Fluid dynamics

Methods of Experimental Physics VOLUME 18 FLUID DYNAMICS PART A METHODS OF EXPERIMENTAL PHYSICS: L. Marton and C. Mar...

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Methods of Experimental Physics VOLUME 18 FLUID DYNAMICS PART A

METHODS OF EXPERIMENTAL PHYSICS: L. Marton and C. Marton, Editors-in-Chief

Volume 18

Fluid Dynamics PART A

Edited by

R. J. EMRICH Department of Physics Lehigh University Bethlehem, Pennsylvania

1981

ACADEMIC PRESS

@

A Subsidiary of Harcourt Brace jovanovich, Publishers

New York

London

Toronto

Sydney

San Francisco

COPYRIGHT @ 1981, B Y ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTt:M, WITHOUT PERMISSION IN WRITING FROM THI; PUBLISHER.

ACADEMIC PRESS, INC.

111 Fiflh Avenue, New York, New York 10003

United Kingdom Edition publislied by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1

7DX

Library of Congress Cataloging in Publication Data Main entry under tille: Fluid dynamics. (Methods of' experimental physics ;v. I X ) 1. Fluid dynamic measurements. 2. I l u i d dynamics. I . Emrich, Raymond J a y , Dale. 11. Series TA357.1:683 620.1'064 XO-27697 ISBN 0 - 12-47S96O-2 (v. 1 XA)

PRINTED IN THE UNITED STATES O F AMERICA

81828384

9 8 7 6 5 4 3 2 1

CONTENTS CONTRIBUTORS .............................................. FOREWORD. ................................................. ................................................... PREFACE.

vii ix xi

OF VOLUME18, PARTB . ......................... CONTENTS

xiii

....................

xvii

............................ LISTOF VOLUMESIN TREATISE..

xix

CONTRIBUTORS TO VOLUME18, PARTB . .

1. Measurement of Velocity 1 . 1 . Tracer Methods

......................................

1

by E. F. C. SOMERSCALES List of Symbols ......................................

1

1 . 1 . 1 . Introduction ...................................

2

.......................... 1.1.3. Chronophotography ............................. 1.1.4. Laser Doppler Velocimeter* .....................

6

1.1.2. Flow Tracing Particles

1.2. Probe Methods for Velocity Measurement..

64 93

. . . . . . . . . . . . . 240

1.2.1. Introduction ................................... by R. J. EMRICH 1.2.2. Velocity Measurement by Pitot Probe.. . . . . . . . . . . . by R. J. EMRICH 1.2.3. Propeller and Vane Anemometers ................ by R. J. EMRICH 1.2.4. Hot-wire and Hot-Film Anemometers . . . . . . . . . . . . by RON F. BLACKWELDER 1.2.5. Velocity Measurement by Other Probes. . . . . . . . . . . by R. J. EMRICH 1.2.6. Howmeters ..................................... by R. J. EMRICH

240 242 254 259 315 321

* Section 1.1.4.5, “Fabry-Perot Spectrometer,” is by A . N . Papyrin and R. 1. Soloukhin. V

vi

CONTENTS

1.3. Measurement of Velocity by Analysis of Doppler Shift of Characteristic Radiation ............................... by R. J . EMRICH

341

342 1.3.1. Doppler Shift Formulas ......................... 1.3.2. Method of Measurement of Doppler Shift . . . . . . . . . 343 2. Density Sensitive Flow Visualization by W . MERZKIRCH

.........................................

345

2.2. Refractive Behavior of Fluids ..........................

346

2.1. Introduction

2.2.1. Relation between Fluid Density and Refractive Index ......................................... 2.2.2. Deflection and Retardation of Light in a Density Field .................................. 2.3. Visualization by Means of Light Deflection . . . . . . . . . . . . . . 2.3.1. Shadowgraph Method ........................... 2.3.2. Schlieren Systems .............................. 2.3.3. Fringe Distortion Methods ....................... 2.4. Interferometry

.......................................

346 352 356 356 361 369 374

2.4.1. Reference Beam Interferometers . . . . . . . . . . . . . . . . . 376 383 2.4.2. Shearing Interferometers ........................ 2.4.3. Phase Contrast and Field Absorption . . . . . . . . . . . . . 389 2.5. Evaluation Procedures ................................

392

2.5.1. Axisymmetric Fields ............................ 2.5.2. Three-Dimensional Fields .......................

393 396

2.6. Radiation Emission ................................... 2.6.1. Electron Beam Flow Visualization. . . . . . . . . . . . . . . . 2.6.2. Glow Discharge ................................ AUTHOR INDEX .............................................. SUBJECT INDEX ..............................................

398 399 402 1

13

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin

RON F. BLACKWELDER, Department of Aerospace Engineering, University of Southern California, Los Angeles, California 90007 (259) R . J . EMRICH, Depurtment of Physics, Lehigh University, Bethlehem, Pennsylvania 18015 (240, 315, 341)

W. MERZK~RCH, Znstitut fur Thermo- und Fluiddynamik, Ruhr-Universitat Bochum, 4630 Bochum, Federal Republic of Germany (345) A. N . PAPYRIN, Institute of Theoretical and Applied Mechanics, U S S R Academy of Sciences, Siberian Division, Novosibirsk 630090, U S S R (194) R . I . SOLOUKHIN, Institute of Heat and Muss Transfer, Byelorussian Academy of Sciences, Minsk 220728, U S S R (194)

E. F . C. SOMERSCALES, Department of Mechanical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12181 (1)

vii

This Page Intentionally Left Blank

FOREWORD We know of no one more qualified that Professor Raymond Emrich to have edited this volume of “Methods of Experimental Physics”“Fluid Dynamics.” Together with a number of outstanding and eminent contributors, Professor Emrich has produced a volume that we believe will be of unusual value to the physics community. Because of the central role of fluid phenomena in so many of the subdisciplines of physics as well as in engineering and the life sciences, the usefullness of this volume may be extraordinarily broad. Our gratitude goes to all involved. L. MARTON C. MARTON

ix


PREFACE Fluid dynamics is a somewhat unusual part of physics in the twentieth century. Fluids are so universally used in every experiment in physics that the techniques are considered standard and “well known.” When the fluid dynamic parts of his apparatus misbehave, the physics experimentalist feels that the solution is one to be “left up to the engineers.” There is good justification for this view because plumbing, pressure gages, thermometers, pumps, and fans are so reliable that one takes it for granted that one may simply order what is needed from a scientific supply house catalog. However, there is an active research body, involving people trained as physicists, valiantly searching for an understanding of the “first approximation to nonequilibrium.” Besides the large group in industry, departments of mechanical, aerospace, chemical, and nuclear engineering in universities, as well as chemistry, geology, meteorology, oceanology, and biology departments contain groups who are fully occupied with research in fluid dynamics. Although in the United States and Canada there is only a tiny minority in physics departments, the American Physical Society serves as a rallying point where these diverse groups get together to share their knowledge. This volume, which has been bound as Parts A and B, has been prepared by some devoted members of this group and their friends abroad. It has been written for members of the group who are expert in other fields and for the graduate students in all these disciplines who need to measure liquid and gas velocity, density, temperature, pressure, and composition. The authors’ goal has been to explain the principles of physics employed in making measurements, and to give some practical design information. Often our basic knowledge of an overwhelmingly important aspect of fluid dynamics-turbulence-is so rudimentary that the principles in a measuring system are undefinable. Fluid dynamicists then fall back on an organized guessing method called “dimensional analysis” as a guide for presenting what is known empirically. As physicists read the articles herein, and find the authors appealing to dimensional analysis to try to organize the complex observations of fluid behavior, they may be inclined to conclude that fluid dynamics is the “science of the undetermined constant that isn’t constant.” They are then ready to read the final chapter of Part B, titled “Dimensional Analysis and Model Testing Principles,” xi

xii

PREFACE

which is a method of theoretical physics as well as of experimental physics. I am happy to express my appreciation and thanks to all of the contributors whose time and effort have made this volume available to the scientific community, and especially to C. W. Curtis, W. Merzkirch, R. 1. Soloukhin, and E. F. C. Somerscales. I also thank the late Dr. L. I. Marton and Dr. Claire Marton for proposing the volume and for their support and encouragement in the years of its preparation. Much credit also is due to the staff of Academic Press.

RAYMONDJ. EMRICH

CONTENTS OF VOLUME 18, PART B 3. Measurement of Density by Beam Absorption and Scattering

3.0. Introduction by R. J. EMRICH 3.1. Beam Attenuation Densitometry by R. J. EMRICH 3.2. Analysis of Raman and Rayleigh Scattered Radiation by MARSHALL LAPPA N D C. MURRAY PENNEY 3.3. Measurement of Density by Analysis of Electron Beam Excited Radiation by E . P. MUNTZ 4. Measurement of Temperature

4.1. Probe Methods by W. PAULTHOMPSON 4.2. Measurement of Temperature by Radiation Analysis 4.2.0. Introduction by N . A. GENERALOV 4.2.1. Emitted and Absorbed Radiation by N. A. GENERALOV 4.2.2. Temperature Measurement by Analysis of Scattered Radiation PENNEY by MARSHALL LAPPA N D c. MURRAY 4.2.3. Measurement of Temperature by Analysis of Electron Beam Excited Radiation by E. P. MUNTZ 5. Measurement of Pressure by R. I. SOLOUKHIN, C. W. CURTIS,A N D R. J. EMRICH

5.1. 5.2. 5.3. 5.4. 5.5.

Introduction Gages for Measuring Constant and Slowly Varying Pressures Pressure Measurement in Moving Fluid Time Dependent Pressure Measurements: Preview Gage Characterization ...

XI11

xiv

CONTENTS OF VOLUME

5.6. 5.7. 5.8. 5.9. 5.10.

18,

PART B

Sensors Pressure-Time Recording Dynamic Calibration Diaphragm Gages: Strain by Bending and Stretching Fast Response Gages: Compressional Strain

6. Measurement of Composition by JOHNE. DOVE

6.1. Introduction. 6.2. Analysis of Sampled Fluids 6.3. Analysis of Radiation Absorbed by in Situ fluids 6.4. Analysis of Radiation Emitted by in Situ fluids 6.5. Mass Spectrometry 7. Heat Transfer Gages

by W. PAULTHOMPSON 7.1. Introduction 7.2. One-Dimensional Heat Conduction Relations 7.3. Instrumented Models 7.4. Thin Membrane Calorimeters 7.5. Thick Calorimeters 7.6. Thin Film Gages 7.7. Radiation Heat Transfer Gages 8. Light Sources and Recording Methods by M. HUGENSCHMIDT A N D K. VOLLRATH

8.1. Light Sources 8.2. Recording Methods 9. Apparatus

9.0. Introduction by R. J. EMRICH 9.1. Wind Tunnels and Free Flight Facilities by DANIELBERSHADER 9.2. Shock Tubes and Tunnels by DANIELBERSHADER 9.3. Low Reynolds Number Flows by DANIELBERSHADER 9.4. Apparatus for Rotating Geophysical Fluid Dynamic Studies by ALANJ. FALLER

CONTENTS OF VOLUME

18,

PART B

10. Dimensional Analysis and Model Testing Principles by MAURICE HOLT 10.1. Mathematical Foundations of Dimensional Analysis 10.2. Geometrical and Dynamicd Similarity 10.3. Applications in Fluid Dynamics 10.4. Model Testing Principles

AUTHORINDEX-SUBJECTINDEX

xv


CONTRIBUTORS TO VOLUME 18, PART B DANIELBERSHADER, Department of Aeronautics and Astronautics, Stanford University, Stanford, California 94305 C. W . CURTIS,* Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015

JOHN E. DOVE, Department of Chemistry, University of Toronto, Toronto, Canada M5S 1AI

R. J. EMRICH, Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015

ALANJ . FALLER, Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742 N . A. GENERALOV, Institute of Problems of Mechanics, USSR Academy of Sciences, Moscow A-40, U S S R

MAURICEHOLT, Department of Mechanical Engineering, University of California, Berkeley, California 94720 M. HUGENSCHMIDT, Deutsch-Franzosisches Forschungsinstitut, SaintLouis, 7858 Weil am Rhein, Federal Republic of Germany MARSHALL LAPP,General Electric Research and Development Center, Schenectady, New York 12301 E . P. MUNTZ,Department of Aerospace Engineering, University of Southern California, Los Angeles, California 90007

C. MURRAYPENNEY,General Electric Research and Development Center, Schenectady, New York 12301 R. I . SOLOUKHIN, Institute of Heat and Mass Transfer, Byelorussian Academy of Sciences, Minsk 220728, U S S R

W. PAULTHOMPSON, Advanced Systems Technology Division, The Aerospace Corporation, Los Angeles, California 90009

K . VoLLRAm, Deutsch-Franzosisches Forschungsinstitut, Saint-Louis, 7858 Weil am Rhein, Federal Republic of Germany

* Professor Emeritus. xvii


METHODS OF EXPERIMENTAL PHYSICS Editors-in-Chief L. Marton C. Marton Volume 1. Classical Methods Edited by lmmanuel Estermann Volume 2. Electronic Methods. Second Edition (in two parts) Edited by E. Bleuler and R. 0. Haxby Volume 3. Molecular Physics, Second Edition (in two parts) Edited by Dudley Williams Volume 4. Atomic and Electron Physics-Part A: Atomic Sources and Detectors, Part 6: Free Atoms Edited by Vernon W. Hughes and Howard L. Schultz Volume 5. Nuclear Physics (in two parts) Edited by Luke C. L. Yuan and Chien-Shiung Wu Volume 6. Solid State Physics (in two parts) Edited by K. Lark-Horovitz and Vivian A. Johnson Volume 7. Atomic and Electron Physics-Atomic two parts) Edited by Benjamin Bederson and Wade L. Fite

Interactions (in

Volume 8. Problems and Solutions for Students Edited by L. Marton and W. F. Hornyak Volume 9. Plasma Physics (in two parts) Edited by Hans R. Griem and Ralph H. Lovberg Volume 10. Physical Principles of Far-Infrared Radiation By L. C. Robinson Volume 11. Solid State Physics Edited by R. V. Coleman Volume 12. Astrophysics-Part A: Optical and Infrared Edited b N. Carleton Part 6: adio Telescopes, Part C: Radio Observations Edited by M. L. Meeks

i

Volume 13. Spectroscopy (in two parts) Edited by Dudley Williams xix

xx

METHODS OF EXPERIMENTAL PHYSICS

Volume 14. Vacuum Physics and Technology Edited by G. L. Weissler and R. W. Carlson Volume 15. Quantum Electronics (in two parts) Edited by C. L. Tang Volume 16. Polymers (in three parts) Edited by R. A. Fava Volume 17. Accelerators in Atomic Physics Edited by P. Richard Volume 18. Fluid Dynamics (in two parts) Edited by R. J. Emrich Volume 19. Ultrasonics (in preparation) Edited by Peter D. Edmonds Volume 20. Biophysics (in preparation) Edited by Harold Lecar and Gerald Ehrenstein

1. MEASUREMENT OF VELOCITY 1.l.Tracer Methods" List of Symbolst 18/(u + 0.5) [dimensionless] Area of moving particle image on the emulsion [cm'] 3/[2(u + 0 3 1 [dimensionless] Brightness of the source [W/(cm2 . sr)] size of negative [cm] 9 / [ ( ~ ) " ~+( u 0.5)] [dimensionless] drag coefficient [see Eq. (1.1.3)] [dimensionless] diameter of the limiting circle of confusion [cm]; camera constant [cm] 6H/[a d p d u + 0.5)] = normalized force [cm/s]; duct diameter, see Table I1 [cm] width of camera field of view [cm] width of incident light beam [cm] particle diameter b m ] diameter stationary particle image on the emulsion [pm] luminous flux density incident on the emulsion [W/cm'] monochromatic flux density of light incident on a flow tracing particlet [W/(cmz . pm)] interruption frequency [Hz]; focal length of lens [cm] {O[O + C(lr0/2)"'](B - l)}/{[A + C(a0/2)"']' + [O + C(lr0/2)"*]') {o[A + C(~O/Z)"'](B - l)}/{[A + C(TO/~)"']' + [0 + C(TO/~)"'~} Acceleration of gravity = 981 cm/sz Monochromatic angular scattering cross section [cm2/sr] Ih/(ad'/4) = monochromatic angular scattering coefficient [ST-'] Intensity functions for scattered radiation, subscripts 1 and 2 indicate the planes of polarization; see Eq. (1.1.23) and Fig. 9 [sr] Multiplying factor that allows for departures from the conditions required by Stokes' law; see Table I1 [dimensionless] Wave number = 2a/A of light bm-l] Boltzmann's constant = 1.38046 x [J/K . molecule] Distance of particle from wall [pm] Magnification = image distance/object distance [dimensionless] Mass of particle = ppad3/6 [g] f-number of the lens = focal length/lens diameter [dimensionless] [dimensionless] Stokes number = [Y~/(o~')]'/' Number of particles

* Chapter 1.1 is by E. F. C. Somerscales, except for Section 1.1.4.5, which is by A. N. Papyrin and R. l. Soloukhin. t This list is for Sections 1.1.1 - 1.1.3 and includes only terms not defined in the text or terms that are used frequently in the discussion. 1 METHODS OF EXPERIMENTAL PHYSICS. VOL. I8A

Copyright 0 1981 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 012-475960-2

2

n’

T TA t UP

a

P

PF PP U 7

ll llR

1. MEASUREMENT OF VELOCITY Refractive index of the experimental fluid when the light is incident from a vacuum [dimensionless] Refractive index of the material of the flow tracing particle when the light is incident from a vacuum [dimensionless] nP/nF = refractive index of the material of the flow tracing particle when the light is incident from a medium of refractive index nF [dimensionless] Monochromatic radiant fluxt [W/km] Hydrodynamic resistance of particle [N] Vd/v&laminar flow) = (V)”V/vF(turbulent flow) = relative particle Reynolds number [dimensionless] Temperature [K]; total transmission of the optical system [dimensionless] Monochromatic transmission of the optical system [pm-’] Time [s] Particle velocity; the fluctuating portion of the particle velocity in turbulent flow [cm/sl Fluid velocity; the fluctuating portion of the fluid velocity in turbulent flow [cm/s] up - uF = velocity of the particle relative to the fluid [cm/s] Initial relative velocity of the particle [cm/s] Terminal relative velocity of the particle, i.e., the velocity of the particle when its acceleration is zero [cm/s] 1 + fl)], see Eq. (1.1.13) [rad]; particle size parameter = Phase angle = tan-’ ?rd/A, [dimensionless]; tC(?r)l‘z[(1 - 4A/C)1/Z]in Eq. (1.1.14) and Table VIII Half-angle of field of view [deg]; t C ( r ) ” * [ l- 4A/C)1/2] in Eq. (1.1.14) and Table VIII Camera depth of field [cm] Distance on the emulsion between the first and last of N consecutive images [cm] I/f = time interval between any two adjacent images [s] Added mass coefficient [dimensionless] Wavelength of light [pm] Wavelength of light in a vacuum b m ] Ao/nF = wavelength of light in a medium of refractive index nF [pm] PFVF = dynamic viscosity of the experimental fluid [Pa s] Kinematic viscosity of the experimental fluid [m’/s] Frequency of periodic motion [Hz]; solid angle subtended by the aperture of the observation system at the scattering particle, see Eq. (1.1.22) and Fig. 8 [sr] Density of the experimental fluid [g/cms] Density of the particle material [g/cm3] p p / p F= particle density ratio [dimensionless] t v F / d z= dimensionless time (~ijp[z/~iF~z)l~z = amplitude function for absolute particle motion [dimensionless] (~klz//iiF~z)’~z = amplitude function for relative particle motion [dimensionless]

v2/(

1.1.1. Introduction

1.1.1.l.General. Particle tracking is the most accurate technique of fluid velocity measurement that is available. It involves inferring the velocity of the fluid at a particular point and time from measurements of the mot Total quantities, which have been integrated over all possible wavelengths, are indicated by the elimination of the subscript A.

1.1. TRACER METHODS

3

tion of small particles mixed with the fluid. This avoids the introduction of probes into the fluid, which is an important advantage of the method. In addition, it is more sensitive at low fluid velocities than other measurement methods. A particle tracking fluid velocity measurement system consists of three components, namely, a source of illumination, a tracing particle, and an observation system. The tracing particles are mixed with the moving fluid, and, at some point or region of interest, the measuring volume, they are illuminated by a source of light. The light scattered by the particle is directed into the receiving optics of the observation system. The scattered light is then interpreted or processed in such a way that the output from the observation system provides a measure of the particle motion. Since this may not be the same as the fluid motion, it may be necessary to apply certain corrections to the observation system output in order to obtain the fluid velocity. In addition, the observation system may introduce errors which must also be corrected. The observation of particle motion is undoubtedly the oldest method of fluid velocity measurement.* In spite of this, it is only comparatively recently that it has been supplemented by other methods which do not use flow tracing particles. The available methods of velocity measurement, in addition to particle tracking, may be broadly classified by the nature of the fundamental physical effect that is measured, viz., total pressure, drag, and heat transfer. Of these, techniques dependent on heat transfer measurement, in the form of the hot wire anemometer and hot film anemometer, are probably the most highly developed and widely used. However, the recently introduced laser velocimeter, a particle tracking technique, appears t o be finding increasing use in fluid mechanics, which should make this section on particle tracking particularly timely. The particle tracking method is superior to other methods in being more sensitive at lower velocities and in avoiding the insertion of probes into the fluid. Actually, this last statement is not entirely true. Bubbles and drops, when used as flow tracing particles, have to be introduced into the P. A . G. Monro, Adv. Opt. Electron Microsc. 1, 1 (1966). E. J. Marey, C. R . Hebd. Seunces Acud. Sci. 105, 267 (1882).

* The earliest reference to particle tracking known to the author is given by Monro,' who cites measurements of the velocity of blood cells by van Leeuwenhoek (1689) and Weber (1831, 1838). Because the measurements involved timing the passage of a cell between marks in the field of view of a microscope, they could only be made at comparatively low velocities (say, not more than 2 mm/s). The measurement of higher speeds became possible with the development of chronophotography, which appears to have been due to Marey* in 1893. Since that time various particle tracking methods have been developed (see later in this section).

4

1.

MEASUREMENT OF VELOCITY

flow by means of a structure (fine tubes or wires) that can interfere with the flow field. However, these need not be located at the point of measurement; this is in contrast to the probes required by other methods of fluid velocity measurement, which could introduce spurious motions into the flow field at the point of measurement. The tracing particle itself can, of course, be viewed as a probe, and, as such, it may interfere with the flow by modifying the physical properties of the fluid, such as its thermal conductivity or its viscosity, and thereby affecting the physical phenomena being observed. This is discussed in Section 1.1.2.3. One major limitation of the particle tracking method concerns the nonelectrical nature of the output signal. In general, nonelectrical outputs must be subjected to tedious hand analysis, which can introduce very serious difficulties into the data manipulation, particularly when measurements of turbulent flows are being made. There are, however, some compensating advantages connected with methods which have nonelectrical outputs, and these are discussed in Section 1.1.3. The objective of this section on tracer methods of fluid velocity measurement is to gather together the available information and thereby make it more readily accessible to the experimenter. The presentation is considered to be practical, but the emphasis is on the discussion of principles rather than on the description of hardware. In addition, emphasis has been placed on quantitative estimates of the precision and accuracy to be obtained from a particle tracking measurement system. Sections 1.1.2- 1.1.4 are concerned, respectively, with the flow tracing particles, chronophotographic observation systems, and laser velocimeter observation systems. It is recognized that there are other observation systems besides the two mentioned (see the paper by Somer~cales~), but limitations of space do not allow these to be considered. Their exclusion can be justified by their being less used than the chronophotographic and laser velocimeter methods. 1.1.1.2. Sampling Error. The measurement of fluctuating fluid velocities, particularly turbulent velocities, that are based on the determination of the velocities of individual flow tracing particles are subject to errors associated with temporal and directional variations in the fluid velocity. Randomly fluctuating velocities are usually represented by statistical parameters, such as mean velocities and root mean square velocity fluctuations. For example, consider the determination of the average flow tracing particle velocity from the ensemble average of the measured E. F. C. Somerscales, in “Flow-Its Measurement and Control in Science and Industry,” (R.B. Dowdell, ed.), Vol. 1, p. 795. Instrum. SOC.Am., Pittsburgh, Pennsylvania, 1974.

1.1.

TRACER METHODS

5

velocities of N realizations, when the velocity measurements are restricted to one component; thus (1.1. la)

oj

where ( t i ) is the mean velocity componentj obtained from the ith measurement realization. This relation will give an incorrectly high value for the mean particle velocity. This biased result occurs because if it is assumed that at some point in the measuring volume, the particle number density is uniformly distributed in space, then more flow tracing particles associated with a high velocity will pass through the measuring volume in a given time than low velocity particles. Consequently, the measured mean velocity component Djwill be in error and will exhibit a bias toward higher values. The same conclusion can be drawn for the other statistical parameters .3a The relation that should be used to obtain the average flow tracing particle velocity is (1.1. lb)

where Atf is the time that the particle is in the measuring volume during the ith measurement realization. Equation (1.1. lb) indicates that the correct determination of involves averaging over the total time (2 At1) that particles are actually under observation during the velocity determination. It can also be seen from Eq. (1.1. Ib) that if it is assumed that all the flow tracing particles are in the measuring volume for the same time, i.e., Atf = At = const, then we can obtain Eq. (1.l.la) from Eq. (1.l.lb). However, if we write

oj

Atf = An/lU(tf)(A,n,,

(1.1. lc)

where Afr is the time fc n measurement realizations; (U(tf)l,the magnitude of the velocity U(ff); A,, the projected area of the probe volume looking from the direction of the velocity vector; and n,, the effective number density of flow tracing particles in the measuring volume, then we can see that in general Atr is not a constant if IU(t,)l varies from one measurement realization to another. Furthermore, introduction of Eq. (1.1. lc) into Eq. (1.1. lb), assuming An, A, and n, are constants, gives ( 1.1. Id)

3a

P. Buchhave, W. K . George, Jr., and J . L. Lumley, Rev. Fluid Mech.

11,443 (1979).

6

1. MEASUREMENT OF

VELOCITY

Equation (1.1. Id) was first proposed by McLaughlin and TiedermarP as a practical relation with which to obtain unbiased mean velocity data; a similar relation was proposed for the root mean square of the velocity fluctuations. * However, this result suffers from two important limitations: (a) the projected area A, varies with the trajectory of the flow tracing particle; (b) IU(tt)l is correlated with the sampling rate, so that when there are velocity fluctuations in directions other than that in which 0, is being measured, there will be an overcompensation for the apparent presence of more high velocity flow tracing particles than low velocity flow tracing particles. Other sources of bias can arise from: (a) variations in the particle number density n, due to density fluctuations in the fluid; and (b) rejection of data points because the signal strength is inadequate (photographic record in the case of chronophotography, and electronic signal in the case of the laser Doppler velocimeter). Techniques for correctly determining the statistics of the fluid velocity are still under development at the time of writing (1980), and a good review of the status of this work will be found in Buchhave et 1.1.2. Flow Tracing Particles 1.1.2.1. Introduction. Flow tracing particles are an element of an instrumentation system. Their performance and operational characteristics must therefore be considered in the same way as any other instrument or element of an instrumentation system when assessing their suitability in a particular experimental situation. In the past, this has only been done in an incomplete way, and it is the objective of this section to present information which will allow a proper assessment to be made of particle performance and characteristics. Flow tracing particles can be defined as small solid, liquid, or gaseous bodies with particular optical and dynamic characteristics which allow them to be added to an experimental fluid so as to make visible any motions which may be present in the fluid. The range of particle sizes used in flow tracing extends from about 0.5 km to about 3600 pm (the range of sizes usually associated with particles in general is from about 0.01 p m up to about 10,000 pm). Figure 1 illustrates some typical flow tracing particles and their usual size ranges.3c To provide a scale of reference, this figure also indicates some of the techniques of particle size measurement 3b

D. K . McLaughlin and W. G . Tiederman, Phys. Fluids 16, 2082 (1973).

* McLaughlin and Tiederman discussed bias errors in relation to the laser Doppler velocimeter, but is is clear that this error can be present in all fluid velocity measurements from individual flow tracing particles, so that chronophotography is also subject to the bias error which is accounted for by the application of Eq. ( I . I . Id).

1.1.

7

TRACER METHODS

PARTICLE D I A M E T E R ( p m )

0.1

0.01 1

:ommon Methods of neasuring Particle Size

1 1 1 1 1 1 1

I

I I 1 1 1 1

I

1

I

I

I

,,,,,

Electron microscope

I mm 1000

100 I

I

4

, 1 1 1 1 1

, 1 1 1 1 1

10.000 1

I

Sieving

I-

Microscope

I I I I I I

L

+ m

I Oil Smokes

Typical Flow Tracing Particles

10

I

m

I

I

Tobacco Smokes

Gloss "Eccospheres" Hydrogen Bubbles in Water

Air BubblesinWater

Nozzle Drops Fat Globules in Milk

I Sizes of ~otumlly Occuring Materials

Carbon Black

-

c

Bacteria

I

Hydraulic Nozzle Drops

_Human Hair ,Drizzle

--

Rain

I

t

RedBloodCell Diameter (Adult): 7 . 5 m~t 0 . 3 p m

FIG.1. Size ranges of typical particles used in flow tracing; see Section 1.1.2.2.3for discussion of particle size measurement techniques. (Adapted, with permission, from a similar chart published by the Stanford Research Institute and also appearing in Irani and C a l l i ~ . ~ ~ )

and the size ranges of some naturally occurring particulate materials. There are certain obvious restrictions on the permissible combinations of particle materials and experimental fluids. Thus, solid particles and liquid droplets may be mixed with either gaseous or liquid experimental fluids. However, gas bubbles can only be used as flow tracing particles in a liquid, although gas filled liquid bubbles, e.g., soap bubbles containing helium, have been used in gaseous experimental fluids. The two properties which most strongly affect the choice of a flow tracing particle are its instrumentation characteristics and its optical characteristics. By the former is meant the accuracy and precision with which the fluid velocity can be inferred from their motion, their sensitivity to changes in the fluid velocity, and the reliability with which these characteristics are maintained during the period of measurement. Particle instrument characteristics are dealt with in Sections 1.1.2.2-1.1.2.5 and 1.1.2.8. The optical characteristic of the particles which is of interest in flow tracing applications is the spatial distribution of the scattered light, R . R. Irani and C. F. Callis, "Particle Size: Measurement, Interpretation, and Application." Wiley, New York, 1973.

8

1. MEASUREMENT

OF VELOCITY

which must be sufficient to provide a satisfactory input to the observation system. Section 1.1.2.8 reviews the light scattering properties of small particles. In general, it is not possible to choose a flow tracing particle solely on the basis of its instrumentation performance and its light scattering properties; it is also necessary to consider its generation and dispersal. The aim in choosing a particular technique of tracer particle generation is to produce particles which have known optical and instrumentation characteristics. Unfortunately, as shown in Section 1.1.2.6, where particle generation and dispersal is considered, the possibilities of designing a system which produces particles of predictable characteristics are rather limited. 1.1.2.2. Dynamic Characteristics 1.1.2.2.1. EQUATION OF MOTION.The motion of a particle in a viscous fluid is governed by the so-called Basset-Boussinesq-Oseen (BBO) equation (see Pearcey and Hill,4 Landau and L i f ~ c h i t z and , ~ Fortiera for a derivation of this equation). The equation was deduced independently by Basset,’ Boussinesq,8 and OseenO for the particular case of motion under gravity in a fluid at rest. In this discussion we will use the form due to Tchen’O: T@ du nd3 ~ P P $= H + -PF 6

duF dr

~ d 3 dV 6

xpFdt

The first term H on the right-hand side of this equation is proportional to the external force, e.g., gravity (see Section 1.1.2.5 for other examples), acting on the particle. These forces could be functions of time. The next term is the force on the particle originating in the pressure gradient which is accelerating the fluid. The third term is proportional to the resistance due to setting the fluid itself in motion. The coefficient of the derivative dV/dr is often called the added mass (this is discussed in more detail below). The coefficient R in the fourth term is proportional to the viscous resistance of the fluid to the motion of the particle. The fourth

‘T. Pearcey and G. W. Hill, Ausf. J . Phys. 9, 18 (1956). L. D. Landau and E. M. Lifshitz, “Fluid Mechanics.” Pergamon, Oxford, 1959. A. Fortier, “Mecanique des suspensions.” Masson, Pans, 1967. ’ A. B. Basset, Philos. Trans. R . SOC. London, Ser. A 179, 43 (1888). * J. Boussinesq, “Theorie analytique de la chaleur.” Gauthier-Villars, Pans, 1903. C.. W. Oseen, “Neure Methoden und Ergibnisse in derr Hydrodynamik.” Akad. Verlagsges., Leipzig, 1927. C. M. Tchen, “Mean Values and Correlation Problems Connected with the Motion of Small Particles Suspended in a Turbulent Fluid.” Nijhoff, The Hague, 1947.

1.1.

9

TRACER METHODS

term is in two parts. The integral is a viscous resistance associated with the accelerated motion of the particle. This is sometimes known as the Basset integral, because Basset was one of the first investigators to realize its importance in the theory of particle motion, or the history integral, because it refers to the motion of the particle for all times between the initiation of the particle motion (at time to) and the time t under consideration. For high rates of acceleration, or where the density ppof the material of the particle is substantially smaller than the density pFof the experimental fluid, this term may become very large and substantially increase the resistance experienced by the particle when it is in unsteady motion. The other portion of the fourth term is the viscous resistance of the medium for a constant velocity difference V between the particle and the fluid. If we introduce the velocity V = u p - uF of the particle relative to the fluid into Eq. (1.1.2a), we obtain another form of the equation of motion that will be useful in subsequent discussions:

IT*

+ C rol(~

-

7')"'

dr'

+ D,

(1.1.2b)

where r(= rv,/dL) is a dimensionless time, and the quantities A , B, C, and D are defined in the List of Symbols. It is sometimes useful to rearrange Eq. (1.1.2b) into the following compact form originally suggested by Hjelmfelt"

The form of Eq. (1.1.2c), even though it is not dimensionless, suggests that consideration of the relative magnitudes of the different terms may allow some of them to be eliminated from the equation, resulting in a simplification of the equation of motion. Where the density of the flow tracing particle is substantially greater than the density of the fluid (a>> 1 >> x), as would be the case for a solid or liquid particle immersed in a gas, it is possible to neglect the effect of the added mass and to set the added mass coefficient x to zero. In the same circumstances it is also possible to neglect the history integral term because the coefficient C in Eq. (1.1.2) is small compared to the other coefficients in the equation. The approximate forms of Eqs. (1.1.2) resulting from these assumptions are easier to solve than the full equation. Various combinations of the assumptions may be used to give different approximations, and these are listed in Table I. The terminology types I ,

1. MEASUREMENT

10

OF VELOCITY

TABLEI. Constants in the Approximate Form of the Equation of Motion Approximation

A

B

C

Physical situation where valid

ZI, and ZZZ is due to Hjelmfelt,11.12and a type IV has been added by the author to accommodate the approximation used by Schraub et al. l 3 Hjelmfelt has compared the solutions of the approximate differential equation with the solutions of the complete equation of motion and has confirmed that the added mass and history terms may be ignored with little error for density ratios IT of one thousand or greater. 1.1.2.2.2. STOKES'L A W . The physical laws which govern the hydrodynamic resistance experienced by a single undeformable particle have been extensively investigated, and an excellent review has been published by Torobin and Gau~in.~"-'~The phenomena related to the resistance experienced by a body in motion through a fluid are extremely complex, so a complete theoretical analysis is not possible. However, a number of very reliable empirical correlations and theoretical expressions are available which make use of the following combination of variables: (1.1.3)

The dimensionless group on the left-hand side of this expression is the drag coefficient CD,and the dimensionless group on the right-hand side will be called the relative particle Reynolds number RepR, since the velocity V used in its definition is the velocity of the particle relative to the fluid. The quantity d is the diameter of the flow tracing particle, i.e., it is assumed that the particle is spherical or that its hydrodynamic characteristics can be represented by an equivalent sphere. For steady flows and with Reynolds numbers (RepRup to about 2 x lo5, the hydrodynamic resistance may be divided into three regimes corresponding to streamline, intermediate, and turbulent flow. When streamline flow (RepRd 1) exists around the particle, the drag coefficient may be A. T. Hjelmfelt, Jr., Behavior of a sphere accelerating in a viscous fluid. Ph.D. Thesis, Northwestern University, Evanston, Illinois (1965). l* A. T. Hjelmfelt, Jr. and L. F. Mockros, Appl. Sci. Res., Sect. A 16, 149 (1966). Is F. A. Schraub et al., J . Basic Eng. 87, 429 (1965). I' L. B. Torobin and W. H . Gauvin, Can. J . Chem. Eng. 37, 129 (1959). l5 L. B. Torobin and W. H. Gauvin, Can. J . Chem. Eng. 37, 167 (1959). L. B. Torobin and W. H. Gauvin, Can. J . Chem. Eng. 37, 224 (1959).

1.1.

TRACER METHODS

PARTICLE DIAMETER ( r m )

FIG.2. Relative particle velocity V as a function of particle diameter at different values of the relative particle Reynolds number Rep,: @, 1.0 (19%); @, 0.82 (10%);@, 0.38 (5%); @, 0.074 (1%). Percentage figures in parentheses indicate the error (estimated by Davies'?) in using Stokes' law when the relative particle Reynolds number is equal to or greater than the value shown. The following fluid properties have been assumed for air at 20°C (---) and 0.1 MPa and for water at 20°C (-): Fluid property pF(kg m-' s-')

PF (kg m13)

Air 1.816 x 1.206

Water

1.01 x 10-3 0.998 x 103

represented by the following theoretical result: CD= 24/Re,,.

.

( 1 1.4)

This is due to Stokes and is therefore usually known by his name. Its range of validity is indicated in Fig. 2. Appropriate drag coefficients for higher values of the relative particle Reynolds number will be found in the papers of Torobin and G a u ~ i n . ' ~ - ' ~ On combining Eq. (1.1.3) with Eq. (1.1.4), we obtain

R = 3rpFdV.

(1.1.5)

C. N . Davies, "Symposium on Particle Size Analysis," p. 25. Inst. Chem. Eng., London, 1947.

TABLE11. Modifications to Stokes' Drag Law

Assumption

Fluid medium: Incompressible

Conditions under which modification is required (particle drag changed by at least 10%)

Type

Particle Mach number exceeds unity.

Empirical correlations for drag coefficient

Particle in duct of circular cross section with d / D 3 0.1. Particle sufficiently close to plane wall for l/d < 5 . Interaction among flow tracing particles when particle concentration is greater than 1 : lo00 parts by volume.

Correction factor K in Eq. (1.1.5).

d-h

Correction factor K in Eq. (1.1.5).

i -I

Air" (P = 0.1 MPa, T = 20°C) d

0.1

I 10 PARTICLE DIAMETER (pm)

100

FIG.6. Brownian displacement in one second and sedimentation terminal velocity in air and in water (the assumed conditions and properties are the same as those used in constructing Fig. 2). Numbers on the sedimentation terminal velocity curves represent the specific gravity of the particles relative to water at 4°C. Slip correction included for particles in air.

than Brownian motion effects for all particles except the very smallest. The sedimentation terminal velocity is also plotted in Fig. 6, and for particles in air it can be seen that Brownian motion is only of importance for particles with a diameter of 0.5 p m or less (the conclusion would be different if measurements were completed in shorter times). Since this is close to the smallest size that is usually of interest in flow tracing, it can be concluded that the sensitivity of the particle tracking technique in air is not usually limited by Brownian motion. The Brownian motion defines the limit of sensitivity of the particle tracking technique for fluid velocity measurement. To be detectable, the magnitude of any non-Brownian motion of a flow tracing particle must be at least m times greater than the average particle displacement due to Brownian motion. The multiplying factor m is greater than one (Burton75 proposes m = 3) to ensure that we can distinguish non-Brownian motion fluctuations from Brownian motion fluctuations, both of which may exceed the average value for Brownian motion displacements. Velocity data obtained from particles that are sufficiently small and

1.1.

TRACER METHODS

41

taken in times of the order of 0.01 s, will include random effects due to Brownian motion which limits the precision of the measurement of fluid velocity. A careful study of this, when air is the experimental fluid, has been made by Elrick.BZ He has concluded that the measured velocity up of a flow tracing particle may be represented by up & [ E + (?)l’z] where k E is the uncertainty in the measurement of up, and +(2)1’z is obtained from Eq. (1.1.21) for the time of measurement and represents the contribution of the Brownian motion to the uncertainty in up. 1.1.2.5. Sensitivity to Extraneous Forces. The discussion of the dynamic characteristics of flow tracing particles in Section 1.1.2.2 has been based on the assumption that the only forces acting on the particle are pressure forces and viscous drag forces; but in any practical experimental system there are a number of additional forces which may be present which can result in spurious particle motions which must either be minimized or eliminated. These addition forces are the following:

(a) gravity: important when gas bubbles are used as flow tracing particles; (b) hydrodynamic lift: causes lateral particle motion across the fluid streamlines if there are asymmetric velocity distribution^^-^^ or if the particles are deformablesB(liquid droplets and gas bubbles); (c) interaction between particles and photons: leads to photophoresis; (d) interaction between particles and molecules of the experimental fluid: leads to thermophoresis in temperature gradients and to Brownian motion, for which a temperature gradient need not be present (see Section 1.1.2.4); (e) electric fields: will affect the motion of charged particles; and (f) magnetic fields: will affect the motion of ferromagnetic particles. Because of space limitations, this section will only be concerned with items (a) and (b) of the above list, and we will assume that the conditions which could cause the other forces to act are absent. The motion of a flow tracing particle under the action of gravity in a fluid which is in steady motion (duF/d7 = 0) is given by Eq. (1.1.2), with the dimensionless force term D equal to [ ~ ( c T- l)g]/[vF(a + x)]. The velocity V in the equation of motion refers to the component of the particle velocity in the direction of action of the gravity force and the component of the fluid velocity in the same direction. The equation of motion is most appropriately solved using the Laplace B1 &L

T. V. Starkey, Br. J . A p p l . Phys. 7, 52 (1956). G . Segre and A. Silberberg, J . Fluid Mech. 14, 1 I5 (1962). G . Segre and A. Silberberg, J . Fluid Mech. 14, 136 (1962). H. L. Goldsmith and S. G. Mason, J . Colloid Sci. 17, 448 (1962).

42

1. MEASUREMENT OF VELOCITY TABLEVIII. Sedimentation Velocity of Flow Tracing Particles

exp(a%) e r f c [ a ( ~ ) ~ -.' ~ ]exp(p%) e r f c [ 8 ( ~ ) ~ / ~ ] a(a - P ) P(a - P )

#

1 - 8 ( ~ / a )+~(327 / ~ - 1) exp(l6r) e r f c [ 4 ( ~ ) ~ / ~ ] ' ] (x/y)F[z(~)"~] 1 + exp[(.? - 3)~][(x/y)sin(2qr)- cos(2xy7)] + 4 ~ ( t ) ~ ' a = x + iy = z , p = x - iy = z*, E + iF = exp(z%) e r f c [ z ( ~ ) ~ / ~ ]

a V , is the particle relative terminal velocity = D / A . Vm is the particle relative gravitational terminal velocity = (a- l)gd2/18uF,used in place of V , when the accelerating force is due to gravity.

t r a n ~ f o r m . ~ ~The . ~ ' calculations of Hjelmfelt" are the most complete, and his results" are summarized in Table VIII. Numerical valuest based on these formulas are plotted in Fig. 7. The problem of analyzing the lateral motion of small rigid spherical particles under the action of lift forces is extremely complex and is only partially understood. An excellent review of the problem has been given by Brenner.gO The results of the calculations available in the literature are summarized in Table IX. Experimental work on particle migration in ducts has been comprehensively reviewed by Cox and Mas0n.O' A series of papers by Lee and his a s s o ~ i a t e s deals ~ - ~ ~with the experimental and theoretical aspects of particle migration in boundary layer flows. Although our understanding of the origin and magnitude of the lift force acting on small particles is still limited, a comparison of theoretical and experimental data by Lawler and Liueawould appear to justify superposiL. M. Brush, H. W.Ho,and B. C. Yen, Proc. A m . SOC.Civ. Eng. 90, 149 (1964). R. R. Hughes and E. R. Gilliland, Chem. Eng. Prog. 48,497 (1952). 89 A. B. Basset, "A Treatise on Hydrodynamics." Deighton, Bell, & Co., Cambridge, 1888 (reprinted: Dover, New York, 1961). H. Brenner, Adv. Chem. Eng. 6 , 287 (1966). R. G. Cox and S. G. Mason, Annu. R e v . Fluid Mech. 3, 291 (1971). a B. Otterman and S. L. Lee, Z. Angew. Mafh. Phys. 20, 730 (1%9). 83 S. L. Lee and E. Einav, Prog. Hear Mass Transfer 6 , 385 (1972). ~4 S. Einav and S. L. Lee, Inr. J . Mulriphase Flow 1,73 (1973). 85 S. L. Lee and P. R. di Giovanni, J. Appl. Mech. 41, 35 (1974). gs M. T. Lawler and P. C. Liu, in "Advances in Solid-Liquid Flow in Pipes and Its Applications'' (I. Zandi, ed.), p. 39. Pergamon, Oxford, 1971. 87

B8

* These results also apply to the case of a particle moving under the action of any constant force, e.g., the force experienced by a charged particle in an electric field, and (as noted in Table VII) to the case of a particle released from rest in a fluid moving at a steady velocity. t Hughes and GillilandB8prepared a graph that is similar to Fig. 7, but this was based on Basset's" solution of the problem, which Brush, Ho, and Yens7 contend is incorrect.

1.1.

TRACER METHODS

43

7= t v / d P FIG.7. Velocity-time curves for spheres falling in a viscous fluid at rest ( V , = uT = ppgdZ/18pF)or for spheres accelerating in a fluid moving with a velocity uF(VT = uF). V, = up(r = 0) - uF is the initial particle relative velocity.

tion of the separate effects (see Table IX) obtained by Rubinow and Kellers7 and by Saffmaneeto make an estimate of the order of magnitude of the lift force in a particular experimental system. Such an estimate should be satisfactory for the preliminary design of a flow tracing system, although it would not be advisable to use it for correcting measured fluid velocities. Chaffey et uf.se*lOO and Wohl and Rubinowlo' (see Table IX for the results) have carried out theoretical calculations on the lateral motion of a deformable, neutrally buoyant particle in a fluid flowing with a parabolic velocity profile. The comparison with experimental datasa*lo2is not entirely satisfactory, but in the absence of any other information, it seems appropriate to base estimates of the lift on the results of this theoretical work. 1.1.2.6. Generation and Dispersal of Particles. Velocity measurement by flow tracing requires particles which have known dynamic and optical characteristics, and which meet the requirements of the chosen particle observation system. Many experimental fluids contain naturally occurring particles, such as dust or bubbles from dissolved gases, which have S. I. Rubinow and D. J. Keller, J . Fluid Mech. 11, 447 (1961). P. G . Saffman, J . Fluid Mech. 22, 385 (1965). C. E. Chaffey, H. Brenner, and S. G . Mason, Rheol. Acra 4, 64 (1965). C. E. Chaffey, H. Brenner, and S. G . Mason, Rheol. Acra 6, 100 (1%7). '01 P. R. Wohl and S. I. Rubinow, J . Fluid Mech. 62, 185 (1974). lo* A. Karnis and S. G . Mason, J . Colloid Sci. 24, 164 (1967).

97

44

1. MEASUREMENT OF VELOCITY TABLEIX. Magnitude of Extraneous Forces” Type

Gravityb Lift: rotationb*c Lift: velocity gradientbed Lift: deformable particlese

Magnitude H wd3(o - l)g/6 Td3p~VUp/8

81.2r~V(dU~/dZ)”*(d/2)*/(V~)”* 67rpFVb(We/Rep&F

a 6 is the deviation of the particle from the center line of duct [cm]; F and K are numerical factors that depend on hydrodynamic conditions (see original reference); V is the fluid velocity at the center line duct [cm/s]; the Weber number We = pFYPd/oT [dimensionless]; uT is the interfacial tension of the experimental fluid against the material of the particle; and upis the rotational velocity of the particle [rad/s]. Spherical, rigid particles. S. I. Rubinow and D. J. Keller, J. Fluid Mech. 11, 447 (l%l). P. G. Saf€man, J. Fluid Mech. 22, 385 (1%5). P. R. Wohl and S. I. Rubinow, J. Fluid Mech. 62, 185 (1974).

been used as flow tracing particles by some investigators. However, these do not usually meet all the necessary requirements for flow tracing, so consideration must be given to artificially generated particles. Numerous systems have been devised for the production and distribution of liquid droplets, gas bubbles, and solid particles. Most of these have been applied to the dispersal of liquid fuels in combustion chambers, the spray drying of materials, the formation of aerosols, or the formation of liquid droplets and gaseous bubbles for chemical extraction processes.* Only some of these will be suitable for flow tracing applications. Primary criteria for the selection of a flow tracing particle generation system are (a) the mean size and the range of sizes of the particles produced; (b) the capacity, in number rate of generation; and (c) the spatial distribution of the particles. IoS W. R. Marshall, Jr., “Atomization and Spray Drying.” Chem. Eng. h o g . Monogr. Ser. No. 2, Amer. Inst. Chem. Engrs., New York, 1954. IM C. C. Miesse, Ind. Eng. Chem. 47, 1690 (1955). Io6 J. A. Browning, Adv. Chem. Ser. 20, 136 (1958). N. A. Fuchs and A. G. Sutugin, in “Aerosol Science” (C. N. Davies, ed.), p. 1 . Academic Press, New York, 1966. lo’ E. Seltzer and J. T . Settlemeyer, Adv. Food Res. 2, 399 (1949). Ion E. Giffen and Q.Muraszew, “The Atomization of Liquid Fuels.” Wiley, New York, 1953. R. Kumar and N. R. Kuloor, Adv. Chem. Eng. 8, 255 (1970).

* The theory and practice of particle generation involves an extensive literature,18*31*m*103-1w which it is neither appropriate nor possible to review completely in this section.

45

1 . 1 . TRACER METHODS

Knowing the sizes of the particles produced, the experimenter is in a position to assess their dynamic and their optical characteristics. The rate of particle generation must be known because the number of flow tracing particles observed affects the time required to measure the fluid velocity. Thus, the greater the number of particles observed, the shorter the time required to obtain valid data. Finally, the distribution of particles in the flow field can affect the choice of observation system. Observation systems, such as the laser velocimeter, which approximate single point measurements of the fluid velocity, may operate satisfactorily with particle dispersion systems that have a rather restricted spatial distribution. On the other hand, particles must be widely distributed throughout the flow field when using large scale chronophotography (see Section 1.1.3.6). Other considerations which play a somewhat less important role in the choice of a particle generation and dispersal system are its simplicity of construction and of operation, and its tendency to interfere with the flow field that is to be measured (the latter topic is considered further at the end of this section). Unfortunately, because of the complexity of the physical processes involved in particle generation, it is neither possible to predict the performance nor to make a rational design of a particle generation and dispersal device. The selection of a suitable system must therefore be based on experience, either by the examination of published data or by the conduct of suitable tests. The information available in the literature on particle generation and dispersal systems has been reviewed and is summarized in Table X. This does not include all the available types of systems, since those which are included have been chosen for two reasons, viz., the mean size of the particles generated includes or is within the useful range for flow tracing (0.5-3600 pm), and the capacity of the system is appropriate for flow tracing applications.* For details of the techniques included in the table, see the general references listed earlier and also those references cited in the body of Table X. 110

11' 113

115

B . J . Mason, 0. W. Jayaratne, and J. D. Woods, J . Sci. Instrum. 40, 247 (1%4). A. C. Rayner and H. Hurtig, Science 120, 672 (1954). W. R. Wolf, Rev. Sci. Instrum. 32, 1124 (1961). D. Hasson and J . Mizrahi, Truns. I n s t . C h e m . E n g . 39, 415 (1961). R. W. Tate and W. R . Marshall, Jr., C h e m . Eng. Prog. 49, 169 (1953). N . Dombrowski and P. C. Hooper, J . Fluid Mech. 18, 392 (1964).

* Systems which have not been included are vibrating conical spray,'13 swirl spray,L14 and impact nozzle.115

interrupted jet,"'

TABLEX. Particle Generation and Dispersion Systems Particle size Technique

Mean size d bm)

Range"

References Applications in flow tracing

a. Liquid droplets in a gaseous experimentalfluidb e0.056 Rotating surface 5-3000

Design and construction information

Z

Pneumatic atomization Electrical dispersion

1-1000

20.56

aa

2-300

Uniform

bb

Bubbling Ultrasonic

2-500 2-60

26

Phase change

Combustion

+2d

h cc i,dd,ee

0.1- 1.5

20.56

1

0.2

*0.56

i,k

b. Solid particles in a gaseous experimental fluid' Pneumatic dispersion 0.1-15 Fluidized bed 0.05- 1.O c. Gas bubbles in a liquid experimentalfluid Submerged orifice 1000- l5,OOO Uniform

Electrolysis

h -

13-80

Notes Complicated apparatus. Possibly dangerous. Commercially available perfume sprays are simple and inexpensive. Single streams and conical dispersions of particles. Appears to be capable of producing particles at a very high rate. Appears to be capable of producing particles at a very high rate. The Rappaport- Weinstock generator is superior to the Sinclair-La Mer generator.

-

gg-ii g,n

u r,s,kk

Multiple orifices may be required because particles are produced in a single stream. Hydrogen bubbles generated at the cathode in water are used since they are twice as numerous as the oxygen bubbles at the anode.

d. Liquid droplets in a liquid experimental fluid Submerged orifice: Slow drip mode 800-12,OOO Uniform

Atomization mode Shaking, beating, and mixing

1400-3500 500- lo00

*d

e. Solid particles in a liquid experimental fluid Shaking, beating, and mixing

ji

Multiple orifices may be required because particles are produced in a single stream. -

-

-

u-w

-

X

-

-

Ranges of particle size are estimates based on the author’s examination of the published experimental data on the performance of particle generation and dispersal systems. This approach has been used because a wide variety of incompatible methods have been used to express the range of particle sizes. * These methods may be used to produce and disperse solid particles in a gaseous experimental fluid if the solid material is first dissolved in a liquid which is evaporated after the formation of the particles. If it is necessary to grind the solid flow tracing material before dispersal, a good discussion of the theory and practice of grinding will be found in J. M. Dallavalle, “Micromeritics, the Technology of Fine Particles,” 2nd ed. Pitman, New York, 1948. R. E. Davis, Inr. J . Air Water Pollut. 8, 177 (1964). C. J. Chen and R. J. Emrich, Phys. Fluids 6, 1 (1%3). J. A. Breslin and R. J. Emrich, Phys. Fluids 10, 2289 (1%7). A. Melling and J. H. Whitelaw, Disa I f . No. 15, p. 15 (1973). * 0. M. Griffin and C. W. Votaw, Int. J. Heat Mass Transfer 16, 217 (1973). M. K. Mazumder, B. D. Hoyle, and K. J. Kirsch, Proc. Int. Workshop on Laser Velocimerry (H. D. Thompson and W. H. Stevenson, eds.)., Purdue Univ. E n g . Exp. Stn.. Bull. 144 (1975). R. M. Elrick, I1 and R. J. Emrich, Phys. Fluids 9, 28 (1966). J. P. Yu, E. M. Sparrow, and E. R. G. Eckert, Int. J. Heat Mass Transjer 16, 557 (1972). R. Eichorn, Int. J. Heat Mass Transfer 5 , 915 (1%2). K. Brodowicz and W. T. Kierkus, Arch. Budowy Masz. 12, No. 4, 473 (1%5). “ J. A. Asher, P.F. Scott, and J. C. Wang, “Parameters Affecting Laser Velocimeter Turbulence Spectra Measurements,” Rep. No. SRD-74-021. General Electric Co. 1974. A. Acrivos, L. G. Leal, D. D. Snowden, and F. Pan, J. Fluid Mech. 34, 25 (1968). (Continued)

‘rABLE

x.

(Continued)

R. S. Howes and A. R. PhillipJ. Iron SfeefInst. 162, 392 (1949). G. Birkhoff and T. E. Caywood, J. A p p f . Phys. 20, 646 (1949). W. Davis and R. W. Fox, J. Basic Eng. 89, 771 (1%7). F. A. Schraub, S. J. Wine, J. Henry, P. W. Runstadler, Jr., and A. Littell, J . Basic Eng. 87, 429 (1965). J. E. Caf€yn and R. M. Underwood, Nature (London) 169,239 (1952). ” A. A. Kalinske, Trans. A m . SOC.Civ. Eng. 111, 355 (1946). S. K. A. Naib, Engineer 221, %1 (1966). ID J. P. Sachs and J. H. Rushton, Chem. Eng. Prog. 50, 597 (1954). P. B. Walker, in “Technical Report of the Aeronautical Research Council, 1931-32,” Vol. I, p. 97. HM Stationery Office, London, 1933. ” E. S. R. G o p i , in “Emulsion Science” (.P. Sherman, ed.), p. 1. Academic Press, New York, 1968. D. J. Ryley, J. Sci. Insfrum. 35, 237 (1958). a E. Giffen and Q. Muraszew, “The Atomisation of Liquid Fuels.’’ Chapman & Hallo, London, 1953. bb M. A. Nawab and S. G. Mason, J. Colloid Sci. 13, 179 (1958). J. Stupar and J. B. Dawson,Appf. Upr. 7, 1351 (1968). dd E. Rappaport and S. E. Weinstock, Experientia 11, 363 (1955). ec D. Sinclair and V. K. L a Mer, Chem. Rev. 44,245 (1949). E. B a h t , Aircr. Eng. 25, 161 (1953). L. Dautrebande, W. C. Alford, and B. Highman, J . f n d . Hyg. Toxicol. 30, 108 (1948). D. Sinclair, in “Handbook of Aerosols,” p. 77. U.S.At. Energy Comm., Washington, D.C., 1950. “ L. Silverman, in “Air Pollution Handbook” (P. L. Magill er a [ . , eds.), p. 12-1. McGraw-Hill, New York, 1956. R. Kumar and N. R. Kuloor, Adv. Chem. Eng. 8, 255 (1970). kk D. W. Clutter and A. M. 0. Smith, Aerosp. Eng. 20, 24 (l%l). H.R. Null and H. F. Johnson, AIChE J. 4, 273 (1958).

J I

1.1.

TRACER METHODS

49

It is clear from Table X that the production of particles with a very narrow range of sizes is difficult, but it is possible to limit the range by using filtration after generation but before introduction into the test section. This can be carried out by deposition of the larger particles under the action of gravity,110*116-118 by using electric fields, if the particles are charged,ll9 or by observing inertial effects which arise when the particles are subjected to sudden changes in their direction of motion. In some cases (as indicated under the heading “notes” in Table X) the spatial distribution of particles is limited, but this can be overcome in various ways. First, it may be possible to arrange a number of duplicate particle generation systems in such a way that the desired spatial coverage is obtained. Second, natural or induced turbulence in the experimental fluid may be used to disperse the particles. Finally, generation and dispersion can be separated so that the combination of the respective processes is the most satisfactory for the proposed application. Thus, for instance, liquid particles may be generated by bubbling the gaseous experimental fluid through a mass of the liquid, the particles then being dispersed in the test section through a number of small orifices. Up to this point in the discussion, primary emphasis has been placed on the size and number of the particles generated and the method of dispersion, but there is also a problem of potential interference between the measurement system and the experimental fluid, which also requires consideration. These interference effects are connected, respectively, with the process of particle injection and with the dynamic characteristics of the particle at the time of injection. The interference effects associated with the process of particle injection are of two kinds. First, a wake can be formed that extends downstream from the particle injection structure. Flow tracing particles moving in this wake will not have the velocity of the undisturbed flow. Second, the method used to produce the particles may superimpose spurious fluid motions on the motion of the experimental fluid. A good example of this is provided by the motions which must be generated by the rapidly moving gas jet required in the pneumatic atomization process. It is very difficult in practice to separate the wake effect and the acceleration effect, and although a satisfactory theoretical model based on Eq. (1.1.2a) could be derived for the effects of particle acceleration, the end result would probably not be very useful. A review of experimental data lie

D. C. Blanchard, J . Colloid Sci. 9, 321 (1954). N. A . Dimmock, Nature (London) 116,686 (1950). W. H. Walton and W. C. Prewett, Proc. Phys. Soc. London, Ser. B 62, 341 (1949). J. M. Schneider and C. D. Hendricks, Rev. Sci. Insrrurn. 35, 1349 (1964).

50

1. MEASUREMENT

OF VELOCITY

due to Schraub et ~ 1 . and ’ ~ Grove120on the motion of bubbles downstream from the generatinglinjecting structure suggests that if measurements are made more than two hundred structure characteristic diameters downstream from the structure, the interference effects should be negligible; by structure characteristic diameters would be meant, for example, the injection nozzle diameter. The interference effects that have been described may be overcome by eliminating the apparatus required for the generation and introduction of the particles. This can be achieved by using particles which are “permanently mixed” with the experimental fluid so even though there may be some initial disturbance when they are first added to the experimental fluid, this will have died away before any measurements are made. Of course, in the narrowest sense, no particles remain permanently mixed with the fluid since they are all subject to a greater or lesser extent to forces, (e.g., buoyancy, and inertial deposition on the surfaces of the ducts, which tend to remove them from the fluid. However, particles which remain mixed long enough to ensure that an appropriate amount of meaningful data is obtained can be viewed as permanently mixed. 1.1.2.7. Error Analysis. The results of any measurement should include a statement about the estimated magnitude of the errors of measurement. This section attempts to do this for the errors associated with inferring the fluid velocity from the measured particle velocity; it is not concerned with errors arising from the measurement of the particle velocity. These are discussed in Sections 1.1.3 and 1.1.4. Having ascertained the dynamic characteristics of the flow tracing particles by means of the model described in Section 1.1.2.2, we need to find the residual uncertainty in the determination of the fluid velocity. In doing this, it will be assumed that the following fixed errors are negligible, for the reasons indicated in parentheses: (i) loading errors (particle concentration less than 1: 1000 by volume; see Section 1.1.2.3); (ii) particle motion due to lift forces (by avoiding excessive lateral velocity gradients or the use of deformable particles; see Section 1.1.2.5); and (iii) interference due to the particle injection structure (makes velocity measurements more than two hundred characteristic structure diameters downstream from the structure; see Section 1.1.2.6). In addition to the preceding, the following sources of error, which may not be negligible, can be assumed accounted for as indicated: (i) Brownian motion, as discussed in Section 1.1.2.4; (ii) particle motion due to gravity, A . S. Grove, An investigation into the nature of steady separated Rows at large Reynolds numbers. Ph.D. Thesis, University of California, Berkeley (1963).

1.1. TRACER METHODS

51

by using where possible particles with u = 1, and otherwise by employing the corrections discussed in Section 1.1.2.5; and (iii) distribution of particle diameters and densities (and hence u),which can be taken into account by using an appropriate mean diameter as described in Section 1.1.2.6.

The remaining errors of the measurement of fluid velocity are (i) uncertainty due to the measurement of the parameters of the dynamic model, viz., the particle diameter d, the density ratio (T,and the kinematic viscosity vF of the experimental fluid; and (ii) residual uncertainty associated with the particle dynamic model of Section 1.1.2.2. A completely general error analysis which takes into account the uncertainty in the parameters of the dynamic model leads to considerable algebraic complexity and is, in any case, of doubtful utility because the variations in particle diameter and density caused by the particle generation process are much more significant. (see Section 1.1.2.2.5) indicates that the The work of Mazumder et model for the particle dynamic characteristics is satisfactory within experimental error. Furthermore, an examination of the literature leads to the conclusion that the corrections listed in Table I1 are also satisfactory within experimental error. It will therefore be assumed that all sources of error connected with the model contribute an uncertainty of f 5% to the determination of the fluid velocity. In conclusion, the uncertainty of the fluid velocity measurement will be taken as 2 5% (the residual error of the dynamic model) together with the estimates of the Brownian motion, and the effects of the particle size and density distribution. 1.1.2.8. Optical Characteristics of Flow Tracing Particles 1.1.2.8.1. INTRODUCTION. The motion of flow tracing particles is conveyed to the observation system by the light they scatter as they pass through the illuminated observation volume within the fluid. The strength of this scattered light must therefore be sufficient to provide an input to the observation system which can be unambiguously identified as originating at a single flow tracing particle within the observation volume, rather than from some spurious source, e.g., light scattered by flow tracing particles not within the observation volume. The preliminary assessment of a flow tracing particle's suitability therefore necessitates a quantitative knowledge of the amount of light scattered into the aperture of the observation system. The usual practice heretofore in particle tracking has been to maximize or optimize the strength of the scattered light by trial and error methods in the laboratory. However, light scattering theory is well estab-

52

1.


lished,121-123 so that in principle it should be possible to calculate the light scattered by a candidate flow tracing particle. In fact, such calculations usually are only an estimate, not because of any deficiencies of the theory, but as a result of limitations in the necessary information about the flow tracing particles.* For this reason it will probably be necessary in most cases to base the final selection on laboratory tests, but at least these can be limited to a comparatively short list of candidates. The objective of this section is to review briefly those results of light scattering theory that are useful in assessing the optical properties of flow tracing particles. The discussion will be limited to the case of a single incident beam of light, which is the situation in chronophotography (see Section 1.1.3). The extension to light scattering from two or more incident beams, which is of importance in the laser Doppler technique, is discussed in Section 1.1.4. The fundamental objective of the calculations considered in this section is to estimate the total luminous flux P (in watts) of the scattered light available at the aperture of the observation system. To do this, the directional distribution of this scattered light must be summed over the solid angle w subtended by the aperture at the scattering particle (see Fig. 8). This gives the relationt

P, dh

=

F,,

ZAni n, do dh.

(1.1.22)

H. C. van de Hulst, “Light Scattering by Small Particles.” Wiley, New York, 1957. M. Kerker, “The Scattering of Light.” Academic Press, New York, 1969. lz8 D. Deirmendjian, “Electromagnetic Scattering on Spherical Polydispersions.” Am. Elsevier, New York, 1969. lz1

lz2

* The following sources of error are considered to affect the calculations: (a) errors of interpolation, when using tabulated data; (b) erroneous or inaccurate refractive indices; (c) erroneous or inaccurate distributions of particle size and density (includes both errors of measurement and sampling errors); (d) incorrect light source spectra; and (e) incorrect spectral response of the sensitive element of the observation system. The quantitative effect of these errors on the calculation of light scattering cannot be ascertained because of the complexity of the theory of scattering. However, crude calculations indicate that the errors introduced into the scattering calculations could be substantial, so care should be taken in assigning values to the parameters of a given scattering calculation if a reasonable estimate of the scattering is to be made. t The unit vectors nl and n, refer, respectively, to the directions of the incident light beam and the direction of the normal to the plane of the observation system aperture (see Fig. 8). If, as is usually the case, these two vectors intersect, when extended, at the scattering particle, then their scalar product reduces to the cosine of the polar angle &.. In Eq. (1.1.22) it is further assumed that the incident flux density Fl is uniform over the beam cross section. Since this is not correct, the particle must be assumed to be much smaller than the cross section of the illuminating beam at the observation volume.

1.1.

TRACER METHODS

53

Fic. 8. Geometry of light scattering by flow tracing particles. Note that vectors are indicated by wavy underlines in figures and by boldface type in the text.

The quantity FiA is the monochromatic flux density [W/(cm2 pm)] of the light incident on a unit area of the particle projected into a plane normal to the direction of the assumed collimated beam of incident light. The directional distribution of the scattered light is represented by the monochromatic angular scattering cross section ZA , which has units of square centimeters per ~ t e r a d i a n . ’ ~ The ~ dependence of this quantity on direction is provided by the theory of light scattering or by direct measurement. 125-128 1.1.2.8.2. LIGHTSCATTERING DATA.The scattering of electromagnetic radiation, such as light, is the result of the interaction between the incident electromagnetic waves and the electrons of the material forming the particle. The incident radiation excites the electrons which in turn emit secondary waves, and these latter waves are the scattered radiation. The theoretical calculation, which for spherical particles* is due to Mie R. Penndorf, J. Opr. Soc. Am. 52,402 (1962). F. T. Gucker and J. J. Egan, J. Colloid Sci. 16, 68 (l%l). IZB F. T. Gucker and R. L. Rowell, Discuss. Furuduy Soc. 30, 185 (1960). lZ7 D. T. Phillips, P. J . Wyatt, and R. M. Berkman, J. Colloid Interface Sci. 34, 159 (1970). IzBE. C. Roberson, “The Development of a Flow Visualization Technique” Report NO. R181. National Gas Turbine Establishment, England, 1955. lZ4

IZ5

* Many flow tracing particles are nonspherical, e.g., metal flakes, naturally occurring fibers in the air and in water, distorted gas bubbles, and liquid droplets. Numerical data1z1-1z3 for such particles is much sparser than the corresponding information on spherical

54

1.


(1908), of the directional distribution of this scattered radiation, i.e., the scattering cross section, involves an .application of the electromagnetic theory of radiation. According to the Mie theory and confirmatory experiments, when the incident light is randomly polarized, the light scattered by a small particle consists of two incoherent, plane polarized components with mutually orthogonal planes of polarization. One of the components, indicated by subscript 1, vibrates perpendicularly to the plane of observation (the plane containing the direction of observation and the direction of propagation of the incident beam). The other component, associated with subscript 2, has vibrations parallel to the plane of observation. Mie calculated the dependence of these two components on the direction 8. In terms of the monochromatic angular scattering cross section Z,, , the Mie theory gives the result* (1.1.23) When the particle is illuminated by plane polarized light (subscript l), e.g., light from a laser, the plane of polarization of the scattered light (subscript 2) is perpendicular to the polarization plane for the incident light. The appropriate angular Mie scattering cross section is AiJ2/47r2. The quantities il and iz in Eq. (1.1.23) are called the intensity functions and have units of steradians-'. They are given in terms of complicated infinite series, and for practical application it is necessary to use graphs or tables of these quantities. Lists of such tabulations and plots will be found in van de Hulst,121Kerker,122*12g and H o d k i n s ~ n . ~ ~ As the diameter of the particles increases relative to the wavelength of the incident light, i.e., as a increases, the calculations using the Mie theory become progressively more tedious, and this has served to limit the range of particle sizes covered by the published calculations130to sizes

119

Is)

M.J. Kerker, J . Opt. SOC. Am. 45, 1081 (1955). H.Walter, Oprik (Stuttgart) 16, 401 (1959).

particles. This, together with the limited ability to assess the dynamic characteristics of such particles, makes the use of spherical particles desirable in flow tracing applications. If it is considered necessary to use nonspherical particles, their scattering properties are best ascertained by m e a ~ u r e m e n t ~ *or, ~ - ~better, *~ their suitability can be ascertained directly using the illumination and observation system which is to be employed in the actual velocity measurements. * In Eq. (1.1.23), a is the dimensionless particle size parameter defined as ?zdn,/A,, = ?zd/A,, where 4 and AF are the wavelengths of the incident tight in a vacuum and in the experimental fluid, respectively; nF is the refractive index of the experimental fluid; m' = n' - i d , where K' is nonzero for an,absorptive material; and n' = np/nF.

00 30" 60"90"120"150"180"

0" 30"60"90"120"150"180"

(b) (C) FIG.9. Typical diagrams of the intensity functions il (solid curve), and it (dotted curve) for spherical particles. (a), (b), and (c) The refractive index n = 2, 1.55, and 1.33 spheres, respectively. x = ad/X, = a. The vertical scale is logarithmic, with one division equal to ten units. The horizontal scale shows the polar angle 8, and 0" corresponds to the forward direction of light scattering, which is coincident with the direction of the incident light. The value of the polar angle 8 for backward scattering is 180". The values of il and it for 0 = 0" and 180" are shown in the margin. (Adapted, with permission, from van de Hulst.'*')

56

1. MEASUREMENT OF VELOCITY

smaller than about 50 pm. This is not sufficiently large to allow the estimation of the light scattering by some of the larger particles (d < 3600 pm) that are used for flow tracing. As a consequence of this computational limitation, various approximate methods, which greatly simplify the scattering calculations, have been devised. They are reviewed by H a ~ k s l e y , ' ~ ' .van ' ~ ~de Hulst,'21 and Green and Lane.33 Of the numerical results which have been provided by these approximate methods, the most useful are those due to Hodkinson and green leave^'^^ (who give data on Z/(?rd2/4), called Z in their notation), and those due to Davis'34 (who calculated Z/(d2/4), called G in his notation). The advent of the electronic digital computer has to some extent made the tabulations of scattering data superfluous. Probably it is now more appropriate to use the computer as the information is required and for the values of the parameters that are of interest at that moment. Alternatively, the calculations can be made for the appropriate ranges of the parameters and variables, and then stored in punched cards or magnetic tape for use as required. Examples of typical computer programs for carrying out scattering calculations are provided by Deirmendjiar~'~~ and Gie~e.'~~ An examination of the sample light scattering characteristics given in Fig. 9 shows that these have a very sensitive and complicated dependence on the size a! and the refractive index n. Nevertheless, two general results are that for a! greater than about three, scattering in the forward direction is significantly larger than scattering in any other direction, and this effect becomes greater as the particle size increases. This is even true for perfectly reflecting spheres (mF+ m) due to the light passing around the edges of the particle by refraction. Thus, from the point of view of maximizing the amount of light scattered into the observation system, large flow tracing particles are more desirable than small ones, and observation from the forward direction is superior to observation from an oblique angle (actually, from a near forward direction, in order to avoid the light directly transmitted from the source; see Section 1.1.3 for more details). To apply the numerical scattering data, three pieces of information are required. These are the refractive index m' of the particle material relative to the experimental fluid, the diameter d of the particle, and spectral information on the light source. lS1

lSS IM lss

P. G . W. Hawksley, Er. Coal Util. Res. Assoc., M o n . Bull. 16 (4), 117 (1952). P. G . W. Hawksley, Br. Coal Util. Res. Assoc., M o n . Bull. 16 ( 9 ,181 (1952). J. R. Hodkinson and I . Greenleaves, J. Opt. Soc. Am. 53, 577 (1963). G. E. Davis, J. Opt. Soc. Am. 45, 572 (1955). R. H. Giese, Electron. Rechnenanlagen 3, 204 (1961).

1.1.

TRACER METHODS

57

Refractive index data can be obtained from standard tabulation^'^^ or by direct measurement. For convenience, Table XI lists the indices at one wavelength for a representative selection of particle and experimental fluid combinations that have been reported in the literature on particle tracking. Information on particle diameters will be available to the experimenter either from a knowledge of the characteristics of the particle generation system (see Section 1.1.2.6) or from direct measurement of the particle sizes (see Section 1.1.2.2.3). If the light source is monochromatic (laser or mercury arc light with optical filtering), Eq. (1.1.22) may be used without integration over all frequencies to calculate the light scattered by the particle. Where the light is supplied by such polychromatic sources as incandescent lamps or flash tubes, it is necessary to integrate in Eq. (1.1.22), as indicated, over all frequencies, taking into account the spectral character (see Section 1.1.3.3) of the light source (Fib). Before concluding this section it should be pointed out that it has been assumed that the individual particles are separated by sufficiently large distances so as to ensure that the scattering from a given particle is unaffected by the presence of neighboring particles and that the scattered light experiences no further scattering (multiple scattering) en route to the observation system. Both these effects can be avoided if the concentration of particles is sufficiently dilute (say 1O8-10l8 particles/cm3) and the illuminated observation volume is kept as small as possible. 1.1.2.8.3. LIGHTINCIDENT AT THE OBSERVATION SYSTEM. Since the monochromatic angular scattering cross section I h depends only on the polar angle 6 , we can write Eq. (1.1.22) as

I@,

8"+A812

PA =

TFiA

IA(6)sin 26 do,

(1.1.24)

where A6 is the angle subtended at the scattering particle by the aperture of the observation system, and 6,, the scattering angle as shown in Fig. 8. If tabulated values of the angular scattering cross section I are available, the graphical or numerical evaluation of the integral in Eq. (1.1.24) is straightforward (Hodkinson and green leave^'^^ have tabulated the results of such calculations for the case 6, = 0). However, in view of the tediousness of such calculations, it is more usual in flow tracing applications to assume that over the range of polar angles 6 subtended by the observation system aperture, the scattering can be assumed to be constant and J . A. Dean, ed., "Lange's Handbook of Chemistry," 1 Ith ed. McGraw-Hill, New York, 1973.

TABLEXI. Refractive Indices for Typical Particle-Experimental Fluid Combinations Used in Flow Tracing Refractive index

Particle material Benzene and n-butyl phthalate Benzene and carbon tetrachloride Olive oil and ethylene dibromide Xylene and n-butyl phthalate Olive oil and nitrobenzene White spirit and carbon tetrachloride Kerosene and dibutyl phthalate Hexane Air Polystyrene Polystyrene Octoil S + 20% by weight white titanium dioxide

Experimental fluid Water Water Water Water Water Water Water Water Water Water Water Air

Reference for application

Constituents of mixture relative to vacuum (npp

C

1.5011 1.491I 1so1 1 1.4630 1.4662 1.5379 1.494 (commercial) 1.491 1.4662 1S524 I .420 1.4630 1.440 I .490 1.3749 1.ooo 1.6 1.6 1.486 (dioctyl

c-e,l

f &?

h,i

h j

k k,o,r

m,s n

P

Mixture relativeb to exp'tl fluid Reference W

(n' = np/nF)

1.12

W W

1.12

W W

1.10

W

X

1.12

W W

1.12

W

Y

1.07

W

Z

1.11

aa W

aa bb bb W

I .05 0.750 1.20 1.20 1.486

phthalate) Medicinal mineral oil Xylene and dibutyl phthalate

Air Water

4 t

Hydrogen Dioctyl phthalate

Water Air

U V

.a

1 1.494 (commercial) 1.490 1.Ooo 1.486

CC

X

1.11 1.12

aa aa dd

0.750 1.486

Refractive indices are the value at the wavelength (0.5893 p m ) of the sodium D line and at a temperature of 20°C unless indicated otherwise by a subscript, which identifies the actual temperature. Calculated on the assumption that relative proportions of the constituents of the particle were such that u = 1. (The following values of nF were used: water: 1.3333; air: 1.OOO.) A. A. Kalinske, and C. L. Pien, Ind. Eng. Chem. 36, 220 (1944). A. A. Kalinske, Trans. Am. Soc. Civ. Eng. 111, 355 (1946). ‘A. A. Kalinske, Trans. Am. Soc. Civ. Eng. 105, 1547 (1940). P. B. Walker, in “Technical Report of the Aeronautical Research Council,” p. 97. HM Stationery Office, London, 1933. E. 0. Macagno and H. Rouse, Proc. Am. SOC. Civ. Eng. 87 (EM5), 55 (1961). * S. K. A. Naib, Engineer 221, %I (1966). P. Frenzen, AEC Rep. ANL 6794 (1%3). j J. P. Sachs and J. H. Rushton, Chem. Eng. Prog. 50, 597 (1954). G. Birkhoff and T. E. Caywood, J . Appl. Phys. 20, 646 (1949). J. E. CalTyn and R. M. Underwood, Nature (London) 169,239 (1952). E. F. Winter and J. H. Deterding, Br. J . Appl. Phys. 7 , 247 (1956). ” H . G. Schwartzberg and R. E. Treybal, Ind. Eng. Chem., Fundam. 7 , 1 (1968). R. S. Howes and A. R. Phillip, J . Iron Steel Inst. 162, 392 (1949). C. J. Chen and R. J. Emrich, Phys. Fluids 6, 1 (1963). * J. A. Breslin and R. J. Emrich, Phys. Fluids 10, 2289 (1%7). R. M. Nedderman, Chem. Eng. Sci. 16, 113 (1961). E. C. Roberson, “The Development of a Flow Visualization Technique” Report No. R181. National Gas Turbine Establishment, England, 1955. M. M. Kolpak and P. S. Eagleson, Mass. Inst. Technol., Civ. Eng. Dep., Hydrodyn. Lab. Rep. NO. 118 (1%9). A. C. Tory and K. H . Haywood, Am. Soc. Mech. Eng. [Pap.] No. 71-FE-36. M. K. Mazumder, B. D. Hoyle, and K. J. Kirsch, Purdue Univ. (Indiana)E n g . Exp. Stn. Bull. 144, 234 (1975). J. A. Dean, ed., “Lange’s Handbook of Chemistry,” 11th ed. McGraw-Hill, New York, 1973. C. Marsden and S. Mann, “Solvents Guide,” 2nd ed. Cleaver-Hume, London, 1%3. W. Gardner, “Chemical Synonyms and Trade Names,” 7th ed. Technical Press Ltd., London, 1971. 2 H. S. Bell “American Petroleum Refining,” 3rd ed. Van Nostrand-Reinhold, Princeton, New Jersey, 1945. an R. C. Weast, ed., “Handbook of Physics and Chemistry,” 55th ed. CRC Press, Cleveland, Ohio, 1974. ba N. S. Berman, “Fluid Particle Considerations in the Laser Doppler Velocimeter,” Rep. No. ERC-R-73017, Eng. Res. Cent., Arizona State University, Tucson, 1973. cc G. W. C. Kaye and T. H. Laby, “Tables of Physical and Chemical Constants,” 14th ed. Longmans, Green, New York, 1973. dd A. K . Doolittle, “Technology of Solvents and Plasticizers.” Wiley, New York, 1954. (I

’

‘

y,

60

1.

MEASUREMENT O F VELOCITY

equal to the value at the angle of observation 8,. Then Eq. (1.1.24) becomes P A=

(1.1.25)

FlAzA(~s)~,

where o is the solid angle subtended at the scattering particle by the aperture of the observation system (see Fig. 8). In assessing the optical characteristics of a flow tracing particle, it is more useful to determine the strength of the scattered light at the sensitive element of the observation system (photographic emulsion, the photocathode of a photomultiplier tube, or the retina of an observer's eye) rather than at its aperture, as assumed in Eq. (1.1.25). In particular, we are interested in the luminous flux density F D A (W/cmZ pm) of the image projected by the observation system lens on the sensitive element. From Eq. (1.1.25), we obtain13'

-

FDA = [ T T A / ~ ~ +( M 1)2]Fi~z~(es>,

(1.1.26)

where N is the f-number of the lens and is equal to the ratio of the focal length of the lens to its diameter,* M is the magnification of the optics (equal to the ratio of the image distance to the object distance); Ti, the monochromatic transmission of the optical system?; and Z; , the monochromatic angular scattering coefficient, which is equal to Z/(~d2/4). 1.1.2.9. Particle Selection. As a general rule, the smaller a flow tracing particle, the better are its dynamic characteristics, but the poorer are its optical characteristics.$ This situation necessitates an optimum choice of particle size. However, the dynamic and optical characteristics of a flow tracing particle do not define the optimization problem completely, because there are certain qualitative considerations that should be inL. E. Mertens, "In-Water Photography." Wiley, New York, 1970. W. G. Hyzer, "Engineering and Scientific High-speed Photography." New York, 1962. la@L. M. Myers, "Television Optics." Pitman, London, 1936. lsoa F. Durst, Z.Angew. Math. Phys. 24, 619 (1973). ls7

lS8

Macmillan,

* Where the observation system is a microscope (see Section 1.1.3). it is usual to use the numerical aperture (NA) rather than the f-number(N) of the lens; these are related by NA = M/[ZN(M + l)]. t H y z e F indicates that for most coated lenses the total transmission T has a value of about 0.8. The procedure for more detailed calculations of the transmission is given by Myers.'= t: This is true provided the direction of the incident light and the direction of observation are fixed (see Durstlma). In the case of the laser Doppler velocimeter, coherence reyuirements introduce further complications which would modify the relation between particle size and the amount of scattered light.

1.1.

TRACER METHODS

61

cluded. The choice of a flow tracing particle is therefore a complicated process, involving quantitative and qualitative judgments together with a trial and error examination of the actual performance of the system. The objective in this section is to review, by means of a numerical example, the first trial of the particle selection process. Subsequent trials are the same, but may involve experimental tests as well as a revision of the numerical calculations of the first trial. The example, which is based on a measurement system designed and used by E i ~ h o r n ,requires '~~ the measurement of the velocity distribution in a direction normal to a heated vertical plate having a surface temperature of about 55°C. The plate is enclosed in a chamber filled with air at atmospheric pressure. The chronophotographic technique is to be employed (the design of the illumination and observation system for this particular example will be discussed in Section 1.1.3.10). The anticipated range of fluid velocities is, from the theoretical calculation^,'^^ 21 to 1.5 cm/s. The first step is to decide on the material* and size of the flow tracing particles. The particle material is to be dioctyl phthalate (DOP). This is a liquid, so that the particles will be of uniform density and spherical, which will improve the accuracy with which the dynamic characteristics can be determined. It boils142at temperature (215-23OoC), which allows a Rappaport-Weinstock generator to be used (see Table X). DOP is a commercially used plasticizer, readily available and of moderate cost. Previous experience56suggests that it is safe to use in flow tracing applications. The Rappaport-Weinstock generator will be assumed to be adjusted to give particles with a mean diameter (this is not the dynamic mean discussed in Section 1.1.2.2.3) of 1 pm at a concentration of 10s particles/cm3 of air. The boiling point of DOP is considered to be sufficiently high so as to ensure that particle evaporation, which could result in time dependent dynamic characteristics, can be neglected at the temperatures (less than the plate temperature of 55°C) that will be encountered. The particle half-life and fouling tendencies of DOP in the heated plate system are unknown and will have to be evaluated by experiment. R. Eichorn, Int. J . Heat Mass Transfer 5 , 915 (1962). S. Ostrach, Nut. Advis. Comm. Aeronaut., Rep. 111 (1953). H. R. Simonds and C. Ellis, "Handbook of Plastics." Van Nostrand-Reinhold, New York. 1943. 110

* A list of particles that have been used in previous flow tracing applications has been compiled by the autho?; this may be helpful in selecting a suitable material. See also Table XI.

62

1.


The next step in the particle selection process is to examine the dynamic characteristics. This involves items (a) and (b) discussed below: (a) Check the validity of Stokes' law (Table 11). In this case, the following conditions on Stokes' law can be satisfied: (i) the experimental fluid is incompressible, because the Mach number is less than unity; (ii) no wall correction is required because no particle is observed to approach closer to the heated plate than 50 pm (see the discussion on thermophoresis below), and a wall correction, in this case, is only applicable when the particle approaches within 5 pm of the heated plate; (iii) according to Fig. 2, for 1 pm spherical particles in air the relative particle Reynolds number is much smaller than 0.074 and the resultant error in using Stokes' law is less than 1%; (iv) there are no fluid motions within the liquid droplet because the experimental fluid is a gas; and (v) the droplets are undeformed because they are less than 1000 pm in diameter. The only correction that must be applied to Stokes' law is Cunningham's correction to take account of the inhomogeneity of the air. For particles of 1 p m diameter, a correction factor K in Eq. (1.1.5) of 0.86 should be introduced. (b) Determine the relation between the particle velocity up and the fluid velocity uF (see Section 1.1.2.2.4). For the preliminary assessment being carried out here, it will be assumed that the fluid velocity does not vary along the length of the plate for points at a given distance from the plate.* A type I11 approximation (Table I) to Eq. ( 1 . 1 . I) will be used because the particle to fluid density g/cm3). ratio is about one thousand (pp = 0.98 g/cm3, pF = 1.205 x Including the gravitational force, which acts in the direction opposite to ], K = the particle motion, we obtain up = (uF - uT)[l - exp(- ~ t ) where 18KpF/ppdz = 3.34 x 10-5K S-' [with pF= 1.816 x lop4g/(cm-s)], K = 0.86 is Cunningham's correction, and uT is the gravitational terminal velocity (see Section 1.1.2.5). According to this result, the time for a particle to attain a velocity within 0.1% of its final velocity [uP/(uF - uT) = 0.9991 is 24 ps. The error involved in using these particles is small enough so that no correction need be applied to the measured particle velocity u p to obtain the fluid velocity uF. M. Gilbert, L. Davis, and D. Altman, Jet Propul. 25, 26 (1955).

* A more accurate procedure would be to take into account the longitudinal variation in fluid velocity using Ostrach'sl" theoretical results and then calculate the relation between the particle and fluid velocities using the results obtained by Gilbert ef a/.'"

1.1.

TRACER METHODS

63

The third step in the particle assessment is to estimate the magnitude of the fixed errors due to : (a) the presence of the particles affecting the properties of the experimental fluid, (b) Brownian motion of the particles, and (c) particle motions caused by extraneous forces. If any of these are unacceptably large, the steps that are necessary to minimize them should be investigated. The anticipated particle number concentration of lo6 particles/cm3 of air gives a particle volume concentration (assuming all the particles are spheres of 1 pm) of 1 : lo6, which is much smaller than the volume concentration ( 1 : lo3)at which the fluid properties would be affected by the presence of the particles (see Section 1.1.2.3). Brownian motion cannot be ignored for particles smaller than 0.5 p m in air. Since particles of this size will be present, the best way to eliminate it from consideration is to ensure that the luminous flux density at the photographic emulsion from the smaller flow tracing particles is insufficient to produce a measurable trace (see Section 1.1.3.10). In this case gravity, lift, and thermophoresis are the extraneous forces that affect the particle motion (see Section 1.1.2.5). According to Fig. 6, for pp = 0.98 g/cm3 the gravitational terminal velocity is 0.035 mm/s. This is negligible compared to the anticipated minimum fluid velocity of 1.5 cm/s. The particles may experience lift due to the velocity gradients normal to the heated plate. The particle will then have a component of velocity normal to the plate, as well as a parallel component. The theoretical results of Ostrachlql are used to estimate the maximum velocity gradient as 127 s-l. The corresponding terminal velocity is 0.6% of the component parallel to the plate. To estimate the terminal velocity arising from particle rotation, we follow Tollert14 and take wp = O.S(du,/dz). The velocity ratio is then 1.8 x lo-', which is negligible. The terminal velocity of the particle due to the temperature gradient normal to the wall can be calculated using Epstein's e q ~ a t i o n . ' ~ ~Os*'~~ trach'sl4' results are used to estimate the maximum temperature gradient as 42.0°C/cm. This corresponds to a terminal velocity of 9.8 X cm/s, which is negligible compared to the velocity of the particle parallel to the heated plate. Where it is necessary to apply a correction to the experimental data in order to obtain the fluid velocity, it would be appropriate, as the final step in the assessment process, to estimate the magnitude of the error contrib1

H. Tollert, Chem.-/ng.-Tech. 26, 141 (1954). 54, 537 (1929). '* R. L. Saxton andR. E. Ranz, J . A p p l . Phys. 23,917 (1952).

P. Epstein, Z . Phys.

64

1.


uted to this by the residual error of the dynamic model (see Section 1.1.2.7). However, this is unnecessary here because, as shown previously, the difference between the fluid and particle velocities is small enough to require no correction. 1.1.3. Chronophotography 1.1.3.1. Introduction. The chronophotographic technique of fluid velocity measurement requires simple apparatus which can provide quantitative data on fluid velocities that are directly relatable to the fluid flow patterns in the system. This method of fluid velocity measurement involves photographically recording the trajectory of a flow tracing particle in such a way that it includes a time scale as an intrinsic feature of that trajectory, so that it can then be used to obtain the velocity of the particle. The fundamental performance parameters of the chronophotographic method are (a) precision +- 5%; (b) uP.MAX = 780 m/s, up,MIN = 2.6 x mm/s; and (c) measuring volume 8 x 1oJ to 1.5 mm3. The minimum velocity that has been reported as measured by this method is substantially lower than the accepted minimum for a hot wire anemometer of 4 cm/s. According to the manufacturer's literature, laser velocimeters are capable of measuring particle velocities as low as 1 mm/s (5 pm/s with photon correlation). Although the minimum velocity given above is less than either of these figures, it should be recognized that spurious free convection in the experimental fluid and the effects of particle sedimentation could significantly raise the acceptable minimum velocity for both the chronophotographic and laser methods to a more practical lower limit of, say, 10 mm/s. The maximum velocity of 780 mm/s compares rather unfavorably with the los m/s which Jackson and Pau114' propose as an upper velocity limit for the laser velocimeter. However, velocities of this magnitude would probably only be encountered in rather special circumstances, e.g., under hypersonic flow conditions, and the laser velocimeter and the chronophotographic method can be considered to reach comparable maximum fluid velocities. The dimensions of the smallest chronophotographic measuring volume do not compare very favorably with those for the hot wire anemometer148 of 2.5 x mm3 and for the laser velocimeter of mm3. However, the measurements made by the hot wire anemometer are not local be-

]''D. A. Jackson and D. M. Paul, Phys. Lett. A 32, 77 (1970). 'ls F.

Durst and J . H. Whitelaw, Prog. Heat Mass Transfer 4, 311 (1971).

1.1. TRACER METHODS

65

cause the volume given is that of a cylinder of length 0.5 mm and diameter lov3mm. Fage and Townends2 have made chronophotographic measuremm and diameter 0.25 ments in a cylinder of depth less than 2.5 x mm, which are comparable with the dimensions of the space in which the hot wire anemometer operates. The smallest attainable measuring volume of the laser anemometer is not only smaller than that of the hot wire and of chronophotography , but is approximately spherical in shape so that, of the three techniques, it provides the closest approximation to point measurements. Chronophotography has a number of other desirable performance characteristics that cannot be expressed quantitatively. These are the following: (a) Velocity data from a large region of the flow can be obtained in a single photographic exposure lasting only a fraction of a second. This compares favorably with other methods, such as the laser anemometer, where it may be necessary to take data at many points throughout the test section, with the data gathering process at each station lasting several minutes. Velocity measurements in unsteady flow processes, for example, are therefore very appropriately made chronophotographically . (b) Qualitative and quantitative data on conditions in the flow field can be much more closely related by chronophotography than by other methods. This is because the data represent the paths followed by the fluid in the test section. (c) The data are not directionally ambiguous, as are those obtained from the hot wire anemometer. (d) Chronophotography is capable of providing Lagrangian correlation data in turbulent fluid flow. The technique has some drawbacks. First, data handling is very time consuming, particularly if turbulent flows are involved, and requires substantial human labor. Second, the method cannot be applied if the fluid contains a high concentration of suspended matter, i.e., the particles in a solid fuel rocket exhaust, or the silt in a river. In the course of reviewing the literature while preparing this section, it became clear to the author that there are many similarities in methods and aims between chronophotography in fluid velocity measurement and bubble chamber photography (used as a detector of the nuclear particles produced by high energy accele -ators). In both cases the tracks of small particles are photographed in a three-dimensional space, and the coordinates of that path must be obtained from the photograph. Chronophotography adds only one thing to this, namely, interrupted illumination, which provides the time scale that is necessary in fluid velocity measure-

66

1.


ments. The design of illuminating and photographic systems for bubble chambers has been summarized by Welford 1.1.3.2. The Technique. The trajectory of the flow tracing particle is made visible by appropriate illumination and is recorded photographically. A time scale or time base is added to the trajectory by interrupting the illumination at known intervals. The recorded trajectory consists of dots or dashes with a spacing proportional to the particle velocity.* Determination of the velocity of the flow tracing particle is then reduced to the measurement of a distance and a direction. The relation between the spacing of the images of the flow tracing particle and its speed up is given by up

= f A x / M ( N - 1).

( 1.1.27)

In this equation, Ax is the distance on the emulsion between the first and the last of N consecutive images; f,the interruption frequency (so l/f = Ar is the time interval between any two adjacent images); and M, the magnification of the camera optical system. This equation either assumes that the particle velocity (direction and magnitude) is constant in the time taken to obtain the N consecutive images or, if it is not, gives the average particle velocity in that time (the resulting error is discussed in Section 1.1.3.9). Where this approach to data analysis is inconvenient (with turbulent data), or could introduce substantial error (in the presence of steep velocity gradients), it might be better to represent the particle motion by a time displacement function obtained graphically or by a least squares fit. This method, which is discussed further in Section 1.1.3.8, depends upon having an adequate number N of interruptions visible in the camera field of view. The maximum possible particle velocity up,MAX that can be measured by the chronophotographic method can be obtained from Eq. (1.1.27). The length of the particle trace on the photographic emulsion depends on the magnification M of the camera lens and the characteristic dimension D,of the field of view. This, in turn, for given illumination interruption frequency f,fixes the upper limit of the particle velocity up which is deter140 W. T. Welford, in “Bubble and Spark Chambers” (R. P. Shutt, ed), Vol. 1 , p. 233. Academic Press, New York, 1967.

* If the interruption is too great, it may be difficult to obtain from the trajectory the successive locations of a flow tracing particle. This can be avoided if the particle is subjected to uninterrupted illumination to obtain the trajectory, and on this is superimposed an interrupted light source which increases the light scattered by the particle. The trajectory then appears continuous, with a succession of bright spots whose distance is proportional to the velocity of the particle. It should be noted that this technique may decrease the precision with which the length Ax, in Eq. (1.1.27), is measured.

1.1.

67

TRACER METHODS

mined by the possibility of finding two images of the same particle on the emulsion. Thus, from Eq. (1.1.27), with N = 2 and Ax = DcM (for DL > Dc) or Ax = DLM (for DL< D c ) , ( 1.1.28)

In many cases, e.g., in turbulent flow, the image must include information on the direction of particle motion. This can be done by introducing an irregularity, a timing key,82into the interruption of the light. This serves to identify simultaneous events in the measuring volume, and hence the spatial relation among the flow tracing particles (see Section 1.1.3.3). The chronophotographic technique requires for its practical realization the following three elements: (a) an illumination system to irradiate the measuring volume, so that only the flow tracing particles in that volume can be observed without being obscured by particles in the unilluminated flow field; (b) means for interrupting the light precisely and accurately at given intervals and for a given period of exposure; and (c) a camera for recording the light scattered by the flow tracing particle. Means are also required for determining the velocities of the individual flow tracing particles from measurements on the photographic record. The camera is only capable of recording components of velocity in the plane normal t o the axis of the lens system. Thus the presence of a third velocity component aligned with this axis must be detected in one of two ways: (a) observation from two different directions, i.e., stereoscopic ob~ e r v a t i o n ' ~ ~ or - ' ~(b) ~ ; introducing some means of identifying the motion of the flow tracing particle in the third coordinate direction.lS9 Space does not permit a discussion of these methods, but further details can be found in the indicated references. 1.1.3.3. Interrupted Illumination. The design of a chronophotographic illumination system depends on several important parameters. These are (a) pulse duration, (b) interruption frequency, (c) amount of incident light required, (d) size of the illuminated measuring volume, (e) type of illumilJ0 151

C. Chartier, Publ. Sci. Tech. Minist. Air ( F r . ) ,Bull. Serv. Tech. 114 (1937). J . E. Miller, U.S.Air Force Cambridge R e s . C e n t . , Geophys. R e s . Dir., Geophys. R e s .

Pup. No. 19 (1952). 152

lS

R . M. Nedderrnan, C h e m . Eng. Sci. 16, 113 (1961). L. F. Daws, A. D. Penwarden, and G . T. Waters, J .

Inst.

Heat. Vent. Eng. 33, 24

(1%5).

K . V. S. Reddy, M. C. van Wijk, and D. C. T. Pei, Can. J . C h e m . Eng. 47,85 (1969). H. Bippes, Dtsch. Lufi-Raumfahrt, Forschungsber. DLR-FB-7437 (1974). las J . K . Nieuwenhuizen, C h e m . Eng. Sci. 19, 367 (1964). 15' D. Mehrnel, In!.-Arch. 31, 294 (1962). la8 J. E. Caffyn and R. M. Underwood, Nature (London) 169, 239 (1952). D. A . van Meel and H. Vermij, A p p l . Sci. R e s . , Sect. A 10, 109 (1961). lss

68

1.


nation (dark field or bright field), and (f) certain miscellaneous factors, such as spectral character of the light, luminous efficiency of the source, power and cooling requirements of the source. The first two of these parameters, (a) and (b), are concerned with the method of interruption and its relation to the particle velocity; these will be considered in this section. The final four, (c)-(f), involve the type of light source and the optical arrangements for producing the desired illumination. Of these, item (f) will receive some consideration in this section, and the remainder, items (c)-(e), will be dealt with in the Section 1.1.3.4. Interrupted illumination may be produced by a continuous source of light together with some mechanical device (chopper) for interrupting the light leaving the source. Alternatively, an intermittent source of illumination, such as a flash tube, may be used. Compared to the mechanical method, the intermittent source is capable of delivering larger amounts of luminous energy in each light pulse, it can respond much more rapidly to some initiating trigger, and it is probably more flexible in its ability to vary the duration and frequency of the light pulses.* On the other hand, the mechanical systems are usually much simpler and are capable of operating at much higher interruption frequencies (up to 3 kHz for mechanical interruption, compared to 300 Hz for flash tubes). In some cases it is advantageous to combine both methods; these will be described later. There appears to be only one general contribution in the literature on interrupted light sources, and this is due to Ruddock.180 It is concerned with biological studies of the response of living organisms to intermittent illumination, but the material is directly applicable to chronophotography . The methods of mechanically interrupting a continuous light source to produce intermittent illumination are listed in Table XII. The table lists the important characteristics of the different devices. A column headed “Timing Key” briefly describes methods that may be used to introduce a timing key into the interruption. t The exact form of the various keys that have been used are described in the references listed in the last column of the table; these references will also provide other practical information on the design, construction, and operation of the interrupters. lea K . H. Ruddock, in “Techniques of Photostimulation in Biology” (B. H. Crawford, ed.), pp. 104-143. Am. Elsevier, New York, 1968.

* This would allow the selection of an optimum trace length for a given particle velocity. t C h a r t i e P points out that an irregularity in the interruption of the illumination makes for difficulty in data analysis [see discussion of Eq. (1.1.27)], and proposes that a mark be made on one of the traces by moving the camera lens slightly. It was found that the force of opening the camera shutter (single action type) was sufficient to do this; other methods will suggest themselves to the experimenter.

1.1.

TRACER METHODS

69

The placement of the interrupter in the optical system can have a marked effect on the waveform of the light that is produced. A waveform that approximates a square wave as closely as possible is highly desirable to ensure that the end of each trace is well defined. The interrupter should be placed at a focal point in the light projection system. The beam cross section will then be a minimum, giving minimum rise and fall time at the opening and closing of the interrupter. A small beam cross section also minimizes the opening in the interrupter, which is important for mechanical reasons, particularly in ensuring that the device has a low inertia. The waveform is also dependent on the speed of operation of the shutter. ChartierlS0suggests that a rotating slotted wheel which contains a few slots but rotates at a high speed gives a better approximation to a square wave then does a wheel which has the same interruption frequency but uses more slots rotating at a slower speed. Certain experimental situations may require an interrupted light source that responds with the minimum of delay to a trigger. If the permissible delay is only of the order of a few milliseconds, intermittent light sources are much better than a mechanically interrupted continuous light source. However, where mechanical interruption is usable, the devices that can be used, in order of decreasing desirability, are the (a) vibrating vane, (b). camera shutter, (c) falling plate shutter, and (d) rotating wheel. In the case of the rotating wheel, a drive system incorporating a tension spring can be used to minimize the delay at starting. The characteristics of the continuous light source which are important in the successful realization of a chronophotographic system are its brightness (luminous flux per unit emitting area per unit solid angle), and the uniformity of the distribution of brightness across the emitting area. A bewildering variety of sources are available, but fortunately the experimenter has access to a number of reference^'^^^^^'-'^^ that are very helpful in making a good choice. The paper by Carlson and Clarklsl is probably the best overall review, but it may be somewhat out of date. CannlB2provides information on commercially available lamps, particularly numerical data on their spectral characteristics. Crawford and Nimeroff ls3 are strong on spectral character. The monograph by Hyzer13’ on high speed photography contains good information on lamp life and electrical power requirements. The most up-to-date references on light sources F. E. Carlson and C. N . Clark, A p p l . O p f . Opt. E n g . 1,43 (1965). M . W. P. Cann, Appl. Opt. 8, 1645 (1969). B. H. Crawford and I. Nimeroff, in “Techniques of Photostimulation in Biology” (B. H . Crawford, ed.), p. 19. Am. Elsevier, New York, 1968. lS4 D. McLanahan. f r o c . Tech. Program-Electro-Opt. S y s t . Des. Con$ 3(2), 18 (1971). 16*

TABLEXII. Mechanical Light Interruption Techniques

Technique Rotating slotted wheel

Vibrating vane or tuning fork

Frequency range or exposure duration

Frequency precision

Timing key

Comments

References

0.5 Hz-6 kHz

+O. 1% (constant speed drive) &1% (variable speed drive)

Irregularity in slot spacing

a -f

1 HZ-3 kHz

-C5% (1-50 Hz) 2 1% (50 Hz-3 kHz)

Should be able to introduce an irregularity into motion.

Fairly good approximation to square wave pulses. In Eq. (1.1.27),f= SN, where S is the number of slots in the wheel; N, the rotational speed of the wheel (revolutions per unit time). Square wave chopping pattern if chopped beam is small relative to vane opening.

f-h

Oscillating pendulum

1 Hz

20.5%

By means of multiple slots in the pendulum blade.

Single action shutter: camera shutter

Minimum exposure 1 ms

5 5%

By irregularity in a train of successive operations.

Approximately square wave pulses. Useful for only a limited number of swings because of the effect of air damping. Manufacturers calibration is unreliable, should be checked. Definition of ends of trace poor.”

f.i

fJ

P. B. Walker, in “Technical Report of the Aeronautical Research Council, 1931-32,” Vol. I, p. 97. HM Stationery Office, London, 1933. C. Chartier, Publ. Sci. Tech. Minist. Air ( F r . ) ,Bull. Serv. Tech. 114 (1937). R. Eichorn, I n t . J. Heat Mass Transfer 5, 915 (1%2). * A. Fage and J. H. Preston, Proc. R . Aeron. SOC. 45, 124 (1941). A. Fage and J. H.Preston, Proc. R. SOC.London, Ser. A 178, 201 (1941). K. H.Ruddock, in “Techniques of Photostimulation in Biology” (B. H. Crawford, ed.), p. 109. Am. Elsevier, New York, 1968. a H. G. Lipson and J. R. Littler, Appl. Opt. 5,472 (1966). * R. G. Berry and 0. C. Jones, J. Sci. Instrum. 41, 92 (1964). E. N. da C. Andrade, Proc. Phys. SOC. London 51,784 (1939). A. Schwartz, Appl. Opt. Opt. Eng. 2, 95 (1962). a

’

J

72

1.


are the one of McLanahan,lB4who reviews the characteristics of the latest commercially available lamps and‘Part 8 of this volume which deals with very rapid lighting. Continuous light sources may be classified by the mechanisms of light generation, which are resistive heating of a conductor, electric discharge in a gas or vapor, fluorescence of a phosphor by a gas discharge, and the laser. The characteristics of these different light sources are listed in Table XIII. In the past, the electric discharge lamps have been the most popular in chronophotography. This source has a particularly high brightness (luminous flux per unit of emitting area per unit solid angle). The modem compact arcs which operate in a gas contained by a gas envelope are the brightest, but they also have the greatest nonuniformity across the emitting area, and so are not necessarily the best source to use with imaging optics. The positive crater of the carbon arc and the zirconium arc are much more uniform than the compact arcs. For illumination over large areas, the problem of the uniformity of the light emission is less important. For this reason and others, e.g., shape of the source and power and cooling requirements, filament lamps and fluorescent lamps have been used to some extent in chronophotography, 151,153,165 Short duration, repetitive light sources suitable for chronophotography can be provided by a spark discharge together with an appropriate electrical circuit to ensure repetitive operation. The disadvantage of such light sources is the frequency with which they can be operated repetitively; this is about 300 flashes per second for continuous operation. This limitation is a function of the finite time that is required for the gas in the vicinity of the spark gap to become deionized. For very high frequency, it is therefore necessary to use multiple light sources which operate individually at a lower frequency but are controlled by a timing circuit so that the frequency of the effective source is much higher. The earliest spark sources operated in air, and they are still used today because they can produce very short duration flashes. Durations of the order of 1 p s are easily attained, and durations as short as one tenth of this have been produced. This type of source is also attractive because of its low first cost. Spark gaps have been used as sources of intermittent illumination in chronophotography by Chen,IBBBreslin,lB7and Mellor et al. lB8 The indicated references give some details of the timing and supply cirE. F. Winter and J. H. Deterding, Br. J . Appl. Phys. 7 , 247 (1956). C. J. Chen and R. J. Emrich, Phys. Fluids 6, 1 (1963). J. A. Breslin and R. J. Emrich, Phys. Fluids 10, 2289 (1967). R. Mellor, N . A. Chigier, and J. M. Beer, in “Combustion and Heat Transfer in Gas Turbine Systems” (F.B. Norster, ed.), p. 291. Pergamon, Oxford, 1971.

TABLEXIII.

Characteristics of Continuous Light Sources Type of source

Characteristic

Incandescent

Brightness”

900 Im/(sr . cm*)

Source size

various

Spectral characteristics Efficiency (Im/W) Power requirements

Closely approximates a black body 4-30 ac or dc

Cooling

a

Electric discharge

Fluore scent

Laser

2 x 103-7.5 x 1oJ Im/(sr . cm2) Point: 0.2 mm min diam. Line: 25 mm long Radiates only at characteristic wavelengths

0.685 Im/(sr. cm2)

0.05-1sw

-

Point source: 0.3-1 cm diameter Radiates only at characteristic wavelengths -

16-65

Requires special starting circuit Cooling is nearly always required

“White” 75- 80 Requires special starting circuit

No

Required by high power lasers

To convert from lumens to watts, the ratio 0.00147 Im/W can be used [see R. P. Teele, Appl. Opt. Opt. Eng. 1, 8 (l%S)].

1.

74


cuits. A paper by Whitlowlagdescribes the use of spark gap light sources in photography and contains a good deal of practical information on their use and operation. An interesting feature of the spark gap is its transient, oscillatory nature, and this may be used to provide interrupted illumination for as long as the light is sufficient to produce a satisfactory exposure of the film.'" The light output of atmospheric spark gaps is limited, and for this reason gas filled flash lamps are much more widely used as intermittent light sources. These have been used in chronophotographic applications by Cadle and Wiggins,170Benson et a/.,17'and King.172 Multiple source timing circuits, suitable for chronophotography, are described in the papers of Boyce and B l i ~ k , ' Winter ~~ and Deterding,la5 York and Stubbs,20 and Dombrowski et Good general references on flash tubes have been prepared by A ~ p d e n , " Rutk~wski,'~' ~ H y ~ e r , Rud'~~ dock,'" G. E. Flashtube Data and Carlson and Clark.'" Owing to the light output characteristics of this type of source, the ends of the interrupted traces cannot be located as precisely as allowed by the mechanical methods of interruption. The light output of an electric discharge rises very steeply at first and then tails off from the peak value.165 The traces on the photographic emulsion have the form of tear drops, with the head of the tear pointing in the direction of motion of the particle. This characteristic actually represents an intrinsic timing key. Pyrotechnic light sources suitable for photographic use have been described by H y ~ e rand l~~ applied by Fulmer and W i t - t ~to l ~flow ~ tracing. Mercury vapor arc lamps have a built-in interrupter, which has been used as a source of intermittent illumination in chronophotography by Atkinson et Brodowicz and Kierkus,lBONaib,lsl and Roberson.lZ8 The lamp output varies sinusoidally when connected to an ac supply. The light interruption frequency f is twice the supply frequency. This source

ITo

L. Whitlow, Electron. Eng. 33, 709 (1961). R. D. Cadle and E. J . Wiggins, A M A Arch. Ind. Health 12, 584 (1955). G . M. Benson, M. M . El-Wakil, P. S. Myers, and 0. A. Uyehara, J. A m . Rocket Soc.

30,447 (1960).

R. E. King, Electron. Eng. 32, 294 (1960). M. P. Boyce and E. F. Blick, A m . Soc. Mech. Eng. [Pup.] No. 71-FE-32 (1971). N. Dombrowski, R . P. Fraser, and G . T. Peck, J . Sci. Insfrum. 32, 329 (1955). R. Aspden, "Electronic Flash Photography." Macmillan, New York, 1960. J. Rutkowski, "Stroboscopes for Industry and Research." Pergamon, Oxford, 1966. lT7 "G. E. Flashtube Data Manual," Photo Lamp Dep. No. 281. General Electric Co., Cleveland, Ohio. R . D. Fulmer and D. P. Wirtz. A l A A J . 3, 1506 (1965). B. Atkinson, Z. Kemblowski, and J . M. Smith, AIChE J . 13, 17 (1967). I8O K. Brodowicz and W. T. Kierkus, Arch. Eudowy Musz. 12 (4), 473 (1965). lE1 S. K. A. Naib, Engineer 221, 961 (1966). IT*

1.1. TRACER

75

METHODS

has the disadvantage that the light output also varies sinusoidally, which results in poor definition of the ends of the traces, but these can be clipped by suitably synchronizing a mechanical interrupter with the light variation~~ (see ' the report of Walker'** for a detailed discussion on clipping to improve the definition of the end points of the particle traces). This represents a combination of two light interruption methods. Two light interruption methods can also be combined to increase the operating range of a chronophotographic system. If one interrupted light source operates at a lower frequency than the other, this will permit the simultaneous measurement of particle velocities, if both high and low veTABLEXIV. Techniques for the Calibration of Light Interrupters" Interruption method Calibration technique Comparison with standard frequency source Chronophotography of a body moving at a known speed Revolution counter Stroboscope Temporal photometry of interrupted light (oscilloscope with time marks) Siren effect of rotating wheel

Rotating Vibrating Oscillating Single action Intermittent wheel shutter pendulum shutter light source b -e

X

-

-

-

f

X

X

X

h,i

b

X X

X X

h

-

-

X

c,k

b

-

-

-

-

-

Entries with a reference letter indicate a known application of the calibration technique. Entries with a cross indicate a possible application of the technique. * J. M. Bourot, Publ. Sci. Tech. Minis. Air (Fr.),Bull. Serv. Tech. 226 (1949). R. M. Elrick, I1 and R. J. Emrich, Phys. Fluids 9, 28 (1%6). K. H. Ruddock, in "Techniques of Photostimulation in Biology" (B. H. Crawford, ed.), p. 109. Am. Elsevier, New York, 1%8. V. D. Hopper and T. H. Laby, Proc. R. SOC. London, Ser. A 178, 243 (1941). E. F. Winter and J. H. Deterding, Br. J. Appl. Phys. 7, 247 (1956). O J. L. York and H. E. Stubbs, Trans. ASME 74, 115 (1952). C. Jones and G. Hermges, Br. J . Appl. Phys. 3, 283 (1952). ' E. N. da C. Andrade, Proc. Phys. Soc. London 51, 784 (1939). 3. A. Breslin and R. J. Emrich, Phys. Fluids 10, 2289 (1%7). ' C. J. Chen and R . J. Emrich, Phys. Fluids 6 , I (1%3). (I

la2 P. B. Walker, in "Technical Report of the ARC 1931-32," Vol. 1, p. 97. HM Stationery Office, London, 1933.

76


locities are encountered in the same measuring volume.1so This increases the dynamic range of the chronophotographic apparatus, i.e., it becomes possible to record velocities which vary rapidly over a wide range without manually changing the range of the apparatus. Calibration of the interruption frequency fand the duration of exposure S t are essential for accurate chronophotographic measurements. The various techniques that have been used or those which appear to be applicable to the calibration of interrupted light sources in chronophotography are listed in Table XIV. 1.1.3.4. Dark Field and Bright Field Illumination. The flow tracing particles must produce a usable image on the photographic emulsion, i.e., the traces of the particles in the measuring volume must have sufficient contrast to be distinguishable unambiguously from the images produced by particles out of the measuring volume, dust and dirt in the test section, and general background fog due to reflection of the light from the surfaces in the test section. To ensure this, an adequate amount of illumination must be scattered into the camera lens. The incident illumination can originate from approximately the position of the camera and be reflected back from the flow tracing particles into the camera. However, as the discussion of Section 1.1.2.8 shows, the amount of reflected light is very meager, and satisfactory images of the particles are very difficult to obtain with this front lighting arrangement. In general, transmitted illumination, either dark field or bright field (socalled because of the appearance of the field in view in the camera), is preferable, with the former being the more widely used because the contrast of the images is usually much better. With dark field illumination, the incident rays of light are not allowed to enter the camera, and the particles are made visible by the light which they scatter out of the main beam. The particles then appear as bright spots against a dark background. In bright field illumination, most of the light from the source enters the camera, and the particles scatter most of the light that is incident on them out of the camera lens. The particles appear as dark spots against a light background. Figure 10 shows schematically dark and bright field illumination systems. The choice between dark field and bright field illumination depends on the desired resolution of the optical system, the contrast between the image of the particle on the film and the background, and the effficient use of the incident illumination. In chronophotography, questions of detail revealed by shadows are unimportant, since the details of the surface or shape of the flow tracing particle are not of interest; the aim is to determine the location of the particle. The comparison (see Table XV)of the two types of illumination is therefore reduced to considerations of con-

1.1. TRACER METHODS

77

TEST SECTION WALLS

CAMERA LENS (SMALL ANGLE SCATTERING) FLOW TRACING PARTICLE

CAMERA LENS (90"SCATTERING) (a)

TEST SECTION WALLS ( b)

FIG.10. Schematic arrangement of dark field (a) and light field (b) illumination systems. Light stops and beam defining apertures are not shown. Note the optical system would be satisfactory for a flash tube light source. For a mechanically interrupted light source, an additional lens would be required to focus the beam on the interrupter.

trast and utilization of illumination. Both dark field and bright field illumination have their respective advantages in chronophotography, but it may be difficult in certain circumstances with bright field illumination to obtain a satisfactory image of small particles. This situation can be alleviated to some extent if the light absorbing characteristics of the particles are i n c r e a ~ e d . ~ ~ ~ ~ ~ ~ Although bright field illumination has important disadvantages, it is an attractive technique from the point of view of the high efficiency with which it utilizes the incident illumination. This makes the light requirements of bright field illumination less than those of dark field illumination, and in general the light levels are low enough to permit the use of very slow photographic emulsions in the camera, with a consequent gain in resolution. Recently the performance of bright field illumination systems used in certain types of bubble chambers has been significantly improved by lining the wall of the chamber facing the camera with Scotchlite sheet.183 This acts as a retroreflector for a source of illumination placed W. M. Powell, L. Oswald, G . Griffin, and F. Swart, Rcv. Sci. Insfrum. 34, 1426 (1963).

1. MEASUREMENT

78

OF VELOCITY

TABLEXV. Comparison of Dark Field and Bright Field Illumination Characteristic

Dark field illumination

Contrast

High

Efficient use of illumination Stray light

Low, but forward scattering is the most efficient Very sensitive to stray light, unless light shields are used, light scattering dust and dirt is minimized, and nonreflecting surfaces are present.

Large scale“

c-f,i,?*

( M < 10) Small scale“

g,h,g,bhb,(k-o)*

Bright field illumination Satisfactory, provided the particle is large enough or is sufficiently light absorbing High Very insensitive to stray light, because background illumination is strong, and dust and dirt particles are not large enough to give adequate contrast. P -r

-

(M > 10) Arbitrary classification. In small scale systems, the imaging optics are required to concentrate the illumination (see Section 1.1.3.6). * Small angle scattering. L. F. Daws, A. D. Penwarden, and G. T.Waters,J. Insr. Hear. Vent. Eng. 33,24 (1%5). * H. G. Schwartzberg and R. E. Treybal, Ind. E n g . Chem., Fundam. 7 , 1 (1968). E. F. Winter and J. H. Deterding, Br. J. Appl. Phys. 7 , 247 (1956). K. Brodowicz and W. T.Kierkus, Arch. Budowy Masz. 12 (4), 473 (1965). R. D. Cadle and E. J. Wiggins, AMA Arch. Ind. Health 12, 584 (1955). R. Eichorn, I n ( . J. Hear Mass Transfer 5 , 915 (1962). J. 0. Laws, Trans. Am. Geophys. Union 22,709 (1941). W. T.Welford, in “Bubble and Spark Chembers Principles and Use” (R.P. Shutt, ed.), Vol. I , p. 233. Academic Press, New York, 1%7. J. A. Breslin and R. J. Emrich, Phys. Fluids 10, 2289 (1967). I R. M. Elrick, I1 and R. J. Emrich, Phys. Fluids 9, 28 (1%6). V. D. Hopper and T. H. Laby, Proc. R. SOC.London, Ser. A 178, 243 (1941). W. B. Kunkel and J. W. Hansen, Rev. Sci. Insfrum. 21, 308 (1950). O E. R. Corino and R. S . Brodkey, J. Fluid Mech. 37, 1 (1%9). J. L. York and H. E. Stubbs, Trans. ASME 74, 1157 (1952). q N. Dombrowski and P. C. Hooper, J. Fluid Mech. 18, 392 (1964). R. Mellor, N. A. Chigier, and J. M. Beer, in “Combustion and Heat Transfer in Gas Turbine Systems” (F. B. Norster, ed.), p. 291. Pergamon, Oxford, 1969. a S. M. De Corso, J . Eng. Power 82, 10 (1960). A. L. Chaney, in “Air Pollution” (L. McCabe, ed.), p. 603. McGraw-Hill, New York,

‘

J

1952.

alongside the cameras, and the image in the camera shows the bubble tracks on a bright field. This technique might have some place in chronophotography of flow tracing particles. Dark field illumination may be realized in practice with either small angle scattering or 90”scattering (see Fig. 10). The former makes much

1.1.

TRACER METHODS

79

more efficient use of the incident light than the latter, because the amount of light scattered forward by small particles is many times greater than that scattered at 90" (see Section 1.1.2.8). However, small angle scattering has not been used as much in flow tracing. Table XV provides a list of references in which useful, practical construction details of various illumination systems will be found. A preliminary estimate of the light available at the photographic emulsion is essential in the design of the lighting system. The methods are different for dark field and bright field illumination. In dark field illumination, the luminous flux density FD incident on the photographic emulsion is given by Eq. (1.1.26),with the flux density Fi incident at the particle assuming a focused light source of

Fi= T ~ B ~ T , / ~ M + ( 1)2, M~

(1.1.29)

where Bi is the brightness of the source (W/sr cm2); N i , the f-number of the light projecting optics; M , the overall magnification of the light projecting optics; and r l , the overall transmission of the light projecting optics. In chronophotography, the amount of light scattered onto the emulsion by the particle with dark field illumination is not actually as great as the value computed using Eq. (1.1.26) and (1.1.29). This is a consequence of the relative motion between the emulsion and the particle. Thus, during each light pulse of duration at, the particle will move through an area of (1.1.30) where dD = M d . Imperfect imaging in the case of small f number (fast) lenses or diffraction in the case of larger f number can result in 10- 100 times this area, particularly for particles in the micron range of sizes. With bright field illumination, if the projected image of the light source completely fills the camera entrance pupil, the flux density FD at the emulsion is the same as in Eq. (1.1.26),but replacing FiZ' by B,, the brightness of the source, (1.1.31)

In Eq. (1.1.3 l), the quantities with subscript i refer to the light source and its optical system; the unsubscripted quantities apply to the camera optics. 1.1.3.5. Camera. The possibility of carrying out chronophotographic velocity measurements and the accuracy with which they can be accomplished depends on the following factors which are discussed in this section: (a) the choice of a suitable camera lens, (b) the presence of distortion in the lens and in the test section observation windows, and (c) the selec-

80

1.


tion of a photographic emulsion that has both sufficient sensitivity and sufficient resolution. The choice of a lens depends on the following characteristics: (a) depth of field, (b) field of view, (c) working distance of the lens from the measuring volume, and (d) light gathering ability of the lens. At the same time it is also necessary to consider the camera magnification which interrelates with the four listed factors. The depth of field and the field of view control the size of the measuring volume, which can be defined as that portion of the flow field within which, for a given illumination, the flow tracing particles are visible to the camera with adequate resolution. The depth of field is defined subjectively by the permissible blur (or circle of confusion) in the photographic image. If a point source of light is viewed through a lens while being moved toward the lens, the blur of its image is seen to pass through a minimum at the focal point of the lens. The range of object distances giving an acceptable circle of confusion is referred to as the depth of field A. This is a function of the lens focal lengthy, lens aperture N, lens magnifi~~'~~ cation M, and diameter c of the limiting circle of c o n f ~ s i o n . ' ~Then, assuming that diffraction effects may be ignored,

A = 2Nc(M

+ 1)/M.

(1.1.32)

Useful charts relating these quantities will be found in H y ~ e rand ' ~ ~Loveland.'" The field of view* can be related by simple geometry, and the properties of thin lenses, to the focal lengthfof the lens, the magnification M and the size b of the negative; thus b = 2(1

+ M)ftanP,

(1.1.33)'

where P is half the desired angular field of view (see Merten~'~'and Allen1mfor charts). As mentioned in the previous section, chronophotography can involve measuring volumes of large dimensions (large scale systems), or it may be concerned with measuring volumes that are extremely small (small scale systems). Chronophotography with large scale systems is a fairly R. P. Loveland, "Photomicrography." Wiley, New York, 1970. R. M. Allen, "Photomicrography," 2nd ed. Van Nostrand-Reinhold, Princeton, New Jersey, 1958. lSl

* The field of view will be less than this if the density of the fluid in the test section is greater than the density of the ambient air. This can be overcome by appropriately locating the light source and the camera!'

1.1.

81

TRACER METHODS

straightforward application of press or amateur cameras. There is an extensive literature on photography which should be available to the expenmenter, and there is no point in reproducing this material. However, the lens should have a wide field of view, and because of the limited laboratory and test section dimensions, the capability of focusing down to a fraction of a meter. Small scale systems involve measuring volumes of sufficiently small size that magnifying optics are required. For low magnifications, from unity up to about thirty-five, a simple microscope, consisting of a single objective lens, can be used. At higher magnifications, a compound microscope is required to ensure that the measuring volume is at a practical distance, of the order of several millimeters, from the objective lens of the microscope. The simple microscope is usually a modified camera in which a suitable magnification is obtained by separating the lens from the camera body so that the ratio of the image distance v to the object distance u is greater than unity. This is usually done with an extension tube for small format (35 mm) cameras or bellows for large format (plate o r cut film) cameras. It is advisable to use the camera lens in the reversed position so as to take advantage of its design features. A camera lens is intended to work with an object distance which is greater than the image distance; since this situation is reversed when the magnification is greater than unity, the camera lens should be turned around. L ~ v e l a n d points ~ * ~ out that 16 or 8 mm movie camera lenses make very good simple microscopes, when reversed, because they normally operate with a very small image size. The typical laboratory compound microscope has a magnification range from about thirty-five up to four hundred (higher if oil immersion lenses are used). Usually, the highest magnifications used in chronophotography are slightly in excess of one hundred, but Fage and TownendsZ report on the use of a magnification of two hundred to permit the examination of the flow in the immediate vicinity of the test section wall. Because high magnification systems have small depths of field [see Eq. (1.1.32)], a relay lenss4*lB8 may be required to photograph the flow at high magnifications away from the wall in a test section of large cross section (say, greater than 2 cm diameter). The working distance u of the camera lens depends on the magnification M and the focal lengthf, thus u/f= 1

+

1/M.

G . Vogelphol and D. Mannesmann, NACA Tech. Memo. 1109 (1946).

.

( 1 1 .34)

82

1.


This shows that the working distance* u can never be less than the focal lengthfof the lens, so to change the working distance it may be necessary to change the lens. Lenses with large apertures increase the exposure of the film; ways to increase exposure are needed when small flow tracing particles scatter relatively little light. For high magnification microscopes, it may be difficult to obtain lenses with a sufficiently large aperture, so that the lens system may set an upper limit on the maximum speed and minimum size of the flow tracing particles. At low magnifications, M e r t e n ~ suggests '~~ that lenses with maximum apertures from about f/2.0to f/4.0 represent a reasonable cost-performance compromise. When adequate exposure cannot be obtained with such lenses, it is generally preferable to modify the lighting system, change the flow tracing particles, or use faster film, rather than to obtain a faster, that is, a more expensive lens. Serious distortions and aberrations of the particle images can be produced by the differences in density between the material of the observation windows, the ambient air, and the experimental fluid. These are the following: (a) Chromatic aberration will produce fringes (colored in color photographs) surrounding the image, which degrades the precision of particle track measurement. This can be eliminated by using a monochromatic light source, or by using observation windows of two materials of different color dispersion but the same refractive index.137 (b) Magnification of the particle image will vary across the field of view, as the angle of incidence of the light rays joining the camera lens and the points on the particle trajectory change. Consequently, the trajectories in some portions of the field of view are disproportionately stretched, and at other points they are disproportionately shrunk.t Magnification effects are usually corrected in flow tracing by either correcting the observation windows or by interposing a correction lens between the camera and the test section wall. The latter method can be used when the observation windows are plane.137*187 The distortion produced by observation through the side wall of ducts of circular cross section can be avoided by choosing a working fluid and wall material that have the same index of refraction, and constructing a flat window. Practically this can N . P. Campbell, and I . A . Pless, Rev. Sci. Insfrum. 27, 875 (1956).

* This is actually the distance between the object and the first principal point of the lens. For a compound lens this point may be an appreciable distance behind the front surface of the first lens, SO the practical working distance will be less than that given by Eq. (1.1.34). t If the observation windows are not normal to the camera axis, either unintentionally, or intentionally (as in stereophotography), the distortion will be asymmetrical.

1.1.

83

TRACER METHODS

be realized in two ways. In the first method, the duct is made from a solid block of material such that the surface adjacent to the camera is plane. This can be accomplished by using a block of clear plastic of square cross section.188 The other way is to immerse the tube in a duct of square cross section that is filled with the experimental fluid. The emulsion used in chronophotography should have a fine grain and a high speed. Unfortunately, these are competing characteristics, but it appears138that the grain size of an emulsion may not be as important in practice as its speed. The speed required for an emulsion in dark field photography can be ascertained by identifying the emulsion that produces a trace density of about 0.4 with the flux density Fo given by Eqs. (1.1.26) and (1.1.29), and using the emulsion sensitivity data of Zweig et Although it is the usual practice in chronophotography to use film rather than glass plates, the tendency of the former to distort during development should be recognized. Tests reported by Welfordlgosuggest that the random distortion is equal to & 1.5 pm. 1.1.3.6. Measuring Volume. It is essential to be able to estimate the size of the measuring volume in which the chronophotographic measurements are to be made. This depends on the size of that portion of the flow field illuminated by the interrupted light source, and the field of view and depth of field of the camera.

c

D~

CAMERA

I

(0)

lb)

FIG.1 1 . Measuring volume dimensions: (a) 90" scattering, cases of the illuminated region filling the field of view (DL> Dc) and partially filling the field of view (DLiDc) are shown; (b)small angle scattering. DLis the diameter or thickness of the light beam at the measuring volume; D c , the diameter of the camera field of view; and A, the camera depth of field. K . D. Cooper, G . F. Hewitt, and B. Pinchin, J . Phofogr. Sci. 12, 269 (1964). H . J. Zweig, G . C. Higgins, and D. L. Macadam, J . Opr. SOC. Am. 48, 926 (1958). W. T. Welford, J . Phorogr. Sci. 10, 243 (1962).

84

1.


Consider the case of small scale (M > 10) or point measurements. As a general rule, the width DLof the light beam at the measuring volume is greater than the depth of field A, so the measuring volume can be estimated by DE A or D O c A, depending on the relation between the dimensions of the light beam and the camera field of view (see Fig. 11). When a large region of space is to be examined (M < lo), it can be assumed that the illuminated volume fills the camera field of view, so the size of the measuring volume is given by 02.A. The chief problem with the photography of very large regions is to ensure there is adequate and uniform illumination throughout the measuring volume. Lighting arrangements which have proved satisfactory in such situations have been described by Daws et al. 153 and Winter and Deterding.165 Useful information, in the context of bubble chamber illumination, has been given by W e l f ~ r d . ~ ~ ~ 1.1.3.7. Camera Calibration. The fundamental relation of chronophotography, Eq. (1.1.27), contains the magnification M, a characteristic of the camera which must be known.* A number of techniques for determining the magnification (actually the inner orientation; see footnote) are described by Hallert.lg1 Of these, the most appropriate in flow tracing is to photograph a scale or grid located normal to the camera axis. The scale can be engraved in metala3,1esor on glass (a stage micrometer may be For large scale systems, wires very useful for microscope or string on a frame,lS1and ruled graph paper supported between glass plates have been used. In practice, a knowledge of the magnification is not sufficient to determine the particle velocity from Eq. (1.1.27), because the camera lens introduces fixed errors, due to optical distortion,t into the photograph which must be corrected.138 The distortion appears to displace by 6r the image points (position rM)from their ideal position rl, where 6r = r M rI* 'I

B. Hallert, "Photogrammetry."

McGraw-Hill, New York, 1960.

* The magnification is defined as the ratio of the image distance to the object distance, or, in photogrammetric terminology, the ratio of the camera constant c to the camera height h. In most photogrammetric situations, the camera constant is determined, rather than the magnification, because the camera height varies during a series of measurements. Thus, in aerial photogrammetry, the height of the aircraft above the ground varies. In photogrammetry, information on the camera constant and the location of the camera axis (actually the principal point; see below) is called the inner orientation of the camera. This distinguishes it from the outer orientation, which relates to the alignment of the camera relative to the object space. t Optical distortion is so called to differentiate it from distortion due to other causes, e.g., film shrinkage in development (see Section 1.1.3.5).

1.1. TRACER METHODS

85

The results of lens distortion measurements quoted by Hallertlgl show that typical values for topographical photogrammetric lens, for the measurement of radial displacements from the camera axis, can be better than 2 0.01 mm. However, the potential user of chronophotography should be aware that these represent lenses of exceptional photogrammetric accuracy, and that even very good camera lenses that are not intended for such applications may introduce unacceptably large distortion (see the case described by Benson et ~ 1 . ~ ' ~ ) . The lens distortion can be determined at the same time as the camera magnification, and Hallertlgl describes the process in detail. The location of the measuring volume is a factor of prime importance in fluid velocity measurement. It is located on the optical axis of the camera at the point of focus, so both of these must be determined. The optical axis can be established at the same time as the magnification is determinedlgl by locating its intersection, the so-called principal point, with the camera focal plane. The position of the principal point is given relative to the intersection of the straight lines that join opposite members of a set of four fiducial marks that appear at the edge of the photograph. These are generally made by small tags that are pressed against the film as it lies in the camera film gate. The point of focus is established by focusing the camera on some surfaces2*63~152~1se or point temporarily located at a position corresponding to the desired central point of the measuring volume. Weighted strings, thin rods, or stiff fibers (all preferably painted with alternate black and white ~ * ' been ~ ~ ) used. The cdistripes so as to improve their v i ~ i b i l i t y ~ ' * ~have brating scale or grid for determining the camera magnification could also be used in this way. Careful alignment of the focusing surfaces or points is required, and the gravitational field is very useful in this respect, since weighted strings naturally hang vertically and surfaces can be aligned with a spirit level. To ensure that the camera magnification is maintained and to assist in data analysis,1g3it is advisable to have control points (or fiducial marks), whose position in the object space is known accurately, appear in every photograph. The calibration scale or grid used to obtain the camera magnification might be used for this purpose in certain circumstances. Alternatively, special marks can be introduced into the field of view, e.g., lines engraved on the observation windows of the test section.149 Maintenance of the camera calibration is obviously of importance. This can be ensured by rigidly attaching the camera to a support that is permanently located relative to the test section. The techniques used in '02

J . 0. Laws, Trans. A m . Geophys. Union 27, 709 (1941). E. R . Flynn and P. J. Bendt, Rev. Sci. Instrum. 33, 223 (1962).

86

1.


bubble chamber photography to maintain the calibration of the cameras might be useful in this c~nnection.'~' 1.1.3.8. Data Analysis. The analysis of the chronophotographic record takes place in three steps.* In the first step, the particle displacement-time data are extracted from the record. Second, the particle velocity is determined. Finally, the data are subjected to manipulation relevant to their subsequent application. This last step includes plotting the data as a spatial distribution, and, in the case of turbulent flows, determining various statistics. The negative on which the particle trajectory is recorded is usually quite small, and the images on the negative are, in themselves, small, so that the data must be obtained from a projected, magnified version of the original record,43~1ge~'s4 or the record must be examined by a microscope. The displacement (Ax) measurements can be made directly from the enlarged image, but sometimes it is helpful to trace the trajectories onto plain or divided paper. The latter approach is very useful in large scale systems, particularly when the flow is turbulent, since it helps the process of data analysis by separating judgments about the validity of certain data points (e.g., whether certain images represent particles in or outside the measuring volume) from the measurement process. The data extracted in this way could, for example, be arranged in a single line on the traced record, which could considerably speed the measurement process. If the measurements are made with a microscope, the negative should be arranged so that it can be turned relative to the microscope. The microscope can be a traveling microscope which is carried on a calibrated screw, or the negative can be arranged on a table which is driven by micrometer screws. Sometimes the traces are sufficiently short, relative to the microscope field of view, to allow the measurements of displacement Ax to be made with a micrometer eyepiece. This method can be improved if a shearing or image splitting eyepiece is ~ s e d . ~ ' ~ ~ ~ ~ The particle velocity can be obtained from the time-displacement data by means of Eq. (1.1.27). It was mentioned earlier (Section 1.1.3.1) that another method is the use of a curve or polynomial fitted to the timedisplacement data.'Os This method minimizes the errors arising from the notorious sensitivity of numerical differentiation [which is the basis of Eq. lw

Is

J. A. Lewis and W. H. Gauvin, J . SMPTE 80, 951 (1971). J. Dyson, J . Opt. SOC. A m . 50, 754 (1960). B. P. Selberg and J . A. Nicholls, AIAA J . 6, 401 (1968).

* Many of the methods used in the analysis of chronophotographicrecords have also been used in the analysis of high speed cinematographic films, and the monograph by HyzerIa8is a valuable source of practical information in this area.

1.1. TRACER METHODS

87

(1.1.27)] to errors of measurement. The polynomial approach is also advantageous if the data are to be subjected to manipulation by a digital computer. This avoids the time consuming table look-up procedure which is necessary with the raw data. Stereophotography introduces special difficulties because the two views of a single track have little resemblance to one another. The fact that one coordinate of each point of a given trajectory appears in both views permits the unambiguous identification of that point in both views. Flynn and BendtlD3give a brief discussion, in the context of hydrogen bubble chamber photography, of the method of obtaining a particle trajectory from two 90" stereophotographs. The most difficult and time consuming data treatment is required when the fluid motion is turbulent; by comparison, data treatment with laminar fluid motions is trivial, so we will not consider the latter situation in this section. Typically, the following statistics are of interest (see also Section 1.1.2.2.6):(a) the time mean velocity OF, (b) fluctuating velocity uF, (c) energy spectrum density function FF,(d) Eulerian correlation coefficient R E ,and (e) Lagrangian correlation coefficient. Very few turbulent data, other than items (a) and (b) above, have been obtained by chronophotographic methods, probably because of the tremendous amount of hand labor required, so techniques for calculating turbulence statistics from chronophotographic data are not very highly developed. A discussion of the fundamental relations required to obtain the turbulence statistics from chronophotographic data has been given by Joneslg7and by Komasawa, Kuboi, and Otake.lD8 The formidable quantities of data that must be handled in turbulence studies make the use of digital computers mandatory, and a computer program that would be suitable for handling chronophotographic data has been described by Jones.lQQ Another possible approach is to use optical computing.200 The technique seems attractive because it is well adapted for use with data obtained by hand methods. Apparently, the method has not been used by other experimenters but would seem worthy of wider application.

1.1.3.9. Error Analysis. The following sources of error are considered to affect the performance of the chronophotographic system: fixed errors, i.e., linear approximation to the particle velocity in the measuring volume lg7 B. G. Jones, in "Advanced Heat Transfer" (B. T. Chao, ed.), p. 339. University of Illinois, Urbana, 1969. Ion I . Komasawa, R. Kuboi, and T. Otake, Chem. Eng. Sci. 29, 641 (1974). B . G. Jones, An experimental study of the motion of small particles in a turbulent fluid field using digital techniques for statistical data processing. Ph.D. Thesis, University of Illinois, Urbana (1966). S. L. Soo, C. L. Tien, and V. Kadambi, Rev. Sci. lnsrrum. 30, 821 (1959).


88

[Eq. (1.1.27)], and random errors, including (a) measurement of trace length, (b) value of the camera magnification, and (c) value of the interruption frequency. In addition, if the flow is turbulent, there are sampling errors due to the statistical nature of the data. Schraub et al.13 have provided an elaborate analysis of the linear approximating error. Actually this is for the cinematographic measurement of particle velocity, but the calculations they have carried out for what they call the frame-to-frame or parhline measurement are directly applicable to chronophotography . This rather general case is a consideration of a flow tracing particle with two velocity components (up and vp , the latter being due to the effects of extraneous forces; see Section 1.1.2.5) moving in an unsteady fluid flow field with velocity gradients in the x and z directions. The error in the particle displacement is shown to be

The quantities Ax and Ay are the displacements in time ( N - l)/f, measured along appropriate x and y axes, of the flow tracing particle. The subscript to represents the time at which the particle passes through the midpoint (xo, zo) of the measuring volume. Equation (1.1.35) indicates that the linear approximating error is minimized by restricting the measurements to regions of the flow field with small velocity gradients and by using the highest possible interruption frequency. A number of errors contribute to the random error in the measurement of the trace length Ax. These are (a) random errors (including personal errors) in the measuring device, 2 10 pme2*17ei201; (b) definition of the end of the trace, +3%;202 (c) emulsion shrinkage in the developing, & 0. 1%82*190; (d) residual distortion in the optical system, 2 2%;202and (e) enlargement of the photograph for the purposes of measurement (if used), 2 1%.202 These are estimated to give a combined uncertainty of about f4%.

The measurement of the magnification M introduces an estimated random error of +0.2% into the determination of the particle velocity from Eq. (1.1.27). The uncertainty in the interruption frequency f depends on the interruption method and the method of calibration (see Table XII) * The total contribution of the random errors probably does not exceed 5%. Iol

G. Birkhoff and T. E. Caywood, J . Appl. Phys. 20,646 (1949). H. G.Schwartzberg and R. E. Treybal, Ind. Eng. Chem., Fundam. 7, 1 (1968).

1.1.

89

TRACER METHODS

Turbulence measurements, like all other measurements of statistical quantities, involve sampling errors. These are due to the impossibility of obtaining an infinite number of measurements of the mean velocity, the rms velocity fluctuations, and any other desired statistics. Statistical theory is able to provide a method for estimating errors of this type, provided it is assumed that the population from which the measurements are drawn is normal (Gaussian). Suppose the measured velocities of flow tracing particles in a turbulent flow can be represented by (see Section 1.1.2.2.6; the subscript P has been suppressed here for convenience)

U=O+u. Then, according to statistical theory,203the error sample of N data points is

( 1.1.36) ED

of the mean 0 for a (1.1.37)

where z, is the parameter which depends on the confidence limits that are to be applied to the data, and equals 1.645, 1.96, and 2.58 for 90,95, and 99% confidence limits, and u’ is the rms velocity fluctuation [u’ = (u’)’’~]. Similarly it can be shown that the error E”, in the rms velocity fluctuations is given by cut = ?2u’/(2N)”2.

( 1.1.38)

These equations can be used to estimate the sampling errors in the measurement of turbulent particle velocities or, alternatively, to determine the number of measurements that should be made to attain a sampling error of a given magnitude. Thus, consider how many data points are required to attain a 5% sampling error at the 90% confidence limit in the measurement of the mean and rms particle velocities. If the rms velocity fluctuations are 10% of the mean velocity, then about ten data points are required to obtain the mean. However, about five hundred data points are necessary to measure the rms velocity fluctuations. Clearly, the measurement of turbulence statistics by chronophotography presents a formidable data handling problem, particularly in view of the unavoidable necessity for the intervention of a human operator at the point where the data are extracted from the photographic record. 1.1.3.10. Example of System Design. It is first necessary to ascertain that the anticipated fluid velocities are measurable by chronophotogu)3 A . G . Worthing and J . Geffner, “Treatment of Experimental Data.” York. 1943.

Wiley, New

90

1. MEASUREMENT

OF VELOCITY

raphy. Following this, the amount of light incident on the photographic emulsion should be estimated. This will avoid setting up complicated and expensive equipment which may not be capable, under any circumstances, of providing the desired data. The light flux calculation will only establish the feasibility of the proposed measurements, and, to obtain the best possible trace on the emulsion, will require careful adjustment ,of the system components. The final step in the design is to select the camera parameters so as to obtain the desired measuring volume size and a practical working distance. It is proposed in this section, as in Section 1.1.2.9, to demonstrate the design of a chronophotography system by means of a numerical example. It is based on the measurement system, originally due to Eichorn,'" described in Section 1.1.2.9, and is therefore a continuation of that example. The observation system is a 35 mm camera (film transport) and a movie camera lens which is operated in the reverse position (see Section 1.1.3.5). The lens has an aperture offll.9 and a focal length of 2.54 cm, and is mounted on a 45.2 cm extension tube giving a measured magnification of 17.6. Dark field illumination is to be used with 90"scattering, And the optical axis of the camera is to be normal to the heated plate. The light source is a 100 W high pressure DC mercury arc lamp with a brightness of 73,000 lumens/(sr cm2). The arc size is 0.3 mm diameter. The light projection system, consisting of four identical achromatic lenses of diameter 5.1 cm and focal length 6.35 cm, images the source onto the light interrupter. The light leaving the interrupter is refocused at the observation volume by two achromatic lenses of 3.8 cm diameter with a combined focal length of 3.8 cm. Since velocity gradients are present, the measurements must be as close to pointwise as possible. To meet this condition, the distance normal to the heated plate must be measurable to ? 0.0025 cm. This can be ensured by making the measuring volume sufficiently small and by locating its position, relative to the heated plate, with a precision of better than kO.0025 cm. The latter condition was attained by mounting the light source and camera together, so that they move simultaneously, and measuring this movement on a dial indicator that can be read to +- 0.00025 cm . The maximum required frequency of light interruption, which must be within the range attainable by the methods used in chronophotography (see Table XII), is obtained from Eq. (1.1.28). With Dc = 0.5 mm (allowing for some degradation of the projected image of the arc) and up,MAX = 21 cm/s, we havef = 420 Hz. This can be achieved by a wheel

TABLEXVI. Total Flux Density FDat the Emulsion for 1 p m Particles ~~

A

bun)

F,Jlm/cm*

. pm]

I;[sr-l . pm-,]

(e

=

0.5770 4.03

X

102

6.13 x

0.5461

5.04

X

102

0.4916 5.04

0.4358

0.4047

4.03 x 102

2.67 x 102

2.86 x 10-z

2.31 x 10W

1.82 x lo-*

1.57 x

2.51

2.03

1.28 x

7.31 x lo-*

90”)

FDA[erg/(cmz.s . pm)]

4.31

X

lo-’

X

X

Notes From Eq. (1.1.26) with N = 2.67, M = 1, T = 0.55 (for six element lensa) From Eq. (1.1.23). I; = IA/(mdZ/4). Values of i,, i2 for y = 90”.m = 1.44 at the nearest value of From Eq. (1.1.26) with N = 1.9, M = 17.6, T = 0.8.“ Used 0.00147 Im/W.d

L. M. Myers, “Television Optics.” Pitman, London, 1936. “Tables of Scattering Functions for Spherical Particles,” U.S. Nat. Bur. Stand., Appl. Mathe. Ser. 4. US Gov. Printing Office, Washington, D.C., 1948. W. G. Hyzer, “Engineering and Scientific High-speed Photography.” Macmillan, New York, 1962. * R. P. Teele, Appl. Opt. Opt. Eng. 1, 8 (1965). a

* A. M. Lowan et al.,

92


with sixty slots rotating at 450 rpm (a motor locked to the power line frequency and operating at this speed is available). The calculation of the incident flux density on the emulsion should take account of the spectrum of the light emitted by the mercury arc. To do this, the total luminous flux was divided among the various lines in the visible region according to the spectral information given by Elenbaaszo4 (the 25% contribution of the continuous part of the spectrum was ignored). The calculations for a 1 p m particle are summarized in Table XVI. The sum of the last line in the table is an approximation to the total flux density FD at the emulsion due to a stationary particle in the measuring volume. The motion of the particle is allowed for by multiplying FD by AD St/[dD(upSt + dD/4)] [see Eq. (1.1.30)] to obtaim a modified flux density aD.In the preceding formula, S t is the exposure time for one slot in the rotating wheel light interrupter. With up = 21 cm/sec, dD = Md = 17.6 pm, and 6r = 44 psec (correspoding to slots 7$ in. wide in the light interrupting wheel), we obtain = 87.9 x erg/cm2. According to Zweig et al., this will produce a density of about 0.4, which is satisfactory for measurement purposes, in a fast (ASA 650) emulsion, such as Kodak Royal-X. The corresponding flux for up = 1.5 cm/s is 141 X erg/cmz, for which a Kodak Tri-X (ASA 200) will give an image density of about 0.4. Velocity measurements using 1 p m particles appear to be feasible, and the calculations indicate the type of film with which to start the experimental adjustment of the measuring system. The light scattered by 0.5 p m DOP particles is so small that an extremely sensitive conventional emulsion, such as Kodak type 2485 recording film (ASA 800), or a Polaroid emulsion, such as type 410 (ASA 8000),would be required to produce an adequate trace on the emulsion. However, as pointed out in Section 1.1.2.9, the very small particles are sensitive to Brownian motion, which increases the uncertainty of the velocity measurements, and by using, say, Kodak Tri-X, the traces of such particles will not be visible. The measuring volume will be assumed to be a cylinder of diameter equal to the diameter Dc of the field of view, and length along the camera optical axis equal to the depth of field A. The latter is taken as 0.005 cm, which is twice the desired uncertainty in the measurement of the distance normal to the heated plate. From Eq. (1.1.32), A = 0.005 cm, which corresponds to a maximum circle of confusion c of 200 p m (with M = 17.6 and N = 1.9); so if images on the film larger than this are rejected during the measurement of the trace lengths on the film, we will know the particle position to within f 0.0025 cm. *M W. Elenbaas, in “High Pressure Mercury Vapour Lamps and their Applications” (W. Elenbass, ed.), p. 43. Philips Technical Library, Eindhoven, 1965.

I . 1.

TRACER METHODS

93

The diameter Dc of the field of view is obtained from Eq. (1.1.33). Using f = 2.54 cm, M = 17.6, and b = (1.8 x 2.45)1'2 = 2.1 cm (from Dejager et a/.205), we have Dc = 0.71 cm. This gives a measuring volume of 2.5 mm3. The working distance of the camera lens is calculated as 2.68 cm by Eq. (1.1.34). The latter is adequate, according to O ~ t r a c h ' s 'theoretical ~~ results, to ensure that the lens does not interfere with the flow in the vicinity of the heated plate. The error arising from the linear approximation to the fluid velocity is negligible in this case, according to Eq. (1.1.33, compared to the random errors of f4% (see Section 1.1.3.9). 1.1.4. Laser Doppler Velocimeter LIST O F SYMBOLS*

A 14 .1 A a

B c

D d* 4arl

E E e

F Fl

f G H

h I lo

i

Vector amplitude of electric field vector [V/m] Amplitude of electric field vector [V/m] Effective cross section [m] radius [m], radiant sensitivity of photodetector [mA/W], dimension of rectangular aperture used for division of wave front method for forming interfering beams in homodyne systems [m] Bandwidth of signal processing electronics [rad/s] velocity of light [=2.99793 x lo8 m/s], dimensions of rectangular aperture used for division of wave front method for forming interfering beams in homodyne systems [ml Diameter of lens at photodetector [m] Mean particle diameter [m] Distance between particles that scatter mutually interfering light fields [m] Electric field vector [V/m] Magnitude of electric field vector [v/m] Unit vector aligned in direction of propagation of a plane wave Finesse of Fabry-Perot interferometer (= AAo/6Ao) [dimensionless] Intensity of light incident in the measuring volume [W/m'] Frequency [Hz], focal length of lens [m] Photodetector gain [dimensionless] Random amplitude of the photodetector output current when more than one flow tracing particle is simultaneously present in the measuring volume [A] W . s2/photon Planck's constant = 6.6237 x Time average light intensity =<EE*>, where < > indicates a time average [see Eq. ( 1.1.54a)l. and the superscript asterisk indicates the complex conjugate [W/m'] Maximum value I [W/m'] Unit vector in the x coordinate direction [m]

'05 D. Dejager et u l . , in "SPSE Handbook of Photography" (W. Thomas, Jr., ed.), p. 1140. Wiley, New York, 1973.

* This list is for Section

1.1.4 only.

94

N

NJ n

P PB PJ PSH

P 9

R

Ro

RO

S T TN

1.


Photodetector output current [A] Bessel function of the first kind with argument x (-I)"* Light distribution factor in Eq. (1.1.80) Light distribution factor in Eq. (1.1.83) Propagation vector for electromagnetic field = ke [m-'1 Wave number for electromagnetic field ( = w/c = 2w/A [m-'1, photon multiplier noise factor in Eq ( I . I .78) [dimensionless] Spectral mean wave number [m-'1 Boltzmann's constant = I .38046 x 10-25J/K Distance or effective distance between measuring volume and photodetector [m] Coherence length of light (= c/Aw) [m] Number of autocorrelation channels; total number of lag times in a digital autocorrelator; magnification of lens placed between the effective light source and the measuring volume [dimensionless] Noise power [W]; number of zero crossings in frequency counting and period counting techniques of signal processing; number of flow tracing particles contributing to the signal [see Eq. (1.1.90)]; number of fringes in the measuring volume. Effective number of flow tracing particles in the measuring volume [see Eq. ( I . 1.95b)l Cosine of the angle between thejth components of the vectors es and eo Refractive index [dimensionless]; distribution of flow tracing particles by size n ( r ) and by velocity magnitude n ( U ) [dimensionless] Luminous power of light field at the photodetector [W] Background radiation power [W] Johnson noise power [W] Shot noise power [W] Probability density function of the signal frequency w[s/rad] Electronic charge = 1.6018 x lo-'@A . s Distance from flow tracing particle [m]; load resistance at photodetector output [ohm]; resolving power of Fabry -Perot interferometer ( = A / SAo) [dimensionless] Distance between the flow tracing particle and the photodetector at t = 0 [m] Distance between the flow tracing particle and the center of the photodetector at r = 0 [m] Real part of a complex quantity Position vector [m] Magnitude of r [m]; radius of flow tracing particles [m] Radius of circular aperture used for division of wave front method for forming interfering beams in the homodyne system [m] An optical path difference that depends on the location of the light source relative to the photodetector [m] Signal power [W]; power spectrum of signal [AZ/s . rad] Time constant of photodetector [s]; time constant of output smoothing circuit [s] Actual time between any two points of equivalent phase in the photodetector output signal [s] Time [s]; etalon spacing in Fabry-Perot interferometer [m] Time at which flow tracing particle i[i = p . k , I] enters the measuring volume [s] Velocity vector of flow tracing particle [m/s] Velocity of flow tracing particle [m/s] Component of flow tracing particle velocity in coordinate direction j ( j = 1, 2, 3) [m/sI

1.1.

4

TRACER METHODS

95

Time mean value of the component in the coordinate direction j ( j = I , 2, 3) of the flow tracing particle velocity [m/s] Fluctuating part of UJ(uJ= 17, - DJ)[m/s] UJ V Voltage [V] W Light distribution in the measuring volume (see Table XVIII) [dimensionless] X Fringe spacing = distance between two consecutive light maxima or minima = n/[kosin (a/2)1 [ml X Geometric distance measured in the plane of observation of the spectral output of a Fabry-Perot interferometer [m] Value of x where [ / I o = 0.5, i.e., the geometric distance corresponding to the free XLl dispersion region Aho [m] CY Angle between incident beams at the measuring volume [rad or deg]; p0/& (see Fig. 17) Angle of light separation in dual incident beam homodyne configurations (see P Fig. 18) [rad or deg]; angle of light combination in single incident beam homodyne configurations (see Fig. 18) [rad or deg] Angle of light separation in single incident beam heterodyne configurations (see Y Figure 18) [rad or deg]; angle of light combination in dual incident beam heterodyne configurations (see Fig. 18) [rad or deg] Bandwidth of signal processing electronics [Hz]; filter bandwidth [Hz] Af Ak Half-width in wave numbers at half-maximum of spectral line profile of light emitted by light source [mP] Pressure variation in Fabry-Perot interferometer using chamber pressure to vary AP the etalon spacing [Pa] Interference fringe shift in plane of observation due to Doppler frequency shift [m]; Ax axial length of measuring volume in the x direction [m] Axial length of measuring volume in the y direction [m] AY Axial length of measuring volume in the z direction [m] Az Free dispersion of etalon (= F 6Ao) [m] AAo Average time of passage of flow tracing particle through the measuring volume, or AT signal duration [s] Filter bandwidth [rad/s]; width of probability density function distribution for photoAfJJ detector output signal frequency [rad/s] Difference frequency due to light beating at the photodetector of light scattered by particles k and I [rad/s] Free dispersion range of Fabry-Perot interferometer [rad/s] AWSA Bandwidth of swept oscillator wave analyzer [rad/s] Optical path length difference that depends on the location of the light source relas tive to the photodetector and/or the measuring volume [m] Optical path length difference that depends on the location relative to the photofi detector of the center of the light source ( f s )and/or the center of the measuring volume (R,,) [m] Half-width in terms of wavelength at half-maximum of light spectrum [m] Spectral broadening due to velocity fluctuations and instrumental effects [m] Instrumental half-width of Fabry-Perot interferometer spectrum [m] Instrumental half-width of Fabry-Perot interferometer spectrum [rad/s] Epoch angle of electromagnetic field [rad] Quantum efficiency of photodetector [dimensionless] Phase of photodetector output signal [oDr - cp(r)] [rad] An angle defined by the particle size (0 = a),the particle location (0 = P ) , or the method of light field division of the light source (0 = y ) (see Table XIX) [rad]

96 K

A P 7

cp

4

a, w w OD 0 1 0 1

1. MEASUREMENT OF VELOCITY A constant depending on the input-output characteristics of the Fabry-Perot inter[dimensions depend on definition]; K & / K , ferometer (= V / U , V/Ap, or Ax&) [dimensionless] Wavelength [m] Coherence function [dimensionless] Lag time in autocorrelation [s]; effective temperature of photodetector [“K] Angle between U and @ [rad]; random phase of the photodetector output current when more than one flow tracing particle is present simultaneously in the measuring volume [rad]; angle of incidence of beam in Fabry-Perot interferometer (see Fig. 42a) [rad] Angle between U and eo [rad] Solid angle subtended at the photodetector by the measuring volume (see Fig. 19)[s] Frequency [rad/s] Time mean photodetector output signal frequency = 1/T fl w dr = .f% wp(w) d o [rad/sI Doppler frequency shift [rad/s] Instantaneous signal frequency [see Eq. (1.1.1 IS)] Frequency of ith electric field vector (i = 0, 1, 2) [rad/s] Subscripts

D

S

or A where it indicates the Doppler frequency Photodetector (except in case of o,f, shift) kth particle (k = I , 2, . . . N) lth particle ( I = 1, 2, . . . N) Incident electric field ith incident electric field pth particle Particle p in ith incident beam (p = 1, 2, . . . N ;i = 1, 2) Particle k in ith incident beam (k = 1, 2, . . . N ;i = 1, 2) Particle I in the ith incident beam (I = 1, 2, . . . N;i = 1, 2) Light source Signal

S

Scattered

k 1 0 Oi P Pi ki li S

Superscripts

1.1.4.1. Introduction. The laser velocimeter* technique of fluid velocity measurement is capable of operating in a velocity range from cm/s to 2 x lo8 m/s. The spatial resolution is such that measurements may be confined to a volume of about mm3, which is superior to the comparable figure for a hot wire anemometer. The precision of measurement can be as small as +3%. The output signal is converted from optical to electrical and can be subjected to analog or digital manipulation.

* Durst has proposed that the term velocimerry be restricted to the measurement of the velocity of the flow tracing particles, and anemomerry be applied to those devices that measure fluid velocity (thereby, presumably, including an appropriate allowance for the dynamic characteristics of the flow tracing particles). The former situation is understood in this article, so it is logical to use the term velocimeter.

1.1. TRACER METHODS

97

Although most reported measurements, using the laser Doppler velocimeter, have been concerned with fluid velocities at one point in a flow, there appears to be no reason why measurements could not be made over an extended flow field, provided the hydrodynamics of the situation would allow the velocimeter output to be interpreted meaningfully.2o6 In general, the laser Doppler velocimeter is competitive with the hot wire or hot film anemometer, in that its electrical output signal is well adapted to the handling of the randomly fluctuating signals associated with turbulent flow. In fact, the laser Doppler velocimeter has two advantages over the hot wire/film anemometers in connection with measurements in turbulent flows. It has a linear relation between the output and the velocity of the flow tracing particles, while the hot wire/film anemometer has a nonlinear relation between input and output. This means that the equations used, with the laser Doppler velocimeter, to obtain the mean velocity and the rms of the velocity fluctuations, although resembling those associated with the hot wire/film anemometer, are much simpler (see Section 1.1.4.2). The other advantage of the laser Doppler velocimeter is that a probe does not need to be inserted in the fluid; the interference with the downstream sensor operation due to the wake of the upstream sensor that occurs in spatial correlation measurements with hot wire/film measurements is therefore absent.* The laser Doppler velocimeter employs the shift in frequency (the Doppler shift) of light that is scattered from a moving object. The shift in frequency depends on the velocity of the scattering surface. Therefore, monochromatic light that is scattered by a moving flow tracing particle contains information on the velocity of that particle. If this information can be extracted, knowledge of the particle velocity can be acquired and the velocity of the fluid inferred. The principle of the laser Doppler velocimeter, is not new and, in fact, is an essential element in many radar systems. However, the first practical demonstration of the technique, by Yeh and Cumminsss in 1964, had to await the development of the laser (1960). This event provided a source of visible radiation with sufficient power to ensure appreciable scattering of light from the small flow tracing particle. Furthermore, the light from a laser has enough coherence (see Section 1.1.4.4.3) to allow the detection of the Doppler frequency shift by heterodyning (see Section 1.1.4.3).

Microwaves, as used in radar, are unsuitable for use with flow tracing M . J . Schwar, Nature (London)229, 621 (1971).

* The absence of a probe in particle tracking fluid velocity measurements in general has been discussed in Section 1.1.1.

98

1.


particles, which are typically spheres with diameters of about 1 pm, because their wavelengths are in the range of 1-50 mm. Such wavelengths are too long, because the dimensions of the particles must be of the order of the wavelength of the incident radiation in order to ensure adequate scattering. Only visible light, which has wavelengths between 0.4 and 0.75 pm, is suitable for use in flow tracing systems.* It is not necessary, in principle, to employ a laser light source for the measurement of fluid velocities using the Doppler shift of light scattered by the flow tracing particles (see Section 1.1.4.4.3). Hence, the technique might better be called Doppler velocimetry. However, there are definite practical advantages to using a laser light source; consequently, nonlaser light sources have probably never been used except in experiments designed to explore the characteristics of a nonlaser Doppler velocimeter . The theory of the laser Doppler velocimeter involves physical optics and communication theory. The former deals with the wave aspects of light, as opposed to geometrical optics, which is concerned with ray propagation and image formation. Communication theory can be defined as the theory of transmitting information “from one point in space and time, called the source, to another point, the destination, or user’’.2o7 Probably neither of these topics is familiar to the users of laser Doppler velocimeters, who, typically, are much more likely to have a background in fluid mechanics. However, it is essential to understand the fundamental theory of the operation of the laser Doppler velocimeter if the maximum amount of fluid mechanical information is to be obtained from its application. This requirement was kept in mind in preparing this section. The basic idea involved in the laser velocimeter, namely, the Doppler shift of light scattered from a moving flow tracing particle, is described and related to the particle velocity in Section 1.1.4.2. The purpose of the observation system in the laser Doppler velocimeter is to extract the Doppler shift information from the scattered light. Section 1.1.4.3 concerns the receiving techniques that carry out this extraction and involve the application of the methods of light spectroscopy. Optical mixing is the optical spectroscopic method that is most widely used with the laser Doppler velocimeter. This is discussed in Section 1.1.4.4.4. The extent of the material presented in this section has required it to be divided into a number of subsections.

~0’

A. B . Carlson, “Communication Systems.” McGraw-Hill, New York, 1968.

* Ultraviolet light could be used, but difficulties of aligning the optical system with invisible light preclude its application in practical systems.

1.1. TRACER METHODS

99

The Fabry-Perot spectrometer has also been used as a receiving element in laser Doppler velocimeters, finding its application at very high fluid velocities (in excess of 300 m/s). Section 1.1.4.5 deals with this method. The fundamental theory of using the Fabry-Perot spectrometer in connection with the laser Doppler velocimeter, is considered. In addition, important new developments, the work of the authors, is also described. Section 1.1.4.6 compares the characteristics of the various signal processing methods. Section 1.1.4.7 briefly considers the analysis of the output from the signal processor from the point of view of obtaining fluid mechanical information from the velocimeter. The last two sections are practical in nature. Section 1.1.4.8 summarizes the information of the earlier sections, with a view to assisting the experimenter in selecting and adjusting a laser Doppler velocimeter. Section 1.1.4.9 provides a numerical example of designing a laser Doppler velocimeter. Finally, in presenting this discussion of the laser Doppler velocimeter, it has been found convenient to prepare summarizing tables. It is hoped that these will be helpful to the reader. 1.1.4.2. The Doppler Shift. Light that is scattered by a moving particle undergoes an apparent change in frequency. This frequency change is related to the velocity of the particle and is, as explained earlier, the basis of the laser Doppler velocimeter. The objective of this section is to obtain a quantitative relation between the particle velocity and the frequency shift of the scattered light. According to Maxwell’s electromagnetic theory, light may be considered as energy in transit due to the simultaneous propagation of an electric field and a magnetic field. It is usual to formulate problems in electromagnetic field theory in terms of the electric field. Thus an electric field at a point r associated with a plane electromagnetic wave propagating with angular frequency wo is given by

where c0 is the epoch angle (or initial phase); and 16, a vector specifying the direction of propagation of the wave (k,, = koeo, where ko = 2rr/Xo is the wave number and eo is the unit vector in the direction of propagation of the wave). In the laser Doppler velocimeter, we have the situation shown in Fig. 12. The point P (located at r) represents a flow tracing particle moving with a velocity U. A plane electromagnetic wave, given by Eq. (1.1.39), is incident on the particle, and a spherical wave, the scattered light, leaves the particle. The photodetector receives the scattered light.

1.

100


\ SCATTERED SPHERICAL WAVE

I I

DZ

FIG. 12. Scattering geometry in a laser velocimeter.

The spherical wave is represented by the electric field at a radial distance R from the flow tracing particle by

Es(R) = ( A S ( t ) / Rexp[-j(osi )

+ eS - ksRR)],

(1.1.40)

where usis the frequency of the wave, and eSdepends on the assumption that there is no phase change as a result of the scattering process. Then we have at the flow tracing particle, where R = 0, oot

+ e0 - k,,z

+ 8,

= oSt

so that eS = oot - wst

-

+ c0 - kor,

(1.1.41)

where kor = k,, r. Substituting from Eq. (1.1.41) in Eq. (1.1.40), we have for the electric field at some point Q on the photodetector (if R is the distance between flow tracing particle and photodetector)

ES(R)= ( A S ( t ) / Rexp[-j(oot ) - kSR - k J ?

+ s)].

(1.1.42)

On comparing Eqs. (1.1.39) and (1.1.42), it may appear that the frequency of the scattered light is the same as that of the incident light (coo). However, since the flow tracing particle is moving, the quantities k”, R , and r are functions of time, with r = U eot and R = Ro - U e2t, where eo and esare, respectively, unit vectors in the direction of propagation of the incident plane wave and the direction of the line joining the flow tracing particle P and the point Q on the photodetector. The quantity Ro is the distance between the flow tracing particle (usually assumed located at the center of the measuring volume) and the photodetector at time t = 0. Hence, for a moving flow tracing particle, Eq. (1.1.42) becomes

ES(R)= (AS(r)/R)exp[ -j(ost - kSRo+ cO)],

( 1.1.43)

1.1. TRACER METHODS

101

where the apparent frequency ws of the scattered light is given by us=

00

+ (ks- ko) *

( 1.1.44a)

U

or as - k S *U

=

WO

- ko

(1.1.44b)

U

Equation (1.1.44a) shows that the scattered light experiences a frequency shift wD given by OD =

(1.1.44~)

(kS - ko) * U,

known as the Doppler frequency shift [named for J. C. Doppler (1803-1853), who first studied this phenomenon in connection with sound waves]. The propagation vectors k in Eqs. (1.1.44) combine both direction and spectral (frequency or wavelength) information, since k = (w/c)e = (27r/X)e = ke, where the wave number k = o / c = 2 7 r / A , and e is a unit vector aligned in the direction of propagation of the wave. It is convenient to separate the directional and spectral aspects of Eqs. (1.1.44), so we write wo - ws = (woeo -

uses) (U/c).

Then os= wo[l - eo * (U/c)]/[l - es * > 1 this gives N, = N. In Eq. (1.1.95a), 1/N, . - is associated with the noncoherent contribution to the signal, and P A ~ is related to the coherent contribution. As for the case of single particles, we have for the homodyne configuration, using a photomultiplier with gain G and assuming negligible background light (PB= 0): N

N = 4qaG2kR Af K,

2 p"p.

(1.1.96)

p= 1

Hence, from Eqs. ( I . 1.95a) and (1.1.96) * In Eq. ( I . 1.94), the terms corresponding to Is and I , , and I, and I, in Eq. (1.1.93) have = Pp= PB)in order to simplify the expression. This is considered been consolidated (P,, to be appropriate because the objective of this discussion is to consider in general terms the effects of multiple particles on the photodetector output current, rather than to carry out a detailed numerical calculation of the signal-to-noise ratio.

1.1. TRACER METHODS

SIN

=

2Km/2hfk Af)Ps(l/Ne

+ ,)'AF

151

(1.1.97a)

where p s = (Ne

i:@)/i Pi (1.1.97b)

p=1

p= 1

is equal to the luminous power of the scattered light incident on the photodetector. Equation (1.1.97a) is identical in form to Eq. (1.1.87) except for the multiplying coefficient [( l/N,) + pA2]. Drainzs3has used this expression to investigate the variation of the signal-to-noise ratio with the size of the receiving aperture. For small values of N e t the noncoherent contribution to the photodetector signal dominates, and the signal increases with Ps, which is, in this case, proportional to the receiver aperture (the signal can also be increased by increasing the amount of scattered light Ps),as has been shown earlier in this section for the limiting case of a single particle in the measuring volume. When N, is large, the signal magnitude is controlled by the coherent contribution. If the receiving aperture is very small, practically all the signal power is due to the optical mixing of light scattered by different flow tracing particles in the measuring volume, and the signal strength increases in proportion to the receiving aperture size. After a certain critical aperture diameter Af/dparthas been attained, the coherent contribution ceases to increase with increasing aperture size. The increase is due to the noncoherent contribution to the photodetector signal. The variation of the signal-to-noise ratio with receiving aperture size when N, is large, is similar to the variation described earlier for the heterodyne configuration when there is a single particle in the measuring volume. Hence for large values of N, and large receiver apertures, the signal-to-noise ratio approaches an asymptotic value given by (N, + 00) SIN =

(VK,/~

hfk Af)pbzPs.

(1.1.98)

In the heterodyne configuration, Eq. (1.1.94) becomes (the term Cp,P2,is suppressed, as described in Section 1.1.4.4.2)

where pb2 is the coherence function described earlier, and pD is the coherence function of the optical mixing of the reference beam and light scattered from the flow tracing particles (this has a probabilistic character for

152

1.


the multiple particle case; the details of the calculation will be found in Drainze3). Equation ( 1 . 1 .!B)shows that there are only coherent contributions to the signal in the heterodyne configurations. The noise power N is given by

N = 4qaGZkR A f Kw ( P o +

N

Pi).

(1.1.100)

p= 1

Hence, from Eqs. (1.1.99) and (1.1.100):

Since the reference beam power is significantly greater than the light scattered from the flow tracing particles (Z'_,P;), Eq. (1.1.101) becomes [cf. Eq. (1.1.86)]

The expression shows that the signal-to-noise ratio for the heterodyne configuration is unaffected by the effective number N, of flow tracing particles in the measuring volume, If P is defined as the quantity in parentheses in Eq. (1.1.102) divided by P o , then Eq. (1.1.102) can be written S/N = (r)pbzKw/hfkAf)Ps.

(1.1.103)

In this way the signal-to-noise ratio expression for the heterodyne configuration with multiple flow tracing particles in the measuring volume has been cast into the same form as the corresponding single particle case [Eq. (1.1.86)J. The variation of the multiple particle heterodyne configuration signalto-noise ratio with receiving aperture has the same form, for the same reasons, as the multiple particle homodyne signal-to-noise ratio when N , is large. However, the asymptotic signal-to-noise ratio for the heterodyne configurations is about twice that of the homodyne case. This is the same relation between the two cases as was found earlier for a single particle. The preceding discussion of both the single particle and multiple particle cases suggests that the homodyne configuration should be used where the number density of the flow tracing particles in the measuring volume is small, e.g., in gaseous flows. Under these conditions the signal-to-noise ratio is better than that of the heterodyne configuration. See Table XXII.

TABLEXXII. Signal-to-Noise Ratio with Photomultiplier Configuration Number of particles in the measuring volume

Heterodyne

Homodyne

SNR better in the homodyne configuration because Pi can be increased by increasing the receiving aperature; however, in the heterodyne configuration, the maximum receiving aperature is limited by coherence requirements.

One

More than One

Notes

'))PL2KL

hf k A f (PolP, -+

m)

For N ,

+ 0~

'))PL2KW

Zhfk A f P s

SNR in the heterodyne configuration is twice that in the homodyne case when the effective flow tracing particle number density N , is high. In either case F is unaf€ected by increasing the size of the receiving aperature (see Draine").

154

1.


When the number density of the flow tracing particles in the measuring volume is high, as is usually the case in liquid flows, the signal-to-noise ratio of the heterodyne configuration is superior to that of the homodyne configuration. This is also advantageous, because a large particle number density tends to increase the scattered light and hence to compensate to some extent for limitation on the receiving aperture size imposed by the coherence conditions (see Section 1.1.4.4.3). 1.1.4.4.7. SIGNAL PROCESSING 1.1.4.4.7.1. Introduction. To extract the desired fluid flow data from the output of the photodetector, it must be subjected to appropriate processing and analysis. Processing will be defined here as that manipulation of the photodetector output signal that provides the Doppler frequency shift wD. The term analysis will be restricted to the extraction of information from the Doppler frequency shift oD,i.e., from the output of the signal processor. Analysis will therefore involve, for example, the estimation of the statistical parameters of a turbulent flow. The major part of this section will deal with signal processing, while signal analysis will be considered in Section 1.1.4.7. The processing of the photodetector output is influenced by the manner in which the information on the flow tracing particle velocity is carried, by the temporal character of the signal, and by the noise associated with the signal. The photodetector output is a frequency modulated (FM) carrier, in which the carrier frequency wo is modulated by the Doppler frequency shift oD,with wD ],

(1.1.107~)

p=1

where P p = (P,,P;)1'2for the heterodyne configuration, and P p = Pp for the homodyne configuration. As discussed earlier, this expression involves random variables, and George and Lumley266 and Durrani and GreatedZZ0show that Eqs. (1.1.107), after high pass filtering, can be written i = a(r) cos(oDt)

+ b(r) sin(wDr),

(1.1. 08)

where a(r) and b(t) are Gaussian random variables.* Alternatively, i = H(r) cos[oDt -

&)I,

(1.1. 09)

where H2 = a2(r) + b2(r),and cp = tan-'(b/a). According to this expression, the phase 8 = wDt - cp(t) of the photodetector signal consists of a term wDt related to the velocity of the flow tracing particles and a randomly varying phase function &) having its origin in the random, time dependent character of the signal, as given by items (i) and (ii) of the list of photodetector output current characteristics. The amplitude H of the photodetector output current also varies randomly due to the causes indicated under item (iii) in the previously mentioned list. Because of the random character of the phase and amplitude of the photodetector output, its frequency spectrum t o,even with steady flow, occupies a range Ao of frequency values.$ This is shown in Fig. 22(b), together with the specW. K. George and J . L. Lumley, J . FIuid Mech. 60, 321 (1973).

* This is justified by typical spectra of continuous photodetector signals observed in practice. t The spectrum would be measured by a swept oscillator wave analyzer (see Section 1.1.4.4.7.3). so the displayed information is the probability density function of the photodetector output (a frequency modulated signal). This should not be confused with the power spectral density of the velocity fluctuations, which is obtained from the processed photodetector output [see Fig. 22(c) and Section 1. I .4.7]. $ Spatial variations in fluid velocity can affect wD by varying the velocity of the flow tracing particle as it passes through the measuring volume. Although this has the appearance of noise, it is not intrinsic to the laser velocimeter and reflects an uncertainty in the measurement associated with size of the measuring volume relative to the spatial variation of fluid velocity, i.e., it is a measure of the spatial resolution of the measuring technique (see Section 1.1.4.4.4). It may sometimes be effectively eliminated by appropriately designing the optical system of the velocimeter. Accordingly, this phenomenon will not be included under ambiguity noise.

1.1.

I59

TRACER METHODS

(a)

W

o

I.. I

w

W

D

S

4

t

FIG.22. Spectra at different stages in signal processing (p is the probability density function; a, the power spectral density). (a) Spectrum of signal (light incident on photodetector) before detection. Line at o0is due to the laser, and AOJ is the ambiguity broadening of the spectrum. The broadband small amplitude signal shown is due to optical noise. (b) Spectrum of signal (photodetector output) after detection. Low frequency component (pedestal) is introduced during detection (but not in the single particle case when using heterodyne detection), but is usually filtered out before the signal is processed. The portion of the spectrum (shown dotted) corresponding to negative frequencies is not always exhibited. The broadband small amplitude signal is due to optical and electronic noise. (c) Spectrum of velocity fluctuations w ' , obtained after demodulating the FM signal that has the spectra shown in (a) and (b). The broadband small amplitude signal is due to ambiguity noise Ao in (a) and (b)]. The electronic and optical noise shown in (b) and the pedestal [where observed, see caption to Fig. 22(b)], are removed by filtering.

160

1.


trum [Fig. 22(a)] of the light incident on the photodetector. This broadening of the signal spectrum affects the precision with which the velocity measurements can be made. The measurement precision is also affected by the signal processing, and this is considered at the relevant points in the discussion that follows. Spectral broadening is also observed in Doppler radar measurements, where it is called the Doppler ambiguity, and, probably for this reason, the spectral broadening of the laser Doppler velocimeter photodetector output signal is often called ambiguity broadening or ambiguity noise (the last named terminology will be adopted here). Durrani and GreatedZz0have estimated the magnitude Sw of the spectral broadening due to ambiguity noise from the spectrum of the photodetector output current [Eq. (1.1.107c)l. They obtained the following result: SOD/OD

=

k

CO~(CX/~ SCI, )

( 1.1.110)

where k, which has a value of about 0.2, is a constant that depends on the weighting function W ( t ) and on the function (Fourier transform or power spectral density) used to define the spectrum. This result is only applicable to the dual incident beam homodyne configuration, but it can also apply to the dual incident beam heterodyne configuration, where the dimensions of the measuring volume are assumed to be equal to the illuminated volume formed by the intersection of the illuminating and reference beams. * Apart from the value of the multiplicative numerical factor k, Eq. (1.1.1 10) is identical to Eq. (1.1.76b) for aperture broadening, which was obtained from precision considerations [k = 0.5 in Eq. (1.1.76b)l. This demonstrates that aperture broadening of the spectrum and ambiguity noise have identical physical origins.? The ambiguity noise cannot be filtered; otherwise velocity information in the photodetector output would be lost. However, the broadband noise having its origin in the photodetector and in the optical system (see Section 1.1.4.4.6) may be reduced by filtering (see Section 1.1.4.7.2). 287 R. V. Edwards, J . C. Angus, M. J . French, and J. W . Dunning, J . Appl. Phys. 42, 837 (1971).

* Where the magnitude of the measuring volume depends on the aperture of both the transmitting and receiving optics, presumably an expression equivalent to Eq. ( I . l.76a) could be devised by the methods used to obtain Eq. (1.1.110). t There was some uncertainty regarding this point in the early literature on laser Doppler velocimetry, but in 1971 the identity of aperture and ambiguity broadening was demonusing a rather more sophisticated argument than that employed strated by Edwards et a/.2s7, here.

1.1. TRACER

METHODS

161

Under unsteady flow conditions, e.g., when the flow is turbulent, the Doppler frequency shift wD is time dependent. For turbulent flow, this quantity is often considered to be a random variable and to contribute to the observed broadening of the spectrum of the photodetector output. This contribution is proportional to the root mean square Wb of the velocity fluctuations 0;; however, if this parameter is sought from the data, it must be separated from the ambiguity broadening. It has been assumed that the difference between observed spectral broadening with and without turbulence is equal to the broadening due to t u r b ~ l e n c e . ~ ~ ~ ~ ~ With high speed flows [alarge in Eq. (1.1.1 lo)] the width of the photodetector output spectrum can be predominantly due to the effects of ambiguity noise, so that the broadening attributable to turbulence is a comparatively small number which is equal to the difference between two large numbers, a situation that is well known to introduce large uncertainties into any measurement. Signal processing and signal analysis methods can be broadly classified by the so-called domain in which the processing or analysis is carried out. These domains are the time domain and the frequency domain, and different considerations must apply depending on which domain is being used.* In this subsection, attention will be concentrated on the differences between signal processing in the time and frequency domains. Devices that process signals in the time domain are only sensitive to the phase 8 = wDf - q ( f )of the signal; the signal amplitude H plays no role. t Accordingly, the effective photodetector output has the form [from Eq. (1.1.109)] i = cos[wDf - (o(t)].

288

(1.1.1 11)

R. J. Goldstein and W. F. Hagen, Phys. Fluids 10, 1349 (1967). Penner and T. Jerskey, Annu. Rev. Fluid Mech. 5 , 9 (1973).

ztm S. S.

* There is some confusion in the literature regarding the classification of signal processors and signal analyzers by their domain of operation. A particularly common example is to state that the tracking bandpass filter (see Section 1.1.4.4.7.4) processes signals in the time domain. This device processes (according to the definition of signal processing given earlier) signals in the frequency domain, but the output (essentially the velocity of the flow tracing particles) is analyzed in the time domain. The author suspects that a lack of a clear differentiation between the signal manipulations that are here termed processing and analysis is the source of this confusion. t Actually this statement is not quite correct, since a signal must have a small but finite amplitude in order to register its presence in the device. Furthermore, because the signal always contains unwanted noise (all types, including ambiguity noise), it is necessary to introduce an "amplitude filter" (called a discriminator) so that small amplitude (small being decided by the setting of the discriminator) noise signals do not contribute to the output from the signal processor (see Section 1.1.4.4.7.2).

162

1. MEASUREMENT O F VELOCITY

When the photodetector output is processed in the frequency domain, it is subjected to spectral analysis (Fourier transformed) so as to determine its probability density function. Since the Fourier transform acts on both the amplitude Hand phase 8 of the signal, the spectral analysis is affected by both of these elements of the signal. Accordingly, the effective photodetector output, when processing is in the frequency domain, retains the form of Eq. (1.1.109). Because of the different forms for the effective photodetector output, time domain and frequency domain signal processors have different sensitivities to noise (all types, including ambiguity noise). Thus in time domain processing only those noise effects associated with the phase function cp(r) affect the interpretation of the processor output. By contrast, frequency domain signal processors are sensitive to the effects of noise in both the amplitude and the phase of the photodetector output. In the context of ambiguity noise, we can say that frequency domain processors are sensitive to all forms of ambiguity noise as given by items (i), (ii), and (iii) in the list of ambiguity noise sources given earlier. On the other hand time, domain processors are only affected by the ambiguity noise sources of items (i) and (ii) of the list. The following sections concern the most widely used methods for processing the photodetector output. One of these, the tracking bandpass filter, is restricted to situations involving high concentrations of flow tracing particles, but the other methods can be used with both low and high particle concentrations. 1.1.4.4.7.2. Signal Conditioning. The output from the photodetector is not necessarily suitable for processing without some preliminary manipulation, which will be called signal conditioning. This manipulation is required to: (a) (b) (c) (d) (e)

raise the signal level by preamplification, remove the low frequency component of the signal, shift the apparent signal frequency, suppress large amplitude signals, and minimize broadband input noise at the signal processor.

The photodetector output, actually the voltage across the detector load resistor, is typically of the order of 10 PV.~'O This level is inadequate for most signal processors, so a preamplifier, with sufficient bandwidth (typically 0-200 MHz) to accommodate the anticipated variations in signal frequency, is required to raise the signal level to about 0.1 V. The photodetector signal comprises two components: a low frequency D. T. Davis, ISA Trans. 7, 43 (1968).

1.1.

TRACER METHODS

I63

component or pedestal with a spectrum centered about zero frequency, and higher frequency components (sidebands at positive and negative frequencies symmetrically disposed about zero frequency) that carry information on the velocity of the flow tracing particles [see Fig. 22(b)]. It is not appropriate to process the complete signal, because undesirable signal degradation may result with no improvement in the information content of the processor output. This is because the bandwidth of the processor would have to be sufficiently wide to accommodate the full frequency range of the signal from DC to the maximum anticipated Doppler shift frequency, so a correspondingly wide bandwidth would be provided for noise. To avoid this retention of noise in the signal processor, the low frequency component is removed. High pass filtering is the simplest method for removing the low frequency component of the photodetector output. However, care must be taken to ensure that this does not result in loss of information. For example, with turbulent flows having a large range of velocities (i.e., a large range of Doppler frequency shifts), some of the high frequency information bearing components of the signal may be removed with the low frequency component, because the low frequency end oDMIN of the Doppler shift frequency spectrum may fall below the lower cutoff frequency of the high pass filter. According to Durst and Zare,271this will occur when WDMIN < < ( l h D M A X ) / N , where N is the 'number of fringes in the measuring volume, and oDMAX, the anticipated maximum Doppler frequency shift. From the discussion of the previous paragraph, it is clear that for a fluid flow in which the velocity varies over a wide range of values, it may be necessary to continually adjust the setting of the high pass'filter in order to compensate for the decrease in the dynamic range of the signal processor due to the filtering. This adjustment would probably have to be made by the experimenter and would be impractical where the fluid velocity variations are very rapid; it would also prevent the measurement of velocity profiles by automatic devices that traverse the measuring volume across the fluid stream. The limitations imposed by the need to avoid signal distortion in high pass filtering may be overcome by frequency shifting using the techniques described in Section 1 . 1 . 4 . 4 . 8 . Thus, if the frequency of one of the beams of the velocimeter is shifted by an amount Aw relative to the frequency of the other beam, the higher frequency component of the photodetector output will be shifted away from the lower frequency component by the same amount Au. High pass filtering may then be used to remove the F. Durst and M.Zare, Appl. Opt. 13, 2562 (1974).

164

1.


lower frequency component without the danger of distorting the signal processor output. Frequency shifting is useful for purposes other than avoiding the errors that may be introduced by high pass filtering. Thus, frequency shifting (a) allows the location of the signal within those portions of the processor bandwidth where the distortion due to the processor characteristics is a minimum, (b) avoids the introduction into the processor of signals corresponding to zero velocity (this is known as signal dropout and can introduce errors into the results obtained by certain types of signal processors; see Sections 1.1.4.4.7.4 and 1.1.4.4.7.5), (c) brings signals corresponding to high fluid flow velocities (i.e., high signal frequencies) within the frequency range of the processor, (d) makes the laser Doppler velocimeter directionally sensitive (see Section 1.1.4.4.8), and (e) improves the accuracy of the photon counting correlation technique of signal processing (see Section 1.1.4.4.7.7). Large amplitude signals should not be processed. They may originate with large flow tracing particles which have poor dynamic characteristics. They can also be due to flow tracing particles that do not pass through the measuring volume, or to flow particles in the measuring volume that have a velocity component in a direction other than that being measured. In the first case, there is the danger that invalid data may be processed. In the second case, where the flow tracing particle density is high, the signal-to-noise ratio of the laser velocimeter will be decreased. This is because the signal will have a very poor modulation, since particles which do not pass through the measuring volume only contribute to the low frequency portion of the signal. Likewise, velocity components that are not being measured will add to the low frequency portion without providing additional information. In view of this, certain signal processor^^'^ have facilities for rejecting large amplitude signals. However, it should be noted that it may be possible to decrease the amount of light incident on the photodetector from particles that do not pass through the measuring volume by a pinhole aperture placed in front of the photodetector. This has the disadvantage that the alignment of the optical system becomes more critical, but Bossel et have demonstrated that a phase shift method provides a very good technique for eliminating effects due to particles not passing through the measuring volume. The signal from the photodetector will include broadband noise from optical sources and from the photodetector (see Section 1.1.4.4.6). This could introduce uncertainty into the measurement of the fluid velocity. A filter with a suitable bandpass (this should be variable and lie somewhere J. A. Asher, Prog.

Asironaui. Aeronaut. 34, 141 (1974).

1.1.

TRACER METHODS

165

in the following range: low cutoff frequency from 2 Hz to 50 MHz, and high cutoff frequency from 200 Hz to 200 MHz) should be provided to limit the noise at the processor input. As pointed out earlier, such bandpass filtering will limit the effective (velocity) dynamic range of the signal processor, which may be a disadvantage in measurement situations where rapid variations in fluid velocity over a wide range of magnitudes are anticipated. 1.1.4.4.7.3. Swept Oscillator Wave Analyzer. Spectral analysis is the most obvious technique for processing the output from the photodetector. Analog techniques are used because the signal frequencies are too high (of the order of 1014Hertz) and the number of data (typically in excess of lo5) is greater than current digital techniques can handle. In principle, either a filter bank or a swept oscillator wave analyzer may be employed. In practice, the latter is almost always used because the filter bank is very expensive, whereas the wave analyzer is available as a cathode ray oscilloscope plug-in and is of comparatively low cost. In addition, filter banks are not very flexible with regard to frequency ranges." The wave analyzer (see Fig. 23 for schematic diagram) samples a small fixed bandwidth (determined by the analyzer resolution) with a center frequency that changes linearly with respect to time at a fixed rate called the sweep rate. As the swept filter passes through the frequency range occupied by the signal, its center frequency and the frequency of the signal coincide from time to time (see Fig. 24). On those occasions, a signal with a magnitude proportional to the amplitude H of the photodetector output at the instant of coincidence, together with the corresponding frequency, are registered at the wave analyzer output. Over the time of observation, the output of the wave analyzer represents the cumulative sum of the signal amplitudes, distributed over the various frequencies, of each such individual signal. The output of the swept oscillator wave analyzer is then the probability density function p of the frequency w of the photodetector output within the range of frequencies swept by the filter. This may be exhibited as a function of frequency w on an XY recorder or cathode ray oscilloscope (X = w , Y = p ) . From this display, the experimenter may visually estimate the frequency wM corresponding to the highest value of the probability density function. This frequency may be assumed equal to the average frequency W of the photodetector output. Under steady laminar flow conditions, this would be proportional to the T. S. Durrani and C. A. Greated, Proc. Insr. Elecrr. Eng. 120, 913 (1973).

* Durrani and have described a technique for converting the swept oscillator wave analyzer into a bank of filters. This may minimize the cost of the filter bank approach.

I66


"

~

HIGH PbSS FILTER

*

w- w

+ MIXER

-

-

-Vco

IF FILTER

-

X Y RECORDER

VOLTAGE

CONTROLLED OSCILLATOR

RECTIFYING AND SMOOTHING CIRCUIT

GENERATOR

FIG.23. Swept oscillator wave analyzer. The rectifying and smoothing circuit are an integral part of commercially available oscilloscope plug-in spectrum analyzers.

Doppler frequency oD of the flow tracing particles in the measuring volume, and with unsteady laminar turbulent flow, it would be proportional to the time mean Doppler frequency WD. The width Ao of the probability density function distribution curve may be assumed to be approximately equal to the rms frequency deviation from the Doppler frequency shift wD. If the flow is laminar, this parameter depends, as shown in Section 1.1.4.4.7.1, on the ambiguity noise, and is a measure of the resolution of signal processing by the swept oscillator wave analyzer. If the flow is turbulent, it is usual to assume that it depends on the sum of the mean square frequency fluctuation arising from the ambiguity noise and from fluid turbulence (although the discussion of Section 1.1.4.4.7.1 on this point should be noted). In practice, the preceding resolution estimate must be modified because of the limitations of available wave analyzers, namely: (a) the frequency resolution of the instrument is finite, with a bandwidth AoSAinversely proportional to the correlation time fSA of its output; and (b) the analyzer is inefficient in the sense that the spectrum must be obtained by sweeping the filter over a chosen frequency range (wl - w2) a finite number of times, with a total processing time o f t , (for n sweeps, t, = nt,, where t, is the sweep time). W i l m ~ h u r shas t ~ ~shown ~ that the velocity resolution AU/ U , neglecting noise other than ambiguity noise, is then given by

where K is a constant. According to this result, the resolution of the 274

T. H.Wilmshurst, J . Phys. E 4, 77 (1971).

1.1.

I67

TRACER METHODS

w

t

tsqt

‘s t, = n t s

=I=

- .....-.

’”

0

,

m

FIG. 24. Relation between signal frequency o and filter center frequence o, (shown by dashed line) is a swept oscillator wave analyzer. Point where signal frequency o and filter center frequency coincide is indicated by @ or x . @ is the low sweep rate ( t , > t v ) ; X, the high sweep rate (1, Gr1I3.

For a hot-wire with 2.5-pm diameter at 300 K in air, the Grashof number is approximately 6 x lo-'. Thus, no serious free convection effects will 1e

D. C. Collis and M. J . Williams, J. Nuid Mech. 6, 357 (1959).

* See Part 10, Dimensional Analysis, for the definitions and meanings of the Reynolds, Grashof, and other nondirnensional numbers.

270

1.


be present as long as the Reynolds number is greater than 0.01. For typical hot-wires in air, free convection can be neglected for velocities greater than approximately 10 cm s-l. A similar restriction applies for hot-films because Collis and Williams’ criteria is independent of the diameter of the sensor. The rate of heat transferred by forced convection from hot-wires and hot-films depends primarily upon the velocity and fluid temperature. The heat lost by convection is given by (1.2.20) where h is the convective heat transfer coefficient as defined by the equation and S is the surface area through which the heat is transferred. The forced convection losses are expressed nondimensionally in terms of the Nusselt number, which is the ratio of the heat lost by convection to that lost by conduction. For a cylinder of diameter d, the Nusselt number is NU = h d / k f , where kf is the thermal conductivity of the fluid. The convective heat losses, and hence the Nusselt numbers, depend upon almost every possible parameter of the fluid as well as the properties of the heated element. In nondimensional terms, Corrsin” has suggested that Nu = Nu[Re, Pr, Ma, Gr, Kn, ( l / d ) , uT, y, 01, where Re, Pr, Ma, Gr, and Kn are the Reynolds, Prandtl, Mach, Grashof, and Knudsen numbers, respectively; l / d is the aspect ratio of the probe; uT , the overheat ratio; y, the ratio of specific heats of the fluid; and 8, the angle between the axis of the sensing element and the velocity vector. Heat transfer from cylinders with diameters larger than the Kolmogorov length scale may depend also upon the roughness of the cylinder and the turbulence level in the free stream. It is impossible to consider all of these independent variables simultaneously. This is avoided in the ensuing analysis by making some appropriate approximations. The buoyancy, i.e., Grashof effect, will be negligible for most velocities of interest, as explained earlier. For probes having diameters much greater than the mean free path of the molecules in the fluid, the Knudsen number will not affect the heat transfer. The effects of compressibility can be neglected when Ma < 0.3. Flow fields with higher Mach numbers have been treated extensively by Kovasznayl8 and M o r k o ~ i n and ’ ~ will not be dupliI’S. Corrsin, “Handbuch der Physik,” p. 524. Springer-Verlag. Berlin and New York, 1%3. lo L. S. G. Kovasznay, J . Aerosp. Sci. 17, 565 (1950). M. V. Morkovin, AGARDograph 24 (1956).

1.2.

PROBE METHODS FOR VELOCITY MEASUREMENT

27 1

TABLE11. Experimental Values of the Coefficients in Eq. (1.2.22)

Collis and Williams” Hilpert*

McAdamsP

Reynolds number

A

B

n

m

Re < 40 Re > 40 1 1. After the signal due to the pulse has decayed to zero, i.e., M d Ae,/dt = 0 , the am-

FIG. 16. Dynamic response of a constant temperature hot-wire anemometer to a 1-kHz square wave. Above: underdamped; center: critically damped; below: overdamped. The sensor is a 2 . 5 - p n platinum wire 1.3 m m long with aR = 0.5.

300

1.


plitude of the square wave remains and is nonzero, i.e., Aet(r) # 0, and thus the signals do not return completely to their average value. The oscillograms in Fig. 16 show that the frequency response can be set by observing the constant temperature anemometer output on an oscilloscope when the system is perturbed by a square wave. Because the critically damped state has a flat broad spectral response, it is the preferred mode of operation. Freymuth3' has shown that if the anemometer is operated in an overdamped condition, nonlinear errors are introduced into the output signal which are especially significant in higher-order statistics. Thus if a smaller frequency range is desired to decrease the noise, an external filter should be used. 1.2.4.5.4. HIGHER-ORDER SYSTEM RESPONSE.The previous analysis assumed that the elements in the bridge were purely resistive and that there was no additional filtering in the system. Although this situation is highly desirable and quite often gives a very good approximation to the operating system, there may be enough stray capacitance or inductance in long cables or other reactive elements so that they must also be considered in determining the frequency response. Any cable or circuit with stray capacitance or inductance will always affect the frequency response in some range. Usually this range is above the limitf, determined in the previous section and does not affect the anemometer at frequencies below f,.

One technique for coping with reactive elements is to introduce other compensating reactive elements into the bridge itself. This common technique is mathematically described by writing the balance equation [Eq. (1.2.47)] in complex notation for the reactive bridge as where Z is the complex impedance in each leg. Any stray reactance in the bridge leg containing the sensor can be compensated for by introducing another reactance into an opposite leg of the bridge to balance the bridge equation. In practice, stray capacitance and inductance in the sensor leg is often due to the cables leading to the sensing element. For example, coaxial cables can introduce 10-50 pF/m. Although small, these values illustrate that the user should avoid cables longer than necessary. Stray inductance may also be due to wire wound potentiometers in the circuit. To achieve the highest frequency response possible, a unity bridge with equal elements in opposite legs is used so that the reactive elements of the cable leading to the probe can be compensated for in a neighboring leg by using an identical cable terminated with a constant resistor chosen so that the resistance yields the desired overheat of the sensing element.

1.2.


sensor.

301

-

The reactance of the cable can be modeled as a capacitor in parallel and an inductor in series with the sensing element. F r e y m ~ t hhas ~ ~discussed the effect of the capacitance, and the inductance has been modeled by Perry and and Wood.39 Wood also includes the effect of a compensating inductor. The effects of the distributed capacitance and inductance usually become important at different frequencies, as seen below. A bridge circuit with a capacitive element in parallel with the sensor is shown in Fig. 17. The third-order response of the system will depend upon the additional time constant 7 = C R R , / ( R + R J . An inductor in series with the sensor yields another time constant of T = L / ( R + R,). The importance of these effects can be seen by comparing the relative values of the three time constants associated with the amplifier, sensor, and cable: p , M, and 7. For example, if a 3 m cable with L = 1 p h and C = 500 p F is used in a bridge with R = 10 fi and R, = 40 0, the time constant due to the inductance is L / ( R + R,) = 20 ns, and that due to the capacitance is R R I C / ( R-I R,) = 4 ns. These values are so small that they can be neglected for frequencies less than a 1 MHz, which is the upper frequency limit of anemometers today. It is easily verified that quite large values of inductance or capacitance must be introduced before a third-order analysis is necessary. An extreme example is R = 100 a, R1 = 400 0,and a cable of 1-pF capacitance giving a time constant of 80 ps. Since this is larger than nominal values of p , the anemometer may behave as a third-order system. In most applications, the time constant attributed to the sensing element, M , will be larger than any of the other M. R. Davis, J . P h y s . E 3, 15 (1970). E. Perry and G . L. Morrison, J . Nuid M e c h . so N. B. Wood, J . Fluid M e c h . 61,169 (1975). 37a

A.

47, 577 (1971).

302

1.


time constants. If p los for values of the orifice ratio m ranging from 0.1 to 0.7. Other charts for other shapes of orifices are given in Hengstenberg et al.Oz A consideration that may be of importance, particularly in large installations, is the pressure drop that occurs in any flowmeter. In the orifice meter, the frictional nature of the flow with shear stresses at the walls is evidenced in part by the pressure loss p i - ph displayed schematically in Fig. 23. This pressure loss is associated with a failure to regain the pressure p i on expansion to the cross section A l again, which would occur if the Bernoulli formula truly applied. In designing or choosing a flowmeter, p 1 - p z must be large enough so that it can be accurately measured. However, a large pi - p2 is accompanied by a large p i - pk or pressure loss, so attention must be paid to whether the pressure loss is too large to be acceptable on economic grounds. The term Q ( p ; - pk) representspower lost, i.e., power that would otherwise not have to be supplied by pumps to maintain the flow rate Q. When high pressure gases, e.g., steam, are moving through pipes rapidly, the pressure loss becomes quite significant, and rounded edge orifices and nozzles are used in preference to sharp-edged orifices. Some of these do not have as large a pressure loss for a given working pressure; these nozzles are also used when the pressure drop becomes an appreciable fraction of the entering pressure, and compressibility of the gas plays a role. For an ideal gas undergoing adiabatic expansion and con-

@*J. Hengtenberg, B. Sturm, and 0. Winkler, eds., “Messen und Regeln in her chemischen Technik,” pp. 208-210. Springer-Verlag, Berlin and New York, 1957.

1.2.


329

a 0.8

0.75

07

F i c . 24. Discharge coefficient a(m, Re) for sharp-edged orifice. (a) Plate dimensions. D is the pipe diameter. The orifice diameter d may have any value between 0.320 and 0.840. (b) Chart for determining a for various orifice ratios m = Ao/AI = ( d / 0 2 , and various Reynolds numbers Re = DUl/u, where D is the pipe diameter; U1, the average fluid speed in full sized pipe; and Y , the kinematic viscosity of the metered fluid. Flow rate Q = an(8/4)[(2/p)(pI - p2)]’’* (in m3 s-’); p is the fluid mass density (in kg m-3); and p 1 - p z , the pressure difference (in Pa). This chart and formula are applicable to incompressible fluids; air may be considered incompressible at M 5 0.2.

traction in a nozzle, the Bernoulli formula is different from Eq. (1.2.84), as discussed in Sections 1.2.2.1 and 1.2.2.2. EXTENSION OF FLOWMETERING FORMULAS TO COMPRESSIBLE FLUIDS. Making the same assumptions of uniform steady flow as those which led to Eq. (1.2.86), but utilizing the compressible Bernoulli formula for an isentropic ideal gas,03 we find that the muss per second flowing through the pipe is

[2PiPi Y

():”’

1 P

[ 1 (:)”-‘)’] -

]’”,

(1.2.92)

93 L. D. Landau and E. M. Lifshitz, “Fluid Mechanics,” paragraphs 80, 90. Pergamon, Oxford, 1959.

330

1.


where, as before, rn = Ao/A1, and y is the ratio of specific heats of the gas. With nonuniform velocity profiles, shear stresses at the walls, and uncalculable pressure corrections, Eq. (1.2.92) can have coefficients inserted; instead an empirical equation recognizing that compressible flow through a constriction depends on p 2 / p 1in a more complicated way is employed: (1.2.93) Q = a ~ A 0 [ ( 2 / ~ 1 ) ( P1 ~2)1”~, where a is the same discharge coefficient introduced into Eq. (1.2.90), while E , which is always less than one, depends on p 2 / p 1 ,in, and y for a given nozzle or orifice plate geometry. For certain standard nozzle shapes, and for rn S 0.4, the theoretical formula in Eq. (1.2.92) adequately describes the mass flow rate if the values of a for incompressible fluids and ( 1 - m2)l12are used as additional factors on the right-hand side.s4 SONICNOZZLE; CHOKEDFLOW. If the expression for Qmassin Eq. (1.2.92)is plotted as a function of p2/p1going from 0 to 1, it is seen that it vanishes at both 0 and 1 and has a maximum at some value between. At the maximum, the gas has expanded and cooled so that the sound speed is equal to the gas speeds3 and the flow is said to be choked. The value of p 2 / p 1at the maximum Qmassis called the critical pressure ratio (P2/pl)crit. If it is arranged that p1is held fixed, and the downstream pressure is gradually reduced starting at p l , Qmaswill increase until (p2/pl)cr,t is reached. If the downstream pressure is made still smaller, p 2 at the nozzle “throat” will not decrease because no signal from downstream can propagate upstream to influence the gas at the throat of the nozzle since the flow is supersonic; the mass flow through the nozzle is therefore fixed. A nozzle used in this way is called a sonic nozzle. It meters a flow independently of all conditions beyond the throat so long as the pressure is below the critical value. The values of (P2/pl)critfor air and the standard nozzle shape are 0.527, 0.532, 0.548, and 0.581 at values of m = 0, 0.2, 0.4, and 0.6. The factors a(1 - m2)l12 added to Eq. (1.2.92) give the flow through a sonic nozzle satisfactorily at m values 5 0.4, when the a values for incompressible nozzle flow, which are within ? 5% of 1.00 for the standard shaped are employed. A plate orifice or other nonstandard nozzle also shows choking, but the effective size of the throat due to the vena contracta and boundary layers is altered, and each needs to be calibrated for a given gas. Data on some conical and other nozzles are given in the l i t e r a t ~ r e . Impure ~~ gases containing condensible vapors J. Hengstenberg, B. Sturm, and 0. Winkler, eds., “Messen und Regeln in der chemischen Technik,” pp. 219-221, Springer-Verlag, Berlin and New York, 1957. 95 R. B. Dowdell, ed., “Flow-Its Measurement and Control in Science and Industry,” Vol. 1 , Part 1 , pp. 231-297. Instrument Society of America, Pittsburgh, Pennsylvania, 1974.

1.2.


33 I

may cause deposits to form on the nozzle and invalidate its calibration; the temperature of air at its sonic value, when expanding from 300 K, is 250 K.03 VENTURIMETER.Up to this point in Section 1.2.6.3, it has been implied that the constriction placed in the pipe for flowmetering is a rather short one, extending less than a pipe diameter along the pipe. This is the type of constriction that has been shown to be most practical. Some industrial installations employ gradually narrowing and gradually widening inserts, usually referred to as Venturi meters. A small wall pressure tap in the throat and another wall tap upstream before the pipe starts to taper are used to develop the working pressure difference. At certain flow rates, for which separation of the flow from the wall of the downstream widening part is avoided, a Venturi meter has a smaller pressure loss for the same working pressure than either an orifice plate or a nozzle. Usually this advantage does not justify the additional space in piping required, since the advantage disappears at high flow rates. 1.2.6.4. Variable-Area Meters; Float Meter. A precisely machined flanged body, in the interior of a precision tapered tube carrying the fluid to be metered and held vertical with the outward flare upward,Jloars at a steady height, which is a measure of the fluid flow rate. The fluid is passing upward through the small clearance between the flange of the body and the wall of the tapered tube, and, as in the case of the orifice meter described in Section 1.2.6.3, a pressure loss occurs over the small clearance opening. The pressure difference acts on the surface of the float so that, together with its weight, it is subject to zero net force and remains at rest. If the flow rate increases, the pressure loss across the same clearance area would increase and the body would rise; however, in rising the clearance area would increase, so that a larger pressure difference due to the higher flow rate would not exist. Since a net force either upward or downward will cause the body to change its position, it seeks that position at which the pressure loss produces a force which just balances its weight. Made with great precision in mass production assembly lines, these gages are supplied by instrument sales agents and chemical supply houses. They can be ordered from catalogs in various ranges and are supplied with calibration charts; they can easily be placed in service, usually by making connection with flexible tubing. They are easily read with 2-3% accuracy, and if restricted to use with completely dust and vapor-free gases, or liquids without entrained particulates, may continue to give flow rates accurate to this order. Actually, as anyone who has washed windows in his house knows, deposits from room air and most other gases are made on surfaces even though the gas appears to be clear. The clearances in the variable area meters are very small, and deposits on the flange of the float and on the walls of the tapered tube alter the clear-

332


ance between calibrations, and an instrument may be in error by more than 50% while appearing to be as clean as new. They can only be relied upon for accurate measurement when they are used in situations where experience with frequent calibrations has shown that they maintain such accuracy. They work equally well for liquids and gases. Variable-area meters are also constructed with variations, such as providing a central guide on which the float moves, and arranging for springs to supply a greater downward force than that due to the weight alone. Most commonly, the outer tapered tube is constructed of glass, and the height of the float is read by noting its position relative to a scale painted or etched on the outside of the tube. In other metal constructions, magnetic or inductive sensing of the position of the float is transmitted to an electrically operated indicator or re~order.’~ 1.2.6.5. Weir and Flume for Open Channel Liquid Metering. The methods of measuring the volume of a liquid flowing per second under its own weight in a canal or partially filled pipe are known to hydraulics engineers and civil engineers. As with orifice flow metering in a filled pipe, an idealized view of the way an incompressible, inviscid laminar fluid might behave is useful, but the departure of real flows from the ideal is even greater, and no rational and orderly way of combining correction factors has been developed and agreed upon. We will briefly outline the idealized picture, which helps to introduce the terminology, but then recommend use of a textbook or handbook for practical a n ~ w e r s . ~ ~ , ~ * Again the Bernoulli formula is used. In an open channel, a streamline lying in the surface separating the liquid from the air has a constant pressure; the term involving the weight, which was omitted from Eq. (1.2.2) in Section 1.2.2.1 because it was taken into account as a hydrostatic correction to the pressure, is no longer omitted. However, since there is a constant (atmospheric) pressure on the surface, by measuring pressures relarive to armospheric, the pressure at the surface is zero. Equation (1.2.1) of Section 1.2.2.1 thus becomes, on a streamline below the surface, (p/p)

+ $ u z + gz = const,

where z is the vertical coordinate of any point on the streamline where the pressure is p and the fluid speed is u . In hydraulics, it is customary to divide the equation by g and to use the symbol y for the vertical coordinate K . W. Bonfig, “Technische Durchflussmessung.” p. 76. Vulkan-Verlag, Essen, 1977. Measurement Structures.” International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands, 1976. H. W. King and E. F. Brater, “Handbook of Hydraulics,” 5th ed. McGraw-Hill, New York. 1963. OB

@’ M. G. Bos, ed., “Discharge

1.2.


333

ofrhe surfuce. The idealized flow is also assumed to be irrotational (i.e.,

potential flow) so that the same constant pertains to every streamline, and Bernoulli’s formula becomes [(Plpg) +

ZI

+ v2/2g

=y

+ u2/2g

=

y,,

(1.2.94)

where y , referred to as the head* of the fluid, has dimensions of length, and Bernoulli’s formula is now read, “head + velocity head = total head.” At a point below the surface, where the pressure p is not zero, the quantity p/(pg) + z is referred to as that point’s piezomerric head. Note that Eq. (1.2.94)says that where the speed is large, the surface is depressed, and also note that the reference level from which head is measured is not specified. Now consider steady uniform flow of a liquid in a long straight level channel with uniform rectangular cross section of breadth b. Let the head be measured from the bottom of the level channel. The principle of conservation of matter states that in the steady state, the volume flow at any point, Q = uyb, is the same as at any other point since no matter can collect in time between the two points. For a given ul and y l , from Eq. (1.2.94)y , has a given value; if u = Q/(yb) is substituted into Eq. (1.2.94), we obtain yt = y

+ Q2/2gb2y2.

(1.2.95)

This is a cubic equation in y if Q, 6 , and y , are fixed, and in general it has two roots, i.e., another pair of values u, and y , also can provide the same Q and y , as the original u1 and y l . One pair is a large v and a small y , and the other pair is a small 21 and a large y . The flow in a given channel can be either subcritical for the small u (tranquil) or supercritical for the large u (shooting), and it can transform from one form to the other. For a given channel and a given Q ,other pairs of values u and y can exist, but they correspond to other values of y,. Not all values of y, are possible because the cubic equation [Eq. (1.2.991for too small a value of yt will have no solution whatever. There is a critical value of y, for which Eq. (1.2.95) has only one solution. The Q ,b, and yt for which this solution prevails specify a critical flow with only one y and one u. This set is such that l.l

= (gY)”2,

(1.2.96)

a relation which can be derived by differentiating Eq. (1.2.95) and setting

* The symbol y is also often referred to as energy or heud energy in hydraulics. It is gravitational potential energy per unit weight, and Eq. (1.2.94)has the appearance of a conservation equation. Its relation to energy in the thermodynamic sense is tenuous.

334

1.


the derivative dy,/dy equal to zero. It is also interesting to note that in a level tray of nonflowing fluid, U = (gy)”2 is the speed of a small gravity wave relative to the fluid,99where y is the height of water in the tray. If we now consider the long straight level channel to have a slowly varying breadth, and inquire how u and y, for a given flow rate Q , will adjust themselves, continuing to satisfy Eqs. (1.2.94) and (1.2.95),*we will see the condition that is sought in a tlowmeter. Note that Eq. (1.2.94) does not involve the breadth b . For a given Q , as b changes, the Eq. (1.2.95) condition for a single y, i.e., for critical flow, will arise even though yt does not change. If b were to continue to get smaller, no steady solution could exist, The aim in flowmetering is to arrange for critical flow to be achieved in a channel at a constriction, whereupon Q can be determined by only ( 1 ) the geometry of the channel and constriction and (2) measurement of the height of the surface upstream. What actually happens is that the liquid backs up in a transient, increasing y t , until critical flow at the constriction exists, after which a steady state prevails and the critical nature of the flow at the constriction assures that no change downstream of the constriction will affect the flow upstream. This can be understood in terms of waves; no wave from downstream can move upstream through the constriction where the liquid speed is equal to the wave speed. To achieve this, the flow after the constriction, where b increases again, needs to be supercritical. This is arranged by allowing the liquid to fall freely (waterfall), or to shoot out to a lower channel where it reverts to subcritical flow at a hydraulic jump. The constriction and subsequent lower channel is called a weir if a waterfall is produced, and a flumeif the side walls narrow and the channel slopes downward gradually after the constriction. One finds that if he considers steady flow in a channel whose cross section is gradually reduced by raising the bottom of the channel, keeping the breadth b constant, then the same critical value u, at the same cross sectional area A, results. The height of the liquid surface above the former reference level is, at the critical cross section, the same, and the same value of the total head yt pertains. This situation is arranged for in practice by inserting a weir block in the channel with a level upper surface, which becomes the bottom surface of the critical cross section. It is called a broad crested weir. Since the Bernoulli formula [Eq. (1.2.94)] may have any choice of reference level, and the total volume of flow Q is L. D. Landau and E. M. Lifshitz, “Fluid Mechanics,” paragraph 13. Pergamon, Oxford, 1951.

* Even the idealized conditions of inviscid, irrotational flow would not permit uniform velocity over a cross section to exist if there were an abrupt change in breadth b.

1.2.


335

uJA, = vcycb if y is measured from the weir crest, the new reference level is taken at the weir crest. Then, calling the total head above the weir crest H , Eq. (1.2.94) becomes at the weir

H Using Q

=

v,y,b and v,

=

yc

+ vf/2g.

= (gy,)1/2, we

Q

=

have

g+g)1'2bH3'2.

If we measure the liquid surface height hl upstream where the velocity head ufl(2g) is small so that the head above the level of the weir crest is essentially the total head H , we can compute the volume flow Q . This idealized formula is valid regardless of whether the channel upstream has the width b or not. The control section should have a constant width b and a level base over a long enough distance so that the flow streamlines are parallel and the velocity uniform. The broad crested weir has been described because it seems to have best met the idealizations of (i) one-dimensional flow (no curved streamlines), (ii) irrotational flow (no deviation from uniform velocity profile), and (iii) no shear stresses and no turbulence. In fact, these conditions are not met, and actual calibrations show that the effects of departures must be accounted for by a discharge coefficient CDwhich is tabulated and graphed. When the condition for no curved streamlines in the control section is best met, C D= 0.848. Another correction for neglecting the velocity head in the approach channel C , can be computed from the ratio of areas at the approach and control section. The working equation is, finally, (1.2.97) where hl is the measured head in the approach section, i.e., the height of the surface above the weir crest. Where a high degree of accuracy is desired in laboratory work, standard specifications for precisely constructed sharp-crested weirs are usually employed, and calibration charts peculiar to the particular installation are available. The flow separates from the sharp edge, and provision must be made to aerate the pocket that forms under the falling sheet of liquid. For a specified geometric shape of the control section with a weir height P above the channel floor, the dimensional quantities in the accompanying tabulation are involved in open channel metering. These are seven variables, and dimensional analysis (see Part 10) says that four dimensionless numbers have a functional relationship for the particular geome-

336 Q/b

hl h, g p

p

1.


Volume flow rate per unit width Height of upstream water surface relative to weir crest Elevation of downstream water surface relative to weir crest Weight per unit mass Mass density of fluid Fluid viscosity

try. Led by the idealized relations leading to Eq. (1.2.97),it is common in hydraulics to write

The first of the dimensionless variables on the right is a Reynolds number; the second is the drowning ratio and is ignored if critical flow is known to exist at the crest, and the waterfall is aerated in the case of a sharp-edged weir; and the third takes into account the flow geometry. These elementary relations may be useful to keep in mind when exploring the hydraulics literature, but many more than the two coefficients CDand C, in Eq. (1.2.97) are used and considered as functions of variable combinations other than those listed in Eq. (1.2.98). Only rectangular control sections and horizontal weirs have been mentioned in the preceding discussion. A partially filled circular conduit, or a V-notched weir, for example, require additional consideration. Equation (1.2.97) is useful for rough metering of an open rectangular channel liquid flow in a laboratory, however. 1.2.6.6. Flowmetering by a Bundle of Capillaries. Laminar steady flow in a straight pipe is governed by shear stresses alone at Reynolds numbers pOD/p < 2000. The volume flow rate Q is related to pipe diameter D, dynamic viscosity coefficient p, and the shear stress at the wall expressed in terms of the pressure drop Ap over a length of pipe L as r = Ap D/(4L). The relation is Poiseuille's formula

Q = (?rD3/32p)r= ?r Ap D4/128pL.

(1.2.99)

While in principle this looks like a very convenient method of metering a fluid of known viscosity-to merely measure the pressure drop Ap over a length L in a pipe of known diameter-practically, it is seldom convenient or accurate. For any but very small capillaries, the mean flow speed t, is very small when Re < 2000, so Q is very small. The Poiseuille formula does not apply in the ends of pipes-end effects are still quite prominent at fifty diameters in from an end. Tubing with a preci-

1.2.


337

sion bore is difficult to manufacture and is quite expensive, and the percentage accuracy in D must be small because Q depends on D4. Flowmeters have been made by bundling thousands of small capillary tubes in a close packed arrangement and measuring the pressure drop. The very small openings between the exteriors of the tubes, combined with the end corrections, lead to deviations from the Poiseuille formula that can be taken into account.100 One big advantage of this flowmeter is its linear relation between Ap and Q. Its chief disadvantage is the ease with which entrained solids block the small capillaries and invalidate the calibration. 1.2.6.7. Acoustic and Electromagnetic Flowmeters. Both these flowmeter types are attractive in that they can often be installed without extensive additions to piping, and under adverse conditions. Electromagnetic flowmeters are quite reliable and accurate in installations where conditions are predictable. Both types respond rapidly and are sensitive to changes in a flow pattern or flow rate and can be used for monitoring and control. The principles on which they work have been outlined in Section 1.2.5, and here we just indicate how they are arranged for flowmetering. ULTRASONIC FLOWMETER. All practical acoustic flowmeters use ceramic or quartz crystal piezoelectric senders and receivers, arranged in one of three ways: pulse timing, beam deflection, or Doppler reflection from entrained scatterers. These are schematically diagrammed in Fig. 25. The simplest, in Fig. 25a, uses just one piezoelement on each side of the channel; a pulse is timed as sent from sender # I and received on receiver X1, and then the roles of sender and receiver are reversed as a pulse is timed in the sense opposing the flow. If the velocity across the channel is fairly uniform with an average 0,the measurements can be used to find the flow rate from the cross sectional area and

(1.2. loo) where a is the angle between the sonic beam and the flow direction, and d is the separation of the sender and receiver. Equation (1.2.100) is derived from Eq. (1.2.77) of Section 1.2.5.1 under the assumption that the ratio of fluid speed to sound speed a / c > 1). The grating interferometer, as originated by K r a ~ s h a a ruses , ~ ~ a diffraction grating as the interferometer unit, and it corresponds to a ratio of d / D = 1 . The principal arrangement is similar to that of a schlieren system with parallel light through the test object (Fig. 22). A diffraction grating placed in the focal point of the first lens or spherical mirror M 1separates the incident light into several diffraction orders. Two orders, say, s* W. Kinder, Optik 1, 413 (1946).

U. Grigull and H. Rottenkolber, J. Opr. SOC. A m . 57, 149 (1967). R. Kraushaar, J. Opr. SOC.Am. 40,480 (1950). The original concept was published by C. Barus, Carnegie Ins?. Washington. Pub/. 149, I(191 I), II(1912), lII(1914). 40

2.4.

381

INTERFEROMETRY test object test beom

/

sourcie grating I

'\

/

comero

I

film

grating 2

reference beam

FIG.22. Diffraction grating interferometer. Numbers indicate diffraction orders.

zero and one, are collimated by M , to propagate as parallel beams behind MI. The transparent test object is placed into one of the parallel beams whose action is that of the test beam, while the second remains undisturbed and plays the role of the reference beam. The lens (or schlieren head) M 2refocuses the two beams onto a second grating, where these beams are again separated into several diffraction orders. The second grating is aligned so as to provide that the ith diffraction order of the test beam overlaps with the (i + 1)th order of the reference beam. Interference is established between the light of the overlapping orders if the conditions of optical coherence are fulfilled. It is obvious that not only one but a series of identical and somewhat overlapping interferograms appear, from which the most intense interference pattern must be selected by means of an appropriate aperture. This grating interferometer is simple in construction, but it is associated with a strong loss of light intensity, since only two diffraction orders are used at each grating. By a suitable choice of the grating characteristics, the distribution of the diffracted light intensity can be such that a great portion of the total intensity is concentrated in one or two single orders.41 By utilizing a laser light source, one can abandon the first diffraction grating, thus achieving a gain in light intensity. Holographic i n t e r f e r ~ m e t r yis~ ~another way of producing reference beam interferograms. The spatial separation d between test and reference beam in a conventional interferometer is replaced here by a separation in time. Holographic interferograms usually are taken by means of a double exposure on the same holographic plate. One exposure is made in the absence of the flow in the test facility and designates the reference beam. The second exposure is made in the presence of the compressible flow and constitutes the test beam. Upon illumination of the double exposed hologram with the reconstruction light of the holographic system, 'I

'*

A. R. Maddox and R. C. Binder, Appl. Opt. 8,2191 (1969). L. 0. Heflinger and R. F. Wuerker, J . Appl. Phys. 37, 642 (1%6).

2.

382

DENSITY SENSITIVE FLOW VISUALIZATION

one reproduces a light wave pattern which is the superposition of the two light waves of the individual recordings. The principal difference between a conventional and a holographic interferometer is, therefore, that the two interfering beams-test and reference beam-exist simultaneously but are spatially separated in the former case, whereas they coincide in space but are separated in time in the case of a holographic interferometer. Of the great number of holographic interferometers described in the literature for use in flow studies, only a few representative publications can be listed It appears that holographic interferometers are gradually taking over the role which Mach-Zehnder interferometers have played in experimental fluid dynamics. There are two factors which make the new technique superior to classical interferometers. The first factor is that optical disturbances resulting from imperfections of test chamber windows are eliminated due to the spatial coincidence of the two beams. For the same reason, it is possible to observe the flow through test models made of a transparent material and so to measure the fluid density in corner flow regimes, which normally are not accessible to investigations with conventional interferometric methods.4e The second factor is that a holographic interferogram made with diffuse light can be observed under different viewing directions. If the interferogram is obtained from a three-dimensional density field, the observed fringe pattern depends on the viewing direction.50 Thus, the holographic interferogram contains information on the three-dimensional nature of the flow field. This information can be decoded with the aid of appropriate evaluation procedures5' (see Chapter 2.5). For the purpose of completeness, it is worth mentioning two more methods which produce reference beam type interferograms, but which are only occasionally used for experimental flow studies. The radiul shear or aperture reducing interfer~rneter~~ (Fig. 2 3 ) uses a special beam splitter which separates a certain portion of light from the principal beam. L. H. Tanner, J . Sci. Instrum. 43, 81 (1966). H. J. Raterink and C. W. Lamberts, Proc. I n t . Congr. High-speed Photogr., 9th, 1970 p. 30 (1970). ld

W. Aung and R. O'Regan, R e v . Sci. Instrum. 42, 1755 (1971). A. B. Witte, J. Fox, and H. Rungaldier, AIAA J . 10, 481 (1972). P. Smigielski, A. Hirth, and C. Thery, IEEE Trans. Aerosp. Electron. Syst. aes-8, 751

(1972).

J. Delery, J. Surget, and J.-P. Lacharme, Rech. AProsp. No. 1977-2, p. 89 (1977). A. G. Havener, AIAA J . 15, 592 (1977). A. B. Witte and R. F. Wuerker, AIAA J . 8, 581 (1970). R. D. Matulka and D. J. Collins, J . Appl. Phys. 42, 1109 (1971). I* L. H. Tanner, J . Sci. Instrum. 43,878 (1966).

2.4. INTERFEROMETRY

3 83

reference beam

Object object

source

-_ I

special beam splitter splitter beam

beam expander

FIG.23. Idealized radial shear interferometer. Beam splitter and beam expander only affect the light of the reference beam.

The separated light traverses in the form of a beam of extremely reduced diameter and very small angular spread through the test section and acts as the reference beam, while the wide principal beam is the test beam in this arrangement. An appropriate apparatus provides that the reference beam is expanded after the test section to overlap and interfere with the light of the test beam. Tanners2has shown that the reference beam diameter can be made extremely small with the aid of a laser light source. It is of course very useful to find for the reference beam a position in the test field where the fluid density varies only weakly or not at all. If it becomes essential to have the reference beam completely separated from the test beam, this interferometer is rather of the Mach-Zehnder type, as described in Fig. 21. The second arrangement to be mentioned here is Erdmann's field absorption method, which exhibits a very high sensitivSince it is more convenient to describe this principle in connection with the phase contrast method, the reader is referred to Section 2.4.3. 2.4.2. Shearing Interferometers

In this two-beam interferometer, both light beams traverse through the test object, where they are separated or sheared by a small lateral distance d. In contrast to the aforementioned radial shear interferometer^,^^ these instruments are named lateral shearing interferometers. With D being the diameter of the field of view or of the test object, only such cases are of interest here which are described by a ratio d / D r > r,-l, can be derived by linear interpolation between the respective values R(r,) and R(r,-l). Complete computer programs are now available for performing the described procedure which applies, as shown, equally well to reference beam and shearing interferograms. Unfortunately, such computer programs are only contained in special institute report^.^^,^^ The accuracy of the method increases, of course, with the value of N. Particular difficulties arise if the test field contains conical or cylindrical discontinuity surfaces, e.g., shock waves. In this case, one has to split the data function into a singular portion which accounts for the discontinuous density jump across such surface, and into a "reduced" portion describing the continuous density change. This method has been described by Wey1,'O and has been successfully applied in many cases where the object was to study the supersonic flow around a sphere or a circular c ~ n e . ~ O * ~ ~ 2.5.2. Three-Dimensional Fields The principal idea of decoding the two-dimensional information (interferogram) obtained from a three-dimensional density field was already developed by Schardin.15 As mentioned earlier, such procedures always require taking several interferograms of the object, each under a different viewing direction. Recently considerable progress in this field has been made, owing to the availability of holographic interferometry and of appropriate computer techniques. The principle underlying all evaluation procedures will be explained with a simplified analogy. Assume that the object can be subdivided into N segments (Fig. 32), and that the density function R of Eq. (2.5.1)is constant within each segment, say R = Ri in the ith segment. The problem is then to determine N unknown parameters R1 . . . RN, which requires defining and solving a set of N equations. A light ray which traverses the test object carries information on the Ri values of n segments, with n C N. This information is expressed "

L. Oudin and M. Jeanmaire, Report 21/70. Institut Franco-Allemand de Recherches,

St. Louis, France, 1970.

2.5.

EVALUATION PROCEDURES

397

FIG. 32. Subdivision of a three-dimensional object into segments with constant values R, of the density function R . The information obtained from the six light rays shown in the figure delivers six equations for the evaluation procedure (see text).

by a value D, of the data function D(x, y). Equation (2.5.1) written for this one light ray is then just one equation of the aforementioned set of N equations. For this simplified discussion it might be assumed that the integral in (2.5.1) can be approximated by a summation, so that the first equation of the system reads m

D1= const

Risi, i=

1

where m is the number of segments through which light ray no. 1 passes in the object, and si is the length covered by the light ray in the ith segment. Additional equations arise in expressing the paths of other parallel light rays which traverse the object at different positions. From Fig. 32 one may conclude that one beam of parallel light rays cannot yield sufficient information to complete the system of N equations. It is therefore necessary to utilize the information from other light rays which pass the test object in a different direction, i.e., to measure the test field with the interferometer under different viewing directions. In the case of an arbitrary three-dimensional density field, it is necessary to observe the object over an angular range of 180". This value and the number of necessary viewing directions decrease if the object field has some kind of symmetry. This means, in terms of the presented simplified description, that some segments have the same Ri values, so that the number df unknowns and the number of required equations are reduced. The major problems in performing such three-dimensional evaluation procedures are finding an appropriate subdivision of the object field into segments, replacing the integral in Eq. (2.5.1) by a summation or series of suitable analytic functions, and solving the respective set of equations. The precision of the method increases with the number N of segments, but at the same time increases the complexity of the computational process. The choice of N will therefore be a compromise between the de-

398

2.


sired accuracy and the capacity of the computing system. One usually starts the procedure with the equation for a light ray which traverses close to the edge of the test field. An optimum situation is achieved if each new equation introduces only one new coordinate into the system of equations. This had been the case for the procedure applied to the evaluation of axisymmetric test fields. Finally, it should be emphasized that Fig. 32 is only a two-dimensional representation of the actual situation. The evaluation of three-dimensional object fields appears to be the most important problem to be solved for the interferometric testing methods; and only when a reliable and feasible solution of this problem is found will optical methods be generally superior to the testing methods using probes. Several different approaches have been described recently. Matulka and Collins75use a set of orthogonal functions to represent the density function R in the integral of Eq. (2.5.1). The equation becomes integrable, since the applied series of functions is subject to an integral inversion, and the unknown coefficients of the expansion of R are found from the orthogonality relation. Sweeney and Vest76describe a transformation of the integral by means of Fourier transforms, which also allow for a direct inversion of the integral. The procedure proposed by Belotserkov~ky'~ is most similar to the simplified model described in the beginning of this subsection, i.e., the object field is subdivided into segments of constant density values. A way of testing the accuracy of such methods is to prescribe a certain density field, calculate the resulting pattern of interference fringes, and use this pattern for reconstructing the density field by means of the appropriate evaluation procedure.

2.6. Radiation Emission The optical visualization methods which make use of the refractive behavior of the gas flow to be studied exhibit a certain sensitivity limit if the average level of the gas density becomes too low. It is in this range of low-density or rarefied gas flows that a visualization of the flow can be achieved by making use of the radiative characteristics of the gas. By means of an appropriate energy release, the molecules of the flowing gas are excited to emit a characteristic radiation. The intensity of this radiation increases with the value of the local gas density, so that it becomes

I6

R. D. Matulka and D. J . Collins, J . Appl. Phys. 42, 1 1 0 9 (1971). D. W. Sweeney and C. M. Vest, Appl. Opr. 12, 2649 (1973). M. Belotserkovsky, Proc. Int. Congr. High-speed Phorogr., 8th. 1968 p. 410 (1%9).

2.6.

3 99

RADIATION EMISSION

possible to detect regimes of an elevated density level, e.g., behind a shock wave. The emitted radiation also includes information on other gas parameters, particularly on temperature, which can be evaluated quantitatively. This is discussed in Sections 3B and 4B, and the present aim is only to demonstrate the possibility of extending density sensitive flow visualization methods into the regime of low density gas flows. 2.6.1. Electron Beam Flow Visualization* A narrow beam of high energy electrons traverses the gas flow under study; owing to inelastic collisions between the fast electrons and the gas molecules, some gas molecules are excited and subsequently return to the ground state with emission of a characteristic radiation. The light emission can be prompt, or it may occur from an excited metastable state. The prompt radiation is emitted more or less at the same place where the gas is excited, i.e., at the position of the electron beam in the flow. The electron beam appears, therefore, as a column of bright fluorescent light, which is often called ajluorescence probe. Under certain conditions, the intensity of the direct radiation is proportional to the local gas density. If one moves the electron beam with constant speed in a particular plane through the gas flow, one obtains a representation of the density distribution in this plane by taking a photographic time exposure while the beam is moving. On the other hand, the lifetime of an excited metastable state is relatively long. The transition into the ground state takes place after the molecule is swept a certain distance by the flow. The associated radiation is emitted at some point in the flow downstream of the original beam position and is called the afterglow radiation. The luminescence of the afterglow radiation is also appropriate for visualizing density changes in the gas flow. It is then not required to move the electron beam, but the intensity of this radiation is much smaller than that of the direct radiation. Beyond the application for pure flow visualization, the electron beam technique can be used for quantitative temperature and density measurements if it is combined with spectroscopic analysis of the electron beam radiation.'* The intensities of a single line or a band in the radiation spectrum are proportional to the number density of the test gas particles, the factor of proportionality depending on both vibrational and rotational

'* E. P. Muntz, The electron beam fluorescence technique. AGARDogruph 132 (I%@. See also section 3C. * See also L. Marton, D. C. Schubert, and S. R . Mielczarek, Natl. Monogr. 66 (1963).

Bur. Srund.

(U.S.),

400

2.


temperature of the gas. The measurement of line and band intensities allows one, therefore, to determine vibrational and rotational temperatures, and the concentration rates or partial densities of the active gas species as well. The discussion of this section, however, is restricted to the sole purpose of producing density sensitive pictures of a rarefied gas flow. The test gas most studied is of course air, but only the interaction between fast electrons and nitrogen molecules accounts for the visualization of air flows by means of the fluorescence probe. Most of the N2molecules undergoing a collision are ionized and simultaneously excited; the Provided that the kinetic energy resulting state may be denoted by N:*. of the electrons is high enough, the most preferred transition is to a level 18.7 eV above the ground level of N2. The predominant subsequent emission is caused by a spontaneous transition to a level 3.1 eV below the N:* state, equivalent to the first negative emission system of Nt. The most intense radiation of this transition is the (0, 0) band at a wavelength of 3914 A. A first-order analysis shows that the intensity of the radiation emitted per unit length of the electron beam and at constant electric current of the beam is proportional to the number density of the gas molecules in the respective beam section. The proportionality factor is of the type of a collision cross section. Such analysis, however, suffers from several simplifying assumptions, Collision cross sections for all possible transitions which contribute to the total radiation are not known; the measured radiation also contains contributions from collisions of the gas particles with secondary electrons, while the theory cannot account for those electrons which excite metastable states and 'do not contribute to the direct radiation. These and other error sources, e.g., beam broadening, electron scattering, and quenching collisions, increase at higher gas densities, so that the electron beam technique must be restricted to the investigation of low density gas flows. Electron beam flow visualization will remain a qualitative method, allowing one just to discriminate between regimes of reduced or increased gas density. A thin and narrow electron beam of about 1 mm in diameter has to be produced by an appropriate electron gun. Usual values for voltage and current are 20 kV and 1 mA. The test chamber of the wind tunnel and the attached electron gun form one evacuated system. The beam can be moved either mechanically parallel to itself7e or by means of deflection coils to cover a certain angular sector.8o The speed of motion depends on D.E. Rothe, AIAA J . 3, 1945 (1965). S. Lewy, Rech. Aerosp. No. 1970-3, p 155 (1970).

2.6.

RADIATION EMISSION

40 I

FIG. 33. Supersonic low density flow over a spherical test model as visualized by the electron beam technique. Direction of the moving electron beam is from above to below. A shadow is therefore seen below the sphere. [Courtesy of Dr. S. Lewy, ONERA, Chltillon, France.]

the available test time of the wind tunnel flow. Facilities producing a stationary flow allow a slow movement of the beam, and exposure times up to 60 s have been used. In order to prevent the production of secondary electrons, the electron beam must be received by a graphite target. Test models in the wind tunnel should be metallic to avoid fluorescence from body surfaces, and the models should be connected to the ground so that no electric charges are built up at these bodies. The direct radiation allows one to visualize supersonic flow fields at a density level which is one or two orders of magnitude below the sensitivity limit of a schlieren system (Fig. 33). The excitation of metastable states in the test gas can cause a noticeable afterglow radiation downstream of the electron beam. This afterglow is less intense than the direct radiation, and only cold flows of nitrogen and argon and mixtures of nitrogen and noble gases yield an afterglow which is intense enough for taking flow pictures. Flow regimes with an increased gas density can be discriminated due to a more intense afterglow radiation. The afterglow disappears almost completely in air due to inelastic collisions (quenching) between excited N2 molecules and nonexcited O2molecules. The mechanism of the transition from the metastable states and the associated emission of radiation is not yet fully understood in this case, and no analysis is available for deriving quantitative data

402

2.


from the flow pictures. An additional difficulty in interpreting the visualized pattern is that the intensity of the afterglow radiation decreases with increasing distance from the electron beam. The sole advantage of this method is that it is not necessary to move the electron beam through the flow field under study. 2.6.2. Glow Discharge

The electric discharge in gases at low pressures is accompanied by the emission of light. Since the intensity of this radiation depends on the density of the gas in the control volume, one may adapt this method to the visualization of rarefied gas flows. The processes in the glow discharge are similar to those of the electron beam technique. Free electrons and ions which are in the test volume are accelerated by the external electric field and can produce a cascade of secondary electrons and ions due to collisions with neutral gas molecules. The primary and secondary electrons and ions excite gas molecules which subsequently emit radiation upon spontaneous transition into the ground state. This radiating regime in the electric discharge is called the positive column. The emission intensity of the positive column is a function of the gas density. In a certain density range, the emitted light intensity increases with the number of exciting collisions and therefore with the level of the gas density. This, however, holds only up to a particular value of the gas density where the free path length of the electrons becomes too small, and the electrons gain insufficient energy between collisions for excitation. This useful range of radiation is usually at values of about of the density at normal conditions. In order to visualize the compressible flow in a low density wind tunnel, the test model is made one of the electrodes for the discharge, and a certain portion of the wind tunnel wall may serve as the second electrode. By a suitable choice of the geometry of the electrodes, the field of the positive column can be varied so as to cover the desired portion of the flow field. The potential required between the electrodes depends on the test gas. Appropriate voltages are 1000 V for air or nitrogen and 300 V for helium flows. In order to obtain a uniform luminosity of the positive column, one uses an ac rather than a dc voltage. With the voltage applied between the electrodes, the flowing and radiating gas can be observed or photographed. Density changes in the flow appear as a change in intensity and sometimes in color of the emitted radiation. This method has been applied to visualizing flows of nitrogene1and of helium.@ Air is not W. J. McCroskey, S. M. Bogdonoff, and J . G . McDougall, AIAA J . 4, 1580 (1966). C. C. Horstman and M. I. Kussoy, AIAA J . 6,2364 (1%8).

2.6.

RADIATION EMISSION

403

very appropriate for study with the positive column, since the electric discharge in air is followed by a great degree of afterglow radiation. The origin of this afterglow can be the excitation of metastable states, as in the case of the electron beam method. Additionally, the afterglow can be caused by slow chemical reactions between different chemical constituents of gases with the associated emission of radiation. Such chemiluminescence can again be a means for flow visualization.


AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that an author's work is referred t o although his name is not cited in the text.

A

Balint, E., 46cff), 48 Ballik, E. A., 131 Barber, N. F., 34 Barker, S. J., 173, 175(284) Basset, A. B., 8, 42 Bateman, H., 13(r), 14 Bauer, N., 19 Beams, J. W., 361 Bearman, P. W., 314 Beer, J. M., 72, 78 Beguier, C., 312 Bell, H. S., 58(z), 59 Belotserkovsky, M.,398 Belousov, P. Ya., 214, 217(369) Bendat, J. S., 230(e), 231 Bendt, P. J., 85, 87 Benedek, G. B., 343 Bennett, F. D., 377, 394, 3%(71) Bensen, R. S., 272 Benson, G. M., 16, 22(y), 23,74, 85 Berg, R. H., 22(nn), 24, 25 Bergdolt, V. E., 394, 396(71) Berkman, R. M., 53, 54(127) Berman, N . S., 31, 58(bb), 59 Berry, R. G., 70(h), 71 Bershader, D., 351, 379 Betchov, R.,305 Beyer, G. L., 17 Binark, H., 22(cc), 24 Binder, R. C., 368, 381 Bippes, H.,67 Birch, A. D., 184-186, 230(h), 231 Birchenough, A., 124 Birkhoff, G., 31, 46(q), 47(9), 48, 58(k), 59,

Abbiss, J. B., 186, 193 Abrams, F. R., 143, 146(261) Acrivos, A., 13(s), 14, 46(o), 47 Adler, C. R.,22(yq), 24, 27 Adrian, R. J., 170 Agrawal, J. K., 27 Ahmadi, G., 38 Akers, A. B., 26 Alcaraz, A., 309, 31066) Alford, W. C., 46(Rg), 48 Ali, S. F., 313 Alkhimov, A. P., 199, 203(354-356), 204(354,355), 205(355,356), 206,208, 210(355,365), 222, 225 Allen, R. M., 80 Allen, T., 16(27), 17, 18(27), 22(y), 24, 26. 86(27) Alpher, R. A,, 350 Altman, D., 33, 62 Anderson, D. M., 27 Anderson, J. H. B., 350 Andrade, E. N. da C., 31, 71, 75 Asher, J. A., 46(n), 47, 164, 180, 230(g), 23 1 Aspden, R., 74 Astier, M., 312 Atkinson, B., 74, 88(179) Aung, W., 382 Avidor, J. M., 199, 217(360), 226

B

88

Baker, J. L. L., 13, 14 Baker, R. C., 321 Baker, R. J . , 141

Black, W. Z., 387 Blackwelder, R. F., 313 Blake, K. A., 175, 192, 193 I

2

AUTHOR INDEX

Blanchard, D. C., 49 Blank, E. W., 19 Blick, E. F., 74 Bogdonoff, S . M.,402 Bolko, V. M.,199, 203(356), 205(356) Bonfig, K. W., 258, 315, 321, 322, 324, 327, 332, 339, 340(101)

Born, M.,110, 198(230), 199(230) Bossel, H. H., 143, 164, 192, 193, 233 Bourke, P. J., 104, 230(a), 231 Bourot, J. M.,75 Boussinesq, J., 8 Bouyer, R., 389 Boyce, M.P., 74 Bradley, D. J., 197, 199(350) Bradshaw, P., 230(d), 231 Brandt, 0.. 31 Brater, E. F., 332 Brayton, D. B., 22(rr), 24, 124, 180, 235 Brenner, H., 12(d), 13(d), 14, 38, 42, 43 Breslin, J. A., 46Cf). 47, 58(q), 59,72,75,78 Brinkman, H. C., 12(i), 14 Brodkey, R. S . , 34, 78 Brodowicz, K., 46(m), 47, 74, 76(180), 78 Browand, F. K., 280, 281, 290, 298 Brown, C. G., 104 Browning, J. A., 26, 44 Brundrett, G. W., 272 Brush, L. M.,33, 42 Bryer, D. W.,250 Bryngdahl, O., 387 Buchhave, P., 5, 6, 119, 141, 191 Buhrer, C. F., 188 Bures, J., 209 Burgers, J. M.,277 Burton, E. F., 39, 40 Buzzard, R. D., 368 C Cadle, R. D., 16(25), 17, 18(25), 22(aa), 24, 26, 27, 74, 78

CafTyn, J. E., 47(r), 48, 58(1), 59, 67 Callis, C. C., 22G), 24 Callis, C. F., 6(3c), 7, 16(3c), 18(3c) Calvert, J. R.,308 Campbell, N. P., 82 C a m , M.W. P., 69 Carlson, A. B., 98

Carlson, D. J., 12(a), 14 Carlson, F. E., 69, 74 Cam, W. W., 387 Carslaw, H. S., 282 Carter, W. C., 394, 396(71) Cassel, H. M.,21, 22(dd), 24 Cattolica, R.,343 Caywood, T. E., 31, 46(q), 47(q), 48, 58(k), 59, 88

Chaf€ey, C. E., 43 Chaikovskii, A. P., 222 Champagne, F. H., 305, 306, 308, 309, 31 1 Chan, J. H. C., 131 Chaney, A. L.. 78 Chao, B. T., 13, 14, 33, 35, 36(69), 38 Charman, W. N., 22(s), 23 Chartier, C., 67-69, 70(&), 71, 389 Chen, C. H. P., 313 Chen, C. J., 33, 46(e), 47, 58@), 59, 72, 74(166), 75

Chevalerias, R.,387 Chevray, R.,303, 313 Chiba, T., 141 Chigier, N. A., 72, 78 Choudhury, A. P. R.,22(11), 24 Chuang, S. C., 23(zz), 24, 38 Chue, S. H., 248-252, 254 Clark, C. N., 69, 74 Cline, V. A., 180, 186 Clutter, D. W., 46(kk), 48 Cole, M.,22(r), 23 Collins, D. J., 382, 398 Collis, D. C., 269, 271 Compte-Bellot, G., 303, 309-31 1 Connes, P., 195, 199(347) Considine, D. M.,324 Cooper, E. E., 31 Cooper, K. D., 83 Corcoran, V. J., 119 Corino, E. R.,34, 78 Corrsin, S., 35, 270, 280, 305, 308, 314 Corson, D., 319 Cox, R. G., 42 Crawford, B. H., 69 Crosswy, F. L., 180, 193, 194 Crowe, C. T., 12(b), 14 Curnmins, H. Z., 34, 97, 107, 109, 116, 168, 195

Curtis, A. S. G., 22(i), 23 Cushing, V., 320

AUTHOR INDEX

D

Dandliker, R., 167, 189, 191 Dahm, M.,309 Dallavalle, J. M.,17, 22(z), 24, 27, 46(c), 47

Dautrebande, L., 46(gg), 48 Davies, C. N., 11, 12(n), 13(n, w ) , 14, 20 Davies, P. 0. A. L., 305 Davies, R., 22(00), 23(00), 24, 25 Davis, D. T., 162, 235 Davis, G. E., 56 Davis, L., 33, 62 Davis, M. R., 301 Davis, R. E., 13(s), 14, 46(d), 47 Davis, W., 33, 46(r), 48 Daws, L. F., 67, 72(153), 78, 84 Dawson, J. B., 46(cc), 48 DeCorso, S. M.,22(d), 23, 27, 78 Deighton, M. 0.. 174 Deirmendjian, D., 52, 53( 123), 56 Dejager, D., 93 de Lange, 0. E., 107, 116(217), 173 D&ry, J., 382 Denison, E. B., 31, 191 Dennis, R., 27 Deterding, J. H., 58(rn), 59, 72, 74, 75, 78, 84

DeVelis, J. B., 226s). 24 Diamant, L. M.,208 di Giovanni, P. R.,42 Dimmock, N. A.. 49 Dodd, E. E., 22(m), 23 Dombrowski, N., 17, 22(c), 23, 26, 27, 44(31), 45, 74, 77(115), 78 Donegan, J. J., 34 Doolittle. A. K., Sl(dd), 59 Dowling, J. A., 141 Doyle, G. J . , 31 Drain, L. E.. 104, 145, 149, 150(263), 151-153, 188. 191 Dryden, H. L., 13(r), 14, 277, 283 Dumas, R., 312 Dunn, E. T., Jr., 21, 86(43) Durrani, T. S., 108, 109, 132, 133, 156, 158, 160, 165, 170, 171, 184, 190, 232 Durst, F., 60, 64, 103, 108(209), 109, 117, 148, 163, 181, 186, 188, 191, 195, 23O(u), 231, 233, 235 Dyson, J., 86

3 E

Eagleson, P. S., 58(t), 59 Eckelmann, H., 312 Eckert, E. R. G., 46(k), 47 Edwards, D. F., 186 Edwards, R. V., 160 Egan, J. I., 53, 54(125) Eggins, P. L., 203 Ehrmantraut, H. C., 26, 27 Eichorn, R., 46(1), 47, 61, 70(c), 71, 78, 85(140), 90,237 Einav, E., 42 Einstein, A., 39 Elbern, A., 344 Elenbaas, W., 92, 132 Ellis, C., 61 Elrick, R. M., 11, 39, 41, 46(j), 47, 67(82), 75, 78, 88(82)

El-Wakil, M. M.,16, 22(g), 23, 74, 85(171) Emrich, R. J., 33, 39, 41, 46(e, f. j ) , 47, 58(p, 4),59, 67(82), 72, 74(166), 75, 78, 88(82)

Engez, S. M., 12(g), 14 Epstein, P., 63 Erdmann, S. F., 383, 391 Erdmann, W., 369, 395 Evans, H. D., 16 Ezekiel, S., 194

F Fage, A., 34, 65, 7qd, c ) , 71, 80(62), 81, 84(63), 85(62, 63)

Fairs, G. L., 21, 22@. 4),23, 27 Farmer, W. M.,22(rr-w). 23(uu), 24, 192, 193

Faure, J., 33 Feller, W. V., 175 Feynman, R. P., 319 Fidleris, V., 12cf), 14 Fisher, M. B., 143, 146(261) Fisher, M.I., 305 Fletcher, H., 39 Florent, P., 308 Flynn, E. R.,85, 87 Fog, C., 186 Foord, R.,183, 185, 188 Foremann, J. W., Jr., 101, 108(221), 109, 173

4

AUTHOR INDEX

Forrester, A. T., 107 Fortier, A., 8, 33 Fox, J., 382 Fox, R. W., 31, 33, 46(r), 48, 191 Francon, M.,110 Frankle, J. T., 170 Fraser, R. P., 74 Frenzen, P., 58(i), 59 Freund, H., 31 Freymuth, P., 290, 293, 300, 301 Fridman, J. D., 174 Fried, D. L., 142 Friedlander, S. K., 33, 35 Friehe, C. A., 308 Fuchs, N . A., 13(v), 14, 22cf), 24, 25.44, 84(50) Fujita, H., 308 Fulrner, R. D., 74

Greated, C. A., 104, 108. 109, 125, 132, 133, 156, 158, 160, 165, 170, 171, 184, 190, 230(a), 231, 232 Green, H. L., 17, 22@p), 24, 44(33), 56 Greenleaves, I., 56, 57 Greenstein, T., 12(e), 14 Greytak, T. J., 343 Griffin,G., 77 Griffin, 0. M.,46(h), 47 GriguU, U.,346, 361(3), 380 Grodzovsky, G. L., 221 Grosch, C. E., 33 Grossin, R., 373 Grove, A. S., 50 Gucker, F. T., Jr., 22(aua), 24, 31, 53, 54(125, 126) Gudmundsen, R. A., 107 Gupta, A. K., 313 Gutti, S. R.,33

G Gagn6, J. M., 209 Gardner, S., 142 Gardner, W., 58(y), 59 Garon, A. M., 175 Gaster, M., 23ocf), 231 Gauvin, W. H., 10, 11, 12(q, x . y). 13(q), 14, 31, 86 Geffner, J., 89 George, E. W., 108(221), 109 George, W. K., Jr., 5 , 6(3a), 141(3a), 158, 232(266) Gerjuoy, E., 107 Gerke, R. H., 22(ee), 24, 25. 84(49) Gessner, F. B., 306 Giese, R. H., 56 Giffen, E., 44,46(ua), 48 Gilbert, M., 33, 62 Gilliland, E. R.,42 Gilmore, D. C., 309 Girard, A., 105 Glastonbury, J. R.,33 Goldschmidt, V. W., 23(xx, zz), 24, 25, 27, 38 Goldsmith, H. L., 41, 43(86) Goldstein, R. J., 108(222), 109, 161, 175, 232(268) Gopal, E. S. R.,47(y), 48 Gorenflo, R.,394, 395, 396(72) Grant, G. R., 193

H Hagen, W. F., 161, 232(268) Hallermeier, R. J., 193 Hallert, B., 84, 85 Hansen, J. W., 22(k), 23, 25, 75(47), 78, 85(47) Hanson, S., 139 Happel, J., 12(d. e ) , 13(d), 14, 38 Hargreaves, R. A., 387 Harper, J. F., 13(f), 14 Hasson, D., 22(mm), 24,45 Hauf, W., 346, 361(3) Havener, A. G., 382 Hawksley, P. G. W.,17, 19, 20, 56 Haywood, K. H., 58(u), 59 Heard, H. G., 197, 199(349) Heflinger, L. O., 381 Heidmann, E., 31 Hendricks, C. D., 49 Henry, J., 33, 46(s), 48 Hercher, M., 195, 196(348), 197(348), 199(348), 202(348) Herdan, G., 16, 17, 18(24), 22(h), 23, 26, 21 Hermges, G.. 75 Hewitt, G. F., 83 Heywood, H., 20, 220, ii), 23, 24 Higgins, G. C., 83, 92(189)

AUTHOR INDEX

5

Highman, B., 46(gg), 48 Jackson, J. D., 342 Hill, G. W., 8 Jacquinot, P., 105 Hill, W. G., 367, 370(21) Jaeger, J. C., 282 Hiller, W. J., 143, 164(259) Jakeman, E., 183, 185, 186 Hilpert, R.,271 Jannot, M., 373 Hinglais, J. R.,317 Jayaratne, 0. W., 45, 49(110) Hintz, E., 344 Jeanmaire, M., 3% 33, 36, 253, 308 Hinze, J. 0.. Jernqvist, L. F., 31, 191 Hioki, R.,191 Jerskey, T., 161, 175, 232(269) Hirth, A., 382 Johansson, T. G., 31, 191 Hjelmfelt, A. T., Jr., 9, 10, 31, 33, 42 Johnson,C.C., 191 Ho, H. W., 33, 42 Johnson, H. F., 47(//), 48 Hodkinson, J. R., 17, 22(rr), 24, 25(35), 54, Johnson, P. O., 107 56, 57 Jones, A. R.,384 Hogland, R.,221 Jones, B. G., 87 Hoglund, R. F., 12(a), 14 Jones, C., 75 Holder, D. W., 346, 361(1), 368, 370 Jones, 0. C., 70(h), 71 Hoole, B. F., 308 Jones, R. T., 306 Hooper, P. C., 22(c), 23, 45, 77(115), 78 Joyce, J. R.,22(kk), 24 Hopkins, H. H., 110, 129(233), 132 Hopper, V. D., 22(/), 23, 25, 75, 78 Hornkohl, J. O., 193, 194 K Horstman, C. C., 402 Houghton, G., 31 Kadambi, V., 87 Householder, M. K., 38 Kalb, H. T., 180, 186 Howes, R. S . , 46@), 48, 58(0), 59 Kalinske, A. A., 47(u), 48, 58(c-e), 59 Hoyle, B. D., 30, 31, 32(56), 46(i), 47, Kantrowitz, A., 368 51(56), 58(v), 59, 61(56) Kaplan, R. E., 313 Huber, M. C. E., 351 Karnis, A., 43 Huffaker, R. M., 132, 133, 192 Kastrinakis, E. G., 312 Hughes, R. R., 42 Katzenstein, J., 220 Hurtig, H., 45 Kawall, J. G., 313 Huss, C. R.,34 Kaye, G. W. C., 58(cc), 59 Hutchinson, P., 141 Keffer, J. F., 313 Hvidberg, I., 21 Keller, D. J., 43, 44 Hyzer, W. G . , 60, 69, 74, 80, 83(138), Kelley, J. G., 387 84(138), 86, 91 Kemblowski, Z., 74, 88(179) Kerker, M., 52, 53(122), 54 I Kessler, D. P., 16, 22cf), 23 Kessler, T. J., 367, 370(21) Irani, R. R.,6(3c), 7, 16(3c), 18(3c), 22(jj), Khairullina, A. Ya., 222 24 Kharchenko, A. V.,199, 203(357), 205(357), Iskol'dsky, A. M., 218 207(357) Iten, P. D., 167, 174, 189, 191 Khosla, P. K., 15, 33, 44(18) Ivanov, A. P., 222 Kierkus, W. T., 46(m), 47, 74, 76(180), 78 Kiesskalt, S., 22(hh), 24 J Kinder, W., 380 King, H. W., 332 Jackson, D. A , , 64, 139, 141, 199, 203, King, L. V.,271, 305 206(248), 217(359) King, R. E., 74

6

AUTHOR INDEX

Kinnard, K. F., 174 Leal, L. G., 46(0), 47 Kirsch, K. J., 30, 31, 32(56), 46(i), 47, Lederman, S., 15, 33, 44(18) 51(56), 58(v), 59, 61(56) Lee, H. M., 12(g), 14 Klapper, J., 170 Lee, S. L., 42 Klebanoff, P. S., 258, 259 Lehmann, B., 109, 117 Klein, M. V., 110 Leighton, R. B., 319 Kline, S. J., 33, 46(s), 48 Levins, D. M., 33 KnBOs, S.,373 Lewin, S. Z., 19 Knollenberg, R. G., 23(ww), 24 Lewis, J. A., 86 Koch, B., 360 Lewis, R. D., 101, 108(221), 109 Koenig, W., 33 Ltwy, s.,400 Kolpak, M. M., 58(r), 59 Libby, P. A., 284, 313 Komasawa, I., 87 Lifshitz, E. M., 8, 246, 247, 329, 330(93), Kovasznay, L. S. G., 265, 270, 274, 287, 331(93), 334 289,290, 303, 308, 312, 360 Lipson, H. G., 7Mg), 71 Kraushaar, R.,380 Littell, A., 33, 46(s), 48 Kravtchenko, J., 33 Littler, J. R., 7%). 71 Kreid, D. K., 108(222), 109 Liu, B. Y. H., 27 Kruglyakov, E. P., 208, 209(366) Liu, P. C., 42 Kuboi, R., 87 Liu, v. c., 33, 35 Kuethe, A. M., 277, 283 Livingston, P. M., 141 Kuloor, N. R., 44,4Q). 4 7 b ) , 48 Lohmann, A., 131 Kumar, R.,44,46(j), 47cu’), 48 Lorrain, P., 319 Kunkel, W. B., 22(k), 23, 25, 75(47), 78, Loveland, R. P., 22(u), 23, 80, 81 85(47) Lowan, A. M., 91, 239 Kuriger, W. L., 206 Lucero, J. A., 184, 186, 230(i), 231 Kussoy, M. I., 402 Lukasik, S. J., 33 Lumley, J. L . , 5 , 6(3a), 35, 141(3a), 158, L 232(266), 290 Laberge, N., 209 M Laby, T. H., 22(1), 23, 25, 58(cc), 59, 75, 78 Lacharme, J. P., 382 Macadam, D. L., 83, 92(189) Ladenburg, R. W., 379 McAdams, W. H.,271 Lading, L., 143, 181, 190 Macagno, E. O., 58(s), 59 Lai, R. Y. S.,12, 14 McCroskey, W. J., 402 Lamb, C. G., 22(11), 24 McDougall, J. G., 402 Lamb, H.,12(m), 13(m), 14 McLanahan, D., 69, 72 Lamberts, C. W., 382 McLaughlin, D. K., 6 La Mer, V. K., 46(ee), 48 McMichael, J. M., 258, 259 Landau, L. D., 8, 246, 247, 329, 330(93), McNowan, J. S., 12(g), 14 331(93), 334 McPherson, M. B., 12(g), 14 Lane, W. R., 17, 19, 22@p), 24, 44(33), 56 Maddox, A. R.,368, 381 Langevin, P., 39 Magill, P. L., 26, 27 Lanz, O., 191 Mann, S.,58(x), 59 LaRue, J. C., 284 Mannesmann, D., 81, 84(186), 85(186) Latron, Y.,387 Marey, E. J., 3 Laufer, J., 313 Mark, A. M., 22(9q), 24, 27 Lawler, M. T., 42 Marsden, C., 58(x), 59 Laws, J. O., 78, 85 Marshall, W. R., Jr., 22(qq), 24, 27, 44, 45

7

AUTHOR INDEX

Mason, B. J., 45, 49(110) Mason, J. S., 124 Mason, S. G., 41-43,46(bb), 48 Mastner, J., 174 Mathes, W., 190 Matulka, R. D., 382, 398 Mayo, W. T., Jr., 184 Mazumder, M. K., 30-32, 46(i), 47, 51, 58(v), 59, 61(56), 109, 117, 138, 139 Meadows, D. M., 231 Mehmel, D., 67 Meier, G. E. A,, 143, 164(259) Meister, K., 174 Melchior, H., 143, 146 Melling, A., 4%). 47, 195 Mellor, R., 72, 78 Meneely, C. T., 186 Mertens, L. E., 60, 80, 82 Merzkirch, W., 241, 346, 361(4), 369, 384, 387, 395 Meyers, J. F., 175

Miesse, C. C., 44 Miller, J. E., 67. 72(151), 84(151) Miller, L. T., 290 Mitchell, C. J., 197, 199(350) Mitsuta, Y., 316 Miyaki, M., 317 Mizrahi, J., 22(mm), 24, 45 Mockros, L. F., 10, 12, 14, 33 Moller, G. L., 306 Monro, P. A. G., 3 Moore, C. T., 184, 185 Morgan, B. B., 22(i), 23 Morikawa, S., 191 Morkovin, M. V., 270 Morrison, G. L., 301 Morse, H. L., 199, 203(358), 206(358), 226(358) Morton, G. A,, 185 Morton, J. B., 138 Moss, B. C., 188, 191 Mowbray, D. E., 356 Mugele, R. A., 16 Munday, G., 17, 26, 27, 44(31) Muntz, E. P., 399 Muraszew, Q., 44, 46(uu), 48 Murnaghan, F. D., 13(r), 14 Myers, L. M., 60.91 Myers, P. S., 16, 22(s), 23, 74, 85(171)

N Naib, S. K. A., 47(v), 48, 58(h), 59, 74 Nawab, M. A., 46(bb), 48 Nedderman, R. M., 58(r), 59, 67, 85(152) Neisch, W. E., 23(ww), 24 Nesterikhin, Yu. E., 218 Newell, G. W., 26, 27 Nicholl, A. A., 26, 27 Nicholls, J. A., 86 Nieuwenhuizen, J. K., 67 Nimeroff, I., 69 North, R. J., 346, 361(1), 368, 370 Null, H. R., 47(//), 48 0

Oertel, H., 387 Oliver, C. J., 183, 186 Olsen, G. J., 313 Oppenheim, A. K., 378, 387(36) O'Regan, R., 382 Orloff, K . L., 193, 233 O n , C . , Jr., 16, 17, 18(26), 22(z), 24, 27 Oseen, C. W., 8 Ossofsky, E., 289 Ostrach, S., 61, 63, 93 Ostrovskaya, G. V., 195, 199(346), 220 Ostrovsky, Yu. I., 195, 199(346) Oswald, L., 77 Otake, T., 87 Otterman, B., 42 Oudin, L., 396

P Page, R. H., 387 Paizis, S. T., 313 Pan, F., 46(0), 47 Pankhurst, R. C., 250 Panofsky, W. K. H., 319 Papoulis, A., 201, 202(361) Papyrin, A. N., 199,203(354,356), 204(354), 205(356)

Parent, R. J., 22(qq), 24, 27 Parkins, W. E., 107 Patterson, H. S., 19 Paul, D. M., 64, 139, 141, 199, 203(248), 206(248), 217(359)

8

AUTHOR INDEX

Pearcey, T., 8 Peck, G. T., 74 Pei, D. C. T., 67 Pellet, M. M.,317 Penndorf, R. 53 Penner, S . S., 161, 175, 232(269) Penwarden, A. D., 67, 72(153), 78, 84(153) Peronneau, P. A., 317 Pemn, J. B., 39 Perry, A. E., 301 Persin, A., 197, 198(351), 199(351) Peters, C. J., 188 Petjanov, J., 22cfJ‘), 24, 25, 84(50) Heifer, H.J., 31, 360 Philbert, M.,389, 391(68) Phillip, A. R., 46@), 48, 58(0), 59 Phillips, D. T., 53, 54(127) Phillips, M.,319 Pien, C. L., 58(c), 59 Piersol, A. G., 230(e), 231 Pikalov, V. V., 221, 222(376) Pike, E. R., 185, 186 Pilcher, J. M.,26 Pinchin, B., 83 Pless, I. A., 82 Powell, W. M.,77 Prandtl, L., 306 Pratt, W. K., 107 Predein, A. L., 199, 203(354), 204(354) Preobrazhensky, N. G., 221, 222(376) Preston, J. H., 70(d, e ) , 71 Prewett, W. C., 49 Pusey, P. N., 183

Q Quick, A , , 190

R Ranz, J. E., 22(cc), 24, 63 Rappaport, E., 46(dd), 48 Rasmussen, G. G., 309 Raterink, H. J., 382 Rayner, A. C., 45 Reddy, K. V. S., 67 Rey, C., 312

Reynolds, G. O., 22(ss), 24 Ribner, H. S., 256 Riebold, W., 190 Rinkevicius, B. S., 195, 199, 203(357), 205(357), 206, 207(357), 220(342)

Rizzo, J. E., 174 Robben, F., 343 Roberson, E. C., 53, 54( 128), 58(s), 59, 74 Roberts, J. M.,23ocf), 231 Robinson, G., 34 Rolfe, E., 174 Ronchi, V., 370 Rose, D. G., 22(aaa), 24 Ross, M.,107, 142(216) Rothe, D. E., 400 Rottenkolber, H., 380 Rouse, H., 58(g), 59 Rowe, P. N., 12(/), 14 Rowell, R. L., 53, 54(126) Royer, H., 373 Rubinow, S. I . , 43, 44 Rudd, M. J., 109, 116, 117(227), 119(227), 125(227), 146, 149

Ruddock, K. H., 68, 7ocf), 71, 74, 75 Rudinger, G., 225 Rungaldier, H., 382 Runstadler, P. W., Jr., 33, 46(s), 48 Rushton, J. H.,47(w), 48, 58(j), 59 Rutkowski, J., 74 Ryley, D. .I., 12(0), 14, 46(z), 48

S Sachs, J. P., 47(w), 48, SSg’), 59 Saffman, P. G., 43,44 Samples, W. R., 27 Sandbom, V. A , , 262, 264 Sands, M.,319 Saw, S., 221 Saxton, R. L., 63 Sayle, E. A , , 174 Saylor, C. P., 22(v), 23 Schardin, H., 361, 363, 371, 394, 3% Schneider, J. M.,49 Schotland, R. M.,316 Schraub, F. A , , 10, 33, 46(s), 48, 50, 88 Schubauer, G. B., 276

AUTHOR INDEX

Schultz, H., 21, 22(dd), 24 Schwar, M. J . , 97, 108(206), 109, 131, 369, 384

Schwartz, A., 71 Schwartzberg, H. G., 58(n), 59, 78, 88

Schwarz, W. H.. 308, 313 Schweer, B., 344 Scott, P. F., 46(n), 47 Sears, W. R., 306 Segre, G., 41 Selberg, B. P., 86 Self, S . A , , 143 Seltzer, E., 44 Sernas, V. A , , 387 Settlemeyer, J. T., 44 Settles, G. S., 368 She, C. Y.,184, 186, 230(i), 231 Shercliff, J. A., 320, 321(84), 340 Siddon, T. E., 256 Siegman, A. E., 126 Silberberg, A , , 41 Silverman, L., 27, 46(ii), 48 Simonds, H. R., 61 Sinclair, D., 46(ee. hh), 48 Skokov, I. V., 198, 199(353) Sleicher, C. A., 305, 306(42), 308(42), 309(42), 3 11

Small, R. D., 387 Smart, A. E., 184, 185 Smeets, G., 389 Smigielski, P., 382 Smith, A. M. 0.. 46(kk), 48 Smith, J. M., 74, 88(179) Snowden, D. D., 46(0), 47 Solignac, J. L., 394, 396(70) Soloukhin, R. I., 218 Somerscales, E. F. C., 4, 38, 61(3) Soo, S. L., 12(h), 14, 33, 87 South, R., 394 Sparrow, E. M., 46(k), 47 Stairmand, C. J., 22(n, o), 23 Stanford, R. A , , 313 Starkey, T. V., 41 Stevens, W. F., 22(1/), 24 Stevenson, W. H., 31, 191 Stolzenburg, W. A,, 367, 370(20) Stone, B. R. D., 19 Strohl, A., 309, 310(56), 311

9

Stubbs, H. E., 16, 22(e), 23, 74, 75, 77(20), 78

Stupar, J., 46(cc), 48 Suchorukich, W. S., 363 Sullivan, J. P., 194 Sumner, C. G., 22(x), 24 Surget, J., 382 Sutugin, A. G., 44 Suzuki, T., 191, 387 Swart, F., 77 Sweeney, D. W., 398 Swinney, H. L., 107, 116, 168 T

Tam, C. K. W., 12(j), 14 Tanner, L. H., 190, 377, 382, 383, 387(52) Tate, R. W., 45 Taylor, C. A., 126 Taylor, G. I., 313 Taylor, L. S., 355 Tchen, C. M.,8, 33 Teele, R. P., 73, 91 Thatcher, G., 321 Thkry, C., 382 Thiolet, G., 308 Thomas, R. E., 26 Thompson, B. J., 110, 126 Thompson, D. H., 190 Thornton, J. R., 101 Tiederman, W. G., 6 Tien, C. L., 12(h), 14, 87 Titterton, P. J., 173 Tolansky, S., 198, 199(352) Tolkachev, A. V., 199, 203(357), 205(357), 206, 207(357)

Tollert, H., 63 Tompkins, E. E., 337 Torobin, L. B., 10, 11, 12(q, x , y ) , 13(q), 14, 31

Tory, A. C., 58(u), 59 Townend, H. C. H., 34, 65, 80(62), 81, 85(62)

Treybal, R. E., 58(n), 59, 78, 88 Trimpi, R. L., 368 Tritton, D. J., 308 Trolinger, J. D., 369 Tunstall, E. B., 31 Tutu, N. K., 313

I0

AUTHOR INDEX

U Uberoi, M. S., 265, 360 Underwood, R. M., 47(r), 48, 58(1), 59, 67 Urtiew, P. A., 378, 387(36) Uyehara, 0. A., 16, 22(g), 23, 74, 85(171)

V van de Hulst, H. C., 52, 53(121), 54-56 van der Ziel, A., 143, 144(260) van Meel, D. A., 67 van Paasen, C. A. A., 23(yy), 24, 25 van Wijk, M. C., 67 Vasilenko, Yu. G., 195 Vasudeva, B. R., 290 Vkret, C., 364, 366, 387, 389(19), 391(19) Vermij, H., 67 Vest, C. M., 398 Viannay, S., 373 Villat, H., 33 Vogelphol, G., 81, 84(186), 85(186) von Srnoluchowski, M., 39 von Stein, H. D., 31, 360 Votaw, C. W., 46(h), 47 Vukicevik, D., 197, 198(351), 199(351) Vukoslaveevic, P., 312

W Walker, P. B.,47(x), 48,58(f), 59,7O(a), 71, 75

Wall, L. S., 142 Wallace, J. M., 312 Walter, H., 54 Walton, W. H., 49 Wang, C. P., 110, 131, 168 Wang, J. C., 46(n), 47, 178 Wankum, D. L., 109, 117, 138, 139 Waters, G. T., 67, 72(153), 78, 84(153) Watrasiewicz, B. M., 141, 149 Watson, H. H., 27 Watson, H. J., 101 Webster, C. A. G., 308 Wehrmann, 0. H., 305, 306(42), 308(42),

Weinstock, S. E., 46(dd), 48 Welch, N. E., 192 Welford, W. T., 66, 78, 83, 84, 85(149), 86(149), 88(190)

Wells, P. V., 22(ee), 24, 25, 84(49) Weske, J. R., 289 Weyl, F. J., 353, 3% Whiffen, M. C., 231 Whitby, K. T., 17, 22(w, g g ) , 23, 24 White, D. R., 350 Whitelaw, J. H., 4%). 47,64, 103, 108(209), 117, 141, 181, 195, 230(a), 231, 233, 235 Whitlow, L., 74, 109 Whitrnore, R. C., 12(f), 14 Whittaker, E. T., 34 Whytlaw-Gray, R.,19 Wick, C. J., 387 Wiggins, E. J., 74, 78 Willard, M. L., 19 Williams, M. J., 269, 271 Wilmshurst, T. H., 166, 170, 174, 175 Winnikow, S., 387 Winter, E. F., 58(m), 59, 72, 74, 75, 78, 84 Wiolrnarth, W. W., 312 Wirtz, D. P., 74 Witte, A. B., 382 Wohl, P. R., 43, 44 Wolf, E., 110, 198(230), 199(230) Wolf, W. R., 45 Wolter, H., 361, 373 Won, W. D., 26 Wood, N. B., 301 Woods, J. D., 45, 49(110) Work, L. T., 17, 22(w), 23 Worthing, A. G., 89 Wright, F. H., 33 Wuerker, R. F., 381, 382 Wyatt, P. J., 53, 54(127) Wyngaard, J. C., 265, 290, 311, 312 X

Xhaard, M. C., 317

309(42)

Weidrnan, P. D., 280, 281, 290, 298 Weinberg, F. J., 346, 361(2), 369, 370, 378, 384, 387(36)

Y Yanenko, N. N., 223

AUTHOR INDEX

Yanta, W. J., 12(c), 14, 21, 22(bb), 24, 31, 230(a), 231 Yeh, Y., 34, 97, 109 Yen, B. C., 33, 42 Yokozeki, S ., 387 York. J . L., 16, 22(e,f), 23, 74, 75, 77(20), 78 Yu, J. P., 46(k), 47

11 2

Zagorodnikov, S. P., 218 Zaidel, A. N . , 195, 199(346), 220 Zare, M . , 163, 186, 188, 191 Ziegler, M., 287, 289 Zuber, N., 12(k), 14 Zweig, H. J . , 83, 92


SUBJECT INDEX This is a combined index for Parts A and B of Volume 18. A

Abel inversion, 394, 742 Absorption of radiation atomic attenuation coefficient, 406 for chemical composition, 621-634 for density measurement, 405-408 linear attenuation coefficient, 406 mass attenuation coefficient, 406 Absorptivity, spectral, 465, 472 -475 Acoustic anemometer, 315-318 Acoustic Doppler velocimeter, 317 Acoustic Bowmeter, 337-340 Adiabatic wall temperature, 458-459, 665 Aerodynamic force principle, 254-256 on vane anemometer, 254-258 on whirling arm anemometer, 256-259 Aerodynamic noise, study in wind tunnel, 779 Ambiguity noise, 104, 126, 136, 160, 162, I66 effect on LDV signal processing, 162 Anechoic chamber, 778 Antenna theorem, 126 Anti-Stokes Raman line, 422, 723 Aperture broadening, see Antenna theorem Apparatus, for fluid dynamic research, 755-819 Arc-plasma tunnel, 462, 784

B Ballistic range, 779-781 Bar gage for pressure measurement, 593, 602 Barium titanate pressure gage sensor, 542 Basset-Boussinesq-Oseen (BBO) equation, 8 Beam, see Light source

Beam absorption densitometry, 405-408 Beam splitter, 379 Beer’s law, 407, 475, 623-624, 629, 632 Bellows gage for pressure measurement, 51 1 Bernoulli formula, 243-245, 325, 333, 776 Bernoulli pressure, 503 Bias, in particle tracking, 5 , 174, 195 Biot number, 668-669 Blackbody radiation, 465-466, 690, 699 Blast wave solutions, by self-similarity, 835-842 Blow-down wind tunnel, 758 Boltzmann distribution, 464, 473-474, 480, 640-641 Bond (chemical) density, 413, 419 Boundary layer recovery factor for temperature probe, 459 study in wind tunnel, 766-768 Bourdon gage for pressure measurement, 51 1

Bow shock wave, 358 BOXCARS (variant of CARS), 432 Bragg cell, for frequency shifting in LDV, 193-194 Brehmsstrahlung, 407, 486, 706 Brightness of light source, 689 Brightness temperature, 465-466 Brillouin scattering, 415-417 Broad crested weir, 334-336 Brownian motion, effect on tracer method, 39 C

Calibration camera, in chronophotography, 84-86 electron beam fluorescence system, 45045 1, 453 flowmeter, 322

14

SUBJECT lNDEX

Calibration (continued) heat transfer gage, 677, 680, 682, 684 hot-wire anemometer, 285, 297 pressure gage, 504, 507, 509, 512, 514, 555, 592, 606,610 Raman scattering diagnostic system, 429 Calorimetry applied to heat transfer measurement, 664, 670-679 capacitance calorimeter, 672-674 tangential conduction error, 670-671 Canal mechanism of spark formation, 696 Candela, 689 Capacitance sensor diaphragm gage, 570-572 method, 540-542 Capillary correction to manometer, 507 Capsule gage for pressure measurement, 51 1 CARS (coherent anti-Stokes Raman scattering), 43 1-433, 489 Cavitation, dimensional analysis, 843, 848 Centrifugal force, 803 Ceramic capacitor spark light source, 700 Channel flow metering, 332-336 Chapman-Jouguet condition, 838 Chemical composition measurement, see Composition measurement Chemical kinetics, study in shock tube, 656-659, 792-795 Chemiluminescence, use in composition measurement, 641-643 Choked flow, 330, 772 Chromatography, for composition of sampled fluid, 617-621 Chrono-interferometer, 551, 607 Chronophotography calibration of camera, 84-86 camera requirements, 79-83 compared with other velocimeters, 64-66 dark and bright field illumination, 76-79 data analysis, 86-87 definition, 64,66 directional information, method, 67 error analysis, 87-89 illustration of system design, 89-93 interrupted illumination, 67-76 measuring volume, 83-84 rotating flow apparatus, 818-819 system elements, 67 Cinematography, high speed, 726, 732-739

Clausius-Mosotti relation, 348 Coal mine dust explosions, 796 Coherence lateral, 125 spatial, 406, 707, 710 temporal, 125, 129, 707, 710 Coherence function, 118-131, see also Heterodyne efficiency Coherence length definition, 71 1 light source, 130-132, 715 measurement, 71 1 Coherence time definition, 130, 711 measurement, 71 1 Color interferometry, 742-743 Color schlieren, 367, 370 Combustion driver, shock tube, 788 Composition, method of description, 61 1 Composition measurement absorbed radiation by in situ fluid, 630637 absorption spectrophotometry of sampled fluid, 621-630 analysis of emitted radiation by in siru fluid, 637-643 analysis of sampled fluids, 616-630 classification of methods, 613-616 electron beam fluorescence, 434, 445 mass spectrometer, 645-661 methods, 611-661 sampling methods, 616-617 species concentration by molecular scattering, 408-433, 643-645 Compressible flow, in wind tunnel, 759-761 Compressible flow field, density by light refraction, 346 Compton effect, 407 Conrad probe, 254 Constant current anemometer, hot-wire or hot-film basic circuitry, 277 calibration, 285 compensation, 283 square wave test, 285 Constant temperature anemometer, hot-wire or hot-film basic circuitry, 290-292 calibration, 297 characteristic frequency, 295

15

SUBJECT INDEX

cutoff frequency, 292-293, 301 damping coefficient, 295 higher-order system response, 300 linearization of signal, 302 offset voltage, 291 square wave test, 297 unbalance parameter, 292, 294, 296 Convection of heat at surface, 667 role in hot-wire and hot-film anemometer, 269 Conversion of units, 823-825 Coriolis force, 803 Couette viscometer, 797 Cranz-Schardin camera, 737 Critical flow liquid in channel, 333 nozzle throat, 330, 772

D Data analysis chronophotography, 86-86 interferometry, 205-398 Dead weight pressure gage, 512 Decibel, 508 Density gradient, by Raman scattering, 424 Density measurement beam absorption technique, 405-408.705 electron beam excited radiation, 434-455 interferometer technique, 345-403 Raman scattering technique, 418-433 Rayleigh scattering technique, 414-418 schlieren method, 363 Depth of modulation, 121 Detonation, 838-839 Detonation wave, temperature measurement, 470 Diaphragm pressure gage, see Pressure gage, diaphragm Differential pressure flowmeter, 324-331 Diffraction, effect on schlieren method, 364 Diffraction grating interferometer, 380 Diffraction-limited point light source, 710 Diffuser orifice flowmeter, 326 wind tunnel, 758, 760, 772, 784 Dilatational pressure gage,'see Pressure gage, dilatational Dimensional analysis

examples, 832-842 mathematical foundations, 821 -828 nature, 821 Dimensional and dimensionless quantities, 822-825 Dimensional homogeneity, 826 Dimensionless numbers in fluid dynamics, 829-831 Dimensions, 822-823 Directional ambiguity in LDV, removal of, 186-190 Direct spectrum analysis, in LDV illustrations of use, 220-227 image converter use, 208-209 method, 194-227 streak camera use, 217-218 synchronous detection use, 205-208 Discharge coefficient, flowmeter, 328-330, 335 Distortion of solid, measurement by laser speckle, 713 Division, of amplitude or wave front, LDV configuration, 124 Doppler ambiguity, see Ambiguity noise Doppler bursts, 154, 175, 180, 190 Doppler shift formulas, 99-104, 342 Doppler velocimeter, acoustic, 317-318 Drag coefficient, sphere, 8, 10-15,759, 800 Drag force, measurement, 768-769 Dropout, see Signal dropout Drum camera, 733 Dust, acceleration by shock wave, 31-33, 795 Dye, marker for flow visualization, 819 Dye laser applications, 715-716, 746 pump lamp, 693-694 Dynamical similarity, 828-829 Dynamic pressure, 247, 503 Dynamic response, see Frequency response

E EBF, see Electron beam fluorescence Ekman boundary layer, 805-806, 810, 817 Electromagnetic anemometer, 3 18-32 1 Electromagnetic flowmeter, 337, 340 Electron beam fluorescence beam generation, 452 beam spreading, 452

I6

SUBJECT INDEX

Electron beam fluorescence (continued ) calibration of system, 450-451 chemical composition measurement, 434, 445 compared with laser light scattering technique, 435 density measurements, 450-451 flow visualization, 399, 437, 453 general description, 434-438 intensity relation to gas density, 441 -450 role of gas motion, 438, 446-447 role of secondary electrons, 443-445 selection rules, 438-441 temperature measurement, 489-497 Electron density, measurement, 698, 704, 705, 742, 750-753, 794 Electron gun, EBF system, 451-452 Electron spin resonance, for species concentrations, 634-637 Electron temperature, 464, 472, 475, 705 Electro-optical shutter, 727-732 Elliptic flow equation, 760 Emittance of light sources, 465, 689 Emitted characteristic radiation, for velocity measurement, 341-345 Equations of state, 612 Equivalent surface conductance, 667 Error analysis chronophotography, 87-89 particle tracking methods, 50 Error functions, definitions, 666,668 Etalon, Fabry-Perot, Fizeau-Tolansky, 198, 202, 214, 708, 717, 740 Excitation cross section, electron beam, 438-447 Explosion diagnostics, X-ray flash, 408 Exposure times, photographic, 692-702

F Fabry-Perot etalon, use in laser, 708, 717, 740 Fabry-Perot filter, see Direct spectrum analysis, in LDV Fabry-Perot interferometer, 105, 195-227, 708, 717, 740 Faraday shutter, use in high speed photography, 731-732 Fast luminous fronts, 6% Fast response pressure gages, 576-610

Fermat’s principle, 352 Field absorption as visualization method, 389-392 Finesse, 197 Fizeau-Tolansky interferometer, use in LDV, 198-199, 214 Flame composition by emission spectroscopy, 641-643 by mass spectrometry, 659-660 Flame front velocity, measured by schlieren method, 369 Flame temperature, 421-425, 466-470 Flash lamp characteristics, 692-695 dye laser pump, 719 Flash radiography, 408 Flight testing apparatus, 779-781 heat transfer, 664 Flow disturbance by electron beam fluorescence diagnostics, 451, 453 by hot-wire probe, 308 tracer particles, 38, 41, 49-51 by Pitot probe, 243, 250 by Raman scattering diagnostics, 419 Flowmeter acoustic, 337-340 bundle of capillaries, 336-337 calibration, 322 definition, 241 electromagnetic, 337, 340 float meter, 331 flume, 332, 334 orifice, 324-330 positive displacement, 323-324 power loss, 328 sonic nozzle, 330 turbine, 324 variable area, 331 Venturi, 324, 331 weir, 332-336 wet-gas, 323 Flow straightener, 326 Flow tracing particles advantages and disadvantages, 3-4, 97 definition, 2, 6 dynamic characteristics, 32-34, 221-222, 225 effect on flow field (loading error), 38-41

SUBJECT INDEX

effect of sedimentation, 41-43 equation of motion, 8 generation and dispersal, 43-50 hydrodynamic resistance, 8- 15, 795 interaction effects, 223-224 light scattering, 52-60, 64 limit of sensitivity in velocity measurement, 38-41 location in measuring volume, effect on LDV, 126 motion of, effect of size and density, 15-16

optical characteristics, 51 -60 refractive index data, 53-54 response time determination, 26-32, 795 response time effect on turbulence measurements, 34-38 selection of, illustration, 60-64 size and density, measurement of, 16-26 size effect on LDV performance, 124 system for velocity measurement, 2-4, 6-7, 201-206, 235-240 use in rotating flow apparatus, 818-819 Flow visualization -1ectric glow discharge, 402 : x t r o n beam fluorescence, 399, 437 .iele -Shaw apparatus, 798 -799 high speed photography, 725-753 infrared, 750-753 interferometer, 377 jet, 359 light source, 694 phase contrast, 389 radiation emission, 398 rotating flow apparatus, 818-819 schlieren, 365 shadowgraph, 358 shock waves, 365, 386 smoke, 6-8, 770 tufts, 241, 770 wind tunnel, 769-771 Fluid, definition, 501 Fluid dynamic equations in rotating coordinate system, 802-806 Flume, 332, 334 Fluorescence dye laser, 715-716 infrared sensor, 672, 751 meaning, 411-412 quench;&, 412

I7

relation to resonance scattering, 413 use for density, temperature, composition diagnosis, 410-414 Fluorescent lacquer, visualize transition to turbulence, 771 Fluorescent radiation Doppler shift to measure velocity, 343 Force balance, aerodynamic model in wind tunnel, 768-770 Force balance, aerodynamic model in wind tunnel, 768-770 Forcing function, role in hot-wire and hot-film circuit response, 278 Fourier heat conduction equation, 665 Fourier number, 668-669, 675 Fourier transform spectra, for species concentrations, 629 Framing camera application, 696, 733-734 light source, 705, 715 Franck-Condon factors, 491-492 Free flight apparatus, 779-781 Free-molecule flow, 763 Frequency counting as LDV signal processing technique, 174-175 Frequency domain signal processing in LDV, 161 Frequency response, see ulso Response time calibrator for pressure gage, 555 density measurement by Rayleigh scattering, 418 diaphragm pressure gage, 562 flow tracing particles, 32-34, 221, 225 function, 527 hot-wire and hot-film probe, 273, 278, 282, 293, 297 pressure bar gage, 599 Raman scattering diagnostics, 420, 42 1 Rayleigh scattering diagnostics, 418 stub pressure gage, 587 vane anemometer, 257-259 Frequency shifting in LDV, 163-164, 185, 187-189, 193-194 Fringe anemometer, 109, see also Optical heterodyne detection Fringe distortion methods, 369 Fringe interpretation, of LDV operation, 117

18

SUBJECT INDEX

Froude number definition, 830 role in ship fluid dynamics, 848 G

Gage factor, resistance sensor, 538 Gardon heat transfer gage, 672-675 Gaussian line profile, distortion by Brillouin scattering, 414-416 Geometric similarity, 828, 843, 844 Geophysical flow apparatus, 801-819 Geostrophic flow, 803-806 Gladstone-Dale constant electron gas, 350 ionized gas, 350 Gladstone-Dale relation, 348, 351 Grashof number definition, 830 hot-wire and hot-film convection, 269

H Hagen-Poisseuille formula, 336, 798 Head (of fluid), meaning, 333 Heat conduction relations, one-dimensional, 665 -670 Heat transfer coefficient, definition, 459 Heat transfer gage asymptotic type, 672-675 balanced heat removal type, 664 calorimeter type, 664 capacitance calorimeter, 672-674 construction and principles, 663-685 Gardon type, 672-675 high heat flux, 676-677,681 high temperature gas flows, 683-685 infrared bolometer, 679 membrane calorimeter, 664 multilayer gage, 666 radiation type, 683-686 sandwich type, 664 shock tube and shock tunnel, 666, 677-679, 682-685, 792 thick film type, 677-678 thin film type, 664, 679-683 thin membrane calorimeter, 672-675 use in arc-plasma tunnel, 784 use in free-flight model, 781 Heat transfer measurement conceptual methods, 664 gages, 663-685

shock tube technique, 791 wind tunnel technique, 674, 778 Heat transfer hypersonic tunnel, 762 radiation loss in arc-plasma tunnel, 674-677, 784 Hele-Shaw apparatus, 788-789 Heterodyne detection, see Optical heterodyne detection Heterodyne efficiency, 119 High speed recording methods, 725-753 High temperature gases, produced in shock tube, 787-790 Hold time definition, 530 diaphragm pressure gage, 568 free surface motion pressure gage, 606 pressure bar gage with end sensor, 603 stub pressure gage, 590 Holographic interferometry, 381, 550, 746-750 Holography combined with interferometer method, 38 1 combined with schlieren method, 368 methods, 715, 743-750 principle, 744-746 reconstruction, 550, 743-746, 747, 748, 750 Homodyne detection, see Optical homodyne detection Hook method, 351 Hopkinson pressure bar, 596 Hot-film anemometer, see also Constant current anemometer; Constant temperature anemometer; hot-wire anemometer calibration, 258 -289, 297-303 compensating circuit, 283 construction, 266-268 external coating, 282 frequency response, 282 heat conduction, 273, 382 linearized energy balance equation, 275 physical characteristics, 267-268 resistance, 268 Reynolds number, 269 substrate, 267, 273 temperature sensitivity, 313-314, 461 theory, 268-276 thickness, 267 time constant, 281-297

19

SUBJECT INDEX

Hot-wire anemometer, see also Constant current anemometer; Constant temperature anemometer aspect ratio, 271, 304, 307, 309 calibration, 285-289, 297-303 compensating circuit, 283 constant current, 276-289 constant temperature, 289-303 construction, 261 diameter, 261, 284 directional dependence, 306-308 effective cooling velocity, 306 effect of temperature variation along length, 303 flow interference effects, 308 frequency response, 278, 283, 297 heat conduction, 272 linearized energy balance equation, 275 multiple probe arrays, 310-313 physical characteristics, 260-267 resistance of wire, 268 Reynolds number, 269 sheath, 261, 265 supports, 265 temperature sensitivity, 313-3 14, 461 theory, 268-276 time constant, 278-297 X-probe, 310 Hydrostatic correction, 244, 332 Hydrostatic law, 505 Hydrostatic pressure, 501 Hyperbolic flow equation, 760 Hypersonic apparatus, 781 -784, 791, 793 Hypersonic atmosphere entry, 664 I

Image converter camera description, 732, 735-737 use in LDV, 208-209 Image converter streak camera, 720 Image dissection camera, 734-735 Image intensifier camera, 735-737 Impact pressure, 247, 503 Implosion, strong, 841 -842 Index of refraction, see Refractive index Inductance sensor, use with diaphragm gage, 570 Induction wind tunnel, 758 Infrared interferometer diagnostic method, 750-753 Infrared pyrometer, 672

Instrumented heat gage models thin wall, 670-671 thick wall, 671 surface temperature mapping, 671-672 Intensifying screen, for x-ray detector, 407 Interference fringe visibility, 71 1 Interferometry diffraction grating, 380 evaluation procedures, 392-398 high speed recording, 739-743 holographic, 381, 550, 746-750 infrared, 750-753 measurement of electron density, 698 multiple-beam resolving power, 196 principles, 196, 374-392, 739-743 reference beam, 375 shearing, 375 two color, 742-743, 794 Ionization chamber, 407 Ionization rate, study in shock tube, 792-794 Ionized gas heat transfer measurement, 673-679 Irradiance by light sources, 689 Isotope dilution analysis, for species concentrations, 630

J Jet open, use as wind tunnel, 759, 764 shadowgraph visualization, 359 Jitter, spark triggering, 702-703

K Kerr cell, 717, 723, 729-731 Kiel probe, 249 Kinematic viscosity, 797 King's law, 271, 297 Kirchhoff radiation law, 465, 467, 471, 690, 707 Knudsen number, 763, 831 Kolmogorov length scale, 265 1

Lagrangian and Eulerian mean square velocities, 38 Lambert-Beer relation, see Beer's law

20

SUBJECT INDEX

Laminar boundary layer heat transfer, 667 in hypersonic tunnel, 783-784 similarity solution, 832-834 study in wind tunnel, 766-767 Laser active modulation, 718-719 argon ion, 716 carbon dioxide, 714-716, 750-753, 794 coherence properties, 710-712, 717 continuous emission, 715-716 dye, 715-725 energy output, 718, 750-753 fundamental properties, 707-708 gasdynamic, 796 generation of harmonics, 722-725 giant pulses, 717-718 helium-neon, 714-716 inversion of levels, 707-708 line shape, 708-709, 718 mode spectrum, 708-710 mode-locked pulses, 718-720 neodymium-doped glass, 714-716 nitrogen, 721 nonlinear optical methods, 721-725 parametric amplifier, 724-725 pumping by flash lamp, 692-693, 707 Q-switch, 717-718 Raman, 722-724 recording interferometry, 739-743 relaxation pulses, 716-717 NbY, 714-716, 717, 725, 746 saturable absorber, 7 17-7 19 speckle, 707, 712-714 spectral ranges, 714-716 superradiant, 720-72 I YAG, 714-716, 725 Laser anemometer, see Laser Doppler velocimeter Laser Doppler anemometer, see Laser Doppler velocimeter Laser Doppler velocimeter characteristics of, % choice of technique, 232-235 combined with Raman scattering diagnostics, 43 1 combined with schlieren method, 369 compared with probe methods, 97 design calculation, illustration of, 235-240 illustrations of signal, 155

optical configurations, 108-1 10 optimization of performance, 235 photodetector output current, 110-1 15 principle, 97 rotating flow apparatus, 819 signal analysis, 229-232 signal processing methods, 227-228 Laser triggered spark gap, 703 Lava1 nozzle, in wind tunnel, 772 LDA, see Laser Doppler velocimeter LDV, see Laser Doppler velocimeter Lift force, aerodynamic, 768-769 Light beating, see Optical mixing Light distribution function, in LDV measuring volume, 119-120 Light gas gun, 780 Light path lengths, effect on LDV performance, 131 Light recording methods, 725-753 Light scattering, see also Raman scattering; Rayleigh scattering by flow tracing particles, 52-60, 64,410 Light sensor Image intensifier camera, 735-737 photographic material, 689, 715, 726-727 phototube, 689 spectral response, 689, 751 Light source absorptivity, 690 beam, 406 broad source, 406 chemical explosive, 705 coherence length, 130-132 diffraction-limited point source, 710 duration, 692 -695, 700- 702, effect of size on LDV performance, 124-126 energy, flash lamp, 693-695 exploding wire, 703-705 flash lamps, 692-695 flow visualization, 694, 695-703 general, 687-725 infrared, 750 laser, 707-725 laser pump lamps, 693-694 laser spectral ranges, 714-716 luminous efficiency, 691, 692, 698-703, 705 nonlinear optical methods, 721-725 physical and photometric aspects, 688-689

21

SUBJECT INDEX

plasma focus, 705-706 point source, 356, 405, 710 Raman scattering diagnostics, 419, 425 shadowgraph, 694, 695-703 short duration pulse, 710, 717-721, 727, 750 measurement, 720 spark, 695-703 triggering, 702-703 spatial coherence, 125, 406 spectral characteristics, 688, 691 spectral luminance, 690 spectral output of lamps, 691-695 of spark, 699 temporal coherence, 125, 129 thermal, 688-707 units of output, 688-689 xenon flash lamp, 693-694 Light spectroscopy, application in LDV, 105-106 Line reversal method of temperature measurement, 466-470 Liquid crystal, infrared sensor, 751 -753 Liquid manometer, 505 Low density gas flows, see Rarefied gas flows Low density wind tunnel, 784-785 Ludwieg tube, 774 Luminance, 689 Luminous efficiency, 691,692,698-703.705

M Mach number definition, 246, 830 measurement, 776-777 wind tunnel, 758, 759-761, 771-779, 844-847 Mach-Zehnder interferometer, 377, 379, 740 McLeod gage, 509 Magnetohydrodynamic flow studies, in shock tube, 795 Marx surge generator, for nitrogen laser, 72 1 Mass spectrometry advantages in composition diagnostics, 645-646 composition measurement, 645-661

detectors, 652-653 flame composition measurement, 659-660 fragmentation, 646-647 free radicals, 648-649 ion sources, 646-649 Kantrowitz-Grey molecular beam inlet, 654-656 quadrupole, 650 sampling systems, 653-656 time-of-flight , 649-650 use in shock tube, 656-659 Measuring volume, see also Spatial resolution dimensions in LDV, 64, 83-84, 96, 126, 131-134 distribution of light in, in LDV, 119- 120 Membrane calorimeter, 664 Membrane pressure gage, see Disphragm pressure gage Metering nozzle, 324-331 Michelson interferometer diagnostic uses, 606-610, 740 infrared, 752-753 measure coherence time, 71 1 Micromanometer, 506 Mie theory of light particle scattering, 54 Mode-locked laser, 718-720 Model testing principles, 821-848 Molecular light scattering, advantages of diagnostics, 409, 418-421 Multiple-beam interferometer (FabryPerot), 195-227, 708, 717, 740 Multiple spark camera, 737-739 Multiplexing, 555 Mutual coherence function, 712-714

N Nanolight, 700, 702 Negative absorption, 406, 707, 720-725 Neutron absorption, for density measurement, 705 Newtonian fluid, 502, 797-800 Newton’s law of cooling, 665 Noise, see also Signal-to-noise ratio Johnson, 143 optical, 141 photodetector in LDV, 142-143 shot, 143, 185

22

SUBJECT INDEX

Nonequilibrium system composition measurements, 631 -634, 637-643, 645-649 level population by Raman scattering, 420, 424, 430 temperature measurements, 463-465, 472-497 Nozzle, wind tunnel, 758, 760, 764, 772 Nullpoint calorimeter, 676-677 Nusselt number, 460, 461 0

Open channel liquid metering, 332-336 Optical characteristics of flow tracing particles, 51-60 Optical delay line, 738-739 Optical filter, multiple-beam interferometer, 105, 201, 195-227, 708, 717, 740 Optical heterodyne detection in LDV, 109, see also Heterodyne efficiency Optical homodyne detection, in LDV, 116-118 Optical interferometer, see Interferometry Optical mixing, 106-1 15 Optical multichannel detector, 429 Optical radiation absorbed, 405-408, 621-634 emitted, 341-345, 641 -645, 687-725 scattered, 408-433, 643-645 Optical sensor for surface displacement or velocity, 549, 606 Orifice flowmeter, 324-331 Overheat ratio, hot-wire and hot-film probes, 269 P Paint, temperature indicating, 671 -672, 771 Parametric oscillations, laser, 724-725 Partial pressure, use in description of composition, 61 1 Particle tracking, see Flow tracing particles Particulates, acceleration by flow, 8-15,795 Partition function, 427 Pascal (pressure unit), 508 Pebble-bed storage heater, 782 Pedestal, in LDV signal, 114, 121, 156, 159, 184

Period counting, LDV signal processing, 174- 1 80 Phase contrast as visualization method, 389 Phase object, 345 Photodetector, see also Light sensor output, LDV statistical character, 154 Photoelectric effect, 407 Photography high speed, 725-753 recording material, 726-727 short duration light sources, 727 Photometric aspects of light sources, 688-689 Photomultiplier, 104-107, 407 Photon counting correlation, LDV signal processing, 180-186 Piezoelectric scanning interferometer, 201 Piezoelectric sensor, 542 Piezometric head, 333 Pitch, aerodynamic, 768-769 Pi theorem, 826-828 Pitot probe angular sensitivity, 248 calibration, 243, 248, 253, 515-524 corrections for turbulence, 252 corrections for viscous effects, 251 general, 240, 242-254, 515-524 principle, 243 tube construction, 248 use in low density wind tunnel, 784 use in velocity gradient, 250 use near wall, 250 wind tunnel, 776-777 Pitot-static probe, 249 Pitot tube, see Pitot probe Planck radiation law, 465-466, 690 Pockels cell, 717, 728-729 Point light source, 356, 405, 710 Point source explosion, 839-841 Poiseuille formula, 336, 798 Polarizability electronic, 347 infrared measurement, 750-753 molecular, 41 1 Polarization vector, 347 Poled ceramic pressure gage sensor, 542 Polyvinylidene fluoride (PVF,) piezoelectric sensor, 542 Positive displacement flowmeter, 323-324 Prandtl number definition, 459, 830

SUBJECT INDEX

wind tunnel, 777, 844 Pressure relation to stress tensor, 501 units, 508 Pressure bar gage, 593 Pressure concept extension by thermodynamics, 503 kinetic theory, 502 mechanical, 500, 801 Pressure gage bar gage, 593, 602 Bourdon tube type, 51 1 calibration, dynamic, 555 calibration at gigapascal range, 592, 606 calibration at kilopascal and megapascal range, 507, 512, 557 calibration at 100 gigapascals, 610 calibration below 10 pascals, 509, 515 calibration standards, 504, 514 capsule type, 511 characterization, 527 deformation type, 510 diaphragm, 559-576 diaphragm below 10 pascals, 515 diaphragm types, 570 dilatational gage, 604 dynamic calibration, 555 fast response, 576 free surface sensor, 606 frequency response function, 527 hold time, 527, 531, 557 holographic method of recording many diaphragm gages, 575 Hopkinson bar, 596 inductance sensor, 548 McLeod, 509 meaning, 504 miniature bar gage, 602 miniature capacitance, 571, 573, 575 miniature probe, 592 miniature stub, 588 optical sensor, 549 peak pressure, 535 piezoelectric sensor on bar gage, 593, 603 on stub gage, 588 piston and cylinder, 512 probe, 591 range, 53 I , 557 recording methods, 552-555 reluctance sensor, 548, 571

23

resonant period, 527, 531 response, to step function, 527 response characteristics of diaphragm gage, 567-570 response time, 527, 531, 557 sensitivity, 531, 557 diaphragm type, 561, 562, 566 sensors, 534-552 slab type, 588 standards, 504 static calibration, 504, 514 steady or slowly varying pressure, 505-515

stub type, 588 theory of diaphragm gage, 559-570 of fast response gage, 579-588 thin polymer piezoelectric, 573 types, 505, 526, 577 use in free-flight model, 781 use in non-Newtonian flow, 801 U-tube manometer, 505 wall taps, 516-518, 801 Pressure measurement above 100 kilopascals, 512 below 10 pascals, 515 general, 499-610 in moving fluid, 515 static probe, 516, 518, 521 Pressure probe in moving fluid, 515 Pressure recovery, wind tunnel, 758, 760 Pressure-time recording, 552 Pressure transducers, see Pressure gage, sensors Probe gage for dynamic pressure measurement, 591 Probe methods for pressure measurement, 515-516, 518, 521, 591 for temperature measurement, 457-463 for velocity measurement, 240-341 Propeller anemometer, 254 Pulsed Doppler ultrasonic velocity meter, 317-318 Pyroelectnc temperature sensor, 685

0 Q-switched laser, 717-718 Quartz piezoelectric pressure gage sensor, 542

24

SUBJECT INDEX

Quenching role in electron beam fluorescence, 438, 447-448 insensitivity of Raman scattering, 413

R Radiant energy of light sources, 689, 706-707, 718-719 Radiation boundary condition, 665, 667, 669 Radiation constants, blackbody, 466, 468 Radiation detectors, 407 Radiation source, see Light source Radiative heating, study in hypersonic tunnel, shock tube, 679, 683-685, 762, 791-792 Radiative loss, temperature sensor, 461 Radiography, 407 Raman laser, 722-724 Raman scattering advantages over other density measurement techniques, 414, 418-421 basic features, 412-414 calibration, 429 density, temperature, composition diagnosis, 408-455, 643-645 light sources, 419 line intensity use for density, concentration measurement, 428, 431 meaning, 41 1-414 molecular rotation, 41 1 molecular vibration, 41 1 nitrogen vibrational line contour, 421 -428 pulsed laser illumination, 419-421, 425 rotational line contribution, 425-430, 488 scattering amplitudes, 413 Stokes and anti-Stokes line, meaning, 422 temperature effects on density measurement, 421 -425 Raman shift, 413,see also Raman scattering Rarefied gas flow density measurement by EBF, 434-455 visualization, 398 wind tunnel, 762-763, 784-785 Rayleigh scattering advantages over emission and absorption spectroscopy, 414 basic features, 412-414 line intensities used for diagnostics, 417-418

line shape used for diagnostics, 414-417 meaning, 411-414 Real gas effects, in hypersonic apparatus, 782 Receiving aperture size, effect on LDV performance, 138, see olso Antenna theorem Recording methods infrared, 750-753 light, 725-753 pressure gage, 552-555 wind tunnel, 768-769 Recovery factor, temperature Couette flow, 459 definition, 459 flat plate boundary layer, 459 wind tunnel, 777-778 Recovery pressure, wind tunnel, 776 Reference beam interferometer, 375-383 Refractive behavior of fluids, 346, 347, 41 I Refractive index density dependence, 348 flow tracing particle materials, 58-59 gas mixture, 349 Reradiation after interaction with medium, 407, 411-412 Resistance, temperature coefficient, 262, 264 Resistance thermometer, 3 13-3 14, 461 Resisting vane anemometer, 254, 258 Resistivity, hot-wire material, 262, 264 Resolving power of multiple-beam interferometer, 196 Resonance scattering, 413 Response time, see also Frequency response dilatational pressure gage, 604 flow tracing particles, 26-32, 38 free surface motion pressure gage, 609 heat transfer gage, 668-669, 674-678 hot-wire anemometer, 293-295 liquid manometer, 508 Pitot probe, 252-253 pressure bar gage, 599 with end sensor, 603 radiation scattering diagnostics, 409 tracer particle, 26-32, 38 Reynolds number definition, 830 hot-wire and hot-film convection, 269 low, apparatus, 796-801

SUBJECT INDEX

temperature sensor, 461 wind tunnel, 758-759, 773-774, 844-847 Rheological fluid, 801 Rise time, see Response time Rochelle salt pressure gage sensor, 547 Roll, aerodynamic, 768-769 Ronchi schlieren, 370 Rotameter, 322, 331 Rotating flow apparatus construction, 806-809 data transmission and photography, 817-819 examples of studies, 801-819 experimental configurations, 81 1-813 moving boundaries, 814-816 precision and control requirements, 809-812 pumping, 816-817 Rotating mirror camera, see Framing camera; Streak camera Rotational temperature EBF technique, 436, 493-497 measurement, 457,464,466,480,493-497 spectral emission technique, 480 Ruby high pressure gage, 610 Ruby laser giant pulse, 717-720 properties, 714-716, 725, 746 pump lamp, 693-694

S

Sabot, 780 Sampling error, 4-6 Sandwich heat transfer gage, 664 Scale effect, 846 Scanning multiple-beam interferometer, 201 -220 Schlieren interferometer, 384 Schlieren method combined with holography, 368 combined with laser Doppler, 369 dephasing schlieren system, 373 effects of diffraction, 364 sharp focusing, 368 Schlieren systems color, 367 construction and principles, 361 -374 double knife edge, 367 double pass, 367

25

infrared, 750 schlieren head, 361 spark light source, 695-703 Toepler, 361 Scintillating crystal radiation detector, 407 Sedimentation, effect on flow tracing particles, 41 Selection rules, in electron beam excitation, 438-440 Self-absorption, effect on intensity of emitted radiation, 640 Self-similarity , 828 Sensor light, 689, 715, 726, 735 temperature probe, 460-463 pressure gage, 534-552 Servo frequency tracking, applied to LDV, 169-174 Servo multiple-beam interferometer, applied to LDV, 21 1-220 Settling plenum, wind tunnel, 758 Shadowgraph high speed frames, 738 light source, 694, 695-703 method, 355, 356-360, 407, 738 Shearing interferometer, 375, 383-389, 740 Ship flow dynamics, dimensional analysis, 847 -848 Shock strength, 787 Shock tube chemical kinetic studies, 792-795 combustion driver, 788 composition measurement in reactions, 632-634, 656-659 description as research apparatus, 785-796 electric driver, 790 gas and sound speed measurement by ultrasound, 339 gas temperature measurement methods, 467-487 measurement of surface heat transfer, 667, 677-685 modified as shock tunnel, 791-792 pressure gage test and calibration, 557 production of high temperature gases, 787-790 reflected shock region, 787-788 x - f diagram, 786 Shock tunnel, research apparatus, 462,666, 791 -792

26

SUBJECT INDEX

Shock wave recorded by interferometer, 741 by schlieren method, 365 by shadowgraph, 358, 738, 768 by shearing interferometer, 386 Short duration light sources, 717-718 Shrouded thermocouple stagnation temperature probe, 463 Shutter, for single exposure photography, 727-732

SI system of units (Systeme International), 822

Signal analysis classification, 161 definition, 154 Signal conditioning, 162-165 Signal dropout in LDV, 157-164, 172-173, 175

Signal filtering, 163 Signal processing classification, 161 definition, 154 effect of signal-to-noise ratio, 155 heat transfer gage, 666,670 hot-wire anemometer linearizer, 302 Signal spectra, in LDV, 159 Signal-to-noiseratio effect of refractive index variations, in LDV, 141 heterodyne configuration, 144 homodyne configuration, 145 multiple particle effects in LDV, 147-154 photodetector, 140, 146 requirements in LDV, ,140-154, 234, 235 Similarity, 828-829 Similarity solution, examples, 832, 834, 835-842

Sing-around type flowmeter, 339 Skimmer, for molecular beam, 654-656 Slab pressure gage, 588 Slip flow, 763 Slug calorimeter, 676 Sonic anemometer, 315-318 Sonic flow, 760 Sonic nozzle flowmeter, 330 Sound speed, 246, 315-317 Spark discharge, electrical and fluid dynamical parameters, 695-698, 752-753

Spark formation, mechanism, 695-698 Spatial coherence, see Light source Spatial resolution, see also Measuring

volume dimensions density measurement by Rayleigh scattering, 418 in LDV, 134, 137-138, 235 pressure gages, 525, 573-576 in radiation scattering diagnostics, 409, 420

Species concentration, see also Composition measurement electron beam fluorescence technique, 434,451

mass spectrometer, 645-661 Raman scattering diagnostics, 428, 643 -645

Speckle, laser, 712-714 Spectral broadening, effect in LDV resolution, 133-140, see also Ambiguity noise Spectral line shape distortion by Brillouin scattering, 415-417

Raman and Rayleigh scattering, 412 use for temperature measurement, 414-417.490

Spectral radiance, 690 Spectral response of the eye, 688-689, 692 Spectral width of light, effect on LDV performance, 137 Spectrometer slit function effect on spectral line shape, 412 Raman scattering analysis, 427-428 Spectrum-scanned LDV, requirements, 201 -203

Spectrum scanning, applied to LDV, 154, 159-162, 165-174, 201-211

Spin-down, 809, 814-815 Spontaneous emission requirements in EBF, 438, 441, 442-447 temperature measurement, 465-482 Ssuare wave test hot-wire and hot-film probe, 279, 285, 297-299

pressure probe, 563-570 system frequency response, 528-531 whirling vane anemometer, 259 Stagnation enthalpy, 245, 458, 788 Stagnation pressure meaning, 244, 246, 503, 515 produced for shock tunnel, 791 ratio across shock, 247 wind tunnel, 776 Stagnation temperature

27

SUBJECT INDEX

hypersonic apparatus, 782 meaning, 245, 457, 665 measurement, 460-463 produced for shock tunnel, 791 relation to gas speed and temperature, 458 in shock tube flow, 788 Static pressure, meaning, 247, 516 Static pressure probe in steady flow, 243, 518 in unsteady flow, 521 Static temperature, meaning, 457 Static vents on airplane, 520 Stefan-Boltzmann constant, 690 Step function hot-wire and hot-film response testing, 279, 285 loading bar pressure gage, 598, 603 diaphragm pressure gage, 563 dilatational pressure gage, 605 response function, 527 Stewartson layer, 806, 813 Sting, wind tunnel mount, 765 Stokes drag formula, 10-15, 759, 798, 799 Stokes number, 28 Stokes Q-branch, 421 Stokes Raman diagnostics for temperature measurement, 422-425 Stokes Raman line, 422, 723 Strain pulse dispersion, 580 reflection at end or interface, 585 theory of one-dimensional wave, 581 Strain sensitivity of diaphragm, 561, 562, 566 Strain sensor, 536-540, 542-549, 572-574 Stratified fluid, optical visualization, 355 Streak camera image converter recording, 696 light source, 705, 715 use with interferometer, 795 using rotating mirror, 733-734 Stress tensor, 500 Strouhal number, definition, 830 Stub pressure gage, 588 Superradiant light sources, 720-721 Supersonic flow, 330, 760 Supersonic wind tunnels, 771 -779 Surface temperature sensor fluorescent paint, 672 infrared pyrorneter, 672

light transmitting paint, 671 thermocouple-thermopile, 666,672-673, 675 thin fllm resistance, 664, 666, 672-685 Swept oscillator wave analyzer, LDV signal processing, 165

T Taylor-Proudman theorem, 803, 805 Temperature fluctuations, measurement by Rayleigh scattering, 417 Temperature gradient measurement, Raman scattering, 424 Temperature in moving fluid, 457-460 Temperature measurement by analysis of emitted and absorbed radiation, 465-487, 698-699, 704 behind detonation front, 470 by Doppler broadened line shape, 481-482, 490 electron beam fluorescence, 489-497 emittance on two paths, 475-478 hot-wire probe method, 313, 461 infrared pyrometer, 672 line reversal methods, 466-470 method of absorption in two spectral regions, 472-475 molecular scattering of radiation, 409, 414-418, 421-428 in moving fluid by probe, 457-463 by paint transparency and phosphorescence, 671-672 probe methods, 457-463 radiation analysis methods, 463-497 Raman scattering, 487-489 by Rayleigh scattered spectral lineshape, 412, 414-417, 482-485, 487-489 relative intensities, 478-481 by simultaneous detection of radiation emission and absorption, 470-472 simultaneous with velocity measurement, 313 in sparks, 698-699, 704 two path absorption in thin foils of x-rays, 485 -486 vibrational temperature by analysis of emitted radiation, 473-474, 481 Temperature sensors probe type, 460-463 resistance film, 460, 677-685 thermocouple, 460-461, 671-676

28

SUBJECT INDEX

Temporal coherence, see Coherence, temporal Test section, wind tunnel, 758, 764 Thermal conductivity, 665-666 Thermal diffusivity, 665 Thermal wind relation, 804, 810 Thermocouple, 460-461, 671, 672, 675-676 Thin film heat transfer gage, 664, 666-685 Time constant hot-wire and hot-film anemometers, 293 -295 Raman scattering diagnostics, 420, 421 -425 Time dependent response of pressure gage, 527 Time domain signal processing, LDV, 161 Time resolution, see Frequency response Time response, see Hold time; Response time Toepler schlieren system, 361 Torr, 508 Total enthalpy, see Stagnation enthalpy Total head (fluid), 333 Total pressure, see Stagnation pressure Total temperature, see also Stagnation temperature probe, 462-463 Towing tank, dimensional analysis, 843, 847-848 Townsend mechanism of spark formation, 695-696 Tracer particle tracking, see Flow tracing particles Tracking bandpass filter, LDV signal processing, 169- 174 Tracking multiple-beam interferometer for LDV signal, 211-220 Transducer, see Sensor Transition probability, in electron beam fluorescence, 438-441 Translational temperature, 457, 464,490 Transonic wind tunnels, 771 -779 T-tube, electrically driven shock tube, 790 Tufts, for visualization, 241, 770 Tunnel wall corrections, 773, 847 Turbulence level, wind tunnel, 764, 774 Turbulence application of LDV for measurement, 103-104 effect on LDV spectrum, 161, 166, 221

effect on photon counting correlation in LDV, 184 effect on pressure probe, 252-253 temperature fluctuations in flame, 417 visualize transition, 771 wind tunnel flow, 764 Turbulent boundary layer, 459, 767 heat transfer, 666

U Ultrasonic flowmeter, 337-338 Unbalance parameter, hot-wire and hot-film probe compensation, 294, 296 Units conversion, 823-824 photometric, 688-689 pressure, 508 SI (Systeme International), 822

V Vane anemometer, principle, 255-257 Variable area flowmeter, 331 Velocity components measurement by chronophotography, 67 measurement by hot-wire probe, 306-31 1 measurement by LDV, 190-195 Velocity head, 333 Velocity gradient, measurement by hot-wire probe, 312 Velocity measurement by chronophotography, 64-93, 818-819 direction by chronophotography , 67 direction by hot-wire anemometer, 306-312 direction by LDV, 190-195 direction by Pitot probe, 254 direction, 241 by Doppler shift of emitted characteristic radiation, 341-345 of scattered light, 93-240, 342 of scattered sound from tracers, 317-318 electromagnetic method, 318-321 fluorescent radiation Doppler shift, 343 by Hall voltage, 318 by heat loss probe method, 259-314 hot-wire and hot-film probes, 259-314

SUBJECT INDEX

laser Doppler from tracing particles, 96-240 by laser Doppler velocimeter, 93-240 LDV with direct spectrum analysis, 194-227 Pitot probe, 242-254 by pressure probe, 242-254 probe methods, 240-341 propeller anemometer, 254, 256 resonant absorption of Doppler shifted radiation, 344 rotating flow apparatus, 817-819 sensitivity of measurement using tracer methods, 36 simultaneous with temperature measurement by hot-wire probe, 313 by timed sound pulses, 315-318 tracer methods, 1-240 tracer particle loading error, 38 vane anemometer, 254-259 Ventilated wall, wind tunnel, 773, 775 Venturi flowmeter, 324, 331 Vibrational spectral line contour analysis, 425 -428 Vibrational temperature, 436, 457,464,466, 473-474, 481, 491-493 Virial equation of state, 61 1-612 Virtual fringes, in LDV, I18 Viscometry, 798-801 Viscosity of a particulate suspension, 38 Viscous fluid, 502, 797-800 Visualization, see Flow visualization Vortex generator, 816 Vorticity meter, 241, 312

W Wall temperature discontinuity, effect on heat transfer measurement, 673 Wave machine, 796 Weir, 332, 334 Weir block, 334

29

Wet-gas meter, 323-324 Whirling cup anemometer, 256-257 Wien displacement law, 690 Wien radiation formula, 466 Wind tunnel blockage interference, 765 classification, 758-764 dimensional analysis applied to, 844-847 heat transfer techniques, 670-672 flow visualization, 769-771 lift interference, 765 low speed, 764-771 model testing principles, 843-849 open-jet test section, 764 research apparatus, 756-785 supersonic, 771-779 transonic, 771-779 turbulence, 764 wall corrections, 773, 847 Wollaston prism shearing interferometer, 387 Working section, wind tunnel, 758

X X-probe, hot-wire anemometer, 310 X-ray radiation, 407-408, 705 x - f diagram, shock tube, 786 Xenon flash lamp, 693-694

Y Yaw aerodynamic moment, 768-769 card, 780 hot-wire probe correction, 306-308 meaning, 248 Pitot probe correction, 248-249 total temperature probe correction, 462 Young's experiment, measure spatial coherence, 710-71 1


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