﬀerent spatial resolutions. In addition, a sequence of three-component measurements was performed just after the vortex creation at ﬁner azimuth steps of 0.056◦ in order to generate a three-dimensional volumetric data set of the blade tip vortex. The inﬂuence of the PIV image analysis parameters on the vortex parameters derived -in particular sampling window size and window overlap- has been investigated. The measurements presented are part of the HOTIS (HOver TIp vortex Structure) project. Within the HOTIS project velocity ﬁeld measurements were conducted using two-component (2C) and three-component (3C) PIV on a four-bladed rotor in hover condition in ground eﬀect inside the rotor preparation hall of the Institute of Flight Systems at DLR Braunschweig. The aging process of the blade tip vortex and the inﬂuence of the rotor parameters on the vortex characteristics were investigated at diﬀerent vortex ages from 1◦ to 150◦ for several rotor parameters: rotation speed (200, 540 and 1041 rpm) and thrust (from 0 N to 3500 N). The blade tip vortex was measured with diﬀerent spatial resolutions: low and very high spatial resolution in case of 2C-PIV measurements and with high spatial resolution for 3C-PIV. Figure 9.58 shows an example of the vortex development measured with the very resolution 2C-PIV

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Fig. 9.58. Averaged velocity and vorticity maps of a young blade tip vortex measured with 2C-PIV.

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system. The spatial resolution, deﬁned by the ﬁeld of view and the size of the interrogation window, is an important parameter when looking for vortex properties such as maximum swirl velocity or core radius [361]. In addition to these measurements, 3C measurements were performed for vortex ages between 3.4◦ and 7◦ using very ﬁne age increments of 0.056◦ in order to generate an averaged volumetrically resolved velocity data set of the vortex. The Experimental Setup The rotor model, which has been described earlier (9.11.2) was installed in the center of the 12 m × 12 m × 8 m rotor testing hall of DLR Braunschweig. The rotor was operated in ground eﬀect - the hub center located 2.87 m above the ground - at diﬀerent rotation speeds of 200, 540 and 1041 rpm, corresponding to tip Mach numbers of MH = 0.122, 0.329 and 0.633, and with diﬀerent thrust coeﬃcients varying from CT /σ = 0 to 0.072. Due to the closed hall, recirculation existed and generated an inherently unsteady ﬂow ﬁeld. The illumination source of the PIV setup consisted of a double oscillator, frequency-doubled Nd:Yag laser (320 mJ/pulse at 532 nm) and light sheet forming optics which were bolted to the ground below the rotor. The light sheet was oriented vertically upward and parallel to the trailing edge of the rotor blade and had a waist thickness of 1–2 mm at the measurement plane and a width of around 30 cm. Three thermo-electrically cooled CCD-cameras, one regular (1280×1024 pixel) and two intensiﬁed PCO cameras (1360×1076 pixel) were used. One camera for 2C PIV and for recording the position of the blade tip and the two other cameras in stereo arrangement for 3C PIV. The cameras were mounted on a support structure consisting of standard optical rails which was bolted to the wall of the testing hall. The complete PIV setup is shown on ﬁgure 9.59. The camera support can be seen on the left hand side of the ﬁgures. The cameras were located at 4.5 m from the rotor hub and the stereo cameras were mounted with a stereo viewing angle of 47◦ . Laser and camera were synchronized according to the one per revolution signal given by the reference blade of the rotor. This signal was delayed using a phase-shifter in order to measure at a desired blade azimuth angle.

25.5°

4500

21.5°

Fig. 9.59. Sketch of the PIV setup and of the rotor model.

9.11 Stereo and Volume Approaches to Helicopter Aerodynamics

323

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Fig. 9.60. The diﬀerent ﬁelds of view.

Two component PIV measurements were performed using the middle camera which was ﬁrst equipped with a f = 85 mm lens and later with a f = 600 mm yielding ﬁeld of views of 281 mm × 357 mm (low spatial resolution) and 58 mm × 45 mm (high spatial resolution). The stereo system was equipped with a pair of f = 300 mm lenses with a common ﬁeld of view of 126 mm × 96 mm. Fig. 9.60 shows the diﬀerent ﬁelds of view measured during the campaign. The time delay between the two laser pulses was between 2 and 40 μs depending of the size of the ﬁeld of view and of the velocity to be measured. Around 200 measurements were taken for each setting. The ﬂow seeding was introduced to the measurement area by Laskin nozzle particle generators ﬁlled with DEHS ﬂuid producing particles with a mean diameter below 1 μm. Evaluation and Analysis Dewarping coeﬃcients were extracted from calibration images in order to map the stereo recordings onto a common grid. A so-called disparity correction was applied as well using the actual PIV recordings in order to account for possible misalignment between the calibration target and the light sheet plane. This correction is performed by cross-correlating simultaneously recorded images of the two views (upper and lower camera). The residual misalignment was found to be on the order of 100 pixel corresponding to a few millimeters in the object space. The resulting vector map is then used to correct the original mapping coeﬃcients which were used to dewarp the raw PIV images prior to PIV interrogation. The images obtained during the measurement were preprocessed using high pass ﬁlter (= 3 pixel), then binarized and ﬁnally lowpass ﬁltered (= 0.7 pixel) in order to increase the signal to noise ratio. A multi-grid PIV algorithm based on pyramid grid reﬁnement and full image deformation was applied to process the image starting with large interrogation windows on a coarse grid and reﬁning the windows and the grid with each pass. A sampling window of 64 × 64 pixel was used in the initial step gradually reﬁning down to 24×24 pixel as ﬁnal window size at 75% sample overlap which represent an interrogation area of 2.3 mm × 2.3 mm for the 3C measurements

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and 1.05 mm × 1.05 mm for the 2C measurements with very high resolution. Sub-pixel peak position estimation was performed by means of Whittaker reconstruction. All the processing was done using the PIVview software which makes use of the advanced algorithms presented in section 5.4.4. The primary aim of these measurements was to gain a better understanding of the development of the blade tip vortex, especially in its early stages of development. The velocity vector ﬁelds are used to extract vortex characteristics such as the maximum swirl velocity, the core radius or the peak of vorticity. Prior to the full processing of the PIV image, the inﬂuence of the PIV interrogation window size and overlap on these characteristics was investigated. Sampling window size: One of the most important PIV parameters is the size of the interrogation or sampling window which, in terms of other measurement techniques, deﬁnes the probe volume. In case of vortex characteristic investigation, the decrease of window size results in an increase of maximum swirl velocity and a decrease of vortex radius [358, 361]. Sampling window overlap: Numerical investigations of the eﬀect of correlation window overlap on vortex characteristics (maximum swirl velocity and core radius) were performed in [361] using a Vatistas model [362]. The same investigation was reproduced using a real PIV image (ψ = 5◦ , Ω = 56.55 rad/s and thrust T = 550 N). First this image was processed with 96 × 96 pixel and 128 × 128 pixel windows sizes with overlap between 2 pixel (98% overlap) to the window size value (0% overlap) in x direction whereas the overlap in y direction was kept constant at 50% of the window size. Figure 9.61 shows the maximum tangential (swirl) velocity and core radius extracted from the horizontal velocity proﬁle (one dimensional analysis). The curves obtained are in good agreement with the numerical simulation mentioned before. They converge to diﬀerent values depending of the window 28

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9.11 Stereo and Volume Approaches to Helicopter Aerodynamics

325

size and the oscillations decrease with increasing overlap. The swirl velocity values are always equal or below the value obtained with the maximum sampling whereas the core radius oscillates around it. Maximum swirl velocity is reached when the center point of an interrogation window falls onto the maximum in the velocity proﬁle which has an increased probability as the overlap is increased. In a second step the same image was processed using window sizes of 48, 64, 96, and 128 pixel size with overlap between 2 pixel and the window size value. The maximum swirl velocity and core radius were computed by averaging circularly the tangential velocity proﬁle over r (two dimensional analysis), the radial distance from the vortex center. This radial averaging method is known to be more robust. The curves obtained are presented on ﬁgure 9.62. The eﬀect of the window size is still noticeable but the curves are smoother and the oscillations which were observed in ﬁgure 9.62 are nearly completely damped. This investigation shows that the overlap parameter can play an important role when looking for vortex characteristics and that in order to avoid random eﬀects an overlap as large as possible should be used in order to avoid these sampling artifacts. While the minimum interrogation window size is limited by experimental parameters like seeding distribution, in-plane loss of image pairs (mainly compensated by multi-grid algorithms) and image background noise, the overlap parameters do not have such limitation. The only limitation for the use of large sample overlap is the processing time and the size of the resulting data set. In eﬀect the processing time and the size of the resulting data set increase by a factor of 4 when the overlap is increased by 50%. The processing on a Pentium IV (3.0 GHz, 1GB RAM) of a 1360 × 1076 pixel 2C-PIV image using multi-grid algorithm with 64 × 64 pixel initial window size and 24 × 24 pixel ﬁnal window size with an overlap of 22 pixel (91% overlap) requires 2.5 min and generates ﬁle of 14MB. Under these conditions it is not realistic to use such parameters for a large number of images: the processing of the images recorded during HOTIS campaign would require few months requiring nearly one Tera Byte to store the result. In order to overcome this problem and to be 32 0.3 30 +

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326

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able to use larger sample overlap a new feature was implemented within the processing software which allows using multi-region interrogation parameters based on physical properties. The procedure consists of: 1. Processing of the PIV image using a large window size and small overlap, for example 64 × 64 pixel window size with 50% overlap, the vector ﬁeld obtained is used to compute diﬀerential operators. In case of vortex ﬂow: the vorticity and the λ2 operator. 2. The vorticity or/and the λ2 operator are used to estimate the location of the vortex center which is used to deﬁne the position of the new region of interest. This region is then processed using ﬁner window size and larger overlap, for example 24 × 24 pixel window size with 90% overlap. 3. The vortex characteristics are extracted from this region (solid curve in Fig 9.63. In the example, Fig. 9.63, less than a sixth of the PIV image area was processed using small window size optimizing in this way the time and storage requirements. In addition to azimuth steps of 1◦ to 10◦ , sequences of 3C measurements were performed with vortex age increments of about 0.056◦ for vortex ages from 3.4◦ to 7◦ for diﬀerent rotor conditions. These small increments allow reconstructing a 3D volume of the vortex due to the azimuthal axisymmetry of the ﬂow ﬁeld. The resulting volume allows the computation of the two missing components of the vorticity vector (ωx and ωy ), based on out-of-plane derivatives of u, v and w. The reconstruction of the volume can only be done using averaged results because every plane forming the volume was recorded at diﬀerent instants of time. After conditional averaging, the orientation of each plane was corrected in order to take the step angle into account. Figure 9.64 shows the three vorticity components after merging all the 3C PIV vector ﬁelds. The vortex tube as well as the blade wake is clearly visible on these graphs.

Fig. 9.63. Multi region principle for vortex ﬂow analysis.

9.12 Stereo PIV Applied to a Transonic Turbine

327

ωx ωy ωz Fig. 9.64. Contour plot of the three vorticity components.

Conclusions Helicopter model tests were performed with a 40% Mach scaled rotor of the Bo105 in a hover testing chamber and in a large low-speed wind tunnel. The blade tip vortices were traced from their creation up to half a revolution with small age increments. A parametric study of the two main PIV parameters - the correlation window size and the overlap - was done. It shows that the overlap should be as large as possible in order to avoid random eﬀects and to improve the accuracy of vortex characteristics which can be extracted from the result. The results show that a high spatial resolution is required in order to resolve the vortex characteristics and particularly the core radius of young vortices properly. A three-dimensional reconstruction of the blade tip vortex has been applied for vortex ages between ψ = 3.4◦ and 7◦ using a very ﬁne age increment step of Δψv = 0.056◦ . The volume reconstruction allows computing all the diﬀerential quantities which can not be obtained by single plane 3CPIV. A ﬁrst post-processing and analysis shows that the vortex characteristics - swirl velocity, core radius and axial velocity - seem to be independent of the blade tip speed at least in the range measured during the tests (0.122 ≤ MH ≤ 0.633).

9.12 Stereo PIV Applied to a Transonic Turbine Contributed by: J. Woisetschl¨ ager, H. Lang, E. G¨ ottlich

Stereoscopic recording techniques were used to perform detailed studies of the unsteady rotor-stator interaction in a transonic turbine. The schematic of the test section is given in ﬁgure 9.65. A total number of 24 stator blades and 36 rotor blades were used in this turbine with rotational speeds between 9600 and 10600 rpm. The inlet ﬂow temperature was between 360 and 403 K. The turbine’s control system (Bently Nevada) provided 12 TTL and one analog

328

9 Examples of Application

Fig. 9.65. Test section (measures in mm).

pulse per revolution. By combining both, a high resolution trigger signal was obtained. 9.12.1 Optical Conﬁguration Most challenging about this application is the highly turbulent, high-speed and unsteady ﬂow where neither the observation window nor the light sheet probe nor the seeding pipe is allowed to cause disturbances. Large amounts of seeding particles have to be used and ﬁnally, a careful calibration of the imaging geometry has to be done due to a curved observation window. The beneﬁt of PIV to immediately obtain unsteady ﬂow ﬁeld data outweighed these diﬃculties. The recording parameters for this ﬂow are given in table 9.16. The details of the camera arrangement are shown in ﬁgure 9.66, with one camera axis being perpendicular to the light sheet and the second camera axis being inclined by 27◦ to the ﬁrst. This angle was limited by the geometry of the Table 9.16. PIV recording parameters for the transonic turbine ﬂow. Flow geometry Maximum in-plane velocity Field of view (both cameras) Interrogation volume Observation distance Recording method Recording medium Recording lens Illumination Pulse delay Seeding material a

frequency doubled

perpendicular to trailing edges Umax ≈ 450 m/s 47.5 × 37.5 mm2 1.2 × 1.2 × 2 mm3 (H × W × D) 260 mm dual frame / single exposure 1280 × 1024 pixel, progressive scan CCD f = 60 mm, f# = 2.8 dual cavity Nd:YAG lasera , 120 mJ/ pulse Δt = 0.7 μs DEHS, Palas-AGF 5.0D (dp ≈ 0.3 . . . 0.7 μm)

9.12 Stereo PIV Applied to a Transonic Turbine

329

turbine casing. The cameras were mounted according to the Scheimpﬂug condition (see ﬁgure 9.66). The light sheet probe was ﬁxed to the cameras’ base plate while a ﬂexible silicon tube was used to seal the air inside the turbine. This rigid system eased focusing and calibration. The laser light was guided through an articulated arm to the light sheet probe. A high-temperature resin was used to glue the single elements, especially the cover glass (three component resin, R&G GmbH). This glass shielded the aligned optical prism against the ﬂuctuating pressure caused by the ﬂow. The light sheet was observed through a plan-concave quartz glass window (HERASIL; anti-reﬂection coated) with dimensions 123 × 75 × 15 mm3 and a R = 264 mm curvature on one side.

light-sheet probe

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Fig. 9.66. Optical setup (all measures in mm).

Seeding was provided approximately 500 mm upstream the stator blades through a specially formed pipe (S-shape, 7 mm inner diameter). In the most upstream part of the pipe a large number of holes with 1.8 mm diameter were drilled (“shower head”). This part with a length of 130 mm was aligned in a tangential direction upstream the light sheet.

9.12.2 Results Since the ﬂow through a turbine is highly directional with only the secondary ﬂow eﬀects (vortex shedding, rotor interaction) being of special interest, an image shifting technique was applied in the main ﬂow direction (up to 11 pixel). Additionally, a background image with no seeding was subtracted from each of the single recordings in order to improve the contrast of the tracer particles.

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9 Examples of Application

In some cases a peak locking eﬀect was observed from the images (see section 5.5.2). A slight defocusing using a step motor driven device for focusing the camera lenses helped to avoid this eﬀect. For each rotor-stator position investigated, approximately 200 recordings (dual frame) were captured. A cross-correlation technique with 64 × 64 pixel interrogation size and 50% overlap was then applied resulting in the individual vector ﬁelds. A standard two-dimensional Gauss ﬁt in a 3 × 3 pixel matrix was used for sub-pixel resolution. The individual vector ﬁelds were validated using the peak height, a maximum and minimum displacement range and a moving average ﬁlter (5 × 5 nodes). This ﬁlter compares mean value and standard deviation for the center vector with the values of the surrounding vectors rejecting vectors whose deviation is too high. Due to the high turbulence sometimes only a relatively small number of vectors were validated (50%). Then, a higher number of recordings must be used for the averaging. Additionally, the wake region is oftentimes poorly seeded, due to the fact that the vortex shedding from the trailing edge of the turbine blade contains boundary layer ﬂuid without seeding. After averaging a number of recordings at a given phase of the vortex shedding, the core of the vortices will contain less validated vectors compared to the main ﬂow. In these areas this might increase the uncertainty in the estimation of the mean value by 10% or even more compared to the rest of the ﬁeld. Since the light sheet plane was observed through a curved window section a higher order approach had to be used for de-warping the two camera images [212]. After the recordings were taken, the window section was ﬁxed to the base plate of the two cameras (with the light sheet probe attached). So all parts were removed from the turbine together to enable calibration outside the turbine test rig. Since the shock areas are most pronounced only at mid-span, focusing the particles in the shock area is less a problem. Unfortunately, the displacement gradients happen to inﬂuence the results, especially when the calibration target is misaligned. This eﬀect depends on the strength of the gradients and the misalignment of the two camera projections due to a misalignment of the calibration target. When back-projecting the two vector sets to the measurement plane, one camera will then provide the displacement value from upstream the shock for a given position, while the other projection provides the displacement value from downstream the shock for the same position in the light-sheet plane. This will lead to a measurement error which can be easily recognized by an exceedingly large out-of-plane component. This eﬀect can be seen in ﬁgure 9.67. Finally we ended up with following error estimates:

9.12 Stereo PIV Applied to a Transonic Turbine

331

Fig. 9.67. Error in the out-of-plane component in the vicinity of the shock due to a misalignment during the calibration procedure.

• ±2–4 m/s per recording (main ﬂow section) with 10–12 pixel particle displacement, 0.1 pixel uncertainty (Gauss Fit) and more than 10 particles per interrogation area, • ﬁve times less sensitivity of the out-of-plane component compared to the in-plane-component (due to only 27◦ between the camera axes), • additional ±3 m/s in the vicinity of the shock (5 pixel misalignment) • and after averaging ±1 m/s statistical error (in-plane) and ±5 m/s outof-plane with a minimum of 100 validated vectors per interrogation area (outside the wake). In a ﬁnal step the in-plane and out-of-plane components have to be transformed into axial, tangential and radial components to provide the full data set for comparison to data obtained by computational ﬂuid dynamics. Two results for the mean velocity (all three components) and the vorticity (from the two in-plane components) are presented in ﬁgure 9.68. When looking at the vorticity one can observe seven phases of vortex shedding during one period of blade passing. This means the vortex shedding frequency is about 40 kHz. A detailed comparison with interferometric measurements indicated that the tracer particles used started to act as lowpass ﬁlter at about 80 kHz. Therefore, only the ﬁrst harmonic of the vortex movement can be found in the PIV results. While high resolution CFD methods predict various shapes of vortices, PIV results showed vortices of more or less circular type because of the band-pass ﬁltering. On the other hand, PIV provided the unique possibility to investigate the interaction between shocks, shock reﬂections, vortex shedding and wake-wake interaction in these

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9 Examples of Application

Fig. 9.68. Velocity (upper image) and vorticity (lower image) at mid-span in a transonic turbine stage (10600 rpm).

turbulent and transonic ﬂows. Further discussion of the results can be found in [365, 366, 367, 368, 369].

9.13 PIV Applied to a Transonic Centrifugal Compressor Contributed by: M. Voges, C. Willert

In the present application PIV was chosen to analyze the complex ﬂow phenomena inside vaned diﬀuser of a new generation transonic centrifugal com-

9.13 PIV Applied to a Transonic Centrifugal Compressor

333

Table 9.17. PIV recording parameters for transonic centrifugal compressor Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Recording medium Recording lens Illumination Acquisition rate Pulse delay Seeding material a

only 1100(V) lines utilized

b

M a = 0.4 − 1.2 parallel to light sheet Umax ≈ 300 − 700 m/s 48 × 33 mm2 1 × 1 × 1 mm3 (H × W × D) DSR ≈ 80 : 1 DVR ≈ 200 : 1 ≈ 500 mm dual frame / single exposure 1600(H) × 1200(V ) pixela , progressive scan CCD f = 105 mm, f# = 4.0 dual cavity Nd:YAG laserb , 120 mJ/ pulse 15 Hz (188 image pairs per sequence) Δt = 1.5 − 2.5 μs paraﬃn oil (dp ≈ 0.3 . . . 1.2 μm) frequency doubled

pressor, as this planar technique is capable of detecting unsteady ﬂow structures and to resolve even high velocity gradients as well as unsteady shock conﬁgurations previously undetectable with point-wise techniques such as laser2-focus velocimetry (L2F) [370]. Measurements were carried out at rotational speeds between 35,000 and 50,000 rpm. The operating conditions are summarized in table 9.18. The compressor stage was designed for a pressure ratio of 6 : 1. Due to the advanced impeller geometry the diﬀuser section has a conical shape with a constant passage height of 8.1 mm (ﬁgure 9.69). The conical shape of the diﬀuser plane required specialized engineering solutions concerning the laser light delivery as well as on the imaging side. Oﬀ-the-shelf light sheet probes commonly have a 90◦ beam deﬂection and generally are not actively cooled or heat resistant. For this application a special periscope probe was speciﬁcally designed that precisely matched to the geometrical conditions of the diﬀuser and casing. The light sheet probe including the internal beam path is shown in ﬁgure 9.70. The periscope probe allowed for adjustment of the light sheet in rotational and axial position and angle relative to the chord of the vane proﬁle. Together with the probe support in the diﬀuser casing it was possible to adjust the light sheet to the three Table 9.18. Operating conditions for PIV investigation inside the diﬀuser passage Rotational speed 35,000 rpm Pressure ratio 2.5 : 1 Mass ﬂow 1.4 kg/s Mean temperature 110◦ C PIV pulse separation Δt 2.5 μs

44,000 rpm 4.0 : 1 2.15 kg/s 175◦ C 2.0 μs

50,000 rpm 50,000 rpm 5.6 : 1 5.3 : 1 2.6 kg/s 2.83 kg/s 245 − 255◦ C 230 − 235◦ C 1.5 μs 1.5 μs

334

9 Examples of Application

Fig. 9.69. Radial compressor test rig; cross-sectional view (right).

vane span locations chosen for ﬂow investigation: center plane (50% span), one plane close to the hub (19% span) and one plane close to the tip (74% span). The probe support close to the diﬀuser outlet was not perpendicular to the outer machine casing, but inclined to match the conical diﬀuser area. The free beam path inside the probe had a diameter of 6 mm. A pair of cylindrical lenses inside the probe formed the light sheet with a thickness of 1 mm and a divergence angle of about 6◦ . At the outlet of the periscope probe a mirror deﬂected the light sheet with an angle of 97◦ , thus to support the adjustment in the diﬀuser vane passage. Before entering the light sheet probe the beam diameter of the PIV laser has to be reduced to pass through the periscope probe without striking the surface of the metal inner tube. Here a pair of spherical lenses was used in a telescopic set up. The periscope was also continuously purged with compressed, dry air to prevent deposition of seeding particles as well as to cool the probe with its optical components with air existing at the open delivery end. To provide suﬃcient optical access for planar PIV measurements a relatively large quartz window was needed in the diﬀuser casing. The prepared access port provided a camera observation area of one complete diﬀuser vane passage, including the impeller exit region (ﬁgure 9.71). A quartz window and a metal window supporting brace were manufactured with considerable eﬀort to precisely match the inner contour of the diﬀuser casing, thereby minimizing

9.13 PIV Applied to a Transonic Centrifugal Compressor

335

Fig. 9.70. Setup of periscope light sheet probe.

disturbances of the near-wall ﬂow. While the outer surface of the glass was ﬂat, the inner surface was CNC-milled according to the conical diﬀuser shape and subsequently surface polished by hand. A bulky design of the window was necessary in order to withstand the high temperature and pressure strains during compressor operation and also reduces the likelihood of glass fracture.

Fig. 9.71. Observation window and camera positions.

336

9 Examples of Application

To reduce compressive stresses and to provide a reliable seal between the vane passages a silicone sealing was applied to the contact surface of the vanes in the window area. For ﬂow observation both the frequency-doubled, dual-cavity Nd:YAG laser and the thermo-electrically cooled, double-shutter CCD camera (1600 × 1200 pixel at 14 bit/pixel) were operated at a frame rate of 15 Hz. Compared to previously available PIV-cameras, the roughly ﬁve-fold increase in frame rate signiﬁcantly reduced overall measurement time, thus reducing operating costs of the test facility. In addition facility seeding is only required for a reduced time period, which resulted in signiﬁcantly decreased window contamination. The camera itself was mounted on a Scheimpﬂug adapter to optimize alignment of the camera optics with the laser light sheet. To further increase the camera’s spatial resolution in the investigated diﬀuser passage the camera was traversed resulting in two camera positions, one position observing the impeller exit, the other position observing the ﬂow downstream in the diﬀuser throat (ﬁgure 9.71). Both measurements were performed in succession for each operating condition and light sheet position respectively. Calibration and light sheet alignment was performed by means of a target that could reproducibly positioned between the vanes of the diﬀuser passage. To account for refraction eﬀects due to the thick glass window it was necessary to perform the calibration with the window installed. The target was made of a thin aluminum plate thickness with a precise 2.0 × 2.0 mm2 dot grid applied to its surface. Adjustment of the grid within the vane passage was achieved with three small set-screws. All of PIV equipment was mounted on a separate rigid support in order to minimize the inﬂuence of rig vibration. An articulated laser guide arm delivered the laser light to light sheet forming shaping optics and light sheet probe which were rigidly attached to the compressor rig. Seeding particles were introduced upstream of a contraction in front of the impeller. Here a circumferential traverse supported four seeding probes with diﬀerent radial positions which allowed for a nearly uniform seeding distribution across a given sector of the pipe ﬂow. Droplet-based seeding was produced by a battery of three Laskin-nozzle generators ﬁlled with paraﬃn oil. Although the oil evaporation temperature is near 200◦ C, it should be noted that the evaporation temperature increases with increased pressures, which in part explains the fact the seeding particles remained visible in the diﬀuser at temperature above 250◦C. The use of solid particle seeding was also considered but deemed too risky without additional investigations. However Wernet reports successful use of solid particle seeding is previous turbomachinery investigations [371, 372]. The size of the seeding particles was limited to 0.3−0.8 μm through the use of an impactor downstream of the Laskin-nozzle generators. At higher temperatures during compressor operation, the smallest particles might evaporate while larger particles survive longer in the ﬂow ﬁeld, although they have reduced their size when reaching the investigated ﬂow area. By switching oﬀ the impactor the particle size distribution in the seeded ﬂow could be increased to 0.8 − 1.2 μm. This had a signiﬁcant positive eﬀect on the PIV signal during

9.13 PIV Applied to a Transonic Centrifugal Compressor

337

measurements at design conditions of the compressor stage, but also did not result in window contamination. For phase-stationary acquisition of PIV images a phase shifter was triggered by the 1/rev-trigger of the compressor. PIV image sequences of 188 images each were obtained at 16 equally spaced phase angles on one particular main-splitter-main blade passage of the impeller. While this number of images is considered to be suﬃcient for the calculation of phase-average velocities, it certainly is insuﬃcient to reach convergence for the estimation of statistical quantities such as RMS values or Reynolds stresses. Here an estimated 1000 images per phase angle may have been more adequate. For this application a compromise was made between the detailed investigation of the ﬂow phenomena occurring in the advanced compressor stage on the one hand and the precise analysis of the various parameters characterizing the diﬀuser ﬂow on the other hand. Thus the limited operation time interval on the compressor rig was used to perform detailed investigation of the diﬀuser ﬂow ﬁeld with respect to the various operating conditions. Evaluation of the PIV image data was performed after pre-processing with high pass ﬁlter, subtraction of background image and masking image areas without velocity information (e.g. diﬀuser casing, window support or shadowed areas). Transformation from the CCD sensors coordinates to physical space was performed using the calibration grid. Fortunately distortion of the particle images (blurring) as well as geometrical distortion (lensing eﬀect) by the curved inner contour of the window was insigniﬁcant due to the proximity of the light sheet plane to the window surface. Therefore a separate nonlinear mapping procedure of the images was not required. The PIV processing was based on an adaptive, grid reﬁning cross-correlation scheme with continuous image deformation as described in section 5.4.4. At the ﬁnal resolution the algorithm used interrogation sample sizes of 32 × 32 pixels (1.0 × 1.0 mm2 ) at 50% overlap (ﬁnal grid size: 0.5×0.5 mm2 ) and sub-pixel peak position estimation using Whittaker reconstruction. Outlier detection employed normalized median ﬁltering [275], followed by linear interpolation of rejected vectors. A correlation plane signal-to-noise ratio of 50 or better could be achieved; the number of spurious vectors was below 3% for all imaged planes. The re-combination of the obtained velocity ﬁelds for both camera views could be easily performed during post-processing of the PIV data with the help of the common calibration grid, as both camera views overlap in one area. With a mean pixel shift varying from 13 pixels at 35,000 rpm up to 20 pixels at 50,000 rpm, the corresponding relative measurement error was estimated at 0.5 − 0.8% (2.7 − 3.5 m/s). Given a ﬁnal size of 1.0 × 1.0 mm2 for the interrogation area results in structure passing frequencies between 600 kHz and 1.4 MHz in the measured velocity range of 300 − 700 m/s. Here the size of the particles has an important inﬂuence on the obtained velocity data. As the response time of particles about 1 μm in size is on the order of 10 μs (see section 2.1.1), the particles behave like a low pass ﬁlter with a cut oﬀ frequency of 100 kHz applied to the ﬂow. Given a blade passing frequency around 20 kHz suggests

338

9 Examples of Application

Mabs at 50,000 rpm: 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 74 % 50 %

0.8 19 %

0.6

z/b

0.4

0 -40

-20

x [mm]

0

y[

-60

mm ]

20 0.2

-20 20

40

Fig. 9.72. Mach number distribution calculated from the averaged velocity ﬁelds at 19%, 50% and 74% diﬀuser passage height (operating conditions: 50,000 rpm, mass ﬂow = 2.6 kg/s).

that only large scale structures are faithfully captured, while smaller scales are damped out. Here the use of sub-micron particles may be considered, but this would have the eﬀect of a signiﬁcant decrease in light scattering eﬃciency of the particles (Rayleigh scattering regime). In this context it should be noted

Fig. 9.73. Stream lines at 19% (hub) nicely follow the deﬂection imposed by the diﬀuser vane, with the passage core ﬂow passing straight through; at 74% (tip region) the stream lines show evidence of the tip clearance ﬂow as the stream traces turn in the opposite direction compared to the hub ﬂow.

9.14 PIV in Reacting Flows

Mabs: 0.4

0.6

0.8

1.0

339

1.2

Phase angle 360° 20

y [mm]

10

0

-10

ω

S

-20

M

M -30 -60

-50

-40

-30

-20

-10

0

10

20

30

x [mm] Fig. 9.74. Instantaneous Mach number distribution at 50% span, 50,000 rpm, for one phase angle, characterizing the splitter passage ﬂow; ﬂow patterns of preceding impeller passages are visible in the diﬀuser throat (1/12th of total vectors displayed).

that PIV is only capable of capturing a certain portion of the spatial energy spectrum, limited by the wave numbers corresponding to the largest scales (given by the ﬁeld size) and the smallest scales respectively (interrogation window size). A detailed analysis on the eﬀect of the velocity spectrum captured with PIV on the measurement accuracy is given by Foucault et al. [80]. Figures 9.72 through 9.74 provide some samples of the experimental data obtained with PIV in the radial compressor diﬀusor passage. A closer discussion of these PIV results, more detailed information on the compressor rig as well as further investigations using stereo PIV can be found in VOGES et al. [373].

9.14 PIV in Reacting Flows Contributed by: C. Willert

Application of PIV in reacting ﬂows is associated with a number of additional challenges not found in typical aerodynamic applications. Most importantly the seed material must withstand the high temperatures without evaporat-

340

9 Examples of Application

Table 9.19. PIV recording parameters for reactive ﬂow in pressurized combustor Flow geometry Maximum in-plane velocity Field of view Interrogation volume Dynamic spatial range Dynamic velocity range Observation distance Recording method Ambiguity removal Recording medium Recording lens Illumination Pulse delay Seeding material

swirling ﬂow with strong out-of-plane component near nozzle Umax ≈ 70 m/s 70 × 40 mm2 (W × H) 1.7 × 1.7 × 1.0 mm3 (W × H × D) DSR ≈ 40 : 1 DVR ≈ 120 : 1 z0 ≈ 500 mm dual frame/single exposure frame separation (frame-straddling) full frame interline transfer CCD 1280 × 1024 pixels (770 illuminated lines) f = 55 mm f# = 8 Freq. doubled Nd:YAG laser 120 mJ/pulse at 532 nm Δt = 4 μs Si2 O3 and Al2 O3 (dp ≈ 200 − 800 nm)

ing or chemically interacting with the ﬂow under investigation. Metal oxide powders such as silica, alumina or titanium oxide are generally well suited for this purpose due to their high melting point and availability. These powders are best introduced into the ﬂow using ﬂuidized bed seeders as described in section 2.2.2. Another diﬃculty arises due to ﬂame luminosity which generally increases with higher pressures and higher fuel-air ratios and is mainly caused due to glowing soot. This ﬂame luminosity can be reduced by placing a narrowbandwidth interference ﬁlter tuned to the wavelength of the PIV laser in front of the sensor or collecting lens. Due to the rather long sensor exposure of the second image frame in modern PIV cameras, this ﬁlter may however be insuﬃcient in the suppression of the ﬂame luminosity. Here fast acting electromechanical or electro-optical shutters [375, 377] are required. Alternatively a pair of CCDs could be used, each synchronized to one of the two PIV laser pulses [111]. The pressurized single sector combustor, schematically shown in ﬁgure 9.75, can be operated at up to 20 bar with air preheating of up to 850 K at mass ﬂow rates of 1.5 kg/s. The nozzle plenum is supplied with preheated primary air downstream of a critical throttle. The primary air is split in a 2:1 ratio and guided into the double swirl nozzle and the inner window cooling slits, respectively. The pressure and mass ﬂow in the chamber is controlled through a sonic oriﬁce at the exit. Jets in cross-ﬂow arrangement at a roughly mid-length position of the 250 mm long combustor provide preheated mixing air and conﬁne the primary zone to a roughly cubic volume.

9.14 PIV in Reacting Flows

341

Fig. 9.75. Single sector pressurized combustion facility. Right: cross-sectional axial view.

Optical access is granted from three sides through windows consisting of a thick pressure window and a thin liner window. The gap between the windows is purged with cooling air while the inside of the liner window is ﬁlm-cooled using a portion of the plenum air. For PIV the light sheet was aligned with the burner axis with the camera arranged in a classical light sheet normal viewing arrangement. By allowing the laser light sheet to pass straight through the combustor the amount of laser ﬂare on imaging windows and walls could be kept at an acceptable level. Seeding consisting of amorphous silicon dioxide particles was introduced to the plenum upstream of the burner through a porous annular tube. As the window ﬁlm-cooling is supplied directly with seeded air from the plenum, the windows are unfortunately subjected to an accelerated build-up of seeding deposits. Preferably the ﬁlm-cooling air should have been separated from the main burner air. Fired with kerosene the combustor provided PIV images exhibiting strong Mie scattering oﬀ the kerosene spray as well as strong ﬂame luminosity. A corresponding PIV result obtained at lean operating conditions with less kerosene spray is provided in ﬁgure 9.76. Image enhancement as shown in ﬁgure 9.77 was applied prior to PIV processing and reduced the inﬂuence of droplet velocities on the air ﬂow velocity estimates by equalizing the droplet image intensities with the much weaker intensity of the seeding particles [377].

342

9 Examples of Application Vorticity [sec-1]

50 m/s

y [mm]

30

20

10

0

Vorticity [sec-1]

50 m/s -40000

-20000

0

20000

40000

y [mm]

30

20

10

0

-30

-20

-10

0 x [mm]

10

20

30

Fig. 9.76. Velocity map (top) and vorticity map (bottom) obtained at 3 bar.

The application of stereo PIV in facilities of this type is not trivial even if optical access seems suﬃcient. Aside from the loss of common viewing area caused by the oblique views of two cameras through a common window, further problems are introduced by the reﬂections of laser ﬂare from the light sheet entering the test section through an orthogonal window. Hence the most desirable arrangement is the ‘classical’ normal view of the light sheet through a window that itself is parallel to it. For stereoscopic viewing the second camera will suﬀer from the reﬂection and occlusion eﬀects described before such that reconstructed 3C velocity data will be available in a reduced area. One solution to the limited access problem of multi-camera PIV imaging is to combine standard 2C PIV with Doppler global velocimetry (DGV) which

9.14 PIV in Reacting Flows

(a)

(d)

(b)

343

(c)

(e)

Fig. 9.77. Portion (120 × 100 pixel) of PIV recordings - inverted for clarity - and processed PIV data: a) fuel droplets, b) brightened version of a) with seeding visible, c) pre-processed image, d) ﬂow ﬁeld after standard PIV analysis, e) ﬂow ﬁeld obtained after enhancement of PIV images (from [377]).

has a sensitivity to the out-of-plane component [468, 471]. The so-called DGVPIV method was applied to recover the velocity ﬁeld inside the dilution zone of the single sector combustor [378]. Since the dilution zone could only be observed through two windows opposite to each other the light sheet was introduced through the top of the combustion chamber. Traversal of the entire acquisition system allowed the recovery of time-averaged volume resolved data sets of the dilution zone at pressures of up 10 bars [378]. In this particular application DGV and PIV were applied in succession due to technical limitations with the lasers used for illumination. Truly simultaneous DGV-PIV measurements could be demonstrated on a free jet experiment by Wernet [472].

344

9 Examples of Application

9.15 A High-Speed PIV Study on Trailing-Edge Noise Sources Contributed by: A. Schr¨ oder, U. Dierksheide, M. Herr, T. Lauke

Table 9.20. PIV recording parameters for high-speed PIV (HS-PIV) measurements at the trailing-edge of a ﬂat plate. Flow geometry Maximum velocity Field of view Interrogation volume Dynamic velocity range Observation distance Recording method Recording medium Recording lens Illumination Pulse delay Seeding material

perpendicular to trailing-edge 50 m/s 140 × 37 mm2 4 × 4 × 0.7 mm3 0 – 50 m/s 1.2 m double frame/single exposure/4 kHz CMOS-camera f = 105 mm, f# = 1.8 Nd:YLF laser, 7 mJ per pulse Δt = 20 s DEHS (dp ≈ 1 μm)

9.15.1 Introduction Airframe noise is essentially due to the interaction of unsteady, mostly turbulent ﬂow with the structure of the airplane, particularly caused by vortical ﬂows around edges or over open cavities. A classical problem in this ﬁeld is the trailing edge noise, which involves diﬀerent noise generating mechanisms. Extensive investigations have been conducted on airfoil- and on ﬂat plate trailing edges. According to [380, 381, 382] the major noise contribution is provided by the span-wise component of vorticity, the corresponding dipole (“principal edge noise dipole”) is the sc. perturbed Lamb vector being perpendicular to the plane of the plate. A numerical simulation of trailing edge noise can be performed, based on such HS-PIV input data. 9.15.2 Setup, Measurements and Procedure A ﬂat plate (chord based Re = 5.3 × 106 and 6.6 × 106 ) with proﬁled leading and trailing edges was mounted vertically in the Aeroacoustic Wind Tunnel Braunschweig (AWB), which is an open jet anechoic test facility (see Figures 9.78). The ﬂat plate boundary layer was tripped at the leading edge, reaching a thickness of δ = 0.03 m on each side of the trailing edge, corresponding to free stream velocities of U = 40 m/s to 50 m/s and a chord length of 2 m. Towards the trailing edge the plate is slightly and symmetrically convergent (5◦ taper), but no ﬂow separation occurs. A full description of the experimental setup is provided in [379, 383]. The PIV measurement volume was located at the trailing edge in a x-y-plane within the turbulent boundary layer in order to track the ﬂow structures with a spatial resolution of 256 pixel in y- and

9.15 A High-Speed PIV Study on Trailing-Edge Noise Sources

345

Fig. 9.78. Setup of the high-speed-PIV system at AWB (left) and the directional microphone focused at the measurement volume at the trailing-edge (right).

1024 pixel (corresponding to 135 mm) in x-direction. The used high-speed PIV system consists of a New Wave Pegasus PIV, dual cavity Nd:YLF laser with an output beam wavelength of 527 nm, a pulse length of 135 ns and 2 × 10 mJ at 1 kHz and approximately 2 × 7 mJ at 4 kHz for each resonator, optics to produce a light sheet and a HighSpeedStar4 (HSS4) CMOS camera with a spatial resolution of max. 1024 × 1024 pixel at 2 kHz frame rate. In this application, a double frame rate of 4 kHz was achieved, thus yielding images at 8 kHz with a spatial resolution of 1024 × 256 pixel and a 10 bits gray-scale dynamics. 2.6 GB camera memory inside the camera housing allowed to capture 4096 double-images per run. The camera lens was a Nikon 105 mm with an aperture of f# = 1.8. The evaluation of the particle images was performed with a cross correlation scheme using standard FFT with multi-pass (four iteration steps), image deformation, interrogation window shift and a ﬁnal window size of 32 × 32 pixel, with 75% overlap, corresponding to a resolution down to 3 mm in both directions. The Whittaker reconstruction was used for the deformation scheme and peak detection was achieved by a three point Gaussian ﬁt. For post-processing, a median ﬁlter was used to remove outliers. As an example, ﬁgure 9.79 shows an instantaneous velocity ﬁeld out of a run of 4096 velocity vector ﬁelds measured in one second. A “straight forward” method which should result in a direct calculation of the source terms and therefore a reconstruction of the whole sound ﬁeld was applied: As the major vortex source term, the perturbed Lamb vector, i. e. the source term of the acoustic analogies of [380, 381, 382], was directly computed from the measured HS-PIV velocity ﬁeld quantities (namely the instantaneous velocity, vorticity and the mean ﬂow). After linear interpolation onto the body-

346

9 Examples of Application

Fig. 9.79. Instantaneous velocity vector ﬁeld (v scalar ﬁeld gray leveled) of a 4 kHz run in the trailing edge region (top) and the corresponding directional microphone signal (actual value at the arrow position).

ﬁtted block-structured grid for the trailing edge, these source term values were fed into the subsequent computational aeroacoustic (CAA) simulations. Assuming a mean ﬂow at rest (Ma = 0) the computations were performed by the DLR acoustic code PIANO (Perturbation Investigation of Aerodynamic Noise), which in this case solves the acoustic perturbation equations.

Fig. 9.80. Frequency spectra at diﬀerent ﬂow velocities measured by the directional microphone (left) and the pressure wave calculations by CAA code PIANO on the basis of HS-PIV data (right).

9.16 Volume PIV

347

9.15.3 Conclusions The example presented is one of the ﬁrst applications of highly time-resolved PIV to a classical aeroacoustic problem at industrially relevant Reynolds numbers. A new method for the prediction of trailing edge noise was suggested with the future aim to compute the noise ﬁeld and directivity by using PIV data, namely the aeroacoustic source quantities, as input for a CAA calculation. Both high-speed PIV and acoustic experiments (for a later validation of the suggested method) were performed on a ﬂat plate model in an aeroacoustic wind tunnel. The PIV data-set was captured at a double-frame rate of 4 kHz with a suﬃciently large ﬁeld of view and enough spatial resolution to resolve all main features of the sound generating ﬂow. Time resolved PIV allows the non-intrusive quantiﬁcation of the relevant ﬂow parameters and helps to investigate vortex- structure interactions. In terms of the aeroacoustic optimization of existing aircraft components such an ”optical” detection of aeroacoustic source terms will be beneﬁcial, since a huge amount of (at least low-speed) problems could be investigated at lower costs without the need of quiet test facilities.

9.16 Volume PIV Contributed by: A. Schr¨ oder, G.E. Elsinga, F. Scarano, U. Dierksheide

The formation and development of a turbulent spot in a laminar boundary layer ﬂow is governed by the self-organization of generic substructures like hairpin-vortices and spanwise alternating wall bounded low- and high-speed streaks which eﬀectively produce new turbulent ﬂow structures out of the oncoming laminar ﬂow reaching the spot at the rear edge. Although the main ﬂow structures are identiﬁed and the exchange topologies in wall normal directions inside the spot are depicted roughly the principle growth mechanism of the turbulent spot itself is still not fully understood [242]. The analysis of this growing process would also help to understand the function of very similar structures for the turbulent exchange in fully developed turbulent boundary layer ﬂows [384]. In this feasibility study the volume (tomographic) PIV technique [230] has been applied to time resolved PIV recordings in order to capture the development of the ﬂow structures especially the rapid formation process of hairpin vortices at the rear edge of the spot in a time series of a whole volume of the boundary layer ﬂow. This measurement method oﬀers the unique possibility to determine the complete three-dimensional velocity gradient- tensor within the measurement volume.

348

9 Examples of Application

Fig. 9.81. Experimental setup at a ﬂat plate boundary layer ﬂow at a free stream velocity of U = 7 m/s consisting of four Photron APX-RS CMOS cameras enabling time resolved tomographic PIV measurements of turbulent spots at the open test section of the 1m-wind tunnel of DLR Gttingen.

Four high speed CMOS cameras are imaging tracer particles which were illuminated in a volume inside a boundary layer ﬂow at 5 kHz by using two high repetitive Nd:YAG pulse lasers (see ﬁgure 9.81). The instantaneously acquired single particle images of these cameras have been used for a three dimensional tomographic reconstruction of the light intensity distribution of the particle images in a volume of voxels (volume elements) virtually representing the measurement volume. Each of two subsequently acquired and reconstructed particle image distributions are cross-correlated in small interrogation volumes using iterative multi-grid schemes with volume-deformation in order to determine a time series of instantaneous 3D-3C velocity vector ﬁelds. The measurement volume with a size of ∼ 34 × 19 × 30 mm3 was located near the wall in a ﬂat plate boundary layer ﬂow with zero pressure gradients and downstream of a local disturbance source. At a free stream velocity of U = 7 m/s a turbulent spot grows downstream inside a laminar ﬂat plate boundary layer ﬂow after introducing a short initial ﬂow injection and convects through the measurement volume. The time resolved tomographic PIV method enables capturing the spatio-temporal development of the complete ﬂow structures and in particular the rapid formation process of hairpin-like vortices at the trailing-edge of the spot (see ﬁgure 9.82). This step to a full 3D-3C and time resolved PIV measurement technique demonstrates the applicability of optical measurement techniques as an im-

9.16 Volume PIV

349

Fig. 9.82. Three instantaneous 3D-3C- velocity vector ﬁelds at y = 5.6, 6.1 and 6.6 mm of the same turbulent spot at T = 87.4 (top), 87.8 (center) and 88.2 ms (bottom) after initial disturbance showing rapid growth of Q2 events or hairpin-like vortices at the TE (u∞ = u − 6.6 m/s, V y gray scaled.).

350

9 Examples of Application

portant tool and complementary data source to CFD for the understanding of wall bounded turbulence at high Reynolds numbers and more in general of complex and unsteady phenomena in ﬂuid mechanics.

9.17 Supersonic PIV Measurements on a Space Shuttle Model Contributed by: J.T. Heineck, E.T. Schairer, S.M. Walker

Table 9.21. PIV recording parameters for transonic ﬂow above a NACA0012 airfoil. Recording geometry Maximum cross-plane velocity Maximum in-plane velocity Field of view Interrogation volume Observation distance Recording method Recording medium Recording lens Illumination Seeding material

a

light sheet swept 15◦ from normal to the stream Umax = 585 m/s Vmax = 130 m/s 870 × 380 mm2 20 × 10 × 4 mm3 (H × W × D) 1.8 m – 3.2 m dual frame - single exposure CCD 1386 × 1024 px f = 35 mm, f# = 2.0 and 50 mm, f# = 1.2 two Nd:YAG lasera 250 mJ/pulse oil droplets (dp ≈ 0.3 μm – 0.5 μm)

frequency doubled

Before the Space Shuttle could return to ﬂight after the loss of Columbia, NASA was required to validate the computational ﬂuid dynamics (CFD) codes that it uses to predict the trajectories of debris that may be shed from the vehicle during launch. To meet this and other requirements, NASA conducted two tests of a 3% scale model of the Shuttle ascent conﬁguration in the NASA Ames 9 × 7 ft2 Supersonic Wind Tunnel (9 × 7 SWT). In these tests, Dual Plane Particle Image Velocimetry (PIV) was used to measure the three components of velocity upstream of the Orbiter wings where debris shed from the External Tank (ET) would be convected downstream. The measurements were made in four cross-stream vertical planes located at diﬀerent axial positions upstream of the Orbiter and above the ET. The measurements were made at

9.17 Supersonic PIV Measurements on a Space Shuttle Model

351

Fig. 9.83. Sample plot of normalized axial velocity in the most upstream measurement plane M a∞ = 2.5, α = 0◦ , β = 0◦ .

two Mach numbers (1.55 and 2.5) over a range of model attitudes. The high stream velocity necessitated the use of the dual plane technique. These measurements revealed a complex network of interacting shock waves and a region of turbulent, separated ﬂow on the ET just upstream of the Orbiter-to-ET attach point (“bipod”), where foam broke loose during Columbia’s ﬁnal ﬂight. Figure 9.83 shows average axial velocities in the most upstream measurement plane for a typical case. Higher spatial-resolution measurements were made in a single vertical plane in the separated-ﬂow region above the Intertank section of the ET. More than 7000 samples were acquired at a single test condition to allow computing turbulence statistics. Figure 9.84 shows a plot of the overall velocities measured at this position. Figure 9.83 clearly shows the bow shock-wave from the nose of the External Tank (ET). The data are not laterally symmetric because the measurement plane was not perpendicular to the ﬂow (it was yawed 15◦ ). In addition, the cable tray on the starboard side of the ET ogive (ﬁgure 9.83) probably induced ﬂow asymmetry. Figure 9.84 shows the shock wave from the ﬂange at the upstream edge of the Intertank. As in ﬁgure 9.83, the measurement plane is yawed with respect to the freestream direction. The lower-speed region near the surface is a separation bubble. Turbulence statistics were derived from this dataset.

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Fig. 9.84. Contours of mean axial velocity with vectors indicating mean spanwise and vertical velocities ( m/s). M a∞ = 2.5, α = 0◦ , β = 0◦ .

An adjustable dual-plane laser projection system was required to allow the position of the second plane to be remotely moved downstream during testing operations. This was necessary because the Mach number range was too large to allow a ﬁxed downstream separation of the ﬁrst and second laser sheets. Each sheet was produced from separate laser heads, with each head providing 250 mJ per pulse. The laser heads were rotated 90◦ with respect to each other to allow the beams to be combined using a polarized beamsplitter cube. The beam from the second laser was reﬂected into the cube by a mirror mounted on a high-resolution translation stage. With this arrangement, the separation of the laser sheets corresponded to the readout of the translation stage controller. The plate that supported both lasers and all of the optics was carried by a linear traverse that provided one meter of displacement in the streamwise direction. This traverse permitted remote control of the streamwise locations of the measurement planes.

9.18 Multiplane Stereo PIV

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9.18 Multiplane Stereo PIV Contributed by: C.J. K¨ ahler

9.18.1 Introduction Particle Image Velocimetry (PIV) has become a widely applied technique whenever the spatial distribution of the velocity together with its derivatives helps to understand the physics of the ﬂow. However, quite often the distribution of the velocity within one single plane, captured at one instant in time, does not yield the information required to answer ﬂuid-mechanical questions. To overcome these limitations a stereoscopic PIV based technique has been developed, which is well suited to determine many ﬂuid-mechanical quantities with high accuracy and spatial resolution at any ﬂow velocity [200, 390, 392, 393]. This technique is reliable, robust and easy to handle. Furthermore it is based on standard PIV equipment and evaluation procedures so that available PIV systems can be easily extended. The multiplane stereo PIV system, developed for applications in air ﬂows in particular, consists of a four-pulse laser system delivering orthogonally polarized light, two pairs of high resolution progressive scan CCD cameras in an angular imaging conﬁguration with Scheimpﬂug correction, two high reﬂectivity mirrors and a pair of polarizing beam-splitter cubes according to ﬁgure 9.85. After the illumination of the tracer particles with orthogonally linearly polarized light, the polarizing beam-splitter cube (7) separates the incident

α1 α2 6 5

7

8 3

4

2

s-polarized light p-polarized light

1

Fig. 9.85. Schematic setup of the recording system. 1-4 digital cameras, 5 lens, 6 mirror, 7 polarizing beam-splitter cube with dielectric coating between the two right-angle prisms, 8 absorbing material, α opening angle.

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9 Examples of Application 4 2

7

1 5 6

3

12

12

7b

11 8a

5

12

10 9

11 8c 8b

Fig. 9.86. Four-pulse four frequency Doppler laser system. 1 Pump cavity, 2 Full reﬂective mirror, 3 Partially transmitting mirror, 4 Pockels cell, 5 λ/4 retardation plate, 6 Glan-Laser polarizer, 7 Mirror, 8 Dielectric polarizer, 9 Dichroic mirror, 10 Frequency doubler crystal with phase angle adjustment, 11 λ/2 retardation plate, 12 Beam dump.

wave-front scattered from the particles into two parts according to the state of polarization. The separation based on polarization works perfectly as long as the radius of the spherical particles is comparable to the wavelength of the laser light (see also section 2.1 and [72, 73] for the generation of appropriate particles) and the observation direction is properly aligned relative to the direction of the polarization vector. In the case that these requirements cannot be fulﬁlled a frequency based multiplane stereo PIV approach can be applied but in this case the laser system has to be modiﬁed to generate the required frequency shift [395]. For the illumination of the tracer particles the beams of four independent laser-oscillators need to be combined in such a way that the linearly polarized light sheets can be positioned independently with respect to each other. This can be easily and precisely done by means of the four pulse system shown in ﬁgure 9.86. The appropriate method of adjusting the displacement between the orthogonally polarized light-sheets depends on the desired distance [389, 390]. Small separations between the orthogonally polarized light-sheet pairs (up to a few millimeters) can be generated by a simple rotation of mirror 8c in the re-combination optics around the axis perpendicular to the laser-beam plane [393]. For a wider range of light-sheet spacings (up to a few cm) and independent positioning of both beam-pairs, it is useful to remove mirror 7b along with the beam dump (12) such that two spatially separated laser beams with orthogonally polarized radiation emerge. Using two separate light-sheetoptics (one for each polarization) each with a 45◦ mirror behind the last lens, all positions are possible by moving the mirrors [389]. Once calibrated, the actual position of each pair of light-sheets is determined with a micrometer scale. The multiple plane stereo system is well suited to determine diﬀerent ﬂuidmechanical quantities without perspective error simply by changing the time sequence or light sheet position. For constant pulse separation (Δt = t2 − t1 = t3 − t2 = t4 − t3 ) and overlapping light sheets (see ﬁgure 9.87), a time

Intensity

9.18 Multiplane Stereo PIV t1

t2

t3

355

t4

t

z

Fig. 9.87. Timing diagram for the temporally separated determination of all three velocity components. Diﬀerent shading of the light-sheet proﬁle indicates diﬀerent states of polarization.

Intensity

sequence of three velocity ﬁelds can be measured at any ﬂow velocity by cross-correlating the ﬁrst acquired gray-level distribution with the second, the second with the third and the third with the gray-level distribution from the last illumination. In this mode it is also possible to increase the accuracy of the velocity measurement by using the Multiframe PIV evaluation approach outlined in section 9.7 and [323]. By increasing the time delay between the second and third illumination (Δt = t2 − t1 = t4 − t3 < t3 − t2 ) the ﬁrst order estimation of the acceleration ﬁeld in its Lagrangian and Eulerian form can be calculated to study the dynamic behavior and the interaction processes of moving ﬂow structures [388]. For large time delays between a pair of images being acquired (Δt = t2 − t1 = t4 − t3 t3 − t2 ), time correlations can be measured for instance [389]. When the light sheet pairs with equal polarization are spatially separated, as indicated in ﬁgure 9.88, further important information about the ﬂow ﬁeld can be achieved. For small separations or partially overlapping light-sheets, the spatial distribution of all three vorticity vector components can be measured when the orthogonal polarized light-pulses are ﬁred simultaneously (t1 = t3 and t2 = t4 ). In addition, all components of the velocity gradient tensor can be estimated along with the invariants of this tensor [390, 396]. This allows vortex identiﬁcation to be made more reliably in combination with topological ﬂow analysis. By increasing the distance between the light sheet pairs, the spatial correlation tensor at diﬀerent locations within the ﬂow ﬁeld

t

Δz

z

Δt

Fig. 9.88. Timing diagram for the simultaneous determination of all three velocity components in spatially separated planes.

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can be measured [394]. Furthermore the orientation of a vortex crossing the planes can be determined precisely which is of major importance for aircraft wake-vortex investigations for example. Time-correlations can also be deduced from the data by varying the time between the horizontal and vertical pairs of illuminations [389]. 9.18.2 Application To demonstrate the capabilities of the technique a ﬂat-plate boundary layer ﬂow was examined in stream-wise span-wise planes located at y + = 10, y + = 20 and y + = 30. The experimental investigation was performed 18 m behind the leading edge of the ﬂat plate placed in the temperature-stabilized closed circuit wind tunnel at the Laboratoire de M´ecanique de Lille (LML), see [386] for details. The ﬂow and recording parameter are listed in table 9.22. Table 9.22. Relevant parameters for the characterization of the experiment performed 18 m behind the leading edge of the ﬂat plate in the xz-plane (stream-wise span-wise) of the turbulent boundary layer ﬂow. Reθ 7800 74000 Reδ 3.6 × 106 Rex 3 U∞ 0.121 uτ δ 0.37 3000 δ+ t+ = tu2τ /ν ﬁeld of view 67 × 35 ﬁeld of view 0.18 × 0.09 ﬁeld of view 544 × 284 spatial resolution 1.42 × 0.6 × 1.42 spatial resolution 11.5 × 5.0 × 11.5 pulse separation Δt 200 1.00 to 9.330 dynamic range at y + = 10 1.37 to 11.74 dynamic range at y + = 20 2.61 to 11.84 dynamic range at y + = 30 vectors per sample 7936 number of samples 4410

[1] [1] [1] [ m/s] [ m/s] [ m] [1] [1] [ mm2 ] [δ 2 ] [Δx+ Δz + ] [ mm] [Δx+ Δy + Δz + ] [ μs] [ pixel] [ pixel] [ pixel]

For phenomena associated with the production of turbulence, the cross correlation between the ﬂuctuating stream-wise u and the wall-normal v velocity components, measured simultaneously at diﬀerent y + -locations, is very important, as Rvu reﬂects the size and shape of the coherent ﬂow structures being responsible for the transport of relatively low-momentum ﬂuid outwards

9.18 Multiplane Stereo PIV Ru(y+=20) v(y+=10)

Rv(y+=30) u(y +=20)

Ru(y+=30) v(y+=20)

Rv(y +=20) (+u)(y+=10))

Rv(y+=30) (+u)(y+=20))

Δ z+

Δ z+

Δ z+

Rv(y+=20) u(y +=10)

357

Δ x+

Δ x+

Fig. 9.89. Two-dimensional spatial cross-correlation function of ﬂuctuating streamwise (u) with wall-normal (v) velocity components measured at Reθ = 7800. Top: Rv(y + =20)u(y + =10) (left) and Ru(y + =20)v(y + =10) (right). Centre: Rv(y + =30)u(y + =20) (left) and Ru(y + =30)v(y + =20) (right). Bottom: Rv(y + =20)u(y + =10) with u > 0 (left) and Rv(y + =30)u(y + =20) with u > 0 (right).

into higher speed regions and for the movement of high-momentum ﬂuid toward the wall and into lower speed regions. Figure 9.89 displays the statistical relation between the ﬂuctuating velocity components. The top row shows the cross-correlation Rv(y+ =20)u(y+ =10) (left) and Ru(y+ =20)v(y+ =10) (right). It can be stated from the negative sign of the correlation, indicated by the dashed lines, that the ejection and sweeps must be the predominant processes and the diﬀerent size, shape and intensity of the functions (Rv(y+ =20)u(y+ =10) > Ru(y+ =20)v(y+ =10) ) imply the dominance of ejection at these wall locations. In addition, it can be estimated from the location of the maximum in the upper left graph that the low momentum region appears as a shear layer in the y-direction while the strong positive side

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peaks in the same ﬁgure indicate that the outﬂow of low-momentum ﬂuid is associated with a secondary motion. The center row of ﬁgure 9.89 reveals the same functions but measured at y + = 20 and y + = 30 e.g. Rv(y+ =30)u(y+ =20) (left) and Ru(y+ =30)v(y+ =20) (right). The bottom row of ﬁgure 9.89 yields the conditional cross-correlation Rv(y+ =20)u(y+ =10) with u > 0 (left) and Rv(y+ =30)u(y+ =20) with u > 0 (right). Especially the amplitude of Rvu in the lower right plot should be noted. To estimate the convection velocity of the various coherent ﬂow structures, and further dynamical aspects which are not accessible by using standard, stereoscopic or holographic PIV techniques, space-time correlations were measured in addition. See [389, 390, 391] for details. 9.18.3 Conclusion The Multiplane Stereo PIV system is very reliable, robust and well suited for all kinds of applications, purely scientiﬁc as well as for industrially motivated investigations in large wind tunnels where acquisition time, optical access and observation distances are constrained. Furthermore, it is based on the conventional PIV equipment and the familiar evaluation procedure so that available PIV systems can easily be expanded. The advantage of this measurement system with respect to other imaging techniques lies in its ability to determine a variety of fundamentally important ﬂuid-mechanical quantities with high accuracy (no perspective error), simply by changing the time sequence or light sheet position. Due to the advantage of this technique relative to conventional PIV systems, it is increasingly applied by other leading research groups all over the world [385, 387, 396].

9.19 Microscale PIV Wind Tunnel Investigations Contributed by: M. Raﬀel, C. Rondot, D. Favier, K. Kindler Detailed studies of the boundary layer proﬁle and the characteristics of the ﬂow velocity distribution close to the leading edge of a helicopter blade proﬁle were conducted using 2C-PIV. The relatively small scales of ﬂow structures related to dynamic stall, the study in which the ﬂow ﬁeld has been measured with a relatively high spatial resolution. The feasibility of μ-PIV measurements utilizing a mirror telescope in a wind tunnel has been demonstrated successfully. The spatial resolution of approximately 50 μm allowed for an assessment of the diﬀerent turbulence models and damping coeﬃcients for the improvement of CFD predictions.

9.19 Microscale PIV Wind Tunnel Investigations

359

Table 9.23. PIV recording parameters for microscale wind tunnel investigations. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Observation distance Recording method Recording medium Recording lens Illumination Pulse delay Seeding material a

Boundary layer on an OA209 blade tip model Umax = 10 mm/s 1.658 × 1.358 mm2 96 × 96 pixel z0 = 35.5 cm double frame/single exposure CCD camera, 1280 × 1024 pixel Mirror objective lens, M = 4.86, f# = 6 Nd:YAG lasera 2 × 200 mJ/pulse Δt = 10 ms oil droplets (dp 1 μm)

frequency doubled

9.19.1 Introduction Over the past decade, considerable progress has been made in the development of performance prediction capabilities for isolated helicopter components. Modern CFD methods deliver promising results for moderate operation conditions. The prediction of high-speed and high-load cases still needs more intensive experimental investigations of the unsteady viscous ﬂow phenomena, such as the dynamic stall at the retreating side of the rotor and the complex mechanism of the stall in the vicinity of the blade tip. Overall ﬂow ﬁeld measurements on pitching airfoils, pitching ﬁnite blade models and on rotating blades in hover chambers and wind tunnels have been successfully performed at diﬀerent places. In this section we focus on measurements, which have been obtained for a steady incidence angle of 11.5◦ , since the ﬂow phenomena involved are best understood and documented. This incidence angle corresponds to the point where maximum lift is obtained, shortly below the incidence angle where massive ﬂow separation occurs. The ﬂow around the OA209 proﬁle for this range of Reynolds number, span wise location and incidence angle is determined by the transition of the boundary layer and the ﬂow separation on the suction side which results in the generation of vorticity dominating the wake ﬂow and the performance of the wing. This holds in a similar way for ﬁnite wings and 2D-airfoil proﬁles. For the present case of moderate Reynolds numbers and high incidence angles, laminar ﬂow separation occurs shortly behind the leading edge and transition to turbulent ﬂow conditions occurs immediately after separation. The resulting turbulence intensity forces a reattachment of the ﬂow within a short distance, resulting in a signiﬁcantly increased maximum lift with respect to the low Reynolds number cases. The separation together with the re-attachment forms a laminar separation bubble containing a recirculation region, which has only a few millimeter extension in chord wise direction (see ﬁgure 9.90). The turbulent boundary layer behind the bub-

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9 Examples of Application

Fig. 9.90. Sketch of the separation bubble.

ble allows the ﬂow to stay attached even at relatively high adverse pressure gradients. Detailed investigations of the turbulence intensity, the size and the temporal development of the ﬂow structure at the leading edge are required in order to validate CFD-codes which are under development for a more accurate prediction of the dynamic stall cases. Therefore, stereoscopic PIV and pressure measurements have been performed to quantize the overall ﬂow features close to the tip of a rotor blade, both in steady cases and during pitching motion. Two-components PIV measurements with an observation ﬁeld size of 1 mm and 50 μm resolution, have been performed further inboard in order to resolve the relevant ﬂow features in the phase shortly before the stall onset in steady and unsteady cases. The results of the steady case have been compared with ELDV (Embedded Laser Doppler Velocimetry see [431]) and CFD data. 9.19.2 The Test Setup The laser system used had 2×200 mJ pulse energy at 532 nm and was equipped with conventional light sheet optics. The cameras had a resolution of 1280 × 1024 pixel. The setup used for the PIV measurements with a very high resolution is similar to the one described in [432], but has been used under relatively rough conditions in a wind tunnel. The microscope lens used for the test was a mirror objective lens QM100 of Questar Corporation. It is optimized for working distances G, ranging from G = 150 mm to G = 380 mm. It has an aperture angle of ω = arctg(D/2G) with D being the aperture diameter. The numerical aperture for the working distance of G = 355 mm, which has been chosen for the experiment, is A = n · sin ω = 0.083 with n being the refractive index of air. The F-number was f# = 1/2A = 6 and the magniﬁcation M = 4.86 resulting in a calibration factor of 754 pixel/mm. The diﬀraction limited minimum image diameter was ddif f = 2.44 f# (M + 1) = 45.6 μm and the estimated depth of focus δZ = 2f# ddif f (M + 1)/M 2 = 136 μm. The light

9.19 Microscale PIV Wind Tunnel Investigations

361

sheet thickness was also 400 μm. Particle image diameters observed were between 50 and 130 μm (8–20 pixel) which is similar to observations made in [283, 284]. The development of the boundary layer, the reverse ﬂow region and the shear layer towards the outer ﬂow can clearly be seen in the results presented below. The experiments have been performed in the S2 Luminy wind tunnel of the Laboratoire d’A´erodynamique et de Biom´echanique du Movement LABM of the French research center CNRS at the University of Marseille. The PIV measurements were performed close to the leading edge of an OA209 blade tip model, in a plane orthogonal to the span in a distance of approximately 200 mm. The majority of the μ-PIV measurement has been made at a steady incidence angle of 11.5◦ and has been compared with CFD and ELDV results at the same condition. 9.19.3 Results and Discussion The image quality obtained with the mirror telescope objective, allows for the analysis of the unsteady ﬂow features at a spatial resolution of 50 μm. The number of outliers is of the order of 5%. The relative accuracy, as compared to conventional PIV recordings, is slightly lower due to the fact that the particle images are approximately ﬁve times larger. However, the uncertainty due to noise is assumed to be in the order of 0.1 m/s and the wall distance of each measurement location can be determined very precisely, since the surface is visible in each recording. Therefore, the development of the boundary layer, the reverse ﬂow region and the shear layer towards the outer ﬂow can clearly be seen in the result ﬁgure 9.91.

Fig. 9.91. PIV results obtained with a mirror telescope objective at α = 11.55◦ , steady case. The magnitude of the velocity has been plotted by gray levels. The coordinates x and y are given in millimeters. The origin is placed on the model surface (y), 5% chord length behind the leading edge (x).

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9 Examples of Application

Fig. 9.92. CFD, ELDV and PIV results of the laminar separation bubble at α = 11.5◦ and x/c = 0.05; steady case.

Figure 9.92 depicts tangential velocity proﬁles through the laminar separation bubble obtained by CFD with SA- and SST turbulence models and by ELDV, PIV and μPIV. It can be seen that the agreement of most of the diﬀerent methods in the outer regions (20 mm and above) is relatively high (∼98%) with respect to the free stream velocity). However, the more detailed presentation in ﬁgure 9.92 shows that the diﬀerences between the diﬀerent experimental results as well as of the diﬀerent CFD results becomes evident. The reason for the diﬀerences of the PIV and the μ-PIV results can easily be explained by the weak spatial resolution of the recordings made with the 100 mm-lens. The diﬀerences between the ELDV measurements and the μ-PIV measurements are more signiﬁcant and diﬀer not as much in the measured ﬂow velocity, but in the size of the separation bubble in wall normal direction. One reason for this might be a small disagreement of the incidence angle adjustment of both tests. However, the conclusion, the conclusion which turbulence model resolves the ﬂow ﬁeld in the separation bubble best, can easily be drawn in favor for the SA-turbulence model. 9.19.4 Conclusions Discrepancies concerning the size of the separation bubble on a helicopter blade proﬁle have been observed as well as an acceptable agreement of the velocity magnitudes found by the diﬀerent measurements. The present results can be considered to be a good data basis for the validation of numerical

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363

codes. However, the ﬁnely structured vortices observed at high incidences and their complex evolution during some phases of the pitching motion is not yet suﬃciently predicted by two-dimensional computations. It has been demonstrated that instantaneous velocity ﬁelds determined by PIV at high spatial resolutions can be used to choose turbulence models and numerical damping coeﬃcients. The strength, scale, and distribution of the laminar separation bubble - measured for the ﬁrst time with such a high resolution in a wind tunnel experiment - are essential for the validation of numerical simulation techniques.

9.20 Micro-PIV I Contributed by: M. Oshima, H. Kinoshita

9.20.1 Application of PIV to Microscopic Flow Microﬂuidics that deals with ﬂuid ﬂow phenomena in microscale is becoming a signiﬁcant research subject with the development of microﬂuidic devices or systems [405]. In the ﬁelds of chemistry, biochemistry, and biology, various types of microﬂuidic devices have been designed and demonstrated with the aim of developing the miniature-sized bio/chemical analysis apparatuses that are usually referred to as micro total analysis systems (μTAS) or laboratoryon-a-chip [397, 409]. Downsizing of analytical apparatuses brings us many potential proﬁts, such as reduced amount of sample and reagent, high sensitivity, short analysis time, automation of processes, parallel processing, etc [399, 401, 402]. In order to make eﬀective use of those advantages and work out a hydrodynamically-designed system or geometry, the key issue is better understanding of the ﬂuid ﬂow phenomena taking place in such a microﬂuidic device. Over the last several years, microscopic PIV (μPIV) has been studied and developed for use as a diagnostic tool of microscopic ﬂow ﬁelds. The idea of that is very simple: an optical microscope is used for imaging of a particleseeded ﬂow in microscale instead of photographic lenses in macroscale. Nanosized ﬂuorescent beads are suspended as the tracers within working ﬂuid, and then the distribution patterns of the tracer particles are imaged sequentially with an epiﬂuorescent microscope and a video camera. The captured particle images are analyzed in the same way as standard macroscale PIV. 9.20.2 Examples of Micro-PIV Santiago et al. [410] developed a micro PIV system using an epiﬂuorescent microscope with a high numerical aperture (NA) objective lens and an inten-

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9 Examples of Application 50 μm/s

100

100

80

80

60

60

40

40

20

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0 0 a

20

50 μm/s

120

μm

μm

120

40

60 μm

80

0

100

0 b

20

40

60 μm

80

100

Fig. 9.93. Vector ﬁelds of a surface-tension driven Hele-shaw ﬂow around a 30 μm wide obstacle (left: instantaneous, right: time-averaged) [410].

siﬁed CCD camera. The system employed a continuous-illumination mercury lamp and a low frame-rate camera to capture the particle images, so that the time interval between images was relatively large. They successfully measured the low-speed Hele-Shaw ﬂow with a velocity of approximately 50μ m/s using their micro PIV system (see ﬁgure 9.93). The combination of continuous light and CCD video camera is suitable for low-velocity ﬂow measurement, as in the case of electroosmotic ﬂow [411]. Meinhart et al. [406] achieved the frame-straddling micro PIV system using reﬂected-light microscopy with a double-pulsed Nd:YAG laser by improving the lighting system. Figure 9.94 shows a standard μPIV system using a reﬂected-light illumination with a high-power double-pulsed Nd:YAG laser [403]. The dynamic range of PIV can be extended by applying the framestraddling method, which is often used in conventional PIV. Frame-straddling μPIV can be also conducted by illuminating the ﬂow ﬁeld directly from the outside with a pulsed Nd:YAG laser beam rather than using reﬂected-light illumination. PTV technique can also be applied to the microscopic particle images [412] instead of PIV algorithm based on the image correlation method. In microscale, tracking algorithm such as PTV makes a signiﬁcant contribution to investigation of not only ﬂuid ﬂows but also the behavior of solid particles such as cells since PTV enables us to detect the movements of individual particles in time series.

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365

Fig. 9.94. Schematic of a micro PIV system. A high-power pulsed Nd: YAG laser is used to illuminate ﬂuorescent tracer particles through an epi-ﬂuorescent microscope [403].

9.20.3 Diﬀerences from Macroscale PIV Micro-PIV diﬀers from standard macroscale PIV in a few signiﬁcant respects. One of the diﬀerences is the eﬀect of Brownian motion of tracer particles. In micro-PIV, the eﬀect of Brownian motion is not negligible because the diameter of tracer particles is smaller than 1 μm. Those sub-micron particles show relatively large random movements due to Brownian motion under a microscope. The Brownian motion of tracer particles signiﬁcantly aﬀects the basis of velocity estimation in PIV since the PIV method itself is based on the assumption that the tracers follow the ﬂuid motion rigorously. In order to reduce the eﬀect of Brownian motion, the time averaging procedure such as time-averaged correlation method [398, 407] is often conducted. Although time averaging method is useful for reduction of the measurement errors associated with Brownian motion, it is not available for unsteady ﬂow phenomena. Another major diﬀerence lies in the illumination method when taking pictures of tracer particles. In conventional macroscale PIV, the seeding particles in the ﬂow are usually illuminated by a thin planar light sheet so as to visualize the cross-sectional cut plane of the measurement volume. Under a microscope, on the other hand, the sheet lighting proves to be impractical because the ﬂow channel and observation region are certainly smaller than 1 mm and it is diﬃcult to produce the light sheet with the thickness of less than 100 μm and to align the sheet precisely with the focal plane of microscope. For this reason, almost all micro-PIV systems employ the ﬂuorescent microscopy with volume illumination method such as reﬂected- or

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transmitted-light illumination instead of the sheet lighting. Therefore, the depth-of-ﬁeld (DOF) of the microscope plays a signiﬁcant role in micro-PIV measurement. Meinhart et al. [408] particularly deﬁned the measurement depth (MD) as the out-of-plane measurement resolution of micro-PIV, which depends on the tracer particle diameter and NA of objective lens. Only the particles within the MD aﬀect the PIV analysis, whereas the out-of-focus particles do not contribute to the evaluation. Consequently, the obtained velocity data is regarded as two-dimensional projected planar velocity ﬁeld in the focal plane. 9.20.4 Advanced Technique: Confocal Micro-PIV Recently, a new micro-PIV technique has been developed with the aim of resolving the problem associated with the volume illumination and DOF of micro-PIV. It is “confocal micro-PIV”. Confocal microscopy [414] is an advanced technique that oﬀers some advantages over conventional optical microscopy, such as shallow DOF and optical cutoﬀ of out-of-focus light. Application of confocal imaging technique to micro-PIV allows us to slice the seeded ﬂuid volume as if using a thin light sheet in macroscale PIV. Confocal microscopy had scarcely ever been used for micro-PIV till lately because of Desktop PC

Piezo driver

Inverted microscope

X

Y

(Gigabit ethernet)

Imager control, Data transfer

Z

Measurement targer: Microchannel

Objective 40× /1.25 on piezo positioner um ( Z-resolution:0.1 Frequency: < 10 kH ) Z

High-speed camera × 600 pixels, 12-bit ( 800 ) 2000 fps

Confocal scanner CSU22 Optical fiber

ms/frame (Scan speed: 0.5 =2000 fps ) CW laser (532 nm green ) Power: < 1.5 W

Fig. 9.95. Schematic diagram of confocal micro-PIV system. The system consists of an inverted-type microscope, a confocal scanner, a high-speed camera, CW laser, and PC. The target device is ﬁxed on the microscope and observed from the bottom side.

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367

its too slow scanning rate. Although PIV needs at least two subsequent exposures of the seeded ﬂuid ﬂow at a short time interval from nanoseconds to milliseconds, conventional confocal microscopy requires time from seconds to minutes to complete the scanning of a whole planar area. But recently, a highspeed confocal scanner has been developed [413], which is capable of scanning a single cross-sectional plane only in 0.5 ms at 2000 frames per second. Figure 9.95 shows the schematic diagram of confocal micro-PIV system. The measurement target is ﬁxed on the mechanical stage of the invertedtype microscope and the operator observes it from the bottom side. Confocal imaging unit consists of a high-speed confocal scanner, a CW diode laser and a high-speed camera, which is assembled with the side camera port of the inverted-type microscope. The confocal images produced by the confocal scanner are recorded on the high-speed and high-sensitive camera with 800 × 600 pixel, 12-bit monochrome CMOS image sensor. In this case, the frame rate is ﬁxed at 2000 frames per second and the exposure time of each

Fig. 9.96. Velocity distribution relative to the moving velocity of a moving droplet.

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frame is 0.5 ms. This system enables us to measure cross-sectional velocity distributions of micro ﬂows in the region of 228 × 171 μm2 with the confocal depth of 1.8 μm. The confocal depth indicates the out-of-plane measurement resolution of the confocal micro-PIV system, which has been measured actually by means of imaging the actual tracer particles at diﬀerent focus positions. The key advantage of confocal micro-PIV lies in the fact that the light from out-of-focus particles is cut oﬀ optically and only the particles in quite shallow depth centering on the focal plane are visualized at high contrast level. Confocal micro-PIV has been applied to the internal ﬂow of a small droplet that is transported in a square microchannel [404]. Figure 9.96 shows the instantaneous velocity distributions in each cross-section. In order to elucidate the ﬂow phenomenon inside the droplet, the velocities relative to the moving speed of the droplet are estimated and mapped in Fig. 9.96. The axisymmetrical circulation ﬂow is observed in any cross-section, although, its direction diﬀers for the top/bottom wall region and the center of the channel. This result suggests that the ﬂuid inside a closed droplet circulates in the threedimensions by the drag force on the contact surface with the surrounding channel walls when the droplet passes through a square microchannel.

9.21 Micro-PIV II Contributed by: S.T. Wereley and C.D. Meinhart

9.21.1 Flow in a Microchannel No ﬂow is more fundamental than the pressure driven ﬂow in a straight channel of constant cross section. Since analytical solutions are known for most such ﬂows, they prove invaluable for gauging the accuracy of μPIV. Analytical Solution to Channel Flow Although the solution to ﬂow through a round capillary is the well-known parabolic proﬁle, the analytical solution to ﬂow through a capillary of a rectangular cross section is less well known. Since one of the goals of this section is to illustrate the accuracy of μPIV by comparing to a known solution, it is useful here to brieﬂy discuss the analytical solution. The velocity ﬁeld of ﬂow through a rectangular duct can be calculated by solving the Stokes equation (the low Reynolds number version of the Navier-Stokes equation ), with no-slip velocity boundary conditions at the wall [6] using a Fourier Series

9.21 Micro-PIV II

z

369

2W x

Measurement plane

y

Flow direction

x

2H

Fig. 9.97. Geometry of the microchannel. The microchannel is 2H high and 2W wide, and is assumed inﬁnitely long in the axial direction. The measurement plane of interest is orientated in the x − z plane and includes the microchannel wall at z = 0. The centerline of the channel is at y = 0. The microscope objective images the test section from below, in the lower part of the ﬁgure.

approach. Figure 9.97 shows a rectangular channel in which the width W is much greater than the height H. Suﬃciently far from the wall (i.e., Z H) the analytical solution in the Y direction (for constant Z), converges to the well-known parabolic proﬁle for ﬂow between inﬁnite parallel plates. In the Z direction (for constant Y ), however, the ﬂow proﬁle is unusual in that it has a very steep velocity gradient near the wall (Z < H) which reaches a constant value away from the wall (Z H). Experimental Measurements A 30 × 300 × 25 μm3 glass rectangular microchannel, fabricated by Wilmad Industries, was mounted ﬂush to a 170 μm thick glass coverslip and a microscope slide. By carefully rotating the glass coverslip and the CCD camera, the channel was oriented to the optical plane of the microscope within 0.2◦ in all three angles. The orientation was conﬁrmed optically by focusing the CCD camera on the microchannel walls. The microchannel was horizontally positioned using a high-precision x − y stage, and veriﬁed optically to within ∼ 400 nm using epi-ﬂuorescent imaging and image enhancement. The experimental arrangement is sketched in ﬁgure 9.97. The ﬂow in the glass microchannel was imaged through an inverted epiﬂuorescent microscope and a Nikon Plan Apochromat oil-immersion objective lens with a magniﬁcation M = 60 and a numerical aperture NA = 1.4. The object plane was placed at approximately 7.5 ± 1 mm from the bottom of the 30 μm thick microchannel . The Plan Apochromat lens was chosen for the experiment because it is a high quality microscope objective designed with low curvature of ﬁeld, low distortion, and is corrected for spherical and chromatic aberrations. Since deionized water (refractive index nw = 1.33) was used as the working ﬂuid but the lens immersion ﬂuid was oil (refractive index ni = 1.515), the

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eﬀective numerical aperture of the objective lens was limited to NA ≈ nw /ni = 1.23 [14]. A ﬁltered continuous white light source was used to align the test section with the CCD camera and to test for proper particle concentration. During the experiment, the continuous light source was replaced by the pulsed Nd:YAG laser. A Harvard Apparatus syringe pump was used to produce a 200 ml/h ﬂow through the microchannel . The particle-image ﬁelds were analyzed using a custom-written PIV interrogation program developed speciﬁcally for microﬂuidic applications. The program uses an ensemble-averaging correlation technique to estimate velocity vectors at a single measurement point by (1) cross-correlating particle-image ﬁelds from 20 instantaneous realizations, (2) ensemble averaging the crosscorrelation functions, and (3) determining the peak of the ensemble-averaged correlation function. The signal-to-noise ratio is signiﬁcantly increased by ensemble averaging the correlation function before peak detection, as opposed to either ensemble averaging the velocity vectors after peak detection, or ensemble averaging the particle-image ﬁeld before correlation. The ensembleaveraging correlation technique is strictly limited to steady, quasi-steady, or periodic ﬂows, which is certainly the situation in these experiments. This process is described in detail in section 5.4.2. For the current experiment, 20 realizations were chosen because that was more than a suﬃcient number of realizations to give an excellent signal, even with a ﬁrst interrogation window of only 120 × 8 pixels. The signal-to-noise ratio resulting from the ensemble-average correlation technique was high enough that there were no erroneous velocity measurements. Consequently, no vector validation postprocessing was performed on the data after interrogation. The velocity ﬁeld was smoothed using a 3 × 3 Gaussian kernel with a standard deviation of one grid spacing in both directions. Figure 9.98(a) shows an ensemble-averaged velocity-vector ﬁeld of the microchannel. The images were analyzed using a low spatial resolution away from the wall, where the velocity gradient is low, and using a high spatial resolution near the wall, where the wall-normal velocity gradient is largest. The interrogation spots were chosen to be longer in the streamwise direction than in the wall-normal direction. This allowed for a suﬃcient number of particle images to be captured in an interrogation spot, while providing the maximum possible spatial resolution in the wall-normal direction. The spatial resolution, deﬁned by the size of the ﬁrst interrogation window, was 120 × 40 pixels in the region far from the wall, and 120 × 8 pixels near the wall. This corresponds to a spatial resolution of 13.6 μm × 4.4 μm and 13.6 μm × 0.9 μm, respectively. The interrogation spots were overlapped by 50% to extract the maximum possible amount of information for the chosen interrogation region size according to the Nyquist sampling criterion. Consequently, the velocity vector spacing in the wall-normal direction was 450 nm near the wall. The streamwise velocity proﬁle was estimated by line-averaging the measured ve-

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Fig. 9.98. (a) Large area view of ensemble-averaged velocity-vector ﬁeld measured in a 30 × 300 × 25 μm3 channel. The spatial resolution, deﬁned by the interrogation spot size of the ﬁrst interrogation window, is 13.6 μm × 4.4 μm away from the wall, and 13.6 μm × 0.9 μm near the wall [406]. (b) Near wall view of boxed region from (a).

locity data in the streamwise direction. Figure 9.99 compares the streamwise velocity proﬁle estimated from the PIV measurements (shown as symbols) to the analytical solution for laminar ﬂow of a Newtonian ﬂuid in a rectangular channel (shown as a solid curve). The agreement is within 2% of full-scale resolution. Hence, the accuracy of μPIV is at worst 2% of full-scale for these experimental conditions. The bulk ﬂow rate of the analytical curve was determined by matching the free-stream velocity data away from the wall. The wall

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Fig. 9.99. Velocity proﬁle measured in a nominally 30 μm × 300 μm channel. The symbols represent the experimental PIV data while the solid curve represents the analytical solution [406].

position of the analytical curve was determined by extrapolating the velocity proﬁle to zero near the wall. Since the microchannel ﬂow was fully developed, the wall-normal component of the velocity vectors is expected to be close to zero. The average angle of inclination of the velocity ﬁeld was found to be small, 0.0046 rad, suggesting that the test section was slightly rotated in the plane of the CCD array relative to a row of pixels on the array. This rotation was corrected mathematically by rotating the coordinate system of the velocity ﬁeld by 0.0046 rad. The position of the wall can be determined to within about 400 nm by direct observation of the image because of diﬀraction as well as the blurring of the out-of-focus parts of the wall. The precise location of the wall was more accurately determined by applying the no-slip boundary condition, which is expected to hold at these length scales for the combination of water ﬂowing through glass, and extrapolating the velocity proﬁle to zero at 16 diﬀerent streamwise positions (ﬁgure 9.98a). The location of the wall at every streamwise position agreed to within 8 nm of each other, suggesting that the wall is extremely ﬂat, the optical system has little distortion, and the PIV measurements are very accurate. Most PIV experiments have diﬃculty measuring velocity vectors very close to the wall. In many situations, hydrodynamic interactions between the particles and the wall prevent the particles from traveling close to the wall, or background reﬂections from the wall overshadow particle images. By using 200 nm diameter particles and epi-ﬂuorescence to remove background reﬂections, we have been able to make accurate velocity measurements to within about 450 nm of the wall; see ﬁgure 9.98.

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9.21.2 Flow in a Micronozzle The utility of these new imaging and processing algorithms along with the μPIV technique itself can be demonstrated by measuring the ﬂow through a micronozzle. The micronozzles were designed to be operated with supersonic gas ﬂows. In the initial stages of this investigation, however, they were operated with a liquid in order to assess the spatial resolution capabilities of the μPIV technique without having to push the temporal envelope simultaneously. Consequently, the converging-diverging geometry of the micronozzle served as a very small venturi. The micronozzles were fabricated by Bayt & Breuer (now at Brown University) at MIT in 1998. The two-dimensional nozzle contours were etched using DRIE in 300 μm thick silicon wafers. The nozzles used in these experiments were etched 50 μm deep into the silicon wafer. A single 500 μm thick glass wafer was anodically bonded to the top of the wafer to provide an end wall. The wafers were mounted to a macroscopic aluminum manifold, pressure sealed using O-rings and vacuum grease, and connected with plastic tubing to a Harvard Apparatus syringe pump. The liquid (de-ionized water) ﬂow was seeded with relatively large 700 nm diameter ﬂuorescently-labeled polystyrene particles (available from Duke Scientiﬁc). The particles were imaged using an air-immersion N A = 0.6, 40× objective lens, and the epi-ﬂuorescent imaging system described in Chapter 8. A ﬂow rate of 4 ml/h was delivered to the nozzle by the syringe pump. Figure 9.100 is the velocity ﬁeld inside a nozzle with a 15◦ half angle and a 28 μm throat. The velocity ﬁeld was calculated using the CDI technique with image overlapping (10 image pairs) and image correction, as explained above. The interrogation windows measured 64 × 32 pixel in the x and y directions, respectively. When projecting into the ﬂuid, the correlation windows were 10.9 μm × 5.4 μm in the x and y directions, respectively. The interrogation spots were overlapped by 50% in accordance with the Nyquist criterion, yielding a velocity-vector spacing of 5.4 μm in the streamwise direction and

Fig. 9.100. Velocity ﬁeld produced from 10 overlapped image pairs. The spatial resolution is 10.9 μm in the horizontal direction and 5.4 μm in the vertical direction. For clarity only every ﬁfth column of measurements is shown [299]. Copyright 2002, AIAA. Reprinted with permission.

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2.7 μm in the spanwise direction. The Reynolds number, based upon bulk velocity and throat width, is Re = 22. Turning now from a converging geometry to a diverging geometry, we can explore whether instabilities well predicted by the Reynolds number at macroscopic length scales are indeed as well predicted by the Reynolds number at small length scales. The diﬀuser has a throat width of 28 μm and a thickness of 50 μm. The divergence half angle is quite large, 40◦ . The expected behavior for this geometry would be that at low Reynolds number s the ﬂow would be entirely Stokes ﬂow (i.e., no separation), but at larger Reynolds numbers, where inertial eﬀects become important, separation should appear. Indeed, this is just what happens. At a Reynolds number of 22, the ﬂow in the diverging section of the nozzle remains attached to the wall (not shown), while at a Reynolds number of 83, the ﬂow separates as shown in ﬁgure 9.101. This ﬁgure is based on a single pair of images and as such represents an instantaneous snapshot of the ﬂow. The interrogation region size measured 32 × 32 pixel2 or 5.4 × 5.4 μm2 . A close inspection of ﬁgure 9.101 reveals that the separation creates a stable, steady vortex standing at the point of separation. After the ﬂow has dissipated some of its energy in the vortex, it no longer has suﬃcient momentum to exist as a jet and it reattaches to the wall immediately downstream of the vortex. This is arguably the smallest vortex ever measured. Considering that the Kolmogorov length scale is frequently on the order of 0.1−1.0 mm, μPIV has more than enough spatial resolution to measure turbulent ﬂows at, and even signiﬁcantly below, the Kolmogorov length scale. The example shown has 25 vectors measured across the 60 μm extent of the vortex. 9.21.3 Flow Around a Blood Cell A surface tension driven Hele-Shaw ﬂow with a Reynolds number of 3 · 10−4 was developed by placing deionized water seeded with 300 nm diameter polystyrene particles between a 500 μm thick microscope slide and a 170 μm

y [? m]

x [? m]

Fig. 9.101. Recirculation regions in a microdiﬀuser with spatial resolution of 5.4 × 5.4 mm2 : (a) only every fourth column and every second row are shown for clarity; and (b) close up view of vortex region with all rows and columns of data shown [299]. Copyright 2002, AIAA. Reprinted with permission.

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coverslip. Human red blood cells, obtained by autophlebotomy, were smeared onto a glass slide. The height of the liquid layer between the microscope slide and the coverslip was measured as approximately 4 μm by translating the microscope objective to focus on the glass surfaces immediately above and below the liquid layer. The translation stage of the microscope was adjusted until a single red blood cell was visible (using white light) in the center of the ﬁeld of view. This type of ﬂow was chosen because of its excellent optical access, ease of setup, and its 4 μm thickness, which minimized the contribution of out-of-focus seed particles to the background noise. Also, since red blood cells have a maximal tolerable shear stress, above which hemolysis occurs, this ﬂow is potentially interesting to the biomedical community. The images were recorded in a serial manner by opening the shutter of the camera for 2 ms to image the ﬂow and then waiting 68.5 ms before acquiring the next image. Twenty-one images were collected in this manner. Since the camera is exposed to the particle reﬂections at the beginning of every video frame, each image can be correlated with the image following it. Consequently, the 21 images recorded can produce 20 pairs of images, each with the same time between exposures, Δt. Interrogation regions, sized 28 × 28 pixel, were spaced every 7 pixel in both the horizontal and vertical directions for a 75% overlap. Although technically, overlaps greater than 50% over sample the images, they eﬀectively provide more velocity vectors to provide a better understanding of the velocity ﬁeld. Two velocity ﬁelds are shown in ﬁgure 9.102. Part (a) is the result of a forward diﬀerence interrogation (FDI) and part (b) is the result of a central diﬀerence interrogation (CDI). The diﬀerences between these two ﬁgures will be discussed below while the commonalities will be considered now. The ﬂow exhibits the features that we expect from a Hele-Shaw ﬂow. Because of the disparate length scales in a Hele-Shaw ﬂow, with the thickness much smaller than the characteristic length and width of the ﬂow, an ideal Hele-Shaw ﬂow will closely resemble a two-dimensional potential ﬂow [1]. However, because a typical red blood cell is about 2 μm, while the total height of the liquid layer between the slides is 4 μm, there is a possibility that some of the ﬂow will go over the top of the cell instead of around it in a Hele-Shaw conﬁguration. Since the velocity ﬁeld in ﬁgure 9.102 closely resembles that of a potential ﬂow around a right circular cylinder, we can conclude that the ﬂow is primarily a Hele-Shaw ﬂow. Far from the cylinder, the velocity ﬁeld is uniformly directed upward and to the right at about a 75◦ angle from the horizontal. On either side of the red blood cell, there are stagnation points where the velocity goes to zero. The velocity ﬁeld is symmetric with respect to reﬂection in a plane normal to the page and passing through the stagnation points. The velocity ﬁeld diﬀers from potential ﬂow in that near the red blood cell there is evidence of the no-slip velocity condition. These observations agree well with the theory of Hele-Shaw. For both algorithms, measurement regions that resulted in more than 20% of the combined area of the ﬁrst and second interrogation windows inside of the

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Fig. 9.102. Flow around a single human red blood cell: (a) forward diﬀerence adaptive window oﬀset analysis; and (b) central diﬀerence adaptive window oﬀset analysis. Cross symbols (×) indicate points that cannot be interrogated because they are too close to the cell [169].

blood cell are eliminated and replaced with a ’×’ symbol because they will tend to produce velocity measurements with serious errors. The central diﬀerence interrogation (CDI) scheme is able to accurately measure velocities closer to the surface of the cell than the forward diﬀerence interrogation (FDI) scheme. The CDI scheme has a total of 55 invalid measurement points, equivalent to 59.4 μm2 of image area that cannot be interrogated, while the FDI has 57 invalid measurement points or 61.6 μm2 . Although this diﬀerence of two measurement points may not seem signiﬁcant, it amounts to the area that the FDI algorithm cannot interrogate being 3.7% larger than the area the CDI algorithm cannot interrogate. Furthermore, the distribution of the invalid points is signiﬁcant. By carefully comparing ﬁgure 9.102a with ﬁgure 9.102b, it is apparent that the FDI has three more invalid measurement points upstream of the blood cell than does the CDI scheme while the CDI scheme has one more invalid point downstream of the blood cell. This diﬀerence in distribution of invalid points translates into the centroid of the invalid area being nearly twice as far from the center of the blood cell in the FDI case (0.66 μm, or 7.85%, of the cell diameter) versus the CDI case (0.34 μm, or 4.05%, of the cell diameter). This diﬀerence means that the FDI measurements are less symmetrically distributed around the blood cell than the CDI measurements are. In fact, they are biased toward the time the ﬁrst image was recorded. Computing the average distance between the invalid measurement points and the surface of the blood cell indicates how closely to the blood cell surface each algorithm will allow the images to be accurately interrogated. On average, the invalid measurement points bordering the red blood cell generated using the

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FDI scheme are 12% farther from the cell than the invalid measurement points generated with the CDI scheme. Consequently, the adaptive CDI algorithm is more symmetric than the adaptive FDI algorithm and also allows measuring the velocity ﬁeld nearer the cell surface. 9.21.4 Flow in Microﬂuidic Biochip Microﬂuidic biochips are microfabricated devices that are used for delivery, processing, and analysis of biological species (molecules and cells). Gomez et al. [400] successfully used μPIV to measure the ﬂow in a microﬂuidic biochip designed for impedance spectroscopy of biological species. This device was studied further in [415]. The biochip is fabricated in a silicon wafer with a thickness of 450 μm. It has a series of rectangular test cavities ranging from tens to hundreds of microns on a side, connected by narrow channels measuring 10 μm across. The whole pattern is deﬁned by a single anisotropic etch of single crystal silicon to a depth of 12 μm. Each of the test cavities has a pair or series of electrodes patterned on its bottom. The array of test cavities is sealed with a piece of glass of about 0.2 mm thick, allowing optical access to the ﬂow. During the experiment, water-based suspensions of ﬂuoresceinlabeled latex beads with a mean diameter of 1.88 μm were injected into the biochip and the ﬂow was illuminated with a constant intensity mercury lamp. Images are captured with a CCD camera through an epi-ﬂuorescence microscope and recorded at a video rate (30 Hz). One of the PIV images covering an area of 542 × 406 μm2 on the chip with a digital resolution of 360 × 270 pixel is shown in ﬁgure 9.103. The ﬂow in a rectangular cavity of the biochip is determined by evaluating more than 100 μPIV recording pairs with the ensemble correlation method, CDI, and the image correction technique, and the results are given in ﬁgure 9.104. An interrogation window of 88 pixel is chosen

Fig. 9.103. Digital image of the seeded ﬂow in the cavities and channels of the biochip (360 × 270 pixel, 542 × 406 μm2 ) [415]. Copyright 2001, AIAA. Reprinted with permission.

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15 mm/s 250

200

150

Y [μm] 100

50

0

0

50

100

150

200

250

X [μ m]

Fig. 9.104. PIV measurement results in a rectangular cavity of the biochip with a spatial resolution of 12 × 12 μm2 [299]. Copyright 2001, AIAA. Reprinted with permission.

for the PIV image evaluation, so that the corresponding spatial resolution is about 12 × 12 μm2 . The measured velocities in the cavity range from about 100 mm/s to 1600 μm/s. This biochip can function in several diﬀerent modes, one of which is by immobilizing on the electrodes antibodies speciﬁc to an antigen being sought. The rate at which the antibodies capture the antigens will be a function of the concentration of the antigens in the solution as well as the solution ﬂux past the electrodes. Consequently, knowledge of the velocity ﬁeld is critical to characterizing the performance of the biochip. Since this is an example of the μPIV technique, the reader is directed to [400] for more details about the biochip performance characterization.

9.22 Nano-PIV Contributed by: M. Yoda

9.22.1 Background The volume illumination used in μPIV typically gives images with low contrast, especially in the near-wall region where both reﬂections from the wall

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Table 9.24. Nano-PIV recording parameters for electroosmotic ﬂow through microchannels. Flow geometry Maximum in-plane velocity Field of view Interrogation volume Observation distance Recording method Recording medium Recording lens Illumination Time interval between images Seeding material

plane parallel and adjacent to wall Umax = 300 μm/s 115 μm × 15 μm 28 × 15 × 0.3 μm3 (H × W × D) z0 < 1.5 mm multi frame, single exposure intensiﬁed CCD camera microscope objective (M = 63, NA = 0.7) camera adaptor (NA = 0.5) argon-ion laser P 0.2 W Δt = 5.6 ms ﬂuorescent polystyrene spheres (dp = 100 nm)

and light scattered by particles beyond the focal plane contribute to the background. An alternative approach for microscale velocimetry in near-wall ﬂows is evanescent-wave illumination. When a beam of light is incident upon a planar refractive-index interface between glass and water, for example, at an angle of incidence θ exceeding the critical angle (θ 63◦ for a glass-water interface), the beam will undergo total internal reﬂection (TIR) in the glass. Light in the form of an “evanescent wave” that is, the electromagnetic waves with complex wave number will, however, also be transmitted into the water, propagating along a direction parallel to the refractive-index interface. The intensity of these inhomogeneous waves I ∝ exp{−z/zp} where z is the distance normal to the interface, and zp , the (intensity-based) penetration depth, is somewhat less than the illumination wavelength [416]: zp =

-−1/2 λ0 , 2 2 n sin Θ − n21 . 4π 2

(9.5)

Here, n1 and n2 are the refractive indices of the water and glass, respectively, with n1 < n2 . For illumination at visible wavelengths, the evanescent waveillumination is therefore essentially negligible within a few hundred nanometers of the interface. Evanescent waves can be used to illuminate the same colloidal polystyrene ﬂuorescent spheres used in μPIV to obtain the two velocity components parallel to the wall. The resultant velocimetry technique, which we term nano-PIV (nPIV) [422], interrogates only the tracers within a very small distance – as little as 250 nm, based on tracer center position – of the wall. The spatial resolution of the technique along the optical axis (i.e., normal to the image plane) is determined by the illumination (and the camera sensitivity), and is typically signiﬁcantly less than that of typical μPIV near-wall measurements which have a resolution along z of about 900 nm. Since the illumination is

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restricted to such a small region, the evanescent-wave images also have less background noise than volumetrically illuminated images (cf. ﬁgure 9.105). Nano-PIV is, however, limited by the nature of evanescent-wave illumination to near-wall (vs. bulk) ﬂow studies and hence complements the capabilities of μPIV. The near-wall capabilities of nPIV are illustrated here by a study of microscale electroosmotic ﬂow. 9.22.2 Nano-PIV Studies of Microscale Electroosmotic Flow Electroosmotic ﬂow (EOF), where a conducting liquid is driven by an external electric ﬁeld, is an important technology for ”pumping” ﬂuids through microand nanochannels. Strictly speaking, the electric ﬁeld drives the ﬂuid in a thin screening layer of counterions, known as the electric double layer (EDL), that electrostatically shields the charged wall. The characteristic length scale of the EDL, the Debye length λD , is less than 100 nm for most aqueous solutions. For a monovalent electrolyte solution at 298K [417]: 0.3 nm . λD ≈

C/(1 M)

(9.6)

The Debye length is therefore about 1 nm in a 0.1 M aqueous solution. It can be shown that the velocity proﬁle is nearly uniform in a fully-developed,

Fig. 9.105. Sketch showing the microchannel cross-section and generation of the evanescent wave using the prism method (top) and a typical image of 100 nm ﬂuorescent polystyrene spheres illuminated by an evanescent wave at λ0 = 488 nm and zp ≈ 100 nm (bottom). The ﬂow direction x in the top image is out of the page. The ﬁeld of view of the bottom image is about 115 μm (x) × 12.3 μm (y).

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one-dimensional and steady electroosmotic ﬂow except for a very thin region next to the wall that corresponds to the EDL, the only region of the ﬂow with nonzero charge. The extent of this “boundary layer”, δ, deﬁned as the distance from the wall where the velocity recovers to 99% of its bulk value UEO , is about 4.6λD . The length scales required to resolve ﬂows within the EDL have greatly limited experimental studies in this region. Both atomic force microscopy and surface forces apparatus have been used to probe the structure of the EDL in quiescent electrolyte solutions [418, 419], but such intrusive techniques cannot at present be used in most microscale EOF situations. Although the presence of a ﬂow will not aﬀect the EDL per se, surface adsorption of diﬀerent electrolyte species as they are convected past the microchannel wall can greatly aﬀect the EDL structure and hence the ﬂow within the EDL. Nano-PIV has been recently used to measure velocities within the EDL for very dilute aqueous solutions. Monovalent electrolyte solutions were driven by DC electric ﬁelds up to 3 kV/m through fused silica microchannels of roughly trapezoidal cross-section with nominal dimensions of 47 μm (y) × 23 μm (x) (ﬁgure 9.105). The evanescent wave was generated by the TIR of an argonion laser beam (λ0 = 488 nm, output power ≤ 0.2 W) at the interface between the fused silica cover slip sealing the channel and the working ﬂuid. The penetration depth z = 100 ± 10 nm, and the z-extent of the imaged region was 2.5zp , or about 250 nm. For this steady and fully-developed ﬂow, the velocity in the imaged region at a given z-location should be uniform in both x and y. The working ﬂuid was an aqueous sodium tetraborate solution (N a2 B4 O7 · 10H2 O in nano-pure water) seeded with 0.07 vol% 100 nm diameter polystyrene ﬂuorescent spheres. The ﬂuorescent tracers were imaged using an inverted epiﬂuorescent microscope through a 63×, 0.55 NA objective and a 0.5× camera adaptor onto a CCD camera with on-chip gain at framing rates up to 178 Hz and exposures of 1 ms. A FFT correlation-based interrogation algorithm and a Gaussian peak-ﬁnding algorithm based upon a surface ﬁt [421, 418] were used to obtain tracer displacements. Given that the ﬂow was essentially uniform in the image plane, the interrogation windows for the second image in each pair were as large as 160 (x) × 68 (y) pixel, vs. image dimensions of 653 (x) × 70 (y) pixel. Both window shift and cross-correlation averaging over windows as small as 16 × 10 pixel were tested on representative data in an attempt to improve the nPIV processing, but the resultant displacements were within 2% of those obtained without window shift or cross-correlation averaging. In all cases, 1000 consecutive images were processed to obtain displacements from 999 image pairs. The results were then temporally averaged to reduce Brownian eﬀects and, given the uniformity of the ﬂow (based, for example, on the results from cross-correlation averaging over much smaller interrogation windows), spatially averaged as well over the entire ﬁeld of view to obtain an average velocity U .

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Figure 9.106 shows this average velocity as a function of external electric ﬁeld E for sodium tetraborate solutions at molar concentrations C = 0.02 mM (), 0.037 mM (), 0.19 mM (), 1.9 mM (♦), 3.6 mM (), 18.4 mM () and 36 mM (×) [420]. The error bars on these data denote one standard deviation. As expected, the average velocity varies linearly with electric ﬁeld, and the slope of these data is the (constant) electroosmotic mobility. For C ≥ 0.19 mM, equation (9.6) gives λD ≤ 22 nm, corresponding to δ < 100 nm, or less than half the spatial resolution of the nPIV data along z. The average velocity for these higher concentration values is therefore essentially the bulk velocity UEO . At C = 0.02 and 0.037 mM, however, δ = 310 and 230 nm, however, suggesting that these data are obtained within the EDL, and the resultant velocities should therefore be less than UEO . The nPIV results cannot be compared with known analytical predictions of the velocity proﬁle within the EDL because of Brownian eﬀects. The electroosmotic mobilities from the nPIV data are used instead, to validate the technique. Figure 9.107 shows μex (), the mobility obtained from the data in ﬁgure 9.106 using linear regression, as a function of molar salt concentration C (the error bars on these results represent 95% conﬁdence intervals) and analytical model predictions of the bulk electroosmotic mobility μex () [420]. For C ≥ 0.19 mM, μex and μ∞ are within about 8% of each other. The values obtained from the nPIV data at the two lowest concentrations are, however,

Fig. 9.106. Average velocity as a function of driving electric ﬁeld for aqueous sodium tetraborate solutions at concentrations of C = 0.02 ( ), 0.037 ( ), 0.19 (), 1.9 (♦), 3.6 (), 18.4 () and 36 (×) mM. The error bars represent one standard deviation.

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Fig. 9.107. Mobility as a function of molar concentration for aqueous sodium tetraborate solutions based on the nPIV data of ﬁgure 9.106 ( ), analytical model predictions for the bulk EOF ( ), and model predictions corrected for the nonuniform velocities within the EDL (; only at the two lowest C). The error bars represent 95% conﬁdence intervals for μex .

signiﬁcantly less than μ∞ , and the discrepancy is greater at C = 0.02 mM, as expected. The displacements and velocities from nPIV are, strictly speaking, the tracer (vs. ﬂuid) velocity. If the particle follows the ﬂow with good ﬁdelity, the average nPIV velocity U shown in ﬁgure 9.106 is the ﬂuid velocity uz averaged over the particle diameter of 100 nm. The eﬀect of the nonuniform velocity proﬁle within the EDL sampled by the nPIV technique on the mobility can then be estimated by averaging U over the z-extent of the region interrogated by the evanescent-wave illumination. An unweighted average then gives a correction factor: 1 H 1 z+0.5d uex = u(z )dz dz (9.7) u∞ H 0 d z−0.5d where d = 100 nm is the tracer diameter and H = 250 nm is the z-extent of the region sampled by the nPIV tracers. The shaded symbols at C = 0.02 and 0.037 mM in ﬁgure 9.107 are values for μ∞ corrected using equation (9.7) (using the same analytical model to predict the velocity proﬁle within the EDL).

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The corrected model value and μex diﬀer by 3% and 15% at C = 0.02 mM and 0.037 mM, respectively. The nPIV data (unlike the model predictions) are aﬀected by tracer electrophoresis, Brownian eﬀects, and particle-wall EDL interactions (among other phemomena). Moreover, the nPIV tracers probably non-uniformly sample velocities within the region illuminated by the evanescent wave. Given these considerations, we consider the agreement between the experimental data and the model predictions to be acceptable. In summary, these results illustrate some of the capabilities of evanescent wave-based particle-image velocimetry. Nano-PIV has been used to obtain what is believed to be the ﬁrst experimental veriﬁcation of the reduction in near-wall mobility due to the presence of the diﬀuse electric double layer in electroosmotic ﬂow of an aqueous monovalent solution. The nPIV results are supported by analytical predictions. The asymmetric tracer diﬀusion and nonuniform illumination inherent in this technique present interesting challenges in terms of improving the accuracy and robustness of this technique.

9.23 Micro-PIV in Life Science Contributed by: A. Delgado, H. Petermeier, W. Kowalczyk

Table 9.25. PIV recording parameters for micro-PIV in life science. Flow geometry Maximum in-plane velocity Field of view Interrogation area Observation distance Recording method Recording medium Recording lens Illumination Pulse delay Seeding material

vicinity of the surface of Granular Activated Sludge (GAS) Umax = 40 μm/s 602 × 505 μm2 12 × 6 μm2 z0 = 100–300 μm double frame/single exposure CCD camera objective Zeiss “Epiplan” 20 × /0.40 HD objective Zeiss “Epiplan” 10 × /0.20 HD light build in microscope 40 ms yeast cells Saccharomyces cervisiae dp = 3–10 μm)

9.23.1 Introduction Most of the biological processes in Life Sciences are connected to convective phenomena on the micro-scale. For making progress towards a better under-

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standing of natural phenomena being optimized by evolution in nature, a biologically oriented ﬂuid mechanics has to overcome totally new challenges imposed by the new studies of biological eﬀects on the micro-scale. In this context the availability of powerful optical whole ﬁeld systems such as PIV or PTV is of crucial importance. This is especially the case in microorganismic ﬂows, that is ﬂows induced by living microorganisms are studied and discussed in the present contribution. However, any measuring method has to strictly fulﬁl the requirements imposed by its biocompatibility with respect to the biological system in aspects according to the prevention of systematic errors. On the other hand this imposes considerable restrictions on the experimental setup and the ﬂow evaluation methods. Till this date in the literature this topic has been treated poorly, hence this contribution focuses on the corresponding aspects which must be kept in very similar ways to a large number on application in Life Sciences. The microorganismic ﬂow studied here by using Micro-PIV and MicroPTV is generated by the peritriche ciliate Opercularia Asymmetrica (typical dimension of zooids amounts approx. 30 μm), which plays a vital role not only in nature but also in bioﬁlm reactors used for water puriﬁcation. These ciliates generate a ﬂow ﬁeld to access the nutrients in the surrounding ﬂuid by ciliary beating, see ﬁgure 9.108. The generated ﬂow pattern shows a vortex ring inducing strong elongational and shear eﬀects in the ﬂuid, see ﬁgure 9.109. This inﬂuences the mass transport to the bioﬁlm and thus also the nutrition of the bioﬁlm [424], [425]. The main components of the experimental test rig used are an invert phase contrast microscope (magniﬁcation factors 10-, 20- and 40-fold) with microscope object holder consisting of glass plates both without and with cover plates (distance 200 and 300 μm) as well as a CCD camera with a macrozoom objective allowing a maximum speed of 500 frames/s, which are recorded on the computer. Obtained frames have a resolution of 860 × 1024 pixel (i.e. 602 μm × 505 μm). The determination of the ﬂow ﬁeld is carried out basically with the PIVview2C which employs a statistical correlation algorithm. The frames are interrogated with a Fast Fourier Transform accelerated inter-

Fig. 9.108. Photographs (40× magniﬁcation) of the ciliary kinematics of the investigated ciliates (Opercularia Symmetrica) at three diﬀerent horizontal planes. The distance between them amounts about 5 μm.

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Fig. 9.109. Illustration of the biological milieu of the ciliates (left) as well as a detailed picture of single ciliates (middle; magniﬁcation 20) and the corresponding evaluation of the ﬂow ﬁeld (right). The latter show the model based GPUreconstruction of the velocity ﬁeld after a regularization based on a predictorcorrector method [429].

rogation algorithm. The interrogation window size is 32 × 32 pixel and the interrogation grid is 16 × 16 pixel. 9.23.2 Biocompatible μPIV/μPTV Biocompatibility means avoiding from any changes in the environment of the ciliates, since the ciliates react very sensitive to them. As a consequence of this a large number of measures are required which may result in a peculiar design of the image delivering system and, even, in inﬂuencing negatively the experimental determination of the ﬂow ﬁeld. For the microorganismic motion generated by the ciliates biocompatibililty can only be achieved by employing biocompatible tracer particles, an illumination not altering the microbiological milieu and, last but not least, an adequate observation volume with negligible impact on the microorganismic ﬂow ﬁeld. Additionally, it must be taken into consideration that the microorganismic ﬂow generated is dominated by viscous forces (Stokes region). Therefore, special consideration must be devoted to the viscous eﬀects on the seeding particles for avoiding departure from the ﬂow path due to possible proper motion. Hence, it is obvious that the choice of seeding fulﬁlling the wide spectrum of restrictions imposed by biocompatibility is of essential importance. In fact, it has been observed, that usual artiﬁcial seeding (polystyrene particles of 4.8 μm diameter from Microparticles GmbH, Germany) is detected (obviously by chemotaxis) and rejected by the ciliates. It results in their artiﬁcial motion and, thus, in systematic deviations from the natural motion to be observed. These systematic deviations can be avoided by using biotic particles which are accepted by the ciliates as natural nutrient. However, this imposes the demand to operate a bioreactor [423] as the seeding source.

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Comprehensive studies have shown yeast cells (Saccharomyces cerevisiae) to have the best suitability. They have an ellipsoidal shape with a typical length scale of 3–10 μm, a density deviating only slightly from that of water (ρ = 1050 kg/m3) with good refractive properties. Thus, they fulﬁl basic requirements for seeding. But additionally, in order to keep biocompatibility, i.e. to prevent from an excessive feeding of the ciliates, only a moderate yeast cell concentration is suitable. As a consequence of the minimum spatial information density required by usual PIV-correlation algorithms is often not achieved. Thus, using PTV may represent an adequate alternative. However, as a rule, the natural milieu of the ciliate contains biotical and abiotical structures which increase substantially the diﬃculties of evaluating the ﬂow images provided the optical system. This in connection to the required sparse seeding leads to image artefacts. In [428, 429] a system based on a neuronumerical hybrid based on the synergetic use of an Artiﬁcial Neuronal Network supported by a priori knowledge applying a numerical approach is suggested for suppressing images artefacts and correcting spurious velocity vectors automatically. In this context the model based reconstruction of the ﬂow ﬁeld on the graphical processor unit (GPU) and the interactive visualization of the ﬂow topology results in considerable advancements [429, 430]. Regarding the viscous eﬀects on the seeding particles mentioned above the most prominent are Brownian motion, a motion as a consequence of a deviation from sphericity as well as orbital drift due to the availability of a shear or extensional gradient, a curvature of the velocity proﬁle and sedimentation of the particles. Fortunately, these inﬂuences seem to be negligible for the yeast cells employed [426]. The impact of illumination on the microorganismic ﬂow induced by the ciliates is not totally understood, yet. However, the sensitivity of microbiological systems on light radiation (due to phototaxis, photokinetics or, even, scotophobotaxis) is well described in literature. Thus, this prohibits the use of a laser light sheet as done in PIV generally. At ﬁrst glance, ﬂuorescence microscopy, that is marking the yeast cell by genetic modiﬁcation delivers an alternative as the intensity of the illumination could be kept at low level. But, on the other hand, the ciliates have been proven to be very sensitive to the wave lengths (ultraviolet) of the ﬂuorescence exciting radiation required [426]. More crucial, even employing light transmission microscopy appears to stress the ciliates. In contrast to this, the used invert phase contrast microscopy seems to have the smallest impact on biocompatibility. Hereby, the depth of sharpness (approx. 11 μm) determined the thickness of the measuring plane. A photorefractive novelty ﬁlter microscope (NFM) [427, 428] has been also proven to fulﬁl the restrictions imposed by biocompatibility in an excellent manner. As mentioned above, realizing biocompatibility required even detailed considerations on an adequate observation volume which prevent the viscous forces exerted by the walls of the microscope object holders from inﬂuencing the behaviour of the ciliates and distorting the ﬂow ﬁeld generated by

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them. In this context the ratios of the typical length dimension of the zooid and ﬂow ﬁeld in relation to the distance of the glass plates of the object holder are of essential importance. From a comparison of the results obtained by using microscope object holders with and without a glass plate cover, it can be concluded, that the inﬂuence of the viscous forces on the biological system becomes negligible when the distance of the glass plate achieves the order of magnitude of the wave length of the vortex ring induced by the microorganisms which takes on nearly the 10-fold value of the typical dimension of the zooid (i.e. approx. 300 μm). 9.23.3 Experimental Results The most prominent features of the microorganismic motion found are (a) three diﬀerent scenarios of motion and rest, (b) pairs of motion eddies with a wave length which corresponds to the distance between two neighboring ciliates, and (c) a synergistically emphasized transport of nutrients by two ciliates which are intermittently active. Figure 9.109 illustrates the ﬂow induced by the ciliates in their active phase, the biological milieu available and the determination of the velocity ﬁeld by Micro-PIV. A more detailed evaluation of the ﬂow ﬁeld shows that dissipation eﬀects of the microorganismic convection induced by the ciliates takes on an order of magnitude of only 1% of the kinetic energy of the ﬂow. This indicates that the natural evolution has conducted to a strategy of highly eﬃcient energy conversion. Thus, survival of the ciliates is guaranteed by an incredibly favorable balance of the energy they employed to get access to the nutrient and the energy contain of the latter. Furthermore, the intermittent ciliary activity of neighboring individuals increases substantially mixing eﬀect and, thus, the probability to access nutrients from the surroundings.

10 Related Techniques

As noted previously (chapter 1), PIV developed from Laser Speckle Interferometry. Therefore, one of the early names for this technique was ‘Laser Speckle Velocimetry’ before ‘Particle Image Velocimetry’ was established. The Laser Speckle Interferometry (or Laser Speckle Photography) was mainly developed for the determination of displacement and strain in engineering structures. The laser speckles are created due to random interference of scattered light from an optically rough surface illuminated by coherent light. In a pioneering publication Burch & Tokarski [436] showed that, when two such speckle patterns of an object, recorded with and without displacement, are optically transformed, fringes representing the local displacement can be obtained. These fringes, the Young’s fringes described in section 5.3, represent the squared intensities of the Fourier Transform of the speckle patterns, which is the power spectrum. An inverse Fourier Transform yields the correlation function of the original image a method which is still the basis for nearly all modern “relatives” of speckle photography. In contrast to PIV, where lasers are advantageous due to their ability to generate thin and intense light sheets, speckle deformation measurements frequently favor white light illumination. White light speckles instead of laser speckles oﬀer two major advantages: Since laser speckle ﬁelds tend to change their appearance when the displacement to be measured exceeds a certain range, the image pairs might de-correlate and the evaluation becomes increasingly diﬃcult. Secondly, the illumination of large and highly three-dimensional objects requires small recording apertures and, as a result of that, powerful and expensive lasers for illumination. Therefore, white light illumination is more frequently used since the 1980’s. For this method a speckle pattern, which simply is a random dot pattern sprayed, painted or projected on a background, is generated. The dots should have the highest possible contrast and a spatial frequency that is as high as possible and as small as necessary to be imaged with suﬃcient contrast. Retroﬂective paint, consisting of small suspended glass beads for example was used by Asundi & Chiang [433] for contrast enhancement. The technique of pattern projection for wind tun-

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nel model deformation measurements has successfully been used by van der Draai et al. [441]. Multi-camera systems can be used in order to resolve complex object shapes for the analysis of their topology. The use of pulsed white light sources allows for the observation of moving surfaces. The list of references at the end of the book contains some of the numerous publications on such deformation measurement techniques. From all the various names, the Digital Image Correlation (DIC) apparently became the most common one within the past decade. Therefore, we decided to use the name DIC deformation measurement in the remainder of the book. However, not only deformations can easily be measured by means of acquisition and evaluation systems similar to PIV, but also density gradients in transparent media. This approach has probably been known since a long time, but it has not drawn much attention until a few years ago. The technique has been referred to as Synthetic Schlieren [438], and Background Oriented Schlieren (BOS) [454] by the diﬀerent authors. In contrast to the conventional schlieren methods the BOS technique does not require any complex optical devices for illumination. The only optical part needed is an objective lens mounted for instance on a video camera. The camera is focused on a random dot pattern in the background, which generates an image quite similar to a particle image or speckle pattern shown as in ﬁgure 10.4. For this reason we refer to this approach as background oriented. Compared to conventional schlieren techniques this procedure results in signiﬁcantly reduced eﬀorts during its application. However, the optical paths over which the density eﬀects are integrated, are divergent with respect to each other. This can result in a clear disadvantage when large viewing angles have to be used, but is of little inﬂuence for recording distances of more than 30 meters used for the tests to be described in section 10.2. In an extension towards Background Oriented Optical Tomography (BOOT), which was proposed by Raffel et al. [455], the divergence of the optical paths can be compensated by the evaluation algorithms [443, 447]. Both, the measurement of density gradients in a ﬂow and the detection of the deformation and position of a model in the ﬂow are frequently of interest and can be obtained easily based on PIV imaging hard- and software.

10.1 Deformation Measurement by Digital Image Correlation (DIC) One of the main motivations for measurements in ﬂuid dynamics is the determination of forces and moments on structures resulting from their interaction with the ﬂuid. Those ﬂuid dynamic forces frequently lead to model deformations and displacements of parts of the setup. Since scaling and shape factors are important experimental parameters, the shape and position of a model have to be monitored repeatedly during the test. Point-wise methods are commonly used for this purpose but are sometimes quite laborious and could miss critical regions. Whole ﬁeld optical methods can be used for the non-intrusive

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measurement of model deformations and displacements. The Moir´e Interferometry is one such techniques, which allows obtaining highly accurate results over large ﬁelds at once. The disadvantages of the Moir´e techniques are their experimental complexity and the fact that the evaluation software, like in many other cases of interferometry, can not always be run fully automatically. Therefore, starting in the 1970’ties correlation based procedures for deformation, displacement, and strain analysis have been developed and applied more and more frequently. 10.1.1 Deformation Measurement in a High-Pressure Facility Due to the high costs and a limited feasibility of complex laser measurements at full-scale test conditions, most experimental studies on the aerodynamics of high-speed trains are conducted on sub-scales in wind tunnel facilities. However, in most cases the Reynolds and Mach numbers of the model investigations do not match the full-scale vehicle at the same time. Most modern low-speed wind tunnels reach appropriate Mach numbers of 0.1 < M a < 0.3. However, if the relative air speed is approximately the same for both the model and the full-scale train, the Reynolds number in the wind tunnel will obviously be much smaller than that of the full-scale vehicle. For aerodynamically welldesigned conﬁgurations, the resulting mismatch in Reynolds number leads to a certain discrepancy of measured drag and moment values. This brings facilities into play, which are specialized for high Reynolds number investigations like high-pressure facilities. Those facilities allow for the realization of the relevant Mach and Reynolds numbers at reduced scales. Since loads on the model increases with increasing pressure, model deformations and deﬂections have to be monitored carefully during tests. The high-pressure wind tunnel of DNW (HDG), shown in ﬁgure 10.1, is a closed circuit low-speed wind tunnel, which can be pressurized up to 100 bar. The dimensions of the test section are 0.6 × 0.6 m2 with a length of 1 m. 1 : 50 and 1 : 66 models are usually used in order to realize a blockage ratio below 10%. With the maximum speed of 35 m/s and a maximum pressure of 100 bar the achieved Reynolds number is of the same order of magnitude as the fullscale one (e.g. Re = O[107 ]). The ﬂow remains incompressible over the whole range of Reynolds numbers. The model of the lead car and the ﬁrst part of the trailer of a train shown in ﬁgure 10.2, are designed to carry a six-component internal balance. The strain gage balance is relatively compact and measures forces up to approximately 1000 N. All components of force and moment are available and have been measured for a range of yaw angles between −30◦ < β < +30◦ during these tests. The lead vehicle and the trailer were mounted sidewards onto the sting. Digital Image Correlation (DIC) has been applied to generic high-speed train models in order to measure model deﬂections and model deformations due to wind load. The stiﬀness of models in high-pressure wind tunnels is generally more critical since their dimensions are smaller and the wind loads are

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Fig. 10.1. Sketch of the high pressure wind tunnel (HDG) of DNW.

higher as compared to conventional wind tunnels. As mentioned previously, DIC is basically an image processing technique that calculates the displacement of a random dot pattern – which is somehow attached or projected onto the object under investigation – by using correlation techniques. Today, the digital correlation algorithms used for DIC, BOS and PIV are robust and oﬀer small relative errors below 0.1%. In the case of the model deformation measurements presented here, DIC allows for the determination of small deformations with standard deviations of approximately 0.1 pixel of the CCD-sensor

Fig. 10.2. Sketch of the train model conﬁguration in the test section of the HDG.

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393

Fig. 10.3. Instantaneous DIC result, showing the local displacement vectors gray leveled by their magnitude (left) and the measured positions of the random dots at α = 30◦ reference recording (straight line) and the positions measured under wind load at P0 = 30 bar and U∞ = 20 m/s (dotted line) (right).

used. This corresponds to a 0.01 mm accuracy for this measurement of the position of the train model in the HDG wind tunnel. The model – on which a random dot pattern of black ink has been painted – was mounted on a sting and equipped with an internal six component balance (ﬁgure 10.2). An additional ground plate was mounted in order to cut oﬀ the wind tunnel boundary layer and to ensure well-deﬁned boundary conditions. The plate was parallelized to the wind tunnel ﬂoor by measuring the pressure distribution in the center (at yawing angles of β = 0◦ ). Force measurements were performed in the high-pressure wind tunnel and the DIC technique was applied in order to correct the yawing angle. It can easily be seen in ﬁgure 10.3, that in spite of the very stiﬀ sting and support, the yawing angle of the model was signiﬁcantly changing depending on the free stream velocity (e.g ±1.3◦ at 20 m/s). Figure 10.3 shows an example of instantaneous pattern correlations on the left hand side. One hundred vector ﬁelds were used to process the mean angle shown on the right of ﬁgure 10.3. It can be seen that the displacement increases linearly due to sting bending. Additional model deformations, which would result in more complex displacement patterns, were not observed. In the way the DIC technique was applied here, only deformations in the x − zplane could be determined. However, generally this technique is well-suited for three-dimensional measurements if two or more cameras are used [24].

10.2 Background Oriented Schlieren Technique (BOS) 10.2.1 Introduction Optical density visualization techniques such as schlieren photography, shadowgraphy or interferometry are well known and have been used world wide for

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many decades. The techniques are sensitive to changes of the refractive index caused by the variations in the ﬂuid density. Even though some researchers have performed large-ﬁeld and focusing schlieren photography and shadowgraphy in the great outdoors (e.g. [435], [459], [466]), most of these techniques are conﬁned to laboratories or to wind tunnels and they are less suitable for large- or full-scale applications. Nevertheless, full-scale measurements are desirable if the ﬂow is strongly dependent on Mach and Reynolds numbers. This section describes a technique which determines density gradients without using any sophisticated optical equipment. The technique has been successfully applied to a helicopter in hovering ﬂight and to the transonic ﬂow behind a cylinder. 10.2.2 Principle of the BOS Technique The Background Oriented Schlieren technique is based on the relation between the refractive index of a gas, n, and the density, ρ, given by the Gladstone-Dale equation, (n−1)/ρ = const. It can best be compared with laser speckle density ¨ pf et al. [449] and photography as described by Debrus et al. [439] and Ko in improved versions by Wernekinck & Merzkirch [465] and Viktin & Merzkirch [464]. Like interferometry, the laser speckle density photography relies on an expanded parallel laser beam, which crosses through a transonic ﬂow ﬁeld or – in more general terms – through an object of varying refractive index (i.e. a phase object). However, in contrast to interferometry, laser speckle patterns are generated instead of interference fringes. Since white light based techniques for the determination of ﬂuid density gradients are frequently called schlieren1 techniques [19], the technique described in this chapter will be referred to as the background oriented schlieren technique. Compared to the optical techniques mentioned above, the BOS method simpliﬁes the recording. The speckle pattern, usually generated by the expanded laser beam and ground glass, is replaced by a random dot pattern on a surface in the background of the test volume. This pattern has to have a high spatial frequency that can be imaged with high contrast. Usually, the recording has to be performed as follows: ﬁrst a reference image is generated by recording the background pattern observed through air at rest in advance or subsequent to the experiment. In the second step, an additional exposure through the ﬂow under investigation (i.e. during the wind tunnel run) leads to a displaced image of the background pattern. The resulting images of both exposures can then be evaluated by correlation methods. Without any further eﬀort existing evaluation algorithms, which have been developed and optimized for example for particle image velocimetry (or other forms of speckle photography) can then be used to determine speckle displacements. As stated previously, the deﬂection of a single beam contains 1

The German word “Schliere” designates a local optical inhomogeneity in a transparent medium.

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Background plane

Volume with different density

εy

Image plane Lens

y ZD ZA ZB Zi

Fig. 10.4. Sketch of a BOS setup.

information about the spatial gradient of the refractive index integrated along the line of sight (see ﬁgure 10.4). Details on the theory of ray tracing through gradient-index media could be found in [461] and [440]. The idea of the BOS method is to simplify the optical arragement by replacing the laser speckle with a random dot pattern, which may simply consist of ink or paint droplets, splashed onto a background surface (see ﬁgure 10.4). The background pattern may also be generated by a print of a single exposed image of tracer particles. Assuming paraxial recording and small deﬂection angles, a formula for the image displacement Δy can be derived, which is valid for density speckle photography as well as for the BOS technique. Δy = ZD M εy

(10.1)

with the magniﬁcation factor of the background, M = zi /ZB , the distance between the dot pattern and the density gradient, ZD , and 1 ∂n εy = dz (10.2) n ∂y The image displacement Δy can thus be rewritten as ZD Δy = f ZD + ZA − f

(10.3)

with ZA being the distance from the lens to the object and the focal length of the lens, f . Since the imaging system has to be focused onto the background; we note: 1 1 1 = + (10.4) f zi ZB Equation (10.4) shows that a large image displacement can be obtained for a large ZD and small ZA . The maximum image displacement for ZD approaches Δy = f · εy . On the other hand certain constraints in the decrease of ZA have

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Background

di

ZA

Zi

Fig. 10.5. BOS focusing position and image blur.

to be fulﬁlled in order to image the ﬂow ﬁeld suﬃciently sharply. The optical system has to be focused on the background in order to obtain maximum contrast at high spatial frequencies for later interrogation, and equation (10.4) applies. On the other hand, the sharp imaging of the density gradients would be best at zi with 1 1 1 = + f zi ZA

(10.5)

By introducing the aperture diameter dA and the magniﬁcation of the density gradient imaging M = zi /ZA a formula for the blur di (see also ﬁgure 10.5) of a point at ZA reads: 1 (10.6) di = dA 1 + M (f − ZA ) . f Since correlation techniques average over the interrogation window area, the image blur di does not lead to a signiﬁcant loss of information, as long as di is considerably smaller than the interrogation window size. 10.2.3 Application of the BOS to Compressible Vortices Rotor Tip Vortices The background oriented technique was successfully applied to study diﬀerent types of ﬂows [454, 438]. In this section two experiments are presented: A helicopter blade tip vortex investigation and an investigation of the wake behind a cylinder. First tests were performed in order to verify the feasibility of BOS for large-scale aerodynamic investigations. The subject helicopter – a Eurocopter BK117 – departed from the ground of DLR in G¨ ottingen. Progressive scan CCD cameras, with a resolution of 1280 × 1024 pixel at 8 frames per second, were previously mounted in a window of a building at a horizontal distance of 32 m from the helicopter and 11.2 m above the ground. A random dot pattern was generated by splashing tiny droplets of white wall paint (between 1 and 10 mm diameter) with a brush onto the concrete ground. More than 50 digital images were recorded within 20 seconds of hover ﬂight.

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Fig. 10.6. An example of a BOS reference-data image pair at the blade tip of a helicopter in hover ﬂight, see the text for details.

The exposure time was set to 100 μs. The reference recordings were made directly after the departure of the helicopter. Figure 10.6 shows an example of an image pair. The left-hand side shows the reference image, which was recorded after the departure of the helicopter. The right-hand side of ﬁgure 10.6 shows the corresponding BOS image with the blade tip (moving at approximately 280 m/s). Even though acceptable results could be obtained by using a standard cross-correlation displacement measurement software, as developed for PIV, more sophisticated programs helped to adapt the peak-ﬁtting routine to the size of the dot images. The best results were obtained by using an iterative Levenberg-Marquardt ﬁt to a 10 × 10 pixel area, where the correlation values are weighted according to the Fisher transform (for details see [155]). The size of the interrogation window was 20 × 20 pixel, corresponding to 2.4 by 1.8 cm in the rotor plane, localized in a window of 64 × 64 pixel. The evaluation led to the vector plots shown in ﬁgure 10.7, which were obtained by a massive oversampling using a ﬁve-pixel step-width, resulting in an improved visibility of the ﬂow structures under investigation. In ﬁgure 10.7 (left-hand side), the young vortex shedding from the blade that has just passed the observation area can easily be detected as well as the vortices generated by previous blades. After their generation, the blade tip vortices do not dissipate or diﬀuse for many rotor revolutions. The perspective projection led to an elliptical appearance of the vortex axes in the image plane. The line of lower density, which can be deﬁned as the vortex center, is also clearly visible on the zoom view (right hand side). Since the cameras were not synchronized with the helicopter rotor blades, the blades do not appear on

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Fig. 10.7. BOS results showing vectors of the measured displacement ﬁeld which are proportional the density gradients in the helicopter tip vortices.

all recordings and it is therefore impossible to distinguish between the vortex generated by the blade just passing and the vortices from previous blades. If the vortices are overlapping on the camera’s ﬁeld of view, the BOS technique does not allow us to obtain a deﬁned path. Cylinder Wake Flows The experiment described next was set up in order to study compressible vortex ﬂows involved in the blade vortex interaction (BVI) phenomena of helicopter rotors in more detail. Therefore, the vortex shedding on a cylinder with a diameter of d = 25 mm in a transonic wind tunnel – the VAG of DLR in G¨ ottingen – has been investigated by simultaneous velocity and density gradient measurements at diﬀerent free-stream Mach numbers. This information was complemented by additional measurements of the unsteady pressure ﬂuctuations at diﬀerent locations along the wind tunnel walls. These data enable a more detailed analysis of compressible vortices than successive measurements of single quantities. In addition, a detailed description of compressible vortices plays a key role in numerical simulations, which aim at further improvements in the prediction of helicopter noise emissions. The measurement of the velocity induced by the vortex is needed, since the amplitude of the pressure ﬂuctuations which are emitted from the interaction of the vortex with a blade is proportional to the circulation, Γ , of the vortex. In the past, the velocity information has been derived by simultaneous pressure and density measurements (see e.g. [451]). Therefore, one had to assume the vortex to move at a constant convection velocity, that it is symmetric with respect to its axis and that the vortex can be described by a solution of the stationary Euler equations, that is, disregarding the eﬀects of dissipation. Even if thoseassumptions were justiﬁed, they deﬁnitely limit the accuracy of

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Fig. 10.8. Density gradients obtained with a long exposure time at Ma=0.79.

Fig. 10.9. Density gradients obtained with a short exposure time at Ma=0.79.

the experimental results. Furthermore, the spatio-temporal derivatives of the pressure signal, which have to be computed in order to derive the induced velocities, amplify the noise and uncertainty of the data. The situation can be improved by simultaneous measurements of pressure, density and velocity ﬁelds. The setup needed for the BOS measurements was composed of one camera looking through the test section and a light source, which illuminates a background paper containing a dot pattern (ﬁgure 10.4). The dot pattern was generated by printing a single exposure PIV recording on a laser printer. The size of the dots was approximately 1 mm and the size of

400

10 Related Techniques synchronization laser stroboscope

time

white light from stroboscope flow light sheet

polarized beam splitter PIV camera

BOS camera

Fig. 10.10. Optical setup for simultaneous PIV and BOS measurements.

the interrogation window backprojected into the observation area was 25 mm2 on average. Two diﬀerent light sources were used: a continuous white light and a stroboscope light synchronized with the camera. The results which are shown ﬁrst represent the averaged density ﬁeld and were obtained with continuous light. Figure 10.8 shows displacement ﬁelds using continuous light at Ma = 0.79. The region of the shear layers above and below the cylinder can clearly be detected on the density gradient ﬁeld for a Reynolds number of Re = 335000. Intermittent compression shock waves (expected at Ma=0.6 . . . 0.8), however, could not be observed due to smearing eﬀects of long exposures. However, the strong density gradients and the expected decrease of density behind the cylinder can clearly be seen at this Reynolds number (cp. ﬁgure 10.11). The unsteadiness of the vortex wake for this set of parameters becomes visible when comparing averaged and instantaneous results (Fig. 10.8 and Fig. 10.9). This demonstrates the need of instantaneous measurements and made simultaneous velocity and density gradient ﬁelds desirable, which will be described further below.

Fig. 10.11. BOS density gradient distributions (left) and their integration to density distributions (right).

10.2 Background Oriented Schlieren Technique (BOS)

401

The displacement ﬁeld can be integrated in order to obtain a distribution which is proportional to the density distribution assuming a two-dimensional ﬂow ﬁeld. Two integration methods can be used: either by solution of the Poisson equation [456] or by line integration. The second method which is the simplest to implement, has the disadvantage of producing line noise. Figure 10.11 shows the line extracted (left hand side) from the time averaged displacement ﬁelds for diﬀerent Reynolds numbers at x = 1.5 cm and the result after integration (right hand side). It can be seen that the displacement increases while the density decreases with Reynolds number and shows an almost perfect symmetry in relation to the center of the cylinder with a lowest density for y = 4 cm. The cylinder diameter has been chosen in order to restrict the ratio between cylinder diameter, d, and its span, s, to d/s = 1/4. As can be seen by comparing averaged and instantaneous results (ﬁgure 10.8 and 10.9), the vortex wake for this set of parameters is highly unsteady. This demonstrates the need for instantaneous measurements and makes simultaneous velocity and density gradient ﬁelds desirable. The advantage of the BOS technique is that it can very easily be coupled with PIV. Figure 10.10 shows the setup needed to perform BOS and PIV measurements at the same time, which allows for obtaining simultaneous velocity and density data. It is composed of two cameras, one used for PIV and the other for BOS. Both cameras have the same ﬁeld of view and are looking through a polarized beam splitter, which blocks the light from the laser sheet for the BOS camera. The PIV camera was focused on the laser light sheet plane, whereas the BOS camera was focused onto the background dot pattern. The stroboscope light was synchronized with the second pulse of the laser. The background of the second image is therefore brighter than the ﬁrst, but the quality of the correlation data was not signiﬁcantly reduced. Figure 10.12 shows a zoom of a region where a vortex is visible: in the BOS result, the vectors are diverging from the center of the vortex, which corresponds to an area of lower density. 10.2.4 Conclusions The measurements demonstrate the feasibility of the BOS technique for diﬀerent applications – even large-scale ones – by visualizing the blade tip vortices of a helicopter in ﬂight. It is expected that geometry parameters such as the location of the vortex relative to the rotor plane, the orientation of the vortex axis in space and its strength can be derived [448]. In spite of the diﬃcult experimental conditions density gradient data were obtained, which allows for the visualization of density ﬁelds with quite a promising spatial resolution. Compared to previous measurements, the time needed for the setup and for data acquisition can be considerably decreased. However, since a single camera system is capable of measuring only two components of the spatial density gradient integrated along the optical path, further information on the position or orientation of the vortex axis in three-dimensional space cannot be derived

402

10 Related Techniques

5

Y [cm]

Y [cm]

5

4

3

4

3

5

6 X [cm]

7

8

5

6 X [cm]

7

Fig. 10.12. Sample result of simultaneous BOS (left) and PIV (right) data of a cylinder wake ﬂow.

without changing the viewing direction. Assuming a radially symmetric density distribution in the vortex, the use of a second camera in a stereoscopic arrangement would allow the determination of the location of the density gradient in space. After having demonstrated the feasibility of the concept by its application to a technologically relevant but ﬂuid mechanically complex problem, more detailed studies of vortices behind a cylinder were performed in order to reduce the complexity of the vortical structures under investigation and to perform simultaneous velocity measurements. Those measurements will allow a more accurate modeling of vortices in future aero-acoustic prediction codes for helicopters. Furthermore, BOS renders full-scale in-ﬂight rotor tip vortex morphology studies possible by relying on fairly simple sensor units in combination with natural formation backgrounds. Moreover, tomographic BOS data evaluation enables airborne vortex core density estimations [447].

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A Mathematical Appendix

A.1 Convolution with the Dirac Delta Distribution For a real function f of the variable x, and a given vector xi , we have: f (x − xi ) = f (x) ∗ δ(x − xi ) where ∗ denotes convolution and δ the Dirac delta function.

A.2 Particle images For inﬁnite small geometric particle images the particle image intensity distribution (intensity proﬁle) is given by the point spread function τ (x), which has been assumed to have a Gaussian shape [84]: 8 |x|2 τ (x) = K exp − 2 dτ with K=

8 τ0 . πdτ 2

(A.1)

A.3 Convolution of Gaussian Image Intensity Distributions If we assume Gaussian image intensity distributions as given in equation (A.1) the product of two displaced images yields:

2 2 τ (x − xi ) τ (x − xi + s) = K 2 exp −8 (|x − xi | + |x − xi + s| )/dτ 2 .

438

A Mathematical Appendix

For two vectors a and b, it can be shown that: 2

2

2

2

|a| + |a + b| = |b| /2 + 2|a + b/2| . Hence: 4 |s|2 τ (x − xi ) τ (x − xi + s) dx = exp − 2 dτ aI 2 K 2 exp(−16|x − xi + s/2| /dτ 2 ) dx × aI

8 |s|2 = exp − √ 2 2 dτ × τ 2 (x − xi + s/2) dx . aI

A.4 Expected Value We deﬁned fl (X) = V0 (X)V0 (X+D) in equation (3.8). Here we will determine N E fl (Xi ) . i=1

The sum must be considered as a function of N random variables X1 , X2 ...XN . Hence: N N N 1 fl (Xi ) = E {fl (Xi )} = fl (Xi ) dXi E V i=1 i=1 i=1 F

⇒

E

N i=1

fl (Xi )

N = VF

fl (X) dX . VF

B List of Symbols

a a CI CII CR cI cII c 1 , c2 D D DI Da Dmax DPhoto d d d ddiﬀ dmax dmin dopt dp dr ds dshift dτ E E{} e Fi

aperture radius local acceleration vector spatial auto-covariance spatial cross-covariance constant factor of the correlation function spatial correlation coeﬃcient spatial cross-correlation coeﬃcient constant factors for outlier detection diﬀusion coeﬃcient particle displacement within ﬂow ﬁeld interrogation area aperture diameter maximum particle displacement photographic emulsion density particle image displacement mean value of measured image displacement approximation of the image diameter diﬀraction limited imaging diameter maximum particle image displacement minimum particle image displacement optimum particle image displacement particle diameter diﬀerence between real and ideal particle image diameter diameter of the Airy pattern particle image displacement due to the rotating mirror system particle image diameter exposure during recording expected value resolution limit of a microscope in-plane loss of correlation

440

B List of Symbols

Fo f f# g g(x, y) I I I0 (Z) Ilnc Itrans Iz I+ ˆ Iˆ I, JB Jn K lw M ˜ TF (r ) M N mp N NA N NI nexp n n0 nw Pr QE QL r Ra RC RD RD+ RD− RF RI RII RP Rτ s

out-of-plane loss of correlation lens focal length lens f -number acceleration due to gravity gray value distribution image intensity ﬁeld of the ﬁrst exposure image intensity ﬁeld of the second exposure light sheet intensity proﬁle in the Z direction light intensity for interrogation of photographic recordings light intensity locally transmitted through a photographic recording maximum intensity of the light sheet correlation of the intensity ﬁeld with itself Fourier transforms of I and I light ﬂux Bessel function of ﬁrst kind Boltzmann’s constant imaging depth magniﬁcation factor modulation transfer at a certain spatial freqency (r ) total number particle mass total number numerical aperture particle image density (per unit area) number of particle images per interrogation window number of exposures per recording refractive index refractive index of glass refractive index of water Prandtl number quantum eﬃciency number of quantization levels spatial frequency Rayleigh number mean background correlation displacement correlation peak positive displacement correlation peak negative displacement correlation peak noise term due to random particle correlations spatial autocorrelation spatial cross-correlation particle image self-correlation peak correlation of a particle image separation vector in the correlation plane

B List of Symbols

sD so Ta T (x, y) t t t te tf Δt Δtmin Δttransfer U, V U Ug Umax Umean Umin Un (u, v) Up Us Ushift Uτ U∞ u, v V0 (xi ) VF VI Vn (u, v) vI W W0 (X, Y ) X, Y, Z Xm Xp Xv , X v x, y, z x , y , z

x Δx0 , Δy0 ΔX0 , ΔY0 Δxstep , Δystep

displacement vector in the correlation plane object distance absolute temperature local varying intensity transmittance of photographic emulsion time of the ﬁrst exposure time of the second exposure time of the third exposure frame transfer time pulse length exposure time delay minimum time delay charge transfer time in CCD sensor in-plane components of the velocity U ﬂow velocity vector gravitationally induced velocity maximum ﬂow velocity in streamwise direction mean ﬂow velocity in streamwise direction minimum ﬂow velocity in streamwise direction Lommel function velocity of the particle velocity lag shift velocity

friction velocity, τw /ρ free stream velocity dimensionless diﬀraction variables intensity transfer function for individual particle images ﬂuid volume that has been seeded with particles interrogation volume in the ﬂow Lommel function interrogation area (image plane) out-of-plane component of the velocity U interrogation window function back projected into the light sheet ﬂow ﬁeld coordinate system distance between rotating mirror and optical axis particle position within ﬂow ﬁeld point in the virtual light sheet plane image plane coordinate system mirror coordinate system point in the image plane, x = x(x, y) interrogation area dimensions horizontal, vertical interrogation area dimensions within light sheet distance between two interrogation areas

441

442

B List of Symbols

ΔX, ΔY Z0 Zm z0 zcorr ΔZ0

grid spacing in object plane distance between object plane and lens plane distance between object plane and mirror axis distance between image plane and lens plane depth of correlation light sheet thickness

Greek symbols δ(x) δZ ε tot bias sys resid resid thresh U γ Γ λ λ0 μ μI ν ωm ρ ρm ρp σ σI θ τ (x) τs ω ωx , ωy , ωz

Dirac delta function at position x depth of focus cutoﬀ of image intensity total displacement error displacement bias error systematic error residual (nonsystematic) error residuals from stereo PIV vector reconstruction threshold for outlier detection velocity measurement uncertainty photographic gamma state of the ensemble wavelength of light vacuum wavelength of light dynamic viscosity spatial average of I kinematic viscosity, μ/ρ angular velocity of the rotating mirror ﬂuid density spatial resolution limit during recording particle density width parameter of Gaussian bell curve spatial variance of I half angle subtended by the aperture point spread function of imaging lens relaxation time vorticity vector vorticity components

Abbreviations and Acronyms Mod pixel, px

image modulation picture element

B List of Symbols

443

rms Tu

root mean square turbulence level in a ﬂow

BOS CBC CCD CCIR CIV CMOS CW DLR

background oriented schlieren correlation-based correction [139, 141] charge coupled device video transmission standard correlation image velocimetry complementary metal-oxide-semiconductor continuous wave Deutsches Zentrum f¨ ur Luft- und Raumfahrt (= German Aerospace Center) di-ethyl-hexyl-sebacate (liquid for droplet seeding CAS-No.122-62-3) Doppler global velocimetry digital image correlation digital particle image velocimetry digital stereo particle image velocimetry dynamic spatial range [54] dynamic velocity range [54] fast Fourier transform ﬁeld of view Fourier transform holographic particle image velocimetry image pattern correlation technique local ﬁeld correction [150] laser Doppler anemometry, - velocimetry laser speckle velocimetry laser transit anemometry, laser-2-focus anemometry modulation transfer function National Television System Committee (refers to a video standard) phase alternating line (video standard) particle image displacement velocimetry (early name for PIV) particle image velocimetry phase transfer function particle tracking velocimetry planar Doppler velocimetry (= DGV) quantum eﬃciency signal-to-noise ratio stereo particle image velocimetry symmetric phase-only ﬁltering [171] time-resolved particle image velocimetry

DEHS DGV DIC DPIV DSPIV DSR DVR FFT FOV FT HPIV IPCT LFC LDA, LDV LSV LTA, L2F MTF NTSC PAL PIDV PIV PTF PTV PDV QE SNR SPIV SPOF TR-PIV

Index

λ2 operator, 208, 305 1/4 rule, 137 absorption, 29 Airy disk, 48 Airy function, 81 aliasing displacement, 137 autocorrelation, 84, 143 axial modes, 31 B-splines, 154 backward scatter, 19 bandpass ﬁlter, 139 Bessel function, 81 bias error, 137, 164 binary image, 139, 172 Biochip, 377 Blood Cell, 374 Brownian motion, 244, 252 camera calibration, 218 camera model, 219 coplanar calibration, 221 capillaries, 242 capillary, 368 CBC, 141 CCD, 69 back-thinned, 70 dynamic range, 74 frame transfer, 104 full-frame, 102 full-frame interline transfer, 107 interlaced interline transfer, 105

linearity, 73 microlens array, 71 progressive scan, 107 read noise, 72 responsivity, 73 CDI, 376 center diﬀerences, 192 central peak, 84 centroid, 159 Channel Flow, 368 charge coupled device, 69 circulation, 200 CMOS, 69 complementary metal-oxide semiconductor, 69 contact copy, 130 convolution of the geometric image, 81 convolution of the mean intensity, 84 coplanar calibration, 221 correlation circular, 137 correlation peak aliased, 144 correlation theorem, 135 correlation-based correction, 141 cross-correlation circular, 137 discrete, 133 linear, 133 periodic, 137 cross-correlation coeﬃcient, 138 cross-correlation function, 87

446

Index

dark current, 72 dark current noise, 72 data validation dynamic mean value operator, 183 global histogram operator, 181 gradient ﬁlter, 184 median test, 185 minimum correlation ﬁlter, 187 normalized median test, 185 peak height ratio ﬁlter, 187 reconstruction residuals, 187, 215 signal-to-noise ﬁlter, 187 deconvolution, 133 depth of correlation, 47 depth of ﬁeld, 245 detection probability, 170 DGV-PIV, 342 diﬀerential estimate and curvature, 199 and interrogation window size, 199 and oversampling, 198 spatial resolution, 199 uncertainty, 198 diﬀerential estimation, 191 diﬀerentiation ﬂow ﬁeld, 190 Diﬀraction, 244 diﬀraction limited imaging, 48, 51 displacement estimation bias error, 137 one-quarter rule, 137 displacement histogram, 168 Doppler global velocimetry, 342 dual-plane, 227, 285 energy level diagram, 29 energy states, 29 ensemble correlation, 141, 370 enstrophy, 207, 305 error due to background noise, 174 due to displacement gradient, 175 due to image quantization, 172 residual, 164 systematic, 164 FDI, 376 FFT, see Fourier transform, 136 ﬁll factor, 70

ﬁnite diﬀerences, 191 least-squares approximation, 192 ﬁxed pattern noise (FPN), 73 ﬂow ﬁeld diﬀerentiation, 190 ﬂuctuating noise, 84 ﬂuidized bed, 26 forward diﬀerences, 192 forward scatter, 19 Fourier transform data windowing, 137 digital, 127 fast, 127, 136 output format, 163 symmetry properties, 136 frame straddling, 105, 106, 121 Fraunhofer diﬀraction, 48 full-well capacity, 70 gradient error, 175 treatment, 175 grain noise, 130 gross fog, 67 Hele-Shaw ﬂow, 374 high-pass ﬁlter, 139 high-speed video, 242 histogram stretching, 140 image intensity ﬁeld, 81 image blooming, 70 image deformation, 149 image enhancement, 139 image interpolation, 154 image modulation, 52 inkjet, 244 integration circulation, 200 mass ﬂow, 201 intensity capping, 140 interlacing, 75 interpolation Whittaker reconstruction, 154 interrogation areas, 79 multiple pass, 146 spots, 79 volumes, 79 windows, 79

Index interrogation window and diﬀerential estimation, 199 oﬀset, 144, 170, 199 Knudsen number, 253 Kolmogorov, 374 laser, 28 laskin nozzle, 23 lens aberrations, 52 LFC, 157 light scattering, 18 light sheet, 44 Lommel functions, 244 loss of pairs in-plane, 170 minimization, 144 out-of-plane, 170 low Reynolds number, 374 low-pass ﬁlter, 140 measurement uncertainty, 164 measurement volume, 79 median test, see data validation microchannel, 369, 370 microfabricated, 241 microﬂuidic, 241, 377 Micronozzle, 373 min/max ﬁlter, 140 minimum image diameter, 49 modulation transfer functions, 81 molecular tagging velocimetry, 242 Monte Carlo simulation, 164, 165 motion out-of-plane, 176 multiplane stereo PIV, 191, 353 multiple peak detection, 161 multiscattering, 20 Navier–Stokes equation, 189 Navier-Stokes equation, 368 normal strain, 191 NTSC, 75 Nyquist theorem, 137 one-quarter rule, 137 optical ﬂow, 157 out-of-plane motion, 176 outliers, 179

oversampling error due to, 198 particle image numerically generated, 165 optimum diameter, 166 optimum displacement, 169 particle image density, 170 particle image diameter, 50 particle path curvature, 199 particle tracking, 158 particle visibility, 249 peak detection, 158 peak ﬁt 2-D Gaussian, 160 centroid, 159 Gauss ﬁt, 159 three-point ﬁt, 159 Whittaker reconstruction, 161 peak locking eﬀect, 167 perspective error, 57 projection, 55 phase noise, 130 phase-only ﬁltering, 140 photoelectric eﬀect, 70 photographic density, 66 pixel, 61 point spread function, 48, 81 Poisson equation, 203 population inversion, 30 potential well, 70 pressure driven ﬂow, 368 pressure ﬁeld, 205 pressure sensitive paint, 206, 301 PSP, 301 PTV, 158 pump mechanism, 30 quantum eﬃciency, 73 random error, 164 Rayleigh criterion, 52 relaxation time, 16 resolution limit, 52 Richardson extrapolation, 192, 197 scalar image velocimetry, 242 Scheimpﬂug condition, 211

447

448

Index

seeding cyclone, 25 ﬂuidized bed, 26 generator, 23, 26 laskin nozzle, 23 particles, 21, 22, 36 self correlation peak, 84 separation vector, 84 shear strain estimation, 196 in-plane, 190 sinc-based interpolation, 154 sine wave test, 157 slide scanner, 144 spatial resolution increasing, 147 spectral ﬁltering, 140 spectral sensitivity, 68 spontaneous emission, 29 stereo PIV, 280 camera calibration, 218 camera misalignment, 221 multiplane, 191, 353 reconstruction residuals, 215 self-calibration, 218 stimulated emission, 29 Stokes theorem, 194 strain out-of-plane, 191 strain tensor, 190 super-resolution PIV, 158

toe, 67 tomographic PIV, 210 transfer function, 81 translation imaging, 211 transverse mode, 31

three-point estimators, 159

zero padding, 137

uncertainty diﬀerential estimate, 198 velocity, 198 validation, see data validation velocity diﬀerentials, 192 velocity gradient tensor, 190 velocity lag, 15 video interlacing, 75 standards, 75 visibility of a particle, 81 volume illumination, 248 volume PIV, 210 vortex detection, 207, 305 vorticity estimation, 194 out-of-plane, 190 spatial resolution, 199 uncertainty, 193, 195 vorticity equation, 189 vorticity vector, 190 Whittaker reconstruction, 154, 161 Young’s fringes, 126